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+{"text":"\\section{Introduction}\n\n\\subsection{Overview}\n\nTwo multi-kilometer interferometric gravitational-wave (GW) detectors\nare presently in operation: LIGO\\footnote{http:\/\/www.ligo.caltech.edu}\nand Virgo\\footnote{http:\/\/www.virgo.infn.it}. They are sensitive to\nGWs produced by the coalescence of two neutron stars to a distance of\nroughly 30 Mpc, and to the coalescence of a neutron star with a\n$10M_{\\odot}$ black hole to roughly 60 Mpc.  Over the next several\nyears, these detectors will undergo upgrades which are expected to\nextend their range by a factor $\\sim 10$.  Most estimates suggest that\ndetectors at advanced sensitivity should measure at least a few, and\npossibly a few dozen, binary coalescences every year (e.g.,\n\\citealt{koppa08,ligo_rates}).\n\nIt has long been argued that neutron star-neutron star (NS-NS) and\nneutron star-black hole (NS-BH) mergers are likely to be accompanied\nby a gamma-ray burst \\citep{eichler89}.  Recent evidence supports the\nhypothesis that many short-hard gamma-ray bursts (SHBs) are indeed\nassociated with such mergers (\\citealt{fox05}, \\citealt{nakar06},\n\\citealt{berger07}, \\citealt{perleyetal08}).  This suggests that it\nmay be possible to simultaneously measure a binary coalescence in\ngamma rays (and associated afterglow emission) and in GWs\n\\citep{dietz09}.  The combined electromagnetic and gravitational view\nof these objects will teach us substantially more than what we learn\nfrom either data channel alone.  Because GWs track a system's global\nmass and energy dynamics, it has long been known that measuring GWs from a coalescing binary allows us to determine, in the\nideal case, ``intrinsic'' binary properties such as the masses and spins\nof its members with exquisite\naccuracy (\\citealt{fc93}, \\citealt{cf94}).  As we describe in\nthe following subsection, it has also long been appreciated that GWs\ncan determine a system's ``extrinsic'' properties\n\\citep{schutz86} such as location on the sky and distance to the\nsource.  In particular, the amplitude of a binary's GWs directly\nencodes its luminosity distance.  Direct measurement of a coalescing\nbinary could thus be used as a cosmic distance measure: Binary\ninspiral would be a ``standard siren'' (the GW equivalent of a\nstandard candle, so-called due to the sound-like nature of GWs) whose\ncalibration depends only on the validity of general\nrelativity~\\citep{HolzHughes05,dalaletal}.\n\nUnfortunately, GWs alone do not measure extrinsic parameters as\naccurately as the intrinsic ones. As we describe in more detail in the\nfollowing section, GW observation of a binary measures a complicated\ncombination of its distance, its position on the sky, and its\norientation, with overall fractional accuracy $\\sim\n1\/\\mbox{signal-to-noise}$.  As distance is degenerate with these\nangles, using GWs to measure absolute distance to a source requires a\nmechanism to break the degeneracy. Associating the GW coalescence\nwaves with a short-hard gamma-ray burst (SHB) is a near-perfect way to\nbreak some of these degeneracies.\n\nIn this paper we explore the ability of the near-future advanced\nLIGO-Virgo detector network to constrain binary parameters (especially\ndistance), when used in conjunction with electromagnetic observations\nof the same event (such as an associated SHB). We also examine how\nwell these measurements can be improved if planned detectors in\nWestern Australia (AIGO\\footnote{http:\/\/www.gravity.uwa.edu.au}) and\nin Japan's Kamioka mine\n(LCGT\\footnote{http:\/\/gw.icrr.u-tokyo.ac.jp:8888\/lcgt\/}) are\noperational. This paper substantially updates and improves upon\nearlier work \\citep[hereafter DHHJ06]{dalaletal}, using a more\nsophisticated parameter estimation technique. In the next section we\nreview standard sirens, and in Sec.\\ {\\ref{sec:dalaletal}} we briefly\nsummarize DHHJ06. The next subsection describes the\norganization and background relevant for the rest of the paper.\n\n\\subsection{Standard sirens}\n\\label{sec:sirens}\n\nIt has long been recognized that GW inspiral measurements could be\nused as powerful tools for cosmology.  {\\cite{schutz86}} first\ndemonstrated this by analyzing how binary coalescences allow a direct\nmeasurement of the Hubble constant; {\\cite{markovic93}} and\n{\\cite{fc93}} subsequently generalized this approach to include other\ncosmological parameters. More recently, there has been much interest\nin the measurements enabled when GWs from a merger are accompanied by\na counterpart in the electromagnetic spectrum~(\\citealt{bloometal09},\n\\citealt{phinney09}, \\citealt{kk09}).  In this paper we focus\nexclusively on GW observations of binaries that have an independent\nsky position furnished by electromagnetic observations.\n\nWe begin by examining gravitational waves from binary inspiral as\nmeasured in a single detector.  We only present here the lowest order\ncontribution to the waves; in subsequent calculations our results are\ntaken to higher order (see Sec.\\ \\ref{sec:gwform}).  The leading\nwaveform generated by a source at luminosity distance $D_L$,\ncorresponding to redshift $z$, is given by\n\\begin{eqnarray}\nh_+ &=& \\frac{2(1 + z){\\cal M}}{D_L}\\left[\\pi(1 + z){\\cal\nM}f\\right]^{2\/3}\\times\\nonumber\\\\\n& & \\qquad\\qquad\\qquad\\left(1 + \\cos^2\\iota\\right)\\cos2\\Phi_N(t)\\;,\n\\nonumber\\\\\nh_\\times &=& -\\frac{4(1 + z){\\cal M}}{D_L}\\left[\\pi(1 + z){\\cal\nM}f\\right]^{2\/3}\\cos\\iota \\sin2\\Phi_N(t)\\;,\n\\nonumber\\\\\n\\Phi_N(t) &=& \\Phi_c - \\left[\\frac{t_c - t}{5(1 + z){\\cal\nM}}\\right]^{5\/8}\\;,\\qquad f \\equiv \\frac{1}{\\pi}\\frac{d\\Phi_N}{dt}\\;.\n\\label{eq:NewtQuad}\n\\end{eqnarray}\nHere $\\Phi_N$ is the lowest-order contribution to the orbital phase,\n$f$ is the GW frequency, and ${\\cal M} = m_1^{3\/5} m_2^{3\/5}\/(m_1 +\nm_2)^{1\/5}$ is the binary's ``chirp mass,'' which sets the rate at\nwhich $f$ changes.  We use units with $G = c = 1$; handy conversion\nfactors are $M_\\odot \\equiv GM_\\odot\/c^2 = 1.47\\,{\\rm km}$, and\n$M_\\odot \\equiv GM_\\odot\/c^3 = 4.92 \\times 10^{-6}\\,{\\rm seconds}$.\nThe angle $\\iota$ describes the inclination of the binary's orbital\nplane to our line-of-sight: $\\cos\\iota = \\mathbf{\\hat\nL}\\cdot\\mathbf{\\hat n}$, where $\\mathbf{\\hat L}$ is the unit vector\nnormal to the binary's orbital plane, and $\\mathbf{\\hat n}$ is the\nunit vector along the line-of-sight to the binary.  The parameters\n$t_c$ and $\\Phi_c$ are the time and orbital phase when \n$f$ diverges in this model. We expect finite size effects to impact\nthe waveform before this divergence is reached.\n\nA given detector measures a linear combination of the polarizations:\n\\begin{equation}\nh_{\\rm meas} = F_+(\\theta, \\phi, \\psi) h_+ + F_\\times(\\theta, \\phi,\n\\psi) h_\\times\\;,\n\\label{eq:hmeas}\n\\end{equation}\nwhere $\\theta$ and $\\phi$ describe the binary's position on the sky,\nand the ``polarization angle'' $\\psi$ sets the inclination of the\ncomponents of $\\mathbf{\\hat L}$ orthogonal to $\\mathbf{\\hat n}$.  The\nangles $\\iota$ and $\\psi$ fully specify the orientation vector\n$\\mathbf{\\hat L}$.  For a particular detector geometry, the antenna\nfunctions $F_+$ and $F_\\times$ can be found in~\\cite{300yrs}.  In\nSec.\\ \\ref{sec:gwmeasure} we give a general form for the gravitational\nwaveform without appealing to a specific detector, following the\nanalysis of \\citealt{cf94} (hereafter abbreviated CF94).\n\nSeveral features of Eqs.\\ (\\ref{eq:NewtQuad}) and (\\ref{eq:hmeas}) are\nworth commenting upon.  First, note that the phase depends on the {\\it\nredshifted}\\\/ chirp mass.  Measuring phase thus determines the\ncombination $(1 + z){\\cal M}$~\\citep{fc93}, not ${\\cal M}$ or $z$\nindependently.  To understand this, note that ${\\cal M}$ controls how\nfast the frequency evolves: using Eq.\\ (\\ref{eq:NewtQuad}), we find\n$\\dot f \\propto f^{11\/3}{\\cal M}^{5\/3}$.  The chirp mass enters the\nsystem's dynamics as a timescale $\\tau_c = G{\\cal M}\/c^3$.  For a\nsource at cosmological distance, this timescale is redshifted; the\nchirp mass we infer is likewise redshifted.  Redshift and chirp mass\nare inextricably degenerate.  This remains true even when higher order\neffects (see, e.g., \\citealt{blanchet06}) are taken into account:\nparameters describing a binary impact its dynamics as timescales which\nundergo cosmological redshift, so we infer redshifted values for those\nparameters. {\\it GW observations on their own cannot directly\ndetermine a source's redshift.}\n\nNext, note that the amplitude depends on $(1 + z){\\cal M}$, the angles\n$(\\theta, \\phi, \\iota, \\psi)$, and the luminosity distance $D_L$.\nMeasuring the amplitude thus measures a combination of these\nparameters.  By measuring the phase, we measure the redshifted chirp\nmass sufficiently well that $(1 + z){\\cal M}$ essentially decouples\nfrom the amplitude.  More concretely, matched filtering the data with\nwaveform templates should allow us to determine the phase with\nfractional accuracy $\\delta\\Phi\/\\Phi \\sim 1\/[(\\mbox{signal-to-noise})\n\\times (\\mbox{number of measured cycles})]$; $(1 + z){\\cal M}$ should\nbe measured with similar fractional accuracy.  NS-NS binaries will\nradiate roughly $10^4$ cycles in the band of advanced LIGO, and NS-BH\nbinaries roughly $10^3$ cycles, so the accuracy with which phase and\nredshifted chirp mass can be determined should be exquisite\n(\\citealt{fc93}, CF94).\n\nAlthough $(1 + z){\\cal M}$ decouples from the amplitude, the distance,\nposition, and orientation angles remain highly coupled.  To determine\nsource distance we must break the degeneracy that the amplitude's\nfunctional form sets on these parameters.  One way to break these\ndegeneracies is to measure the waves with multiple detectors.  Studies\n{\\citep{sylvestre, cavalieretal, blairetal, fairhurst09, wenchen10}}\nhave shown that doing so allows us to determine source position to\nwithin a few degrees in the best cases, giving some information about\nthe source's distance and inclination.\n\nPerhaps the best way to break some of these degeneracies is to measure\nthe event electromagnetically.  An EM signature will pin down the\nevent's position far more accurately than GWs alone.  The position\nangles then decouple, much as the redshifted chirp mass decoupled.\nUsing multiple detectors, we can then determine the source's\norientation and its distance.  This gives us a direct,\ncalibration-free measure of the distance to a cosmic event.  The EM\nsignature may also provide us with the event's redshift, directly\nputting a point on the Hubble diagram. In addition, if\nmodeling or observation give us evidence for beaming of the SHB\nemission, this could strongly constrain the source inclination.\n\n\\subsection{This work and previous analysis}\n\\label{sec:dalaletal}\n\nOur goal is to assess how well we can determine the luminosity\ndistance $D_L$ to SHBs under the assumption that they are associated\nwith inspiral GWs.  We consider both NS-NS and NS-BH mergers as\ngenerators of SHBs, and consider several plausible advanced detector\nnetworks: the current LIGO\/Virgo network, upgraded to advanced\nsensitivity; LIGO\/Virgo plus the proposed Australian AIGO; LIGO\/Virgo\nplus the proposed Japanese LCGT; and LIGO\/Virgo plus AIGO plus LCGT.\n\nThe engine of our analysis is a probability function that describes\nhow inferred source parameters $\\boldsymbol{\\theta}$ should be\ndistributed following GW measurement.  (Components $\\theta^a$ of the\nvector $\\boldsymbol{\\theta}$ are physical parameters such as a\nbinary's masses, distance, sky position angles, etc.; our particular\nfocus is on $D_L$.)  Consider one detector which measures a datastream\n$s(t)$, containing noise $n(t)$ and a GW signal\n$h(t,{\\boldsymbol{\\hat\\theta}})$, where $\\boldsymbol{\\hat\\theta}$\ndescribes the source's ``true'' parameters.  In the language of\n\\cite{finn92}, we assume ``detection'' has already occurred; our goal\nin this paper is to focus on the complementary problem of\n``measurement.''\n\nAs shown by {\\cite{finn92}}, given a model for our signal\n$h(t,\\boldsymbol{\\theta})$, and assuming that the noise statistics are\nGaussian, the probability that the parameters $\\boldsymbol{\\theta}$\ndescribe the data $s$ is\n\\begin{equation}\np(\\boldsymbol{\\theta} | s) =\np_0(\\boldsymbol{\\theta})\\exp\\left[-\\left(( h(\\boldsymbol{\\theta}) - s)\n|( h(\\boldsymbol{\\theta}) - s )\\right)\/2\\right]\\;.\n\\label{eq:likelihood}\n\\end{equation}\nThe inner product $(a|b)$ describes the noise weighted\ncross-correlation of $a(t)$ with $b(t)$, and is defined precisely\nbelow.  The distribution $p_0(\\boldsymbol{\\theta})$ is a {\\it prior\nprobability distribution}; it encapsulates what we know about our\nsignal prior to measurement.  We define $\\boldsymbol{\\tilde\\theta}$ to\nbe the parameters that maximize Eq.\\ (\\ref{eq:likelihood}).\n\nDHHJ06 did a first pass on the analysis we describe here.  They\nexpanded the exponential to second order in the variables\n$(\\boldsymbol{\\theta} - \\boldsymbol{\\hat\\theta})$; we will henceforth\nrefer to this as the ``Gaussian'' approximation (cf.\\\n\\citealt{finn92}):\n\\begin{eqnarray}\n\\label{eq:gaussian}\n& &\\exp\\left[-\\left( h(\\boldsymbol{\\theta}) - s | h(\\boldsymbol{\\theta})\n  - s\\right)\/2\\right] \\simeq\n\\nonumber\\\\\n& &\\qquad\\qquad\\qquad\n\\exp\\left[-\\frac{1}{2}\\left(\\frac{\\partial h}{\\partial\\theta^a} \\Biggl|\n  \\frac{\\partial h}{\\partial\\theta^b}\\right)\\delta\\theta^a\n  \\delta\\theta^b\\right]\\;,\n\\end{eqnarray}\nwhere $\\delta\\theta^a = \\theta^a - \\hat\\theta^a$.  In this limit,\n$\\boldsymbol{\\tilde\\theta} = \\boldsymbol{\\hat\\theta}$ (at least for\nuniform priors).  The matrix\n\\begin{equation}\n\\Gamma_{ab} \\equiv \\left(\\frac{\\partial h}{\\partial\\theta^a} \\Biggl|\n  \\frac{\\partial h}{\\partial\\theta^b}\\right)\n\\label{eq:Fisherdef}\n\\end{equation}\nis the {\\it Fisher information matrix}.  Its inverse $\\Sigma^{ab}$ is\nthe covariance matrix.  Diagonal entries $\\Sigma^{aa}$ are the\nvariance of parameter $\\theta^a$; off-diagonal entries describe\ncorrelations.\n\nThe Gaussian approximation to Eq.\\ (\\ref{eq:likelihood}) is known to\nbe accurate when the signal-to-noise ratio (SNR) is large.  However,\nit is not clear what ``large'' really means {\\citep{vallis}}.  Given\ncurrent binary coalescence rate estimates, it is expected that most\nevents will come from $D_L \\sim \\mbox{a few} \\times 100\\,{\\rm Mpc}$.\nIn such cases, we can expect an advanced detector SNR $\\sim 10$.  It\nis likely that this value is not high enough for the ``large SNR''\napproximation to be appropriate.\n\nIn this analysis we avoid the Gaussian approximation.  We instead\nuse Markov-Chain Monte-Carlo (MCMC) techniques (in particular, the\nMetropolis-Hastings algorithm) to explore our parameter distributions.\nA brief description of this technique is given in Sec.\\\n\\ref{sec:estimate}, and described in detail in \\cite{lewis02}.  We\nfind that the Gaussian approximation to Eq.\\ (\\ref{eq:likelihood}) is\nindeed failing in its estimate of extrinsic parameters (though it\nappears to do well for intrinsic parameters such as mass).\n\n\\subsection{Organization of this paper}\n\nWe begin in Sec.\\ {\\ref{sec:inspiralwaves}} by summarizing how GWs\nencode the distance to a coalescing binary.  We first describe the\npost-Newtonian (PN) gravitational waveform we use in Sec.\\\n{\\ref{sec:gwform}}, and then describe how that wave interacts with a\nnetwork of detectors in Sec.\\ {\\ref{sec:gwmeasure}}.  Our discussion\nof the network-wave interaction is heavily based on the notation and\nformalism used in Sec.\\ 4 of CF94, as well as the analysis of\n\\cite{abcf01}.  Section {\\ref{sec:gwmeasure}} is sufficiently dense\nthat we summarize its major points in Sec.\\\n{\\ref{sec:gwmeasure_summary}} before concluding, in Sec.\\\n{\\ref{sec:detectors}}, with a description of the GW detectors which we\ninclude in our analysis.\n\nWe outline parameter estimation in Sec.\\ \\ref{sec:estimate}.  In Sec.\\\n{\\ref{sec:formaloverview}} we describe in more detail how to construct\nthe probability distributions describing parameter measurement.  We\nthen give, in Sec.\\ \\ref{sec:selectionandpriors}, a brief description\nof our selection procedure based on SNR detection thresholds.  This\nprocedure sets physically motivated priors for some of our parameters.\nThe Markov-Chain Monte-Carlo technique we use to explore this function\nis described in Sec.\\ {\\ref{sec:mhmc}}.  How to appropriately average\nthis distribution to give ``noise averaged'' results and to compare\nwith previous literature is discussed in Sec.\\\n{\\ref{sec:averagedPDF}}.\n\nIn Sec.\\ {\\ref{sec:valid}} we discuss the validation of our code.  We\nbegin by attempting to reproduce some of the key results on distance\nmeasurement presented in CF94.  Because of the rather different\ntechniques used by Cutler \\& Flanagan, we do not expect exact\nagreement.  It is reassuring to find, nonetheless, that we can\nreconstruct with good accuracy all of the major features of their\nanalysis.  We then examine how these results change as we vary the\namplitude (moving a fiducial test binary to smaller and larger\ndistances), as we vary the number of detectors in our network, and as\nwe vary the source's inclination.\n\nOur main results are given in Sec.\\ {\\ref{sec:main_results}}.  We\nconsider several different plausible detector networks and examine\nmeasurement errors for two ``fiducial'' binary systems, comprising\neither two neutron stars (NS-NS) with physical masses of $m_1 = m_2 =\n1.4\\,M_{\\odot}$, or a neutron star and black hole (NS-BH) system with\nphysical masses $m_1 = 1.4\\,M_{\\odot}$ and $m_2 = 10\\,M_{\\odot}$.\nAssuming a constant comoving cosmological density, we distribute\npotential GW-SHB events on the sky, and select from this distribution\nusing a detection threshold criterion set for the entire GW detector\nnetwork.  We summarize some implications of our results in Sec.\\\n\\ref{sec:summary}.  A more in-depth discussion of these implications,\nparticularly with regard to what they imply for cosmological\nmeasurements, will be presented in a companion paper.\n\nThroughout this paper, we use units with $G = c = 1$.  We define\nthe shorthand $m_z = (1 + z)m$ for any mass parameter $m$.\n\n\\section{Measuring gravitational waves from inspiraling binaries}\n\\label{sec:inspiralwaves}\n\nIn this section we review the GW description we use, the formalism\ndescribing how these waves interact with a network of detectors, and\nthe properties of the detectors.\n\n\\subsection{GWs from inspiraling binaries}\n\\label{sec:gwform}\n\nThe inspiral and merger of a compact binary's members can be divided\ninto three consecutive phases.  The first and longest is a gradual\nadiabatic {\\it inspiral}, when the members slowly spiral together due\nto the radiative loss of orbital energy and angular momentum.\nPost-Newtonian (PN) techniques (an expansion in gravitational\npotential $M\/r$, or equivalently for bound systems, orbital speed\n$v^2$) allow a binary's evolution and its emitted GWs to be modeled\nanalytically to high order; see \\cite{blanchet06} for a review.  When\nthe bodies come close together, the PN expansion is no longer valid,\nand direct numerical calculation is required.  Recent breakthroughs in\nnumerical relativity now make it possible to fully model the\nstrong-field, dynamical {\\it merger} of two bodies into one; see\n\\cite{pretorius05}, \\cite{su06}, and \\cite{etienne08} for discussion.\nIf the end state is a single black hole, the final waves from the\nsystem should be described by a {\\it ringdown} as the black hole\nsettles down to the Kerr solution.\n\nIn this work we are concerned solely with the inspiral, and will\naccordingly use the PN waveform to describe our waves.  In particular,\nwe use the so-called ``restricted'' PN waveform; following CF94, the\ninspiral waveform may be written schematically\n\\begin{equation}\nh(t) = \\mathrm{Re}\\left(\\sum_{x,m}\nh^x_m(t)e^{im\\Phi_{\\mathrm{orb}}(t)}\\right) \\, .\n\\label{eq:hPN}\n\\end{equation}\nHere $x$ indicates PN order [$h^x$ is computed to $O(v^{2x})$ in\norbital speed], $m$ denotes harmonic order (e.g., $m = 2$ is\nquadrupole), and $\\Phi_{\\rm orb}(t) = \\int^t \\Omega(t') dt'$ is\norbital phase [with $\\Omega(t)$ the orbital angular frequency].  The\n``restricted'' waveform neglects all PN amplitude terms beyond the\nleading one, and considers only the dominant $m = 2$ contribution to\nthe phase.  The phase is computed to high PN order.\n\nLet the unit vector $\\hat{\\bf n}$ point to a binary on the sky (so\nthat the waves propagate to us along $-\\hat{\\bf n}$), and let the unit\nvector $\\hat{\\bf L}$ denote the normal along the binary's orbital\nangular momentum.  The waveform is fully described by the two\npolarizations:\n\\begin{eqnarray}\nh_+(t) & = & \\frac{2{\\mathcal M}_z}{D_L}\n\\left[\\pi{\\cal M}_z f(t)\\right]^{2\/3}\n[1+(\\mathbf{\\hat{L}}\\cdot \\mathbf{\\hat{n}})^2]\\cos[\\Phi(t)]\\;,\n\\nonumber\\\\\n&\\equiv& \\frac{4{\\mathcal M}_z}{D_L}\n\\left[\\pi{\\cal M}_z f(t)\\right]^{2\/3}\n{\\cal A}_+(\\hat{\\bf n},\\hat{\\bf L})\\cos[\\Phi(t)]\\;;\n\\label{eq:hplus}\\\\\nh_{\\times}(t) & = & - \\frac{4{\\mathcal M}_z}{D_L}\n[\\pi{\\cal M}_z f(t)]^{2\/3}\n(\\mathbf{\\hat{L}}\\cdot \\mathbf{\\hat{n}}) \\sin[\\Phi(t)]\\;,\n\\nonumber\\\\\n&\\equiv& \\frac{4{\\mathcal M}_z}{D_L}\n\\left[\\pi{\\cal M}_z f(t)\\right]^{2\/3}\n{\\cal A}_\\times(\\hat{\\bf n},\\hat{\\bf L})\\sin[\\Phi(t)]\\;.\n\\label{eq:hcross}\n\\end{eqnarray}\nEquations (\\ref{eq:hplus}) and (\\ref{eq:hcross}) are nearly identical\nto those given in Eq.\\ (\\ref{eq:NewtQuad}); only the phase $\\Phi(t)$\nis different, as described below.  ${\\cal M}_z$ is the binary's\nredshifted chirp mass, $D_L$ is its luminosity distance, and we have\nwritten the inclination angle $\\cos\\iota$ using the vectors $\\hat{\\bf\nn}$ and $\\hat{\\bf L}$.  The functions ${\\cal A}_{+,\\times}$ compactly\ngather all dependence on sky position and orientation.  In Sec.\\\n{\\ref{sec:gwmeasure}} we discuss how these polarizations interact\nwith our detectors.\n\nIn these forms of $h_+$ and $h_\\times$, the phase is computed to\n2nd-post-Newtonian (2PN) order \\citep{bdiww95}:\n\\begin{eqnarray}\n\\Phi(t) &=& 2\\pi \\int f(t')\\,dt' = 2\\pi \\int \\frac{f}{df\/dt}df\\;,\n\\\\\n\\frac{df}{dt} &=& \\frac{96}{5}\\pi^{8\/3}\n\\mathcal{M}_z^{5\/3}f^{11\/3}\\left[1 -\n\\left(\\frac{743}{336} + \\frac{11}{4}\\eta\\right)(\\pi M_z f)^{2\/3}\n\\right.\n\\nonumber\\\\\n& &\\quad\\left. +\n4\\pi(\\pi M_z f) \\right. \\nonumber \\\\\n& &\\quad \\left. +\n\\left(\\frac{34103}{18144} + \\frac{13661}{2016}\\eta +\n\\frac{59}{18}\\eta^2 \\right)(\\pi M_z f)^{4\/3}\\right] \\;.\n\\label{eq:freqchirp}\n\\end{eqnarray}\nHigher order results for $df\/dt$ are now known (\\citealt{bij02,\nbfij02, blanchet04}), but 2PN order will be adequate for our purposes.\nSince distance measurements depend on accurate amplitude\ndetermination, we do not need a highly refined model of the wave's\nphase.  The rate of sweep is dominantly determined by the chirp mass,\nbut there is an important correction due to $\\eta = \\mu \/M = m_1\nm_2\/(m_1 + m_2)^2$, the reduced mass ratio.  Note that $\\eta$ is not\nredshifted; both $\\mu$ and $M$ (the reduced mass and total mass,\nrespectively) acquire $(1 + z)$ corrections, so their ratio is the\nsame at all $z$.  Accurate measurement of the frequency sweep can thus\ndetermine both ${\\cal M}_z$ and $\\eta$ (or ${\\cal M}_z$ and $\\mu_z$).\n\nWe will find it useful to work in the frequency domain, using the\nFourier transform ${\\tilde h}(f)$ rather than $h(t)$:\n\\begin{equation}\n{\\tilde h}(f) \\equiv \\int_{-\\infty}^{\\infty}\\, e^{2\\pi i f t}h(t)\\, dt\\;.\n\\label{eq:fourierT}\n\\end{equation}\nAn approximate result for ${\\tilde h}(f)$ can be found using\nstationary phase \\citep{fc93}, which describes the Fourier transform\nwhen $f$ changes slowly:\n\\begin{eqnarray}\n\\tilde{h}_+(f) &=& \\sqrt{\\frac{5}{96}}\\frac{\\pi^{-2\/3} {\\mathcal\nM}_z^{5\/6}}{D_L}{\\cal A}_+ f^{-7\/6} e^{i\\Psi(f)} \\, ,\n\\label{eq:freqdomainhp}\n\\\\\n\\tilde{h}_\\times(f) &=& \\sqrt{\\frac{5}{96}}\\frac{\\pi^{-2\/3} {\\mathcal\nM}_z^{5\/6}}{D_L}{\\cal A}_\\times f^{-7\/6} e^{i\\Psi(f) - i\\pi\/2} \\, .\n\\label{eq:freqdomainhc}\n\\end{eqnarray}\n``Slowly'' means that $f$ does not change very much over a single wave\nperiod $1\/f$, so that $(df\/dt)\/f \\ll f$.  The validity of this\napproximation for the waveforms we consider, at least until the last\nmoments before merger, has been demonstrated in previous work\n{\\citep{droz99}}.  The phase function $\\Psi(f)$ in Eqs.\\\n(\\ref{eq:freqdomainhp}) and (\\ref{eq:freqdomainhc}) is given by\n\\begin{eqnarray}\n\\Psi(f) &=& 2\\pi f t_c - \\Phi_c - \\frac{\\pi}{4} + \\frac{3}{128}(\\pi\n{\\mathcal M} f)^{-5\/3}\n\\times\\nonumber\\\\\n& &\\left[1 + \\frac{20}{9}\\left(\\frac{743}{336} +\n\\frac{11}{4}\\eta\\right)(\\pi M_z f)^{2\/3} \\right.\n\\nonumber\\\\\n& &\\left. -16\\pi( \\pi M_z f)\\right.\n\\nonumber\\\\\n& &\\left.+ 10\\left(\\frac{3058673}{1016064} +\n\\frac{5429}{1008}\\eta + \\frac{617}{144}\\eta^2 \\right)(\\pi\nM_z f)^{4\/3}\\right] \\, .\n\\nonumber\\\\\n\\label{eq:PNpsi}\n\\end{eqnarray}\nAs in Eq.\\ (\\ref{eq:NewtQuad}), $t_c$ is called the ``time of\ncoalescence'' and defines the time at which $f$ diverges within the PN\nframework; $\\Phi_c$ is similarly the ``phase at coalescence.''  We\nassume an abrupt and unphysical transition between inspiral and merger\nat the innermost stable circular orbit (ISCO), $f_{\\rm ISCO}=(6\n\\sqrt{6} \\pi M_z)^{-1}$.  For NS-NS, $f_{\\rm ISCO}$ occurs at high\nfrequencies where detectors have poor sensitivity.  As such, we are\nconfident that this abrupt transition has little impact on our\nresults.  For NS-BH, $f_{\\rm ISCO}$ is likely to be in a band with\ngood sensitivity, and better modeling of this transition will be\nimportant.\n\nIn this analysis we neglect effects which depend on spin.  In general\nrelativity, spin drives precessions which can ``color'' the waveform\nin important ways, and which can have important observational effects\n(see, e.g., \\citealt{v04}, \\citealt{lh06}, \\citealt{vandersluys08}).\nThese effects are important when the dimensionless spin parameter, $a\n\\equiv c|{\\bf S}|\/GM^2$, is fairly large.  Neutron stars are unlikely\nto spin fast enough to drive interesting precession during the time\nthat they are in the band of GW detectors.  To show this, write the\nmoment of inertia of a neutron star as\n\\begin{equation}\nI_{\\rm NS} = \\frac{2}{5}\\kappa M_{\\rm NS} R_{\\rm NS}^2\\;,\n\\end{equation}\nwhere $M_{\\rm NS}$ and $R_{\\rm NS}$ are the star's mass and radius,\nand the parameter $\\kappa$ describes the extent to which its mass is\ncentrally condensed (compared to a uniform sphere).  Detailed\ncalculations with different equations of state indicate $\\kappa \\sim\n0.7$--$1$ [cf.\\ \\cite{cook94}, especially the slowly rotating\nconfigurations in their Tables 12, 15, 18, and 21].  For a neutron\nstar whose spin period is $P_{\\rm NS}$, the Kerr parameter is given by\n\\begin{eqnarray}\na_{\\rm NS} &=& \\frac{c}{G}\\frac{I_{\\rm NS}}{M_{\\rm NS}^2}\n\\frac{2\\pi}{P_{\\rm NS}}\n\\nonumber\\\\\n&\\simeq& 0.06 \\kappa \\left(\\frac{R_{\\rm\nNS}}{\\rm{12\\,km}}\\right)^2\\left(\\frac{1.4\\,M_\\odot}{M_{\\rm NS}}\\right)\n\\left(\\frac{10\\,{\\rm msec}}{P_{\\rm NS}}\\right)\\;.\n\\end{eqnarray}\nAs long as the neutron star spin period is longer than $\\sim10$ msec,\n$a_{\\rm NS}$ is small enough that spin effects can be neglected in our\nanalysis.  We {\\it should}\\\/ include spin in our models of BH-NS\nbinaries; we leave this to a later analysis.  Van der Sluys et al.\\\n(2008) included black hole spin effects in an analysis which did not\nassume known source position. They found that spin-induced modulations\ncould help GW detectors to localize a source.  This and companion\nworks (\\citealt{raymond09}, \\citealt{vandersluys09}) suggest that, if\nposition is known, spin modulations could improve our ability to\nmeasure source inclination and distance.\n\nOur GWs depend on nine parameters: two masses ${\\cal M}_z$ and\n$\\mu_z$, two sky position angles (which set $\\hat{\\bf n}$), two\norientation angles (which set $\\hat{\\bf L}$), time at coalescence\n$t_c$, phase at coalescence $\\Phi_c$, and luminosity distance $D_L$.\nWhen sky position is known, the parameter set is reduced to seven: $\\{\n{\\cal M}_z, \\mu_z, D_L, t_c, \\cos \\iota, \\psi, \\Phi_c \\}$.\n\n\\subsection{Measurement of GWs by a detector network}\n\\label{sec:gwmeasure}\n\nWe now examine how the waves described in Sec.\\ {\\ref{sec:gwform}}\ninteract with a network of detectors.  We begin by introducing a\ngeometric convention, which follows that introduced in CF94 and in\n\\cite{abcf01}.  A source's sky position is given by a unit vector\n$\\hat{\\bf n}$ (which points from the center of the Earth to the\nbinary), and its orientation is given by a unit vector $\\hat{\\bf L}$\n(which points along the binary's orbital angular momentum).  We\nconstruct a pair of axes which describe the binary's orbital plane:\n\\begin{equation}\n\\hat{\\bf X} = \\frac{\\hat{\\bf n}\\times\\hat{\\bf L}}{|\\hat{\\bf\nn}\\times\\hat{\\bf L}|}\\;,\\quad\n\\hat{\\bf Y} = -\\frac{\\hat{\\bf n}\\times\\hat{\\bf X}}{|\\hat{\\bf\nn}\\times\\hat{\\bf X}|}\\;.\n\\label{eq:XYvectors}\n\\end{equation}\nWith these axes, we define the {\\it polarization basis tensors}\n\\begin{eqnarray}\n{\\bf e}^+ &=& \\hat{\\bf X} \\otimes \\hat{\\bf X} - \\hat{\\bf Y} \\otimes\n\\hat{\\bf Y}\\;,\n\\label{eq:plusbasis}\n\\\\\n{\\bf e}^\\times &=& \\hat{\\bf X} \\otimes \\hat{\\bf Y} + \\hat{\\bf Y}\n\\otimes \\hat{\\bf X}\\;.\n\\label{eq:timesbasis}\n\\end{eqnarray}\nThe transverse-traceless metric perturbation describing our source's\nGWs is then\n\\begin{equation}\nh_{ij} = h_+ e^+_{ij} + h_\\times e^\\times_{ij}\\;.\n\\label{eq:wavetensor}\n\\end{equation}\n\nWe next characterize the GW detectors.  Each detector is an $L$-shaped\ninterferometer whose arms define two-thirds of an orthonormal triple.\nDenote by $\\hat{\\bf x}_a$ and $\\hat{\\bf y}_a$ the unit vectors along\nthe arms of the $a$-th detector in our network; we call these the $x$-\nand $y$-arms.  (The vector $\\hat{\\bf z}_a = \\hat{\\bf x}_a \\times\n\\hat{\\bf y}_a$ points radially from the center of the Earth to the\ndetector's vertex.)  These vectors define the {\\it response tensor}\\\/\nfor detector $a$:\n\\begin{equation}\nD^{ij}_a = \\frac{1}{2}\\left[(\\hat{\\bf x}_a)^i (\\hat{\\bf x}_a)^j -\n(\\hat{\\bf y}_a)^i (\\hat{\\bf y}_a)^j\\right]\\;.\n\\label{eq:detresponse}\n\\end{equation}\nThe response of detector $a$ to a GW is given by\n\\begin{eqnarray}\nh_a &=& D^{ij}_a h_{ij}\n\\nonumber\\\\\n&\\equiv& e^{- 2 \\pi i ({\\bf n}\\cdot{\\bf r}_a) f} (F_{a,+}h_+ +\nF_{a,\\times}h_\\times)\\;,\n\\label{eq:measuredwave}\n\\end{eqnarray}\nwhere $\\bf{r}_a$ is the position of the detector $a$ and the factor\n$({\\bf n}\\cdot {\\bf r}_a)$ measures the time of flight between it and\nthe coordinate origin.  The second form of Eq.\\\n(\\ref{eq:measuredwave}) shows how the antenna functions introduced in\nEq.\\ (\\ref{eq:hmeas}) are built from the wave tensor and the response\ntensor.\n\nOur discussion has so far been frame-independent, in that we have\ndefined all vectors and tensors without reference to coordinates.  We\nnow introduce a coordinate system for our detectors following\n\\cite{abcf01} [who in turn use the WGS-84 Earth model\n{\\citep{althouse_etal}}].  The Earth is taken to be an oblate\nellipsoid with semi-major axis $a = 6.378137 \\times 10^6$ meters, and\nsemi-minor axis $b = 6.356752314 \\times 10^6$ meters.  Our coordinates\nare fixed relative to the center of the Earth.  The $x$-axis (which\npoints along ${\\bf i}$) pierces the Earth at latitude $0^\\circ$ North,\nlongitude $0^\\circ$ East (normal to the equator at the prime\nmeridian); the $y$-axis (along ${\\bf j}$) pierces the Earth at\n$0^\\circ$ North, $90^\\circ$ East (normal to the equator in the Indian\nocean somewhat west of Indonesia); and the $z$-axis (along ${\\bf k}$)\npierces the Earth at $90^\\circ$ North (the North geographic pole).\n\nA GW source at $(\\theta,\\phi)$ on the celestial sphere has sky\nposition vector $\\hat{\\bf n}$:\n\\begin{equation}\n\\hat{\\bf n} = \\sin\\theta\\cos\\phi{\\bf i} + \\sin\\theta\\sin\\phi{\\bf j} +\n\\cos\\theta{\\bf k}\\;.\n\\end{equation}\nThe {\\it polarization angle}, $\\psi$, is the angle (measured clockwise\nabout $\\hat{\\bf n}$) from the orbit's line of nodes to the source's\n$\\hat{\\bf X}$-axis.  In terms of these angles, the vectors $\\hat{\\bf\nX}$ and $\\hat{\\bf Y}$ are given by {\\citep{abcf01}}\n\\begin{eqnarray}\n\\hat{\\bf X} &=& (\\sin\\phi\\cos\\psi - \\sin\\psi\\cos\\phi\\cos\\theta) {\\bf\ni}\\nonumber\\\\\n& & - (\\cos\\phi\\cos\\psi + \\sin\\psi\\sin\\phi\\cos\\theta){\\bf j}\n+\\sin\\psi\\sin\\theta {\\bf k}\\;,\n\\nonumber\\\\\n\\label{eq:Xvector2}\\\\\n\\hat{\\bf Y} &=& (-\\sin\\phi\\sin\\psi - \\cos\\psi\\cos\\phi\\cos\\theta) {\\bf\ni}\\nonumber\\\\\n& & + (\\cos\\phi\\sin\\psi - \\cos\\psi\\sin\\phi\\cos\\theta){\\bf j} +\n\\cos\\psi\\sin\\theta {\\bf k}\\;.\n\\nonumber\\\\\n\\label{eq:Yvector2}\n\\end{eqnarray}\nThe angle $\\phi$ is related to right ascension $\\alpha$ by $\\alpha =\n\\phi + {\\rm GMST}$ (where GMST is the Greenwich mean sidereal time at\nwhich the signal arrives), and $\\theta$ is related to declination\n$\\delta$ by $\\delta = \\pi\/2 - \\theta$ (cf.\\ \\citealt{abcf01}, Appendix\nB).  Combining Eqs.\\ (\\ref{eq:Xvector2}) and (\\ref{eq:Yvector2}) with\nEqs.\\ (\\ref{eq:plusbasis})--(\\ref{eq:wavetensor}) allows us to write\n$h_{ij}$ for a source in coordinates adapted to this problem.\n\nWe now similarly describe our detectors using convenient coordinates.\nDetector $a$ is at East longitude $\\lambda_a$ and North latitude\n$\\varphi_a$ (not to be confused with sky position angle $\\phi$).  The\nunit vectors pointing East, North, and Up for this detector are\n\\begin{eqnarray}\n{\\bf e}^{\\rm E}_a &=& -\\sin\\lambda_a{\\bf i} + \\cos\\lambda_a{\\bf j}\\;,\n\\label{eq:eastunitvec}\n\\\\\n{\\bf e}^{\\rm N}_a &=& -\\sin\\varphi_a\\cos\\lambda_a{\\bf i} -\n\\sin\\varphi_a\\sin\\lambda_a{\\bf j} + \\cos\\varphi_a{\\bf k}\\;,\n\\label{eq:northunitvec}\n\\\\\n{\\bf e}^{\\rm U}_a &=& \\cos\\varphi_a\\cos\\lambda_a{\\bf i} +\n\\cos\\varphi_a\\sin\\lambda_a{\\bf j} - \\cos\\varphi_a{\\bf k}\\;.\n\\label{eq:upunitvec}\n\\end{eqnarray}\nThe $x$-arm of detector $a$ is oriented at angle $\\Upsilon_a$ North of\nEast, while its $y$-arm is at angle $\\Upsilon_a + \\pi\/2$.  Thanks to\nthe Earth's oblateness, the $x$- and $y$-arms are tilted at angles\n$\\omega^{x,y}_a$ to the vertical.  The unit vectors $\\hat{\\bf x}_a$,\n$\\hat{\\bf y}_a$ can thus be written\n\\begin{eqnarray}\n\\hat{\\bf x}_a &=& \\cos\\omega^x_a \\cos\\Upsilon_a{\\bf e}^{\\rm E}_a +\n\\cos\\omega^x_a\\sin\\Upsilon_a{\\bf e}^{\\rm N}_a + \\sin\\omega^x_a{\\bf\ne}^{\\rm U}\\;,\n\\nonumber\\\\\n\\label{eq:detectorxhat}\n\\\\\n\\hat{\\bf y}_a &=& -\\cos\\omega^y_a\\sin\\Upsilon_a{\\bf e}^{\\rm E}_a +\n\\cos\\omega^y_a\\cos\\Upsilon_a{\\bf e}^{\\rm N}_a + \\sin\\omega^y_a{\\bf\ne}^{\\rm U}\\;.\n\\nonumber\\\\\n\\label{eq:detectoryhat}\n\\end{eqnarray}\nCombining Eqs.\\ (\\ref{eq:detectorxhat}) and (\\ref{eq:detectoryhat})\nwith Eq.\\ (\\ref{eq:detresponse}) allows us to write the response\ntensor for each detector in our network.\n\n\\subsection{Summary of the preceding section}\n\\label{sec:gwmeasure_summary}\n\nSection {\\ref{sec:gwmeasure}} is sufficiently dense that a brief\nsummary may clarify its key features, particularly with respect to the\nquantities we hope to measure.  From Eq.\\ (\\ref{eq:measuredwave}), we\nfind that each detector in our network measures a weighted sum of the\ntwo GW polarizations $h_+$ and $h_\\times$.  Following \\cite{cutler98},\nwe can rewrite the waveform detector $a$ measures as\n\\begin{equation}\nh_a = \\frac{4{\\cal M}_z}{D_L}{\\cal A}_p\\left[\\pi {\\cal M}_z\nf(t)\\right]^{2\/3} \\cos\\left[\\Phi(t) + \\Phi_p\\right]\\;,\n\\label{eq:measuredwave2}\n\\end{equation}\nwhere we have introduced detector $a$'s ``polarization amplitude''\n\\begin{equation}\n{\\cal A}_p = \\sqrt{ \\left(F_{a,+}{\\cal A}_+\\right)^2 +\n\\left(F_{a,\\times}{\\cal A}_\\times\\right)^2}\\;,\n\\label{eq:polamp}\n\\end{equation}\nand its ``polarization phase''\n\\begin{equation}\n\\tan\\Phi_p = \\frac{F_{a,\\times}{\\cal A}_\\times}{F_{a,+}{\\cal A}_+}\\;.\n\\label{eq:polphase}\n\\end{equation}\nThe intrinsic GW phase, $\\Phi(t)$, is a strong function of the\nredshifted chirp mass, ${\\cal M}_z$, the redshifted reduced mass,\n$\\mu_z$, the time of coalescence, $t_c$, and the phase at coalescence,\n$\\Phi_c$.  Measuring the phase determines these four quantities,\ntypically with very good accuracy.\n\nConsider for a moment measurements by a single detector.  The\npolarization amplitude and phase depend on the binary's sky position,\n$(\\theta,\\phi)$ or $\\hat{\\bf n}$, and orientation, $(\\psi,\\iota)$ or\n$\\hat{\\bf L}$. [They also depend on detector position, $(\\lambda_a,\n\\varphi_a)$, orientation, $\\Upsilon_a$, and tilt, $(\\omega^x_a,\n\\omega^y_a)$.  These angles are known and fixed, so we ignore them in\nthis discussion.]  If the angles $(\\theta,\\phi,\\psi,\\iota)$ are not\nknown, a single detector cannot separate them, nor can it separate the\ndistance $D_L$.\n\nMultiple detectors can, at least in principle, separately determine\nthese parameters.  Each detector measures its own amplitude and\npolarization phase.  Combining their outputs, we can fit to the\nunknown angles and the distance.  Various works have analyzed how well\nthis can be done assuming that the position and orientation are\ncompletely unknown (\\citealt{sylvestre, cavalieretal, blairetal}).\nVan der Sluys et al.\\ (2008) performed such an analysis for\nmeasurements of NS-BH binaries, including the effect of orbital\nprecession induced by the black hole.  This precession effectively\nmake the angles $\\iota$ and $\\psi$ time dependent, also breaking the\ndegeneracy among these angles and $D_L$.\n\nIn what follows, we assume that an electromagnetic identification pins\ndown the angles $(\\theta,\\phi)$, so that they do not need to be\ndetermined from the GW data.  We then face the substantially less\nchallenging problem of determining $\\psi$, $\\iota$, and $D_L$.  We\nwill also examine the impact of a constraint on the inclination,\n$\\iota$.  Long bursts are believed to be strongly collimated, emitting\ninto jets with opening angles of just a few degrees.  Less is known\nabout the collimation of SHBs, but it is plausible that their emission\nmay be primarily along a preferred axis (presumably the progenitor\nbinary's orbital angular momentum axis).\n\n\\subsection{GW detectors used in our analysis}\n\\label{sec:detectors}\n\nHere we briefly summarize the properties of the GW detectors that we\nconsider.\n\n\\noindent\n{\\it LIGO}: The Laser Interferometer Gravitational-wave Observatory\nconsists of two 4 kilometer interferometers located in Hanford,\nWashington (US) and Livingston, Louisiana (US).  These instruments\nhave achieved their initial sensitivity goals.  An upgrade to\n``advanced'' configuration is expected to be completed around 2014,\nwith tuning for best sensitivity to be undertaken in the years\nfollowing\\footnote{http:\/\/www.ligo.caltech.edu\/advLIGO\/scripts\/summary.shtml}.\nWe show the anticipated noise limits from fundamental noise sources in Fig.\\\n{\\ref{fig:aligonoise}} for a broad-band tuning {\\citep{ligo_noise}}.\nThis spectrum is expected to be dominated by quantum sensing noise\nabove a cut-off at $f < 10$ Hz, with a contribution from thermal noise\nin the test mass coatings in the band from 30--200 Hz.\n\n\\begin{figure}\n\\centering \n\\includegraphics[angle=90,width=0.98\\columnwidth]{fig1.eps}\n\\caption{Anticipated noise spectrum for Advanced LIGO\n({\\citealt{ligo_noise}}; cf.\\ their Fig.\\ 3).  Our calculations assume\nno astrophysically interesting sensitivity below a low frequency\ncut-off of 10 Hz.  The features at $f \\simeq 10$ Hz and a few hundred\nHz are resonant modes of the mirror suspensions driven by thermal\nnoise.}\n\\label{fig:aligonoise}\n\\end{figure}\n\n\\noindent\n{\\it Virgo}: The Virgo detector \\citep{acernese08} near Pisa, Italy has slightly shorter\narms than LIGO (3 kilometers), but should achieve similar advanced\nsensitivity on roughly the same timescale as the LIGO\ndetectors\\footnote{http:\/\/www.ego-gw.it\/public\/virgo\/virgo.aspx}.  For\nsimplicity, we will take Virgo's sensitivity to be the same as LIGO's.\n\nOur baseline detector network consists of the LIGO Hanford and\nLivingston sites, and Virgo; these are instruments which are running\ntoday, and will be upgraded over the next decade.  We also examine the\nimpact of adding two proposed interferometers to this network:\n\n\\noindent\n{\\it AIGO}: The Australian International Gravitational Observatory \\citep{barriga10} is\na proposed multi-kilometer interferometer that would be located in\nGingin, Western Australia.  AIGO's proposed site in Western Australia\nis particularly favorable due to low seismic and human activity.\n\n\\noindent\n{\\it LCGT}: The Large-scale Cryogenic Gravitational-wave Telescope \\citep{kuroda10} is\na proposed multi-kilometer interferometer that would be located in the\nKamioka observatory, 1 kilometer underground.  This location takes\nadvantage of the fact that local ground motions tend to decay rapidly\nas we move away from the Earth's surface.  They also plan to use\ncryogenic cooling to reduce thermal noise.\n\nAs with Virgo, we will take the sensitivity of AIGO and LCGT to be the\nsame as LIGO for our analysis.  Table {\\ref{tab:detectors}} gives the\nlocation and orientation of these detectors, needed to compute\neach detector's response function.  It's worth mentioning that more\nadvanced detectors are in the early planning stages.  Particularly\nnoteworthy is the European proposal for the ``Einstein Telescope,''\ncurrently undergoing design studies.  It is being designed to study\nbinary coalescence to high redshift ($z \\gtrsim 5$) {\\citep{sathya09}}.\n\n\\begin{widetext}\n\n\\begin{deluxetable}{lccccc}\n\\tablewidth{17.5cm}\n\\tablecaption{GW detectors (positions and orientations).}\n\\tablehead{\n\\colhead{Detector} &\n\\colhead{East Long.\\ $\\lambda$} &\n\\colhead{North Lat.\\ $\\varphi$} &\n\\colhead{Orientation $\\Upsilon$} &\n\\colhead{$x$-arm tilt $\\omega^x$} &\n\\colhead{$y$-arm tilt $\\omega^y$}\n}\n\n\\startdata\n\nLIGO-Han & $-119.4^\\circ$ & $46.5^\\circ$ & $126^\\circ$ & $ (-6.20 \\times 10^{-4})^\\circ$ & $(1.25 \\times 10^{-5})^\\circ$ \\\\\nLIGO-Liv & $-90.8^\\circ$ & $30.6^\\circ$ & $198^\\circ$ & $ (-3.12 \\times 10^{-4})^\\circ$ & $(-6.11 \\times 10^{-4} )^\\circ$ \\\\\nVirgo & $10.5^\\circ$ & $43.6^\\circ$ & $70^\\circ$ & $0.0^\\circ$ & $0.0^\\circ$ \\\\\nAIGO & $115.7^\\circ$ & $-31.4^\\circ$ & $0^\\circ$ & $0.0^\\circ$ & $0.0^\\circ$ \\\\\nLCGT & $137.3^\\circ$ & $36.4^\\circ$ & $25^\\circ$ & $0.0^\\circ$ & $0.0^\\circ$ \\\\\n\n\\enddata\n\n\\label{tab:detectors}\n\\end{deluxetable}\n\\end{widetext}\n\n\n\\section{Estimation of binary parameters}\n\\label{sec:estimate}\n\n\\subsection{Overview of formalism}\n\\label{sec:formaloverview}\n\nWe now give a brief summary of the parameter estimation formalism we\nuse.  Further details can be found in \\cite{finn92}, \\cite{krolak93},\nand CF94.\n\nAssuming detection has occurred, the datastream of detector $a$,\n$s_a(t)$, has two contributions: The true GW signal\n$h_a(t;{\\boldsymbol{\\hat \\theta}})$ (constructed by contracting the GW\ntensor $h_{ij}$ with detector $a$'s response tensor $D^{ij}_a$; cf.\\\nSec.\\ {\\ref{sec:gwmeasure}}), and a realization of detector noise\n$n_a(t)$,\n\\begin{equation}\n\\label{eq:sig}\ns_a(t) = h_a(t; {\\boldsymbol{\\hat \\theta}}) + n_a(t)\\;.\n\\end{equation}\nThe incident gravitational wave strain depends on (unknown) true\nparameters ${\\boldsymbol{\\hat \\theta}}$.  As in Sec.\\\n\\ref{sec:dalaletal}, $\\boldsymbol{\\hat\\theta}$ is a vector whose\ncomponents are binary parameters.  Below we use a vector ${\\bf s}$\nwhose components $s_a$ are the datastreams of each detector.\nLikewise, ${\\bf h}$ and ${\\bf n}$ are vectors whose components are the\nGW and noise content of each detector.\n\nWe assume the noise to be stationary, zero mean, and Gaussian.  This\nlets us categorize it using the spectral density as follows.  First,\ndefine the noise correlation matrix:\n\\begin{eqnarray}\nC_n(\\tau)_{ab} &=& \\langle n_a(t + \\tau) n_b(t) \\rangle -\n\\langle n_a(t + \\tau) \\rangle \\, \\langle n_b(t) \\rangle\n\\nonumber\\\\\n&=& \\langle n_a(t + \\tau) n_b(t) \\rangle\\;,\n\\label{eq:autocov}\n\\end{eqnarray}\nwhere the angle brackets are ensemble averages over noise\nrealizations, and the zero mean assumption gives us the simplified\nform on the second line.  For $a = b$, this is the auto-correlation of\ndetector $a$'s noise; otherwise, it describes the correlation between\ndetectors $a$ and $b$.  The (one-sided) power spectral density matrix\nis the Fourier transform of this:\n\\begin{equation}\n\\label{eq:sn_def}\nS_n(f)_{ab} = 2 \\int_{-\\infty}^{\\infty} d \\tau \\, e^{2 \\pi i f \\tau}\nC_n(\\tau)_{ab}\\;.\n\\end{equation}\nThis is defined for $f > 0$ only.  For $a = b$, it is the spectral\ndensity of noise power in detector $a$; for $a \\ne b$, it again\ndescribes correlations between detectors.  From these definitions, one\ncan show that\n\\begin{equation}\n\\label{eq:noisestats}\n\\langle {\\tilde n}_a(f) \\, {\\tilde n}_b(f^\\prime)^* \\rangle = {1 \\over 2}\n\\delta(f - f^\\prime) S_n(f)_{ab}.\n\\end{equation}\nFor Gaussian noise, this statistic completely characterizes our\ndetector noise.  No real detector is completely Gaussian, but by using\nmultiple, widely-separated detectors non-Gaussian events can be\nrejected.  For this analysis, we assume the detectors' noises are\nuncorrelated such that Eq.\\ (\\ref{eq:noisestats}) becomes\n\\begin{equation}\n\\label{eq:noisestatsunocrrelated}\n\\langle {\\tilde n}_a(f) \\, {\\tilde n}_b(f^\\prime)^* \\rangle = {1 \\over\n2} \\delta_{ab} \\delta(f - f^\\prime) S_n(f)_a.\n\\end{equation}\nFinally, for simplicity we assume that $S_n(f)_a$ has the universal\nshape $S_n(f)$ projected for advanced LIGO, shown in Fig.\\\n\\ref{fig:aligonoise}.\n\nMany of our assumptions are idealized (Gaussian noise; identical noise\nspectra; no correlated noise between interferometers), and will\ncertainly not be achieved in practice.  These idealizations greatly\nsimplify our analysis, however, and are a useful baseline.  It would\nbe useful to revisit these assumptions and understand the quantitative\nimpact that they have on our analysis, but we do not expect a major\nqualitative change in our conclusions.\n\nThe central quantity of interest in parameter estimation is the\nposterior probability distribution function (PDF) for\n${\\boldsymbol{\\theta}}$ given detector output {\\bf s}, which is\ndefined as\n\\begin{equation}\n\\label{eq:postPDF}\np({\\boldsymbol \\theta} \\, | \\, {\\bf s}) = {\\cal N} \\, p^{(0)}\n({\\boldsymbol{\\theta}}) {\\cal L}_{\\rm TOT} ({\\bf s} \\, |\n\\,{\\boldsymbol{\\theta}}) \\,.\n\\end{equation}\n${\\cal N}$ is a normalization constant,\n$p^{(0)}({\\boldsymbol{\\theta}})$ is the PDF that represents the prior\nprobability that a measured GW is described by the parameters\n$\\boldsymbol{\\theta}$, and ${\\cal L}_{\\rm TOT} (\\bf{s} \\, | \\,\n{\\boldsymbol \\theta} )$ is the total {\\it likelihood function} (e.g.,\n\\citealt{mackay03}).  The likelihood function measures the relative\nconditional probability of observing a particular dataset $\\bf{s}$\ngiven a measured signal ${\\bf h}$ depending on some unknown set of\nparameters $\\boldsymbol{\\theta}$ and given noise ${\\bf n}$.  Because\nwe assume that the noise is independent and uncorrelated at each\ndetector site, we may take the total likelihood function to be the\nproduct of the individual likelihoods at each detector:\n\\begin{equation}\n\\label{eq:totLike}\n{\\cal L}_{\\rm TOT} ({\\bf s} \\, | \\,{\\boldsymbol \\theta}) = \\Pi_{a}\n{\\cal L}_a (s_a \\, | \\,{\\boldsymbol \\theta})\\;,\n\\end{equation}\nwhere ${\\cal L}_a$, the likelihood for detector $a$, is given by\n\\citep{finn92}\n\\begin{equation}\n\\label{eq:Like}\n{\\cal L}_a \\, (s \\, | \\,{\\boldsymbol \\theta}) = \\, e^{ -\n\\big( h_a({\\boldsymbol \\theta}) - s_a \\, \\big| \\, h_a({\\boldsymbol\n\\theta}) - s_a \\big)\/2 } \\, .\n\\end{equation}\nThe inner product $\\left( \\ldots | \\ldots \\right)$ on the vector space\nof signals is defined as\n\\begin{equation}\n(g|h) = 2 \\int_0^{\\infty} df \\frac{\\tilde{g}^*(f)\\tilde{h}(f) +\n\\tilde{g}(f)\\tilde{h}^*(f)}{S_n(f)} \\, .\n\\label{eq:innerproduct}\n\\end{equation}\nThis definition means that the probability of the noise $n(t)$\ntaking some realization $n_0(t)$ is\n\\begin{equation}\n\\label{eq:noise_distribution}\np(n = n_0) \\, \\propto \\, e^{- \\left( n_0 | n_0 \\right) \/ 2 }.\n\\end{equation}\n\\noindent \n\nFor clarity, we distinguish between various definitions of SNR.  The\n{\\it true}\\\/ SNR at detector $a$, associated with a given instance of\nnoise for a measurement at a particular detector, is defined as (CF94)\n\\begin{eqnarray}\n\\left({S\\over N}\\right)_{a, {\\rm true}} & = & { \\left( h_a \\, |\n\\, s_a \\right) \\over \\sqrt{ \\left( h_a \\, | \\, h_a\\right) } }\\;.\n\\label{eq:snr_true}\n\\end{eqnarray}\nThis is a random variable with Gaussian PDF of unit variance. For an\nensemble of realizations of the detector noise $n_a$, the {\\it\naverage} SNR at detector $a$ is given by\n\\begin{equation}\n\\label{eq:snr_ave}\n\\left({S\\over N}\\right)_{a, {\\rm ave}} = {{(h_a | h_a)}\\over\n{ {\\rm rms}\\ (h_a|n_a)}} = (h_a|h_a)^{1\/2}.\n\\end{equation}\nConsequently, we can define the combined {\\it true} and {\\it average}\nSNRs of a coherent network of detectors:\n\\begin{eqnarray}\n\\left({S\\over N}\\right)_{{\\rm true}} & = & \\sqrt{\\sum_a \\left({S\\over\nN}\\right)^2_{a, {\\rm true}}}\\ \\ ,\n\\label{eq:snr_total}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\left({S\\over N}\\right)_{{\\rm ave}} & = & \\sqrt{\\sum_a \\left({S\\over\nN}\\right)^2_{a, {\\rm ave}}}\\ \\ .\n\\label{eq:snr_tot_ave}\n\\end{eqnarray}\n\nEstimating the parameter set ${\\boldsymbol{\\theta}}$ is often done\nusing a ``maximum likelihood'' method following either a Bayesian\n(\\citealt{loredo89}, \\citealt{finn92}, CF94, \\citealt{poisson95}) or\nfrequentist point of view (\\citealt{krolak93}, CF94).  We do not\nattempt to review these philosophies, and instead refer to Appendix A2\nof CF94 for detailed discussion.  It is worth noting that, in the GW\nliterature, the ``maximum likelihood'' or ``maximum a posterior'' are\noften interchangeably referred to as ``best-fit'' parameters.  The\nmaximum a posterior is the parameter set\n$\\boldsymbol{\\tilde\\theta}_{\\rm MAP}$ which maximizes the full\nposterior probability, Eq.\\ (\\ref{eq:postPDF}); likewise, the maximum\nlikelihood is the parameter set $\\boldsymbol{\\tilde\\theta}_{\\rm ML}$\nwhich maximizes the likelihood function, Eq.\\ (\\ref{eq:totLike}).\n\nFollowing the approach advocated by CF94, we introduce the Bayes\nestimator ${\\tilde \\theta}_{\\rm BAYES}^i({\\bf s})$,\n\\begin{equation}\n\\label{eq:Bayes}\n{\\tilde \\theta}_{\\rm BAYES}^i({\\bf s}) \\equiv \\int {\\theta}^i\\,\np(\\boldsymbol{\\theta} \\, | \\, {\\bf s}) d\\boldsymbol{\\theta}\\;.\n\\end{equation}        \nThe integral is performed over the whole parameter set\n$\\boldsymbol{\\theta}$; $d\\boldsymbol{\\theta} = d\\theta^1d\\theta^2\\dots\nd\\theta^n$. Similarly, we define the rms measurement errors\n$\\Sigma_{\\rm BAYES}^{ij}$\n\\begin{equation}\n\\label{eq:sigma_bayes}\n\\Sigma_{\\rm BAYES}^{ij} = \\int ({\\theta}^i - {\\tilde \\theta}^i_{\\rm\nBAYES}) \\, ({\\theta}^j - {\\tilde \\theta}^j_{\\rm BAYES}) \\,\np(\\boldsymbol{ \\theta} \\, | \\, {\\bf s}) d\\boldsymbol{\\theta} .\n\\end{equation}\nTo understand the meaning of ${\\tilde\\theta}_{\\rm BAYES}^i({\\bf s})$,\nconsider a single detector which records an arbitrarily large ensemble\nof signals.  This ensemble will contain a sub-ensemble in which the\nvarious $s(t)$ are identical to one another.  Each member of the\nsub-ensemble corresponds to GW signals with different true parameters\n$\\boldsymbol{\\hat \\theta}$, but have noise realizations $n(t)$ that\nconspire to produce the same $s(t)$.  In this case, ${\\tilde\n\\theta}_{\\rm BAYES}^i({\\bf s})$ is the expectation of $\\theta^i$\naveraged over the sub-ensemble.  The principle disadvantage of the\nBayes estimator is the computational cost to evaluate the\nmulti-dimensional integrals in Eqs.\\ (\\ref{eq:Bayes}) and\n(\\ref{eq:sigma_bayes}).\n\nFor large SNR it can be shown that the estimators\n$\\boldsymbol{\\tilde\\theta}_{\\rm ML}$, $\\boldsymbol{\\tilde\\theta}_{\\rm\nMAP}$, and $\\boldsymbol{\\tilde\\theta}_{\\rm BAYES}$ agree with one\nanother (CF94), and that Eq.\\ (\\ref{eq:postPDF}) is well-described by\na Gaussian form [cf.\\ Eq.\\ (\\ref{eq:gaussian})].  However, as\nillustrated in Sec.\\ IVD of CF94, effects due to prior information and\nwhich scale nonlinearly with $1\/\\mbox{SNR}$ contribute significantly\nat low SNR.  The Gaussian approximation then tends to underestimate\nmeasurement errors by missing tails or multimodal structure in\nposterior distributions.\n\nWe emphasize that in this analysis we do not consider systematic\nerrors that occur due to limitations in our source model or to\ngravitational lensing effects.  A framework for analyzing systematic\nerrors in GW measurements has recently been presented by \\cite{cv07}.\nAn important follow-on to this work will be to estimate systematic\neffects and determine whether they significantly change our\nconclusions.\n\n\\subsection{Binary Selection and Priors}\n\\label{sec:selectionandpriors}\n\nWe now describe how we generate a sample of detectable GW-SHB events.\nWe assume a constant comoving density (\\citealt{peebles93},\n\\citealt{hogg99}) of GW-SHB events, in a $\\Lambda$CDM Universe with\n$H_0=70.5\\ \\mbox{km}\/\\mbox{sec}\/\\mbox{Mpc}$, $\\Omega_{\\Lambda}=0.726$,\nand $\\Omega_{m}=0.2732$ \\citep{komatsu09}.  We distribute $10^6$\nbinaries uniformly in volume with random sky positions and\norientations to redshift $z = 1$ ($D_L \\simeq 6.6$ Gpc).  We then\ncompute the average SNR, Eq.\\ (\\ref{eq:snr_ave}), for each binary at\neach detector, and use Eq.\\ (\\ref{eq:snr_tot_ave}) to compute the\naverage total SNR for each network we consider.  We assume prior\nknowledge of the merger time (since we have assumed that the inspiral is\ncorrelated with a SHB), so we set a threshold SNR for the {\\it total}\ndetector network, $\\mbox{SNR}_{\\rm total} = 7.5$ (see discussion in\nDHHJ06).  This is somewhat reduced from the threshold we would set in\nthe absence of a counterpart, since prior knowledge of merger time and\nsource position reduces the number of search templates we need by a\nfactor $\\sim 10^{5}$ (\\citealt{kp93}, \\citealt{Owen96}).  Using the\naverage SNR to set our threshold introduces a slight error into our\nanalysis, since the true SNR will differ from the average. Some events\nwhich we identify as above threshold could be moved below threshold\ndue to a measurement's particular noise realization.  However, some\nsub-threshold events will likewise be moved above threshold, and the net\neffect is not expected to be significant.\n\nOur threshold selects detectable GW-SHB events for each detector\nnetwork.  We define ``total detected binaries'' to mean binaries which\nare detected by a network of all five detectors---both LIGO\nsites, Virgo, AIGO, and LCGT.  Including AIGO and LCGT substantially\nimproves the number detected, as compared to just using the two LIGO\ndetectors and Virgo.  Assuming that all binary orientations are\nequally likely given an SHB (i.e., no beaming), we find that a LIGO-Virgo\nnetwork detects \n$50\\%$ of the total detected binaries; LIGO-Virgo-AIGO detects $74\\%$\nof the total; and LIGO-Virgo-LCGT detects $72\\%$ of the total.  Figure\n\\ref{fig:detected_binaries} shows the sky distribution of detected binaries for\nvarious detector combinations.\nNetworks which include LCGT tend to have rather uniform sky coverage.\nThose with AIGO cover the quadrants $\\cos\\theta > 0$, $\\phi > \\pi$ and\n$\\cos\\theta < 0$, $\\phi < \\pi$ particularly well.\n\n\\begin{figure}\n\\centering \n\\includegraphics[width=0.9\\columnwidth]{fig2.eps}\n\\caption{Detected NS-NS binaries for our various detector networks as\na function of sky position $(\\cos\\theta,\\phi)$.  The lower right panel\nshows the binaries detected by a five-detector network (both LIGO\nsites, Virgo, AIGO, and LCGT).  We find that LIGO plus Virgo (our\n``base'' network) only detects $50\\%$ of the five-detector events;\nLIGO, Virgo, and AIGO detect $74\\%$ of these events; and LIGO, Virgo,\nand LCGT, detect $72\\%$ of these events.  Detections are more\nuniformly distributed on the sky in networks that include LCGT; AIGO\nimproves coverage in two of the sky's quadrants.  Our coordinate\n$\\phi$ is related to right ascension $\\alpha$ by $\\phi=\\alpha-$GMST,\nwhere GMST is Greenwich Mean Sidereal Time; $\\theta$ is related to\ndeclination $\\delta$ by $\\theta = \\pi\/2 - \\delta$.}\n\\label{fig:detected_binaries}\n\\end{figure}\n\nOur selection method implicitly sets a prior distribution on our\nparameters.  For example, the thresholding procedure results in a\nsignificant bias in detected events toward face-on binaries, with\n$\\mathbf{\\hat L}\\cdot \\mathbf{\\hat n} \\rightarrow \\pm 1$.  Figure\n{\\ref{fig:marg2DDLcosinc}} shows the distribution of detectable NS-NS\nbinaries for the parameters $\\left(\\cos\\iota, D_L\\right)$.  Since we\nuse an unrealistic mass distribution $(1.4\\,M_\\odot$--$1.4\\,M_\\odot$\nNS-NS and $1.4\\,M_\\odot$--$10\\,M_\\odot$ NS-BH binaries), instead of a\nmore astrophysically realistic distribution, the implicit mass prior\nis uninteresting.  Figure \\ref{fig:SNRvsDL} shows the average total\nSNR versus the true $D_L$ of our sample of detectable NS-NS and NS-BH\nbinaries for our ``full'' network (LIGO, Virgo, AIGO, LCGT).  Very few\ndetected binaries have SNR above 30 for NS-NS, and above 70 for NS-BH.\nIt is interesting to note the different detectable ranges between the\ntwo populations: NS-BH binaries are detectable to over twice the\ndistance of NS-NS binaries.\n\n\\begin{figure}\n\\centering \n\\includegraphics[width=0.98\\columnwidth]{fig3.eps}\n\\caption{The 2-D marginalized prior distribution in luminosity\ndistance $D_L$ and cosine inclination $\\cos \\iota$.  Each point\nrepresents a detected NS-NS binary for a network comprising all five\ndetectors.  Notice the bias toward detecting face-on binaries\n($\\cos\\iota \\to \\pm 1$)---they are detected to much larger distances\nthan edge-on ($\\cos\\iota \\to 0$).}\n\\label{fig:marg2DDLcosinc}\n\\end{figure}\n\n\\begin{figure}\n\\centering \n\\includegraphics[width=0.98\\columnwidth]{fig4.eps}\n\\caption{Average network SNR versus luminosity distance of the total detected\nNS-NS and NS-BH binaries. This assumes an idealized network consisting\nof both LIGO detectors,\nVirgo, AIGO, and LCGT.  Left panel shows all detected NS-NS binaries\n(one point with SNR above 100 is omitted); right panel shows all\ndetected NS-BH binaries (one point with SNR above 350 is omitted).\nNotice the different axis scales: NS-BH binaries are detected to more\nthan twice the distance of NS-NS. The threshold SNR for the {\\it total}\ndetector network is 7.5, $\\mbox{SNR}_{\\rm total} = 7.5$.}\n\\label{fig:SNRvsDL}\n\\end{figure}\n\nWe are also interested in seeing the impact that prior knowledge of\nSHB collimation may have on our ability to measure these events.  To\ndate there exist only two tentative observations which suggest that\nSHBs may be collimated (\\citealt{grupeetal06},\n\\citealt{burrowsetal06}, \\citealt{soderbergetal06}); we therefore\npresent results for moderate collimation and for isotropic SHB\nemission.  To obtain a sample of beamed SHBs, we assume that the burst\nemission is collimated along the orbital angular momentum axis, where\nbaryon loading is minimized.  Following DHHJ06, we use a distribution\nfor $\\cos\\iota \\equiv v$ of $dP\/dv \\propto \\exp[-(1 -\nv)^2\/2\\sigma_v^2]$, with $\\sigma_v=0.05$.  This corresponds to a\nbeamed population with $68\\%$ of its distribution having an opening\njet angle within roughly $25^\\circ$.  We construct a beamed subsample\nby selecting events from the total sample of detected events such that\nthe final distribution in inclination angle follows $dP\/dv$.  Joint\nmeasurements of SHBs and GW-driven inspirals should enable us to\nconstrain beaming angles by comparing the measured rates for these two\npopulations.\n\n\\subsection{Markov-Chain Monte-Carlo approach}\n\\label{sec:mhmc}\n\nThe principle disadvantage of the Bayes estimators\n$\\tilde\\theta^i_{\\rm BAYES}$ and $\\Sigma^{ij}_{\\rm BAYES}$ is the high\ncomputational cost of evaluating the multi-dimensional integrals which\ndefine them, Eqs.\\ (\\ref{eq:Bayes}) and (\\ref{eq:sigma_bayes}).  To\nget around this problem, we use Markov-Chain Monte-Carlo (MCMC)\nmethods to explore the PDFs describing the seven parameters $\\{ {\\cal\nM}_c, \\mu, D_L, \\cos \\iota, \\psi, t_c, \\Phi_c \\}$.  MCMC methods are\nwidely used in diverse astrophysical applications, ranging from high\nprecision cosmology (e.g.\\ \\citealt{dunkley09}, \\citealt{sievers09})\nto extra-solar planet studies (e.g.\\ \\citealt{ford05},\n\\citealt{winn07}).  They have seen increased use in GW measurement and\nparameter estimation studies in recent years (e.g.,\n\\citealt{stroeer06}, \\citealt{wickham06}, \\citealt{cornish07},\n\\citealt{porter08}, \\citealt{rover07}, \\citealt{vandersluys08}).\n\nMCMC generates a random sequence of parameter states that sample the\nposterior distribution, $p(\\boldsymbol{\\theta} | \\mathbf{s})$.  Let\nthe $n$th sample in the sequence be $\\boldsymbol{\\theta}^{(n)}$.  If\none draws a total of $N$ random samples, Eqs.\\ (\\ref{eq:Bayes}) and\n(\\ref{eq:sigma_bayes}) can then be approximated as sample averages:\n\\begin{eqnarray}\n{\\tilde\\theta}^i_{\\rm BAYES} &\\simeq& \\frac{1}{N} \\sum_{n = 1}^N\n(\\theta^i)^{(n)}\\;,\n\\label{eq:approxBayes}\\\\\n\\Sigma^{ij}_{\\rm BAYES} &\\simeq& \\frac{1}{N} \\sum_{n = 1}^N\n\\left(\\tilde\\theta^i_{\\rm BAYES} - (\\theta^i)^{(n)}\\right)\n\\left(\\tilde\\theta^j_{\\rm BAYES} - (\\theta^j)^{(n)}\\right)\\;.\n\\nonumber\\\\\n\\label{eq:approxsigma_bayes}\n\\end{eqnarray}\nThe key to making this technique work is drawing a sequence that\nrepresents the posterior PDF.  We use the Metropolis-Hastings\nalgorithm to do this (\\citealt{metropolis53}, \\citealt{hastings70});\nsee \\cite{neal93}, \\cite{gilks96}, \\cite{mackay03}, and \\cite{cml04}\nfor in-depth discussion.  The MCMC algorithm we use is based on a\ngeneric version of CosmoMC\\footnote{See\nhttp:\/\/cosmologist.info\/cosmomc\/}, described in \\cite{lewis02}.\n\nAppropriate priors are crucial to any MCMC analysis.  We take the\nprior distributions in chirp mass ${\\cal M}_z$, reduced mass $\\mu_z$,\npolarization angle $\\psi$, coalescence time $t_c$, and coalescence\nphase $\\Phi_c$ to be {\\it flat} over the region of sample space where\nthe binary is detectable according to our selection procedure.  More\nspecifically, we choose\n\\begin{itemize}\n\n\\item $p^{(0)}({\\cal M}_z) = {\\rm constant}$ over the range\n$[1\\,M_\\odot$, $2\\,M_\\odot]$ for NS-NS; and over the range\n$[2.5\\,M_\\odot$, $4.9\\,M_\\odot]$ for NS-BH.  (The true chirp masses in\nthe binaries' rest frames are $1.2\\,M_\\odot$ for NS-NS and\n$3.0\\,M_\\odot$ for NS-BH.)\n\n\\item $p^{(0)}(\\mu_z) = {\\rm constant}$ over the range\n$[0.3\\,M_\\odot$, $2\\,M_\\odot]$ for NS-NS; and over the range\n$[0.5\\,M_\\odot$, $3.5\\,M_\\odot]$ for NS-BH.  (The true reduced masses\nin the binaries' rest frames are $0.7\\,M_\\odot$ for NS-NS and\n$1.2\\,M_\\odot$ for NS-BH.)\n\n\\item $p^{(0)}(\\psi) = {\\rm constant}$ over the range $[0,\\pi]$.\n\n\\item $p^{(0)}(t_c) = {\\rm constant}$ over the range $[-100\\,{\\rm\nsec}, 100\\,{\\rm sec}]$.  Since we assume that $t_c$ is close to the\ntime of the SHB event, it is essentially the time offset between the\nsystem's final GWs and its SHB photons.  We find that the range in\n$t_c$ we choose is almost irrelevant, as long as the prior is flat and\nincludes the true value.  No matter how broad we choose the prior in\n$t_c$, our posterior PDF ends up narrowly peaked around $\\hat t_c$.\n\n\\item $p^{(0)}(\\Phi_c) = {\\rm constant}$ over the range $[0, 2\\pi]$.\n\n\\end{itemize}\nThe prior distribution for $D_L$ is inferred by taking the density of\nSHBs to be uniform per unit comoving volume over the luminosity\ndistance range [0, 2 Gpc] for NS-NS binaries, and over the range [0, 5\nGpc] for NS-BH binaries.  For our sample with isotropic inclination\ndistribution, we put $p^{(0)}(\\cos \\iota) = {\\rm constant}$ over the\nrange $[-1,1]$.  When we assume SHB collimation, our prior in\n$\\cos\\iota \\equiv v$ is the same as the one that we used in our\nselection procedure discussed previously:\n\\begin{equation}\n\\frac{dp^{(0)}}{dv}(v) \\propto e^{-(1 -v)^2\/2\\sigma_v^2}\\;,\n\\end{equation}\nwith $\\sigma_v = 0.05$.\n\nWe then map out full distributions for each of our seven parameters,\nassessing the mean values [Eq.\\ (\\ref{eq:Bayes})] and the standard\ndeviations [Eq.\\ (\\ref{eq:sigma_bayes})].  We generate four chains\nwhich run in parallel on the CITA ``Sunnyvale'' Cluster.  Each chain\nruns for a maximum of $10^7$ steps; we find that the mean and median\nnumber of steps are $\\sim 10^5$ and $\\sim 10^4$, respectively. Each\nevaluation of the likelihood function takes $\\sim0.3$ seconds.  We use\nthe first 30\\% of a chain's sample states for ``burn in,'' and\ndiscard that data.  Our chains start at random offset parameter\nvalues, drawn from Gaussians centered on the true parameter values.  We\nassess convergence by testing whether the multiple chains have\nproduced consistent parameter distributions.  Following standard\npractice, we use the Gelman-Rubin convergence criterion, defining a\nsequence as ``converged'' if the statistic $R < 1.1$ on the last half\nof our samples; see \\cite{gr92} for more details. We use convergence\nas our stopping criterion.  Each simulation for every binary runs for\nan hour to forty-eight hours; the mean and median runtime are eight\nand three hours, respectively.\n\n\\subsection{The ``averaged'' posterior PDF}\n\\label{sec:averagedPDF}\n\nCentral to the procedure outlined above is the use of the datastream\n${\\bf s} = {\\bf h}(\\boldsymbol{\\theta}) + {\\bf n}$ which enters the\nlikelihood function ${\\cal L}_{\\rm TOT}({\\bf s} |\n\\boldsymbol{\\theta})$.  The resulting posterior PDF, and the\nparameters one infers, thus depend on the noise ${\\bf n}$ which one\nuses.  One may want to evaluate statistics that are in a well-defined\nsense ``typical'' given the average noise properties, rather than\ndepending on a particular noise instance.  Such averaging is\nappropriate, for example, when forecasting how well an instrument\nshould be able to measure the properties of a source or process.  We\nhave also found it is necessary to average when trying to compare our\nMCMC code's output with previous work.\n\nAs derived below, the averaged posterior PDF takes a remarkably simple\nform: It is the ``usual'' posterior PDF, Eq.\\ (\\ref{eq:postPDF}) with\nthe noise ${\\bf n}$ set to {\\it zero}.  This does not mean that one\nignores noise when constructing the averaged PDF; one still relates\nsignal amplitude to typical noise by the average SNR, Eq.\\\n(\\ref{eq:snr_ave}).  As such, the averaged statistics will show an\nimprovement in measurement accuracy as SNR is increased.\n\nTo develop a useful notion of averaged posterior PDF, consider the\nhypothetical (and wholly unrealistic) case in which we measure a\nsignal using $M$ different noise realizations for the same event.  The\njoint likelihood for these measurements is\n\\begin{equation}\n{\\cal L}^{\\rm joint}_{\\rm TOT}({\\bf s}_1, {\\bf s}_2, \\ldots {\\bf\ns}_M | \\boldsymbol{\\theta} ) = \\prod_{i=1}^M {\\cal L}_{\\rm TOT} ({\\bf s}_i | \\boldsymbol{\\theta})\\;.\n\\label{eq:pjointdef}\n\\end{equation}\nLet us define the ``average'' PDF as the product of the prior\ndistribution of the parameters multiplied by the geometric mean of the\nlikelihoods which describe these measurements:\n\\begin{equation}\np_{\\rm ave}(\\boldsymbol{\\theta} | {\\bf s}) \\equiv  {\\cal N} \\, p^{(0)} \n{\\cal L}^{\\rm joint}_{\\rm TOT}({\\bf s}_1, {\\bf s}_2, \\ldots {\\bf\ns}_M| \\boldsymbol{\\theta})^{1\/M}\\;.\n\\label{eq:pavedef}\n\\end{equation}\nExpanding this definition, we find\n\\begin{eqnarray}\np_{\\rm ave} (\\boldsymbol{\\theta} | \\bf{s}) & \\equiv &   {\\cal N} \\,\np^{(0)} \\prod_{i=1}^{M}\\, \\left[  {\\cal L}_{\\rm TOT} ({\\bf s}_i |\n  \\boldsymbol{\\theta}) \\right]^{1\/M}\\;, \n\\label{eq:avepostPDF}\n\\end{eqnarray}\nwhere the subscript $i$ denotes the $i$th noise realization in our set\nof $M$ observations.  The ``ensemble average likelihood function'' can\nin turn be expanded as\n\\begin{eqnarray}\n\\prod_{i=1}^{M} \\left[{\\cal L}_{\\rm TOT} ({\\bf s}_i |\n{\\boldsymbol{\\theta}})\\right]^{1\/M} & = & \\prod_{a} \\prod_{i=1}^{M}\n\\left[{\\cal L}_{a} (s_{a,i} \\, | \\,{\\boldsymbol \\theta})\\right]^{1\/M}\n\\nonumber\\\\\n&=& \\prod_{a} \\prod_{i=1}^{M} e^{ - \\big(h_a({\\boldsymbol\n \\theta}) - s_{a,i} \\, \\big| \\, h_a({\\boldsymbol \\theta}) - s_{a,i}\n \\big)\/2M }\n\\nonumber\\\\\n& = & \\prod_{a} e^{-\\big( h_a({\\boldsymbol \\theta}) -\n h_a(\\boldsymbol{\\hat{\\theta}}) \\, \\big| \\, h_a({\\boldsymbol \\theta})\n - h_a (\\boldsymbol{\\hat{\\theta}})\\big)\/2}\n\\nonumber\\\\\n&\\times&\n\\prod_{i=1}^M \\exp\\left[\\frac{1}{M}\\left( n_{a,i} \\, \\bigg|\n h_{a}(\\boldsymbol{\\theta}) - h_{a}(\\boldsymbol{\\hat\\theta})\\right)\n \\right]\n\\nonumber\\\\\n&\\times&\n\\prod_{i=1}^M \\exp\\left[-\\frac{1}{2M}\\left( n_{a,i} \\, \\bigg|\nn_{a,i}\\right)\\right]\\;.\n\\label{eq:multi_obs_likelihood}\n\\end{eqnarray}\nBy taking $M$ to be large, the last two lines of Eq.\\\n(\\ref{eq:multi_obs_likelihood}) can be evaluated as follows:\n\\begin{eqnarray}\n& &\\prod_{i=1}^M \\exp\\left[\\frac{1}{M}\\left( n_{a,i} \\, \\bigg|\n h_{a}(\\boldsymbol{\\theta}) - h_{a}(\\boldsymbol{\\hat\\theta})\\right)\n \\right] \n\\nonumber\\\\\n& &\\qquad\\qquad = \\exp\\left[\\frac{1}{M}\\sum_{i = 1}^M \\left(\n  n_{a,i} \\, \\bigg| h_{a}(\\boldsymbol{\\theta}) -\n  h_{a}(\\boldsymbol{\\hat\\theta})\\right) \\right]\n\\nonumber\\\\\n& &\\qquad\\qquad \\simeq \\exp\\left[\\left\\langle \\left( n_{a} \\,\n \\bigg| h_{a}(\\boldsymbol{\\theta}) -\n h_{a}(\\boldsymbol{\\hat\\theta})\\right)\\right\\rangle \\right]\n\\nonumber\\\\\n& & \\qquad\\qquad = 1\\;.\n\\end{eqnarray}\nHere, $\\langle \\ldots \\rangle$ denotes an ensemble average over noise\nrealizations (cf.\\ Sec.\\ {\\ref{sec:formaloverview}}), and we have used\nthe fact that our noise has zero mean.  Similarly, we find\n\\begin{eqnarray}\n\\prod_{i=1}^M \\exp\\left[-\\frac{1}{2M}\\left( n_{a,i} \\, \\bigg|\n n_{a,i}\\right)\\right] &=& \\exp\\left[-\\frac{1}{2M}\\sum_{i = 1}^M\n \\left( n_{a,i} \\, \\bigg| n_{a,i}\\right) \\right]\n\\nonumber\\\\ &\\simeq&\n \\exp\\left[-\\frac{1}{2}\\left\\langle \\left( n_{a} \\, \\bigg|\n n_{a}\\right)\\right\\rangle \\right]\n \\nonumber\\\\ &=& e^{-1}\\;.\n\\end{eqnarray}\nThis uses $\\langle (n_a | n_a) \\rangle = 2$, which can be proved using\nthe noise properties (\\ref{eq:autocov}), (\\ref{eq:sn_def}), and\n(\\ref{eq:noisestats}).\n\nPutting all this together, we finally find\n\\begin{equation}\np_{\\rm ave}(\\boldsymbol{\\theta} | {\\bf s}) = {\\cal N}\n p^{0}(\\boldsymbol{\\theta}) \\prod_{a} e^{ -\\big( h_a({\\boldsymbol\n \\theta}) - h_a (\\boldsymbol{\\hat{\\theta}}) \\, \\big| \\,\n h_a({\\boldsymbol \\theta}) - h_a (\\boldsymbol{\\hat{\\theta}}) \\big)\/2\n }\\;,\n\\label{eq:avepostPDF_final2}\n\\end{equation}\nwhere we have absorbed $e^{-1}$ into the normalization ${\\cal N}$.\nThe posterior PDF, averaged over noise realizations, is simply\nobtained by evaluating Eq.\\ (\\ref{eq:postPDF}) with the noise ${\\bf\nn}$ set to zero.\n\n\\section{Results I: Validation and Testing}\n\\label{sec:valid}\n\nWe now validate and test our MCMC code against results from CF94.  In\nparticular, we examine the posterior PDF for the NS-NS binary which was\nstudied in detail in CF94.  We also explore the dependence of distance\nmeasurement accuracies on the detector network and luminosity\ndistance, focusing on the strong degeneracy that exists between $\\cos\n\\iota$ and $D_L$.\n\n\\subsection{Comparison with CF94}\n\\label{sec:cf}\n\nValidation of our MCMC results requires comparing to work which goes\nbeyond the Gaussian approximation and Fisher matrix estimators.  In\nSection IVD of CF94, Cutler \\& Flanagan investigate effects that are\nnon-linear in $1\/\\mbox{SNR}$.  They show that such effects\nhave a significant impact on distance measurement accuracies for low\nSNR.  In particular, they find that Fisher-based estimates understate\ndistance measurement errors for a network of two LIGO detectors\nand Virgo.\n\nBecause they go beyond a Fisher matrix analysis, the results of CF94\nare useful for comparing to our results.  Their paper is also useful\nin that they take source position to be known.  Our approach is\nsufficiently different from CF94 that we do not expect perfect\nagreement, however.  The most important difference is that we directly map out\nthe posterior PDF and compute sample averages using Eqs.\\\n(\\ref{eq:Bayes}) and (\\ref{eq:sigma_bayes}), for the full parameter\nset $\\{ {\\cal M}_z, \\mu_z, D_L, \\cos \\iota, \\psi, t_c, \\Phi_c \\}$.  In\ncontrast, CF94 estimate measurement errors only for $D_L$, using an\napproximation on an analytic Bayesian derivation of the marginalized\nPDF for $D_L$.  Specifically, Cutler \\& Flanagan expand the\nexponential factor in Eq.\\ (\\ref{eq:postPDF}) beyond second order in\nterms of some ``best-fit'' maximum likelihood parameters.  Their\napproximation treats strong correlations between the parameters $D_L$\nand $\\cos \\iota$ that are non-linear in 1\/SNR.  However, other\ncorrelations between $D_L$ and $(\\psi, \\phi_c)$ are only considered to\nlinear order. They obtain an analytic expression for the posterior PDF\nof the variables $D_L$ and $\\cos \\iota$ in terms of their ``best-fit''\nmaximum-likelihood values $\\tilde{D}_L$ and $\\cos \\tilde{\\iota}$ [see\nEq.\\ (4.57) of CF94]. The marginalized 1-D posterior PDFs for $D_L$\nare then computed by numerically integrating over $\\cos \\iota$.  The\n1-D marginalized PDF we compute in parameter $\\theta_i$ is\n\\begin{equation}\n\\label{eq:margPDF}\np_{\\rm marg}(\\theta_i | {\\bf s}) = \\int \\dots \\int\np(\\boldsymbol{\\theta} | {\\bf s}) d\\theta_1 \\dots d\\theta_{i-1}\\;\nd\\theta_{i+1} \\dots d\\theta_N\n\\end{equation}\nwhere $p(\\boldsymbol{\\theta} | \\bf{s})$ is the posterior PDF given by\nEq.\\ (\\ref{eq:postPDF}) and $N$ is the number of dimensions of our\nparameter set.\n\nIn addition to this rather significant difference in techniques, there\nare some minor differences which also affect our comparison:\n\n\\begin{itemize}\n\n\\item We use the restricted 2PN waveform; CF94 use the leading\n``Newtonian, quadrupole'' waveform that we used for pedagogical\npurposes in Sec.\\ \\ref{sec:sirens}.  Since distance is encoded in the\nwaveform's amplitude, we do not expect that our use of a higher-order\nphase function will have a large impact.  However, to avoid any easily\ncircumvented mismatch, we adopt the Newtonian-quadrupole waveform for\nthese comparisons.  This waveform does not depend on reduced mass\n$\\mu$, so {\\it for the purpose of this comparison only}, our parameter\nspace is reduced from 7 to 6 dimensions.\n\n\\item We use the projected advanced sensitivity noise curve shown in\nFig.\\ (\\ref{fig:aligonoise}); CF94 use an analytical form [their Eq.\\\n(2.1)\\footnote{Note that it is missing an overall factor of $1\/5$ (E.\\\nE.\\ Flanagan, private communication).}] based on the best-guess for\nwhat advanced sensitivity would achieve at the time of their analysis.\nCompared to the most recent projected sensitivity, their curve\nunderestimates the noise at middle frequencies ($\\sim 40$--$150$ Hz)\nand overestimates it at high frequencies ($\\gtrsim 200$ Hz).  We adopt\ntheir noise curve for this comparison.  Because of these differences,\nCF94 rather seriously overestimates the SNR for NS-NS inspiral.  Using\ntheir noise curve, the average SNR for the binary analyzed in their Fig.\\\n10 is 12.4\\footnote{CF94 actually report an SNR of 12.8.  The\ndiscrepancy is due to rounding the parameter $r_0$ in their Eq.\\\n(4.28).  Adjusting to their preferred value (rather than computing\n$r_0$) gives perfect agreement.}; using our up-to-date model for\nadvanced LIGO, it is 5.8.  As such, the reader should view the numbers\nin this section of our analysis as useful {\\it only} for validation\npurposes.\n\n\\item The two analyses use different priors. As extensively discussed\nin Sec.\\ {\\ref{sec:mhmc}}, we set uniform priors on the chirp mass\n${\\cal M}_z$, on the time $t_c$ and phase $\\Phi_c$ at coalescence, and\non the polarization phase $\\psi$.  For this comparison, we assume\nisotropic emission and set a flat prior on $\\cos\\iota$.  We\nassume our sources are uniformly distributed in constant comoving\nvolume.  However, our detection threshold depends on the total network\nSNR, and effectively sets a joint prior on source inclination and\ndistance. CF94 use a prior distribution only for the set $\\{ D_L, \\cos\n\\iota, \\psi, \\Phi_c \\}$ that is flat in polarization phase,\ncoalescence phase, and inclination.  They assume a prior that is\nuniform in volume, but that cuts off the distribution at a distance\n$D_{L,{\\rm max}} \\simeq 6.5\\,{\\rm Gpc}$.\n\n\\end{itemize}\n\nOur goal here is to reproduce the 1-D marginalized posterior PDF in\n$D_L$ for the binary shown in Fig.\\ 10 of CF94.  We call this system\nthe ``CF binary.''  Each NS in the CF binary has $m_z = 1.4\\,M_\\odot$ and\nsky position $(\\theta, \\phi) = (50^{\\circ}, 276^{\\circ})$;\nthe detector network comprises LIGO Hanford, LIGO Livingston and\nVirgo.  CF94 report the ``best-fit'' maximum-likelihood values\n($\\tilde{D}_L$, $\\cos\\tilde{\\iota}$, $\\tilde{\\Psi}$) to be ($432\\,{\\rm\nMpc}$, $0.31$, $101.5^{\\circ}$), where $\\Psi = \\psi + \\Delta \\psi\n({\\bf n})$, and where $\\Delta \\psi ({\\bf n})$ depends on the preferred\nbasis of ${\\bf e}^{\\times}$ and ${\\bf e}^{\\times}$ set by the detector\nnetwork [see Eqs.\\ (4.23)--(4.25) of CF94\\footnote{Note that Eq.\\\n(4.25) of CF94 should read $\\tan(4\\Delta\\psi) =\n2\\Theta_{+\\times}\/(\\Theta_{++} - \\Theta_{\\times\\times})$. In addition,\n$\\tilde\\Psi = 56.5^{\\circ}$ should read $\\tilde\\Psi = 101.5^{\\circ}$\nunder the caption of Fig.\\ 10.  (We have changed notation from\n$\\bar\\psi$ in CF94 to $\\Psi$ to avoid multiple accents on the best fit\nvalue.)  We thank \\'Eanna Flanagan for confirming these\ncorrections.}].  To compare our distribution with theirs, we assume\nthat $\\boldsymbol{\\hat\\theta} = \\boldsymbol{\\tilde\\theta}_{\\rm ML}$\nfor the purpose of computing the likelihood function ${\\cal\nL}(\\boldsymbol{\\theta} | {\\bf s})$.  This is a reasonable assumption\nwhen the priors are uniform over the relevant parameter space.  As\nalready mentioned, for this comparison we use their advanced detector\nnoise curve and the Newtonian-quadrupole waveform.  Finally, we\ninterpret the solid curve in Fig.\\ 10 of CF94 as the marginalized 1-D\nposterior PDF in $D_L$ for an average of posterior PDFs of parameters\n(given an ensemble of many noisy observations for a particular event).\nWe compute the average PDF as described in Sec.\\\n{\\ref{sec:averagedPDF}}, and then marginalize over all parameters\nexcept $D_L$, using Eq.\\ (\\ref{eq:margPDF}).\n\nThe left-hand panels of Fig.\\ \\ref{fig:CFbinary} show the resulting\n1-D marginalized PDF in $D_L$ and $\\cos \\iota$.  Its shape has a broad\nstructure not dissimilar to the solid curve shown in Fig.\\ 10 of CF94:\nThe distribution has a small bump near $D_L \\approx 460\\,{\\rm Mpc}$, a\nmain peak at $D_L \\approx 700\\,{\\rm Mpc}$, and extends out to roughly\n1 Gigaparsec.  Because of the broad shape, the Bayes mean ($\\tilde\nD_{L,\\rm{BAYES}} = 694\\,{\\rm Mpc}$) is significantly different from\nboth the true value ($\\hat D_L = 432\\,{\\rm Mpc}$ in our calculation)\nand from the maximum likelihood ($\\tilde D_{L,\\rm{ML}} = 495\\,{\\rm\nMpc}$).  Thanks to the marginalization, the peak of this curve does\nnot coincide with the maximum likelihood.\n    \n\\begin{figure}\n\\centering \n\\includegraphics[width=1.00\\columnwidth]{fig5.eps}\n\\caption{1-D and 2-D marginalized posterior PDFs for $D_L$ and\n$\\cos \\iota$ averaged over noise (as described in Sec.\\\n\\ref{sec:averagedPDF}) for the ``CF binary.''  Our goal is to\nreproduce, as closely as possible, the non-Gaussian limit summarized\nin Fig.\\ 10 of CF94.  Top left panel shows the 1-D marginalized\nposterior PDF in $D_L$ (the true value $\\hat D_L = 432\\,{\\rm Mpc}$ is\nmarked with a solid black line); bottom left panel illustrates the 1-D\nmarginalized posterior PDF in $\\cos \\iota$ (true value $\\cos \\hat\n\\iota = 0.31$ likewise marked).  The right-hand panel shows the 2-D\nmarginalized posterior PDF for $D_L$ and $\\cos \\iota$; the true values\n($\\hat D_L = 432\\,{\\rm Mpc}, \\cos \\hat \\iota = 0.31$) are marked with a cross.  The contours\naround the dark and light areas indicate the 68 and 95\\% interval\nlevels, respectively.  The true values lie within the 68\\% interval.\nThe Bayes mean and rms measurement accuracies are (694.4 Mpc, 0.70)\nand (162 Mpc, 0.229) for ($D_L$, $\\cos \\iota$), respectively.}\n\\label{fig:CFbinary}\n\\end{figure}\n\nWe further determine the 2-D marginalized posterior PDFs in $D_L$ and\n$\\cos \\iota$ for the CF binary.  Figure \\ref{fig:CFbinary} illustrates\ndirectly the very strong degeneracy between these parameters, as\nexpected from the form of Eqs.\\ (\\ref{eq:hplus}) and\n(\\ref{eq:hcross}), as well as from earlier works (e.g.,\n\\citealt{markovic93}, CF94). It's worth noting that, as CF94 comment,\nthis binary is measured particularly poorly.  This is largely due to\nthe fact that one polarization is measured far better than the other,\nso that the $D_L$--$\\cos\\iota$ degeneracy is essentially unbroken.\nThis degeneracy is responsible for the characteristic tail to large\n$D_L$ we find in the 1-D marginalized posterior PDF in $D_L$, $p(D_L |\n\\bf{s})$, which we investigate further in the following section.\n\n\\subsection{Test 1: Varying luminosity distance and number of detectors}\n\\label{sec:cf_vary}\n\nWe now examine how well we measure $D_L$ as a function of distance to\nthe CF binary and the properties of the GW detector network.  Figures\n\\ref{fig:CFbinaryvaryingDLa} and \\ref{fig:CFbinaryvaryingDLb} show the\n1-D and 2-D marginalized posterior PDFs in $D_L$ and $\\cos \\iota$ for\nthe CF binary at $\\hat{D}_L = \\{100$, $200$, $300$, $400$, $500$,\n$600\\}$ Mpc.  For all these cases, we keep the binary's sky position,\ninclination, and polarization angle fixed as in Sec.\\ \\ref{sec:cf}.\nThe average network SNRs we find for these six cases are (going from\n$\\hat D_L = 100\\,{\\rm Mpc}$ to $600\\,{\\rm Mpc}$) 53.6, 26.8, 17.9,\n13.4, 10.7, and 8.9 (scaling as $1\/\\hat D_L$).  Interestingly, the\nmarginalized PDFs for both distance and $\\cos\\iota$ shown in Figs.\\\n{\\ref{fig:CFbinaryvaryingDLa}} and {\\ref{fig:CFbinaryvaryingDLb}} have\nfairly Gaussian shapes for $\\hat D_L = 100$ and 200 Mpc, but have very\nnon-Gaussian shapes for $\\hat D_L \\ge 300\\,{\\rm Mpc}$.  This can be\nconsidered ``anecdotal'' evidence that the Gaussian approximation for\nthe posterior PDF breaks down at ${\\rm SNR} \\lesssim 25$ or so, at\nleast for this case.  For lower SNR, the degeneracy between\n$\\cos\\iota$ and $D_L$ becomes so severe that the 1-D errors on these\nparameters become quite large.\n\n\\begin{figure}\n\\centering \n\\includegraphics[width=1.1\\columnwidth]{fig6.eps}\n\\caption{1-D and 2-D marginalized PDFs for $D_L$ and $\\cos \\iota$,\naveraged (as described in Sec.\\ \\ref{sec:averagedPDF}) over noise\nensembles for the ``CF binary'' at different values of true luminosity\ndistance $\\hat{D}_L$: [100 Mpc, 200 Mpc, 300 Mpc] (top to bottom).\nTrue parameter values are marked with a solid black line or a black\ncross.  The Bayes means and rms errors on luminosity distance are\n[101.0 Mpc, 212.1 Mpc, 411.2 Mpc] and [3.6 Mpc, 21.4 Mpc, 110.0 Mpc],\nrespectively.  The corresponding means and errors for $\\cos \\iota$ are\n[0.317, 0.357, 0.562] and [0.033, 0.089, 0.247].  The dark and light\ncontours in the 2-D marginalized PDF plots indicate the 68 and 95\\%\ninterval levels, respectively.  The true value always lies within the\n68\\% contour region of the 2-D marginalized area at these distances.}\n\\label{fig:CFbinaryvaryingDLa}\n\\end{figure}\n\n\\begin{figure}\n\\centering \n\\includegraphics[width=1.1\\columnwidth]{fig7.eps}\n\\caption{Same as Fig.\\ {\\ref{fig:CFbinaryvaryingDLa}}, but for true\nluminosity distance $\\hat{D}_L =$ [400 Mpc, 500 Mpc, 600 Mpc] (top to\nbottom).  True parameter values are marked with a solid black line or a black\ncross. In this case, the Bayes means and rms errors for luminosity\ndistance are [627.17 Mpc, 857.3 Mpc, 1068 Mpc] and [148.8 Mpc, 198.1\nMpc, 262.2 Mpc], respectively.  The means and errors for $\\cos \\iota$\nare [0.686, 0.745, 0.746] and [0.237, 0.209, 0.218].  The dark and\nlight contours in the 2-D marginalized PDF plots indicate the 68 and\n95\\% interval levels, respectively.  The true value lies within the\n68\\% contour region for $D_L = 400$ Mpc, but moves outside this region\nfor larger values.}\n\\label{fig:CFbinaryvaryingDLb}\n\\end{figure}\n\nNext, we examine measurement accuracy versus detector network.  For\nthe CF binary, adding detectors does not substantially increase the\ntotal SNR.  We increase the average total SNR from 12.4 to 14.6\n(adding only AIGO), to 12.4 (adding only LCGT; its contribution is so\nsmall that the change is insignificant to the stated precision), or to\n14.7 (adding both AIGO and LCGT).  The average SNR in our detectors is\n8.23 for LIGO-Hanford, 8.84 for LIGO-Livingston, 2.91 for Virgo, 8.71\nfor AIGO, and 1.1 for LCGT.  This pathology is an example of a fairly\ngeneral trend that we see; it is common for the SNR to be quite low in\none or more detectors.\n\nIn the case of the CF binary, we find that adding detectors does not improve the\nmeasurement enough to break the $D_L$--$\\cos\\iota$ degeneracy.  The\nmarginalized PDFs as functions of $D_L$ and $\\cos\\iota$ remain very\nsimilar to Fig.\\ {\\ref{fig:CFbinary}}, so we do not show them.  As a\nconsequence, even with additional detectors, the distance errors\nremain large and biased.  The bias is because we tend to find\n$\\cos\\iota$ to be larger than the true (relatively edge-on) value (cf.\\ lower left-hand\npanel of Fig.\\ {\\ref{fig:CFbinary}}).  Thanks to the\n$D_L$--$\\cos\\iota$ degeneracy, we likewise overestimate distance.\n\n\n\\subsection{Test 2: Varying source inclination}\n\\label{sec:faceon_cf}\n\nOne of the primary results from the CF binary analysis is a strong\ndegeneracy between $\\cos \\iota$ and $D_L$.  As Fig.\\\n\\ref{fig:CFbinary} shows, this results in a tail to large distance in\nthe 1-D marginalized posterior PDF $p(D_L | \\bf{s})$, with a Bayes\nmean $\\tilde D_L = 694\\,{\\rm Mpc}$ (compared to $\\hat D_L = 432\\,{\\rm\nMpc}$).  Such a bias is of great concern for using binary sources as\nstandard sirens.\n\nThe CF binary has $\\cos\\hat\\iota = 0.31$, meaning that it is nearly\nedge-on to the line of sight.  Hypothesizing that the large tails may\nbe due to its nearly edge-on nature, we consider a complementary\nbinary that is nearly face on: We fix all of the parameters to those\nused for the CF binary, except for the inclination, which we take to\nbe $\\cos\\hat\\iota = 0.98$.  We call this test case the ``face-on'' CF\nbinary.  Changing to a more nearly face-on situation substantially\naugments the measured SNR; the average SNR for the face-on CF binary\nmeasured by the LIGO\/Virgo base network is 24.3 (versus 12.4 for the\nCF binary).  We thus expect some improvement simply owing to the\nstronger signal.\n\nFigure \\ref{fig:SDbinary} shows the 1-D and 2-D marginalized posterior\nPDFs in $D_L$ and $\\cos \\iota$.  As expected, these distributions are\ncomplementary to those we found for the CF binary.  In particular, the\npeak of the 1-D marginalized posterior PDF in $D_L$ is shifted to\nlower values in $D_L$, and the Bayes mean is much closer to the true\nvalue: $\\tilde D_L = 376.3\\,{\\rm Mpc}$.  The shape of the 1-D\nmarginalized posterior PDF in $\\cos \\iota$ is abruptly cut off by the\nupper bound of the physical prior $\\cos\\iota \\le 1$, and the tail extends to\nlower distances (the opposite of the CF binary).  The Bayes mean\nfor the inclination is $\\cos\\tilde\\iota = 0.83$.\n\n\\begin{figure}\n\\centering \n\\includegraphics[width=1.1\\columnwidth]{fig8.eps}\n\\caption{Same as Fig.\\ {\\ref{fig:CFbinary}}, but for the ``face-on''\nCF binary.  The Bayes mean and rms errors are (376.3 Mpc, 0.83) and\n(51.3 Mpc, 0.12) for ($D_L$, $\\cos \\iota$), respectively.  Top left\nshows the 1-D marginalized posterior PDF in $D_L$ ($\\hat D_L =\n432\\,{\\rm Mpc}$ is marked with a solid black line); bottom left shows\nthe marginalized PDF in $\\cos \\iota$ (solid black line marks $\\cos\n\\hat \\iota = 0.98$).  The right panel shows the 2-D marginalized\nposterior PDF; the cross marks the true source parameters ($\\hat D_L =\n432\\,{\\rm Mpc}$, $\\cos \\hat\\iota = 0.98$).  As\nwith the CF binary, the true values lie within the 68\\% region.}\n\\label{fig:SDbinary}\n\\end{figure}\n\nJust as we varied distance and detector network for the CF binary, we\nalso do so for the face-on CF binary, with very similar results.  In\nparticular, varying network has little impact on the marginalized 1-D\nPDFs in $D_L$ and $\\cos\\iota$.  Varying distance, we find that the\nmarginalized 1-D PDFs are nearly Gaussian in shape for small\ndistances, but become significantly skewed (similar to the left-hand\npanels of Fig.\\ {\\ref{fig:SDbinary}}) when $\\hat D_L > 200$ Mpc.  The\ndistributions in $\\cos\\iota$ are particularly skewed thanks to the\nhard cut-off at $\\cos\\iota = 1$.  Interestingly, in this case we tend\nto infer a value of $\\cos\\iota$ that is smaller than the true value.\nWe likewise find a Bayes mean $\\tilde D_L$ that is smaller than $\\hat\nD_L$.\n\n\\subsection{Summary of validation tests}\n\\label{sec:testing_discuss}\n\nThe main result from our testing is that the posterior PDFs we find\nhave rather long tails, with strong correlations between $\\cos\\iota$\nand $D_L$.  Except for cases with very high SNR, the 1-D marginalized\nposterior PDF in $\\cos\\iota$ tends to be rather broad.  The Bayes mean\nfor $\\cos\\iota$ thus typically suggests that a binary is at\nintermediate inclination.  As such, we tend to {\\it underestimate}\n$\\cos\\iota$ for nearly face-on binaries, and to {\\it overestimate} it\nfor nearly edge-on binaries.  Overcoming this limitation requires us\nto either break the $D_L$--$\\cos\\iota$ degeneracy (such as by setting\na prior on binary inclination), or by measuring a population of\ncoalescences.  Measuring a population will make it possible to sample\na wide range of the $\\cos\\iota$ distribution, so that the\nevent-by-event bias is averaged away in the sample.\n\n\\section{Results II: Survey of standard sirens}\n\\label{sec:main_results}\n\nWe now examine how well various detector networks can measure an\nensemble of canonical GW-SHB events.  We randomly choose events from\nour sample of {\\it detected}\\\/ NS-NS and NS-BH binaries (where the\nselection is detailed in Sec.\\ \\ref{sec:selectionandpriors}).  We set\na total detector network threshold of 7.5.  Crudely speaking, one\nmight imagine that this implies, on average, a threshold per detector\nof $7.5\/\\sqrt{5}=3.4$ for a five detector network.  Such a crude ``per\ndetector threshold'' is useful for getting a rough idea of the range\nto which our network can measure events.  Averaging Eq.\\\n(\\ref{eq:snr_ave}) over all sky positions and orientations yields\n(DHHJ06)\n\\begin{eqnarray}\n\\label{eq:snr_ave_sky_orien}\n\\left({S\\over N}\\right)_{a,\\ {\\rm sky-ave}} &=& \\frac{8}{5}\n\\sqrt{\\frac{5}{96}} \\frac{c}{D_L} \\frac{1}{\\pi^{2\/3}} \\left(\\frac{G\n{\\cal M}_z}{c^3}\\right)^{5\/6}\\times\n\\nonumber\\\\\n& &\\qquad \\int_{f_{\\rm low}}^{f_{\\rm ISCO}}\n\\frac{f^{-7\/3}}{S_h(f)} df\\;,\n\\end{eqnarray}\nFor total detector network threshold of 7.5, a five detector network\nhas an average range of about $600\\,{\\rm Mpc}$ for NS-NS events, and\nabout $1200\\,{\\rm Mpc}$ for NS-BH events.  If SHBs are associated with\nface-on binary inspiral, these numbers are increased by a factor\n$\\sqrt{5\/2} \\simeq 1.58$.  (This factor is incorrectly stated to be\n$\\sqrt{5\/4} \\simeq 1.12$ in DHHJ06.)\n\nLet us assume a constant comoving rate of 10 SHBs Gpc$^{3}$ yr$^{-1}$\n\\citep{nakar06}.  If these events are all NS-NS binary mergers, and\nthey are isotropically oriented, we expect the full\nLIGO-Virgo-AIGO-LCGT network to measure 6 GW-SHB events per year.  If\nthese events are instead all NS-BH binaries, the full network is\nexpected to measure 44 events per year.  If these events are beamed,\nthe factor $1.58$ increases the expected rate to 9 NS-NS or 70 NS-BH\nGW-SHB events per year.  We stress that these numbers should be taken\nas rough indicators of what the network may be able to measure.\nNot all SHBs will be associated with binary inspiral.  Those events\nwhich are will likely include both NS-NS and NS-BH events, with\nparameters differing from our canonical choices.  We also do not\naccount for the fraction of SHBs which will be missed due to\nincomplete sky coverage.\n\nIn all cases we build our results by constructing the posterior\ndistribution for an event given a unique noise realization at each\ndetector.  We keep the noise realization in a given detector and for a\nspecific binary constant as we add other detectors.  This allows us\nto make meaningful comparisons between the performance of different\ndetector networks.\n\n\\subsection{NS-NS binaries}\n\nWe begin by imagining a population of six hundred detected NS-NS\nbinaries, either isotropically distributed in inclination angle or\nfrom our beamed subsample, using a network with all five\ndetectors. Figure \\ref{fig:DeltaDLNSNS} shows scatter plots of the\ndistance measurement accuracies for our unbeamed (blue crosses) and\nbeamed events (black dots), with each panel corresponding to a\ndifferent detector network.  The distance measurement error is defined\nas the ratio of the rms measurement error with the true\nvalue\\footnote{Our definition differs from that given in CF94, their\nEq.\\ (4.62).  Their distance measurement error is described as the\nratio of the rms measurement error with the Bayes mean.  We prefer to\nuse Eq.\\ (\\ref{eq:measaccdefn}) as we are interested primarily in the\nmeasurement error given a binary at its true luminosity distance.}\n$\\hat{D}_L$:\n\\begin{equation}\n\\frac{\\Delta D_L}{\\hat{D}_L} = \\frac{\\sqrt{\\Sigma^{D_L\nD_L}}}{\\hat{D}_{L}}\\;.\n\\label{eq:measaccdefn}\n\\end{equation}\n$\\Sigma^{D_L D_L}$ is computed using (\\ref{eq:approxsigma_bayes}).  We\nemphasize some general trends in Fig.\\ \\ref{fig:DeltaDLNSNS} which\nare particularly relevant to standard sirens:\n\n\\begin{itemize}\n\n\\item {\\it The unbeamed total sample and the beamed subsample separate\ninto two distinct distributions.}  As anticipated, the beamed\nsubsample improves measurement errors in $D_L$ significantly, by\ngreater than a factor of two or more.  This is due to the beaming\nprior, which constrains the\ninclination angle, $\\cos \\iota$, to $\\sim 3\\%$, thereby breaking the\nstrong $D_L$--$\\cos \\iota$ degeneracy.  By contrast, when no beaming\nprior is assumed, we find absolute errors of $0.1$--$0.3$ in\n$\\cos\\iota$ for the majority of events.  The strong $D_L$--$\\cos\n\\iota$ degeneracy then increases the distance errors.  A significant\nfraction of binaries randomly selected from our sample have $0.5\n\\lesssim |\\cos\\hat\\iota| < 1$.  As discussed in Sec.\\\n{\\ref{sec:selectionandpriors}}, this is due to the SNR selection\ncriterion: At fixed distance, face-on binaries are louder and tend to\nbe preferred.\n\n\\item {\\it Beamed subsample scalings.}  We fit linear scalings\nto our beamed subsample:\\\\\n$\\Delta D_L\/\\hat{D}_L \\simeq \\hat{D}_L\/(2.15\\,\\rm{Gpc})$ for LIGO + Virgo\\\\\n$\\Delta D_L\/\\hat{D}_L \\simeq \\hat{D}_L\/(2.71\\,\\rm{Gpc})$ for LIGO + Virgo + AIGO\\\\\n$\\Delta D_L\/\\hat{D}_L \\simeq \\hat{D}_L\/(2.38\\,\\rm{Gpc})$ for LIGO + Virgo + LCGT\\\\\n$\\Delta D_L\/\\hat{D}_L \\simeq \\hat{D}_L\/(2.82\\,\\rm{Gpc})$ for LIGO + Virgo + AIGO + LCGT\n\n\\item {\\it When isotropic emission is assumed, we find a large scatter\nin distance measurement errors for all events, irrespective of network\nand true distance.}  We find much less scatter when we assume a\nbeaming prior.  This is illustrated very clearly by the upper-right\npanel of Fig.\\ {\\ref{fig:DeltaDLNSNS}}.  In that panel, we show the\nscatter of distance measurement error versus true distance for the\nLIGO, Virgo, AIGO detector network, comparing to the\nFisher-matrix-derived linear scaling trend found in DHHJ06.  For the\nunbeamed case, our current results scatter around the linear trend;\nfor the beamed case, most events lie fairly close to the trend.  This\ndemonstrates starkly the failure of Fisher methods to estimate\ndistance accuracy, especially when we cannot set a beaming prior.\n\n\\item {\\it Adding detectors to the network considerably increases the\nnumber of detected binaries, but does not significantly improve the\naccuracy with which those binaries are measured.}  The increase we see\nin the number of detected binaries is particularly significant for\nGW-SHB standard sirens.  For instance, an important application is\nmapping out the posterior PDF for the Hubble constant, $H_0$.  As the\nnumber of events increases, the resulting joint posterior PDF in $H_0$\nwill become increasingly well constrained.  Additional detectors also\nincrease the distance to which binaries can be detected.  This can be\nseen in Fig.\\ \\ref{fig:DeltaDLNSNS}: for the LIGO and Virgo network,\nour detected events extend to $\\hat D_L \\sim 600\\,{\\rm Mpc}$; the\nlarger networks all go somewhat beyond this.  Interestingly, networks\nwhich include the AIGO detector seem to reach somewhat farther out.\n\n\\end{itemize}\n\nIt is perhaps disappointing that increasing the number of detectors\ndoes not greatly improve measurement accuracy.  We believe this is due\nto two effects.  First, a larger network tends to detect more weak\nsignals.  These additional binaries are poorly constrained. Second,\nthe principle limitation to distance measurement is the\n$D_L$--$\\cos\\iota$ degeneracy.  A substantial improvement in distance\naccuracy on individual events would require breaking this degeneracy.\nWe find that adding detectors does not do this, but the beaming prior\ndoes.\n\n\\begin{figure}\n\\centering \n\\includegraphics[width=1.0\\columnwidth]{fig9.eps}\n\\caption{Distance measurement errors versus true luminosity distance\nfor our sample of NS-NS binaries.  Colored crosses assume isotropic\nemission; black points assume our beaming prior.  The dashed lines\nshow the linear best-fit to the beamed sample (see text for\nexpressions).  In the LIGO+Virgo+AIGO panel we also show the\nFisher-matrix-derived linear scaling given in DHHJ06: $\\Delta\nD_L\/\\hat{D}_L \\simeq \\hat{D}_L\/(4.4\\,\\rm{ Gpc})$ assuming beaming (solid), and\n$\\Delta D_L\/\\hat{D}_L \\simeq \\hat{D}_L\/(1.7\\,\\rm{Gpc})$ for isotropic\nemission (dotted).}\n\\label{fig:DeltaDLNSNS}\n\\end{figure}\n\n\\subsection{NS-BH binaries}\n\nWe now repeat the preceding analysis for six hundred detected NS-BH\nbinaries.  Figure~\\ref{fig:DeltaDLNSBH} shows scatter plots of\nmeasurement accuracies for unbeamed and beamed NS-BH binaries.  We\nfind similar trends to the NS-NS case:\n\n\\begin{itemize}\n\n\\item {\\it The unbeamed and beamed samples separate into two distinct\ndistributions.}  Notice, however, that outliers exist in measurement\nerrors at high $D_L$ for several beamed events for all networks.  This\nis not too surprising, given that we expect beamed sources at higher\nluminosity distances and lower SNR.  Such events are more likely to\ndeviate from the linear relationship predicted by the Fisher matrix.\n\n\\item {\\it We see substantial scatter in distance measurement,\nparticularly when isotropic emission is assumed}.  As with the NS-NS case, the\nscatter is not as severe when we assume beaming, and in that case lies\nfairly close to a linear trend, as would be predicted by a Fisher\nmatrix.  This trend is shallower in slope than for NS-NS binaries,\nthanks to the larger mass of the system.\n\n\\item {\\it We do not see substantial improvement in distance\nmeasurement as we increase the detector network.}  As with NS-NS binaries,\nadding detectors increases the range of the network; AIGO appears\nto particularly add events at large $\\hat D_L$ (for both the isotropic\nand beamed samples).  However, adding detectors does not break the\nfundamental $D_L$--$\\cos\\iota$ degeneracy, and doesn't improve errors.  From our\nfull posterior PDFs, we find absolute errors of $0.1$--$0.3$ in $\\cos\\iota$,\nwhich is very similar to the NS-NS case.\n\n\\item {\\it Beamed subsample scalings.} The linear scalings\nfor our beamed subsample are:\\\\\n$\\Delta D_L\/\\hat{D}_L \\simeq \\hat{D}_L\/(4.83\\,\\rm{Gpc})$ for LIGO + Virgo\\\\\n$\\Delta D_L\/\\hat{D}_L \\simeq \\hat{D}_L\/(6.14\\,\\rm{Gpc})$ for LIGO + Virgo + AIGO\\\\\n$\\Delta D_L\/\\hat{D}_L \\simeq \\hat{D}_L\/(5.20\\,\\rm{Gpc})$ for LIGO + Virgo + LCGT\\\\\n$\\Delta D_L\/\\hat{D}_L \\simeq \\hat{D}_L\/(6.76\\,\\rm{Gpc})$ for LIGO + Virgo + AIGO + LCGT\\\\\n\n\\end{itemize}\n\n\\begin{figure}\n\\centering \n\\includegraphics[width=1.0\\columnwidth]{fig10.eps}\n\\caption{Distance measurement errors versus true luminosity distance\nfor our sample of NS-BH binaries.  Colored crosses assume isotropic\nemission; black points use our beaming prior.  The lower right-hand\npanel shows the sample detected by our ``full'' network\n(LIGO+Virgo+AIGO+LCGT).  Upper left is LIGO+Virgo; upper right is\nLIGO+Virgo+AIGO; and lower left is LIGO+Virgo+LCGT. The dashed lines\nshow the linear best-fit to the beamed sample (see text for\nexpressions).}\n\\label{fig:DeltaDLNSBH}\n\\end{figure}\n\n\\section{Summary discussion}\n\\label{sec:summary}\n\nIn this analysis we have studied how well GWs can be used to measure\nluminosity distance, under the assumption that binary inspiral is\nassociated with (at least some) short-hard gamma ray bursts.  We examine two\nplausible compact binary SHB progenitors, and a variety of plausible\ndetector networks.  We emphasize that {\\it we assume sky position is\nknown}.  We build on the previous study of DHHJ06, which used the\nso-called Gaussian approximation of the posterior PDF.  This\napproximation works well for large SNR, but the limits of its validity are\npoorly understood.  In particular, since the SNR of events measured by ground-based\ndetectors is likely to be of order 10, the Gaussian limit may be inapplicable. \nWe examine the posterior PDF for the parameters of observed\nevents using Markov-Chain Monte-Carlo techniques, which do not rely on\nthis approximation.  We also introduce a well-defined noise-averaged\nposterior PDF that does not depend solely on a particular noise\ninstance. Such a quantity is useful to predict how well a detector\nshould be able to measure the properties of a source.\n\nWe find that the Gaussian approximation substantially underestimates\ndistance measurement errors.  We also find that the main limitation\nfor individual standard siren measurements is the strong degeneracy\nbetween distance to the binary and the binary's inclination to the line of\nsight; similar discussion of this issue is given in a recent analysis\nby \\cite{ajithbose09}.  Adding detectors to a network only slightly\nimproves distance measurement for a given single event.  When we\nassume that the SHB is isotropic (so that we cannot infer anything\nabout the source's inclination from the burst), we find that Fisher\nmatrix estimates of distance errors are very inaccurate.  Our\ndistributions show large scatter about the Fisher-based predictions.\n\nThe situation improves dramatically if we assume that SHBs are\ncollimated, thereby giving us a prior on the orientation of the\nprogenitor binary.  By assuming that SHBs are preferentially emitted\ninto an opening angle of roughly $25^\\circ$, we find that the\ndistance--inclination correlation is substantially broken.  The Fisher\nmatrix estimates are then much more reasonable, giving a good sense of\nthe trend with which distances are determined (albeit with a moderate\nscatter about that trend).  This illustrates the importance of\nincorporating prior knowledge, at least for individual measurements.\n\nOur distance measurement results are summarized by\nFig.\\ {\\ref{fig:DeltaDLNSNS}} (for NS-NS SHB progenitors) and\nFig.\\ {\\ref{fig:DeltaDLNSBH}} (for NS-BH).  Assuming isotropy, we find\nthe distance to NS-NS binaries is measured with a fractional error of\nroughly $20$--$60$\\%, with most events in our distribution clustered\nnear $20$--$30$\\%.  Beaming improves this by roughly a factor of two,\nand eliminates much of the high error tail from our sample.  NS-BH\nevents are measured somewhat more accurately: the distribution of\nfractional distance errors runs from roughly $15$--$50$\\%, with most\nevents clustered near $15$--$25$\\%.  Beaming again gives roughly a\nfactor of two improvement, elimating most of the high error tail.\n\nIt is worth emphasizing that these results describe the outcome of\n{\\it individual siren measurements}.  When these measurements are used\nas cosmological probes, we will be interested in constructing the\njoint distribution, following observation of $N$ GW-SHB events.\nIndeed, preliminary studies show that our ability to constrain $H_0$\nimproves dramatically as the number of measured binaries is\nincreased.  In our most pessimistic scenario (the SHB is assumed to be\na NS-NS binary, with no prior on inclination, and measured by the\nbaseline LIGO-Virgo network), we find that $H_0$ can be measured with\n$\\sim 13\\%$ fractional error with $N = 4$, improving to $\\sim 5\\%$ for\n$N = 15$.  This is because multiple measurements allow us to sample\nthe inclination distribution, and thus average out the bias\nintroduced by the tendency to overestimate distance for edge-on binaries,\nand underestimate it for face-on binaries.  Details of this analysis\nwill be presented in a followup paper.\n\nIncreasing the number of measured events will thus be crucial for\nmaking cosmologically interesting measurements.  To this end, it is\nimportant to note that increasing the number of detectors in our\nnetwork enables a considerable increase in the number of detected\nbinaries.  This is due to increases in both the sky coverage and in the\ntotal detection volume.  Going from a network which includes all four\ndetectors (LIGO, Virgo, AIGO, and LCGT) to our baseline network of\njust LIGO and Virgo entails a $\\sim$~50\\% reduction in the number of\ndetected binaries. Eliminating just one of the proposed detectors\n(AIGO or LCGT) leaves us with $\\sim$~75\\% of the original detected\nsample.\n\nAside from exploring the cosmological consequences, several other\nissues merit careful future analysis.  One general result is\nthe importance that priors have on the posterior PDF.  We plan to\nexamine this in some detail, identifying the parameters which particularly\ninfluence the final result, and which uncertainties can be ascribed to\nan inability to set relevant priors.  Another issue is the importance of\nsystematic errors in these models.  We have used the second-post-Newtonian\ndescription of a binary's GWs in this analysis, and have ignored all\nbut the leading quadrupole harmonic of the waves (the ``restricted''\npost-Newtonian waveform).  Our suspicion is that a more complete\npost-Newtonian description of the phase would have little impact on\nour results, since such effects won't impact the $D_L$--$\\cos\\iota$\ndegeneracy.  In principle, including additional (non-quadrupole)\nharmonics could have an impact, since these other\nharmonics encode different information about the inclination angle\n$\\iota$.  In practice, we expect that they won't have much effect on\nGW-SHB measurements, since these harmonics are measured with very low\nSNR (the next strongest harmonic is roughly a factor of 10 smaller in\namplitude than the quadrupole).\n\nAs discussed previously, we confine our analysis to the inspiral.\nInspiral waves are terminated at the innermost stable circular orbit\nfrequency, $f_{\\rm ISCO}=(6^{3\/2} \\pi M_z)$.  For NS-NS binaries,\n$f_{\\rm ISCO} \\simeq 1600\\,{\\rm Hz}$.  At this frequency, detectors\nhave fairly poor sensitivity, so we are confident that terminating the\nwaves has little impact on our NS-NS results.  However, for our\nassumed NS-BH binaries, $f_{\\rm ISCO} \\simeq 400\\,{\\rm Hz}$.\nDetectors have good sensitivity in this band, so it may be quite\nimportant to improve our model for the waves' termination in this\ncase.\n\nPerhaps the most important follow-up would be to include the impact of\nspin.  Although the impact of neutron star spin is likely to be small,\nit may not be negligible; and, for NS-BH systems, the impact of the\nblack hole's spin is likely to be significant.  Spin induces\nprecession which makes the orbit's orientation, $\\bf{\\hat L}$,\ndynamical.  That makes the observed inclination dynamical, which can\nbreak the $D_L$--$\\cos\\iota$ degeneracy.  In other words, with spin\nprecession the source's orbital dynamics may break this degeneracy.\nVan der Sluys et al.\\ (2008) have already shown that spin precession\nphysics can improve the ability of ground-based detectors to determine\na source's position on the sky.  We are confident that a similar\nanalysis which assumes known sky position will find that measurements\nof source distance and inclination can likewise be improved.\n\n\\acknowledgments\n\nIt is a pleasure to acknowledge useful discussions with K.\\ G.\\ Arun,\nYoicho Aso, Duncan Brown, Curt Cutler, Jean-Michel D{\\'e}sert,\nAlexander Dietz, L.\\ Samuel Finn, Derek Fox, \\'Eanna Flanagan, Zhiqi\nHuang, Ryan Lang, Antony Lewis, Ilya Mandel, Nergis Mavalvala,\nSzabolcs M\\'arka, Phil Marshall, Cole Miller, Peng Oh, Ed Porter,\nAlexander Shirokov, David Shoemaker, and Pascal Vaudrevange.  We are\ngrateful to Neil Cornish in particular for early guidance on the\ndevelopment of our MCMC code, to Michele Vallisneri for careful\nreading of the manuscript, and to Phil Marshall for his detailed\ncomments on the ensemble averaged likelihood function. We also are\ngrateful for the hospitality of the Kavli Institute for Theoretical\nPhysics at UC Santa Barbara, and to the Aspen Center for Physics,\nwhere portions of the work described here were formulated.\nComputations were performed using the Sunnyvale computing cluster at\nthe Canadian Institute for Theoretical Astrophysics, which is funded\nby the Canadian Foundation for Innovation.  SAH is supported by NSF\nGrant PHY-0449884, and the MIT Class of 1956 Career Development Fund.\nHe gratefully acknowledges the support of the Adam J.\\ Burgasser Chair\nin Astrophysics.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nWithin the thousands of brown dwarfs discovered in wide-field surveys over the last 20 years lies a population of young, free-floating, brown dwarfs \\citep[e.g.][]{2006ApJ...639.1120K,2008AJ....136.1290R}. These brown dwarfs show redder near- and mid-infrared colors than their older field counterparts and display peculiar spectral features indicative of youth \\citep[e.g.][]{2008ApJ...689.1295K, 2009AJ....137.3345C, 2013ApJ...772...79A, 2013AJ....145....2F}. Many of them have their youth confirmed by being kinematically linked to local young moving groups \\citep{2013ApJ...777L..20L,2014ApJ...783..121G,2016ApJS..225...10F} which helps constrain their ages, temperatures, and masses.  Around the same time as the discovery of young field brown dwarfs, the first directly-imaged exoplanets (2M 1207 b, HR 8799 bcd, and $\\beta$ Pic b) were discovered around young (10\u2013100 Myr) stars \\citep{2005A&A...438L..25C, 2008Sci...322.1348M,2009A&A...493L..21L}. Brown dwarfs with similar absolute magnitudes exhibit prominent near-infrared (hereafter near-IR) methane absorption bands but these exoplanets do not display the blue near-IR colors indicative of cloud-free, methane-rich atmospheres.\n\nObservations of free-floating young brown dwarfs and directly-imaged gas giant planets have revealed a number of important similarities between the two populations including very red near-IR colors compared to the older ($\\sim$Gyr) field brown dwarfs, a triangular-shaped H-band continuum, lower effective temperatures than field brown dwarfs of the same spectral type, and peculiar mid-infrared colors relative to field brown dwarfs \\citep{2017AJ....153..182C,2013ApJ...772...79A, 2008Sci...322.1348M, 2013ApJ...777L..20L,2014ApJ...786...32M}.  Given the relative ease of observing free-floating young brown dwarfs, they make ideal analogs with which to study the atmospheric properties of directly-imaged planets.\n\nThe 3 to 4 $\\mu$m  wavelength range is a particularly important spectral region because it is very sensitive to the various physical processes that control the emergent spectrum.  In particular, it contains the $\\nu_3$  fundamental band of methane which is very sensitive to both disequilibrium chemistry due to vertical mixing within the atmosphere and variations in the cloud properties \\citep{2003IAUS..211..345S,2012ApJ...753...14S}.  However, observing these wavelengths from the ground is difficult because of the strong telluric absorption and high thermal background and so there are relatively few brown dwarf spectra, and in particular young brown dwarf spectra, at these wavelengths. \n\nIn this paper, we present $L$-band spectra of a sample of eight young brown dwarfs. We discuss the sample and observations in Section \\ref{sec:obs}, and go into detail about the reduction process in Section \\ref{sec:data}. In Section \\ref{sec:analysis} we analyze the effects of spectral type on the $L$-band spectra, fit our sample to various model suites, and build a sequence of objects from the same young moving group.\n\n\n\n\\section{Sample and Observations} \\label{sec:obs}\n\n\\begin{table*}\n\\caption{Our Sample of Young L Dwarfs (All parallaxes are from \\citet{2016ApJ...833...96L})} \\label{tbl:samp}\n\\begin{tabular}{cllcccr}\n\\toprule[1.5pt]\n Name &\n SpT &\n SpT &\n YMG &\n Mass Range &\n Refs &\n Parallax \\\\\n  &\n Opt &\n NIR &\n Membership &\n $M_\\mathrm{Jup}$ &\n  &\n mas\\\\\n\\toprule[1.5pt]\n2MASS J00452143+1634446\t&\tL2 $\\beta$\t & L2  \\textsc{vl-g} &\tArgus   &\t20--29 &\tC09, AL13, L16, F16\t&\t65.9\t$\\pm$\t1.3\t\\\\\nWISEP J004701.06+680352.1\t&\tL7~pec\t & L7  \\textsc{int-g} &\tAB~Dor& 9--15\t&\tG15, F16\t&\t82.3\t$\\pm$\t1.8\t\\\\\n2MASS J010332.03+1935361\t&\tL6 $\\beta$\t & L6 \\textsc{int-g} &\tArgus?&\t5--21&\tF12, AL13, F16\t&\t46.9\t$\\pm$\t7.6\t\\\\\n2MASS J03552337+1133437\t&\tL5 $\\gamma$\t & L3  \\textsc{vl-g} &\tAB Dor\t& 15--27  &\tC09, F13, AL13, F16\t&\t109.5\t$\\pm$\t1.4\t\\\\\n2MASS J05012406$-$0010452\t&\tL4 $\\gamma$\t & L3  \\textsc{vl-g} &\t\\nodata & 9--35\t&\tC09, AL13, F16\t&\t48.4\t$\\pm$\t1.4\t\\\\\nG~196$-$3B\t&\tL3 $\\beta$\t & L3  \\textsc{vl-g} &\t\\nodata\t&   22--53  &\tC09, AL13, F16\t&\t49.0\t$\\pm$\t2.3\t\\\\\nPSO J318.5338$-$22.8603\t&\t\\nodata\t & L7 \\textsc{vl-g} &\t$\\beta$~Pic\t& 5--7 &\tL13, A16, F16\t&\t45.1\t$\\pm$\t1.7\t\\\\\n2MASS J22443167+2043433\t&\tL6.5~pec\t & L6 \\textsc{vl-g} &\tAB~Dor\t& 9--12&\tK08, A16, V18\t&\t58.7\t$\\pm$\t1.0\t\\\\\n\\end{tabular}\n\\tablerefs{(A16)~\\citet{2016ApJ...819..133A}, (AL13)~\\citet{2013ApJ...772...79A},\n(C09)~\\citet{2009AJ....137.3345C}, \n(F12)~\\citet{2012ApJ...752...56F}, (F13)~\\citet{2013AJ....145....2F},\n(F16)~\\citet{2016ApJS..225...10F},\n(G15)~\\citet{2015ApJ...799..203G},\n(K08)~\\citet{2008ApJ...689.1295K},\n(L16)~\\citet{2016ApJ...833...96L}, (L13)~\\citet{2013ApJ...777L..20L},\n(V18)~\\citet{2018MNRAS.474.1041V}\n}\n\\end{table*}\n\nOur sample consists of eight young L dwarfs with spectral types ranging from L2 to L7. Table \\ref{tbl:samp} gives the full identifications (hereafter we abbreviate the full designations of 2MASS and WISE sources as HHMM+DD), spectral types (red optical and near-IR), young moving group memberships where applicable, estimated masses, and parallaxes. The near-IR spectra of all our objects exhibit features of low gravity, including weaker alkali and FeH features, triangular-shaped $H$-band spectra, and redder near-IR colors than field brown dwarfs of the same spectral type (See Figure \\ref{fig:cmd}). As further evidence of youth, six members of our sample also are kinematically linked to young moving groups (e.g. Argus, AB Dor, and $\\beta$ Pic) with ages ranging from 12 to 125 Myr.\n\n\\begin{figure}\n\\includegraphics[trim = {.55cm .95cm 1.35cm 2.25cm},clip,width = 3.35in]{J-KDirectlyImagedFinal.pdf}\n\\centering\n\\caption{A J$-$K color-magnitude diagram showing the difference in the near-IR color of older field dwarfs \\citep{2012ApJS..201...19D}, and young objects. Included are young brown dwarfs from the UltracoolSheet \\citep{Best}, directly imaged exoplanets from the NASA Exoplanet Archive, and our sample.}\\label{fig:cmd}\n\\end{figure}\n\n\nWe obtained $L$-band spectra of seven of these young brown dwarfs between 2014 October 27 and December 8 using the Gemini Near-InfraRed Spectrograph \\citep[GNIRS;][]{2006SPIE.6269E..4CE} at the Gemini North Observatory on Maunakea. Details of our observations are listed in Table \\ref{tbl:obs}. For these seven objects, we used the 10.44~lines~mm\\textsuperscript{$-$1} grating with the 0\\farcs05~pix\\textsuperscript{--1} camera resulting in a continuous 2.98--3.96~$\\mu$m spectrum. For most of our targets, we used a slit with a 0\\farcs3 width, resulting in a spectral resolving power ($R = {\\lambda}\/{\\Delta \\lambda}$) varying linearly with wavelength from $\\sim$500 at 2.98~$\\mu$m to $\\sim$675 at 3.96~$\\mu$m with an average of $R\\approx 590$.   For our two faintest targets, we used a 0\\farcs6750-wide slit, resulting in $R\\sim $225 to $\\sim$300 from 2.98 to 3.96~$\\mu$m with an average of $R\\approx 260$.  For our science target observations, we used a maximum exposure time of 10~sec and 5~sec for observations taken with the 0\\farcs3 and 0\\farcs675 slits respectively, and took 10--60 coadds to reach the integration times listed in Table \\ref{tbl:obs}.  An ABBA nodding pattern was used to enable dark, bias, and sky subtraction.  Observations of telluric standard stars with spectral types ranging from B9 V to A0.5V were taken adjacent to observations of our targets. \n\nOur sample also includes  2MASS~J22443167+2043433, a young brown dwarf with a published $L$-band spectrum \\citep[3.0$-$4.1 $\\mu$m, $R\\approx$ 460;][]{2009ApJ...702..154S} obtained with the Gemini Near-InfraRed Imager and Spectrograph \\citep[NIRI:][]{2003PASP..115.1388H}.\n\\begin{table*}\n\\caption{GNIRS Observations} \\label{tbl:obs}\n\\begin{tabular}{llrrlr}\n\\toprule[1.5pt]\nName &\nDate &\nSlit &\nN $\\times$ Int. Time &\nTelluric Standard &\n$\\Delta$ Airmass\\\\\n &\n(UT) &\n(arcsec) &\n(sec) &\n &\n \\\\\n\\toprule[1.5pt]\n2MASS~0045+16& 2014 Nov 28 & 0.300&8$\\times$120 &HIP 117927& 0.025\\\\\nWISE~0047+68 & 2014 Nov 30 & 0.300&8$\\times$300 &HIP 8016  & 0.080\\\\\n2MASS~0103+19& 2014 Nov 25 & 0.675&8$\\times$300 &HIP 117927& 0.076\\\\\n2MASS~0103+19& 2014 Dec 01 & 0.675&8$\\times$300 &HIP 117927& 0.043\\\\\n2MASS~0355+11& 2014 Oct 27 & 0.300&8$\\times$100 &HIP 18907 & 0.013\\\\\n2MASS~0501$-$00& 2014 Oct 27 & 0.300&8$\\times$300 &HIP 24607 & 0.028\\\\\n2MASS~0501$-$00& 2014 Dec 08 & 0.300&8$\\times$300 &HIP 24607 & 0.185\\\\\nG~196$-$3B   & 2014 Dec 08 & 0.300&8$\\times$300 &HIP 51697 & 0.164\\\\\nPSO 318.5$-$22& 2014 Nov 28 & 0.675&10$\\times$300& HIP 108542& 0.131\\\\\nPSO 318.5$-$22& 2014 Nov 30 & 0.675&12$\\times$300& HIP 108542& 0.210\\\\\n\\end{tabular}\n\\end{table*}\n\n\\section{Data Reduction} \\label{sec:data}\nWe reduced our data using the \\texttt{REDSPEC}\\footnote{\\url{https:\/\/www2.keck.hawaii.edu\/inst\/nirspec\/redspec}} package, modified for use with GNIRS.  \nFor spatial map creation, we used the \\texttt{spatmap} procedure on the bright telluric standard, which allowed us to reorient the spectra (of both the science target and telluric standard) along the detector rows. To create a spectral map and wavelength calibrate our data, we compared sky emission spectra from our telluric standard observations to a model sky emission spectrum\\footnote{\\url{http:\/\/www.gemini.edu\/sciops\/telescopes-and-sites\/observing-condition-constraints\/ir-background-spectra}}.\n\nWe found that the brightness of the sky background made fitting individual sky emission lines challenging, and thus chose to fit the entire 2.98--3.96~$\\mu$m spectrum simultaneously.  We first Gaussian-smoothed the model sky to match our observed spectral resolving power.  We then ran both the observed and model sky spectra through a high-pass filter to remove thermal emission.  We established a 3rd order polynomial wavelength solution by specifying the wavelengths of four evenly-spaced (in the spectral direction) anchor pixels.  We then used IDL's \\texttt{AMOEBA} to iteratively solve for the wavelengths of the anchor pixels that minimize the difference between our observed (with a wavelength solution determined from a 3rd order fit to our four anchor pixels) and model sky emission spectra.  This was done for five spatial slices of the observed sky, which allowed us to align each column with a specific wavelength.  After creating the spatial and spectral maps, we rectified all raw frames so that the spectral direction ran along image rows and the spatial (along the slit) dimension was aligned with image columns.\n\nA pair of spectra were extracted for each AB pair with the \\texttt{redspec} process. This process first collapses the rectified images in the spectral direction to create a  spatial profile of the A$-$B subtracted image.  We fit a pair of Gaussian curves centered on the positive and negative peaks in the spatial profile.  Residual sky background was subtracted by fitting a line (at each spectral column) to regions of the spatial profile containing no object flux.  We extracted the spectra by summing the flux within an aperture equal to 1.4$\\times$ the FWHM of the Gaussian fit to the spatial profile, as we found that this value provided the highest signal-to-noise ratio.  The noise for each pixel in our spectra was determined by tracking Poisson noise contributions throughout the reduction and extraction process.\n    \nThese extracted spectra were combined using a robust weighted mean in \\texttt{xcombspec} \\citep{2004PASP..116..362C}, and then were corrected for telluric absorption and flux calibrated using the \\texttt{xtellcor} package \\citep{2003PASP..115..389V}. Though our telluric correction is excellent overall, we removed regions of the spectra where atmospheric transmission is modeled to fall below 20\\%, as the signal-to-noise ratio in these regions is low and the telluric correction less reliable.  This consists of the region around the 3.3 $\\mu$m $Q$-branch of methane. Three of our targets had spectra taken on two separate nights.  We reduced each night's data individually, and then combined the final telluric-corrected spectra using \\texttt{xcombspec} with a weighted mean. \n\n\n\\begin{figure*}\n\\includegraphics[trim = {.85cm .8cm 6.05cm 12.5cm},clip,width=.65\\textwidth]{2021LbandStacked.pdf}\n\\centering\n\\caption{The reduced $L$-band spectra of our objects, normalized and offset to the dotted line. The highlighted region marks the $Q$-branch of methane which appears in our objects at a spectral type of L6. The gap in our spectra are cuts made where atmospheric transmission is below 20\\% and our telluric correction is less reliable.}\\label{fig:LBandAll}\n\\end{figure*}\n\nWe calibrated the absolute flux our spectra using published Spitzer\/IRAC Channel 1 (3.6 $\\mu$m) photometry (Table \\ref{tbl:irac} in Appendix \\ref{app:irac}), as the IRAC Channel 1 bandpass sits comfortably inside the spectral range of our GNIRS observations.  PSO 318.5$-$22 does not have a [3.6] magnitude, so we used its spectral types, WISE W1 magnitudes \\citep{2013ApJ...772...79A,2013ApJ...777L..20L} and a spectral type vs.~W1$-$[3.6] color relation to compute a [3.6] magnitude of $12.892\\pm0.035$~mags for PSO 318.5$-$22. The spectral type vs.~W1$-$[3.6] color relation was created from a linear least-squares fit to the values for young L dwarfs listed in Table \\ref{tbl:irac}, which includes some previously unpublished photometry. The photometry, fit, and covariance matrix for the fit can be found in Appendix \\ref{app:irac}. With these [3.6] magnitudes and the zero-magnitude flux for [3.6], we were able to calculate the scaling factor needed to convert our spectra to absolute units of $\\mathrm{W~m}^{-2}~\\mu \\mathrm{m}^{-1}$ using the process found in \\citet{2005PASP..117..978R}. Our final reduced spectra are presented in Figure \\ref{fig:LBandAll}.  \n\n\\section{Analysis}\\label{sec:analysis}\n\n\\subsection{The Young, L-band Spectral Sequence}\n\nIn general, the $L$-band spectra of our targets do not show the deep atomic and molecular absorption features that are prominent in near-IR spectra of brown dwarfs  \\citep[e.g.][]{2005ARA&A..43..195K, 2017ApJ...838...73M}. There are some weak water features around 3.0-3.2 $\\mu$m, but the strongest absorption feature observed in our spectra is the $Q$-branch of the $\\nu_3$ fundamental band of methane at 3.3 $\\mu$m, and only in our objects of spectral types L6 or later. Moving later in spectral type also corresponds with the spectrum getting redder, shifting from a slightly negative slope with respect to wavelength at L2 to a nearly flat spectrum at L7 when using units of $f_\\lambda$ (Figure \\ref{fig:LBandAll}). \nThis reddening, along with the onset of methane occurring somewhere between L3 and L6, is consistent with tendencies observed in the spectra of older field dwarfs \\citep{2000ApJ...541L..75N, 2008ApJ...678.1372C, 2019RNAAS...3c..52J}. \n\n\n\\subsection{Model Comparisons: Best-Fit Parameters}\nFitting spectra to the predictions of atmospheric models has long been a fruitful method for illuminating the physical and chemical processes that occurs in stellar and substellar atmospheres \\citep[e.g.][]{2010ApJS..186...63R}. Unfortunately, the medium resolution $L$-band spectra of L dwarfs lack deep features (excluding the $Q$-branch of methane), so a broad range of model parameters have been shown to give statistically good fits when fitting just the $L$-band spectra alone \\citep[e.g.][]{2008ApJ...678.1372C}. As such, we chose to combine our $L$-band spectra with the published near-IR spectra listed in Table \\ref{tbl:nirsamp}, and then fit both the combined spectrum and just the near-IR spectra in order to gauge how the addition of the $L$-band spectra affects the best-fit parameters for young L dwarfs.\n\n\\begin{table}\n\\caption{IRTF\/SpeX Near-IR Spectra} \\label{tbl:nirsamp}\n\\begin{tabular}{lccr}\n\\toprule[1.5pt]\nName &\nWavelength &\n$<{\\lambda}\/{\\Delta \\lambda}>$ &\nReferences\\\\\n &\nRange ($\\mu$m) &\n &\n\\\\\n\\toprule[1.5pt]\n2MASS~J0045+16&  0.939$-$2.425& 750 & \\citet{2013ApJ...772...79A}\\\\\nWISEP~J0047+68 &  0.643$-$2.550& 120 &\\citet{2012AJ....144...94G}\\\\\n2MASSI~J0103+19&  0.659$-$2.565& 120 & \\citet{2004yCat..51262421C}\\\\\n2MASS~J0355+11&  0.735$-$2.517& 120 & \\citet{2013AJ....145....2F}\\\\\n2MASS~J0501$-$00& 0.850$-$2.502& 120 & \\citet{2010ApJ...715..561A}\\\\\nG~196$-$3B   &  0.644$-$2.554& 120& \\citet{2010ApJ...715..561A}\\\\\nPSO~J318.5$-$22&  0.646$-$2.553& 100& \\citet{2013AJ....145....2F}\\\\\n2MASS J2244+20&  0.651$-$2.564& 100&\\citet{2008ApJ...686..528L}\\\\\n\\end{tabular}\n\\end{table}\n\nTo combine the $L$-band and near-IR spectra, we performed an absolute flux calibration on the near-IR spectra using 2MASS $J$,$H$, and $K_S$ photometry \\citep{2003tmc..book.....C} with the zero-point fluxes from \\citet{2003AJ....126.1090C}. We determined the scaling factor necessary to convert the spectra to units of $\\mathrm{W~m}^{-2}~\\mu \\mathrm{m}^{-1}$ for each of the three filters, and then used the weighted average of those factors to scale and stitch the near-IR to the $L$ band. PSO 318.5$-$22 lacked 2MASS photometry, so we used MKO $J$,$H$, and $K$ band photometry \\citep{2013ApJ...777L..20L}, with the zero-points coming from \\citet{2005PASP..117..421T}. The average percent difference between the three values and the weighted mean was always under 5\\%, and often under 1\\%.\n\nSeveral grids of atmospheric models were used, allowing us to compare how various approaches to 1-D atmospheric models fare. The models we included were:\n\\begin{itemize}\n\n\\item BT-Settl CIFIST and BT-Settl AGSS Models \\citep{2012RSPTA.370.2765A}: These models include clouds simulated via detailed dust micro-physics, and an estimation of the diffusion process based on 2D hydrodynamic simulations. These models are in chemical equilibrium, though two different chemical abundances were used for AGSS09 and CIFIST11 \\citep[][respectively]{2009ARA&A..47..481A,2011SoPh..268..255C}. \n\nAGSS Parameter ranges: \\teff~ranges from 1000--2600 K at 100 K intervals, \\logg{} from 3.5--5.5 [cm\/s$^{2}$] at 0.5 dex intervals, and [Fe\/H] = 0.0. Above 2000 K, \\logg{} extends to $-$0.5, and [Fe\/H] ranges from +0.5 to $-$4 in 0.5 dex increments, and includes [Fe\/H] = +0.3. \n\nCIFIST Parameter ranges: \\teff~ranges from 1000--2900 K at 50 K intervals, \\logg{} from 3.5--5.5 [cm\/s$^{2}$] at 0.5 dex intervals, and [Fe\/H] = 0.0.\n\n\\item Tremblin Models \\citep{2017ApJ...850...46T}: These models do not include clouds, instead recreating the effects attributed to clouds by changing the adiabatic index in the layer above the convective zone, artificially heating this portion of the atmosphere and simulating convective fingering. They also include a \\kzz~mixing parameter that keeps the model from coming to chemical equilibrium. A higher \\kzz~value models more vigorous mixing.\n\nParameter ranges: \\teff~ranges from 1200--2400 K at 200 K intervals, all with \\logg{} = 3.5 [cm\/s$^{2}$], log \\kzz~= 6 [$\\mathrm{cm}^2 \\mathrm{s}^{-1}$], [Fe\/H] = 0.0, and an effective adiabatic index ($\\gamma$) of 1.03. We also included 4 models that had already been made for the near-IR fits of specific objects (P. Tremblin, private communication), including two from our sample: PSO 318.5$-$22 (\\teff~= 1275 K , \\logg{} = 3.7 [cm\/s$^{2}$], log \\kzz~= 5 [$\\mathrm{cm}^2 \\mathrm{s}^{-1}$], [Fe\/H] = +0.4, and an $\\gamma$ of 1.03) and 2MASS 0355+11 (\\teff~= 1400 K, \\logg{} = 3.5 [cm\/s$^{2}$], log \\kzz~= 6 [$\\mathrm{cm}^2 \\mathrm{s}^{-1}$], [Fe\/H] = +0.2, and an $\\gamma$ of 1.01).\n\n\\item Saumon \\& Marley Models \\citep{2008ApJ...689.1327S}: These models account for clouds using an \\mbox{$f_{\\mathrm{sed}}$}~parameterization, which describes the efficiency of sedimentation in comparison to turbulent mixing, with a lower \\mbox{$f_{\\mathrm{sed}}$}~value implying thicker clouds. The Saumon \\& Marley models also include a \\kzz~mixing parameter to account for disequilibrium chemistry, with a higher \\kzz~once again modeling more vigorous mixing.\n\nParameter ranges: \\teff~ranges from 700--2400 K at 100 K intervals, and down to 500 K at 50 K intervals, \\logg{} ranges from 4.5--5.5 [cm\/s$^{2}$] at approximately 0.5 dex intervals, with log \\kzz~of 0, 2, 4 and 6 [$\\mathrm{cm}^2 \\mathrm{s}^{-1}$], [Fe\/H] = 0.0, and \\mbox{$f_{\\mathrm{sed}}$}~of 1, 2, 3, and 4, along with a cloud-free model (nc). Additionally, there were cloud-free models with [Fe\/H] = +0.3 and +0.5 for 500--600 K, as well as \\logg{} of 5.25 [cm\/s$^{2}$] from 500--700 K and 4.0 [cm\/s$^{2}$] from 1500--1700 K. From 800--1500 K, the \\mbox{$f_{\\mathrm{sed}}$}~= 1 and 2 models included a \\logg{} of 4.0, and 4.25 [cm\/s$^{2}$], and the \\mbox{$f_{\\mathrm{sed}}$}~= 2 models include a \\logg{} of 4.75 [cm\/s$^{2}$].\n\n\\item Drift-Phoenix Models \\citep{2009A&A...506.1367W}: These models simulate clouds using detailed dust micro-physics, with a more robust focus on both the seeding and subsequent growth of these cloud-forming grains, and are allowed to come into chemical equilibrium.\n\nParameter ranges: \\teff~ranges from 1000--2200 K at 100 K intervals, \\logg{} from 3.5--5.5 [cm\/s$^{2}$] at 0.5 dex intervals, and [Fe\/H] = 0.0.\n\n\\item Madhusudhan Models \\citep{2011ApJ...737...34M}: These models include clouds which are parameterized by variable dust grain sizes and various upper altitude cutoffs ranging from a sharp cutoff (E) to extending fully to the top of the atmosphere (A), with AE and AEE ranging between the two. The models are also in chemical equilibrium.\n\nParameter ranges: \\teff~ranges from 700--1700 K at 100 K intervals, \\logg{} = 4.0 [cm\/s$^{2}$], with [Fe\/H] of 0.0 and +0.5, and dust grain sizes of 30, 60, and 100 $\\mu$m. The AE cloud set has some finer \\teff~and \\logg{} spacing, with a 25 K interval from 750--1050 K and \\logg{} ranging from 3.75--4.25 [cm\/s$^{2}$] with a 0.25 dex interval across the whole range.\n\\end{itemize} \nWe found the best fits by first calculating the scaling factor $C$ (where $C$ has the physical analog [Radius$^2$\/Distance$^2$]) that minimized \\gk, a goodness-of-fit statistic \\citep[]{2008ApJ...678.1372C}, for each model spectrum. Then for each model set the synthetic spectrum with the minimum \\gk~is selected as the best-fit model. We report the parameters of these models, for both the near-IR and combined spectra, in Table \\ref{tbl:bestfits}. \n\nFor the sake of compactness, PSO 318.5$-$22 and 2MASS 0355+11 are presented as a representative sample for the rest of the paper, with PSO 318.5$-$22 representing the cooler and later spectral types and 2MASS 0355+11 the hotter and earlier. The best near-IR fits of 2MASS 0355+11 and PSO 318.5$-$22 can be seen in Figures \\ref{fig:0355NIR} and \\ref{fig:PSONIR}, respectively. Note that the $L$-band spectra is included in these figures, but not in this fitting process. The combined best fits for these objects can be seen in Figures \\ref{fig:0355Fits} and \\ref{fig:318Fits}, respectively. The combined fits for the remaining objects can be found in Appendix \\ref{app:data}.\n\nWhen we calculate the best fits to just the published near-IR spectra of our sample, we find that the models fit portions of the near-IR well, though there are definitely a variety of deviations for each set of models, which matches what was seen for young brown dwarfs in \\citet{2014A&A...564A..55M}. However, 8 of the 12 fits are very poor matches to the the $L$ band, as can be seen in Figures \\ref{fig:0355NIR} and \\ref{fig:PSONIR}. The BT-Settl CIFIST and Tremblin models fit the $L$ band well enough for hotter objects like 2MASS 0355+11, but underestimate the $L$-band flux at lower temperatures. Similarly, the Saumon \\& Marley models provide one of the better $L$-band fits for PSO 318.5$-$22, but for the hotter objects they predict strong water features in the $L$ band that are not present. For these same hotter objects the Drift-Phoenix best-fit models predict too much flux emerging out of the $L$ band. Between \\teff~of 1600 K and 1400 K the Drift-Phoenix models transition from being bright in the $J$ and $H$ bands to bright in $K$ and $L$ bands (This transition can be seen between Figures \\ref{fig:0355NIR}, \\ref{fig:0355Fits}, and \\ref{fig:0355_EVO} which show the 1500, 1600, and 1400 K models respectively). This rapid reddening causes most of the cooler objects to be fit by the Drift-Phoenix models at either 1600 or 1500 K, and leads to Drift-Phoenix having the best $L$-band fit of PSO 318.5$-$22, even though the near-IR fit is the worst. All of these discrepancies show that a reasonable fit at near-IR wavelengths does not necessarily mean the same model spectra will also fit the $L$-band region well.\n\n\\begin{figure*}\n\\includegraphics[trim = {.25cm 4.7cm 3.5cm 2.75cm},clip,width=.9\\textwidth]{2021BestFit0355NIR.pdf}\n\\centering\n\\caption{The combined spectrum of 2MASS 0355+11 (black) compared to the model spectra (colored) with parameters that best fit only the shaded near-IR portion of the spectrum.}\n\\label{fig:0355NIR}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[trim = {.25cm 4.7cm 3.5cm 2.75cm},clip,width=.9\\textwidth]{2021BestFit318NIR.pdf}\n\\caption{Same as Fig. \\ref{fig:0355NIR}, except using the cooler PSO 318.5$-$22.}\n\\label{fig:PSONIR}\n\\end{figure*}\n\n\\subsubsection{Overall Summary of the Combined Fits}\nWhen we fit to the combined spectra (See Figures \\ref{fig:0355Fits} and \\ref{fig:318Fits} and Appendix \\ref{app:data}) we find that no single model set fits all the objects well. The Saumon \\& Marley and Tremblin models are the most consistent across the whole spectral range, and the best fits for the later spectral types. For the earlier types, the BT-Settl CIFIST models are often better, but these best-fit models for the later type objects have muted $J$, $H$, and $K$ peaks and shallower troughs between peaks as seen in Figure \\ref{fig:318Fits}. The Drift-Phoenix models tend to fit the $L$ band fairly well, particularly the flux levels between the near-IR and $L$ band, yet the fit to the near-IR is not good. Figure \\ref{fig:318Fits} is a good example of both the flux level match and the near-IR mismatch. The Madhusudhan and BT-Settl AGSS models are not the best fitting model for any of our objects' combined spectra. The Madhusudhan models' poor fits are caused by a humped $L$ band, as well as under-predicting the flux emerging from the $J$ and $H$ bands, which can be seen in both Figures \\ref{fig:0355Fits} and \\ref{fig:318Fits}. For the BT-Settl AGSS models, the poor fits are due to the fact that when the flux levels between the near-IR and $L$-band match the observed spectra, the near-IR spectral morphology is off as seen in Figure \\ref{fig:0355Fits} (often bright in the $K$ band and dim in the $J$ band). \n\n\\subsubsection{Temperature}\n\nWhen we calculate the best fits for the combined spectra, we find the best-fit temperatures are generally $\\sim$100 K colder compared to the fits to just the near-IR, as seen in Table \\ref{tbl:bestfits}. The lower temperature of fits that include the $L$ band can be also seen in Figure \\ref{fig:GK}, which compares the \\gk~values for PSO 318.5$-$22 for a range of model temperatures and surface gravities. The shape of the \\gk~plots for temperature and surface gravity are similar for both the combined and near-IR spectra but including the $L$ band offsets the curve to colder temperatures. This temperature drop is caused in part by the higher temperature models under-predicting the flux ratio between the near-IR and the $L$ band. Even cases where a model's best-fit temperature increased by adding the $L$ band had this 100 K shift (for example the BT-Settl CIFIST fit of PSO 318.5$-$22). In these cases there are two local minima: the hotter minimum with muted troughs and peaks in the near-IR, while the cooler one is less muted. For both cases, the minima shift to cooler temperatures when the $L$ band is added, but the absolute minima switches from the colder minima to the hotter one. \n\nThe combined best-fit temperatures also divides our models into two groups: those models that generally give higher temperature fits for our objects (BT-Settl and  Drift-Phoenix) and those that fit these same objects with lower temperatures (Madhusudhan, Tremblin, and Saumon \\& Marley). The colder-model fits all share a similar feature in their $P\/T$ profile: Moving radially outward, they track with similar models until at some height they have a sharp decrease in pressure over a small temperature range, after which they continue along similar models except with hotter temperatures in the upper layers of the atmosphere (\\citealp[Examples can bee seen in Fig. 3 in][]{2011ApJ...737...34M}\\citealp[, Fig. 4 in][]{2016ApJ...817L..19T}\\citealp[, and Fig. 1 in][]{2010ApJ...723L.117M}). The cause of this thermal perturbation is not the same for all models, as the modified adiabatic index produces the perturbation in the Tremblin models while the existence of clouds produces it in the Madhusudhan and Saumon \\& Marley models. It should be noted that not all the Madhusudhan and Saumon \\& Marley models have this thermal perturbation, only the ones with thick clouds (Type A for Madhusudhan, and at lower \\mbox{$f_{\\mathrm{sed}}$} values for Saumon \\& Marley).\n\n\\subsubsection{Clouds}\nTwo of the model grids allow us to also look at the effects of variations in cloud properties. For both sets of models the thickest clouds were consistently the best fits (the \\mbox{$f_{\\mathrm{sed}}$}~of 1 for Saumon \\& Marley and cloud type A for Madhusudhan), and with the fits getting progressively worse for thinner clouds. Thick clouds cause an increase in opacity which blocks flux coming from the deeper and hotter layers of the atmosphere. As such, the near-IR, which originates below the added cloud deck in cloudless models, is now emerging from a cooler region just above the cloud deck instead. \\citep{2001ApJ...556..872A}. Now that less flux is coming out in the near-IR, to maintain a level of radiation consistent with the \\teff~of this object the upper atmosphere must heat up. Since the $L$ band originates from these upper layers, we get an increase in $L$ band flux, which when combined with the lower near-IR flux results in better fits. However, the thick clouds (combined with a high \\kzz~for Saumon \\& Marley) also create a turndown at the long end of the $L$ band that we do not see in our data. This can be seen in Figure \\ref{fig:318Fits}, and is also an issue for several of the other models. Still, overall it seems that if clouds are the answer to the spectral reddening of our young objects as suggested by \\citet{2014A&A...564A..55M} and \\citet{2016ApJS..225...10F}, then they will need to be thick rather than thin to best fit the full spectrum.\n\n\\begin{landscape}\n\\begin{deluxetable}{lrrrcccr|rrrcccr}\n\\tabletypesize{\\footnotesize}\n\\tablewidth{0pt}\n\\tablecolumns{13}\n\\tablecaption{Atmospheric Model Fits\\label{tbl:bestfits}}\n\\tablehead{\n\\colhead{Model} &\n\\multicolumn{7}{c}{\\underline{Best Fit to Near-IR only}} &\n\\multicolumn{7}{c}{\\underline{Best Fit to Near-IR + $L$ Band}} \\\\\n\\colhead{} &\n\\colhead{\\teff} &\n\\colhead{\\logg{}} &\n\\colhead{[Fe\/H]} &\n\\colhead{\\mbox{$f_{\\mathrm{sed}}$}} &\n\\colhead{\\kzz} &\n\\colhead{Dust Size} &\n\\colhead{\\gk}&\n\\colhead{\\teff} &\n\\colhead{\\logg{}} &\n\\colhead{[Fe\/H]} &\n\\colhead{\\mbox{$f_{\\mathrm{sed}}$}} &\n\\colhead{\\kzz} &\n\\colhead{Dust Size} &\n\\colhead{\\gk}\\\\\n\\cmidrule(lr){2-8}\\cmidrule(lr){9-15}\n\\colhead{} &\n\\colhead{(K)} &\n\\colhead{[cm\/s$^{2}$]} &\n\\colhead{} &\n\\colhead{} &\n\\colhead{[$\\mathrm{cm}^2 \\mathrm{s}^{-1}$]} &\n\\colhead{($\\mu$m)} &\n\\colhead{}&\n\\colhead{(K)} &\n\\colhead{[cm\/s$^{2}$]} &\n\\colhead{} &\n\\colhead{} &\n\\colhead{[$\\mathrm{cm}^2 \\mathrm{s}^{-1}$]} &\n\\colhead{($\\mu$m)} &\n\\colhead{}\n}\n\\startdata\n\\sidehead{\\underline{2MASS 0045+16:}}\n~~BT-Settl AGSS09 &2000&4.50&+0.5& \\nodata& \\nodata& \\nodata & {253710} & 2000&5.00&+0.5& \\nodata& \\nodata& \\nodata & {493972} \\\\\n~~BT-Settl CIFIST11 &1800&5.00& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {184374} & 1800&5.00& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {202512} \\\\\n~~ Drift-Phoenix &1700&4.00& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {217628} & 1800&4.50& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {359250} \\\\\n~~Saumon \\& Marley &1600&4.48& 0.0\\tablenotemark{a}&1&6& \\nodata & {174446} & 1600&4.48& 0.0\\tablenotemark{a}&1&6& \\nodata & {231808} \\\\\n~~Madhusudhan &1700&4.0&0.0& \\nodata & \\nodata &100 & {485212} & 1700&4.0&0.0& \\nodata & \\nodata &100 & {612891} \\\\\n~~Tremblin & 2000&3.5\\tablenotemark{a}& 0.0\\tablenotemark{a}& \\nodata & 6\\tablenotemark{a} & \\nodata & {174357} & 2000&3.5\\tablenotemark{a}&0.0\\tablenotemark{a}& \\nodata & 6\\tablenotemark{a} &\\nodata & {366177} \\\\\n\\sidehead{\\underline{G196$-$3B:}}\n~~BT-Settl AGSS09 &1700&3.50&0.0& \\nodata& \\nodata& \\nodata & {15589} & 1700&4.50&0.0& \\nodata& \\nodata& \\nodata & {42102} \\\\\n~~BT-Settl CIFIST11 &1750&4.00& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {14745} & 1700&3.50& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {40166} \\\\\n~~ Drift-Phoenix &1600&4.50& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {28231} & 1600&4.00& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {43899} \\\\\n~~Saumon \\& Marley &1400&4.25& 0.0\\tablenotemark{a}&1&4& \\nodata & {21832} & 1400&4.25& 0.0\\tablenotemark{a}&1&4& \\nodata & {49006} \\\\\n~~Madhusudhan &1400&4.0&0.0& \\nodata & \\nodata &60 & {47408} & 1400&4.0&0.0& \\nodata & \\nodata &100 & {103123} \\\\\n~~Tremblin & 1600&3.5\\tablenotemark{a}& 0.0\\tablenotemark{a}& \\nodata & 6\\tablenotemark{a} & \\nodata & {15486} & 1600&3.5\\tablenotemark{a}&0.0\\tablenotemark{a}& \\nodata & 6\\tablenotemark{a} &\\nodata & {23985} \\\\\n\\sidehead{\\underline{2MASS 0501$-$00:}}\n~~BT-Settl AGSS09 &2000&5.50&+0.5& \\nodata& \\nodata& \\nodata & {23214} & 1700&4.50&0.0& \\nodata& \\nodata& \\nodata & {105845} \\\\\n~~BT-Settl CIFIST11 &1650&5.00& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {18834} & 1600&5.00& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {106892} \\\\\n~~ Drift-Phoenix &1600&3.50& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {18862} & 1600&3.50& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {35401} \\\\\n~~Saumon \\& Marley &1400&4.00& 0.0\\tablenotemark{a}&1&6& \\nodata & {15106} & 1400&4.25& 0.0\\tablenotemark{a}&1&4& \\nodata & {106145} \\\\\n~~Madhusudhan &1500&4.0&0.0& \\nodata & \\nodata &60 & {53237} & 1400&4.0&0.0& \\nodata & \\nodata &100 & {222405} \\\\\n~~Tremblin & 1800&3.5\\tablenotemark{a}& 0.0\\tablenotemark{a}& \\nodata & 6\\tablenotemark{a} & \\nodata & {12864} & 1600&3.5\\tablenotemark{a}&0.0\\tablenotemark{a}& \\nodata & 6\\tablenotemark{a} &\\nodata & {70538} \\\\\n\\sidehead{\\underline{2MASS 0355+11:}}\n~~BT-Settl AGSS09 &1700&4.50&0.0& \\nodata& \\nodata& \\nodata & {13796} & 1600&3.50&0.0& \\nodata& \\nodata& \\nodata & {75661} \\\\\n~~BT-Settl CIFIST11 &1550&4.00& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {5305} & 1550&4.00& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {21312} \\\\\n~~ Drift-Phoenix &1500&3.50& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {11847} & 1600&5.00& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {94024} \\\\\n~~Saumon \\& Marley &1200&4.25& 0.0\\tablenotemark{a}&1&6& \\nodata & {12298} & 1200&5.47& 0.0\\tablenotemark{a}&1&4& \\nodata & {77846} \\\\\n~~Madhusudhan &1300&3.75&0.0& \\nodata & \\nodata &60 & {17793} & 1200&4.25&0.0& \\nodata & \\nodata &60 & {187057} \\\\\n~~Tremblin & 1400&3.5\\tablenotemark{a}& +0.2\\tablenotemark{a}& \\nodata & 6\\tablenotemark{a} & \\nodata & {7631} & 1400&3.5\\tablenotemark{a}&0.0\\tablenotemark{a}& \\nodata & 6\\tablenotemark{a} &\\nodata & {67283} \\\\\n\\sidehead{\\underline{2MASS 0103+19:}}\n~~BT-Settl AGSS09 &1500&4.50&0.0& \\nodata& \\nodata& \\nodata & {5615} & 1700&4.50&0.0& \\nodata& \\nodata& \\nodata & {24475} \\\\\n~~BT-Settl CIFIST11 &1650&5.00& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {2912} & 1500&5.00& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {15322} \\\\\n~~ Drift-Phoenix &1600&3.50& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {4581} & 1600&4.50& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {11271} \\\\\n~~Saumon \\& Marley &1400&4.25& 0.0\\tablenotemark{a}&1&6& \\nodata & {2551} & 1300&4.00& 0.0\\tablenotemark{a}&1&2& \\nodata & {17097} \\\\\n~~Madhusudhan &1400&4.0&0.0& \\nodata & \\nodata &60 & {7222} & 1300&4.0&0.0& \\nodata & \\nodata &100 & {28273} \\\\\n~~Tremblin & 1800&3.5\\tablenotemark{a}& 0.0\\tablenotemark{a}& \\nodata & 6\\tablenotemark{a} & \\nodata & {4101} & 1600&3.5\\tablenotemark{a}&0.0\\tablenotemark{a}& \\nodata & 6\\tablenotemark{a} &\\nodata & {17127} \\\\\n\\sidehead{\\underline{2MASS 2244+20:}}\n~~BT-Settl AGSS09 &1700&4.50&0.0& \\nodata& \\nodata& \\nodata & {47015} & 1600&3.50&0.0& \\nodata& \\nodata& \\nodata & {109410} \\\\\n~~BT-Settl CIFIST11 &1400&4.50& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {25443} & 1400&4.00& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {58998} \\\\\n~~ Drift-Phoenix &1500&3.50& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {47917} & 1500&3.50& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {63373} \\\\\n~~Saumon \\& Marley &1200&4.00& 0.0\\tablenotemark{a}&1&4& \\nodata & {10359} & 1000&4.00& 0.0\\tablenotemark{a}&1&6& \\nodata & {27231} \\\\\n~~Madhusudhan &1300&4.0&0.0& \\nodata & \\nodata &60 & {44486} & 1200&4.0&0.0& \\nodata & \\nodata &60 & {78465} \\\\\n~~Tremblin & 1400&3.5\\tablenotemark{a}& +0.2\\tablenotemark{a}& \\nodata & 6\\tablenotemark{a} & \\nodata & {15470} & 1400&3.5\\tablenotemark{a}&+0.2\\tablenotemark{a}& \\nodata & 6\\tablenotemark{a} &\\nodata & {40499} \\\\\n\\sidehead{\\underline{PSO 318.5$-$22:}}\n~~BT-Settl AGSS09 &1600&3.50&0.0& \\nodata& \\nodata& \\nodata & {3633} & 1600&3.50&0.0& \\nodata& \\nodata& \\nodata & {54994} \\\\\n~~BT-Settl CIFIST11 &1350&3.50& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {2861} & 1550&3.50& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {11876} \\\\\n~~ Drift-Phoenix &1500&5.00& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {5423} & 1500&5.00& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {9482} \\\\\n~~Saumon \\& Marley &1000&4.25& 0.0\\tablenotemark{a}&1&6& \\nodata & {2122} & 900&5.00& 0.0\\tablenotemark{a}&1&4& \\nodata & {8704} \\\\\n~~Madhusudhan &1100&4.0&0.0& \\nodata & \\nodata &30 & {5440} & 1000&4.0&+0.5& \\nodata & \\nodata &60 & {21581} \\\\\n~~Tremblin & 1275&3.7\\tablenotemark{a}& +0.4\\tablenotemark{a}& \\nodata & 5\\tablenotemark{a} & \\nodata & {1694} & 1200&3.5\\tablenotemark{a}&0.0\\tablenotemark{a}& \\nodata & 6\\tablenotemark{a} &\\nodata & {12564} \\\\\n\\sidehead{\\underline{WISE 0047+68:}}\n~~BT-Settl AGSS09 &1700&4.50&0.0& \\nodata& \\nodata& \\nodata & {23472} & 1600&3.50&0.0& \\nodata& \\nodata& \\nodata & {131724} \\\\\n~~BT-Settl CIFIST11 &1400&4.00& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {13212} & 1500&3.50& 0.0\\tablenotemark{a} & \\nodata& \\nodata& \\nodata & {57278} \\\\\n~~ Drift-Phoenix &1500&3.50& 0.0\\tablenotemark{a}& \\nodata& \\nodata& \\nodata & {25152} & 1500&4.00& 0.0\\tablenotemark{a}& \\nodata& \\nodata& \\nodata & {47773} \\\\\n~~Saumon \\& Marley &1100&4.47& 0.0\\tablenotemark{a}&1&6& \\nodata & {5711} & 1000&4.00& 0.0\\tablenotemark{a}&1&6& \\nodata & {20615} \\\\\n~~Madhusudhan &1200&4.25&0.0& \\nodata & \\nodata &60 & {19659} & 1100&4.0&+0.5& \\nodata & \\nodata &100 & {76802}\\\\\n~~Tremblin & 1400&3.5\\tablenotemark{a}& +0.2\\tablenotemark{a}& \\nodata & 6\\tablenotemark{a} & \\nodata & {5935} & 1275&3.7\\tablenotemark{a}&+0.4\\tablenotemark{a}& \\nodata & 5\\tablenotemark{a} &\\nodata & {25983} \\\\\n\\enddata\n\\tablenotetext{a}{Not a free parameter in the fit}\n\\end{deluxetable}\n\\end{landscape}\n\n\n\\begin{figure*}\n\\includegraphics[trim = {.25cm 4.7cm 3.5cm 2.75cm},clip,width=.9\\textwidth]{2021BestFit2MJ0355+11LBand.pdf}\n\\centering\n\\caption{The combined spectrum of 2MASS 0355+11 (black) compared to the model spectra (colored) with parameters that best fits the entire combined spectrum. For ease of comparison, the near-IR best-fit models are shown in grey.}\\label{fig:0355Fits}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[trim = {.25cm 4.7cm 3.5cm 2.75cm},clip,width=.9\\textwidth]{2021BestFitPSO-318LBand.pdf}\n\\centering\n\\caption{Same as Figure \\ref{fig:0355Fits}, except using the cooler PSO 318.5$-$22}\n\\label{fig:318Fits}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[trim = {0 5cm 1cm 8.5cm}, clip,width=.75\\textwidth]{2022PS0MP.pdf}\n\\centering\n\\caption{The goodness-of-fit parameter (\\gk) for PSO 318.5$-$22 plotted as function of temperature and \\logg{}. Note that the shape does not change drastically when including the $L$ band, but it does move to cooler temperatures ($\\sim$100 K). This held true across all of our objects.}\n\\label{fig:GK}\n\\end{figure*}\n\n\\subsubsection{Mixing Rate}\n\nWe can see how the vertical mixing rate (\\kzz) tends to vary with the inclusion of the $L$ band through the Saumon \\& Marley fits. With the inclusion of the $L$-band spectra, there is a shift to lower \\kzz~values (from the max possible 6 [$\\mathrm{cm}^2 \\mathrm{s}^{-1}$] to 4 or even 2) for the combined spectra, which on face value disagrees with the conclusions from \\citet{2018ApJ...869...18M} finding a need for a high \\kzz~(around $10^8$ $\\mathrm{cm}^2 \\mathrm{s}^{-1}$) to explain $L$-band features. However, when the models' thickest cloud parameter was used (f1), as was the case in the best fit for all of our objects, changing the log \\kzz~value from 6 to 4 had little affect on the model spectra in the near-IR and $L$ band. The effect on \\gk~was at its highest a 0.5 percent increase, and sometimes was as small as 0.01 percent. For some temperatures, the insensitivity of the model to \\kzz~extends down to \\kzz~of 2 [$\\mathrm{cm}^2 \\mathrm{s}^{-1}$], like at the best-fit temperature of 2MASS 0103+19 (1300 K), but often the lowered vertical mixing would start to deepen the $L$-band methane features at this point, making the fit noticeably worse. When a thinner cloud parameter was set, the best-fit \\kzz~values tended to stay high. Overall, we have a low sensitivity between the higher \\kzz~values, but can still say we agree with \\citet{2018ApJ...869...18M} that higher values in general create better fits. \n\n\\subsection{Model Comparisons: Evolutionary Parameters} \\label{EvoParam}\nSome of our objects are members of young moving groups and therefore have ages associated with them. These ages, along with measured $L_{\\mathrm{bol}}$ and evolutionary models, give well-defined evolutionary parameters for five of our objects, 2MASS 0045+16, 2MASS 2244+20, WISE 0047+68, 2MASS 0355+11 \\citep[All from ][]{2016ApJS..225...10F} and PSO 318.5$-$22 \\citep{2016ApJ...819..133A}. The memberships, ages, masses, effective temperature, and surface gravities for all of these objects can be found in Table \\ref{tbl:EVO}. We can us these \\teff~and \\logg{} values to select synthetic spectra that match these parameters, which we'll hereafter refer to as evolutionary effective temperature and surface gravity. For each model set, we took the model spectrum with the effective temperature and surface gravity closest to each object's evolutionary effective temperature and surface gravity, and of these selected the model parameters and $C$ which minimized \\gk. The Madhusudhan and Tremblin models do not fully cover the range of temperatures and gravities needed to encompass all of the objects evolutionary temperatures and gravities. When this occurred, the closest available parameters were used. The comparisons for these objects can be seen in Figures \\ref{fig:0045_EVO} through \\ref{fig:PSO_EVO}. For ease of reference, the combined best-fit model from the previous section is also plotted in grey.\n\n\\begin{table*}\n\\caption{Our Sample of Young L Dwarfs With Known Moving Groups and Ages} \\label{tbl:EVO}\n\\begin{tabular}{lcccccc}\n\\toprule[1.5pt]\nName & YMG & Age & Mass Range & \\teff & \\logg{} & Refs \\\\\n & Membership & (Myrs) & ($M_\\mathrm{Jup}$) & (K) & [$\\mathrm{cm}^2 \\mathrm{s}^{-1}$] & \\\\\n\\toprule[1.5pt]\n2MASS J0045+16\t&\tArgus   &  40 & 20--29& $2059 \\pm 45$& $4.22 \\pm 0.10$ &\tC09, F15, L16, F16\t\\\\\n2MASS J0355+11\t&\tAB Dor\t& 110--130  & 15--27& $1478\\pm 58$  & $4.58^{+0.07}_{-0.17}$ &\tC09, F13, AL13, F16\t\\\\\nWISEP J0047+68 &\tAB~Dor  & 110--130  & 9--15\t& $1230 \\pm 27$& $4.21 \\pm 0.10$ &\tG15, F15 ,F16\\\\\n2MASS J2244+20\t&\tAB~Dor\t& 110--130  & 9--12 &$1184 \\pm 10$& $4.18 \\pm 0.08$ &\tK08, F15, A16 \t\\\\\nPSO J318.5$-$22\t&$\\beta$~Pic& 20--26  & 5--7  & $1127^{+24}_{-26}$ & 4.01 $\\pm$ 0.03 &\tL13, A16, F16\t\n\\end{tabular}\n\\tablerefs{(A16)~\\citet{2016ApJ...819..133A}, (AL13)~\\citet{2013ApJ...772...79A},\n(C09)~\\citet{2009AJ....137.3345C}, \n(F12)~\\citet{2012ApJ...752...56F}, (F13)~\\citet{2013AJ....145....2F},\n(F15)~\\citet{2015ApJ...810..158F}\n(F16)~\\citet{2016ApJS..225...10F},\n(G15)~\\citet{2015ApJ...799..203G},\n(K08)~\\citet{2008ApJ...689.1295K},\n(L16)~\\citet{2016ApJ...833...96L}, (L13)~\\citet{2013ApJ...777L..20L},\n(V18)~\\citet{2018MNRAS.474.1041V}\n}\n\n\\end{table*}\nFor all our objects except 2MASS 0045+16, we see that the Drift-Phoenix and the BT-Settl AGSS models do not fit well when the evolutionary effective temperature and surface gravity are used. The Drift-Phoenix model over-predicts the flux in the $L$ band, with the peaks of the $J$ and $H$ bands being almost non-existent, which can be most clearly seen in Figure \\ref{fig:2244_EVO}. The AGSS model has the opposite problem, as already at 1400 K in Figure \\ref{fig:0355_EVO} it looks remarkably like a T dwarf with its deep methane absorption in the near-IR and $L$ band. The fact that the methane absorption is this strong in the AGSS model at such high temperatures is part of why the best-fit temperatures were hot for our young L dwarfs. The exception is our hottest object 2MASS 0045+16 (Figure \\ref{fig:0045_EVO}), which the AGSS model arguably has the best fit. Meanwhile, the Drift-Phoenix model inverts its color with the $J$ and $H$ band being too strong compared to weaker $L$ and $K$ bands, though not quite as dramatically at this temperature.\n\nThe CIFIST model set is one of the better fits for 2MASS 0355+11 when using the evolutionary effective temperature and surface gravity, where its biggest issues are predicting stronger water absorption features than we observe, both between the near-IR bands and in the $L$ band, and missing the sharp triangle shape of the H band. This issue is there in all the objects, but it is most pronounced for the colder ones like in Figure \\ref{fig:0047_EVO}, where the predicted water absorption becomes even stronger, and is accompanied by strong methane features as well. This increased absorption drops the model's $L$-band flux level too low. In general, the CIFIST BT-Settl model set seems to do comparatively well for young L dwarfs at higher temperatures, but struggles when transitioning to lower temperatures. \n\nThe selected Madhusudhan models have the opposite problem, fitting colder objects better than hotter ones. When using the evolutionary effective temperature and surface gravity of 2MASS 0355+11 and 2MASS 0045+16, we see the differences in the flux levels of the near-IR and $L$ band are much greater in the model than in the observed spectrum. This disparity is not there for the three colder objects, though other issues occur, such as over-estimation of the $K$-band flux and a turndown at the long end of the $L$ band. However, these issues are minor compared to the flux level spread between the bands at higher temperatures, which explains the Madhusudhan model's tendency to give colder best-fit temperatures.\n\nThe Tremblin and Saumon \\& Marley model sets are the most consistently good fits, though both still have some discrepancies. The Saumon \\& Marley models tend to overestimate the flux in the $J$ and $H$ bands, especially for the 2MASS 0045+16 and 2MASS 0035+11. Models with lower temperatures have lower near-IR flux which are closer to the observed values, such as in Figure \\ref{fig:0047_EVO}, which explains why the models' best fits trend towards colder temperatures. The Tremblin models are off in the near-IR about as much as the Saumon \\& Marley models, but with no consistent issues. For the three coldest objects, the Tremblin models matches the $L$-band flux level, yet exhibits strong absorption in the $P$, $Q$, and $R$ branches of methane unseen in the observed spectrum. Both also under-predict the $L$-band flux for 2MASS 0045+16, though this is consistent across all the models.\n\nLooking more generally, the three of the four models without disequilibrium chemistry (both BT-Settl models and  Drift-Phoenix) show an onset of methane at higher temperatures than we observe, such as in Figure \\ref{fig:0355_EVO}. This agrees with what we saw from the best-fit models. This effect is especially notable in the BT-Settl AGSS models, whose spectra at this temperature look more similar to T dwarfs than L dwarfs. The Madhusudhan model with the thickest clouds (Type A) does not have this early appearance of methane, but the less thick AE clouds do, with $Q$-band absorption being strong even at 1400 K. As for the models with disequilibrium chemistry, the $Q$-branch of methane is well fit when using the evolutionary effective temperature and surface gravity, and overall are generally the better fit compared to the equilibrium models. We also find that the cloudless BT-Settl models clouds tend to fit the colder objects worse than those with clouds as their evolutionary temperature, as seen in Figures \\ref{fig:0047_EVO}-\\ref{fig:PSO_EVO}. The exception to the cloudy models fitting better is the Drift-Phoenix models, which are poor past 1500 K due to so little flux coming out in the $J$ and $K$ bands. This is evidence that some combination of clouds and disequilibrium chemistry is a necessary for modeling brown dwarf atmospheres with accurate effective temperatures.\n\n\\begin{figure*}\n\\includegraphics[trim = {.25cm 4.7cm 3.5cm 2.75cm},clip,width=.9\\textwidth]{EVOComp0045.pdf}\n\\centering\n\\caption{The spectra of 2MASS 0045+16 compared to the spectra from each model set that best matches the known evolutionary effective temperature and surface gravity found in Table \\ref{tbl:EVO}. Note that in this case the Tremblin models do not extend to 4.5 [$\\mathrm{cm}^2 \\mathrm{s}^{-1}$] and the Madhusudhan models do not extend to 2000 K. In gray is also the combined best-fit spectra (in the case of the Tremblin models, the best-fit and evolutionary model parameters are the same).}\n\\label{fig:0045_EVO}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[trim = {.25cm 4.7cm 3.5cm 2.75cm},clip,width=.9\\textwidth]{2021EVOComp0355.pdf}\n\\centering\n\\caption{The spectra of 2MASS 0355+11 compared to the spectra from each model set that best matches the known evolutionary effective temperature and surface gravity found in Table \\ref{tbl:EVO}. Note that in this case the Tremblin models do not extend to 4.5 [$\\mathrm{cm}^2 \\mathrm{s}^{-1}$]. In gray is also the combined best-fit spectra (in the case of the Tremblin models, the best-fit and evolutionary model parameters are the same).}\n\\label{fig:0355_EVO}\n\\end{figure*}\n\n\n\n\\begin{figure*}\n\\includegraphics[trim = {.25cm 4.7cm 3.5cm 2.75cm},clip,width=.9\\textwidth]{EVOComp0047.pdf}\n\\centering\n\\caption{The spectra of WISE 0047+68 compared to the spectra from each model set that best matches the known evolutionary effective temperature and surface gravity found in Table \\ref{tbl:EVO}. Note that in this case the Tremblin models do not extend to 4.0 [$\\mathrm{cm}^2 \\mathrm{s}^{-1}$]. In gray is also the combined best-fit spectra.}\n\\label{fig:0047_EVO}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[trim = {.25cm 4.7cm 3.5cm 2.75cm},clip,width=.9\\textwidth]{EVOComp2244.pdf}\n\\centering\n\\caption{The spectra of 2MASS 2244+20 compared to the spectra from each model set that best matches the known evolutionary effective temperature and surface gravity found in Table \\ref{tbl:EVO}. Note that in this case the Tremblin models do not extend to 4.0 [$\\mathrm{cm}^2 \\mathrm{s}^{-1}$]. In gray is also the combined best-fit spectra (in the case of the Madhusudhan models, the best-fit and evolutionary model parameters are the same).}\n\\label{fig:2244_EVO}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[trim = {.25cm 4.7cm 3.5cm 2.75cm},clip,width=.9\\textwidth]{2021EVOComp318.pdf}\n\\centering\n\\caption{The spectra of PSO 318.5$-$22 compared to the spectra from each model set that best matches the known evolutionary effective temperature and surface gravity found in Table \\ref{tbl:EVO}. Note that in this case the Tremblin models do not extend to 1100 K or 4.0 [$\\mathrm{cm}^2 \\mathrm{s}^{-1}$]. In gray is also the combined best-fit spectra (in the case of the Tremblin models, the best-fit and evolutionary model parameters are the same).}\n\\label{fig:PSO_EVO}\n\\end{figure*}\n\n\\subsection{AB Dor Sub-population}\nThree of our targets, 2MASS 2244+20, WISE 0047+68, and 2MASS 0355+13, are members of the AB Doradus young moving group, and thus can be assumed to have similar compositions and ages \\citep[$\\sim$120 Myrs,][]{2013ApJ...766....6B} due to their shared origin and formation history. WISE 0047+68 and 2MASS 2244+20 have approximately the same mass, respectively 9-15 and 9-12 \\mbox{$M_{\\rm{Jup}}$}{}, but 2MASS 0355+13 is almost twice as massive at 15-27 \\mbox{$M_{\\rm{Jup}}$}. This means Figure \\ref{fig:ABDOR}, which shows the spectra of these three objects, is a mass sequence at a fixed age and composition, though admittedly an abbreviated one. 2MASS 0355+13 is more luminous than WISE 0047+68 and 2MASS 2244+20, whose spectra lie nearly on top of each other due to having similar masses (See Table \\ref{tbl:EVO}). Of particular interest is the 3.3 $\\mu$m $Q$-branch methane feature, which has a similar depth in the two cooler objects, even with different infrared spectral types and gravity classification (WISE 0047+68 as L7 INT-G and 2MASS 2244+20 as L6 VL-G). This feature is absent in the more massive (and hotter) 2MASS 0355+13, showing the temperature dependence of the methane feature for a fixed age and composition. AB Dor has several other ultracool members \\citep{2016ApJS..225...10F}, which means it will be possible to expand this mass sequence.  \n\\begin{figure*}\n\\includegraphics[width = \\textwidth]{ABDOR.pdf}\n\\centering\n\\caption{Spectra of three members of the young moving group AB Dor, placed at 10 pc. The two objects of the same mass lie almost on top of each other, and with a $Q$-branch methane feature of similar depth. This feature is absent in the hotter and more massive 2MASS 0355+11.}\\label{fig:ABDOR}\n\\end{figure*}\n\n\\section{Conclusions}\\label{sec:con}\n\nFrom this work, we can see that $L$-band spectra are not particularly strong diagnostic tools on their own for young L dwarfs due to a lack of strong features (with the exception of the $Q$-branch of methane). However, when used in hand with near-IR spectra it is extremely useful for understanding objects' atmospheric properties. The inclusion of the $L$ band highlights where models are falling short, and what physical processes models need to include to match observations. We find adding the $L$ band lowers the best-fit effective temperature of our models by $\\sim$100 K. For these fits, when clouds are included in the models, thick ones are preferred, and when a vertical mixing rate is used (\\kzz), higher values give the best results. We also note that for models to fit the combined spectra well at the published evolutionary effective temperatures for these objects, we need models with disequilibrium chemistry and\/or clouds. Overall, we find the Tremblin and Saumon \\& Marley models fit the full spectrum of the young objects best. These results show the power of wide spectral coverage, matching conclusion from \\citet{2021MNRAS.506.1944B}, and we recommend that our data also be used to enhance future retrievals of these objects to parse their diversity. In particular though, these observations show the value of the $L$ band for understanding the atmospheres of young brown dwarfs, and the giant exoplanets for which they act as proxies and give us a preview of the insights that the James Webb Space Telescope will bring. \n\n\\section*{Acknowledgements}\nBased on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, Tecnolog\u00eda e Innovaci\u00f3n Productiva (Argentina), and Minist\u00e9rio da Ci\u00eancia, Tecnologia e Inova\u00e7\u00e3o (Brazil).  KNA and SAB are grateful for support from the Isaac J. Tressler fund for astronomy at Bucknell University, and the Doreen and Lyman Spitzer Graduate Fellowship in Astrophysics at the University of Toledo. This work benefited from the Exoplanet Summer Program in the Other Worlds Laboratory (OWL) at the University of California, Santa Cruz, a program funded by the Heising-Simons Foundation. We appreciated conversations with Brittney Miles which enhanced this work, and we would like to thank Denise Stephens for contributing the 2MASS 2244+20 spectrum, and Pascal Tremblin for allowing us access to unpublished models. This work also benefited from The UltracoolSheet, maintained by Will Best, Trent Dupuy, Michael Liu, Rob Siverd, and Zhoujian Zhang, and developed from compilations by Dupuy \\& Liu (2012, ApJS, 201, 19), Dupuy \\& Kraus (2013, Science, 341, 1492), Liu et al. (2016, ApJ, 833, 96), Best et al. (2018, ApJS, 234, 1), and Best et al. (2020b, AJ, in press).\n\n\\section*{Data Availability}\nThe combined spectra for all of these objects are available in the online supplementary material for this article.\n\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\nWe have carried out a high statistics ($2 \\times 10^9$ events)\nsearch for ultra-high energy\ngamma-ray emission from the X-ray binary sources Cygnus X-3 and\nHercules X-1.\nUsing data taken with the CASA-MIA detector over a five year period\n(1990-1995), we find no evidence for steady emission from either\nsource.\nThe derived 90\\% c.l. upper limit to the steady integral flux\nof gamma-rays from Cygnus X-3\nis $\\Phi (E > 115\\,{\\rm TeV}) \n< 6.3 \\times 10^{-15}$ photons cm$^{-2}$ sec$^{-1}$,\nand from Hercules X-1 it is $\\Phi (E > 115\\,{\\rm TeV}) \n< 8.5 \\times 10^{-15}$\nphotons cm$^{-2}$ sec$^{-1}$.\nThese limits are more than two orders of magnitude lower than earlier\nclaimed detections and are better than recent experiments\noperating in the same energy range.\nWe have also searched for transient emission on time periods of\none day and $0.5\\,$hr and find no evidence for such emission from\neither source.\nThe typical daily limit on the integral gamma-ray flux\nfrom Cygnus X-3 or Hercules X-1 is\n$\\Phi_{{\\rm daily}} (E > 115\\,{\\rm TeV}) < 2.0 \\times 10^{-13}$\nphotons cm$^{-2}$ sec$^{-1}$.\nFor Cygnus X-3, we see no evidence for emission correlated with the\n$4.8\\,$hr\nX-ray periodicity or with the occurrence of large\nradio flares.\nUnless one postulates that these sources were very active earlier\nand are now dormant,\nthe limits presented here put into question the earlier results,\nand highlight the difficulties that possible future experiments\nwill have in detecting gamma-ray signals at ultra-high energies.\n\n\n\\section{Introduction}\n\\label{sec:intro}\nCosmic ray particles span a remarkable range of energies, from\nthe MeV scale to more than $10^{20}\\,$eV (eV = electron Volt).\nAt energies above $1\\,$TeV ($10^{12}\\,$eV), we know that\ncosmic rays do not originate from local sources in or nearby our\nSolar System.\nTherefore, high energy cosmic rays must come from acceleration sites in the\nGalaxy at large or from outside the Galaxy.\nRemarkably, after many years of research, the exact sites of\nhigh energy cosmic ray acceleration remain unknown.\n\nThere are several difficulties that plague efforts to pinpoint the\norigins of high energy cosmic rays.\nFirst, since the bulk of the cosmic radiation is electrically\ncharged, any source information contained in the directions of the\narriving particles is lost due to deflection in the \nGalactic magnetic field.\nA second difficulty concerns the energetics of \nthe proposed cosmic ray acceleration\nmechanisms.\nFor example, \nalthough models based on shock acceleration in supernova remnants offer\nplausible explanations for the cosmic ray origin up to $10^{14}\\,$eV\n(and perhaps up to $10^{15}\\,$eV), these models become less\nsatisfying and less realistic at energies above $10^{15}\\,$eV.\nSince cosmic ray origins remain\nmysterious, it is natural to search for neutral\nradiation from point sources \nwhich, if seen,\ncould pinpoint possible\nacceleration sites.\nThe question of cosmic ray origin is thus a prime motivation for high\nenergy neutral particle (gamma-ray or neutrino) astronomy.\n\nIn addition to supernova remnants, possible galactic sources of high\nenergy particles include pulsars and compact binary systems.\nGamma-radiation has been unambiguously detected from the Crab\nNebula (a supernova remnant) at energies up to $10\\,$TeV by\nground-based detectors \\cite{ref:Snowmass}, but, historically,\nthe compact binary sources Cygnus X-3 and Hercules X-1 have received\nconsiderably greater attention in the ground-based astronomical community.\nIn the period 1975-1990, literally dozens of \ngamma-ray detections from Cygnus X-3 and Hercules X-1\nwere reported by numerous experiments.\nThe detections spanned\na wide range of energies ($100\\,$MeV to $10^{17}\\,$eV)\n\\cite{ref:Reviews},\nwere generally of low statistical significance\n(typically three to four standard deviations), and episodic in nature.\nOften a statistically significant signal could only be extracted\nas a result of a periodicity analysis, where the\ndata were phase-locked to a known source X-ray periodicity.\nIn spite of these difficulties, the sheer number of reports made\nit difficult to dismiss the detections as being\nentirely due to statistical fluctuation \\cite{ref:Protheroe}.\nIn fact, by the late 1980's, it was generally established that\nCygnus X-3 and Hercules X-1 were powerful\nemitters of high energy gamma-rays (although \ncontrary interpretations of the data existed \\cite{ref:Chardin}).\nA number of new, more sensitive\nground-based air shower arrays were commissioned\nat this time, including the CASA-MIA experiment in Dugway, Utah (USA).\nThis paper describes long-term (1990-1995) observations of Cygnus\nX-3 and Hercules X-1 by CASA-MIA.\n\nIn the following section, we\noutline the experimental techniques of gamma-ray astronomy.\nWe summarize the properties of\nCygnus X-3 and Hercules X-1, and\nreview previous observations\nof these sources and the astrophysical ramifications from the\nobservations.\nWe then turn our attention to the experimental apparatus,\nthe event reconstruction procedures, and the methods used to\nselect gamma-ray candidate events.\nWe present results from a \ndata sample of $2 \\times 10^9$ events, and\nconclude with comparisons of our results to those of\nearlier and contemporaneous experiments.\n\n\\section{Experimental Techniques of Gamma-Ray Astronomy}\n\\label{sec:techniques}\nGamma-ray sources typically exhibit power law spectra;\nfluxes fall rapidly with increasing energy.\nSpace-borne experiments (on satellites or balloons) \ncurrently have sufficient\nsensitivity to detect gamma-rays up to an energy of $\\sim 10\\,$GeV.\nAstronomy at higher energies requires very large collection\nareas available only to ground-based telescopes.\nGround-based instruments rely upon the fact that\nhigh energy gamma-rays interact in the Earth's atmosphere to\nproduce extensive air showers.\nAt energies near $1\\,$TeV, \natmospheric Cherenkov telescopes \nuse optical techniques to\ndetect the Cherenkov radiation in\nthe shower.\nAt higher energies ($\\sim 10\\,$TeV and above),\nthe charged particles in the shower penetrate to ground level.\nHere, air shower arrays \nrecord the arrival times and particle densities of the\ncharged particles.\nAt energies above $10^{17}\\,$eV, there is enough energy in the shower\nto allow the detection of \nnitrogen fluorescence in the atmosphere.\nThe faint near ultraviolet fluorescence signal can be optically detected at\nnight by experiments such as the Fly's Eye.\n\n\\section{Discussion of the Sources and Earlier Results}\n\\label{sec:sources}\n\\subsection{Cygnus X-3}\n\\label{subsec:cygnus}\nCygnus X-3 is one of the most luminous X-ray sources in our\nGalaxy \\cite{ref:Giacconi}.\nThe X-ray emission is characterized by a $4.8\\,$hr \nperiodicity,\nwhich is assumed to be associated with the orbital motion of\na compact object (neutron star or black hole) around its binary\ncompanion.\nThe periodicity has been well studied; a complete ephemeris\nis available for the period from 1970-1995 \n\\cite{ref:Parsignault,ref:vanderKlis1,ref:vanderKlis2,ref:Kitamoto}.\nIn addition to being a powerful X-ray source, Cygnus X-3 is seen\nin the infrared and is a strong and variable radio source.\nRadio flares have been detected in which the output from the source\nincreases by two to three orders of magnitude on the time scale of\ndays \n\\cite{ref:Gregory1,ref:Johnston,ref:Waltman1,ref:Waltman2}.\nThese flares were first detected in 1972 and the\noutbursts have continued through 1994.\nSince\nCygnus X-3 lies in the galactic plane, its optical emission\nis largely obscured by interstellar material.\nThe lack of a strong optical signal makes determination of\nthe distance to Cygnus X-3 difficult, but general\nconsensus places it near $10\\,$kpc \\cite{ref:Lauque}.\n\nThe first published result claiming the detection of\ngamma-ray emission from Cygnus X-3 \ncame in 1977 from the SAS-2 satellite \nat low gamma-ray energies (E$> 35\\,$MeV) \\cite{ref:Lamb1}.\nThis result made use of an apparent correlation between the\ngamma-ray arrival times and the $4.8\\,$hr\nX-ray periodicity.\nLater observations by the COS-B satellite \n\\cite{ref:Bennett}\nwith a much larger\nsource exposure failed to confirm the SAS-2 result, and the COS-B\nauthors argued that the initial detection \\cite{ref:Lamb1}\nwas flawed because of an incorrect treatment of the diffuse\ngamma-ray component \\cite{ref:Hermsen}.\nTo complicate matters, there have been re-examinations\nof both the SAS-2 \\cite{ref:Fichtel} and COS-B \\cite{ref:Li}\ndata sets which claim that \nsignals indeed exist in both cases.\nMost recently, the EGRET experiment on the Compton Gamma-Ray Observatory\n(CGRO) failed to detect gamma-ray emission from Cygnus\nX-3 at a sensitivity level comparable to COS-B\n\\cite{ref:Michelson}.\nTo summarize,\nthere exists some controversy as to whether low energy\ngamma-rays have {\\it ever} been detected from Cygnus X-3.\nRegardless, it can be reasonably concluded\nthat the source is not a strong emitter of gamma-rays in the energy\nrange between $30\\,$MeV and $10\\,$GeV.\n\nThe first published report of very-high energy\ngamma-ray emission from Cygnus X-3\ncame from the Crimean Observatory \nusing an atmospheric Cherenkov telescope at energies above $2\\,$TeV \n\\cite{ref:Neshpor}.\nThis result was based on data taken between 1972 and 1977 and \nthe emission was claimed to be correlated with the\n$4.8\\,$hr\nX-ray period.\nFrom 1980 to 1990, there were numerous additional detections\nof Cygnus X-3 by atmospheric Cherenkov telescopes \n\\cite{ref:Danaher,ref:Lamb2,ref:Dowthwaite1,ref:Cawley1,%\nref:Chadwick,ref:Bhat,ref:Brazier}.\nThe detections were generally episodic in nature and \nusually required the\nuse of the \n$4.8\\,$hr\nperiodicity to extract a signal.\nEvidence for a 12.6 msec gamma-ray pulsar inside the Cygnus X-3 system\nwas claimed on more than one occasion\n\\cite{ref:Chadwick,ref:Brazier,ref:Gregory2}.\n\nAt the higher energies accessible by ground arrays, \nevidence for ultra-high energy gamma-ray emission from\nCygnus X-3 was presented by the Kiel array \\cite{ref:Samorski}\nand subsequently by the Haverah Park experiment \\cite{ref:Lloyd}.\nThese results were based on data taken between 1976 and 1980.\nThe gamma-ray emission was apparently steady over this time\nperiod and was\ncorrelated with the X-ray\nperiodicity.\nAdditional evidence  \nfor gamma-ray emission from Cygnus X-3\nwas later reported by other air shower detectors\n\\cite{ref:Kifune,ref:Alexeenko,ref:Baltrusaitis1,%\nref:Tonwar1,ref:Morello}.\n\nAt extremely-high energies (E $> 5 \\times 10^{17}\\,$eV),\nevidence was presented for neutral particles from the direction\nof Cygnus X-3 by the Fly's Eye \\cite{ref:Cassiday1} and\nby Akeno \\cite{ref:Teshima} groups, based on data taken \nduring the periods 1981-1989 and 1984-1989, respectively.\nThese data apparently indicated steady emission of neutral\nparticles from Cygnus X-3 that was uncorrelated with the X-ray periodicity.\nThe Haverah Park experiment, operating in the same energy range, and\nduring much of the same period in time,\nfound no evidence for such emission\n\\cite{ref:Lawrence}.\n\nThe evidence \nfrom ground-based experiments \nfor gamma-ray emission from Cygnus X-3 from 1975 to 1990\nis shown in Figure~\\ref{fig:OldCyg}.\nHere,\nthe integral gamma-ray fluxes\nare plotted as a function of energy.\nAlso\nshown is a single power law fit of the\nform E$^{-1.1}$.\nThe fact that the gamma-ray fluxes at widely varying energies\ncould be approximately fit by a single power law was taken by\nsome as evidence of a unified acceleration mechanism at the source.\nOne should note, however, that all results\nshown in Figure~\\ref{fig:OldCyg} represent {\\it integral}\nflux measurements by experiments incapable of accurately measuring\ndifferential fluxes.\nSince the detections were generally\nonly marginally statistically significant, the reported fluxes\nequally represent the three to four standard deviation\nsensitivity of each instrument at a fixed energy.\nThe fluxes\ntherefore would naturally fall on an E$^{-1}$ power law,\nif the sensitivities of the experiments scaled linearly\nwith energy (which was approximately true for these\nfirst-generation experiments).\nIt has also been pointed out that even if the source\nmechanism produced emission with a single power law form,\nthe {\\it detected} flux at Earth would have a significant\ndip between $10^{3}$ and $10^{4}\\,$TeV \nbecause of absorption of gamma-rays by\nthe cosmic microwave radiation \\cite{ref:Cawley2}.\n\nStarting with the CYGNUS experiment in 1988 \\cite{ref:Dingus1},\na number of more sensitive ground-based experiments were\nunable to detect gamma-ray emission from Cygnus X-3,\nat levels significantly lower than the earlier reports.\nUpper limits on the flux were reported for experiments using both the\natmospheric Cherenkov technique \\cite{ref:Fegan},\nas well as the ground-array technique\n\\cite{ref:Cassiday2,ref:Alexandreas1}.\nUsing parts of the eventual CASA-MIA detector, some\nof us reported limits for data taken in 1988-1989\n\\cite{ref:Ciampa} and in 1989 \\cite{ref:Cronin}.\nThe general trend of a ``quiet'' Cygnus X-3\ncontinued into the early 1990's, although\nthere were several publications claiming gamma-ray emission based\nlargely on data that had been taken in the previous decade\n\\cite{ref:Muraki,ref:Bowden,ref:Tonwar2}.\n\n\\subsection{Hercules X-1}\n\\label{subsec:hercules}\nLike Cygnus X-3, Hercules X-1 is a powerful binary X-ray \nsource \\cite{ref:Tananbaum}.\nThe X-ray emission is modulated on a time scale of 1.7 days\nwhich is assumed to result from the\norbital motion of the binary pair.\nUnlike Cygnus X-3, Hercules X-1 is not seen in radio, but\nhas been observed for many years in the optical \nrange \\cite{ref:Jones} and a $5.8\\,$kpc distance\nto the source has been determined \\cite{ref:Forman}.\nIn addition, Hercules X-1 contains an X-ray pulsar\nwith a period of $1.24\\,$sec \\cite{ref:Tananbaum}, but\nwhose ephemeris is relatively poorly determined because of\nunpredictable variations in the spin-up rate \\cite{ref:Deeter}.\nHercules X-1 has not been detected by space-borne gamma-ray instruments.\n\nThe first evidence from a ground-based observatory for gamma-ray emission\nfrom Hercules X-1 was reported in 1984 by the Durham group using\nthe atmospheric Cherenkov technique \\cite{ref:Dowthwaite2}.\nThe reported gamma-ray emission came in the form of a short\nburst ($\\sim 3$ minute duration) that exhibited 1.24 sec\nperiodicity.\nFollowing this report, additional pulsed emission was claimed by\nCherenkov detectors operating at ultra-high energies (E $> 500\\,$TeV)\n\\cite{ref:Baltrusaitis2} and at TeV energies\n\\cite{ref:Gorham1,ref:Gorham2}.\n\nThe most intriguing evidence for gamma-ray emission from\nHercules X-1 came from data taken in 1986 by three\nexperiments.\nData taken between April and July of 1986 by \nthe Haleakala \\cite{ref:Resvanis}\nand Whipple \\cite{ref:Lamb3} telescopes operating\nnear $1\\,$TeV, and by the CYGNUS experiment \\cite{ref:Dingus2}\nin July of 1986 operating at energies above\n$50\\,$TeV, all indicated evidence for gamma-ray emission from Hercules\nX-1 in the form of short bursts of approximately 0.5 hr duration.\nIn addition, the emission detected by each experiment exhibited\na common periodicity near $1.2358\\,$sec,\nwhich differed by a significant amount \n($\\sim 0.16\\%$ lower)\nfrom the known X-ray period.\nThe data from the CYGNUS experiment was further puzzling\nbecause the events from the direction of Hercules X-1\nhad a muon content that was similar to the cosmic ray background events,\nwhereas gamma-ray events should have contained\nsignificantly fewer muons.\nLater, two groups with somewhat poorer sensitivity presented\nadditional evidence for gamma-ray emission from Hercules X-1 at\ndifferent times in 1986 \nat TeV \\cite{ref:Vishwanath} and $100\\,$TeV \\cite{ref:Gupta}\nenergies.\n\nSince the advent of upgraded and improved experiments in the early\n1990's, the gamma-ray signals from Hercules X-1 disappeared\nfrom the published literature.\nThe Whipple group, using a more sensitive Cherenkov imaging\ntechnique, failed to detect emission from Hercules X-1, and found \nno statistically significant evidence for gamma-ray\nemission from Hercules X-1 over a six year period, \neven including their data from 1986 \\cite{ref:Reynolds}.\nAn enlarged and improved\nCYGNUS experiment also failed to see gamma-rays from\nHercules X-1 \nin the period between 1987 and 1991\n\\cite{ref:Alexandreas2}.\nUsing data taken in 1989,\nwe reported upper limits on the emission of gamma-rays from\nHercules X-1 using part of the eventual CASA-MIA experiment \\cite{ref:Cronin}.\n\n\\subsection{Theoretical Implications}\n\\label{subsec:theory}\nThe many claims of very high energy gamma-ray\nemission from the binary systems Cygnus X-3 and Hercules\nX-1 fueled great interest in the development of astrophysical\nmodels to explain such emission.\nThere were also non-standard particle physics models put forward\nto explain the observations; these models will not be discussed here.\n \nFor the case of Cygnus X-3, where the gamma-ray emission was\ngenerally observed with a \n$4.8\\,$hr\nperiodicity, the astrophysical\nmodels needed to incorporate the orbital dynamics of the\nbinary system.\nModels in which an energetic pulsar alone served as the power source\nfor the gamma-rays \\cite{ref:Cheng1}\nor in which accretion powered the gamma-rays\n\\cite{ref:Chanmugam}\nwere proposed.\nThese models generally had\ndifficulty\nin producing gamma-rays at energies above $10^{15}\\,$eV.\nMore popular were a \ngeneral class of models in which the gamma-rays were produced from\nthe decays of $\\pi^0$ mesons made in hadronic collisions\n\\cite{ref:Vestrand,ref:Eichler1,ref:Kazanas}.\nThe hadronic beam \nresulted from\ndiffusive shock acceleration,\nperhaps near the neutron star, and possible beam targets included\nthe\natmosphere of the companion star or material in the accretion disk.\nSuch ``beam-dump'' models were capable of explaining\ngamma-rays at ultra-high\nenergies and were also able to accommodate the observed periodicities of\nthe gamma-ray signals.\nSeveral authors recognized\nthat in order to explain\nthe ultra-high energy gamma-ray fluxes initially seen,\nthe required luminosity of Cygnus X-3 \nwould also be sufficient to account for\na substantial fraction of the high energy cosmic ray flux\n\\cite {ref:Wdowczyk}.\nHillas pointed out that if Cygnus X-3 consisted of a $10^{17}\\,$eV\naccelerator with a luminosity of $\\sim 10^{39}\\,$ergs\/sec,\nonly one such object like it would be required to explain the origin\nof cosmic rays above $10^{16}\\,$eV \\cite{ref:Hillas}.\n\nUnlike Cygnus X-3, the gamma-ray emission from Hercules X-1 was not\nseen to be correlated with the orbital motion of the binary system,\nbut instead with the pulsar periodicity.\nThis observation, along with evidence that the emission appeared in\nthe form of short bursts,\nled naturally to models in which the pulsar itself was the \npower source.\nIn such models,\nthe gamma-rays were produced by the interaction\nof a charged particle beam with the accretion disk\n\\cite{ref:Eichler2,ref:Gorham3}.\nMore difficult to explain were the 1986 observations of\ngamma-ray emission at a slightly shorter period than the X-ray period.\nThe anomalous gamma-ray periodicity was\nexplained by the presence of matter in the accretion disk\nwhich periodically obscured the gamma-ray interaction region\n\\cite{ref:Cheng2,ref:Slane}.\n\nIn summary, although theoretical difficulties existed\nin explaining the apparent signals of gamma-rays from\nCygnus X-3 and Hercules X-1, the signals were\ntantalizing because of the possibility that they\nrevealed important sources of the ultra-high\nenergy cosmic ray flux.\n\n\\section{Experimental Procedure}\n\\label{sec:experiment}\n\\subsection{The CASA-MIA Experiment}\n\\label{subsec:casamia}\nThe CASA-MIA experiment is located in Dugway, Utah, USA \n($40.2^\\circ\\,$N, $112.8^\\circ\\,$W) at an altitude\nof $1450\\,$m above sea level ($870\\,$g\/cm$^2$ atmospheric depth).\nCASA-MIA consists of two major components: the Chicago\nAir Shower Array (CASA), a large surface array of scintillation detectors,\nand the Michigan Array (MIA), a buried array of scintillation counters\nsensitive to the muonic component of air showers.\n\nCASA consists of 1089 scintillation detectors placed on a \n$15\\,$m square grid and enclosing an area of $230,400\\,$m$^2$.\nConstruction on the array started in 1988 and a small portion\n(5\\%) of the experiment operated in 1989. A more\nsubstantial portion ($\\sim$50\\%) of it was completed by early 1990.\nData collection with this portion \nstarted on March 1, 1990.\nAdditional detectors were added in 1990 \nto complete the construction.\n\nMIA consists of 1024 scintillation counters located beneath CASA in\n16 groupings (patches).\nThe total active scintillator area is $2,500\\,$m$^2$ and the counters\nare buried beneath $\\sim 3.5\\,$m of soil.\nThis depth corresponds to a muon threshold energy of approximately $0.8\\,$GeV.\nParts of MIA were operational as early as 1987, 50\\% of the experiment\nwas completed by early 1990, and the entire array was working\nby early 1991.\nThe CASA-MIA\nexperiment was turned off temporarily in 1991 for repair due to\nlightning damage, but has operated essentially uninterrupted since\nthat time.\nTable~\\ref{tab:array} summarizes the size and detector makeup of \nCASA-MIA as a function of time.\n\n\\begin{table}\n\\begin{center}\n\\caption{\nSize and makeup of CASA-MIA experiment as a function of time\nfor the data sample used in this analysis.\nA range of values indicates that the experiment was\nbeing enlarged during this period of time.\nData taken after August 1995 are not used in this analysis.}\n\\label{tab:array}\n\\vspace{10pt}\n\\begin{tabular}{|cccc|}\\hline\nEpoch & Enclosed Area (m$^2$) & CASA Detectors & MIA Counters \\\\\n\\hline\nMar. 1990 -- Oct. 1990 &         108,900 &      529 & 512 \\\\\nOct. 1990 -- Apr. 1991 & 108,900--230,400 & 529--1089 & 512--1024 \\\\\nJan. 1992 -- Aug. 1993 &         230,400 &     1089 &     1024 \\\\\nAug. 1993 -- Aug. 1995 &         216,225 &     1056 &     1024 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nFigure~\\ref{fig:array} shows a plan view of the experimental site.\nIn addition to CASA-MIA, there are other installations\nat the same site.\nThe other equipment used in this analysis is an array\nof five tracking Cherenkov telescopes.\nOne telescope is located at the center of CASA-MIA, and the other\nfour are 120$\\,$m away from the center along the major axes.\nEach telescope consists of a $35\\,$cm diameter mirror which focuses\nCherenkov radiation onto a single $5.1\\,$cm photomultiplier tube (PMT).\nThe signals from the PMTs are digitized to record the amplitude and\ntime of arrival of the Cherenkov wavefront at each telescope location.\nThe shower direction is reconstructed\nby fitting the Cherenkov arrival times to a conical wavefront.\n\nA complete description of the CASA-MIA experiment can be found\nelsewhere \\cite{ref:NIMPaper}; here we briefly describe some aspects\nof the experiment that are relevant for this analysis.\nEach CASA station consists of four scintillation counters connected\nto a local electronics board.\nA station is {\\it alerted} when at least two of the four counters\nfire within a $30\\,$nsec window and it is {\\it triggered} when at least\nthree of the counters fire.\nIf three or more stations trigger within a time period of approximately\n$3\\,\\mu$sec, an {\\it array trigger} is said to have occurred.\nThe array trigger rate depends on operating conditions\n(e.g. atmospheric pressure), but is\n$\\sim$20 Hz for the full CASA-MIA experiment.\n\nUpon an array trigger, the Universal Time (UT) is latched and\nrecorded by either a GOES Satellite Receiver Clock (1990-1993)\nwith a precision of $\\pm 1\\,$msec, or by a Global Positioning System (GPS)\nclock (1993-1995) with a precision of $\\pm 100\\,$nsec.\nFor each array trigger,\na command is broadcast to the array\ninstructing each alerted CASA station to digitize\nand record its data, and\na signal is generated to stop time-to-digital\nconverters (TDCs) on each MIA counter.\nThe TDCs have a range of $4\\,\\mu$sec and \na least bit precision of $4\\,$nsec.\nThe data from each CASA station consist of the arrival times and pulse-height\namplitudes of the pulses from each scintillation counter, as well\nas the arrival times of pulses from the four nearest neighbor stations.\nThe data from each MIA counter consist of the \ntime of arrival of the array trigger\nrelative to the passage of a muon through the counter.\nThe CASA-MIA data and the Universal Time recorded as the result\nof an array trigger\ncorrespond to a single air shower {\\it event}.\n\n\\subsection{Event Reconstruction}\n\\label{sec:recon}\nWe briefly summarize some of the important aspects of the\nCASA-MIA event reconstruction; full details can be found\nelsewhere \\cite{ref:NIMPaper}.\nThe data from the experiment are accumulated in runs of six hours\nduration.\nAll calibrations and offsets are \ndetermined for each run separately.\nAt the start of a run, the timing constants associated with the CASA\nstation electronics are calibrated by an internal oscillator.\nTiming constants are corrected for the effects of temperature\nby studying the constants over the span of a week.\nThe CASA counter particle gains are determined for each run from\nthe abundant cosmic ray air showers.\nThe counter gains are found from the PMT amplitude distributions\nof those counters hit in stations with two out of four counters hit.\nA statistical correction of $\\sim 20\\%$\naccounts for the fact that\non average slightly more than one particle passes through a counter\nin this situation.\nThe CASA cable and electronic delays \nare determined from the zenith angle distributions of the detected\nevents.\nThe relative delay between the CASA and MIA trigger systems is\ndetermined by centering the peak of the muon arrival time distribution\nrelative to the position of the CASA trigger time.\n\nWe estimate the shower\n{\\it core position} \nby the location on the ground with the highest particle density.\nThe total number of particles in the shower, or \n{\\it shower size}, is determined\nby fitting the density samples obtained from the CASA stations to\na lateral distribution of fixed form.\nThe mean number of alerted CASA stations is 19 and the\nmean shower size is $\\sim 25,000$ equivalent minimum ionizing\nparticles.\n\nThe {\\it shower direction} is determined from the timing information\nrecorded by CASA.\nThe relative times between pairs of adjacent alerted CASA stations\nare determined.\nEach relative time gives a measure of the shower\ndirection along one axis of the experiment.\nThe times are weighted by an empirical function of the\nlocal particle density and distance to the shower core, and\nare fit to a wavefront which accounts for\nthe conical shape of the shower front.\nThe cone slope is approximately $0.07\\,$nsec\/m.\nThe shower direction in local coordinates is defined by two\nangles.\nThe zenith angle, $\\theta$, is measured with respect to the vertical\ndirection and the azimuthal angle, $\\phi$, is measured with respect\nto East in a counter-clockwise manner.\n\nIn order to be confident of any astronomical results, it is essential\nto measure the  {\\it angular resolution} of the experiment.\nThe resolution has two parts.\nThe statistical part largely derives from\nthe intrinsic \nfluctuations in the arrival times of the\nshower particles\nand from the timing resolution of the CASA counters and electronics.\nThe systematic contribution derives primarily from the\naccuracies of the experiment survey and of\nthe calculation of timing delays and offsets.\n\nThe statistical contribution to the angular resolution\nis determined by three different techniques.\nFirst, on an event-by-event basis, we divide the array into\ntwo overlapping sub-arrays and compare the\nshower directions that are reconstructed by each sub-array.\nUsing an air shower and detector simulation, we estimate\nthe statistical correction required to derive the angular\nresolution from the sub-array direction comparison.\nSecond, we compare the shower direction as determined by CASA with\nthe direction determined by the five Cherenkov telescopes for\nevents in which both CASA and the telescopes triggered.\nBy statistically removing the angular resolution of the telescopes\nfrom this comparison, we estimate the CASA resolution.\nThird, we have detected the shadow that the Moon casts in the cosmic\nrays \\cite{ref:Moon}.  \nFor data taken between 1990 and 1995, the Moon \nshadow is shown in Figure~\\ref{fig:moon}.\nBy deconvolving the size of the Moon from the measured shadow,\nwe obtain another estimate of the angular resolution.\nFigure~\\ref{fig:resolution} shows the resolution estimates\nfrom these different techniques.\nThe agreement between the various methods is good, which allows\nus to determine a single parametric form for the resolution as\na function of the number of alerted CASA stations.\n\nThe systematic contribution to the angular resolution \nof the experiment has been checked by two different\ntechniques.\nFirst, for data taken in coincidence with the tracking\nCherenkov telescopes, we examine \nthe angular difference between the directions determined by CASA\nand by the telescopes.\nThe distribution of these differences indicates\nthat the systematic offset between\nCASA and the telescope array is very small ($< 0.1^\\circ$).\nThe alignment of the telescope array has been verified by the\nobservation of a number of stars.\nA second check on the pointing accuracy of CASA comes from the\nMoon shadow.  The center of the Moon shadow image is within \n$0.1^\\circ$ of the known position of the Moon.\nWe conclude that\nthe pointing uncertainty of CASA is negligible\nin comparison with the experiment's angular resolution.\n\nThe {\\it muon content} of the shower is determined from the\ndata recorded by MIA.\nSince MIA records \nthe times of counter hits over an \ninterval of $4\\,\\mu$sec, it is sensitive to muons\nproduced by showers arriving at any location of the\narray and from any direction.\nDuring the same time interval, MIA\nalso records accidental counter hits produced by PMT noise and by\nnatural radioactivity in the ground.\nThe average number of accidental hits is approximately \nsixteen per event,\nwhile\nthe average number of real muons associated with air showers \nis approximately nine per event.\n\nReal muons arrive within 100$\\,$nsec of the shower front arrival, while\nthe accidental hits occur randomly over the $4\\,\\mu$sec interval.\nWe greatly reduce the acceptance for accidental muon hits by narrowing the\ntime window for accepting muons.\nThe width of the window is determined from the distribution of\nmuon times for each six hour run.\nWe set the width to encompass 95\\% of the real shower muons;\non average, it is $\\sim 150\\,$nsec.\nThe position of the window is found on an event-by-event basis by\nmeans of a clustering algorithm.\nThe algorithm searches for the cluster of three or more muons within an\ninterval of $40\\,$nsec.\nIn approximately 25\\% of the events, no cluster is found and the\nwindow position is placed at the center of the muon time distribution\nas determined for the entire run.\nAs a result of tightening the time window for muon hit acceptance,\nthe average number of real muons recorded is 8.5 per event,\nwhile the average number of accidentals is 0.63 per event.\n\nThe CASA-MIA data undergo several stages of processing and compression.\nIn the most highly compressed format upon which this analysis is based,\nthe data records are 26 Bytes per event and\ninclude the following information for each event:\nUniversal Time (UT), number of alerted CASA stations, number of in-time\nmuon hits, core location, arrival direction, shower size,\nand muon shower size (not used here).\n\n\\section{Analysis}\n\\label{sec:anal}\n\\subsection{Data Sample}\n\\label{subsec:sample}\nThe data used in this analysis were taken between March 4, 1990 and\nAugust 10, 1995, with a gap of 255 days in 1991.\nThe experiment had usable data on 1627 days\nwith the remainder of the days lost \nlargely because of power\noutages at the site and computer problems.\nThe experiment has an instrumental deadtime of approximately\n5.4\\% which is due to a number of effects, including \ndata acquisition computer latency and\nthe time needed to digitize the CASA station data.\nCalibration runs of approximate length of six minutes taken\nat the start of data runs,\nlosses due to $8\\,$mm tape failures,\nand downtime from array maintenance led\nto an additional reduction in the live time to a total of \n1378.4 days (84.7\\% of the total).\nAfter the reduction and processing of the data, the final data sample\nconsisted of $2.0878 \\times 10^9$ reconstructed events.\n\nThe size of the CASA-MIA data sample is unprecedented in air shower physics.\nTo ensure data integrity, we\nimpose a comprehensive\nset of data quality cuts.\nThe cuts are tailored separately for the data sample in which we\nonly use information from the surface array \n({\\it all-data}) and the sample in which we use information from both the\nsurface and muon arrays ({\\it muon-data}).\nFor each of these samples, we make quality cuts on an event-by-event basis\nand on a run-by-run basis.\nCuts are applied to runs and events only in the cases\nthere there is evidence of an instrumental bias.\nThe efficiencies of the \ncuts are summarized in Table~\\ref{tab:events}.\n\n\\begin{table}\n\\begin{center}\n\\caption{Quality cut efficiencies and event totals for \nthe CASA-MIA data sample.\nThe muon-data quality cuts are applied after the all-data quality\ncuts.\nThe data sets (all and muon) are described in the text.}\n\\label{tab:events}\n\\vspace{10pt}\n\\begin{tabular}{|ccc|}\\hline\n  Category &  All-Data Sample  &  Muon-Data Sample \\\\\n\\hline\nInitial Event Total  &  2087.8M  & 1925.8M \\\\\nRun Cut Efficiency   &   0.935   &  0.929  \\\\\nEvent Cut Efficiency &   0.986   &  0.896  \\\\\n\\hline\nOverall Efficiency   &   0.922   &  0.832  \\\\\n\\hline\nFinal Event Total    &  1925.8M  & 1602.7M \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nFor the all-data sample, the run and event cuts have a combined efficiency\nof 92.2\\%, which yields a final sample of $1.9258\\times 10^9$ events.\nThe most restrictive run cut requires\na minimum fraction of the CASA stations to be working reliably and\nremoves 2.2\\% of the data, largely because of instances in which\nisolated parts of the array failed.\nFor the muon-data sample, the run cuts have an efficiency of 92.9\\%.\nA cut which requires a sufficient fraction of the muon\ncounters to be working removes 4.8\\% of the data.\nThe event cuts have an additional efficiency of\n89.6\\%.\nThe most restrictive event cut eliminates 3.2\\% of the events\nbecause they have no muon information due to deadtime of the\nMIA data acquisition system.\nThe overall efficiency of the muon-data cuts is 83.2\\%, which yields\na final sample of $1.6027\\times 10^9$ events.\n\n\\subsection{Gamma-Ray Selection}\n\\label{subsec:select}\nFrom prior observations of Cygnus X-3 and Hercules X-1, \nwe expect that gamma-ray\nfluxes, if present, will be\nsmall in comparison with the isotropic cosmic ray flux.\nTherefore, we need to enhance the presence of a possible gamma-ray signal\nby eliminating as many cosmic ray air showers as possible, while\nkeeping a high fraction of the gamma-ray air showers.\nTo do this, we select\nthose showers with a reconstructed direction\nconsistent with the position of the sources (within the angular resolution\nof the experiment) and with a muon content\nconsistent with that expected from a gamma-ray primary.\n\n\\subsubsection{Angular Search Bin}\n\\label{subsec:search}\nWe define a circular search bin whose size is based on the estimated angular\nresolution of the experiment.\nFor a sufficiently large number of events, the\nbin which optimizes the signal-to-noise has a size equal to\n$1.59$ times the angular resolution and contains 72\\% of the signal.\nThe CASA-MIA angular resolution depends on\nthe number of alerted CASA stations in an event, and therefore we use\na variable-sized search bin which scales with the number of alerts.\nFor simplicity, we use seven different bin sizes that range from\n$2.45^\\circ$ radius for showers with the least number of alerts,\nto $0.41^\\circ$ radius for showers with the largest number of alerts.\nThese bin sizes are shown in Table~\\ref{tab:bins}, along with the\nfraction of events in each alert range.\n\n\\begin{table}\n\\begin{center}\n\\caption{Angular search bin sizes and event fractions \nas a function of the number of CASA alerts.\nThe search bin is a circular region in equatorial coordinates whose\nradius is equal to $1.59$ times the angular resolution.}\n\\label{tab:bins}\n\\vspace{10pt}\n\\begin{tabular}{|ccc|}\\hline\nAlert Range & Event Fraction & Search Bin Radius \\\\\n\\hline\n{\\ 3 - 10 } & 0.331 & $2.45^\\circ$ \\\\\n{ 11 - 15 } & 0.224 & $1.88^\\circ$ \\\\\n{ 16 - 20 } & 0.121 & $1.40^\\circ$ \\\\\n{ 21 - 30 } & 0.150 & $1.05^\\circ$ \\\\\n{ 31 - 40 } & 0.064 & $0.78^\\circ$ \\\\\n{ 41 - 60 } & 0.058 & $0.60^\\circ$ \\\\\n{ $>60$   } & 0.052 & $0.41^\\circ$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsubsection{Muon Content}\n\\label{subsec:muon}\nAir showers created by gamma-ray primaries are expected to\ncontain far fewer muons than showers initiated by cosmic ray\nnuclei.\nThis expectation results because the\ncross section for\nphoto-pion production\nis much smaller than the cross section for electron-positron\npair production \\cite{ref:HERA}.\nTherefore,\nthe interaction of a high energy gamma-ray in the atmosphere\nis much more likely to produce an electromagnetic cascade\nin the atmosphere than it is to create a hadronic cascade.\nConversely,\ncosmic ray nuclei preferentially\ninteract to create hadronic cascades.\nShowers initiated by gamma-rays are thus expected to\ncontain far fewer\nhadrons than those initiated by cosmic rays.\nSince air shower muons are predominantly produced from the\ndecays of pions and kaons in the hadronic cascade,\ngamma-ray air showers should contain far fewer muons as well.\nSimulations have been done to estimate the muon content of\nair showers\n\\cite{ref:Chatelet,ref:Halzen}.\nOur own simulation indicates that an air shower initiated by a \n$100\\,$TeV gamma-ray contains, on average, 3-4\\% of the number of muons\nin a shower initiated by a proton of the same energy.\n\nThe muon content of showers should in principle\nbe a powerful tool in rejecting cosmic ray background events.\nIn our experiment, the rejection capability is limited\nby the collection area of the muon array and, to a lesser\nextent, by the presence of a small amount of accidental\nmuon hits.\nThe muon array (MIA) is significantly larger than any other\nair shower muon detector built to date, \nbut its active area still corresponds to only\n$\\sim 1\\%$ of the enclosed area of the experiment.\nAs shown in Figure~\\ref{fig:in_time_muons}, the average \nof the distribution of the number of in-time muons\nis $\\sim$8.5, but the shape of the distribution\nis such that its mode is three, and a substantial fraction of events\nhave zero muons.\nIn Figure~\\ref{fig:in_time_muons}, we also show the estimated number of\nmuons for showers initiated by gamma-rays, including the contribution\nfrom accidental\nmuon hits.\nFor gamma-ray showers, we expect, on average, 0.28 real muons per event\nand 0.63 accidental muons per event.\n\nIn order to enhance a possible gamma-ray signal,\nwe wish to select {\\it muon-poor} events, i.e. events that\nhave fewer muons than the average expected number.\nTo do this, we make the assumption that any gamma-ray\nsignal in the data is much smaller than the flux of cosmic rays.\nWe can therefore use\nthe muon information from the detected events\nto describe the muon content of the background, and \nour simulation to describe the muon content of the gamma-ray\nsignal.\n\nThe number of muons in a shower depends on a number of observable\nquantities, for example,\nthe number of alerted CASA\nstations, shower zenith angle, and core position.\nWe develop a parameterization for the average number of muons\nas a function of these quantities\nby examining a large ensemble of actual showers.\nWe then determine the relative muon content of a specific shower\nby comparing the observed muon number, $({\\rm n}_\\mu)_{{\\rm obs}}$,\nto the expected number of muons, $<{\\rm n}_\\mu>_{{\\rm exp}}$,\nfor showers\nhaving similar zenith angles, core positions, and numbers of alerts.\nThe relative muon content, ${\\rm r}_\\mu$, is\ndefined by:\n\n\\begin{equation} \n{\\rm r}_\\mu \\ \\ \\equiv\\ \\ {\\rm Log_{\\rm 10}} \\\n\\Biggl[\n{  {  ({\\rm n}_\\mu)_{\\rm obs} } \\over\n   {  <{\\rm n}_\\mu>_{\\rm exp} }        }\n\\Biggr] \\ .  \n\\label{eq:rmu}\n\\end{equation}\n\n\\noindent Figure~\\ref{fig:rmu} shows the \ndistributions of ${\\rm r}_\\mu$ for \nobserved events and for simulated\ngamma-ray events.\nMuon-poor events are defined as those having \n${\\rm r}_\\mu$ values less than some cut value.\nThe position of the cut is chosen to reject as many background\nevents as possible, while keeping a high fraction of the gamma-ray\nevents.\nThe cut value depends weakly on the number of CASA alerts because\nthe separation between the signal and background \n${\\rm r}_\\mu$ distributions improves\nas the showers get larger.\n\nTable~\\ref{tab:rmucut}\nshows the ${\\rm r}_\\mu$ cut values for various samples of data\nalong with the fractions of signal and background events retained,\nand the sensitivity improvement achieved from making a cut.\nFor the entire data set, the sensitivity is improved by a factor\nof 2.94 by cutting on the shower muon content.\nThe quality factor increases to 29.7 for events having more than\n80 alerted CASA stations.\n\n\\begin{table}\n\\begin{center}\n\\caption{\nQuantities associated with the selection of muon-poor events.\nMuon-poor events are those having a relative muon content,\n${\\rm r}_\\mu$ (defined in the text),\nless than a cut value.\nThe cut values\nare given in the\nsecond column, and\nthe third and fourth columns give the efficiencies for\npassing the cut for gamma-ray signal events and\nfor hadronic background events, respectively.\nThe fifth column gives the quality factor, Q, or the\nimprovement in flux sensitivity from making the cut.}\n\\label{tab:rmucut}\n\\vspace{10pt}\n\\begin{tabular}{|ccccc|}\\hline\nData Set    &  ${\\rm r}_\\mu$ cut Value & Signal $\\epsilon$ &\nBackground $\\epsilon$ & Q \\\\\n\\hline\nAll  &  $-0.75$  & 0.72              & 0.0600 & 2.94 \\cr\n$\\le 10$ Alerts &  $-0.50$  &  0.69  & 0.1644 & 1.70 \\cr\n$ > 10$  Alerts &  $-0.75$  &  0.76  & 0.0362 & 3.99 \\cr\n$ > 40$  Alerts &  $-1.00$  &  0.71  & $1.77\\times 10^{-3}$  &  16.9\\cr\n$ > 80$  Alerts &  $-1.00$  &  0.77  & $0.67\\times 10^{-3}$ & 29.7 \\cr\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsection{Background Estimation}\n\\label{subsec:back}\nWe select gamma-ray candidate events \n({\\it on-source} events)\nbased on their\nreconstructed arrival direction \nin equatorial coordinates (right ascension, $\\alpha$, and \ndeclination, $\\delta$) and on their muon content.\nIn order to derive the significance of a possible gamma-ray signal,\nwe need to determine the expected number of background cosmic ray\nevents \n({\\it off-source} events)\nthat would arrive from the same direction in the sky as the source\nand would have a similar muon content as gamma-ray events.\nAgain,\nwe make the assumption that the detected air showers\nare predominantly caused by background cosmic ray events.\nWe thus use the detected events themselves to estimate the\nexpected background.\n\nA common method to estimate the expected number of background events\nis to use off-source bins having the same declination as the source,\nbut having different right ascension values.\nThis method, which assumes a uniform experiment\nexposure over declination,\nwas satisfactory for earlier smaller experiments.\nHowever, given our\nlarge event sample, \nthis technique is not reliable for CASA-MIA\nbecause of small, but non-negligible, systematic biases\n(e.g. diurnal variations).\nFor the CASA-MIA data sample,\na source at a declination of $40^\\circ$\noccupies an angular bin with\n$\\sim 1.8 \\times 10^6$ events.\nThe fractional statistical uncertainty corresponding to \none standard deviation in the number of events is\n0.075\\%.\nIn order to accurately estimate the number\nof background events, the relative systematic uncertainty must\nbe well below this level.\nAs a result, an accurate and robust way to determine\nthe expected background is needed.\nSeveral methods have been developed \nby other groups \n\\cite{ref:Cassiday1,ref:Alexandreas3}\nand the method that we use \nis similar to these.\n\nThe detection rate of an air shower array \ntriggering on cosmic rays \nis determined by the properties of the cosmic ray flux\nand by the properties\nof the array itself.\nAssuming that the cosmic ray parameters do not change with time,\nany variation in the detection rate is caused only by\nchanges in the detector or in the atmospheric conditions.\nOver short intervals of time ($\\sim 1\\,$hour),\nthe relative detection efficiency as a function of the shower \ndirection in local coordinates, ($\\theta$,$\\phi$),\nis largely determined by the array geometry (placement of detectors,\nuniformity of terrain, etc.) and is almost constant, and\nthe time variation of the detector response may be estimated from\nthe trigger rate.\nTherefore,\nwe separate the detection rate per unit solid\nangle in local coordinates, $ N (\\theta,\\phi,t)$, into\ntwo terms:\n\n\\begin{equation} \nN (\\theta,\\phi,t) \\ \\ = \\ \\ D (\\theta,\\phi) \\cdot  R (t) \\ \\ ,\n\\label{eq:rate}\n\\end{equation}\n\n\\noindent \nwhere $ D (\\theta,\\phi) $  is the efficiency per unit solid angle\nof detecting a shower from a given direction in the sky, and\n $R (t) $ is the trigger rate as a function of time.\nThe factor $ D (\\theta,\\phi) $ is determined by maps made\nfrom the arrival directions of cosmic ray showers over\ngiven periods of time.\nThe time dependent term, $R(t)$, is determined from\nthe arrival times of the actual events.\nThe expected number of events for a given bin in the sky \nis then determined by integrating $ N (\\theta,\\phi,t)$\nover the time interval in question.\nTo determine the expected number of events for a bin in\nequatorial coordinates, ($\\alpha,\\delta$), we integrate \n$ N (\\theta,\\phi,t)$ over the time interval and over local\ncoordinate space.\n\nMore explicitly, the background estimation\nis done by the following procedure.\nFor intervals of $4,200\\,$sec,\nwe accumulate the arrival directions of \ncosmic ray events into 2,700 bins segmented in local coordinate space\n(30 bins in $\\theta$, 90 bins in $\\phi$).\nWe use the binned data to construct maps of the relative acceptance\nof any point in the sky over this time interval.\nSeparate maps are calculated for each data sample\nused in the source search (e.g. all-data and muon-poor data).\nTo generate simulated background data,\nwe discard the directional information of an event\nand associate the event time with a local coordinate direction\nobtained by sampling from the appropriate sky map.\nWe then\ncompute artificial values for the\nequatorial coordinates \nand determine if this simulated event falls into a search\nbin of a source.\nBy sampling more than once from the sky map for each event time,\nwe increase the statistics on the \nsimulated data sample.\nNegligible\nsystematic bias is introduced by such oversampling.\nFor this work, we oversample by a factor of ten, an amount that \nis limited only by computational resources.\n\nWe have checked that our background estimation method is free from\nbias by comparing the detected numbers of events in an\nangular bin to our expected number for bins that do not contain\nCygnus X-3 and Hercules X-1.\nFor each bin, we compute the statistical significance of any\nexcess or deficit in the number of detected events relative\nto the number we predict.\nThe distribution of these significances is in close agreement\nwith that expected from statistics, which, because of \nbackground oversampling,\nis dominated by the statistical uncertainty on the number\nof detected events.\n\n\\subsection{Energy Response}\n\\label{subsec:energy}\nAir shower arrays trigger on the shower size, i.e.\nthe number of charged particles in the shower at ground level.\nFor each shower, we determine a shower size\nfrom the particle densities measured in the CASA stations.\nFor astrophysical interpretations of flux measurements or flux limits,\nhowever, it is necessary to translate from the measured shower\nparameters (size and zenith angle) to an estimate of the energy of\nthe primary particle.\nSince there are large fluctuations in shower size \nfor showers initiated by particles at fixed energy,\nit is difficult for air shower experiments to measure accurately\n{\\it differential} primary spectra.\nTraditionally, therefore, \nflux measurements\nhave been quoted as {\\it integral} intensities above\na fixed energy point.\nAlthough to some degree the energy value at which to quote\nthe intensity is arbitrary, we desire to use\nan energy at which the experiment has a significant\ndegree of sensitivity.\nWe chose to quote flux measurements at the {\\it median} energy\nof the experiment which reduces the dependence of the flux on\nthe assumed spectral index \n\\cite{ref:Ciampa,ref:Gaisser1}.\n\nWe estimate the energy response of the experiment by the constant\nintensity method, which has been used by other experiments\n\\cite{ref:Nagano}, as well as by our own group \\cite{ref:McKay}.\nThe constant intensity procedure is described in more detail\nelsewhere \\cite{ref:Newport}. \nBriefly,\nwe determine the relationship between shower size and\nenergy by comparing the detected flux of showers above a given size\nto an assumed form of the all-particle cosmic ray spectrum.\nThe comparison is done on a run-by-run basis to account for\nchanges in the detector response.\nThe cosmic ray flux is\nderived from\nmeasurements made by other space-borne \n\\cite{ref:Asakimori,ref:Swordy}\nand ground-based \n\\cite{ref:Nagano}\nexperiments.\nThe assumed integral cosmic ray intensity above $100\\,$TeV is\n$6.57 \\times 10^{-9}\\,$particles cm$^{-2}$ s$^{-1}$ sr$^{-1}$.\n\nWe use the relationship between energy and shower size to\ndetermine the most likely energy for each shower coming from the\ndirection of Cygnus X-3 or Hercules X-1 in a angular bin of fixed\nradius.\nThe medians of the energy distributions determine\nthe median energies for \ncosmic ray particles from the direction of\nCygnus X-3 and Hercules X-1 that would trigger the experiment and\npass all selection criteria.\nBy normalizing our energy scale to the cosmic ray flux,\nwe make the assumption that the primary particle\nhas the same spectral index as the detected cosmic rays.\nThis assumption is reasonable when dealing with\nsources like Cygnus X-3 and Hercules X-1 in which there are\nare no well established measurements of spectral indices.\nThe median energy of particles from the direction of Cygnus\nX-3 is $114\\,$TeV, and from Hercules X-1, it is $116\\,$TeV.\nSince the difference in the energies for the two sources\nis negligible, we report our measurements at\na common energy of $115\\,$TeV.\n\n\\subsection{Search Strategy}\n\\label{subsec:strategy}\nWe carry out searches for particle emission from a particular source\nby comparing the number of events found\nwithin a circular angular bin\naround the source to the number of events estimated by our background\nprocedure.\nThe angular bin sizes vary as a function of the number of alerted CASA\nstations, as itemized in Table~\\ref{tab:bins}.\nSource positions (J1992) are taken to be \n$(\\alpha,\\delta) = (308.04^\\circ,40.93^\\circ)$ for\nCygnus X-3, and\n$(\\alpha,\\delta) = (254.39^\\circ,35.35^\\circ)$ for\nHercules X-1.\n\nSeparate searches are made based on particle type and energy.\nBy using the {\\it all-data} sample, we are sensitive to any type of\nneutral particle that would create air showers.\nWith the {\\it muon-poor} sample, we are specifically sensitive to\nthe emission of gamma-rays.\nWe carry out three separate searches with various integral cuts on the\nnumber of alerted CASA stations,\nin addition to a search with no cuts.\nThis procedures takes advantage of the correlation between primary\nenergy and size (as represented by the number of alerts),\nand improves our sensitivity to possible emission\nthat might be present at either low or high energies.\nThe data samples selected by cutting on the alert number\nand their corresponding\nmedian energies are shown in Table~\\ref{tab:energycuts}.\n\n\\begin{table}\n\\begin{center}\n\\caption{Data samples selected by integral cuts on the number\nof CASA alerts.\nThe cut values are given in the first column and the fractions\nof events surviving the cut \n(and within the angular search region)\nare shown in the second column.\nThe median energies for events coming from either Cygnus X-3\nor Hercules X-1 are listed in the third column.}\n\\label{tab:energycuts}\n\\vspace{10pt}\n\\begin{tabular}{|crr|}\\hline\n Alert Cut &\\ \\ Event Fraction\\ \\ &\\ \\ Median Energy\\ \\ \\\\\n\\hline\nNone      & 100.00\\%\\ \\ \\  &  $115\\,$TeV\\ \\ \\ \\\\\n\\hline\n$\\le 10$  &  62.87\\%\\ \\ \\  &   $85\\,$TeV\\ \\ \\ \\\\\n$> 40$    &   0.58\\%\\ \\ \\  & $530\\,$TeV\\ \\ \\ \\\\\n$> 80$    &   0.09\\%\\ \\ \\  & $1175\\,$TeV\\ \\ \\ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{Results}\n\\label{sec:results}\nWe search for evidence of neutral \n(gamma-ray or other) particle emission from Cygnus X-3 and\nHercules X-1.\nSeparate searches are carried out for steady and\ntransient emission from either source.\nIn addition, we search for periodic emission from Cygnus X-3\nat the \n$4.8\\,$hr\nX-ray periodicity and for emission from Cygnus\nX-3 that was coincident with the occurrence of large radio flares.\nNo compelling evidence for emission from either source is found\nfor all the different searches, and consequently we\nset upper limits on the fluxes of particles from the sources.\n\n\\subsection{Steady Emission}\n\\label{subsec:steady}\nThe numbers of on-source and background events for the various\nsearches from Cygnus X-3 are shown in Table~\\ref{tab:CygEvents}.\nThe results from similar searches carried out on Hercules X-1\nare shown in Table~\\ref{tab:HerEvents}.\nFor each search, we also calculate\nthe statistical significance of\nany excess or deficit in the number of\nevents observed relative to background by\nthe prescription of Li and Ma \\cite{ref:LiMa}, using\nan oversampling factor of 10.\nNo significant excess is observed for any search from either source.\nTherefore, for each search, we calculate\nan upper limit, $N_{90}$, on the number of excess events\nfrom the source at the 90\\% confidence level\n\\cite{ref:Helene,ref:PDG}.\nEach $N_{90}$ value is converted to a limit on the fractional\nexcess of events from the source, $f_{90}$, \nby dividing by the estimated number of background events, which is\nassumed to represent the background cosmic-ray level.\nSince\nthe $f_{90}$ values are independent of the absolute flux normalization,\nthey are useful in comparing\nresults between different experiments.\n\n\\begin{table}\n\\begin{center}\n\\caption{Steady emission search results for Cygnus X-3 using\nthe all-data sample (top) and muon-poor sample (bottom).\nThe number of events observed on-source and the number\nexpected from background are given in the second the third\ncolumns, respectively.\nThe fourth column gives the statistical significance of\nany excess or deficit.\nThe 90\\% c.l. upper limit on the number of excess events,\n$N_{90}$, and the upper limit on the fractional excess,\n$f_{90}$, are given in the last two columns.\nThe methods used to calculate statistical significances\nand upper limits are outlined in the text.\nThe data samples at $85\\,$TeV, $530\\,$TeV, and $1175\\,$TeV\nare subsets of the data sample at $115\\,$TeV.}\n\\label{tab:CygEvents}\n\\vspace{10pt}\n\\begin{tabular}{|rrrcrc|}\n\\multicolumn{6}{c}{ All-Data Sample} \\\\\n\\hline\nEnergy & On-Source & Background & Signif. & $N_{90}$ &$f_{90}$ \\\\\n\\hline\n  $85\\,$TeV\\ \\ & 1119469\\ \\ & 1119987\\ \\ & $-0.48\\sigma$ & \n        1502.1 &\\ \\ $1.34\\times 10^{-3}$ \\\\\n $115\\,$TeV\\ \\ & 1780594\\ \\ & 1781479\\ \\ & $-0.66\\sigma$ & \n        1774.9 &\\ \\ $9.96\\times 10^{-4}$ \\\\\n $530\\,$TeV\\ \\ &   10286\\ \\ &   10235\\ \\ & $+0.49\\sigma$ &  \n         205.3 &\\ \\ $2.01\\times 10^{-2}$ \\\\\n$1175\\,$TeV\\ \\ &    1583\\ \\ &    1580\\ \\ & $+0.08\\sigma$ &   \n          68.9 &\\ \\ $4.36\\times 10^{-2}$ \\\\\n\\hline\n\\multicolumn{6}{c}{\\hphantom{dummy}} \\\\\n\\multicolumn{6}{c}{ Muon-Poor Sample} \\\\\n\\hline\nEnergy & On-Source & Background & Signif. & $N_{90}$ &$f_{90}$ \\\\\n\\hline\n  $85\\,$TeV\\ \\ & 149676\\ \\ & 149863\\ \\ & $-0.57\\sigma$ & \n        548.1 &\\ \\ $5.90\\times 10^{-4}$ \\\\\n $115\\,$TeV\\ \\ & 121409\\ \\ & 121594\\ \\ & $-0.37\\sigma$ & \n        485.4 &\\ \\ $3.28\\times 10^{-4}$ \\\\\n $530\\,$TeV\\ \\ &   20\\ \\ &   21.0\\ \\ & $-0.21\\sigma$ &  \n         8.2 &\\ \\ $9.47\\times 10^{-4}$ \\\\\n$1175\\,$TeV\\ \\ &    1\\ \\ &    0.6\\ \\ & $+0.44\\sigma$ &   \n          3.5 &\\ \\ $2.67\\times 10^{-3}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\caption{Steady emission search results for Hercules X-1 using\nthe all-data sample (top) and muon-poor sample (bottom).}\n\\label{tab:HerEvents}\n\\vspace{10pt}\n\\begin{tabular}{|rrrcrc|}\n\\multicolumn{6}{c}{ All-Data Sample} \\\\\n\\hline\nEnergy & On-Source & Background & Signif. & $N_{90}$ &$f_{90}$ \\\\\n\\hline\n  $85\\,$TeV\\ \\ & 1058904\\ \\ & 1057583\\ \\ & $+1.12\\sigma$ & \n        2738.1 &\\ \\ $2.59\\times 10^{-3}$ \\\\\n $115\\,$TeV\\ \\ & 1681708\\ \\ & 1681392\\ \\ & $+0.23\\sigma$ & \n        2387.6 &\\ \\ $1.42\\times 10^{-3}$ \\\\\n $530\\,$TeV\\ \\ &   9579\\ \\ &     9532\\ \\ & $+0.46\\sigma$ &  \n         196.5 &\\ \\ $2.06\\times 10^{-2}$ \\\\\n$1175\\,$TeV\\ \\ &    1419\\ \\ &    1459\\ \\ & $-0.98\\sigma$ &   \n          44.0 &\\ \\ $3.02\\times 10^{-2}$ \\\\\n\\hline\n\\multicolumn{6}{c}{\\hphantom{dummy}} \\\\\n\\multicolumn{6}{c}{ Muon-Poor Sample} \\\\\n\\hline\nEnergy & On-Source & Background & Signif. & $N_{90}$ &$f_{90}$ \\\\\n\\hline\n  $85\\,$TeV\\ \\ & 139580\\ \\ & 139670\\ \\ & $-0.24\\sigma$ & \n        577.1 &\\ \\ $6.57\\times 10^{-4}$ \\\\\n $115\\,$TeV\\ \\ & 113360\\ \\ & 113244\\ \\ & $+0.37\\sigma$ & \n        643.4 &\\ \\ $4.62\\times 10^{-4}$ \\\\\n $530\\,$TeV\\ \\ &   14\\ \\ &   16.8\\ \\ & $-0.67\\sigma$ &  \n         6.3 &\\ \\ $7.96\\times 10^{-4}$ \\\\\n$1175\\,$TeV\\ \\ &    0\\ \\ &    0.5\\ \\ & $-0.98\\sigma$ &   \n          2.3 &\\ \\ $1.90\\times 10^{-3}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nFigure~\\ref{fig:CygScan} shows scans in right ascension for\nbands of declination centered on Cygnus X-3 for the all-data\nand muon-poor samples.\nNo significant excess above background \nis seen in either sample for the bin\ncontaining Cygnus X-3.\nThe background estimation \nagrees well with the data in the off-source region.\nSimilar scans for Hercules X-1 are shown in \nFigure~\\ref{fig:HerScan}, and\nagain the background estimation agrees well with the observed data\nand no excesses are seen.\n\n\\subsubsection{Flux Limit Calculation}\n\\label{subsec:flux}\nIn the absence of a statistically significant excess\nfrom either Cygnus X-3 or Hercules X-1, we set upper limits\non the flux of particles from each source.\nSeparate limits are set for neutral and gamma-ray primaries.\nFor gamma-ray primaries, the 90\\% c.l. upper limit, \n$\\Phi_\\gamma (E)$,\non the integral flux is calculated from the measured\nfractional excess by normalizing to the cosmic ray flux:\n\n\\begin{equation}\n\\Phi_\\gamma\\ (E) \\ \\ = \\ \\  \n   {  { f_{90}\\ \\bar{\\Omega} } \\over {\\epsilon\\ R_\\gamma} }\n    \\ J (E)\\ .\n\\label{eq:limit}\n\\end{equation}\n\n\\noindent Here, \n$\\bar{\\Omega}$ is the mean solid angle used in the search,\n$\\epsilon$ is the fraction of events that would pass cuts\nand fall into the search bin,\n$J (E)$ is the integral cosmic ray intensity above energy $E$, \nand $R_\\gamma$ is a factor which accounts for the relative trigger\nefficiency for gamma-rays as opposed to cosmic rays.\nThe value of $\\bar{\\Omega}$ ranges from $5.74\\times 10^{-3}\\,$sr for\nthe lowest energy data set to\n$1.60\\times 10^{-4}\\,$sr for the highest energy data set.\nThe $\\epsilon$ factor accounts for the fraction of gamma-rays that\nwould end up in the angular search bin (0.72) and the fraction\nthat would pass the muon-poor selection\ncriterion (Table~\\ref{tab:rmucut}).\nThe value of $R_\\gamma$ was determined by Monte Carlo\nsimulations to be 1.6.\n\nTo determine an upper limit on the integral\nflux of any neutral particle from\na source, $\\Phi_N (E)$, we use\nEq.~\\ref{eq:limit}, except $\\epsilon$ is now 0.72\nand $R_\\gamma$ is 1.0.\nIn this calculation, we assume that the neutral particle \nwould interact in the atmosphere to create air showers\nin a similar manner to cosmic rays.\n\nTable~\\ref{tab:CygHerLimits} gives the flux limits obtained from the\nvarious searches for steady emission from \nthe two sources.\nLimits are not calculated for the data samples with median energies\nof $85\\,$TeV because these are {\\it not} integral energy samples.\n\n\\begin{table}\n\\begin{center}\n\\caption{Flux limits from searches for steady emission from \nCygnus X-3 (top) and\nHercules X-1 (bottom). The second and third columns give the 90\\% c.l. upper\nlimit on the integral flux of any neutral or gamma-ray particles\nfrom the source, respectively.\nThe units of flux are particles cm$^{-2}$ sec$^{-1}$.}\n\\label{tab:CygHerLimits}\n\\vspace{10pt}\n\\begin{tabular}{|rcc|}\n\\multicolumn{3}{c}{ Cygnus X-3} \\\\\n\\hline\nEnergy & $\\Phi_N\\ (E) $ & $\\Phi_\\gamma\\ (E)$ \\\\\n\\hline\n $115\\,$TeV\\ \\ & $2.20 \\times 10^{-14}$ & $6.26 \\times 10^{-15}$ \\\\\n $530\\,$TeV\\ \\ & $1.43 \\times 10^{-15}$ & $1.21 \\times 10^{-16}$ \\\\\n$1175\\,$TeV\\ \\ & $1.04 \\times 10^{-15}$ & $5.19 \\times 10^{-17}$ \\\\\n\\hline\n\\multicolumn{3}{c}{\\hphantom{dummy}} \\\\\n\\multicolumn{3}{c}{ Hercules X-1} \\\\\n\\hline\nEnergy & $\\Phi_N\\ (E) $ & $\\Phi_\\gamma\\ (E)$ \\\\\n\\hline\n $115\\,$TeV\\ \\ & $3.04 \\times 10^{-14}$ & $8.55 \\times 10^{-15}$ \\\\\n $530\\,$TeV\\ \\ & $2.87 \\times 10^{-15}$ & $9.75 \\times 10^{-17}$ \\\\\n$1175\\,$TeV\\ \\ & $6.91 \\times 10^{-16}$ & $3.56 \\times 10^{-17}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsection{Transient Emission}\n\\label{subsec:transient}\nWe search for transient emission of particles\nfrom Cygnus X-3 and Hercules X-1 on daily (single transit)\ntime scales.\nFor each transit of the source, we compare the number of \non-source events to the number of expected background events\nand calculate a significance based on\nthe prescription of Li and Ma \\cite{ref:LiMa}.\nWe require the live time fraction during the transit to be\nat least 0.20 to remove transits in which the experiment was\noperational for only a small fraction of the time.\nFor each source,\nwe make separate studies of the transit significances for the\nall-data and muon-poor samples, corresponding to possible emission\nfrom any neutral and gamma-ray particles, respectively.\n\nFor Cygnus X-3, the number of good transits in the\nall-data sample is 1500.\nIn the muon-poor sample, it is 1291.\nThe distributions of significances for \nthe two samples of Cygnus X-3 transits\nare shown in Figure~\\ref{fig:Cyg_Trans}.\nEach\ndistribution agrees well with a Gaussian\ndistribution\nof mean zero and unit width.\nThere is no\nevidence for any excess of events at high values of significance\n(either positive or negative).\n\nFor Hercules X-1, there are 1492 good transits in the all-data\nsample, and 1271 good transits in the muon-poor sample.\nThe significance distributions for Hercules X-1 are shown in\nFigure~\\ref{fig:Her_Trans}, and again, no evidence for \nsignificant excesses exists.\n\nBased on the lack of statistically significant excesses,\nwe place limits on the daily\nfluxes of neutral and gamma-ray particles\nfrom Cygnus X-3 and Hercules X-1.\nThese limits are calculated by a similar procedure as used\nfor the steady searches.\nThe limit values depend on the actual statistical significance\nof the search on a given day, and also on the \nepoch of data taking.\nAs shown in Table~\\ref{tab:array}, the size of the experiment has\nchanged with time, and the sensitivity changed accordingly.\nIn Table~\\ref{tab:daily}, we give typical daily flux\nlimits for the two sources for different epochs of the experiment.\nSince the numbers of events detected per transit are the\nsame for the two sources to within 5\\%, the limits for\nCygnus X-3 and Hercules X-1 are virtually identical.\nTypical 90\\% c.l. limits on the integral flux\nusing the full experiment\nare $\\Phi_N ( E > 115\\,{\\rm TeV} ) < 9.7\\times 10^{-13}$\nneutral particles cm$^{-2}$ sec$^{-1}$ and\n $\\Phi_\\gamma ( E > 115\\,{\\rm TeV} ) < 2.0\\times 10^{-13}$\nphotons cm$^{-2}$ sec$^{-1}$.\n\n\\begin{table}\n\\begin{center}\n\\caption{Typical daily\nupper flux limits (90\\% c.l.)\nfor emission of neutral and\ngamma-ray particles from Cygnus X-3 and Hercules X-1.\nThe flux limits are calculated for two different epochs assuming\nthe same number of on-source events as off-source.\nThe third column gives\nthe typical number of\nevents observed on-source during the different epochs.\nEpoch I corresponds to March 1990 to October 1990.\nEpoch II corresponds to January 1992 to August 1993.\nFlux limits for the remaining periods of time of operation\nare close to those for Epoch II.\nUnits of flux are particles cm$^{-2}$ sec$^{-1}$.}\n\\label{tab:daily}\n\\vspace{10pt}\n\\begin{tabular}{|rcrrc|}\n\\multicolumn{5}{c}{Epoch I} \\\\\n\\hline\nEnergy & Particle &\\ Events\\ \\ & $f_{90}\\ \\ $ & $\\Phi_{daily}(E)$ \\\\\n\\hline\n$115\\,$TeV  & Any      & 575    & 0.071  &  $\\ \\ 1.6\\times 10^{-12}$ \\\\\n$530\\,$TeV  & Any      &  3.2   & 1.34   &  $\\ \\ 1.9\\times 10^{-13}$ \\\\\n$1175\\,$TeV & Any      &  0.56  & 4.11   &  $\\ \\ 9.6\\times 10^{-14}$ \\\\\n$115\\,$TeV  & $\\gamma$-ray &  45    & 0.026  &  $\\ \\ 3.6\\times 10^{-13}$ \\\\\n\\hline \n\\multicolumn{5}{c}{\\hphantom{dummy}} \\\\\n\\multicolumn{5}{c}{Epoch II} \\\\\n\\hline\nEnergy & Particle &\\ Events\\ \\ & $f_{90}\\ \\ $ & $\\Phi_{daily}(E)$ \\\\\n\\hline\n$115\\,$TeV  & Any     & 1450    & 0.044  & $\\ \\ 9.7\\times 10^{-13}$ \\\\\n$530\\,$TeV  & Any     &  8.1   & 0.76   &  $\\ \\ 1.1\\times 10^{-13}$ \\\\\n$1175\\,$TeV & Any     &  1.3  &  2.56   &  $\\ \\ 5.9\\times 10^{-14}$ \\\\\n$115\\,$TeV  & $\\gamma$-ray &  96    & 0.015  &  $\\ \\ 2.0\\times 10^{-13}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nWe have also carried out searches for transient emission \non the shorter time scale of \n$0.5\\,$hr.\nHere, we compare the number of events observed on-source\nto the expected background level for ten \n$0.5\\,$hr\ntime intervals on either side of the time of source culmination.\nThe typical number of on-source events, and therefore the flux\nsensitivity, depends strongly on the source zenith angle.\nFor example, for an overhead source\nnear culmination, the experiment observes\n$\\sim 175$ events per \n$0.5\\,$hr,\nwhereas at four hours from\nculmination the rate is $\\sim 15$ events per \n$0.5\\,$hr.\nRegardless of the rate, for each \n$0.5\\,$hr\ninterval, we calculate the significance in\nthe number of on-source events relative to the background\nand combine all such significances into\na single distribution.\nThe resulting\nsignificance distributions are consistent with those expected\nfrom background processes for both sources in both the all-data\nand muon-poor samples.\nThe typical 90\\% c.l. upper limits on the fluxes\nfrom either source are\n$\\Phi_N ( E > 115\\,{\\rm TeV} ) < 3.1\\times \n10^{-12}$ neutral particles cm$^{-2}$ sec$^{-1}$\nand\n$\\Phi_\\gamma ( E > 115\\,{\\rm TeV} ) < 7.1\\times 10^{-13}$\nphotons cm$^{-2}$ sec$^{-1}$\nfor $0.5\\,$hr\nperiods within one hour of culmination.\n\n\\subsubsection{Cygnus X-3 Radio Flares}\n\\label{subsec:radio}\nWe study showers from the direction of Cygnus X-3 during\nthe occurrence of large radio flares at the source.\nWe define large flares\nas those times when the radio\noutput at 8.3 GHz exceeded 2 Jy, a level which is\ntwo orders of magnitude above the typical quiescent level.\nDuring the period of CASA-MIA operations, there\nwere six large flares, as listed in Table~\\ref{tab:flares}\n\\cite{ref:Waltman1,ref:Waltman2}.\n\n\\begin{table}\n\\begin{center}\n\\caption{Large radio flares of Cygnus X-3 from 1990 to 1995,\ncoincident with the operational time of CASA-MIA.\nThe flare number is an arbitrary index used for this work.\nThe peak radio flux values (8.3 GHz)\ncome from \n\\protect\\cite{ref:Waltman1,ref:Waltman2}.\nThe March 1994  flare was actually a prolonged event that\nextended for the ten days following March 1, 1994.}\n\\label{tab:flares}\n\\vspace{10pt}\n\\begin{tabular}{|crr|}\\hline\nFlare & Date\\ \\ \\ \\ \\ \\ \\ \\ & Peak Flux (Jy) \\\\\n\\hline\n1 &\\ \\ Aug. 15, 1990\\ \\ &  7.5\\ \\ \\ \\ \\ \\ \\ \\ \\ \\\\\n2 & Oct. 05, 1990  & 10.2\\ \\ \\ \\ \\ \\ \\ \\ \\ \\\\\n3 & Jan. 21, 1991  & 14.8\\ \\ \\ \\ \\ \\ \\ \\ \\ \\\\\n4 & Sep. 04, 1992  &  4.1\\ \\ \\ \\ \\ \\ \\ \\ \\ \\\\\n5 & Feb. 20, 1994  &  4.9\\ \\ \\ \\ \\ \\ \\ \\ \\ \\\\\n6 & Mar. 09, 1994  &  5.2\\ \\ \\ \\ \\ \\ \\ \\ \\ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nWe examine the daily significances for Cygnus X-3 on\nthe day of each large flare, as well as on the\nday preceeding and following each flare.\nTable~\\ref{tab:flare_results} lists the numbers of\nobserved events, the expected background, and the\nLi-Ma significances\nfor the examined days.\nThere is no\ncompelling evidence for any statistical excess in the \nobserved number of\nevents from Cygnus X-3 for either the all-data or muon-poor samples.\nOn one day (Feb. 20, 1994) the Li-Ma\nsignificance is 2.26$\\sigma$ for the all-data sample.\nThe probability that we would get a day with this level of\nsignificance or greater is 28.2\\% after accounting for the\nfifteen days in which we searched.\nIn addition, on this same day, there is no evidence for any\nexcess in the muon-poor data, while\nwe would expect the statistical\nsignificance to increase by a factor of 3.3 if it were\ndue to a gamma-ray signal.\nTable~\\ref{tab:flare_results} also lists the derived \nfractional excess values, $f_{90}$, as well as the upper limits\nto the integral flux of neutral or gamma-ray particles from\nCygnus X-3 during the flares.\n\n\\begin{table}\n\\begin{center}\n\\caption{CASA-MIA search results for emission from\nCygnus X-3 near the time of large radio flares.\nThe flare numbers are defined in \nTable~\\protect\\ref{tab:flares}.\nThe $-$ and $+$ designations refer to the days preceeding\nand following the flare day, respectively.\nThe significances (columns 4 and 8) are \nstandard deviation values calculated using\nthe prescription of Li and Ma \n\\protect\\cite{ref:LiMa}.\nThe last two columns give the 90\\% c.l. upper limits on the\nintegral flux above $115\\,$TeV\nof any neutral particle and gamma-rays, respectively,\nin units of \n$10^{-12}$ particles cm$^{-2}$ sec$^{-1}$.\nEntries having only a dash indicate the absence of\nany usable data.}\n\\label{tab:flare_results}\n\\vspace{10pt}\n\\begin{tabular}{|c|rrrr|rrrr|cc|}\\hline\n\\ \\ & \\multicolumn{4}{c|}{ All-Data Sample} &\n\\multicolumn{4}{c|}{ Muon-Poor Sample} &\n\\multicolumn{2}{c|}{ Flux Limits}  \\\\\n\\hline\nFlare & On & Back & Signif. & $f_{90}$(\\%) &\n        On & Back & Signif. & $f_{90}$(\\%) &\n        $\\Phi_N(E)$ & $\\Phi_\\gamma(E) $ \\\\\n\\hline\n$-$& 226 & 241.2 & $-0.94$ &  7.8\\ \\ & --\\ \\ & --\\ \\ & --\\ \\ & --\\ \\ & 1.7 & \n  --\\ \\ \\\\\n 1 & 249 & 249.5 & $-0.03$ & 10.8\\ \\ & --\\ \\ & --\\ \\ & --\\ \\ & --\\ \\ & 2.4 & \n  --\\ \\ \\\\\n$+$& 252 & 248.1 & $+0.25$ & 12.1\\ \\ & 22 & 19.4 & $+0.58$ & 5.2\\ \\ & \n2.7 & 0.73 \\\\\n\\hline\n$-$& 850 & 863.1 & $-0.43$ &  4.9\\ \\ & 121 & 113.4 & $+0.67$ & 3.4\\ \\ & \n1.1 & 0.48 \\\\\n 2 & 884 & 849.1 & $+1.13$ &  9.0\\ \\ &  94 & 111.7 & $-1.65$ & 1.5\\ \\ & \n2.0 & 0.21 \\\\\n$+$& 872 & 901.4 & $-0.94$ &  3.9\\ \\ & 103 & 124.5 & $-1.90$ & 1.4\\ \\ & \n0.9 & 0.20 \\\\\n\\hline\n$-$&1279 &1260.1 & $+0.51$ &  5.8\\ \\ & 227 & 230.7 & $-0.23$ & 2.3\\ \\ & \n1.3 & 0.32 \\\\\n 3 &1297 &1267.3 & $+0.80$ &  6.4\\ \\ & 255 & 235.5 & $+1.19$ & 4.0\\ \\ & \n1.4 & 0.56 \\\\\n$+$&1265 &1235.2 & $+0.81$ &  6.6\\ \\ & 216 & 227.2 & $-0.71$ & 1.9\\ \\ & \n1.5 & 0.27 \\\\\n\\hline\n$-$& 884 & 992.8 & $+0.04$ &  5.8\\ \\ &  59 &  50.2 & $+1.15$ & 2.8\\ \\ & \n1.3 & 0.39 \\\\\n 4 & 711 & 756.9 & $-1.61$ &  3.4\\ \\ &  45 &  53.4 & $-1.13$ & 1.4\\ \\ & \n0.8 & 0.20 \\\\\n$+$& 735 & 773.9 & $-1.35$ &  3.7\\ \\ &  30 &  43.4 & $-2.07$ & 0.9\\ \\ & \n0.8 & 0.13 \\\\\n\\hline\n$-$&1708 &1734.0 & $-0.60$ &  3.2\\ \\ & 119 & 126.9 & $-0.68$ & 1.1\\ \\ & \n0.7 & 0.15 \\\\\n 5 &1692 &1596.2 & $+2.26$ &  9.4\\ \\ & 119 & 124.4 & $-0.47$ & 1.2\\ \\ & \n2.1 & 0.17 \\\\\n$+$&1485 &1447.1 & $+0.95$ &  6.4\\ \\ & 127 & 119.1 & $+0.68$ & 2.1\\ \\ & \n1.4 & 0.29 \\\\\n\\hline\n$-$&1382 &1375.2 & $+0.18$ &  4.9\\ \\ &  92 & 102.6 & $-1.02$ & 1.1\\ \\ & \n1.1 & 0.15 \\\\\n 6 &1212 &1225.3 & $-0.36$ &  4.2\\ \\ &  90 &  95.1 & $-0.50$ & 1.4\\ \\ & \n0.9 & 0.20 \\\\\n$+$&1390 &1391.7 & $-0.06$ &  4.4\\ \\ & 108 &  99.1 & $+0.84$ & 2.1\\ \\ & \n1.0 & 0.29 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsection{Periodic Emission}\n\\label{subsec:periodic}\nSeveral previous observations of Cygnus X-3 claimed evidence for\nsteady emission correlated with the \n$4.8\\,$hr\nX-ray periodicity of the source.\nFor this reason, we carry out a search for such emission using the\nentire CASA-MIA data set.\nThe event arrival times (UT) are corrected to the barycenter\nof the solar system using the JPL DE200 planetary ephemeris\n\\cite{ref:Standish}.\nThe corrected times are folded with the \n$4.8\\,$hr\nX-ray ephemeris\nof van der Klis and Bonnet-Bidaud \\cite{ref:vanderKlis2}.\nA slight correction is made for newer X-ray data from the ASCA\nsatellite, as reported by Kitamoto {\\it et al.} \\cite{ref:Kitamoto}.\nEach event is then assigned a phase value\nin the interval (0,1) representing the fraction of a period \nthat the event is from the X-ray minimum. The phase values\nare accumulated in twenty bins of 0.05 phase units each for\nboth the on-source and\ngenerated background events.\n\nFigure~\\ref{fig:CygPhase} shows the \n$4.8\\,$hr\nperiodicity\ndistribution of events from\nthe direction of Cygnus X-3 for the all-data and muon-poor samples.\nAlso shown is the phase distribution expected from the background events.\nNo compelling excesses are seen at any particular phase interval\nfor either sample.\nWe carry out similar periodicity analyses using data\nat higher energies selected by the number of alerted CASA stations.\nThese searches also do not indicate any significant excesses\nat any phase interval.\nIn Table~\\ref{tab:CygPhase}, we list flux limits for the various\nsearches at the phase intervals (0.2,0.3) and (0.6,0.7).\nThese intervals were ones in which\nnumerous earlier experiments had reported detections.\n\n\\begin{table}\n\\begin{center}\n\\caption{CASA-MIA search results for \n$4.8\\,$hr\nperiodic emission from\nCygnus X-3.\nFlux limits are given for selected phase intervals in which\nearlier experiments had reported detections.\nColumns 3 and 4 give the 90\\% c.l. upper limits to the integral\nflux of neutral and gamma-ray particles, respectively, in\nunits of particles cm$^{-2}$ sec$^{-1}$.\nA blank entry corresponds to a data set having\ninsufficient data with which to calculate a limit.}\n\\label{tab:CygPhase}\n\\vspace{10pt}\n\\begin{tabular}{|crcc|}\\hline\nPhase Interval & Energy  & $\\Phi_N(E)$ & $\\Phi_\\gamma(E)$ \\\\\n\\hline\n$0.2 - 0.3 $   & $115\\,$TeV\\ &  $8.9\\times 10^{-14}$ & $2.3\\times 10^{-14}$ \\\\\n               & $530\\,$TeV &  $4.5\\times 10^{-15}$ & $6.9\\times 10^{-16}$ \\\\\n               &$1175\\,$TeV &  $3.5\\times 10^{-15}$ & --\\ \\ \\\\\n\\hline\n$0.6 - 0.7 $   & $115\\,$TeV &  $1.4\\times 10^{-13}$ & $3.5\\times 10^{-14}$ \\\\\n               & $530\\,$TeV &  $3.8\\times 10^{-15}$ & $3.4\\times 10^{-16}$ \\\\\n               &$1175\\,$TeV &  $6.5\\times 10^{-15}$ & --\\ \\ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{Comparison with Other Results}\n\\label{sec:comparison}\nAs described earlier, the many\ndetections of Cygnus X-3 and Hercules X-1 by experiments\noperating between 1975 and 1990 varied greatly in their\ncharacteristics.\nSome results were steady and some were episodic;\nsome exhibited apparent periodicity and others did not.\nWe thus choose to compare the results of this work to a\ngeneralized picture of the earlier results and to more recent\nwork.\n\n\\subsection{Cygnus X-3}\n\\label{subsec:cygnus_comparison}\nIn Figure~\\ref{fig:NewCyg}, we \nplot the flux limits reported here on \nsteady emission from Cygnus X-3.\nWe also show\npublished results from other experiments\nusing data taken at times which overlap our observation period.\nThe other results come from the \nTibet air shower array in Yangbajing, China \n\\cite{ref:Amenomori}, the\nCYGNUS array in New Mexico, USA\n\\cite{ref:Alexandreas4},\nthe EAS-TOP array at Gran Sasso, Italy\n\\cite{ref:Aglietta},\nand the HEGRA experiment on the Canary Island La Palma\n\\cite{ref:Karle}.    \nWe do not show earlier results from data taken by \na portion of our experiment\nin 1989 \\cite{ref:Cronin} or the results of our all-sky survey\nfor northern hemisphere point sources using data taken\nin 1990-1991 \\cite{ref:McKay}.\nIn Figure~\\ref{fig:CygFract}, we show a similar comparison of\nthe limits on the fractional excess of events from Cygnus X-3 relative to\nthe cosmic ray background.\n\nThe data from the recent experiments are consistent; no steady\nemission of ultra-high energy\nparticles (gamma-ray or otherwise) has been\ndetected from Cygnus X-3 at levels which are considerably lower\nthan earlier reports.\nAt TeV energies, the results from the Whipple Telescope\n\\cite{ref:Whipple} are also considerably lower than\nthe earlier reports.\nThe limits presented here are a factor of\n130 lower at $115\\,$TeV, and a factor of 900 at $1175\\,$TeV,\nthan the spectrum \nplotted in Figure~\\ref{fig:OldCyg}.\nOur results are also inconsistent with emission reported by\na smaller experiment using data \ntaken during a time that overlapped our observations \\cite{ref:Muraki}.\n\nThe limits presented here on transient emission from Cygnus X-3\nare lower than, but in agreement with, those reported by\nother air shower experiments.\nThere have been no compelling reports of transient emission of\ngamma-rays from Cygnus X-3 over the period 1990-1995,\nincluding during large radio flares from the source.\nThere was an observation of underground muons from the direction\nof Cygnus X-3 during the January 1991 radio flare \\cite{ref:Thomson}.\nThe reported flux for this observation was $7.5\\times 10^{-10}$\nmuons cm$^{-2}$ sec$^{-1}$,\nfor muon energies above $0.7\\,$TeV.\nIf the muons were produced in air showers by the interaction\nof a hypothetical neutral particle from Cygnus X-3, we would expect\na typical neutral particle energy of $\\sim 10\\,$TeV \n\\cite{ref:Gaisser2}.\nAssuming that the particle spectrum continues to energies\ndetectable by CASA-MIA\n(and conservatively using \na soft spectrum comparable to the cosmic rays),\none derives an expected flux of $\\sim 10^{-11}$\nparticles cm$^{-2}$ sec$^{-1}$ \nfor energies above $115\\,$TeV.\nThis flux is\na factor of 5 to 10 above the flux limits set by CASA-MIA on\nthe emission of any neutral particle during the January 1991 flare \n(Table~\\ref{tab:flare_results}).\n\nWe have also shown that there is no evidence \nfor $4.8\\,$\nperiodic emission from Cygnus X-3.\nThis result is consistent with reports by other experiments\nover the same period of time.\nThe limits on pulsed gamma-ray emission presented here for\nthe phase intervals of 0.2-0.3 and 0.6-0.7 (Table~\\ref{tab:CygPhase})\nare lower at $115\\,$TeV, and considerably lower at $530\\,$TeV,\nthan the fluxes predicted by a recent \ntheoretical paper \\cite{ref:Mitra}.\n\n\\subsection{Hercules X-1}\n\\label{subsec:hercules_comparison}\nThe limits on steady emission of gamma-rays from Hercules X-1\npresented here are in agreement with those from other\nexperiments, as shown in Figure~\\ref{fig:NewHer}.\nGamma-ray emission from Hercules X-1 was typically seen by\nearlier experiments as transient emission over short time\nscales (e.g. the 1986 outbursts).\nWe have no evidence for such emission over the entire period 1990-1995.\nIn Figure~\\ref{fig:HerTran1994}, we compare the \ndaily event totals\nobserved by CASA-MIA from the direction of Hercules X-1 \nto the total expected assuming the flux \nof an earlier reported outburst \\cite{ref:Dingus2}.\nClearly, no evidence for\nemission at even\nmuch weaker levels than this outburst is seen during this time.\nThe flux reported in Ref.~\\cite{ref:Dingus2} was $\\sim 2\\times 10^{-11}$\nparticles cm$^{-2}$ sec$^{-1}$ for minimum energies of $100\\,$TeV.\nThis flux is about a factor of 45 larger than the typical\nlimits placed by CASA-MIA during the early part of operations\nand about a factor of 80 larger than the typical daily gamma-ray\nlimits placed by the full CASA-MIA experiment (Table~\\ref{tab:daily}).\nSince we have no evidence for transient emission from Hercules X-1,\nwe choose not to carry out a periodicity analysis based on the\nX-ray pulsar period of $1.24\\,$sec.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nWe have carried out a high statistics search for ultra-high energy\nneutral and gamma-ray particle emission from Cygnus X-3 and\nHercules X-1 between 1990 and 1995.\nWe have no evidence for steady or transient emission from\neither source, and for Cygnus X-3, we have no evidence for\n$4.8\\,$hr\nperiodic emission or emission correlated with large radio\nflares.\nThese results are in agreement with those from other experiments\noperating during the same period of time, but are in stark\ncontrast to earlier (1975-1990) reports.\n\nThe apparent disappearance of Cygnus X-3 and Hercules X-1 from\nthe ultra-high energy gamma-ray sky can be interpreted in\ntwo ways.\nAn optimistic view \\cite{ref:Protheroe}\nis that the earlier results indicated the presence of\nultra-high energy gamma-rays (or particles) from Cygnus X-3\nand Hercules X-1, and that the sources, which are episodic on long times\nscales, are now dormant.\nA more pessimistic view is that the earlier reported detections were\nlargely, if not entirely, statistical fluctuations, and that no compelling\nevidence exists for ultra-high energy gamma-rays from \nany astrophysical source.\nWe point out that an earlier all-sky survey using a portion of our data\nsample indicates that the northern hemisphere does not contain\nany steady\npoint sources of gamma-rays with fluxes comparable to those\nreported from X-ray binaries in the 1980's\n\\cite{ref:McKay}.\nWe have presented an update on this analysis at a conference \\cite{ref:Nitz}\nwhich are consistent with the absence of bright 100 TeV gamma-ray\npoint sources.\nWe are in the process of completing a final all-sky survey on the\nfive year CASA-MIA data sample.\n\nThe pessimistic interpretation of the ultra-high energy point source\nquestion, if correct,\nhighlights the difficulties in detecting gamma-rays from sources at\nother (high) energies and in detecting neutrinos as well.\nIn addition, without compelling evidence for high energy particle\nacceleration at point sources, the difficulties in explaining\nthe origins of cosmic rays above $10^{14}\\,$eV remain.\n\n\n\n\\vspace{5mm}\n\\noindent {\\bf Acknowledgements}\n\\vspace{3mm}\n\nWe acknowledge the assistance of the command and\nstaff of Dugway Proving Ground, and the University of Utah\nFly's Eye group.\nSpecial thanks go to M. Cassidy.\nWe also wish to thank P.Burke, S. Golwala, M. Galli, J. He, H. Kim, L. Nelson,\nM. Oonk, M. Pritchard, P. Rauske, K. Riley, and Z. Wells \nfor assistance with data processing.\nThis work is supported by the U.S. \nNational\nScience Foundation and the U.S. Department of Energy.\nJWC and RAO wish to acknowledge the support of\nthe W.W. Grainger Foundation.\nRAO acknowledges additional support from the Alfred P. Sloan\nFoundation.\n\n\\vspace{10mm}\n\n\\noindent $^*$ Present Address: Department of Physics, Massachusettts Institute\nof Technology, Cambridge, MA 02139, USA.\n\n\\noindent $^\\dag$ Present Address: Department of Physics and Astronomy,\nIowa State University, Ames, IA 50011, USA.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{physician_system.pdf}\n    \\caption{PClean applied to  Medicare's 2.2-million-row Physician Compare National database. Based on a user-specified relational model, PClean infers a latent database of entities, which it uses to correct systematic errors (e.g. the misspelled \\textit{Abington, MD} appears 152 times in the dataset) and impute missing values.}\n    \\vspace{-5mm}\n    \\label{fig:physicians_results}\n\\end{figure*}\n\nReal-world data is often noisy and incomplete, littered with NULL values, typos, duplicates, and inconsistencies. Cleaning dirty data is important for many workflows, but can be difficult to automate, as it often requires judgment calls about objects in the world (e.g., to decide whether two records refer to the same hospital, or which of several cities called ``Jefferson'' someone lives in).\n\nThis paper presents PClean, a domain-specific generative probabilistic programming language (PPL) for Bayesian data cleaning. Although generative models provide a conceptually appealing approach to data cleaning, they have proved difficult to apply, due to the heterogeneity of real-world error patterns~\\cite{Abedjan2016} and the difficulty of inference. Like some  PPLs (e.g. BLOG~\\cite{Milch2006}), PClean programs encode generative models of relational domains, with uncertainty about a latent database of objects and relationships underlying a dataset. However, PClean's approach is inspired by domain-specific PPLs, such as Stan \\cite{Carpenter2017} and Picture \\cite{Kulkarni2015}: it aims not to serve all conceivable relational modeling needs, but rather to enable fast inference, concise model specification, and accurate cleaning on large-scale problems. It does this via three modeling and inference contributions:\n\\vspace{-2.5mm}\n\\begin{enumerate}\n    \\item PClean introduces a domain-general non-parametric  prior on the number of latent objects and their link structure.  PClean programs customize the prior via a relational schema and via generative models for objects' attributes.\n    \\vspace{-2mm}\n    \\item PClean inference is based on a novel sequential Monte Carlo (SMC) algorithm, to initialize the latent database with plausible guesses, and novel rejuvenation updates to fix mistakes.\n    \\vspace{-2mm}\n    \\item PClean provides a proposal compiler that generates near-optimal SMC proposals and Metropolis-Hastings rejuvenation proposals given the user's dataset, PClean program, and inference hints. These proposals improve over generic top-down PPL inference by incorporating local Bayesian reasoning within user-specified subproblems and heuristics from traditional cleaning systems.\n    \n\\end{enumerate}\n    \\vspace{-2mm}\nTogether, this paper's innovations improve over generic PPL inference techniques, and enable fast and accurate cleaning of challenging real-world datasets with millions of rows.\n \n\n\\subsection{Related work}\n\nMany researchers have proposed  generative models for data cleaning in specific datasets~\\cite{Pasula2003, Kubica2003,  MayfieldJenniferNeville2009, Matsakis2010, Xiong2011, Hu2012,  Zhao2012, Abedjan2016,  De2016, Steorts2016, Winn2017, DeSa2019}. Generative formulations specify a prior over latent ground truth data, and a likelihood that models how the ground truth is noisily reflected in dirty datasets. In contrast, PClean's PPL makes it easy to write short (\\textless 50 line) programs to specify custom priors for new datasets, and yield inference algorithms that deliver fast, accurate cleaning results. \n\n\nThere is a rich literature on Bayesian approaches to modeling relational data~\\cite{Friedman1999}, including `open-universe' models with identity and existence uncertainty~\\cite{Milch2006}. Several PPLs could express data cleaning models \\cite{milch2005, goodman2012church, dippl,  Tolpin2016, Mansinghka2014, bingham2019, Cusumano-Towner2019}, but in practice, generic PPL inference is often too slow. This paper introduces new algorithms that enable PClean to scale better, and demonstrates external validity of the results by calibrating PClean's runtime and accuracy against SOTA data-cleaning baselines~\\cite{Dallachiesat2013,Rekatsinas2017} that use machine learning and weighted logic (typical of discriminative approaches~\\cite{Mccallum2003, Wellner2004, Wick2013}). Some of PClean's inference innovations have close analogues in traditional  cleaning systems; for example, PClean's preferred values from Section~\\ref{sec:hints} are related to HoloClean's notion of domain restriction. In fact, PClean can be viewed as a scalable, Bayesian, domain-specific PPL implementation of the PUD framework from \\cite{DeSa2019} (which abstractly characterizes the HoloClean implementation from~\\cite{Rekatsinas2017}, but does not itself include PClean's modeling or inference innovations).\n\n\n\\section{Modeling}\n\\label{sec:modeling}\nIn this section, we present the PClean modeling language, which is designed for encoding domain-specific knowledge about data and likely errors into concise generative models. PClean programs specify (i) a prior distribution $p(\\mathbf{R})$ over a latent ground-truth relational database of entities underlying the user's dataset, and (ii) an observation model $p(\\mathbf{D} \\mid \\mathbf{R})$ describing how the attributes of entities from $\\mathbf{R}$ are reflected in the observed flat data table $\\mathbf{D}$. Unlike general-purpose probabilistic programming languages, PClean does not afford the user complete freedom in specifying $p(\\mathbf{R})$. Instead, we impose a novel domain-general structure prior $p(\\mathbf{S})$ on the \\textit{skeleton} of the relational database: $\\mathbf{S}$ determines how many entities are in each latent database table, and which entities are related. The user's program specifies $p(\\mathbf{R} \\mid \\mathbf{S})$, a probabilistic relational model over the attributes of the objects whose existence and relationships are given by $\\mathbf{S}$. This decomposition limits the PClean model class, but enables the development of an efficient sequential Monte Carlo inference algorithm, presented in Section~\\ref{sec:inference}.\n\n\n\\subsection{PClean Modeling Language}\n\\label{sec:language}\n\n\n\n\\begin{figure}[t]\n\\includegraphics[width=\\linewidth]{physician_program_tall.pdf}\n\\caption{An example PClean program. PClean programs define: (i) an \\textit{acyclic} relational schema, comprising a set of classes $\\mathcal{C}$, and for each class $C$, sets $\\mathcal{A}(C)$ of attributes and $\\mathcal{R}(C)$ of reference slots; (ii) a probabilistic relational model $\\Pi$ encoding uncertain assumptions about object attributes; and (iii) a query $\\mathbf{Q}$ (last line of program), specifying how latent object attributes are observed in the flat data table $\\mathbf{D}$. \\textit{Inference hints} in gray do not affect the model's semantics.}\n\\label{fig:example-program}\n\\end{figure}\n\nA PClean program (Figure~\\ref{fig:example-program}) defines a set of \\textit{classes}\n$\\mathcal{C} = (C_1, \\dots, C_k)$ representing the types of object \nthat underlie the user's data\n(e.g. Physician, City), as well as a \\textit{query} $\\mathbf{Q}$ that describes how a latent object database informs the observed flat dataset $\\mathbf{D}$.\n\n\\textbf{Class declarations.} The declaration of a PClean class $C$ may include\nthree kinds of statement: \\textit{reference statements} ($Y \\sim C'$), which define a foreign key or reference slot $C.Y$ that connects objects of class $C$ to objects of a target class $T(C.Y) = C'$; \\textit{attribute statements} ($X \\sim \\phi_{C.X}(\\dots)$), which define a new field or \\textit{attribute} $C.X$ that objects of the class possess, and declare an assumption about the probability distribution $\\phi_{C.X}$ that the attribute typically follows; and $\\textit{parameter statements}$ ($\\textbf{parameter } \\theta_C \\sim p_{\\theta_C}(\\dots)$), which introduce mutually independent hyperparameters shared among all objects of the class $C$, to be learned from the noisy dataset. The distribution $\\phi_{C.X}$ of an attribute may depend on the values of a \\textit{parent set} $Pa(C.X)$ of attributes, potentially accessed via reference slots. For example, in Figure~\\ref{fig:example-program}, the \\textit{Physician} class has a \\textit{school} reference slot with target class \\textbf{School}, and a \\textit{degree} attribute whose value depends on \\textit{school.degree\\_dist}. Together, the attribute statements specify a \\textit{probabilistic relational model} $\\Pi$ for the user's schema (possibly parameterized by hyperparameters $\\{\\theta_C\\}_{C \\in \\mathcal{C}}$)~\\cite{Friedman1999}.\n\n\\textbf{Query.} After its class declarations, a PClean program ends with a \\textit{query}, connecting the schema of the latent relational database to the fields of the observed dataset. The query has the form \\textbf{observe} $(U_1 \\textbf{ as } x_1), \\cdots, (U_k \\textbf{ as } x_k) \\textbf{ from } C_{obs}$, where $C_{obs}$ is a class that models the records of the observed dataset (\\textit{Record}, in Figure~\\ref{fig:example-program}), $x_i$ are the names of the columns in the observed dataset, and $U_i$ are dot-expressions (e.g., \\textit{physician.school.name}) picking out an attribute accessible via zero or more reference slots from $C_{obs}$. We assume that each observed data record represents an observation of selected attributes of a distinct object in $C_{obs}$ (or objects related to it), and that these attributes are observed directly in the dataset. This means that errors are modeled as \\textit{part} of the latent relational database $\\mathbf{R}$, rather than as a separate stage of the generative process. For example, Figure~\\ref{fig:example-program} models systematic typos in the \\textit{City} field, by associating each \\textit{Practice} with a possibly misspelled version \\textit{bad\\_city} of the name of the city in which it is located.\n\n\\begin{figure}\n\\begin{algorithmic}[0]\n\\Model{GenerateSkeleton}{$\\mathcal{C}, |\\mathbf{D}|$}\n\\LineComment{Create one $C_{obs}$ object per observed record}\n\\State $\\mathbf{S}_{C_{obs}} := \\{1, \\dots, |\\mathbf{D}|\\}$\n\\LineComment{Generate a class \\textit{after} all referring classes:}\n\\For{class $C \\in \\textsc{TopoSort}(\\mathcal{C} \\setminus \\{C_{obs}\\})$}\n  \\LineComment{Collect references to class $C$}\n  \\State $\\mathbf{Ref}_\\mathbf{S}(C) := \\{(r, Y) \\mid r \\in \\mathbf{S}_{C'}, T(C'.Y) = C\\}$\n  \\LineComment{Generate targets of those references}\n  \\State $\\mathbf{S}_C \\sim \\textsc{GenerateObjectSet}(C, \\mathbf{Ref}_\\mathbf{S}(C))$\n  \\LineComment{Assign reference slots pointing to $C$}\n  \\For{object $r' \\in \\mathbf{S}_C$}\n  \t\\For{referring object $(r, Y) \\in r'$}\n  \t\t\\State $r.Y := r'$\n  \t\\EndFor\n  \\EndFor\n\\EndFor\n\\LineComment{Return the skeleton}\n\\State \\Return $\\{\\mathbf{S}_C\\}_{C \\in \\mathcal{C}}, (r, Y) \\mapsto r.Y$\n\\EndModel\n\n\\Model{GenerateObjectSet}{$C, \\mathbf{Ref}_{\\mathbf{S}}(C)$}\n\\State $s_C \\sim \\textit{Gamma}(1, 1)$; $d_C \\sim \\textit{Beta}(1, 1)$\n\\LineComment{\\parbox[t]{.9\\linewidth}{Partition $\\mathbf{Ref}_{\\mathbf{S}}(C)$ into disjoint co-referring subsets; each represents an object}}\n\\State $\\mathbf{S}_C \\sim CRP(\\mathbf{Ref}_{\\mathbf{S}}(C), s_C, d_C)$\n\\EndModel\n\\end{algorithmic}\n\n\\caption{PClean's non-parametric structure prior $p(\\mathbf{S})$ over the relational skeleton $\\mathbf{S}$ for a schema $\\mathcal{C}$.}\n\\label{fig:skeleton-generator}\n\\end{figure}\n\n\\subsection{Non-parametric Structure Prior $p(\\mathbf{S})$}\n\nA PClean program's class declarations specify a probabilistic relational model that can be used to generate the attributes of objects in the latent database, but does not encode a prior over how many objects exist in each class or over their relationships. (The one exception is $C_{obs}$, the designated observation class, whose objects are assumed to be in one-to-one correspondence with the rows of the observed dataset $\\mathbf{D}$.) In this section, we introduce a domain-general structure prior $p(\\mathbf{S}; |\\mathbf{D}|)$ that encodes a non-parametric generative process over the \\textit{object sets} $\\mathbf{S}_C$ associated with each class $C$, and over the values of each object's reference slots. The parameter $|\\mathbf{D}|$ is the number of observed data records; $p(\\mathbf{S}; |\\mathbf{D}|)$ places mass only on relational skeletons in which there are exactly $|\\mathbf{D}|$ objects in $C_{obs}$ and every object in another class is connected via some chain of reference slots to one of them.\n\nPClean's generative process for relational skeletons is shown in Figure~\\ref{fig:skeleton-generator}. First, with probability 1, we set $\\mathbf{S}_{C_{obs}} = \\{1, \\dots, |\\mathbf{D}|\\}$. (The objects here are natural numbers, but any choice will do; all that matters is the cardinality of the set $\\mathbf{S}_{C_{obs}}$.) PClean requires that the directed graph with an edge $(C, T(C.Y))$ for each reference slot $C.Y$ is acyclic, which allows us to generate the remaining object sets class-by-class, processing a class only after processing any classes with reference slots targeting it.\nIn order to generate an object set for class $C$, we first consider the reference set $\\mathbf{Ref}_\\mathbf{S}(C)$ of all objects with reference slots that point to it:\n$$\\mathbf{Ref}_\\mathbf{S}(C) = \\{(r, Y) \\mid Y \\in \\mathcal{R}(C') \\wedge T(C'.Y) = C\\wedge r \\in \\mathbf{S}_{C'}\\}$$\nThe elements of $\\mathbf{Ref}_\\mathbf{S}(C)$ are pairs $(r, Y)$ of an object and a reference slot; if a single object has two reference slots targeting class $C$, then the object will appear twice in the reference set. The point is to capture all of the places in $\\mathbf{S}$ that will refer to objects of class $C$.\n\nNow, instead of first generating an object set $\\mathbf{S}_C$ and then assigning the reference slots in $\\mathbf{Ref}_\\mathbf{S}(C)$, we  directly model the \\textit{co-reference partition} of $\\mathbf{Ref}_\\mathbf{S}(C)$, i.e., we will partition the references to objects of class $C$ into disjoint subsets, within each of which we will take all references to point to the same target object. To do this, we use the two-parameter Chinese restaurant process $CRP(X, s, d)$, which defines a non-parametric distribution over partitions of its set-valued parameter $X$. The strength $s$ and discount $d$ control the sizes of the clusters.\nWe can use the CRP to generate a partition of all references to class $C$. \\textit{We treat the resulting partition as the object set $\\mathbf{S}_C$}, i.e., each component defines one object of class $C$:\n$$\\mathbf{S}_C \\mid \\mathbf{Ref}_\\mathbf{S}(C) \\sim CRP(\\mathbf{Ref}_{\\mathbf{S}}(C), s, d)$$\nTo set the reference slots $r.Y$ with target class $T(\\mathbf{Class}(r).Y) = C$, we simply look up which partition component $(r, Y)$ (viewed as an element of $\\mathbf{Ref}_{\\mathbf{S}}(C)$) was assigned to. Since we have equated these partition components with objects of class $C$, we can directly set $r.Y$ to point to the component (object) that contains $(r, Y)$ as an element:\n$$r.Y := \\textrm{the unique } r' \\in \\mathbf{S}_{T(\\mathbf{Class}(r).Y)} \\textrm{ s.t. } (r, Y) \\in r'$$\nThis procedure can be applied iteratively to generate object sets for every relevant class, and simultaneously to fill all these objects' reference slots.\n\n\n\n\n\\section{Inference}\n\\label{sec:inference}\n\nPClean's non-parametric structure prior ensures that PClean models admit a sequential\nrepresentation, which can be used as the basis of a resample-move sequential Monte Carlo inference scheme\n (Section~\\ref{sec:smc}). However, if the SMC and rejuvenation proposals are made\nfrom the model prior, as is typical in PPLs, inference will still require\nprohibitively many particles to deliver accurate results. To address this issue, PClean uses\na \\textit{proposal compiler} that exploits conditional independence in the model\nto generate fast enumeration-based proposal kernels for both SMC and MCMC rejuvenation (Section~\\ref{sec:proposals}). Finally, to help users scale these proposals to large data,\nwe introduce \\textit{inference hints}, lightweight annotations in the PClean program that\ncan divide variables into subproblems to be separately handled by the proposal, or direct\nthe enumerator to focus its efforts on a dynamically computed subset of a large discrete domain (Section~\\ref{sec:hints}).\n\n\\begin{figure}\n\\begin{algorithmic}[0]\n\\Model{GenerateDataset}{$\\Pi$, $\\mathbf{Q}$, $|\\mathbf{D}|$}\n\\State $\\mathbf{R}^{(0)} \\leftarrow \\emptyset$ \\Comment{Initialize empty database}\n\\For{observation $i \\in \\{1, \\dots, |\\mathbf{D}|\\}$}\n  \n  \\State $\\Delta_i^\\mathbf{R} \\leftarrow \\textsc{GenerateDbIncr}(\\mathbf{R}^{(i-1)}, C_{obs})$\n  \\State $\\mathbf{R}^{(i)} \\leftarrow \\mathbf{R}^{(i-1)} \\cup \\Delta_i^\\mathbf{R}$\n\n  \\State $r \\leftarrow$ the unique object of class $C_{obs}$ in $\\Delta_i^\\mathbf{R}$\n  \\State $d_i \\leftarrow \\{X \\mapsto r.\\mathbf{Q}(X),\\,\\, \\forall X \\in \\mathcal{A}(\\mathbf{D})\\}$\n\\EndFor\n\\State \\Return $\\mathbf{R} = \\mathbf{R}^{(|\\mathbf{D}|)}, \\mathbf{D} = (d_1, \\dots, d_{|\\mathbf{D}|})$\n\\EndModel\n\\Model{GenerateDbIncr}{$\\mathbf{R}^{(i-1)}$, root class $C$}\n  \\State $\\Delta \\leftarrow \\emptyset$; $r_* \\leftarrow$ a new object of class $C$\n  \\For{each reference slot $Y \\in \\mathcal{R}(C)$}\n  \t\\State $C' \\leftarrow T(C.Y)$\n  \t\\For{each object $r \\in \\mathbf{R}^{(i-1)}_{C'} \\cup \\Delta_{\\mathbf{R}_{C'}}$}\n  \t\t\\State $n_r \\leftarrow |\\{r'\\mid r' \\in \\mathbf{R}^{(i-1)} \\cup \\Delta \\wedge \\exists \\tau, r'.\\tau = r\\}|$\n  \t\\EndFor\n  \t\\State $r_*.Y \\leftarrow r$ w.p. $\\propto {n_r - d_{C'}}$, or $\\star$ w.p. $\\propto {s_{C'} + d_{C'}|\\mathbf{R}^{(i-1)}_{C'} \\cup \\Delta_{\\mathbf{R}_{C'}}|}$\n  \t\\If{$r_*.Y = \\star$}\n  \t\t\\State $\\Delta' \\leftarrow \\textsc{GenerateDbIncr}(\\mathbf{R}^{(i-1)} \\cup \\Delta, C')$\n  \t\t\\State $\\Delta \\leftarrow \\Delta \\cup \\Delta'$\n  \t\t\\State $r_*.Y \\leftarrow $ the unique $r'$ of class $C'$ in $\\Delta'$\n  \t\\EndIf\n  \\EndFor\n  \\For{each $X \\in \\mathcal{A}(C)$, in topological order}\n  \t\\State $r_*.X \\sim \\phi_{C.X}(\\cdot \\mid \\{r_*.U\\}_{U \\in Pa(C.X)})$\n  \\EndFor\n  \\State \\Return $\\Delta \\cup \\{r_*\\}$\n\\EndModel\n\n\\end{algorithmic}\n\\vspace{-4mm}\n\\caption{Sequential model representation.}\n\\label{fig:seqrep}\n\\end{figure}\n\n\\subsection{Per-observation sequential Monte Carlo with per-object rejuvenation}\n\\label{sec:smc}\n\n\\begin{algorithm*}[th!]\n\\caption{Compiling SMC proposal to Bayesian network}\n\\label{alg:increment-bayes-net}\n\\begin{algorithmic}\n\\Procedure{GenerateIncrementBayesNet}{partial instance $\\mathbf{R}^{(i-1)}$, data $d_i$}\n  \\LineComment{Set the vertices to all attributes and reference slots accessible from $C_{obs}$}\n  \\State $U \\leftarrow \\mathcal{A}(C_{obs}) \\cup \\{K \\mid C_{obs}.K \\textrm{ is a valid slot chain} \\} \\cup \\{K.X \\mid X \\in \\mathcal{A}(T(C_{obs}.K))\\}$ \n  \\LineComment{Determine parent sets and CPDs for each variable}\n  \\For{each variable $u \\in U$}\n  \t\\If{$u \\in \\mathcal{A}(C_{obs})$}\n  \t\t\\State Set $Pa(u) = Pa^{\\Pi}(C.u)$\n  \t\t\\State Set $\\phi_{u}(v_u \\mid \\{v_{u'}\\}_{u' \\in Pa(u)}) = \\phi^\\Pi_{C.u}(v_u \\mid \\{v_{u'}\\}_{u' \\in Pa(u)})$\n  \t\\ElsIf{$u = K.X$ for $X \\in \\mathcal{A}(T(C_{obs}.K))$}\n  \t\t\\State Set $Pa(u) = Pa^{\\Pi}(T(C_{obs}.K).X) \\cup \\{K\\} \\cup \\{u'.X \\mid u' \\textrm{ already processed} \\wedge T(C_{obs}.u') = T(C_{obs}.K)\\}$\n  \t\t\\State Set \\[ \\phi_u(v_u \\mid \\{v_{u'}\\}_{u' \\in Pa(u)}) = \\begin{cases} \n      \\mathbf{1}[v_u = v_K.X] & v_K \\in \\mathbf{R}^{(i-1)} \\\\\n      \\phi^\\Pi_{T(C_{obs}.K).X}(v_u \\mid \\{v_{u'}\\}_{u' \\in Pa^\\Pi(T(C_{obs}.K).X)}) & v_K = \\textbf{new}_K \\\\\n      \\mathbf{1}[v_u = v_{u'.X}] & v_K = \\textbf{new}_{u'}, u' \\neq K\n   \\end{cases}\n\\]\n  \t\\Else\n  \t    \\State Set $Pa(u)$ to already-processed slot chains $u'$ s.t. $T(C.u') = T(C.u)$\n  \t    \\State Set domain $V(u) = \\mathbf{R}^{(i-1)}_{T(C.u)} \\cup \\{\\textbf{new}_{u'} \\mid u' \\in Pa(u) \\cup \\{u\\}\\}$\n  \t    \\State Set $\\phi_u(v_u \\mid \\{v_{u'}\\}_{u' \\in Pa(u)})$ according to CRP\n  \t\\EndIf\n  \\EndFor\n\n  \\For{attribute $X \\in \\mathcal{A}(\\mathbf{D})$}\n  \t\\State Change node $\\mathbf{Q}(u)$ to be observed with value $d_i.x$, \\textbf{unless} $d_i.x$ is missing\n  \\EndFor\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm*}\n\n\nOne representation of the PClean model's generative process was given in Section~\\ref{sec:modeling}: a skeleton can be generated from $p(\\mathbf{S})$, then attributes can be filled in using the user-specified probabilistic relational model $p_{\\Pi}(\\mathbf{R} \\mid \\mathbf{S})$. Finally an observed dataset $\\mathbf{D}$ can be generated from $\\mathbf{R}$ according to the query $\\mathbf{Q}$. But a key feature of our model is that it also admits a sequential representation, in which the latent relational database $\\mathbf{R}$ is built in stages: at each stage, a single record is\nadded to the observation class $C_{obs}$, along with any new objects in other classes that it refers to. Using this representation, we can run sequential Monte Carlo on the model, building a particle approximation to the posterior that incorporates one observation at a time.\n\n\\textbf{Database increments.} Let $\\mathbf{R}$ be a database with designated observation class $C_{obs}$. Assume $\\mathbf{R}_{C_{obs}}$, the object set for the class $C_{obs}$, is $\\{1, \\dots, |\\mathbf{D}|\\}$. Then the database's $i^{\\textrm{th}}$ \\textit{increment} $\\Delta_\\mathbf{R}^i$ is the object set\n$$\\{r \\in \\mathbf{R} \\mid \\exists K, \\, i.K = r \\wedge \\forall K', \\forall j < i, j.K' \\neq r\\},$$\nalong with their attribute values and targets of their reference slots. Objects in $\\Delta_\\mathbf{R}^i$ \nmay refer to other objects within the increment, or in earlier increments.\nThat is, the $i^{\\textrm{th}}$ increment of a database is the set of objects referenced by the $i^{\\textrm{th}}$ observation object, but \\textit{not} from any other observation object $j < i$.\n\n\\textbf{Sequential generative process.} Figure~\\ref{fig:seqrep} shows a generative process equivalent to the one in Section~\\ref{sec:modeling}, but which generates the attributes and reference slots of each increment sequentially. Intuitively, the database is generated via a Chinese-restaurant `social network':  Consider a collection of restaurants, one for each class $C$, where each table serves a dish $r$ representing an object of class $C$.\nUpon entering a restaurant, customers either sit at an existing\ntable or start a new one, as in the usual generalized CRP construction.\nBut these restaurants require that to start a new table, customers must first send $\n|\\mathcal{R}(C)|$ friends to \\textit{other} restaurants (one to the target of each reference slot). Once they are seated at these \\textit{parent} restaurants, they phone the original customer to help decide what to order, i.e., how to sample the attributes $r.X$ of the new table's object, informed by \\textit{their} dishes (the objects $r.Y$ of class $T(C.Y)$). \nThe process starts with $|\\mathbf{D}|$ customers at the observation class $C_{Obs}$'s restaurant,\nwho sit at separate tables; each customer who sits down triggers the sampling of one increment.\n\n\\textbf{SMC inference with object-wise rejuvenation.}\nThe sequential representation yields a sequence of intermediate unnormalized target densities $\\tilde{\\pi}_i$ for SMC:\n$$\\tilde{\\pi}_i(\\mathbf{R}) = \\prod_{j=1}^i p(\\Delta_j^\\mathbf{R} \\mid \\Delta_1^\\mathbf{R}, \\dots, \\Delta_{j-1}^\\mathbf{R}) p(d_j \\mid \\Delta_1^\\mathbf{R}, \\dots, \\Delta_j^\\mathbf{R}).$$\nParticles are initialized to hold an empty database, to which proposed increments $\\Delta_i^\\mathbf{R}$ are added each iteration. As is typical in SMC, at each step, the particles are reweighted according to how well they explain the new observed data, and resampled to cull low-weight particles while cloning and propagating promising ones. This process allows the algorithm to hypothesize new latent objects as needed to explain each new observation, but not to revise earlier inferences about latent objects (or delete previously hypothesized objects) in light of new observations; we address this problem with MCMC rejuvenation moves. These moves select an object $r$, and update all $r$'s attributes and reference slots in light of all relevant data incorporated so far. In doing so, these moves may also lead to the ``garbage collection'' of objects that are no longer connected to the observed dataset, or to the insertion of new objects as targets of $r$'s reference slots.\n\n\\subsection{Compiling data-driven SMC proposals}\n\\label{sec:proposals}\nProposal quality is the determining factor for the quality of SMC inference: at each step of the algorithm, a proposal $Q_i(\\Delta_i^\\mathbf{R}; \\mathbf{R}^{(i-1)}, d_i)$ generates proposed additions $\\Delta_i^\\mathbf{R}$ to the existing latent database $\\mathbf{R}^{(i-1)}$ to explain the $i^\\textrm{th}$ observed data point, $d_i$. A key limitation of the sequential Monte Carlo implementations in most general-purpose PPLs today is that the proposals $Q_i$ are not \\textit{data-driven}, but rather based only on the prior: they make blind guesses as to the latent variable values and thus tend to make proposals that explain the data poorly.\nBy contrast, PClean compiles proposals that use exact enumerative inference to propose discrete variables in a data-driven way. This approach extends ideas from \\cite{arora2012gibbs} to the block Gibbs rejuvenation and block SMC setting, with user-specified blocking hints. These proposals are \\textit{locally optimal} for models that contain only discrete finite-domain variables, meaning that of all possible proposals $Q_i$ they minimize the divergence \n$$KL(\\pi_{i-1}(\\mathbf{R}^{(i-1)}) Q_i(\\Delta_i^\\mathbf{R}; \\mathbf{R}^{(i-1)}, d_i) || \\pi_i(\\mathbf{R}^{(i-1)} \\cup \\Delta_i^\\mathbf{R})).$$\nThe distribution on the left represents a perfect sample $\\mathbf{R}^{(i-1)}$ from the target given the first $i - 1$ observations, extended with the proposal $Q_i$. The distribution on the right is the target given the first $i$ data points.\nIn our setting the locally optimal proposal is given by\n\\begin{align*}\nQ_i(\\Delta_i^\\mathbf{R};& \\mathbf{R}^{(i-1)}, d_i) \\propto \\\\\n&p(\\Delta_i^\\mathbf{R} \\mid \\Delta_1^\\mathbf{R}, \\dots, \\Delta_{i-1}^\\mathbf{R})p(d_i \\mid \\Delta_1^\\mathbf{R}, \\dots, \\Delta_{i}^\\mathbf{R}).\n\\end{align*}\nAlgorithm~\\ref{alg:increment-bayes-net} shows how to compile this distribution to a Bayesian network; when the latent attributes have finite domains, the normalizing constant can be computed and the locally optimal proposal can be simulated (and evaluated) exactly. \nThis is possible because there are only a finite number of instantiations of the random increment $\\Delta_i^\\mathbf{R}$ to consider.\nThe compiler generates efficient enumeration code separately for each pattern of missing values it encounters in the dataset, exploiting conditional independence relationships in each Bayes net to yield potentially exponential savings over naive enumeration.\nA similar strategy can be used to compile data-driven object-wise rejuvenation proposals, and to handle some continuous variables with conjugate priors; see supplement for details.\n\n\n\n\\subsection{Scaling to large data with inference hints}\n\\label{sec:hints}\nScaling to models with large-domain variables and to datasets with many rows is a key challenge.\nIn PClean, users can specify lightweight \\textit{inference hints} to the proposal compiler, shown in gray in Figure~\\ref{fig:example-program}, to speed up inference without changing model's meaning.\n\n\\textbf{Programmable subproblems.} First, users may group attribute and reference statements into blocks by wrapping them in the syntax  $\\textbf{subproblem begin}\\dots\\textbf{end}$. This partitions the attributes and reference slots of a class into an ordered list of \\textit{subproblems}, which SMC uses as intermediate target distributions. This makes enumerative proposals faster to compute, at the cost of considering less information at each step; rejuvenation moves can often compensate for short-sighted proposals.\n\n\n{\\bf Adaptive mixture proposals with dynamic preferred values.}\nA random variable within a model may be intractable to enumerate. For example, $\\texttt{string\\_prior(1, 100)}$ is a distribution over all strings between 1 and 100 letters long.\nTo handle these, PClean programs may declare \\textit{preferred values hints}. Instead of $X \\sim d(E,\\dots,E)$, the user can write $X \\sim d(E,\\dots,E) \\textbf{ preferring } E,$ where the final expression gives a list of values $\\xi_X$ on which the posterior mass is expected to concentrate. \nWhen enumerating, PClean replaces the CPD $\\phi_X$ with a surrogate $\\hat{\\phi}_X$, which is equal to $\\phi_X$ for preferred value inputs in $\\xi_X$, but 0 for all other values. The mass not captured by the preferred values, $1 - \\sum_{x \\in \\xi_{X}} \\phi_X(x)$, is assigned to a special $\\textbf{other}$ token.\nEnumeration yields a partial proposal $\\hat{Q}$ over a modified domain; the full proposal $Q$ first draws from $\\hat{Q}$ then replaces $\\textbf{other}$ tokens with samples from the appropriate CPDs $\\phi_X(\\cdot \\mid Pa(X))$. This yields a mixture proposal between the enumerative posterior on preferred values and the prior: when none of the preferred values explain the data well, $\\textbf{other}$ will dominate, causing the attribute to be sampled from its prior. But if any of the preferred values are promising, they will almost certainly be proposed.\n\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.95\\linewidth]{inference_plot.pdf}\n\\vspace{-4mm}\n\\caption{Median accuracy vs. runtime for five runs of alternative inference algorithms on the \\textit{Hospital} dataset \\cite{Chu2013}, with an additional 20\\% of cells artificially deleted so as to test both repair and imputation.}\n\\label{fig:inference-comparison}\n\\end{figure*}\n\n\\section{Experiments}\n\\label{sec:results}\n\n\nIn this section, we demonstrate empirically that (1) PClean's inference works when standard PPL inference strategies fail, (2) short PClean programs suffice to compete with existing data cleaning systems in both runtime and accuracy, and (3) PClean can scale to large real-world datasets. Experiments were run on a  laptop with a 2.6 GHz CPU and 32 GB of RAM.\n\n\n\\textbf{(1) Comparison to Generic PPL Inference.}\nWe evaluate PClean's inference against\nstandard PPL inference algorithms reimplemented\nto work on PClean models, on a popular benchmark from the data cleaning literature (Figure~\\ref{fig:inference-comparison}). We do not compare directly to other PPLs' implementations, because many (e.g. BLOG) cannot represent PClean's non-parametric prior. Some languages (e.g. Turing) have explicit support for non-parametric distributions, but could not express PClean's recursive use of CRPs. Others could in principle express PClean's model, but would complicate an algorithm comparison in other ways: Venture's dynamic dependency tracking is thousands of times slower than SOTA; Pyro's focus is on variational inference, hard to apply in PClean models; and Gen supports non-parametrics only via the use of mutation in its slower dynamic modeling language (making SMC $O(N^2)$) or via low-level extensions that would amount to reimplementing PClean using Gen's abstractions. Nonetheless, the algorithms in Figure~\\ref{fig:inference-comparison} are inspired by the generic automated inference provided in many PPLs, which use top-down proposals from the prior for SMC, MH~\\cite{dippl,ritchie2016c3}, and PGibbs~\\cite{wood2014new,Murray2015,Mansinghka2014}. Our results show that PClean suffices for fast, accurate inference where generic techniques fail, and also demonstrate why inference hints are necessary for scalability: without subproblem hints, PClean takes much longer to converge, even though it eventually arrives at a similar $F_1$ value.\n\n\\begin{table*}[]\n\\centering\n\\begin{tabular}{|lc|ccccc|}\n\\hline\n\\multicolumn{1}{|l}{\\textbf{\\footnotesize{Task}}}    & \\multicolumn{1}{l|}{\\footnotesize{\\textbf{Metric}}} & \\multicolumn{1}{l}{\\textbf{\\footnotesize{PClean}}} & \\multicolumn{1}{l}{\\begin{tabular}[c]{@{}c@{}}\\footnotesize{\\textbf{HoloClean}}\\\\ \\footnotesize{\\textbf{(Unpublished)}} \\end{tabular}\n &\n\\multicolumn{1}{l}{\\footnotesize{\\textbf{HoloClean\n}}  &\\multicolumn{1}{l}{\\footnotesize{\\textbf{NADEEF\n}} & \\multicolumn{1}{l|}{\\begin{tabular}[c]{@{}c@{}}\\footnotesize{\\textbf{NADEEF + Manual}}\\\\ \\footnotesize{\\textbf{Java Heuristics}} \\end{tabular}} \\\\ \\hline \n\\multirow{2}{*}{\\textbf{\\footnotesize{Flights}}}        \n& $F_1$   & \\textbf{0.90}   & 0.64 & 0.41 & 0.07 & \\textbf{0.90}\n\\\\\n & {Time }   & \\textbf{3.1s}  & { 45.4s}  & { 32.6s}  & { 9.1s} & 14.5s\n \\\\\\hline\n\\multirow{2}{*}{\\textbf{\\footnotesize{Hospital}}}       \n& $F_1$    & \\textbf{0.91}  & 0.90  & 0.83 & 0.84 & 0.84  \n\\\\\n & { Time }   & \\textbf{{ 4.5s}}  & {1m 10s}  & { 1m 32s}  & { 27.6s} & 22.8s\n \\\\\\hline\n\\multirow{2}{*}{\\textbf{\\footnotesize{Rents}}} \n& $F_1$    & \\textbf{0.69}   &  0.48 & 0.48 & 0 & 0.51\n\\\\ \n & { Time }  & {1m 20s}  & { 20m 16s}  & {13m 43s}  & { 13s} & \\textbf{{7.2s}}\n\n \\\\\\hline\n\\end{tabular}\n\\caption{Results of PClean and various baseline systems on three diverse cleaning tasks.}\n\\vspace{-1mm}\n\\label{tab:results}\n\\end{table*}\n\\textbf{(2) Applicability to Data Cleaning.} \nTo check PClean's modeling and inference capabilities are good for data cleaning \\textit{in absolute terms} (rather than relative to generic PPL inference), we contextualize PClean's accuracy and runtime against two SOTA data-cleaning systems on three benchmarks with known ground truth (Table~\\ref{tab:results}), described in detail in the supplement. Briefly, the datasets are \\textit{Hospital}, a standard benchmark with artificial typos in 5\\% of cells; \\textit{Flights}, a standard benchmark resolving flight details from conflicting real-world data sources; and \\textit{Rent}, a synthetic dataset based on census data, with continuous and discrete values. The systems are \\textit{HoloClean}~\\cite{Rekatsinas2017}, based on probabilistic machine learning, and \\textit{NADEEF}, which uses MAX-SAT solvers to adjudicate between user-defined cleaning rules~\\cite{Dallachiesat2013}. For HoloClean, we consider both the original code and the authors' latest (unpublished) version on GitHub; for NADEEF, we include results both with NADEEF's built-in rules interface alone and with custom, handwritten Java rules.\n\nTable~\\ref{tab:results} reports \\textit{$F_1$} scores and cleaning speed (see supplement for precision\/recall). We do not aim to anoint a single `best cleaning system,' since optimality depends on the available domain knowledge and the user's desired level of customization. Further, while we followed system authors' per-dataset recommendations where possible, a pure system comparison is difficult, since each system relies on its own rule configuration. Rather, we note that short (\\textless 50-line) PClean programs can encode knowledge useful in practice for cleaning diverse data, and inference is good enough to achieve $F_1$ scores as good or better than SOTA data-cleaning systems on all three datasets, often in less wall-clock time. Additionally, PClean programs are concise, and e.g. could encode in a single line what required 50 lines of Java for NADEEF (see supplement).\n\n\n\\textbf{(3) Scalability to large, real-world data.}\nWe ran PClean on the Medicare Physician Compare National dataset, shown earlier in Figure~\\ref{fig:physicians_results}.  It contains 2.2 million records, each listing a clinician and a practice location; the same clinician may work at multiple practices, and many clinicians may work at the same practice. NULL values and systematic errors are common (e.g. consistently misspelled city names for a practice).\n\nRunning PClean took 7h36m, changing 8,245 values and imputing 1,535,415 missing cells. In a random sample of 100 imputed cells, 90\\% agreed with manually obtained ground truth. We also manually checked PClean's changes, and 7,954  (96.5\\%) were correct. Of these, some were correct normalization (e.g. choosing a single spelling for cities whose names could be spelled multiple ways). To calibrate, NADEEF only changes 88 cells across the whole dataset, and HoloClean did not initialize in 24 hours, using the configuration provided by HoloClean's authors.\n\nFigure~\\ref{fig:physicians_results} shows PClean's real behavior on four rows. Consider the misspelling \\textit{Abington, MD}, which appears in 152 entries. The correct spelling \\textit{Abingdon, MD} occurs in only 42. However, PClean recognizes \\textit{Abington, MD} as an error because all 152 instances share a single practice address, and errors are modeled as happening systematically at the practice level. Next, consider PClean's correct inference that K. Ryan's degree is \\textit{DO}. PClean leverages the fact that her school \\textit{PCOM} awards more DOs than MDs, even though more \\textit{Family Medicine} doctors are MDs than DOs. All parameters enabling this reasoning are learned from the dirty data.\n\n\n\\section{Discussion}\nPClean, like other domain-specific PPLs, aims to be more automated and scalable than general purpose PPLs, by leveraging structure in its restricted model class to deliver fast inference. At the same time, it aims to be expressive enough to concisely solve a broad class of real-world data cleaning problems. \n\nOne direction for future research is to quantify  the ease-of-implementation, runtime, accuracy, and program length tradeoffs that PClean users can achieve, given varying levels of expertise. Rigorous user studies could calibrate these results against other data cleaning, de-duplication, and record linkage systems. One challenge is to account for the subtle differences in the knowledge representation approach between PClean (causal and generative) and most other data cleaning systems (based on  learning and\/or weighted logic)\\footnote{For example, correspondence with some HoloClean authors yielded ways to improve HoloClean's performance beyond previously published results, but did not yield ways for HoloClean to encode all forms of knowledge that PClean scripts can encode.}.\n\nIt may be possible to relax PClean's modeling restrictions without sacrificing inference performance and accuracy. One approach could be to integrate custom open-universe priors with explicit number statements and recursive object-level generative processes\\footnote{See supplement for a discussion of this direction in the context of data cleaning;  many datasets with cyclic links among classes (e.g. people who are friends with other people) can be modeled in PClean by introducing additional latent classes.}, or to embed PClean in a general-purpose PPL such as Gen, to allow deeper customization of the model and inference. Another important direction is to explore learnability of PClean programs, especially for tables with large numbers of columns\/attributes. It seems potentially feasible to apply automated error modeling techniques \\cite{Heidari2019} or probabilistic program synthesis~\\cite{saad-popl-2019,choi2020group} to partially automate PClean program authoring. It also could be fruitful to develop hierarchical variants of PClean that enable parameters and latent objects inferred by PClean programs to transfer across datasets.\n\n\\subsubsection*{Acknowledgements}\nThe authors are grateful to Zia Abedjan, Marco Cusumano-Towner, Raul Castro Fernandez, Cameron Freer, Divya Gopinath, Christina Ji, Tim Kraska, George Matheos, Feras Saad, Michael Stonebraker, Josh Tenenbaum, and Veronica Weiner for useful conversations and feedback, as well as to anonymous referees on earlier versions of this work. This work is supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. 1745302; DARPA, under the Machine Common Sense (MCS) and Synergistic Discovery and Design (SD2) programs; gifts from the Aphorism Foundation and the Siegel Family Foundation; a research contract with Takeda Pharmaceuticals; and financial support from Facebook, Google, and the Intel Probabilistic Computing Center. \n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper, we are concerned with the optimal decay rate\nof global small solution to the Cauchy problem for the compressible Navier-Stokes (CNS) equations with and without external force in three-dimensional whole space.\nThus, our first result is to investigate the optimal decay rate\nfor the CNS equations without external force as follows:\n\\begin{equation}\\label{ns1}\n\\left\\{\\begin{array}{lr}\n\t\\rho_t +\\mathop{\\rm div}\\nolimits(\\rho u)=0,\\quad (t,x)\\in \\mathbb{R}^{+}\\times \\mathbb{R}^3,\\\\\n\t(\\rho u)_t+\\mathop{\\rm div}\\nolimits(\\rho u \\otimes u)+ \\nabla p-\\mu\\tri u-(\\mu+\\lam)\\nabla\\mathop{\\rm div}\\nolimits u = 0,\n\t\\quad (t,x)\\in \\mathbb{R}^{+}\\times \\mathbb{R}^3,\n\\end{array}\\right.\n\\end{equation}\nwhere $\\rho$, $u$ and $p$ represent the unknown density, velocity and\npressure, respectively.\nThe initial data is given by\n\\begin{equation}\\label{initial1}\n(\\rho,u)|_{t=0}=(\\rho_0,u_0)(x),\\quad x\\in  \\mathbb{R}^3.\n\\end{equation}\nFurthermore, as the space variable tends to infinity, we assume\n\\begin{equation}\\label{boundary1}\n\\lim\\limits_{|x|\\rightarrow+\\infty}(\\rho,u)=(\\bar \\rho,0),\n\\end{equation}\nwhere $\\bar\\rho$ is a positive constant.\nThe pressure $p(\\rho)$ here is assumed to be a smooth function\nin a neighborhood of $\\bar\\rho$ with $p'(\\bar\\rho)>0$.\nThe constant viscosity coefficients $\\mu$ and $\\lambda$ satisfy the physical conditions\\begin{equation}\\label{physical-condition}\n\\mu>0,~~~~2\\mu+3\\lambda\\geq 0. \\end{equation}\nThe CNS system \\eqref{ns1} is a well-known model which describes\nthe motion of compressible fluid.\nIn the following, we will introduce some mathematical results related to the CNS equations, including the local and global-in-time well-posedness, large time behavior and so on.\n\nWhen the initial data are away from the vacuum, the local existence and\nuniqueness of classical solutions have been obtained in \\cite{{Serrin-1959},{Nash-1962}}.\nIf the initial density may vanish in open sets, the well-posedness theory also has been studied in \\cite{{Cho-Choe-Kim-2004},{Choe-Kim-2003},{Cho-Kim-2006},{Li-Liang-2014},{Salvi-Straskraba-1993}} when the initial data satisfy some compatibility conditions.\nThe global classical solution was first obtained by Matsumura and Nishida \\cite{Matsumura-Nishida-1980} as initial data is closed to a non-vacuum\nequilibrium in some Sobolev space $H^s$.\nThis celebrated result requires that the solution has small oscillation from a uniform non-vacuum state such that the density is strictly away from vacuum.\nFor general data, one has to face a tricky problem of the possible appearance of vacuum.\nAs observed in \\cite{Li2019,rozanova2008blow,Xin1998,Xin2013}, the strong (or smooth) solution for the CNS equations will blow up in finite time.\nIn order to solve this problem, it is important to study some blow-up criteria of strong solutions, refer to \\cite{Huang2011blowcriter1,{Sun-Wang-Zhang-2011},\nHuang2011blowcriter2,Wen2013}.\nIt is worth noting that the bounds established by the above papers are dependent on time.\nThus, under the assumption that $\\sup_{t\\in \\mathbb R^+}\\|\\rho(t, \\cdot)\\|_{C^\\alpha}\\le  M$ for some $0<\\alpha<1$, He, Huang and Wang \\cite{he-huang-wang} proved global stability of large solution and built the decay rate for the global solution as it tends to the\nconstant equilibrium state.\nLater, Gao, Wei and Yao studied the optimal decay rate for this class of\nglobal large solution itself and its derivatives\nin a series of articles \\cite{{gao-wei-yao-D},{gao-wei-yao-NA},{gao-wei-yao-pre}}.\nIn the presence of vacuum, Huang, Li and Xin\\cite{Huang-Li-Xin-2012} established the global existence and uniqueness of strong solution for the CNS equations in three-dimensional space in the condition that the initial energy is small.\nRecently, Li and Xin \\cite{Li-Xin-2019-pde} obtained similar results for the dimension two,\nin addition, they also studied the large time behavior of the solution for the CNS system with small initial data but allowing large oscillations.\nSome other related results with respect to the global well-posedness theory can be found in \\cite{Li2020,Wen2017}.\n\nThe large time behavior of the solutions to the isentropic compressible Navier-Stokes system has been studied extensively.\nThe optimal decay rate for strong solution to the CNS system was first derived by Matsumura and Nishida \\cite{nishida2}, and later by Ponce \\cite{ponce} for the optimal $L^p(p\\ge 2)$ decay rate.\nWith the help of the study of Green function, the optimal $L^p$ ($1\\le p\\le\\infty$) decay rates in $\\mathbb R^n(n\\ge 2)$ were obtained in \\cite{{hoff-zumbrun},{liu-wang}} when the small initial perturbation bounded in $H^s\\cap L^1$ with the integer $s\\ge[n\/2]+3$.\nAll these decay results mentioned above are restricted to the perturbation framework, that is, if the initial data is a small perturbation of constant equilibrium in $L^1\\cap H^3$,\nthe decay rate of global solution to system \\eqref{ns1} in $L^2$-norm is\n$$\\|\\rho(t)-\\bar\\rho\\|_{L^2}+\\|u (t)\\|_{L^2}\\le C(1+t)^{-\\frac34}.$$\nFurthermore, Gao, Tao and Yao \\cite{gao2016} applied the Fourier splitting method,\ndeveloped by Schonbek \\cite{Schonbek1985}, to establish optimal decay rate for\nthe higher-order spatial derivative of global small solution.\nSpecially, they built the decay rate as follows:\n\\begin{equation}\\label{Decay-Gao-Tao-Yao}\n\\|\\nabla^k(\\rho-\\bar\\rho)(t)\\|_{H^{N-k}}\n+\\|\\nabla^k u(t)\\|_{H^{N-k}}\n\\leq C_0 (1+t)^{-\\f34-\\f k2},\\quad \\text{for}~~ 0\\leq k\\leq N-1,\n\\end{equation}\nif the initial perturbation belongs to $H^N(\\mathbb{R}^3)\\cap L^1(\\mathbb{R}^3)$.\nObviously, the decay rate for the $N-$th order spatial derivative of global solution\nin \\eqref{Decay-Gao-Tao-Yao} is still not optimal.\n\\textbf{Recently, this tricky problem is addressed simultaneously in a series of articles\n\\cite{{chen2021},{Wang2020},{wu2020}} by using the spectrum analysis of\nthe linearized part.}\nIn the perturbation setting, the approach to proving the decay estimate for the solution of the CNS system relies heavily on the analysis of the linearization of the system.\nMore precisely, most of these decay results were proved by combining the linear optimal decay of spectral analysis with the energy method.\nFrom another point of view, under the assumption that the initial perturbation is bounded in $\\dot H^{-s}(s\\in [0, \\frac32))$, Guo and Wang \\cite{guo2012} obtained the optimal decay rate of the solution and its spatial derivatives of system \\eqref{ns1} under the $H^N(N\\ge 3)-$framework by using pure energy method.\nMore precisely, they established the following decay estimate\n\\begin{equation}\\label{Decay-Guo}\n\\|\\nabla^l (\\rho-\\bar{\\rho})(t)\\|_{H^{N-l}}\n+\\|\\nabla^l u(t)\\|_{H^{N-l}}\\le C_0(1+t)^{-(l+s)}, ~{\\rm for~}-s< l \\le N-1.\n\\end{equation}\nThis method in \\cite{guo2012} mainly combined the energy estimates with the interpolation between negative and positive Sobolev norms, and hence do not use the analysis of the linearized part.\nFrom the decay rate of \\eqref{Decay-Guo}, it is easy to see that the decay rate of $N-th$ order derivative of solution $(\\rho-\\bar{\\rho}, u)$ coincides with the lower one.\nHowever, the $N-th$ order spatial derivative of heat equation has the optimal decay rate $(1+t)^{-(N+s)}$ rather than $(1+t)^{-(N-1+s)}$(see Theorem $1.1$ in \\cite{guo2012}).\n\\textbf{\\textit{Thus, the first purpose in this paper is to investigate the optimal decay rate for the quantity $\\nabla^N (\\rho, u)$ as it converges to zero in $L^2-$norm.}}\n\nNow, we state the result of decay rate for the CNS equations that has been established before.\n\n\\begin{prop}\\label{hs1}(\\cite{guo2012})\nAssume that $(\\rho_0-\\bar\\rho,u_0)\\in H^{N}$ for an integer $N\\geq 3$.\nThen there exists a constant $\\delta_0$ such that if\n\t\\[\\|(\\rho_0-\\bar\\rho,u_0)\\|_{H^{[\\f N2]+2}}\\leq \\delta_0, \\]\n\tthen the problem \\eqref{ns1}--\\eqref{boundary1} admits a unique global solution $(\\rho, u)$\n    satisfying that for all $t \\ge 0$\n\t\\begin{equation}\\label{energy-01}\n\t\\|(\\rho-\\bar\\rho)(t)\\|_{H^m}^2\n    +\\|u(t)\\|_{H^m}^2\n    +\\int_0^t(\\|\\nabla\\rho\\|_{H^{m-1}}^2+\\|\\nabla u\\|_{H^m}^2)d\\tau\n    \\leq C (\\|\\rho_0-\\bar\\rho \\|_{H^m}^2+\\|u_0\\|_{H^m}^2),\n\t\\end{equation}\n    where $[\\f N2]+2\\leq m\\leq N$.\n    If further, $(\\rho_0-\\bar\\rho,u_0)\\in \\dot H^{-s}$ for some\n    $s \\in [0, \\f32)$, then for all $t \\ge 0$\n\t\t\\begin{equation}\n\t\t\\|\\Lambda^{-s}(\\rho-\\bar\\rho)(t)\\|_{L^2}^2\n        +\\|\\Lambda^{-s}u (t)\\|_{L^2}^2\\leq C_0,\n\t\t\\end{equation}\n\t\tand\n\t\t\\begin{equation}\\label{sde1}\n\t\t\\|\\nabla^l(\\rho-\\bar\\rho)(t)\\|_{H^{N-l}}^2+\\|\\nabla^l u(t)\\|_{H^{N-l}}^2\n        \\leq C_0 (1+t)^{-(l+s)}, {~\\rm for~}-s<l \\le N-1.\n\t\t\\end{equation}\n\\end{prop}\n\nNow, we state the first result, concerning the optimal decay rate of $N-th$ order spatial derivative of the solution $(\\rho-\\bar\\rho,u)$ of problem \\eqref{ns1}--\\eqref{boundary1}.\n\n\\begin{theo}\\label{hs2}\n\tUnder all the assumptions in Proposition \\ref{hs1}, then the global solution\n    $(\\rho, u)$ of problem \\eqref{ns1}--\\eqref{boundary1} admits the decay estimate for all $t\\ge 0$\n\t\\begin{equation}\\label{sde3}\n\t\t\\|\\nabla^N(\\rho-\\bar\\rho)(t)\\|_{L^2}^2\n        +\\|\\nabla^N u(t)\\|_{L^2}^2\\leq C_0 (1+t)^{-(s+N)}.\n\t\\end{equation}\n\twhere $C_0$ is a constant independent of time.\n\\end{theo}\n\n\\begin{rema}\nThe time decay estimate \\eqref {sde3}, together with the decay estimate \\eqref{sde1}, yields that\n\\begin{equation}\\label{decay-NS}\n\\|\\nabla^l(\\rho-\\bar\\rho)(t)\\|_{H^{N-l}}^2\n+\\|\\nabla^l u(t)\\|_{H^{N-l}}^2\n\\leq C_0 (1+t)^{-(l+s)}, {~\\rm for~}-s<l \\le N.\n\\end{equation}\nThen, the decay estimate \\eqref{decay-NS} provides with optimal decay rate\nfor the global solution $(\\rho-\\bar\\rho, u)$ itself and\nits any spatial derivative converging to zero.\nHere the decay rate of global solution of original problem \\eqref{ns1}--\\eqref{boundary1} converging to equilibrium state is called optimal in the sense that\nit coincides with the rate of solution of the linearized system.\n\\end{rema}\n\\begin{rema}\nCombine the decay rate \\eqref{Decay-Gao-Tao-Yao} and\nthe method as the proof of Theorem \\ref{hs2}, we also have\n\\begin{equation*}\n\\|\\nabla^N(\\rho-\\bar\\rho)(t)\\|_{L^2}+\\|\\nabla^N u(t)\\|_{L^2}\n\\leq C_0 (1+t)^{-\\f34-\\f N2}.\n\\end{equation*}\nThis decay estimate of the $N$-th order spatial derivative is optimal in the sense that it coincides with the rate of solution of linearized part.\n\\end{rema}\n\nNext, we consider the optimal decay rate of solution for the CNS equations\nwith potential force in three-dimensional whole space as follows:\n\\begin{equation}\\label{ns3}\n\\left\\{\\begin{array}{lr}\n\t\\rho_t +\\mathop{\\rm div}\\nolimits(\\rho u)=0,\\quad (t,x)\\in \\mathbb{R}^{+}\\times \\mathbb{R}^3,\\\\\n\t(\\rho u)_t+\\mathop{\\rm div}\\nolimits(\\rho u \\otimes u)+ \\nabla p-\\mu\\tri u-(\\mu+\\lam)\\nabla\\mathop{\\rm div}\\nolimits u +\\rho\\nabla \\phi= 0,\\quad (t,x)\\in \\mathbb{R}^{+}\\times \\mathbb{R}^3,\n\\end{array}\\right.\n\\end{equation}\nwhere $\\rho$, $u$ and $p$ represent the density, velocity and pressure, respectively. And $-\\nabla\\phi$ is the time independent potential force. The constant viscosity coefficients $\\mu$ and $\\lambda$ satisfy the physical conditions \\eqref{physical-condition}.\nThe initial data\n\\begin{equation}\\label{initial-boundary}\n(\\rho,u)|_{t=0}=(\\rho_0,u_0)(x)\\rightarrow (\\rho_{\\infty},0),\\quad |x|\\rightarrow 0,\n\\end{equation}\nwhere $\\rho_{\\infty}$ is a positive constant.\nWe assume that $p(\\rho)$ is a smooth function in a neighborhood of $\\rho_{\\infty}$ with $p'(\\rho_{\\infty})>0$. The stationary solution $(\\rho^*(x),u^*(x))$ for\nthe CNS equations \\eqref{ns3} is given by $(\\rho^*(x),0)$ satisfying\n\\begin{equation}\\label{p}\n\\int_{\\rho_{\\infty}}^{\\rho^*(x)}\\f{p'(s)}{s}ds+\\phi(x)=0.\n\\end{equation}\nThe details of derivation for the stationary solution can be found in \\cite{mat1983}.\nFirst, Matsumura and Nishida \\cite{mat1983} obatined the global existence of solutions to system \\eqref{ns3} near the steady state $(\\rho^*(x),0)$ with initial perturbation \nunder the $H^3-$framework.\nIn addition, they also showed that the global solution \nconverges to the stationary state as time tends to infinity.\nThe first work to give explicit decay estimate for solution was \nrepresented by Deckelnick \\cite{Deckelnick1992}.\nSpecifically, Deckelnick was concerned about the isentropic case and showed that\n\\begin{equation*}\n\\sup_{x\\in\\mathbb{R}^3}|(\\rho(t,x)-\\rho^*(x),u(t,x))|\\leq C(1+t)^{-\\f14}.\n\\end{equation*}\nThis was then improved by Shibata and Tanaka for more general external forces in \\cite{Shibata2003,Shibata2007} to $(1+t)^{-\\f12+\\kappa}$ for any small positive constant $\\kappa$ when the initial perturbation belongs to $H^3\\cap L^{\\f65}$.\nLater, Duan, Liu, Ukai and Yang \\cite{duan2007} investigated the optimal $L^p-L^q$ convergence rates for this system when the initial perturbation is also bounded in $L^p$ with $1\\leq p<\\f65$. Specifically, they established the decay rate as follows:\n\\begin{equation}\\label{highdecayL1}\n\\|(\\rho-\\rho^*,u)(t)\\|_{L^2}\\leq  C(1+t)^{-\\frac32(\\frac1p-\\frac12)},\n\\quad\n\\|\\nabla^k(\\rho-\\rho^*,u)(t)\\|_{L^2}\n\\leq C(1+t)^{-\\frac32(\\frac1p-\\frac12)-\\frac{k}{2}}, ~\\text{for}~~k=1,2,3.\n\\end{equation}\nFor more result about the decay estimate for the CNS equations with potential force, \none may refer to \\cite{{Ukai2006},{duan-ukai-yang-zhao2007},{Okita2014},{Li2011},\n{Wang2017},{Matsumura1992},{Matsumura2001}}.\nObviously, the decay rates of the second and third order spatial derivatives \nin \\eqref{highdecayL1} are only the same as the first one.\n\\textbf{\\textit{In this paper, our second target is to investigate the optimal decay rate for\nthe $k-th$ $(k\\geq2)$ order spatial derivative of solution to the \nCNS equations with potential force.}}\n\nFinally, we aim to investigate the lower bounds of decay rates for the solution\nitself and its spatial derivatives.\nThe decay rate is called optimal in the sense that this rate \ncoincides with the linearized part.\nThus, the study of the lower decay rate, which is the same as the upper one, can\nhelp us obtain the optimal decay rate of solution.\nAlong this direction, Schonbek addressed the lower bound of decay rate for solution\nof the classical incompressible Navier-Stokes equations \\cite{Schonbek1986,Schonbek1991}\n(see also MHD equations \\cite{Schonbek-Schonbek-Suli}).\nBased on so-called Gevrey estimates, Oliver and Titi \\cite{Oliver2000} established\nthe lower and upper bounds of decay rate for the higher order derivatives\nof solution to the incompressible Navier-Stokes equations in whole space.\nFor the case of compressible flow, there are many results of lower bound of\ndecay rate for the solution itself to the CNS equations and related\nmodels, such as the CNS equations \\cite{lizhang2010,Kagei2002},\ncompressible Navier-Stokes-Poisson equations \\cite{limatsumura2010,Zhang2011},\nand compressible viscoelastic flows \\cite{Hu-Wu-2013}.\nHowever, these lower bounds mentioned above only consider the solution itself\nand do not involve the derivative of the solution.\nRecently, some scholars are devoted to studying the lower bound\nof decay rate for the derivative of solution,\nwhich can be referred to \\cite{{chen2021},{wu2020},{gao-lyu-Yao-2019},{gao-wei-yao-pre}}.\n\\textbf{\\textit{Thus, our third target is to establish lower bound of decay rate for\nthe global solution itself and its spatial derivatives.}}\nThese lower bounds of decay estimates, which coincide with the upper ones,\nshow that they are really optimal.\n\nNow, our second result can be stated as follows.\n\n\\begin{theo}\\label{them3}\n\tLet $(\\rho^*(x),0)$ be the stationary solution of initial value problem \\eqref{ns3}--\\eqref{initial-boundary}, if $(\\rho_0-\\rho^*,u_0)\\in H^N$ for $N\\geq3$, there exists a constant $\\delta$ such that the potential function $\\phi(x)$ satisfies\n\t\\begin{equation}\\label{phik}\n\t\\sum_{k=0}^{N+1}\\|(1+|x|)^{k}\\nabla^k\\phi\\|_{L^2\\cap L^\\infty}\\leq \\delta,\n\t\\end{equation}\n\tand the initial perturbation statisfies\n\t\\begin{equation}\\label{initial-H2}\n\t\\|(\\rho_0-\\rho^*,u_0)\\|_{H^{N}}\\leq \\delta.\n\t\\end{equation}\n\tThen there exists a unique global solution $(\\rho,u)$ of initial value problem \\eqref{ns3}--\\eqref{initial-boundary} satisfying\n\\begin{equation}\\label{energy-thm}\n\\begin{split}\n\\|(\\rho-\\rho^*,u)(t)\\|_{H^N}^2\n+\\int_0^t\\big(\\|\\nabla(\\rho-\\rho^*)\\|_{H^{N-1}}^2+\\|\\nabla u\\|_{H^N}^2\\big)ds\n\\leq C\\|(\\rho_0-\\rho^*,u_0)\\|_{H^N}^2 ,\\quad t\\geq0,\n\\end{split}\n\\end{equation}\nwhere $C$ is a positive constant independent of time $t$.\nIf further\n\\begin{equation*}\n\\|(\\rho_0-\\rho^*,u_0)\\|_{L^1}<\\infty,\n\\end{equation*}\nthen there exists constans $\\delta_0>0$ and $\\bar C_0>0$ such that for any $0<\\delta\\leq\\delta_0$, we have\n\\begin{equation}\\label{kdecay}\n\\|\\nabla^k(\\rho-\\rho^*)(t)\\|_{L^2}+\\|\\nabla^k u(t)\\|_{L^2}\n\\leq \\bar C_0(1+t)^{-\\f34-\\f k2},\\quad\\text{for}~~0\\leq k \\leq N.\n\\end{equation}\n\\end{theo}\n\n\\begin{rema}\nThe global well-posedness theory of the CNS equations with potential force\nin three-dimensional whole space was studied in \\cite{duan2007} under the $H^3-$ framework.\nFurthermore, they also established the decay estimate \\eqref{highdecayL1} if\nthe initial data belongs to $L^p$ with $1\\le p < \\frac65$.\nThus, the advantage of the decay rate \\eqref{kdecay} in Theorem \\ref{them3} is that\nthe decay rate of the global solution $(\\rho-\\rho^*,u)$ itself and\nits any order spatial derivative is optimal.\n\\end{rema}\n\n\nFinally, we have the following result concerning the lower bounds of decay rates for solution and its spatial derivatives of the CNS equations with potential force.\n\\begin{theo}\\label{them4}\nSuppose the assumptions of Theorem \\ref{them3} hold on. Furthermore, we assume that $\\int_{\\R^3}(\\rho_0-\\rho^*)(x) d x$ and $\\int_{\\R^3}u_0(x) d x$ are at least one nonzero.\nThen, the global solution $(\\rho,u)$ obtained in Theorem \\ref{them4} has the decay rates for any large enough $t$,\n\\begin{equation}\n\\begin{aligned}\\label{kdecaylow}\n&{c_0}(1+t)^{-\\f34-\\f k2}\\le \\|\\nabla^k(\\rho-\\rho^*)(t)\\|_{L^2}\\le {c_1}(1+t)^{-\\f34-\\f k2};\\\\\n&{c_0}(1+t)^{-\\f34-\\f k2}\\le \\|\\nabla^ku(t)\\|_{L^2}\\le {c_1}(1+t)^{-\\f34-\\f k2};\n\\end{aligned}\n\\end{equation}\nfor all $0\\leq k\\leq N$.\nHere $c_0$ and $c_1$ are positive constants independent of time $t$.\n\\end{theo}\n\n\\begin{rema}\n\tThe decay rates showed in \\eqref{kdecay} and \\eqref{kdecaylow} imply that the $k-th$ $(0\\leq k\\leq N)$ order spatial derivative of the solution converges to the equilibrium state $(\\rho^*, 0)$ at the $L^2-$rate $(1+t)^{-\\f34-\\f k2}$. In other words, these decay rates of the solution itself and its spatial derivatives obtained in \\eqref{kdecay} and \\eqref{kdecaylow} are optimal.\n\\end{rema}\n\n\\textbf{Notation:} Throughout this paper, for $1\\leq p\\leq +\\infty$ and $s\\in\\mathbb{R}$, we simply denote $L^p(\\mathbb{R}^3)$ and $H^s(\\mathbb{R}^3)$ by $L^p$  and $H^s$, respectively.\nAnd the constant $C$ denotes a general constant which may vary in different estimates.\n$\\widehat{f}(\\xi)=\\mathcal F(f(x))$ represents the usual Fourier transform of the function $f(x)$ with respect to $x\\in\\mathbb{R}^3$. $\\mathcal F^{-1}(\\widehat{f}(\\xi))$ means the inverse Fourier transform of $\\widehat{f}(\\xi)$ with respect to $\\xi\\in\\mathbb{R}^3$. For the sake of simplicity, we write $\\int f dx:=\\int _{\\mathbb{R}^3} f dx$.\n\n\nThe rest of the paper is organized as follows. In Section \\ref{approach}, we introduce the difficulties and our approach to prove the results. In Section \\ref{pre}, we recall some important lemmas, which will be used in later analysis. And Section \\ref{h-s} is denoted to giving the proof of Theorem \\ref{hs2}. Finally, Theorem \\ref{them3} and Theorem \\ref{them4} are proved in Section \\ref{rhox}.\n\n\n\\section{Difficulties and outline of our approach}\\label{approach}\nThe main goal of this section is to explain the main difficulties of\nproving Theorems \\ref{hs2}, \\ref{them3} and \\ref{them4} as well as our\nstrategies for overcoming them.\nIn order to establish optimal decay estimate for the CNS equations,\nthe main difficulty comes from the system \\eqref{ns1} or \\eqref{ns3}\nsatisfying hyperbolic-parabolic coupling equations, such that\nthe density only can obtain lower dissipation estimate.\n\nFirst of all, let us introduce our strategy to prove the Theorem \\ref{hs2}.\nIndeed, applying the classical energy estimate, it is easy\nto establish following estimate:\n\\begin{equation}\\label{energy-N-1}\n\\begin{split}\n\\f{d}{dt}\\|\\nabla^N(n,u)\\|_{L^2}^2+\\|\\nabla^{N+1}u\\|_{L^2}^2\n\\leq \\|\\nabla (n, u)\\|_{H^1}^2 \\|\\nabla^N(n, u)\\|_{L^2}^2\n+\\text{some~good~terms}.\n\\end{split}\n\\end{equation}\nIn order to control the first term on the right handside of \\eqref{energy-N-1},\nthe idea in \\cite{guo2012} is to establish the dissipative for the density $\\nabla^N n$,\nwhich gives rise to the cross term $\\frac{d}{dt}\\int \\nabla^{N-1} u\\cdot\\nabla^{N}n dx$\nin energy part.\nThis is the reason why the decay rate of the $N-th$ order spatial derivative of\nsolution of the CNS equations can only attain the decay rate as the $(N-1)-th$ one.\nIn order to settle this problem, our strategy is to apply the time integrability\nof the dissipative term of density rather than  absorbing it by the dissipative term.\nMore precisely, applying the weighted energy method to the estimate\n\\eqref{energy-N-1} and using decay \\eqref{sde1}, it holds true\n\\begin{equation}\\label{energy-N-2}\n\\begin{split}\n&(1+t)^{N+\\s+\\ep_0}\\|\\nabla^N (n,u)\\|_{L^2}^2\n+\\int_{0}^{t}(1+\\tau)^{N+\\s+\\ep_0}\\|\\nabla^{N+1}u\\|_{L^2}^2d\\tau \\\\\n\\leq& \\|\\nabla^N (n_0,u_0)\\|_{L^2}^2\n+\\int_0^t (1+\\tau )^{N-1+\\s+\\ep_0}\\|\\nabla^N (n,u)\\|_{L^2}^2 d\\tau\n+\\text{some~good~terms}.\n\\end{split}\n\\end{equation}\nThus, we need to control the second term on the right handside of \\eqref{energy-N-2}.\nOn the other hand, it is easy to check that\n\\begin{equation}\\label{energy-02}\n\\begin{split}\n\\f{d}{dt}\\mathcal{E}^{N-1}(t)\n+C(\\|\\nabla^{N}n\\|_{L^2}^2+\\|\\nabla^{N}u\\|_{H^1}^2)\\leq 0.\n\\end{split}\n\\end{equation}\nHere $\\mathcal{E}^{N-1}(t)$ is equivalent to $\\|\\nabla^{N-1}(n,u)\\|_{H^1}^2$.\nThe combination of \\eqref{energy-02} and decay estimate \\eqref{sde1} yields directly\n\\begin{equation}\\label{energy-N-3}\n(1+t)^{N-1+\\s+\\ep_0}\\mathcal{E}^{N-1}(t)\n+\\int_0^t(1+\\tau)^{N-1+\\s+\\ep_0}\n\\big(\\|\\nabla^N n\\|_{L^2}^2+\\|\\nabla^N u\\|_{H^1}^2\\big)d\\tau\n\\leq C(1+t)^{\\ep_0}.\n\\end{equation}\nThus, we apply the time integrability of the dissipative term of density in \\eqref{energy-N-3}\nto control the second term on the right handside of \\eqref{energy-N-2}.\nTherefore, we can obtain the optimal decay rate for $\\nabla^N(n, u)$ as it converges to zero.\n\nSecondly, we will establish the optimal decay rate, including in Theorem \\ref{them3},\nfor the higher order spatial derivative of global solution to the CNS equations\nwith external potential force.\nDue to the influence of potential force, the equilibrium state of global solution will\ndepend on the spatial variable. This will create some fundamental difficulties\nas we establish the energy estimates, see Lemmas \\ref{enn-1}, \\ref{enn} and \\ref{ennjc}.\nSimilar to the decay estimate \\eqref{highdecayL1}(cf.\\cite{duan2007}),\none can combine the energy estimate and the decay rate of linearized system\nto obtain the following decay estimates:\n\\begin{equation}\\label{basic-decay-d}\n\\|\\nabla^k (\\rho-\\rho^*)(t)\\|_{H^{N-k}}+\\|\\nabla^k u(t)\\|_{H^{N-k}}\n\\le C(1+t)^{-(\\frac34+\\frac{k}{2})},\\quad k=0,1,\n\\end{equation}\nif the initial data $(\\rho_0-\\rho^*, u_0)$ belongs to $H^N \\cap L^1$.\nTo prove that this decay is true for $k\\in\\{2,\\cdots,N-1\\}$, we are going to do it\nby mathematical induction.\nThus, assume that decay \\eqref{basic-decay-d} holds on for $k=l\\in\\{1,\\cdots,N-2\\}$,\nour target is to prove the validity of \\eqref{basic-decay-d} as $k=l+1$.\nThis logical relationship can be guaranteed by using the classical Fourier splitting\nmethod(cf\\cite{gao2016}). However, similar to the method of the proof of Theorem \\ref{hs2},\nwe guarantee this logical relationship by using the time weighed method,\nsee Lemma \\ref{N-1decay} more specifically.\nSince the presence of potential force term $\\rho \\nabla \\phi$, we can not\napply the time weighted method mentioned above to establish the optimal decay\nrate for the $N-th$ order spatial derivative of global solution.\nMotivated by \\cite{wu2020}, we establish some energy estimate for the quantity\n$\\int_{|\\xi|\\geq\\eta}\\widehat{\\nabla^{N-1}v}\\cdot \\overline{\\widehat{\\nabla^{N}n}}d\\xi$,\nnamely the higher frequency part, rather than $\\int \\nabla^{N-1} u\\cdot\\nabla^{N}n dx$.\nHere $\\widehat{\\nabla^{N-1}v}$ and $\\widehat{\\nabla^{N}n}$ stand for the Fourier part of\n$\\nabla^{N-1}v$ and $\\nabla^{N}n$ respectively.\nThe advantage is that the quantity $\\|\\nabla^{N}(n,v)\\|_{L^2}^2-\\eta_3\\int_{|\\xi|\\geq\\eta}\\widehat{\\nabla^{N-1}v}\\cdot \\overline{\\widehat{\\nabla^{N}n}}d\\xi$\nis equivalent to $\\|\\nabla^{N}(n, v)\\|_{L^2}^2$.\nThen, the combination of some energy estimate and decay estimate can help us build\nthe following inequality:\n\\begin{equation}\\label{highesthigh}\n\\begin{split}\t&\\f{d}{dt}\\Big\\{\\|\\nabla^{N}(n,v)\\|_{L^2}^2-\\eta_3\\int_{|\\xi|\\geq\\eta}\\widehat{\\nabla^{N-1}v}\\cdot \\overline{\\widehat{\\nabla^{N}n}}d\\xi \\Big\\}+\\|\\nabla^{N}v^h\\|_{L^2}^2+\\eta_3\\|\\nabla^{N}n^h\\|_{L^2}^2\\\\\n\t\\leq&  C\\|\\nabla^{N}(n^l, v^l)\\|_{L^2}^2+C(1+t)^{-3-N}.\n\\end{split}\n\\end{equation}\nThen, one has to estimate the decay rate of the low-frequency term\n$\\|\\nabla^{N}(n^l, v^l)\\|_{L^2}^2$.\nIndeed, Duhamel's principle and decay estimate of $k-th$ $(0\\leq k\\leq N)$ order spatial derivative of solution obtained above allow us to obtain that\\begin{equation*}\n\\|\\nabla^N(n^l, v^l)\\|_{L^2}\\leq C\\delta\\sup_{0\\leq\\tau\\leq t}\\|\\nabla^N (n,v)\\|_{L^2}+C(1+t)^{-\\f34-\\f N2}.\n\\end{equation*}\nwhich, together with \\eqref{highesthigh}, by using the smallness of $\\d$,\nwe can obtain the optimal decay rate for the $N-th$ order spatial derivative\nof global solution to the CNS equations with external potential force.\n\nFinally, we will establish the lower decay estimate, coincided with the upper one,\nfor the global solution itself and its spatial derivative.\nIt is noticed that the system in terms of density and momentum is always adopted to establish the lower bounds of decay rates for the global solution and its spatial derivatives in many previous works, one may refer to (\\cite{chen2021,lizhang2010}).\nHowever, the appearance of potential force term(i.e., $\\rho \\nabla \\phi$)\nprevents us taking this method to solve the problem.\nThus, let $(n, v)$ and $(\\tilde{n}, \\tilde{v})$ be the solutions of nonlinear\nand linearized problem respectively.\nDefine the difference: $(n_\\d, v_\\d)\\overset{def}{=}(n-\\widetilde{n}, v-\\widetilde{v})$,\nit holds on\n\\begin{equation*}\n\\|n\\|_{L^2}\\ge \\|\\widetilde{n}\\|_{L^2}-\\|{n}_\\d\\|_{L^2}\n\\quad \\text{and} \\quad\n\\|v\\|_{L^2} \\ge \\|\\widetilde{v}\\|_{L^2}-\\|{v}_\\d\\|_{L^2}.\n\\end{equation*}\nIf these quantities obey the assumptions:\n$\\|(n_\\d,v_\\d)\\|_{L^2}\\leq \\widetilde C\\d(1+t)^{-\\f34}$\nand\n$\\min\\{\\|\\widetilde{n}\\|_{L^2},\\|\\widetilde{v}\\|_{L^2}\\}\\geq \\widetilde{c}(1+t)^{-\\f34}$,\nmoreover, the constant $\\d$ is a small constant and independent of $\\widetilde{c}$,\nthen we can choose the constant $\\d$ to obtain the decay rate\n$\\min\\{\\|{n}\\|_{L^2},\\|{v}\\|_{L^2}\\}\\geq {c_1}(1+t)^{-\\f34}$.\nSimilarly, it is easy to check that the decay \\eqref{kdecaylow} holds true for $k=1$.\nBased on the lower bound of decay rate for first order spatial derivative of solution and the upper bound of decay rate for the solution itself, we can deduce the lower bound of decay rate for $k-th$ $(2\\leq k\\leq N)$ order spatial derivative of solution by using the following Sobolev interpolation inequality:\\begin{equation*}\n\\|\\nabla^k(n,v)\\|_{L^2}\\geq C\\|\\nabla(n,v)\\|_{L^2}^k\\|(n,v)\\|_{L^2}^{-(k-1)},\\quad\\text{for}~~2\\leq k\\leq N.\n\\end{equation*}\nAnd more proof details of Theorem \\ref{them4} can be found in Section \\ref{lower} below.\n\n\n\\section{Preliminary}\\label{pre}\nIn this section, we collect some elementary inequalities, which will be extensively used in later sections.\nFirst of all, in order to estimate the term about $\\bar\\rho(x)$ in the CNS equations with a potential force, we need the following Hardy inequality.\n\\begin{lemm}[Hardy inequality]\\label{hardy}\n\tFor $k\\geq1$, suppose that $\\f{\\nabla\\phi}{(1+|x|)^{k-1}}\\in L^2$, then $\\f{\\phi}{(1+|x|)^{k}}\\in L^2$, with the estimate\n\t\\begin{equation*}\n\t\\begin{split}\n\t\t\\|\\f{\\phi}{(1+|x|)^{k}}\\|_{L^2}\\leq C\\|\\f{\\nabla\\phi}{(1+|x|)^{k-1}}\\|_{L^2}.\n\t\\end{split}\n\t\\end{equation*}\n\\end{lemm}\nThe proof of Lemma \\ref{hardy} is simply and we omit it here. We will use the following Sobolev interpolation of Gagliardo-Nirenberg inequality frequently in energy estimates, which can be found in \\cite{guo2012} more details.\n\\begin{lemm}[Sobolev interpolation inequality]\\label{inter}\n\tLet $2\\leq p\\leq +\\infty$ and $0\\leq l,k\\leq m$. If $p=+\\infty$, we require furthermore that $l\\leq k+1$ and $m\\geq k+2$. Then if $\\nabla^l\\phi\\in L^2$ and $\\nabla^m \\phi\\in L^2$, we have $\\nabla^k\\phi\\in L^p$. Moreover, there exists a positive constant $C$ dependent only on $k,l,m,p$ such that\n\t\\begin{equation}\\label{Sobolev}\n\t\\|\\nabla^k\\phi\\|_{L^p}\\leq C\\|\\nabla^l\\phi\\|_{L^2}^{\\theta}\\|\\nabla^m\\phi\\|_{L^2}^{1-\\theta},\n\t\\end{equation}\n\twhere $0\\leq\\theta\\leq1$ satisfying\n\t\\begin{equation*}\n\t\\f k3-\\f1p=\\Big(\\f l3-\\f12\\Big)\\theta+\\Big(\\f m3-\\f12\\Big)(1-\\theta).\n\t\\end{equation*}\n\\end{lemm}\n\nThen we recall the following commutator estimate, which is used frequently in energy estimates. The proof and more details may refer to \\cite{majda2002}.\n\\begin{lemm}\\label{commutator}\n\tLet $k\\geq1$ be an integer and define the commutator\\begin{equation*}\n       [\\nabla^k,f]g=\\nabla^k(fg)-f\\nabla^kg.\n    \\end{equation*}\n    Then we have\n    \\begin{equation*}\n    \\|[\\nabla^k,f]g\\|_{L^2}\\leq C\\|\\nabla f\\|_{L^\\infty}\\|\\nabla^{k-1}g\\|_{L^2}+C\\|\\nabla^k f\\|_{L^2}\\|g\\|_{L^\\infty},\n    \\end{equation*}\n    where $C$ is a positive constant dependent only on $k$.\n\\end{lemm}\n\nFinally, we conclude this section with the following lemma. The proof and more details may refer to \\cite{chen2021}.\n\\begin{lemm}\\label{tt2}\nLet $r_1,r_2>0$ be two real numbers, for any $0<\\ep_0<1$, we have\n\\begin{equation*}\n\\begin{split}\n\t\\int_0^{\\f t2}(1+t-\\tau)^{-r_1}(1+\\tau)^{-r_2}d\\tau\\leq C& \\left\\{\\begin{array}{l}\n\t\t(1+t)^{-r_1},\\quad \\text{for}~~ r_2>1,\\\\\n\t\t(1+t)^{-r_1+\\ep_0},\\quad~~ \\text{for}~~ r_2=1, \\\\\n\t\t(1+t)^{-(r_1+r_2-1)},\\quad \\text{for}~~ r_2<1,\n\t\\end{array}\\right.\\\\\n\t\\int_{\\f t2}^{t}(1+t-\\tau)^{-r_1}(1+\\tau)^{-r_2}d\\tau\\leq C& \\left\\{\\begin{array}{l}\n\t\t(1+t)^{-r_2},\\quad \\text{for}~~ r_1>1,\\\\\n\t\t(1+t)^{-r_2+\\ep_0},\\quad~~ \\text{for}~~ r_1=1, \\\\\n\t\t(1+t)^{-(r_1+r_2-1)},\\quad \\text{for}~~ r_1<1,\n\t\\end{array}\\right.\n\\end{split}\n\\end{equation*}\nwhere $C$ is a positive constant independent of $t$.\n\\end{lemm}\n\n\\section{The proof of Theorem \\ref{hs2}}\\label{h-s}\n\nIn this section, we study the optimal decay rate of the $N-th$\nspatial derivative of global small solution for the initial value problem \\eqref{ns1}--\\eqref{boundary1}.\nThus, let us write $n\\overset{def}{=} \\rho-\\bar\\rho$, then the original system \\eqref{ns1}--\\eqref{initial1} can be rewritten in the perturbation form as\n\\begin{equation}\\label{ns2}\n\\left\\{\\begin{array}{lr}\n\tn_t +\\bar\\rho\\mathop{\\rm div}\\nolimits u=S_1,\\quad (t,x)\\in \\mathbb{R}^{+}\\times \\mathbb{R}^3,\\\\\n\tu_t+\\gamma \\bar\\rho\\nabla n-\\bar\\mu\\tri u-(\\bar\\mu+\\bar\\lam) \\nabla \\mathop{\\rm div}\\nolimits u=S_2,\\quad (t,x)\\in \\mathbb{R}^{+}\\times \\mathbb{R}^3,\\\\\n\t(n,u)|_{t=0}=(\\rho_0-\\bar\\rho,u_0),\\quad x\\in \\mathbb{R}^3,\n\\end{array}\\right.\n\\end{equation}\nwhere the functions $f(n), g(n)$ and source terms $S_i(i=1,2)$ are defined by\n\\begin{equation*}\\label{fgdefine}\n\\begin{split}\n    &f(n)\\overset{def}{=}\\f{n}{n+\\bar\\rho},\\quad \\quad\n    g(n)\\overset{def}{=}\\f{p'(n+\\bar\\rho)}{n+\\bar\\rho}-\\f{p'(\\bar\\rho)}{\\bar\\rho},\\\\\n\t&S_1 \\overset{def}{=}-n\\mathop{\\rm div}\\nolimits u-u\\cdot\\nabla n,\\\\\n\t&S_2 \\overset{def}{=} -u\\cdot \\nabla u\n         -f(n)\\big(\\bar\\mu\\tri u-(\\bar\\mu+\\bar\\lam) \\nabla \\mathop{\\rm div}\\nolimits u\\big)-g(n)\\nabla n.\n\\end{split}\n\\end{equation*}\nHere the coefficients $\\bar\\mu, \\bar\\lam$ and $\\gamma$\nare defined by\n$\\bar\\mu=\\f{\\mu}{\\bar\\rho}, \\\n\\bar\\lam=\\f{\\lam}{\\bar\\rho}, \\\n\\gamma=\\f{p'(\\bar\\rho)}{\\bar\\rho^2}.$\nDue the the uniform estimate \\eqref{energy-01}, then there exists a positive\nconstant $C$ such that for any $1\\leq k\\leq N$,\n\\begin{equation*}\n\\begin{split}\n\t|f(n)|\\leq C|n|,\\quad |g(n)|\\leq C|n|,\\quad\n\t|f^{(k)}(n)|\\leq C,\\quad |g^{(k)}(n)|\\leq C.\n\\end{split}\n\\end{equation*}\nNext, in order to estimate the $L^2-$norm of the spatial derivatives of $f(n)$ and $g(n)$, we shall record the following lemma, which will be used frequently in later estimate.\n\\begin{lemm}\\label{hrholem}\n\tUnder the assumptions of Theorem \\ref{hs2}, $f(n)$ and $g(n)$ are two functions of $n$ defined by \\eqref{fgdefine}, then for any integer $1\\leq m\\leq N-1$, it holds true\n\t\\begin{equation}\\label{hrho}\n\t\\|\\nabla^mf(n)\\|_{L^2}^2+\\|\\nabla^mg(n)\\|_{L^2}^2\\leq C(1+t)^{-(m+s)},\n\t\\end{equation}\n\twhere $C$ is a positive constant independent of time.\n\\end{lemm}\n\\begin{proof}\n\tWe only control the first term on the left handside of \\eqref{hrho}, and the other one can be controlled similarly.\n\tNotice that for $m\\geq 1$,\n\t\\[\\nabla^mf(n)=\\text{a sum of products}~~f^{\\gamma_1,\\gamma_2,\\cdots,\\gamma_j}(n)\\nabla^{\\gamma_1}n\\cdots\\nabla^{\\gamma_j}n\\]\n\twith the functions $f^{\\gamma_1,\\gamma_2,\\cdots,\\gamma_j}(n)$ are some derivatives of $f(n)$ and $1\\leq \\gamma_{i}\\leq m$, $i=1,2,\\cdots,j$, $\\gamma_1+\\gamma_2+\\cdots+\\gamma_j=m$, $j\\geq 1$. Without loss of generality, we assume that $1\\leq \\gamma_1\\leq\\gamma_2\\leq\\cdots\\leq\\gamma_j\\leq m$. Thus, if $j\\geq2$, we have $\\gamma_{j-1}\\leq m-1\\leq N-2$.\n\tIt follows from the decay estimate \\eqref{sde1} that for $1\\le m\\leq N-1$,\\begin{equation*}\n\t\\|\\nabla^mn\\|_{L^2}\\leq C(1+t)^{-(m+s)},\n\t\\end{equation*}\n\twhich, together with Sobolev inequality and the fact that $j\\geq1$, we deduce that\n\t\\begin{equation*}\n\t\\begin{split}\n\t&\\|f^{\\gamma_1,\\gamma_2,\\cdots,\\gamma_j}(n)\\nabla^{\\gamma_1}n\\cdots\\nabla^{\\gamma_j}n\\|_{L^2}\\\\\n\t\t\\leq & C \\|\\nabla^{\\gamma_1}n\\|_{L^\\infty}\\cdots\\|\\nabla^{\\gamma_{j-1}}n\\|_{L^\\infty}\\|\\nabla^{\\gamma_{j}}n\\|_{L^2}\\\\\n\t\t\\leq &C \\|\\nabla^{\\gamma_1+1}n\\|_{H^1}\\cdots\\|\\nabla^{\\gamma_{j-1}+1}n\\|_{H^1} \\|\\nabla^{\\gamma_j}n\\|_{L^2}\\\\\n\t\t\\leq&C(1+t)^{-\\f{m+j s+(j-1)}{2}}\n\t\t\\leq C(1+t)^{-\\f{m+ s}{2}}.\n\t\\end{split}\n\t\\end{equation*}\n   Consequently, the proof of this lemma is completed.\n\\end{proof}\nNow, based on lemma \\ref{hrholem}, we give the following lemma, which provides the time integrability of $N-th$ order spatial derivate of the solution $(n,u)$.\n\\begin{lemm}\\label{integrability}\nUnder the assumptions in Theorem \\ref{hs2},\nfor any fixed constant $0<\\ep_0<1$, we have\n\\begin{equation}\\label{Enc} (1+t)^{N+s-1}\\|\\nabla^{N-1}(n,u)\\|_{H^1}^2\n+(1+t)^{-\\ep_0}\\int_0^t(1+\\tau)^{N+s+\\ep_0-1}\n\\big(\\|\\nabla^N n\\|_{L^2}^2+\\|\\nabla^N u\\|_{H^1}^2\\big)d\\tau\n\\leq C,\n\\end{equation}\nwhere $C$ is a positive constant independent of $t$.\n\\end{lemm}\n\\begin{proof}\nFrom Lemmas 3.2, 3.3 and 3.4 in  \\cite{guo2012},\nthe following estimates hold on for all $0\\leq k\\leq N-1$,\n\\begin{align}\n\\label{in01} \\f{d}{dt}\\|\\nabla^k(n,u)\\|_{L^2}^2+\\|\\nabla^{k+1}u\\|_{L^2}^2\\leq\n& C\\delta_0 \\|\\nabla^{k+1}(n,u)\\|_{L^2}^2,\\\\\n\\label{in02} \\f{d}{dt}\\|\\nabla^{k+1}(n,u)\\|_{L^2}^2+\\|\\nabla^{k+2}u\\|_{L^2}^2\n\\leq& C\\delta_0 \\big(\\|\\nabla^{k+1}(n,u)\\|_{L^2}^2+\\|\\nabla^{k+2}u\\|_{L^2}^2 \\big),\\\\\n\\label{in03} \\f{d}{dt}\\int \\nabla^ku\\cdot\\nabla^{k+1}n dx+\\|\\nabla^{k+1}n\\|_{L^2}^2\n\\leq& C \\big( \\|\\nabla^{k+1}u\\|_{L^2}^2+\\|\\nabla^{k+2}u\\|_{L^2}^2\\big).\n\\end{align}\nwhere the constant $C$ is a positive constant independent of time.\nBased on the estimates \\eqref{in01}, \\eqref{in02} and \\eqref{in03},\nthen it holds for all $0\\leq k\\leq N-1$,\n\\begin{equation}\\label{e1}\n\\begin{split}\n\\f{d}{dt}\\big(\\|\\nabla^k(n,u)\\|_{H^1}^2\n+\\eta_1 \\int \\nabla^k u \\cdot\\nabla^{k+1}n dx\\big)\n+\\eta_1\\|\\nabla^{k+1}n\\|_{L^2}^2+\\|\\nabla^{k+1}u\\|_{H^1}^2\\leq 0.\n\\end{split}\n\\end{equation}\nHere $\\eta_1$ is a small positive constant.\nTaking $k=N-1$ in inequality \\eqref{e1}, then we have\n\\begin{equation}\\label{e2}\n\\f{d}{dt}\\mathcal{E}^{N-1}(t)\n+\\eta_1\\|\\nabla^{N}n\\|_{L^2}^2+\\|\\nabla^{N}u\\|_{H^1}^2\\leq 0,\n\\end{equation}\nwhere the energy $\\mathcal{E}^{N-1}(t)$ is defined by\n$$\n\\mathcal{E}^{N-1}(t)\\overset{def}{=}\n\\|\\nabla^{N-1}(n,u)(t)\\|_{H^1}^2+\\eta_1 \\int \\nabla^{N-1} u\\cdot\\nabla^{N}n dx.\n$$\nThen, due to the smallness of $\\eta_1$, we have the following equivalent relation\n\\begin{equation}\\label{xih}\nc_1\\|\\nabla^{N-1}(n, u)(t)\\|_{H^1}^2\n\\le \\mathcal{E}^{N-1} (t)\n\\le  c_2\\|\\nabla^{N-1}(n, u)(t)\\|_{H^1}^2,\n\\end{equation}\nwhere the constants $c_1$ and $c_2$ are independent of time.\nFor any fixed $\\ep_0(0<\\ep_0<1)$,\nmultiplying the inequality \\eqref{e2} by $(1+t)^{N+s+\\ep_0-1}$,\nit holds true\n\\begin{equation}\\label{Ek1}\n\\begin{split}\n\\f{d}{dt}\\big\\{(1+t)^{N+s+\\ep_0-1}\\mathcal{E}^{N-1}(t)\\big\\}\n          +(1+t)^{N+s+\\ep_0-1}\n          \\big(\\|\\nabla^{N}n\\|_{L^2}^2+\\|\\nabla^{N}u\\|_{H^1}^2\\big)\n\\leq C (1+t)^{N+s+\\ep_0-2}\\mathcal{E}^{N-1}(t).\n\\end{split}\n\\end{equation}\nThe decay estimate \\eqref{sde1} and equivalent relation\n\\eqref{xih} lead us to get that for $0<k\\leq N-1$,\n\\begin{equation*}\n(1+t)^{N+s+\\ep_0-2}\\mathcal{E}^{N-1}(t)\n\\leq C(1+t)^{N+s+\\ep_0-2}\\|\\nabla^{N-1}(n,u)\\|_{H^1}^2\n\\leq C(1+t)^{-1+\\ep_0},\n\\end{equation*}\n\twhich, together with inequality \\eqref{Ek1}, yields directly\n    \\begin{equation}\\label{Ek}\n\t\\f{d}{dt}\\big\\{(1+t)^{N+s+\\ep_0-1}\\mathcal{E}^{N-1}(t)\\big\\}\n          +(1+t)^{N+s+\\ep_0-1}\n          \\big(\\|\\nabla^{N}n\\|_{L^2}^2+\\|\\nabla^{N}u\\|_{H^1}^2\\big)\n    \\leq C(1+t)^{{-1+\\ep_0}}.\n\t\\end{equation}\n    Integrating the inequality \\eqref{Ek} over $[0, t]$, it holds true\n\t\\begin{equation}\n    \\begin{aligned}\n    &(1+t)^{N+s+\\ep_0-1}\\mathcal{E}^{N-1}(t)\n       +\\int_0^t(1+\\tau)^{N+s+\\ep_0-1}\\big(\\|\\nabla^N n\\|_{L^2}^2\n       +\\|\\nabla^{N}u\\|_{H^1}^2\\big)d\\tau\\\\\n    &\\leq \\mathcal{E}^{N-1}(0)+\\int_0^t(1+\\tau)^{-1+\\ep_0}d\\tau\\leq C(1+t)^{\\ep_0},\n    \\end{aligned}\n    \\end{equation}\n\twhich, together with the equivalent relation \\eqref{xih}, implies that\n\t\\begin{equation*}\n\t(1+t)^{N+s+\\ep_0-1}\\|\\nabla^k(n,u)\\|_{H^1}^2\n     +\\int_0^t(1+\\tau)^{N+s+\\ep_0-1}\n     \\big(\\|\\nabla^{N}n\\|_{L^2}^2+\\|\\nabla^{N}u\\|_{H^1}^2\\big)d\\tau\\leq C(1+t)^{\\ep_0}.\n\t\\end{equation*}\n\tConsequently, we thereupon obtain \\eqref{Enc}.\n\\end{proof}\nFinally, using the time integrability of the dissipative term of the $N-th$ order spatial derivative of $(n,u)$ obtained in Lemma \\ref{integrability}, we can establish the optimal decay of $N-th$ one.\n\\begin{lemm}\\label{ENN}\n\tUnder the assumptions in Theorem \\ref{hs2}, then the global solution\n    $(n, u)$ has the decay estimate\n\\begin{equation}\\label{Enn}\t\n(1+t)^{N+s}\\|\\nabla^N(n,u)\\|_{L^2}^2\n+(1+t)^{-\\ep_0}\\int_0^t(1+\\tau)^{N+s+\\ep_0}\\|\\nabla^{N+1}u\\|_{L^2}^2d\\tau\\leq C,\n\\end{equation}\nwhere $C$ is a positive constant independent of time.\n\\end{lemm}\n\\begin{proof}\nApplying differential operator $\\nabla^N$ to \\eqref{ns2}\nand multiplying the resulting equation by $\\nabla^N (n,u)$, it holds\n\\begin{equation}\\label{nes}\n\\begin{split}\n\t\\f{d}{dt}\\|\\nabla^N(n,u)\\|_{L^2}^2+\\|\\nabla^{N+1}u\\|_{L^2}^2\\leq C\\int \\nabla^N S_1\\cdot \\nabla^Nn dx+C\\int \\nabla^N S_2\\cdot \\nabla^Nu dx.\n\\end{split}\n\\end{equation}\nNext, we estimate two terms on the right handside of \\eqref{nes}.\nThe definition of $S_1$ leads us to get\n\\begin{equation}\\label{s11}\n\\int \\nabla^N S_1\\cdot \\nabla^Nn dx=-\\int \\nabla^N (n\\mathop{\\rm div}\\nolimits u+u\\cdot \\nabla n)\\cdot \\nabla^Nn dx.\n\\end{equation}\nInvoking Sobolev inequality, we deduce\n\\begin{equation}\\label{s12}\n\\begin{split}\n\t\\int \\nabla^N (n\\mathop{\\rm div}\\nolimits u)\\cdot \\nabla^Nn dx\n\t\\leq& C \\|\\nabla^N(n\\mathop{\\rm div}\\nolimits u)\\|_{L^2}\\|\\nabla^N n\\|_{L^2}\\\\\n\t\\leq&C\\big(\\|n\\|_{L^\\infty}\\|\\nabla^N\\mathop{\\rm div}\\nolimits u\\|_{L^2}+\\|\\mathop{\\rm div}\\nolimits u\\|_{L^\\infty}\\|\\nabla^Nn\\|_{L^2}\\big)\\|\\nabla^N n\\|_{L^2}\\\\\n\t\\leq&\\ep \\|\\nabla^{N+1} u\\|_{L^2}+C_{\\ep}\\|\\nabla n\\|_{H^1}^2\\|\\nabla^N n\\|_{L^2}^2+C\\|\\nabla^2u\\|_{H^1}\\|\\nabla^N n\\|_{L^2}^2\\\\\n\t\\leq &\\ep \\|\\nabla^{N+1}u\\|_{L^2}^2\n           +C_{\\ep}(1+t)^{-(1+s)}\\|\\nabla^N(n,u)\\|_{L^2}^2.\n\\end{split}\n\\end{equation}\nIntegrating by parts, then employing Sobolev inequality, Lemma \\ref{commutator}\nand decay estimate \\eqref{sde1}, we find that\n\\begin{equation}\\label{s13}\n\\begin{split}\n\t\\int \\nabla^N (u\\cdot \\nabla n)\\cdot \\nabla^Nn dx\n\t=&-\\int\\mathop{\\rm div}\\nolimits u|\\nabla^Nn|^2 dx+\\int\\big([\\nabla^{N},u]\\cdot\\nabla n\\big)\\nabla^N n dx\\\\\n\t\\leq&C \\big(\\|\\nabla u\\|_{L^\\infty}\\|\\nabla^{N} n\\|_{L^2}+\\|\\nabla^N u\\|_{L^2}\\|\\nabla n\\|_{L^\\infty}\\big)\\|\\nabla^N n\\|_{L^2}\\\\\n\t\\leq&C \\big(\\|\\nabla^2 u\\|_{H^1}\\|\\nabla^{N} n\\|_{L^2}+\\|\\nabla^N u\\|_{L^2}\\|\\nabla^{2} n\\|_{H^1}\\big)\\|\\nabla^N n\\|_{L^2}\\\\\n\t\\leq&C (1+t)^{-(1+\\frac{s}{2})}\\|\\nabla^N(n,u)\\|_{L^2}^2.\n\\end{split}\n\\end{equation}\nSubstituting estimates \\eqref{s12} and \\eqref{s13} into \\eqref{s11}, we have\n\\begin{equation}\\label{S1}\n\\begin{split}\n\t\\int \\nabla^N S_1\\cdot \\nabla^Nn dx\n    \\leq\\ep \\|\\nabla^{N+1}u\\|_{L^2}^2+C_{\\ep}(1+t)^{-1}\\|\\nabla^N(n,u)\\|_{L^2}^2.\n\\end{split}\n\\end{equation}\nNext integrating by parts, and using H{\\\"o}lder inequality, we see that\n\\begin{equation}\\label{S22}\n\\begin{split}\n\t\\int \\nabla^N S_2\\cdot \\nabla^Nu dx\\leq& C \\|\\nabla^{N-1} S_2\\|_{L^2}\\|\\nabla^{N+1}u\\|_{L^2}\\\\\n\t\\leq&C\\big(\\|\\nabla^{N-1}(u\\cdot \\nabla u)\\|_{L^2}+\\|\\nabla^{N-1}\\big[f(n)(\\bar\\mu\\tri u+(\\bar\\mu+\\bar \\lam)\\nabla \\mathop{\\rm div}\\nolimits u)\\big]\\|_{L^2}\\\\\n\t&\\quad+\\|\\nabla^{N-1}\\big(g(n)\\nabla n\\big)\\|_{L^2}\\big)\\|\\nabla^{N+1}u\\|_{L^2}\\overset{def}{=}H_{1}+H_{2}+H_{3}.\n\\end{split}\n\\end{equation}\nWe use Sobolev inequality, H{\\\"o}lder inequality  and Lemma \\ref{commutator}, to find\n\\begin{equation}\\label{unau}\n\\begin{split}\n\tH_1\n\t\\leq&C\\big(\\|u\\|_{L^\\infty}\\|\\nabla^{N} u\\|_{L^2}+\\|[\\nabla^{N-1},u]\\cdot\\nabla n\\|_{L^2}\\big)\\|\\nabla^{N+1}u\\|_{L^2}\\\\\n\t\\leq&C \\big(\\|u\\|_{L^\\infty}\\|\\nabla^{N} u\\|_{L^2}+\\|\\nabla u\\|_{L^\\infty}\\|\\nabla^{N-1} u\\|_{L^2}\\big)\\|\\nabla^{N+1}u\\|_{L^2}\\\\\n\t\\leq&\\ep\\|\\nabla^{N+1}u\\|_{L^2}^2+C_{\\ep}\\|\\nabla u\\|_{H^1}^2\\|\\nabla^{N} u\\|_{L^2}^2+C_{\\ep}\\|\\nabla^2 u\\|_{H^1}^2\\|\\nabla^{N-1} u\\|_{L^2}^2\\\\\n\t\\leq&\\ep\\|\\nabla^{N+1}u\\|_{L^2}^2+C_{\\ep}(1+t)^{-(1+s)}\\|\\nabla^{N} u\\|_{L^2}^2+C_{\\ep}(1+t)^{-{(N+1+2s)}},\n\\end{split}\n\\end{equation}\nwhere we have used the decay estimate \\eqref{sde1} in the last inequality.\nWe employ once again Sobolev inequality and decay \\eqref{sde1}, combined with \\eqref{hrho}, to deduce that\n\\begin{equation}\\label{fn}\n\\begin{split}\n\tH_{2}\n\t\\leq&C \\big(\\|f(n)\\|_{L^\\infty}\\|\\nabla^{N+1} u\\|_{L^2}+\\|\\nabla^{N-1} f(n)\\|_{L^2}\\|\\nabla^{2} u\\|_{L^\\infty}\\big)\\|\\nabla^{N+1}u\\|_{L^2}\\\\\n\t\\leq&C\\big(\\|\\nabla f(n)\\|_{H^1}\\|\\nabla^{N+1} u\\|_{L^2}+\\|\\nabla^{N-1} f(n)\\|_{L^2}\\|\\nabla^{3} u\\|_{L^2}^{\\f12}\\|\\nabla^{4} u\\|_{L^2}^{\\f12}\\big)\\|\\nabla^{N+1}u\\|_{L^2}\\\\\n\t\\leq\n&C\\delta_0\\|\\nabla^{N+1}u\\|_{L^2}^2\n  +C(1+t)^{-\\f{N-1+\\s}{2}}\\|\\nabla^{3} u\\|_{L^2}^{\\f12}\\|\\nabla^{4} u\\|_{L^2}^{\\f12}\\|\\nabla^{N+1}u\\|_{L^2}.\n\\end{split}\n\\end{equation}\nSince the initial data $(\\rho_0-\\bar\\rho,u_0)\\in H^{N}$ with integer $N\\geq 3$,\nthe second term on the right handside of inequality \\eqref{fn}\nshould be estimated in the following two cases.\nIf the integer $N=3$, we apply the Cauchy inequality to obtain\n\\begin{equation}\\label{fn-01}\n\\begin{split}\n\t\t(1+t)^{-\\f{3+s}{2}}\\|\\nabla^{3} u\\|_{L^2}^{\\f12}\\|\\nabla^{4} u\\|_{L^2}^{\\f32}\n\t\t\\leq& \\ep\\|\\nabla^4  u\\|_{L^2}^2+C_\\ep(1+t)^{-2(3+s)}\\|\\nabla^{3} u\\|_{L^2}^2.\n\\end{split}\n\\end{equation}\nIf the integer $N\\geq4$, then we apply  the decay \\eqref{sde1} to get\n\\begin{equation}\\label{fn-02}\n\\begin{split}\n(1+t)^{-\\f{N-1+s}{2}}\\|\\nabla^{3} u\\|_{L^2}^{\\f12}\n\\|\\nabla^{4} u\\|_{L^2}^{\\f12}\\|\\nabla^{N+1}u\\|_{L^2}\n\\leq \\ep\\|\\nabla^{N+1}u\\|_{L^2}^2+C_{\\ep}(1+t)^{-(N+2+2s)}.\n\\end{split}\n\\end{equation}\nSubstituting the estimates \\eqref{fn-01} and \\eqref{fn-02}\ninto \\eqref{fn}, we find that\n\\begin{equation}\\label{fnu}\n\\begin{split}\n\t&H_{2}\n\t\\leq (\\ep+C\\delta_0)\\|\\nabla^{N+1}u\\|_{L^2}^2\n    +C_{\\ep}(1+t)^{-1}\\|\\nabla^{N}u\\|_{L^2}^2+C_{\\ep}(1+t)^{-(N+2+2s)}.\n\\end{split}\n\\end{equation}\nLikewise, employing \\eqref{sde1} and \\eqref{hrho} once again, we calculate\n\\begin{equation}\\label{gnu}\n\\begin{split}\n\tH_{3}\n\t\\leq&C\\Big(\\|g(n)\\|_{L^\\infty}\\|\\nabla^{N} n\\|_{L^2}+\\|\\nabla^{N-1} g(n)\\|_{L^2}\\|\\nabla n\\|_{L^\\infty}\\Big)\\|\\nabla^{N+1}u\\|_{L^2}\\\\\n\t\\leq&\\ep\\|\\nabla^{N+1}u\\|_{L^2}^2+C_{\\ep}\\Big(\\|\\nabla g(n)\\|_{H^1}^2\\|\\nabla^{N} n\\|_{L^2}^2+\\|\\nabla^2 n\\|_{H^1}^2\\|\\nabla^{N-1} g(n)\\|_{L^2}^2\\Big)\\\\\n\t\\leq&\\ep\\|\\nabla^{N+1}u\\|_{L^2}^2+C_{\\ep}(1+t)^{-(1+\\s)}\\|\\nabla^{N}n\\|_{L^2}^2+C_{\\ep}(1+t)^{-{(N+1+2\\s)}}.\n\\end{split}\n\\end{equation}\nSubstituting the estimates \\eqref{unau}, \\eqref{fnu} and \\eqref{gnu} into \\eqref{S22}, then using Young inequality, we thereby obtain that\n\\begin{equation}\\label{S2}\n\\begin{split}\n\t\\int \\nabla^N S_2\\cdot \\nabla^Nu dx\\leq (\\ep+C\\delta_0)\\|\\nabla^{N+1}u\\|_{L^2}^2\n+C_{\\ep}(1+t)^{-1}\\|\\nabla^{N}(n,u)\\|_{L^2}^2\n+C_{\\ep}(1+t)^{-(N+1+s)}.\n\\end{split}\n\\end{equation}\nWe utilize the estimates \\eqref{S1} and \\eqref{S2} into \\eqref{nes}, and choose $\\ep$ and $\\delta_0$ suitably small, to discover that\n\\begin{equation*}\n\\begin{split}\n\t\\f{d}{dt}\\|\\nabla^N(n,u)\\|_{L^2}^2+\\|\\nabla^{N+1}u\\|_{L^2}^2\\leq C(1+t)^{-1}\\|\\nabla^{N}(n,u)\\|_{L^2}^2+C(1+t)^{-(N+1+s)}.\n\\end{split}\n\\end{equation*}\nMultiplying the above inequality by $(1+t)^{{N+s+\\ep_0}}$\nand integrating with respect to $t$, we find\n\\begin{equation}\\label{S21}\n\\begin{split} &(1+t)^{N+s+\\ep_0}\\|\\nabla^N(n,u)\\|_{L^2}^2\n+\\int_{0}^{t}(1+\\tau)^{N+s+\\ep_0}\\|\\nabla^{N+1}u\\|_{L^2}^2d\\tau\\\\\n\\leq &\\|\\nabla^N(n_0,u_0)\\|_{L^2}^2\n     +C\\int_0^t(1+\\tau)^{N+s-1+\\ep_0}\\|\\nabla^{N}(n,u)\\|_{L^2}^2d\\tau\n     +C\\int_0^t(1+\\tau)^{-1+\\ep_0}d\\tau\\\\\n\\leq&C(1+\\|\\nabla^N(n_0,u_0)\\|_{L^2}^2)+C(1+t)^{\\ep_0},\n\\end{split}\n\\end{equation}\nwhere we have used the decay estimate \\eqref{Enc}.\nThus, the estimate \\eqref{S21} yields the estimate \\eqref{Enn} directly.\nTherefore, we complete the proof of this lemma.\n\\end{proof}\n\n\\underline{\\noindent\\textbf{The proof of Theorem \\ref{hs2}.}}\nWith the help of estimate \\eqref{Enn} in Lemma \\ref{ENN}, it holds true\n$$\n\\|\\nabla^N n(t)\\|_{L^2}^2+\\|\\nabla^N u(t)\\|_{L^2}^2 \\leq C(1+t)^{-(s+N)},\n$$\nwhere $C$ is a positive constant independent of time.\nThus, we obtain the estimate \\eqref{sde3} immediately,\nand hence, finish the proof of Theorem \\ref{hs2}.\n\n\n\n\\section{The proof of Theorems \\ref{them3} and \\ref{them4}}\\label{rhox}\n\nIn this section, we will give the proof for the Theorem \\ref{them3}\nthat includes the global well-posedness theory and time decay estimate \\eqref{kdecay}.\nFirst of all, the global small solution of the CNS equations can be proven\njust by taking the strategy of energy method in \\cite{duan2007}\nwhen the initial data is small perturbation near the equilibrium state.\nThus, we assume that the global solution $(\\rho, u)$ in Theorem \\ref{them3}\nexists and satisfies the energy estimate \\eqref{energy-thm},i.e.,\n\\begin{equation}\\label{energy-thm-01}\n\\begin{split}\n\\|(\\rho-\\rho^*,u)\\|_{H^N}^2\n+\\int_0^t\\big(\\|\\nabla(\\rho-\\rho^*)\\|_{H^{N-1}}^2+\\|\\nabla u\\|_{H^N}^2\\big)ds\n\\leq C\\|(\\rho_0-\\rho^*,u_0)\\|_{H^N}^2,\n\\end{split}\n\\end{equation}\nfor all $t\\geq 0$.\nSecondly, similar to the decay estimate \\eqref{highdecayL1}, one can combine\nthe energy estimate and the decay rate of linearized system to obtain the following\ndecay estimates:\n\\begin{equation}\\label{basic-decay}\n\\|\\nabla^k (\\rho-\\rho^*)(t)\\|_{H^{N-k}}+\\|\\nabla^k u(t)\\|_{H^{N-k}}\n\\le C(1+t)^{-(\\frac34+\\frac{k}{2})},\\quad k=0,1,\n\\end{equation}\nif the initial data $(\\rho_0-\\rho^*, u_0)$ belongs to $L^1$ additionally.\nNow, we focus on establishing the optimal decay rate for the higher order spatial\nderivative of solution. In other words, we will prove that the decay rate\n\\eqref{basic-decay} holds on for the case $k=2,...,N$.\nThus, let us set\n$$n(x,t)\\overset{def}{=}\\rho(x,t)-\\rho^*(x),\\quad\n\\bar\\rho(x)\\overset{def}{=} \\rho^*(x)-\\rho_{\\infty}, \\quad\nv\\overset{def}{=}\\f{\\rho_{\\infty}}{\\sqrt{p'(\\rho_{\\infty})}}u,\n$$\nthen \\eqref{ns3}--\\eqref{initial-boundary} can be rewritten in the perturbation form\n\\begin{equation}\\label{ns5}\n\\left\\{\\begin{array}{lr}\n\tn_t +\\gamma\\mathop{\\rm div}\\nolimits v=\\widetilde S_1,\\quad (t,x)\\in \\mathbb{R}^{+}\\times \\mathbb{R}^3,\\\\\n\tv_t+\\gamma\\nabla n-\\mu_1\\tri v-\\mu_2\\nabla\\mathop{\\rm div}\\nolimits v =\\widetilde  S_2,\\quad (t,x)\\in \\mathbb{R}^{+}\\times \\mathbb{R}^3,\\\\\n\t(n,v)|_{t=0}\\overset{def}{=}(n_0,v_0)=(\\rho_0-\\rho^*,\\f{\\rho_{\\infty}}{\\sqrt{p'(\\rho_{\\infty})}}u_0)\\rightarrow (0,0)~~\\text{as}~~|x|\\rightarrow \\infty,\n\\end{array}\\right.\n\\end{equation}\nwhere\n$\\mu_1=\\f{\\mu}{\\rho_{\\infty}}, \\mu_2=\\f{\\mu+\\lam}{\\rho_{\\infty}},\n\\gamma=\\sqrt{p'(\\rho_{\\infty})}$, and\n\\begin{equation*}\n\\begin{split}\n\t\\widetilde S_1&=-\\f{\\mu_1\\gamma}{\\mu}\\mathop{\\rm div}\\nolimits [(n+\\bar\\rho)v],\\\\\n\t\\widetilde S_2&=-\\f{\\mu_1\\gamma}{\\mu}v\\cdot \\nabla v-\\wf(n+\\bar\\rho)\\big(\\mu_1\\tri v+\\mu_2\\nabla\\mathop{\\rm div}\\nolimits v\\big)-\\wg(n+\\bar\\rho) \\nabla n-\\h(n,\\bar\\rho)\\nabla\\bar\\rho.\n\\end{split}\n\\end{equation*}\nHere the nonlinear functions $\\wf, \\wg$ and $\\h$ are defined by\n\\begin{equation*}\n\\begin{split}\n&\\wf(n+\\bar\\rho)\\overset{def}{=}\\f{n+\\bar\\rho}{n+\\bar\\rho+\\rho_{\\infty}},\\quad \\wg(n+\\bar\\rho)\\overset{def}{=}\\f{\\mu}{\\mu_1\\gamma}\\Big(\\f{p'(n+\\bar\\rho+\\rho_{\\infty})}{n+\\bar\\rho+\\rho_{\\infty}}-\\f{p'(\\rho_{\\infty})}{\\rho_{\\infty}}\\Big),\\\\ &\\h(n,\\bar\\rho)\\overset{def}{=}\\f{\\mu}{\\mu_1\\gamma}\\Big(\\f{p'(n+\\bar\\rho+\\rho_{\\infty})}{n+\\bar\\rho+\\rho_{\\infty}}-\\f{p'(\\bar\\rho+\\rho_{\\infty})}{\\bar\\rho+\\rho_{\\infty}}\\Big).\n\\end{split}\n\\end{equation*}\nIn view of the definition of $\\h$ and $\\wg$, it holds true\n\\begin{equation}\\label{hg-relation}\n\\h(n,\\bar\\rho)=\\wg(n+\\bar\\rho)-\\wg(\\bar\\rho),\n\\end{equation}\nwhich will be used in this section.\n\n\\subsection{Energy estimates}\nIn this subsection, we will establish the following differential inequality that\nwill play important role for us to establish the optimal decay rate\nfor the higher order spatial derivative of solution.\nFirst of all, let us define the energy $\\mathcal{E}^N_l(t)$ as\n$$\n\\mathcal{E}^N_l(t)\\overset{def}{=}\n\\sum_{m=l}^{N}\\|\\nabla^m(n,v)\\|_{L^2}^2\n+\\eta_2\\sum_{m=l}^{N-1}\\int\\nabla^{m}v\\cdot\\nabla^{m+1}n dx, \\quad 0\\le l \\le N,\n$$\nwhere $\\eta_2$ is a small positive constant.\nDue to the smallness of parameter $\\eta_2$, we have the following equivalent realtion\n\\begin{equation}\\label{emleq}\nc_3 \\|\\nabla^l(n, v)\\|_{H^{N-l}}^2\n\\le \\mathcal{E}^N_l(t)\n\\le c_4 \\|\\nabla^l(n, v)\\|_{H^{N-l}}^2 .\n\\end{equation}\nwhere $c_3$ and $c_4$ are positive constant independent of time.\nFinally, the relation \\eqref{p} and the condition \\eqref{phik} in Theorem \\ref{them3}\nlead us to obtain\n\\begin{equation*}\n\\begin{split}\n\\sum_{k=0}^{N+1}\\|(1+|x|)^{k}\\nabla^k(\\rho^*-\\rho_{\\infty})\\|_{L^2\\cap L^\\infty}\\leq \\delta.\n\\end{split}\n\\end{equation*}\nThis, together with the Sobolev interpolation inequality, yields\n\\begin{equation}\\label{density-control}\n\\sum_{k=0}^{N+1}\\|(1+|x|)^{k}\\nabla^k(\\rho^*-\\rho_{\\infty})\\|_{L^p}\\leq \\delta,\n\\quad 2\\leq p\\leq+\\infty.\n\\end{equation}\nThis inequality will be used frequently in Section \\ref{rhox}.\nNow we state the main result in this subsection.\n\n\\begin{prop}\\label{nenp}\nUnder the assumptions in Theorem \\ref{them3}, for any $1\\leq l\\leq N-1$, we have\n\\begin{equation}\\label{eml}\n\\begin{split}\t\\f{d}{dt}\\mathcal{E}^N_l(t)+\\eta_2\\sum_{m=l}^{N-1}\\|\\nabla^{m+1}n\\|_{L^2}^2\n+\\sum_{m=l}^{N}\\|\\nabla^{m+1}v\\|_{L^2}^2\\leq 0.\n\\end{split}\n\\end{equation}\nHere $\\eta_2$ is a small positive constant.\n\\end{prop}\nDue to the energy estimate \\eqref{energy-thm-01},\nthere exists a positive constant $C$ such that for any $k\\geq 1$,\n\\begin{equation*}\n\\begin{split}\n\t|\\wf(n+\\bar\\rho)|\\leq C|n+\\bar\\rho|,\\quad |\\wg(n+\\bar\\rho)|\\leq C|n+\\bar\\rho|,\\quad\n\t|\\wf^{(k)}(n+\\bar\\rho)|\\leq C,\\quad |\\wg^{(k)}(n+\\bar\\rho)|\\leq C.\n\\end{split}\n\\end{equation*}\n\nNext we give three lemmas as follows, which is the key to prove Proposition \\ref{nenp}. The first one is the basic energy estimate for $k-th$ $(1\\leq k\\leq N-1)$ order spatial derivative of solution.\n\n\\begin{lemm}\\label{enn-1}\nUnder the assumptions in Theorem \\ref{them3}, for $1\\leq k\\leq N-1$, we have\n\\begin{equation}\\label{en1}\n\\f{d}{dt}\\|\\nabla^k(n,v)\\|_{L^2}^2+\\|\\nabla^{k+1}v\\|_{L^2}^2\n\\leq C\\delta \\|\\nabla^{k+1}(n,v)\\|_{L^2}^2,\n\\end{equation}\nwhere $C$ is a positive constant independent of time.\n\\end{lemm}\n\\begin{proof}\nApplying differential operator $\\nabla^k$ to $\\eqref{ns5}_1$\nand $\\eqref{ns5}_2$, multiplying the resulting equations by $\\nabla^kn$ and $\\nabla^kv$, respectively, then integrating over $\\mathbb{R}^3$, one easily shows that\\begin{equation}\\label{ehk}\n\t\\begin{split}\n\t\t\\f12\\f{d}{dt}\\|\\nabla^k(n,v)\\|_{L^2}^2+\\mu_1\\|\\nabla^{k+1} v\\|_{L^2}^2+\\mu_2\\|\\nabla^k\\mathop{\\rm div}\\nolimits v\\|_{L^2}^2\n\t\t= \\int \\nabla^k\\widetilde{S}_1\\cdot \\nabla^kn dx+\\int \\nabla^k\\widetilde{S}_2\\cdot \\nabla^kv dx.\n\t\\end{split}\n\t\\end{equation}\nIntegrating by part and using the definition of $\\widetilde{S}_1$, it holds\n\\begin{equation}\\label{ks1}\n\\begin{aligned}\n\\int \\nabla^k\\widetilde{S}_1\\cdot \\nabla^kn dx\n\\leq &C\\int \\nabla^{k-1}( v\\cdot\\nabla n )\\cdot \\nabla^{k+1}n dx\n     +C\\int \\nabla^{k-1}(n\\mathop{\\rm div}\\nolimits v )\\cdot \\nabla^{k+1}n dx\\\\\n    & +C\\int \\nabla^{k-1}(v\\cdot\\nabla\\bar\\rho )\\cdot \\nabla^{k+1}n dx\n     +C\\int \\nabla^{k-1}(\\bar\\rho \\mathop{\\rm div}\\nolimits v)\\cdot \\nabla^{k+1}n dx\\\\\n    \\overset{def}{=}&I_1+I_2+I_3+I_4.\n\\end{aligned}\n\\end{equation}\nIt follows from H{\\\"o}lder and Sobolev interpolation inequalities that\n\\begin{equation}\\label{I1}\n\\begin{split}\n|I_1| \\leq C\n\\sum_{0\\leq l\\leq k-1}\\|\\nabla^{k-1-l}v\\|_{L^3}\\|\\nabla^{l+1}n\\|_{L^6}\\|\\nabla^{k+1}n\\|_{L^2}.\n\\end{split}\n\\end{equation}\nThe Sobolev interpolation inequality \\eqref{Sobolev} in Lemma \\ref{inter} yields directly\n\\begin{equation}\\label{nanak}\n\\|\\nabla^{k-1-l}v\\|_{L^3}\\|\\nabla^{l+1}v\\|_{L^6}\n\\leq C\\|\\nabla^{{\\alpha}}v\\|_{L^2}^{\\f{l+2}{k+1}}\\|\\nabla^{k+1}v\\|_{L^2}^{1-\\f{l+2}{k+1}}\n     \\|v\\|_{L^2}^{1-\\f{l+2}{k+1}}\\|\\nabla^{k+1}v\\|_{L^2}^{\\f{l+2}{k+1}},\n\\end{equation}\nwhere\n$${\\alpha}=\\f{k+1}{2(l+2)}\\leq \\f{k+1}{4}\\leq \\f{N}{4}.$$\nSubstituting estimate \\eqref{nanak} into \\eqref{I1}, it holds true\n\\begin{equation}\\label{i1}\n\\begin{split}\n|I_1| \\leq C\\sum_{0\\leq l\\leq k-1}\n\\|\\nabla^{{\\alpha}}v\\|_{L^2}^{\\f{l+2}{k+1}}\\|n\\|_{L^2}^{1-\\f{l+2}{k+1}}\n\\|\\nabla^{k+1}(n,v)\\|_{L^2}^2\n\\leq C\\delta \\|\\nabla^{k+1}(n,v)\\|_{L^2}^2.\n\\end{split}\n\\end{equation}\nSimilarly, it is easy to check that\n\\begin{equation}\\label{i2}\n\\begin{split}\n|I_2| \\leq C\\sum_{0\\leq l\\leq k-1}\n\\|\\nabla^{k-1-l}n\\|_{L^3}\\|\\nabla^{l+1}v\\|_{L^6}\\|\\nabla^{k+1}n\\|_{L^2}\\leq C\\delta \\|\\nabla^{k+1}(n,v)\\|_{L^2}^2.\n\t\\end{split}\n\t\\end{equation}\nThe application of  H{\\\"o}lder inequality implies directly\n\\begin{equation}\\label{ii3}\n\\begin{split}\n|I_3|\\leq C\\sum_{0\\leq l\\leq k-1}\n         \\|(1+|x|)^{l+1}\\nabla^{l+l}\\bar\\rho\\|_{L^3}\n         \\|\\f{\\nabla^{k-1-l}v}{(1+|x|)^{l+1}}\\|_{L^6}\\|\\nabla^{k+1}n\\|_{L^2}.\n\\end{split}\n\\end{equation}\nSince\n\\begin{equation}\\label{vl6}\n\\|\\f{\\nabla^{k-1-l}v}{(1+|x|)^{l+1}}\\|_{L^6}\\leq C \\|\\nabla\\Big(\\f{\\nabla^{k-1-l}v}{(1+|x|)^{l+1}}\\Big)\\|_{L^2}\\leq C \\|\\f{\\nabla^{k-l}v}{(1+|x|)^{l+1}}\\|_{L^2}+C\\|\\f{\\nabla^{k-1-l}v}{(1+|x|)^{l+2}}\\|_{L^2},\n\\end{equation}\none can deduce from \\eqref{ii3} and Hardy inequality that\n\\begin{equation}\\label{i3}\n\\begin{split}\n|I_3|\\leq C\\delta\\sum_{0\\leq l\\leq k-1}\n\\Big(\\|\\f{\\nabla^{k-l}v}{(1+|x|)^{l+1}}\\|_{L^2}\n+\\|\\f{\\nabla^{k-1-l}v}{(1+|x|)^{l+2}}\\|_{L^2}\\Big)\\|\\nabla^{k+1}n\\|_{L^2}\\leq C\\delta\\|\\nabla^{k+1}(n,v)\\|_{L^2}^2.\n\\end{split}\n\\end{equation}\nSimilarly, it is easy to check that\n\\begin{equation}\\label{i4}\n\\begin{split}\n|I_4| \\leq C\\sum_{0\\leq l\\leq k-1}\n\\|(1+|x|)^{k-1-l}\\nabla^{k-1-l}\\bar\\rho\\|_{L^3}\n\\|\\f{\\nabla^{l+1}v}{(1+|x|)^{k-1-l}}\\|_{L^6}\\|\\nabla^{k+1}n\\|_{L^2}\n\\leq C\\delta\\|\\nabla^{k+1}(n,v)\\|_{L^2}^2.\n\\end{split}\n\\end{equation}\nSubstituting \\eqref{i1}, \\eqref{i2}, \\eqref{i3} and \\eqref{i4}\ninto \\eqref{ks1}, it holds\n\\begin{equation}\\label{s1}\t\n\\begin{split}\n\\left|\\int \\nabla^k\\widetilde{S}_1\\cdot \\nabla^kn dx \\right|\n\\leq & C\\delta\\|\\nabla^{k+1}(n,v)\\|_{L^2}^2.\n\\end{split}\n\\end{equation}\nRecall the definition of $\\widetilde{S}_2$,\nwe obtain after integration by part that\n\\begin{equation*}\n\\begin{split}\n\\int \\nabla^k\\widetilde{S}_2\\cdot \\nabla^kv dx\n=&-\\int \\nabla^{k-1}\\Big(\\f{\\mu_1\\gamma}{\\mu}v\\cdot \\nabla v \\Big)\n     \\cdot \\nabla^{k+1}v  dx\n  -\\int \\nabla^{k-1}\\Big\\{ \\wf(n+\\bar\\rho)\\big(\\mu_1\\tri v+\\mu_2\\nabla\\mathop{\\rm div}\\nolimits v\\big)\\Big\\}\n      \\cdot \\nabla^{k+1}v dx\\\\\n&-\\int \\nabla^{k-1}\\Big(\\wg(n+\\bar\\rho) \\nabla n \\Big)\\cdot \\nabla^{k+1}v dx\n-\\int \\nabla^{k-1}\\Big(\\h(n,\\bar\\rho)\\nabla\\bar\\rho\\Big)\\cdot \\nabla^{k+1}v dx\\\\\n\\overset{def}{=}&J_1+J_2+J_3+J_4.\n\\end{split}\n\\end{equation*}\nThe Sobolev inequality and \\eqref{nanak} yield directly\n\\begin{equation}\n\\begin{split}\\label{j1}\n|J_1| \\leq C\\sum_{0\\leq l\\leq k-1}\n       \\|\\nabla^{k-1-l}v\\|_{L^3}\\|\\nabla^{l+1}v\\|_{L^6}\\|\\nabla^{k+1}v\\|_{L^2}\n\t\t\\leq C\\delta \\|\\nabla^{k+1}v\\|_{L^2}^2.\n\\end{split}\n\\end{equation}\nNext, we begin to give the estimate for the term $J_2$.\nNotice that for any $1 \\le l \\le k-1$,\n\\begin{equation}\\label{widef}\n\\begin{split}\n\\nabla^l\\wf(n+\\bar\\rho)=\\text{a sum of products}~~\\wf^{\\gamma_1,\\gamma_2,\\cdots,\\gamma_{i+j}}(n+\\bar\\rho)\n\\nabla^{\\gamma_1}n\\cdots\\nabla^{\\gamma_i}n\\nabla^{\\gamma_{i+1}}\\bar\\rho\\cdots\\nabla^{\\gamma_{i+j}}\\bar\\rho\n\\end{split}\n\\end{equation}\nwith the functions $\\wf^{\\gamma_1,\\gamma_2,\\cdots,\\gamma_{i+j}}(n+\\bar\\rho)$ are some derivatives of $\\wf(n+\\bar\\rho)$ and $1\\leq \\gamma_{{\\beta}}\\leq l$, ${\\beta}=1,2,\\cdots,i+j$; $\\gamma_1+\\gamma_2+\\cdots+\\gamma_i=m$, $0\\leq m\\leq l$ and $\\gamma_1+\\gamma_2+\\cdots+\\gamma_{i+j}=l$.\nThen $J_{2}$ is split up into three terms as follows:\\begin{equation}\\label{J2}\n\\begin{split}\n|J_2|\\leq&C\\int|\\wf(n+\\bar\\rho)||\\nabla^{k+1}v|^2dx\n       +C\\sum_{1\\leq l\\leq k-1}\\sum_{\\gamma_1+\\cdots+\\gamma_j=l}\n       \\int|\\nabla^{\\gamma_1}\\bar\\rho|\\cdots|\\nabla^{\\gamma_j}\\bar\\rho||\\nabla^{k+1-l}v||\\nabla^{k+1}v|dx\\\\\n\t   &+C\\sum_{1\\leq l\\leq k-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\n        \\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l}}\n        \\int|\\nabla^{\\gamma_1}n|\\cdots|\\nabla^{\\gamma_i}n|\n         |\\nabla^{\\gamma_{i+1}}\\bar\\rho|\\cdots|\\nabla^{\\gamma_{i+j}}\\bar\\rho|\n         |\\nabla^{k+1-l}v||\\nabla^{k+1}v|dx\\\\\n\t \\overset{def}{=} & J_{21}+J_{22}+J_{23}.\n\\end{split}\n\\end{equation}\nThen it is easy to verify that\n\\begin{equation*}\n\\begin{split}\n|J_{21}| \\leq C\\big(\\|n\\|_{L^\\infty}+\\|\\bar\\rho \\|_{L^\\infty}\\big)\\|\\nabla^{k+1}v\\|_{L^2}^2\n\t\t \\leq C\\delta \\|\\nabla^{k+1}v\\|_{L^2}^2.\n\\end{split}\n\\end{equation*}\nWe address the second term $J_{22}$ by using Hardy inequality\n\\begin{equation*}\n\\begin{split}\n|J_{22}|\n\\leq& C\\sum_{1\\leq l\\leq k-1}\\sum_{\\gamma_1+\\cdots+\\gamma_j=l}\n\\|(1+|x|)^{\\gamma_1}\\nabla^{\\gamma_1}\\bar\\rho\\|_{L^\\infty}\n\\cdots\\|(1+|x|)^{\\gamma_j}\\nabla^{\\gamma_j}\\bar\\rho\\|_{L^\\infty}\n\\|\\f{\\nabla^{k+1-l}v}{(1+|x|)^{l}}\\|_{L^2}\\|\\nabla^{k+1}v\\|_{L^2}\n\\leq C\\delta\\|\\nabla^{k+1}v\\|_{L^2}^2.\n\\end{split}\n\\end{equation*}\nIn view of the Sobolev interpolation inequality \\eqref{Sobolev}\nin Lemma \\ref{inter} and Hardy inequality, we deduce that\n\\begin{equation*}\n\\begin{split}\n|J_{23}| \\leq& C\\sum_{1\\leq l\\leq k-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l}}\\|\\nabla^{\\gamma_1}n\\|_{L^\\infty}\\cdots\\|\\nabla^{\\gamma_i}n\\|_{L^\\infty}\\|(1+|x|)^{\\gamma_{i+1}}\\nabla^{\\gamma_{i+1}}\\bar\\rho\\|_{L^\\infty}\\cdots\\|(1+|x|)^{\\gamma_{i+j}}\\nabla^{\\gamma_{i+j}}\\bar\\rho\\|_{L^\\infty}\\\\\n\t\t&\\times\\|\\f{\\nabla^{k+1-l}v}{(1+|x|)^{l-m}}\\|_{L^2}\\|\\nabla^{k+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\sum_{1\\leq l\\leq k-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l}}\\|n\\|_{L^2}^{1-\\f{3+2\\gamma_1}{2(k+1)}}\\|\\nabla^{k+1}n\\|_{L^2}^{\\f{3+2\\gamma_1}{2(k+1)}}\\cdots\\|n\\|_{L^2}^{1-\\f{3+2\\gamma_{i}}{2(k+1)}}\\|\\nabla^{k+1}n\\|_{L^2}^{\\f{3+2\\gamma_i}{2(k+1)}}\\|\\nabla^{k+1-m}v\\|_{L^2}\\|\\nabla^{k+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\sum_{1\\leq l\\leq k-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l}}\\|n\\|_{L^2}^{1-\\f{3i+2m}{2(k+1)}}\\|\\nabla^{k+1}n\\|_{L^2}^{\\f{3i+2m}{2(k+1)}}\\|\\nabla^{{\\alpha}}v\\|_{L^2}^{\\f{3i+2m}{2(k+1)}}\\|\\nabla^{k+1}v\\|_{L^2}^{1-\\f{3i+2m}{2(k+1)}}\\|\\nabla^{k+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\|\\nabla^{k+1}(n,v)\\|_{L^2}^{2},\n\\end{split}\n\\end{equation*}\nwhere\n$${\\alpha}=\\f{3i(k+1)}{3i+2m}\\leq \\f35(k+1)\\leq \\f35N\\leq N,$$\nprovided $i\\geq 1$, $m\\geq 1$ and $i\\leq m$.\nThen, substituting the estimates of $J_{21}, J_{22}$ and $J_{23}$\ninto \\eqref{J2}, we have\n\\begin{equation}\\label{j2}\n|J_2|\\leq C\\delta\\|\\nabla^{k+1}(n,v)\\|_{L^2}^{2}.\n\\end{equation}\nIt is easy to check that\n\\begin{equation}\\label{J3}\n\\begin{split}\n|J_3|\n\\leq &C\\int|n+\\bar\\rho||\\nabla^{k}n||\\nabla^{k+1}v|dx\n       +C\\sum_{1\\leq l\\leq k-1}\\sum_{\\gamma_1+\\cdots+\\gamma_j=l}\n       \\int|\\nabla^{\\gamma_1}\\bar\\rho|\\cdots|\\nabla^{\\gamma_j}\\bar\\rho||\\nabla^{k-l}n||\\nabla^{k+1}v|dx\\\\\n\t\t&+C\\sum_{1\\leq l\\leq k-1}\n           \\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\n                  \\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l}}\n         \\int|\\nabla^{\\gamma_1}n|\\cdots|\\nabla^{\\gamma_i}n||\\nabla^{\\gamma_{i+1}}\\bar\\rho|\n         \\cdots|\\nabla^{\\gamma_{i+j}}\\bar\\rho||\\nabla^{k-l}n||\\nabla^{k+1}v|dx\\\\\n\t\t\\overset{def}{=}&J_{31}+J_{32}+J_{33}.\n\\end{split}\n\\end{equation}\nBy Sobolev inequality, it is easy to check that\n\\begin{equation*}\n\\begin{split}\n|J_{31}|\n\\leq C\\|(n+\\bar\\rho)\\|_{L^3}\\|\\nabla^{k}n\\|_{L^6}\\|\\nabla^{k+1}v\\|_{L^2}\n\\leq C\\delta\\|\\nabla^{k+1}(n,v)\\|_{L^2}^2.\n\\end{split}\n\\end{equation*}\nSimilar to estimate \\eqref{vl6}, we apply Hardy inequality to obtain\n\\begin{equation*}\n\\begin{split}\n|J_{32}|\n\\leq&C \\sum_{1\\leq l\\leq k}\\sum_{\\gamma_1+\\cdots+\\gamma_j=l}\\|(1+|x|)^{\\gamma_1}\\nabla^{\\gamma_1}\\bar \\rho\\|_{L^{3}}\\|(1+|x|)^{\\gamma_2}\\nabla^{\\gamma_2}\\bar \\rho\\|_{L^{\\infty}}\\cdots\\|(1+|x|)^{\\gamma_j}\\nabla^{\\gamma_j}\\bar \\rho\\|_{L^{\\infty}}\\|\\f{\\nabla^{k-l}n}{(1+|x|)^{l}}\\|_{L^6}\\|\\nabla^{k+1}v\\|_{L^2}\\\\\n\t\t\\leq& C\\delta\\|\\nabla^{k+1}(n,v)\\|_{L^2}^2 .\n\\end{split}\n\\end{equation*}\nThe application of Sobolev interpolation inequality \\eqref{Sobolev}\nin Lemma \\ref{inter} yields directly\n\\begin{equation*}\n\\begin{split}\n|J_{33}|\\leq& C\\sum_{1\\leq l\\leq k-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l}}\\|\\nabla^{\\gamma_1}n\\|_{L^\\infty}\\cdots\\|\\nabla^{\\gamma_i}n\\|_{L^\\infty}\\|(1+|x|)^{\\gamma_{i+1}}\\nabla^{\\gamma_{i+1}}\\bar\\rho\\|_{L^\\infty}\\cdots\\|(1+|x|)^{\\gamma_{i+j}}\\nabla^{\\gamma_{i+j}}\\bar\\rho\\|_{L^\\infty}\\\\\n\t\t&\\times\\|\\f{\\nabla^{k-l}n}{(1+|x|)^{l-m}}\\|_{L^2}\\|\\nabla^{k+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\sum_{1\\leq l\\leq k-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l}}\\|n\\|_{L^2}^{1-\\f{3+2\\gamma_1}{2(k+1)}}\\|\\nabla^{k+1}n\\|_{L^2}^{\\f{3+2\\gamma_1}{2(k+1)}}\\cdots\\|n\\|_{L^2}^{1-\\f{3+2\\gamma_{i}}{2(k+1)}}\\|\\nabla^{k+1}n\\|_{L^2}^{\\f{3+2\\gamma_i}{2(k+1)}}\\|\\nabla^{k-m}n\\|_{L^2}\\|\\nabla^{k+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\sum_{1\\leq l\\leq k-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l}}\\|n\\|_{L^2}^{1-\\f{3i+2m}{2(k+1)}}\\|\\nabla^{k+1}n\\|_{L^2}^{\\f{3i+2m}{2(k+1)}}\\|\\nabla^{{\\alpha}}n\\|_{L^2}^{\\f{3i+2m}{2(k+1)}}\\|\\nabla^{k+1}n\\|_{L^2}^{1-\\f{3i+2m}{2(k+1)}}\\|\\nabla^{k+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\|(\\nabla^{k+1}n,\\nabla^{k+2}v)\\|_{L^2}^{2},\n\\end{split}\n\\end{equation*}\nwhere\n$${\\alpha}=\\f{(3i-2)(k+1)}{3i+2m}\\leq \\f35(k+1)\\leq \\f35N\\leq N,$$\nprovided $i\\geq 1$, $m\\geq 1$ and $i\\leq m$.\nSubstituting the estimates of term $J_{31}, J_{32}$\nand $J_{33}$ into \\eqref{J3}, it holds\n\\begin{equation}\\label{j3}\n|J_3|\\leq C\\delta \\|\\nabla^{k+1}(n,v)\\|_{L^2}^{2}.\n\\end{equation}\n\tBefore estimating the last term $J_4$, we want to show that $\\h(n,\\bar\\rho)$ and its spatial derivatives can be controlled by a sum of products of some terms of $n$ and its derivatives.\n\tFor this purpose, it follows from Taylor expansion that \\begin{equation*}\n\t\\h(n,\\bar\\rho)=\\f{p'(\\rho_{\\infty})+p''(\\rho_{\\infty})\\rho_{\\infty}}{(\\bar\\rho+\\rho_{\\infty})(n+\\bar\\rho+\\rho_{\\infty})}n+o(|n|).\n\t\\end{equation*}\n\tUsing the fact that $p(\\rho)$ is smooth in a neighborhood of $\\rho_{\\infty}$ with $p'(\\rho_{\\infty})>0$, one obtains that\\begin{equation}\\label{h}\n\t|\\h(n,\\bar\\rho)|\\leq C|n|.\n\t\\end{equation}\n\tNext, let us to deal with the derivatives of $\\h$. In view of the definition of $\\h$ and $\\wg$, it then follows from \\eqref{hg-relation} that for any $l\\geq 1$,\\begin{equation}\\label{wideh}\n\t\\begin{split}\n\t\t&\\nabla^l\\h(n,\\bar\\rho)\n\t\t=\\nabla^l\\wg(n+\\bar\\rho)-\\nabla^l\\wg(\\bar\\rho)\\\\\n\t\t=&\\text{a sum of products}~~\\big\\{\\wg^{\\gamma_1,\\gamma_2,\\cdots,\\gamma_{i+j}}(n+\\bar\\rho)\\nabla^{\\gamma_1}n\\cdots\\nabla^{\\gamma_i}n\\nabla^{\\gamma_{i+1}}\\bar\\rho\\cdots\\nabla^{\\gamma_{i+j}}\\bar\\rho\\\\\n\t\t&\\quad-\\wg^{\\gamma_1,\\gamma_2,\\cdots,\\gamma_{i+j}}(\\bar\\rho)\\nabla^{\\gamma_1}\\bar\\rho\\cdots\\nabla^{\\gamma_{i+j}}\\bar\\rho\\big\\}\n\t\\end{split}\n    \\end{equation}\n\twith the functions $\\wg^{\\gamma_1,\\gamma_2,\\cdots,\\gamma_{i+j}}$ are some derivatives of $\\wg$ and $1\\leq \\gamma_{{\\beta}}\\leq l$, ${\\beta}=1,2,\\cdots,i+j$; $\\gamma_1+\\gamma_2+\\cdots+\\gamma_i=m$, $0\\leq m\\leq l$ and $\\gamma_1+\\gamma_2+\\cdots+\\gamma_{i+j}=l$.\n\tFor the case that $m=0$, we use mean value theorem to find that there exists a $\\xi$ between $\\bar\\rho$ and $n+\\bar\\rho$, such that\\begin{equation*}\n\t\\begin{split}\n\t\t&\\wg^{\\gamma_1,\\gamma_2,\\cdots,\\gamma_{j}}(n+\\bar\\rho)\\nabla^{\\gamma_1}\\bar\\rho\\cdots\\nabla^{\\gamma_{j}}\\bar\\rho-\\wg^{\\gamma_1,\\gamma_2,\\cdots,\\gamma_{j}}(\\bar\\rho)\\nabla^{\\gamma_1}\\bar\\rho\\cdots\\nabla^{\\gamma_{j}}\\bar\\rho\n\t\t=\\wg^{(\\gamma_1,\\gamma_2,\\cdots,\\gamma_{j})+1}(\\xi)n\\nabla^{\\gamma_1}\\bar\\rho\\cdots\\nabla^{\\gamma_{j}}\\bar\\rho,\n\t\\end{split}\n\t\\end{equation*}\n    which, together with \\eqref{wideh}, yields the following estimate\n    \\begin{equation}\\label{h2}\n\t\\begin{split}\n\t\t|\\nabla^l\\h(n,\\bar\\rho)|\\leq C|n|\\sum_{\\gamma_1+\\cdots+\\gamma_{j}=l}|\\nabla^{\\gamma_{1}}\\bar\\rho|\\cdots|\\nabla^{\\gamma_{j}}\\bar\\rho|+ C\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l}}|\\nabla^{\\gamma_1}n|\\cdots|\\nabla^{\\gamma_i}n||\\nabla^{\\gamma_{i+1}}\\bar\\rho|\\cdots|\\nabla^{\\gamma_{i+j}}\\bar\\rho|.\n\t\\end{split}\n\t\\end{equation}\n\tThis, together with the estimate \\eqref{h}, gives\\begin{equation}\\label{j4}\n\t\\begin{split}\n\t\t|J_4|\\leq&C\\int|\\h(n,\\bar\\rho)||\\nabla^{k}\\bar\\rho||\\nabla^{k+1}v|dx+C\\sum_{1\\leq l\\leq k-1}\\int|\\nabla^{l}\\h(n,\\bar\\rho)||\\nabla^{k-l}\\bar\\rho||\\nabla^{k+1}v|dx\\\\\n\t\t\\leq &C\\int|n||\\nabla^{k}\\bar\\rho||\\nabla^{k+1}v|dx+C\\sum_{1\\leq l\\leq k-1}\\sum_{\\gamma_1+\\cdots+\\gamma_{j}=l}\\int|n||\\nabla^{\\gamma_1}\\bar\\rho|\\cdots|\\nabla^{\\gamma_j}\\bar\\rho||\\nabla^{k-l}\\bar\\rho||\\nabla^{k+1}v|dx\\\\\n\t\t&\\quad+C\\sum_{1\\leq l\\leq k-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l}}\\int|\\nabla^{\\gamma_1}n|\\cdots|\\nabla^{\\gamma_i}n||\\nabla^{\\gamma_{i+1}}\\bar\\rho|\\cdots|\\nabla^{\\gamma_{i+j}}\\bar\\rho||\\nabla^{k-l}\\bar\\rho||\\nabla^{k+1}v|dx\\\\\n\t\t\\overset{def}{=}&J_{41}+J_{42}+J_{43}.\n\t\\end{split}\n\t\\end{equation}\n\tBy virtue of Sobolev and Hardy inequalities, it is easy to deduce\n\t\\begin{equation}\\label{j41}\n\t\\begin{split}\n\t\tJ_{41}+J_{42}\\leq& C \\|\\f{n}{(1+|x|)^{k}}\\|_{L^6}\\Big(\\|(1+|x|)^{k}\\nabla^{k}\\bar\\rho\\|_{L^3}+\\sum_{1\\leq l\\leq k-1}\\sum_{\\gamma_1+\\cdots+\\gamma_{j}=l}\\|(1+|x|)^{\\gamma_{1}}\\nabla^{\\gamma_{1}}\\bar\\rho\\|_{L^\\infty}\\\\\n\t\t&\\quad\\cdots\\|(1+|x|)^{\\gamma_{j}}\\nabla^{\\gamma_{j}}\\bar\\rho\\|_{L^\\infty}\\|(1+|x|)^{k-l}\\nabla^{k-l}\\bar\\rho\\|_{L^3}\\Big)\\|\\nabla^{k+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\Big(\\|\\f{\\nabla n}{(1+|x|)^{k}}\\|_{L^2}+\\|\\f{n}{(1+|x|)^{k+1}}\\|_{L^2}\\Big)\\|\\nabla^{k+1}v\\|_{L^2}\\\\\n\t\t\\leq& C\\delta \\|\\nabla^{k+1}(n,v)\\|_{L^2}^2.\n\t\\end{split}\n\t\\end{equation}\n\tIt then follows from Sobolev inequality that\n\t\\begin{equation}\\label{J43}\n\t\\begin{split}\n\t\tJ_{43}\\leq& C\\sum_{1\\leq l\\leq k-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{1+j}=l\\\\\\gamma_1=m\\\\1\\leq m\\leq l}}\\|\\f{\\nabla^{\\gamma_1}n}{(1+|x|)^{k-m}}\\|_{L^6}\\|(1+|x|)^{\\gamma_{2}}\\nabla^{\\gamma_{2}}\\bar\\rho\\|_{L^\\infty}\\cdots\\|(1+|x|)^{\\gamma_{1+j}}\\nabla^{\\gamma_{1+j}}\\bar\\rho\\|_{L^\\infty}\\\\\n\t\t&\\times\\|(1+|x|)^{k+1-l}\\nabla^{k-l}\\bar\\rho\\|_{L^3}\\|\\nabla^{k+1}v\\|_{L^2}+C\\sum_{1\\leq l\\leq k}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l,~i\\geq2}}\\|\\f{\\nabla^{\\gamma_1}n}{(1+|x|)^{k-m}}\\|_{L^6}\\|\\nabla^{\\gamma_2}n\\|_{L^\\infty}\\cdots\\\\\n\t\t&\\times\\|\\nabla^{\\gamma_i}n\\|_{L^\\infty}\\|(1+|x|)^{\\gamma_{i+1}}\\nabla^{\\gamma_{i+1}}\\bar\\rho\\|_{L^\\infty}\\cdots\\|(1+|x|)^{\\gamma_{i+j}}\\nabla^{\\gamma_{i+j}}\\bar\\rho\\|_{L^\\infty}\\|(1+|x|)^{k+1-l}\\nabla^{k-l}\\bar\\rho\\|_{L^3}\\|\\nabla^{k+1}v\\|_{L^2}\\\\\n\t\t\\overset{def}{=}&J_{431}+J_{432}.\n\t\\end{split}\n\t\\end{equation}\n\tThanks to Hardy inequality, one can deduce that\n\t\\begin{equation*}\n\t\\begin{split}\n\t\tJ_{431}\\leq& C\\delta\\sum_{1\\leq l\\leq k-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{1+j}=l\\\\\\gamma_1=m\\\\1\\leq m\\leq l}}\\Big(\\|\\f{\\nabla^{\\gamma_1+1}n}{(1+|x|)^{k-m}}\\|_{L^2}+\\|\\f{\\nabla^{\\gamma_1}n}{(1+|x|)^{k-m+1}}\\|_{L^2}\\Big)\\|\\nabla^{k+1}v\\|_{L^2}\\\\\n\t\t\\leq& C\\delta\\|\\nabla^{k+1}n\\|_{L^2}\\|\\nabla^{k+1}v\\|_{L^2}\\leq C\\delta\\|\\nabla^{k+1}(n,v)\\|_{L^2}^2.\n\t\\end{split}\n\t\\end{equation*}\n\tWith the help of Sobolev interpolation inequality \\eqref{Sobolev}\n\tin Lemma \\ref{inter} and Hardy ineuqlity, one obtains\n\t\\begin{equation*}\n\t\\begin{split}\n\t\tJ_{432}\\leq&C\\delta\\sum_{1\\leq l\\leq k-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l,~i\\geq2}}\\|\\nabla^{k+1-m+\\gamma_1}n\\|_{L^2}\\|n\\|_{L^2}^{1-\\f{3+2\\gamma_2}{2(k+1)}}\\|\\nabla^{k+1}n\\|_{L^2}^{\\f{3+2\\gamma_2}{2(k+1)}}\\cdots\\|n\\|_{L^2}^{1-\\f{3+2\\gamma_i}{2(k+1)}}\\|\\nabla^{k+1}n\\|_{L^2}^{\\f{3+2\\gamma_i}{2(k+1)}}\\|\\nabla^{k+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\sum_{1\\leq l\\leq k-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l,~i\\geq2}}\\|\\nabla^{{\\alpha}}n\\|_{L^2}^{\\theta}\\|\\nabla^{k+1}n\\|_{L^2}^{1-\\theta}\\|n\\|_{L^2}^{1-\\theta}\\|\\nabla^{k+1}n\\|_{L^2}^{\\theta}\\|\\nabla^{k+1}v\\|_{L^2}\n\t\t\\leq C\\delta\\|\\nabla^{k+1}(n,v)\\|_{L^2}^2,\n\t\\end{split}\n\t\\end{equation*}\n\twhere\n\t$$\\theta=\\f{3(i-1)+2(m-\\gamma_1)}{2(k+1)},\\quad{\\alpha}=\\f{3(i-1)(k+1)}{3(i-1)+2(m-\\gamma_1)}\\leq \\f{3}{5}(k+1)\\leq \\f35 N\\leq N,$$\n\tprovided that $i\\geq2$ and $i-1\\leq m-\\gamma_1$.\n    Inserting the estimates of $J_{431}$ and $J_{432}$ into $\\eqref{J43}$, it follows immediately\n\t\\begin{equation}\\label{j43}\n\tJ_{43}\\leq C\\delta\\|\\nabla^{k+1}(n,v)\\|_{L^2}^2.\n\t\\end{equation}\n\tThus, substituting \\eqref{j41} and \\eqref{j43} into \\eqref{j4}, we deduce that\\begin{equation*}\n\t\\begin{split}\n\t\t|J_4|\\leq C\\delta\\|\\nabla^{k+1}(n,v)\\|_{L^2}^2,\n\t\\end{split}\n\t\\end{equation*}\n\twhich, together with \\eqref{j1}, \\eqref{j2} and \\eqref{j3}, gives\n\t\\begin{equation}\\label{s2}\n\t\\begin{split}\n\t\t\\int \\nabla^k\\widetilde{S}_2\\cdot \\nabla^kv dx\n\t\t\\leq C\\delta\\|\\nabla^{k+1}(n,v)\\|_{L^2}^2.\n\t\\end{split}\n\t\\end{equation}\n\tWe finally utilize \\eqref{s1} and \\eqref{s2} in \\eqref{ehk}, to obtain \\eqref{en1} directly.\n\tThus, we complete the proof of this lemma.\n\\end{proof}\n\nWe then derive the energy estimate for $N-th$ order spatial derivative of solution.\n\\begin{lemm}\\label{enn}\n\tUnder the assumptions in Theorem \\ref{them3}, we have\n\t\\begin{equation}\\label{en2}\n\t\\f{d}{dt}\\|\\nabla^{N}(n,v)\\|_{L^2}^2+\\|\\nabla^{N+1}v\\|_{L^2}^2\\leq C\\delta \\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2,\n\t\\end{equation}\n\twhere $C$ is a positive constant independent of time.\n\\end{lemm}\n\\begin{proof}\nApplying differential operator $\\nabla^{N}$ to $\\eqref{ns5}_1$ and $\\eqref{ns5}_2$,\nmultiplying the resulting equations by $\\nabla^{N}n$ and $\\nabla^{N}v$, respectively,\nand integrating over $\\mathbb{R}^3$, it holds\n\\begin{equation}\\label{ehk1}\n\\begin{split}\n\t\t\\f12\\f{d}{dt}\\|\\nabla^{N}(n,v)\\|_{L^2}^2+\\mu_1\\|\\nabla^{N+1} v\\|_{L^2}^2+\\mu_2\\|\\nabla^{N}\\mathop{\\rm div}\\nolimits v\\|_{L^2}^2\n\t\t= \\int \\nabla^{N}\\widetilde{S}_1\\cdot \\nabla^{N}n dx+\\int \\nabla^{N}\\widetilde{S}_2\\cdot \\nabla^{N}v dx.\n\\end{split}\n\\end{equation}\nNow we estimate two terms on the right handside of \\eqref{ehk1} separately.\nIn view of the definition of $\\widetilde{S}_1$, we have\n\\begin{equation}\\label{ks2}\n\\begin{split}\n\\int \\nabla^{N}\\widetilde{S}_1\\cdot \\nabla^{N}n dx\n=\n&C\\int \\nabla^{N}( v\\cdot\\nabla n )\\cdot \\nabla^{N}n dx\n  +C\\int \\nabla^{N}( n\\mathop{\\rm div}\\nolimits v)\\cdot \\nabla^{N}n dx\\\\\n&  +C\\int \\nabla^{N}( v\\cdot\\nabla\\bar\\rho )\\cdot \\nabla^{N}n dx\n  +C\\int \\nabla^{N}( \\bar\\rho \\mathop{\\rm div}\\nolimits v)\\cdot \\nabla^{N}n dx\\\\\n\\overset{def}{=} &K_1+K_2+K_3+K_4.\n\\end{split}\n\\end{equation}\nSobolev inequality and integration by parts yield\\begin{equation*}\n\\begin{split}\nK_1=&C\\sum_{0\\leq l\\leq N}\\int\\nabla^{l+1}n\\nabla^{N-l}v\\nabla^{N}n dx\\\\\n=&C\\int\\nabla n\\nabla^{N}v\\nabla^{N}ndx\n  +C\\sum_{1\\leq l\\leq N-2}\\int\\nabla^{l+1} n\\nabla^{N-l}v\\nabla^{N}n dx\n  +C\\int \\nabla^{N}n\\nabla v\\nabla^{N}ndx-C\\int \\mathop{\\rm div}\\nolimits v|\\nabla^{N}n|^2dx\\\\\n\\leq&C\\|\\nabla v\\|_{L^\\infty}\\|\\nabla^{N}n\\|_{L^2}^2\n     +C\\|\\nabla n\\|_{L^3}\\|\\nabla^{N}v\\|_{L^6}\\|\\nabla^{N}n\\|_{L^2}\n     +C\\sum_{2\\leq l\\leq N-1}\\|\\nabla^ln\\|_{L^6}\\|\\nabla^{N+1-l}v\\|_{L^3}\\|\\nabla^{N}n\\|_{L^2}\\\\\n\\leq&C \\|\\nabla^2 v\\|_{H^1}\\|\\nabla^{N}n\\|_{L^2}^2\n      +C\\|\\nabla n\\|_{H^1}\\|\\nabla^{N+1}v\\|_{L^2}\\|\\nabla^{N}n\\|_{L^2}\n      +C\\sum_{2\\leq l\\leq N-1}\\|\\nabla^ln\\|_{L^6}\\|\\nabla^{N+1-l}v\\|_{L^3}\\|\\nabla^{N}n\\|_{L^2}.\n\\end{split}\n\\end{equation*}\nUsing the Sobolev interpolation inequality \\eqref{Sobolev}\nin Lemma \\ref{inter}, the third term on the right handside\nof above inequality can be estimated as follows\n\\begin{equation}\\label{lnv}\n\\begin{aligned}\n&\\|\\nabla^ln\\|_{L^6}\\|\\nabla^{N+1-l}v\\|_{L^3}\\|\\nabla^{N}n\\|_{L^2}\\\\\n\\leq &C\\|n\\|_{L^2}^{1-\\f{l+1}{N}}\\|\\nabla^{N}n\\|_{L^2}^{\\f{l+1}{N}}\n      \\|\\nabla^{\\alpha} v\\|_{L^2}^{\\f{l+1}{N}}\n      \\|\\nabla^{N+1}v\\|_{L^2}^{1-\\f{l+1}{N}}\\|\\nabla^{N}n\\|_{L^2}\\\\\n\\leq &C\\delta \\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2,\n\\end{aligned}\n\\end{equation}\nhere we denote ${\\alpha}$ that\n$${\\alpha}=\\f{3N}{2(l+1)}+1\\leq \\f{N}{2}+1,$$\nsince $l\\geq 2$.\nThus, we have\n\\begin{equation*}\n|K_1| \\leq C\\delta \\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2.\n\\end{equation*}\nIt follows in a similar way to $K_1$ that\n\\begin{equation*}\n\\begin{split}\nK_2=   &C\\sum_{0\\leq l\\leq N}\\int\\nabla^l n\\nabla^{N+1-l}v\\nabla^{N}n dx\\\\\n  \\leq &C\\Big(\\|n\\|_{L^\\infty}\\|\\nabla^{N+1} v\\|_{L^2}+\\|\\nabla n\\|_{L^3}\\|\\nabla^{N} v\\|_{L^6}\n          +\\sum_{2\\leq l\\leq N-1}\\|\\nabla^ln\\|_{L^6}\\|\\nabla^{N+1-l}v\\|_{L^3}\n          +\\|\\nabla v\\|_{L^\\infty}\\|\\nabla^{N}n\\|_{L^2}\\Big)\\|\\nabla^{N}n\\|_{L^2}\\\\\n  \\leq&C\\delta \\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2,\n\\end{split}\n\\end{equation*}\nwhere we have used \\eqref{lnv} in the last inequality above.\nThanks to Hardy inequality, we compute\n\\begin{equation*}\n\\begin{split}\n|K_3| \\leq& C\\sum_{0\\leq l\\leq N}\\|(1+|x|)^{l+1}\\nabla^{l+1}\\bar\\rho\\|_{L^{\\infty}}\n             \\|\\f{\\nabla^{N-l}v}{(1+|x|)^{l+1}}\\|_{L^2}\\|\\nabla^{N}n\\|_{L^2}\n\t\t\\leq C\\delta \\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2,\\\\\n|K_4| \\leq& C\\sum_{0\\leq l\\leq N}\\|(1+|x|)^{l}\\nabla^{l}\\bar\\rho\\|_{L^{\\infty}}\n            \\|\\f{\\nabla^{N+1-l}v}{(1+|x|)^{l}}\\|_{L^2}\\|\\nabla^{N}n\\|_{L^2}\n\t\t\\leq C\\delta \\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2.\n\\end{split}\n\\end{equation*}\nSubstituting the estimates of term from $K_1$ to $K_4$ into \\eqref{ks2}, we have\n\\begin{equation}\\label{ws11}\n\\begin{split}\n\\left|\\int \\nabla^{N}\\widetilde{S}_1\\cdot \\nabla^{N}n dx \\right|\n\\leq&C \\delta \\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2.\n\\end{split}\n\\end{equation}\nRemebering the definition of $\\widetilde{S}_2$ and integrating by part, we find\n\\begin{equation}\\label{ks22}\n\\begin{split}\n&\\int \\nabla^{N}\\widetilde{S}_2\\cdot \\nabla^{N}v dx\\\\\n=&\\int \\nabla^{N-1}\\Big(\\f{\\mu_1\\gamma}{\\mu}v\\cdot \\nabla v \\Big)\\cdot \\nabla^{N+1}v dx\n+\\int \\nabla^{N-1}\\Big\\{\\wf(n+\\bar\\rho)\\big(\\mu_1\\tri v+\\mu_2\\nabla\\mathop{\\rm div}\\nolimits v\\big) \\Big\\}\\cdot \\nabla^{N+1}v dx\\\\\n&+\\int \\nabla^{N-1}\\Big(\\wg(n+\\bar\\rho) \\nabla n \\Big)\\cdot \\nabla^{N+1}v dx\n+\\int \\nabla^{N-1}\\Big(\\h(n,\\bar\\rho)\\nabla\\bar\\rho\\Big)\\cdot \\nabla^{N+1}v dx\\\\\n\\overset{def}{=}&L_1+L_2+L_3+L_4.\n\\end{split}\n\\end{equation}\nAccording to the Sobolev interpolation inequality \\eqref{Sobolev}\nin Lemma \\ref{inter}, we deduce that\n\\begin{equation}\\label{l1}\n\\begin{split}\n|L_1|\n\\leq  &C\\sum_{0\\leq l\\leq N-1}\\|\\nabla^{N-1-l}v\\|_{L^3}\\|\\nabla^{l+1}v\\|_{L^6}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\\leq  &C\\sum_{0\\leq l\\leq N-1}\\|\\nabla^{{\\alpha}}v\\|_{L^2}^{\\f{l+2}{N+1}}\n      \\|\\nabla^{N+1}v\\|_{L^2}^{1-\\f{l+2}{N+1}}\n      \\|v\\|_{L^2}^{1-\\f{l+2}{N+1}}\\|\\nabla^{N+1}v\\|_{L^2}^{\\f{l+2}{N+1}}\n      \\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\\leq &C\\delta \\|\\nabla^{N+1}v\\|_{L^2}^2,\n\\end{split}\n\\end{equation}\nwhere ${\\alpha}$ is given by\n$$\n{\\alpha}=\\f{N+1}{2(l+2)}\\leq \\f{N+1}{4}.\n$$\nNext, we estimate the term $L_2$ and\ndivide it into two terms as follows\n\\begin{equation}\\label{L2}\n|L_2|\n\\leq C\\int|\\wf(n+\\bar\\rho)||\\nabla^{N+1}v|^2dx\n+C\\sum_{1\\leq l\\leq N-1}\\int|\\nabla^{l}\\wf(n+\\bar\\rho)||\\nabla^{N+1-l}v||\\nabla^{N+1}v|dx\n\\overset{def}{=}L_{21}+L_{22}.\n\\end{equation}\n\tBy Sobolev inequality, one can deduce directly that\\begin{equation}\\label{l21}\n\t\\begin{split}\n\t\tL_{21}\\leq C\\|(n+\\bar\\rho)\\|_{L^\\infty}\\|\\nabla^{N+1}v\\|_{L^2}^2\\leq C\\delta\\|\\nabla^{N+1}v\\|_{L^2}^2.\n\t\\end{split}\n\t\\end{equation}\nIt follows from \\eqref{widef} that $L_{22}$ can be\nsplit up into three terms as follows:\n\\begin{equation}\\label{l22}\n\\begin{split}\nL_{22} \\leq\n&C\\sum_{1\\leq l\\leq N-1}\\sum_{\\gamma_1+\\cdots+\\gamma_i=l}\n \\int|\\nabla^{\\gamma_1}n|\\cdots|\\nabla^{\\gamma_i}n||\\nabla^{N+1-l}v||\\nabla^{N+1}v|dx\\\\\n&+C\\sum_{1\\leq l\\leq N-1}\\sum_{\\gamma_1+\\cdots+\\gamma_j=l}\n \\int|\\nabla^{\\gamma_1}\\bar\\rho|\\cdots|\\nabla^{\\gamma_j}\\bar\\rho||\\nabla^{N+1-l}v||\\nabla^{N+1}v|dx\\\\\n&+C\\sum_{1\\leq l\\leq N-1}\n\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l-1}}\n\\int|\\nabla^{\\gamma_1}n|\\cdots|\\nabla^{\\gamma_i}n|\n    |\\nabla^{\\gamma_{i+1}}\\bar\\rho|\\cdots|\\nabla^{\\gamma_{i+j}}\\bar\\rho|\n    |\\nabla^{N+1-l}v||\\nabla^{N+1}v|dx\\\\\n\\overset{def}{=} & L_{221}+L_{222}+L_{223}.\n\\end{split}\n\\end{equation}\nWe then divide $L_{221}$ into two terms as follows:\n\\begin{equation*}\n\\begin{aligned}\nL_{221}\n\\leq & C\\sum_{1\\leq l\\leq N-1}\n            \\int|\\nabla^{l}n||\\nabla^{N+1-l}v||\\nabla^{N+1}v|dx\\\\\n     &+C\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_i=l\\\\i\\geq2}}\n       \\int|\\nabla^{\\gamma_1}n|\\cdots|\\nabla^{\\gamma_i}n||\\nabla^{N+1-l}v||\\nabla^{N+1}v|dx\\\\\n\\overset{def}{=}&L_{2211}+L_{2212}.\n\\end{aligned}\n\\end{equation*}\nUsing the Sobolev inequality and estimate \\eqref{lnv}, it holds\n\\begin{equation*}\n\\begin{split}\nL_{2211}\n\\leq& C\\Big(\\|\\nabla n\\|_{L^3}\\|\\nabla^{N}v\\|_{L^6}\n       +\\sum_{2\\leq l\\leq N-1}\\|\\nabla^{l}n\\|_{L^6}\\|\\nabla^{N+1-l}v\\|_{L^3}\\Big)\n       |\\nabla^{N+1}v\\|_{L^2}\\\\\n\\leq& C\\Big(\\|\\nabla n\\|_{H^1}\\|\\nabla^{N+1}v\\|_{L^2}\n       +\\sum_{2\\leq l\\leq N-1}\\|n\\|_{L^2}^{1-\\f{l+1}{N}}\\|\\nabla^{N}n\\|_{L^2}^{\\f{l+1}{N}}\n       \\|\\nabla^{\\alpha} v\\|_{L^2}^{\\f{l+1}{N}}\\|\\nabla^{N+1}v\\|_{L^2}^{1-\\f{l+1}{N}}\\Big)\n       \\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\\leq& C\\delta \\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2.\n\\end{split}\n\\end{equation*}\nSince $i\\geq2$ in term $L_{2212}$, then $\\gamma_{\\beta}\\leq N-2$ for ${\\beta}=1,2,\\cdots,i$.\nThus, we can use the Sobolev interpolation inequality \\eqref{Sobolev} in\nLemma \\ref{inter} to find\n\\begin{equation*}\n\t\\begin{split}\n\t\tL_{2212}\\leq&C\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_i=l\\\\i\\geq2}}\\|\\nabla^{\\gamma_1}n\\|_{L^\\infty}\\cdots\\|\\nabla^{\\gamma_i}n\\|_{L^\\infty}\\|\\nabla^{N+1-l}v\\|_{L^2}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_i=l\\\\i\\geq2}}\\|n\\|_{L^2}^{1-\\f{3+2\\gamma_1}{2N}}\\|\\nabla^{N}n\\|_{L^2}^{\\f{3+2\\gamma_1}{2N}}\\cdots\\|n\\|_{L^2}^{1-\\f{3+2\\gamma_{i}}{2N}}\\|\\nabla^{N}n\\|_{L^2}^{\\f{3+2\\gamma_i}{2N}}\\|\\nabla^{N+1-l}v\\|_{L^2}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_i=l\\\\i\\geq2}}\\|n\\|_{L^2}^{1-\\f{3i+2l}{2N}}\\|\\nabla^{N}n\\|_{L^2}^{\\f{3i+2l}{2N}}\\|\\nabla^{{\\alpha}}v\\|_{L^2}^{\\f{3i+2l}{2N}}\\|\\nabla^{N+1}v\\|_{L^2}^{1-\\f{3i+2l}{2N}}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^{2},\n\t\\end{split}\n\t\\end{equation*}\n\twhere ${\\alpha}$ is given by\n\t$${\\alpha}=1+\\f{3iN}{3i+2l}\\leq 1+\\f35N\\leq N,$$\n\tprovided $i\\geq 1$, $m\\geq 1$, $i\\leq m$ and $N\\geq3$. Therefore, it follows from the estimates above that\n\\begin{equation*}\nL_{221}\\leq C\\delta\\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^{2}.\n\\end{equation*}\nThe application of Hardy inequality yields directly\n\\begin{equation*}\n\\begin{split}\nL_{222}\n\\leq &C \\sum_{1\\leq l\\leq N-1}\\sum_{\\gamma_1+\\cdots+\\gamma_j=l}\n        \\|(1+|x|)^{\\gamma_1}\\nabla^{\\gamma_1}\\bar \\rho\\|_{L^{\\infty}}\\cdots\\|(1+|x|)^{\\gamma_j}\\nabla^{\\gamma_j}\\bar \\rho\\|_{L^{\\infty}}\n        \\|\\f{\\nabla^{N+1-l}v}{(1+|x|)^{l}}\\|_{L^2}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\\leq &  C\\delta \\|\\nabla^{N+1}v\\|_{L^2}^2.\n\\end{split}\n\\end{equation*}\nUsing Sobolev interpolation inequality \\eqref{Sobolev} in\nLemma \\ref{inter} again, we have\n\\begin{equation*}\n\\begin{split}\nL_{223}\n\\leq& C\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l-1}}\\|\\nabla^{\\gamma_1}n\\|_{L^\\infty}\\cdots\\|\\nabla^{\\gamma_i}n\\|_{L^\\infty}\\|(1+|x|)^{\\gamma_{i+1}}\\nabla^{\\gamma_{i+1}}\\bar\\rho\\|_{L^\\infty}\\cdots\\|(1+|x|)^{\\gamma_{i+j}}\\nabla^{\\gamma_{i+j}}\\bar\\rho\\|_{L^\\infty}\\\\\n\t\t&\\times\\|\\f{\\nabla^{N+1-l}v}{(1+|x|)^{l-m}}\\|_{L^2}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l-1}}\\|n\\|_{L^2}^{1-\\f{3+2\\gamma_1}{2N}}\\|\\nabla^{N}n\\|_{L^2}^{\\f{3+2\\gamma_1}{2N}}\\cdots\\|n\\|_{L^2}^{1-\\f{3+2\\gamma_{i}}{2N}}\\|\\nabla^{N}n\\|_{L^2}^{\\f{3+2\\gamma_i}{2N}}\\|\\nabla^{N+1-m}v\\|_{L^2}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l-1}}\\|n\\|_{L^2}^{1-\\f{3i+2m}{2N}}\\|\\nabla^{N}n\\|_{L^2}^{\\f{3i+2m}{2N}}\\|\\nabla^{{\\alpha}}v\\|_{L^2}^{\\f{3i+2m}{2N}}\\|\\nabla^{N+1}v\\|_{L^2}^{1-\\f{3i+2m}{2N}}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^{2},\n\t\\end{split}\n\t\\end{equation*}\nwhere ${\\alpha}$ is given by\n$${\\alpha}=1+\\f{3iN}{3i+2m}\\leq 1+\\f35N\\leq N,$$\nprovided $i\\geq 1$, $m\\geq 1$ and $i\\leq m$.\nThen, substituting estimates of terms $L_{221}, L_{222}$ and $L_{223}$\ninto \\eqref{l22}, we have\n\\begin{equation}\\label{L22}\n\\begin{split}\nL_{22}\\leq C\\delta \\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^{2}.\n\\end{split}\n\\end{equation}\nThe combination of estimate \\eqref{L2}, \\eqref{l21} and \\eqref{L22} yields directly\n\\begin{equation}\\label{l2}\n|L_2|\\leq C\\delta \\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^{2}.\n\\end{equation}\nNow we give the estimate for the term $L_3$. Indeed, it is easy to check that\n\\begin{equation}\\label{L3}\n\\begin{split}\n|L_3|\n   &\\leq C\\int|\\wg(n+\\bar\\rho)||\\nabla^{N}n||\\nabla^{N+1}v|dx\n     +C\\sum_{1\\leq l\\leq N-1}\\sum_{\\gamma_1+\\cdots+\\gamma_i=l}\n     \\int|\\nabla^{\\gamma_1}n|\\cdots|\\nabla^{\\gamma_i}n||\\nabla^{N-l}n||\\nabla^{k+2}v|dx\\\\\n\t&\\quad+C\\sum_{1\\leq l\\leq N-1}\\sum_{\\gamma_1+\\cdots+\\gamma_j=l}\n       \\int|\\nabla^{\\gamma_1}\\bar\\rho||\\nabla^{N-l}n|\\cdots\n       |\\nabla^{\\gamma_j}\\bar\\rho||\\nabla^{N-l}n||\\nabla^{N+1}v|dx\\\\\n\t&\\quad+C\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\n            \\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l-1}}\n           \\int|\\nabla^{\\gamma_1}n|\\cdots|\\nabla^{\\gamma_i}n|\n               |\\nabla^{\\gamma_{i+1}}\\bar\\rho|\\cdots|\\nabla^{\\gamma_{i+j}}\\bar\\rho|\n               |\\nabla^{N-l}n||\\nabla^{N+1}v|dx\\\\\n\t&\\overset{def}{=}L_{31}+L_{32}+L_{33}+L_{34}.\n\\end{split}\n\\end{equation}\nBy Sobolev inequality, it is easy to check that\n\\begin{equation}\\label{L31}\n\\begin{split}\nL_{31}\n\\leq C\\|(n+\\bar\\rho)\\|_{L^\\infty}\\|\\nabla^{N}n\\|_{L^2}\\|\\nabla^{N+1}v\\|_{L^2}\n\\leq C\\delta\\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2.\n\\end{split}\n\\end{equation}\nFor $L_{32}$, we divide it into the following two terms: \\begin{equation*}\n\\begin{split}\nL_{32}\\leq& C\\sum_{1\\leq l\\leq N-1}\\int|\\nabla^{l}n||\\nabla^{N-l}n||\\nabla^{N+1}v|dx\\\\\n   &+C\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_i=l\\\\i\\geq2}}\n     \\int|\\nabla^{\\gamma_1}n|\\cdots|\\nabla^{\\gamma_i}n||\\nabla^{N-l}n||\\nabla^{N+1}v|dx\\\\\n\t\\overset{def}{=}&L_{321}+L_{322}.\n\t\\end{split}\n\t\\end{equation*}\nBy Sobolev inequality, it holds\n\\begin{equation*}\n\\begin{split}\nL_{321}\n\\leq& C\\Big(\\|\\nabla n\\|_{L^3}\\|\\nabla^{N-1}n\\|_{L^6}\n           +\\sum_{2\\leq l\\leq N-1}\\|\\nabla^{l}n\\|_{L^6}\\|\\nabla^{N-l}n\\|_{L^3}\\Big)\n          \\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\\leq& C\\Big(\\|\\nabla n\\|_{L^3}\\|\\nabla^{N-1}n\\|_{L^6}\n           +\\sum_{2\\leq l\\leq N-1}\n           \\|n\\|_{L^2}^{1-\\f{l+1}{N}}\n           \\|\\nabla^{N}n\\|_{L^2}^{\\f{l+1}{N}}\n           \\|\\nabla^{{\\alpha}}n\\|_{L^2}^{\\f{l+1}{N}}\n           \\|\\nabla^{N}n\\|_{L^2}^{1-\\f{l+1}{N}}\\Big)\n          \\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\\leq& C\\delta \\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2,\n\\end{split}\n\\end{equation*}\nwhere ${\\alpha}$ is defined by ${\\alpha}=\\f{3N}{2(l+1)}\\leq \\f{N}{2}$.\nFor the term $L_{322}$, the fact $i\\geq2$ implies\n$\\gamma_{\\beta}+2\\leq l+1\\leq N$ for ${\\beta}=1,2,\\cdots,i$.\nThen, it follows from Sobolev interpolation inequality \\eqref{Sobolev} in Lemma \\ref{inter} that\n\\begin{equation*}\n\t\\begin{split}\n\t\tL_{322}\\leq&C\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i}=l\\\\i\\geq2}}\\|\\nabla^{\\gamma_1}n\\|_{L^\\infty}\\cdots\\|\\nabla^{\\gamma_i}n\\|_{L^\\infty}\\|\\f{\\nabla^{N-l}n}{(1+|x|)^{l-m}}\\|_{L^2}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i}=l\\\\i\\geq2}}\\|n\\|_{L^2}^{1-\\f{3+2\\gamma_1}{2N}}\\|\\nabla^{N}n\\|_{L^2}^{\\f{3+2\\gamma_1}{2N}}\\cdots\\|n\\|_{L^2}^{1-\\f{3+2\\gamma_{i}}{2N}}\\|\\nabla^{N}n\\|_{L^2}^{\\f{3+2\\gamma_i}{2N}}\\|\\nabla^{N-m}n\\|_{L^2}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i}=l\\\\i\\geq2}}\\|n\\|_{L^2}^{1-\\theta}\\|\\nabla^{N}n\\|_{L^2}^{\\theta}\\|\\nabla^{{\\alpha}}n\\|_{L^2}^{\\theta}\\|\\nabla^{N}n\\|_{L^2}^{1-\\theta}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^{2},\n\t\\end{split}\n\t\\end{equation*}\nwhere we denote\n$$\\theta=\\f{3i+2l}{2N},\\quad{\\alpha}=\\f{3iN}{3i+2l}\\leq \\f35N\\leq N,$$\nprovided $l\\geq1$, $i\\leq l$ and $i\\geq2$.\nThen, the combination of estimates of terms $L_{321}$ and $L_{322}$ implies directly\n\\begin{equation}\\label{L32}\n\tL_{32}\\leq C\\delta\\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^{2}.\n\\end{equation}\n    For the term $L_{33}$, by Hardy inequality, we obtain\\begin{equation}\\label{L33}\n\t\\begin{split}\n\t\tL_{33}\n\t\t\\leq&C \\sum_{1\\leq l\\leq N-1}\\sum_{\\gamma_1+\\cdots+\\gamma_j=l}\\|(1+|x|)^{\\gamma_1}\\nabla^{\\gamma_1}\\bar \\rho\\|_{L^{\\infty}}\\cdots\\|(1+|x|)^{\\gamma_j}\\nabla^{\\gamma_j}\\bar \\rho\\|_{L^{\\infty}}\\|\\f{\\nabla^{N-l}n}{(1+|x|)^{l}}\\|_{L^2}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\t\\leq& C\\delta \\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2.\n\t\\end{split}\n\t\\end{equation}\n\tIn view of Sobolev interpolation inequality \\eqref{Sobolev} in\n\tLemma \\ref{inter} and Hardy inequality, one deduces that\\begin{equation}\\label{L34}\n\t\\begin{split}\n\t\tL_{34}\\leq& C\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l-1}}\\|\\nabla^{\\gamma_1}n\\|_{L^\\infty}\\cdots\\|\\nabla^{\\gamma_i}n\\|_{L^\\infty}\\|(1+|x|)^{\\gamma_{i+1}}\\nabla^{\\gamma_{i+1}}\\bar\\rho\\|_{L^\\infty}\\cdots\\|(1+|x|)^{\\gamma_{i+j}}\\nabla^{\\gamma_{i+j}}\\bar\\rho\\|_{L^\\infty}\\\\\n\t\t&\\times\\|\\f{\\nabla^{N-l}n}{(1+|x|)^{l-m}}\\|_{L^2}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l-1}}\\|n\\|_{L^2}^{1-\\f{3+2\\gamma_1}{2N}}\\|\\nabla^{N}n\\|_{L^2}^{\\f{3+2\\gamma_1}{2N}}\\cdots\\|n\\|_{L^2}^{1-\\f{3+2\\gamma_{i}}{2N}}\\|\\nabla^{N}n\\|_{L^2}^{\\f{3+2\\gamma_i}{2N}}\\|\\nabla^{N-m}n\\|_{L^2}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l-1}}\\|n\\|_{L^2}^{1-\\theta}\\|\\nabla^{N}n\\|_{L^2}^{\\theta}\\|\\nabla^{{\\alpha}}n\\|_{L^2}^{\\theta}\\|\\nabla^{N}n\\|_{L^2}^{1-\\theta}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^{2},\n\t\\end{split}\n\t\\end{equation}\n\twhere\n\t$$\\theta=\\f{3i+2m}{2N},\\quad{\\alpha}=\\f{3iN}{3i+2m}\\leq \\f35N\\leq N,$$\n\tprovided $i\\geq 1$, $m\\geq 1$ and $i\\leq m$. We substitute \\eqref{L31}-\\eqref{L34} into \\eqref{L3}, to find that\n\t\\begin{equation}\\label{l3}\n\t\\begin{split}\n\t\t|L_3|\\leq C\\delta \\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^{2}.\n\t\\end{split}\n\t\\end{equation}\n\tFor the last term $L_4$, with the aid of the estimates \\eqref{h} and \\eqref{h2} of $\\h$, it is easy to deduce\\begin{equation}\\label{LL4}\n\t\\begin{split}\n\t\t|L_4|\\leq&C\\int|\\h(n,\\bar\\rho)||\\nabla^{N}\\bar\\rho||\\nabla^{N+1}v|dx+C\\sum_{1\\leq l\\leq N-1}\\int|\\nabla^{l}\\h(n,\\bar\\rho)||\\nabla^{N-l}\\bar\\rho||\\nabla^{N+1}v|dx\\\\\n\t\t\\leq &C\\int |n||\\nabla^{N}\\bar\\rho||\\nabla^{N+1}v|dx+C\\sum_{1\\leq l\\leq N-1}\\sum_{\\gamma_1+\\cdots+\\gamma_{j}=l}\\int|n||\\nabla^{\\gamma_1}\\bar\\rho|\\cdots|\\nabla^{\\gamma_j}\\bar\\rho||\\nabla^{N-l}\\bar\\rho||\\nabla^{N}v|dx\\\\\n\t\t&\\quad+C\\sum_{0\\leq l\\leq N-1}\\sum_{\\gamma_1+\\cdots+\\gamma_i=l}\\int|\\nabla^{\\gamma_1}n|\\cdots|\\nabla^{\\gamma_i}n||\\nabla^{N-l}\\bar\\rho||\\nabla^{N+1}v|dx\\\\\n\t\t&\\quad+C\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l-1}}\\int|\\nabla^{\\gamma_1}n|\\cdots|\\nabla^{\\gamma_i}n||\\nabla^{\\gamma_{i+1}}\\bar\\rho|\\cdots|\\nabla^{\\gamma_{i+j}}\\bar\\rho||\\nabla^{N-l}\\bar\\rho||\\nabla^{N+1}v|dx\\\\\n\t\t\\overset{def}{=}&L_{41}+L_{42}+L_{43}+L_{44}.\n\t\\end{split}\n\t\\end{equation}\n\tAccording to Hardy inequality, we obtain immediately\n\t\\begin{equation*}\n\t\\begin{split}\n\t\tL_{41}\\leq& \\sum_{0\\leq l\\leq N-1} \\|\\f{n}{(1+|x|)^{N}}\\|_{L^2}\\|(1+|x|)^{N}\\nabla^{N}\\bar\\rho\\|_{L^\\infty}\\|\\nabla^{N+1}v\\|_{L^2}\\leq C\\delta \\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2,\\\\\n\t\tL_{42}\\leq& \\sum_{0\\leq l\\leq N-1}\\sum_{\\gamma_1+\\cdots+\\gamma_j=l} \\|\\f{n}{(1+|x|)^{N}}\\|_{L^2}\\|(1+|x|)^{\\gamma_1}\\nabla^{\\gamma_1}\\bar\\rho\\|_{L^\\infty}\\cdots\\|(1+|x|)^{\\gamma_j}\\nabla^{\\gamma_j}\\bar\\rho\\|_{L^\\infty}\\\\\n\t\t&\\quad\\times\\|(1+|x|)^{N-l}\\nabla^{k+1}\\bar\\rho\\|_{L^\\infty}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\t\\leq& C\\delta \\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2.\n\t\\end{split}\n\t\\end{equation*}\n\tNext, let us to deal with $L_{43}$. It is easy to deduce that\\begin{equation*}\n    \\begin{split}\n    \tL_{43}\\leq& C\\sum_{0\\leq l\\leq N-1}\\int|\\nabla^{l}n||\\nabla^{N-l}\\bar\\rho||\\nabla^{N+1}v|dx+C\\sum_{0\\leq l\\leq k}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_i=l\\\\i\\geq2}}\\int|\\nabla^{\\gamma_1}n|\\cdots|\\nabla^{\\gamma_i}n||\\nabla^{N-l}\\bar\\rho||\\nabla^{N+1}v|dx\\\\\n    \t\\overset{def}{=}&L_{431}+L_{432}.\n    \\end{split}\n    \\end{equation*}\n    We employ Hardy inequality once again, to get\n    \\begin{equation*}\n    \\begin{split}\n    \tL_{431}\\leq \\sum_{0\\leq l\\leq k} \\|\\f{\\nabla^ln}{(1+|x|)^{N-l}}\\|_{L^2}\\|(1+|x|)^{N-l}\\nabla^{N-l}\\bar\\rho\\|_{L^\\infty}\\|\\nabla^{N+1}v\\|_{L^2}\\leq C\\delta \\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2.\n    \\end{split}\n    \\end{equation*}\n\tThe fact $i\\geq2$ implies that $\\gamma_{\\beta}+2\\leq l+1\\leq N$, for ${\\beta}=1,\\cdots,i$. Applying Sobolev interpolation inequality \\eqref{Sobolev} in Lemma \\ref{inter} and Hardy inequality, we obtain\\begin{equation*}\n\t\\begin{split}\n\t\tL_{432}\\leq&C\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i}=l\\\\i\\geq2}}\\|\\f{\\nabla^{\\gamma_1}n}{(1+|x|)^{N-l}}\\|_{L^2}\\|\\nabla^{\\gamma_2}n\\|_{L^\\infty}\\cdots\\|\\nabla^{\\gamma_i}n\\|_{L^\\infty}\\|(1+|x|)^{N-l}\\nabla\\bar\\rho\\|_{L^2}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i}=l\\\\i\\geq2}}\\|\\nabla^{N-l+\\gamma_1}n\\|_{L^2}\\|n\\|_{L^2}^{1-\\f{3+2\\gamma_2}{2N}}\\|\\nabla^{N}n\\|_{L^2}^{\\f{3+2\\gamma_2}{2N}}\\cdots\\|_{L^2}\\|n\\|_{L^2}^{1-\\f{3+2\\gamma_i}{2N}}\\|\\nabla^{N}n\\|_{L^2}^{\\f{3+2\\gamma_i}{2N}}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\i\\geq2}}\\|\\nabla^{{\\alpha}}n\\|_{L^2}^{\\theta}\\|\\nabla^{N}n\\|_{L^2}^{1-\\theta}\\|n\\|_{L^2}^{1-\\theta}\\|\\nabla^{N}n\\|_{L^2}^{\\theta}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2,\n\t\\end{split}\n\t\\end{equation*}\n\twhere\n\t$$\\theta=\\f{3(i-1)+2(l-\\gamma_1)}{2N},\\quad {\\alpha}=\\f{3(i-1)N}{3(i-1)+2(l-\\gamma_1)}\\leq \\f{3}{5}N\\leq N,$$\n\tprovided that $i\\geq2$ and $i-1\\leq l-\\gamma_1$.\n\tTwo estimates of terms from $L_{431}$ and $L_{432}$ gives\\begin{equation*}\n\tL_{43}\\leq C\\delta\\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2.\n\t\\end{equation*}\n\tThe last term on the right handside of \\eqref{LL4} can be divided into two terms:\n\t\\begin{equation*}\n\t\\begin{split}\n\tL_{44}\\leq&C\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{1+j}=l\\\\\\gamma_1=m\\\\1\\leq m\\leq l-1}}\\|\\f{\\nabla^{\\gamma_1}n}{(1+|x|)^{N-m}}\\|_{L^2}\\|(1+|x|)^{\\gamma_{2}}\\nabla^{\\gamma_{2}}\\bar\\rho\\|_{L^\\infty}\\cdots\\|(1+|x|)^{\\gamma_{1+j}}\\nabla^{\\gamma_{1+j}}\\bar\\rho\\|_{L^\\infty}\\\\\n\t&\\times\\|(1+|x|)^{N-l}\\nabla^{N-l}\\bar\\rho\\|_{L^\\infty}\\|\\nabla^{N+1}v\\|_{L^2}+C\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l-1,~i\\geq2}}\\|\\f{\\nabla^{\\gamma_1}n}{(1+|x|)^{N-m}}\\|_{L^2}\\|\\nabla^{\\gamma_2}n\\|_{L^\\infty}\\cdots\\|\\nabla^{\\gamma_i}n\\|_{L^\\infty}\\\\\n\t&\\times\\|(1+|x|)^{\\gamma_{i+1}}\\nabla^{\\gamma_{i+1}}\\bar\\rho\\|_{L^\\infty}\\cdots\\|(1+|x|)^{\\gamma_{i+j}}\\nabla^{\\gamma_{i+j}}\\bar\\rho\\|_{L^\\infty}\\|(1+|x|)^{N-l}\\nabla^{N-l}\\bar\\rho\\|_{L^\\infty}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\\overset{def}{=}&L_{441}+L_{442}.\n\t\\end{split}\n    \\end{equation*}\n    We use Hardy inequaly, to find\\begin{equation*}\n    \\begin{split}\n    \tL_{441}\\leq C\\delta\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{1+j}=l\\\\1\\leq m\\leq l-1}}\\|\\nabla^{N}n\\|_{L^2}\\|\\nabla^{N+1}v\\|_{L^2}\\leq C\\delta\\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2,\n    \\end{split}\n    \\end{equation*}\n    To deal with term $L_{442}$, by virtue of Sobolev interpolation inequality \\eqref{Sobolev} in Lemma \\ref{inter} and Hardy inequality, we arrive at\n\t\\begin{equation*}\n\t\\begin{split}\n\tL_{442}\\leq&C\\delta\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l-1,~i\\geq2}}\\|\\nabla^{N-m+\\gamma_1}n\\|_{L^2}\\|n\\|_{L^2}^{1-\\f{3+2\\gamma_2}{2N}}\\|\\nabla^{N}n\\|_{L^2}^{\\f{3+2\\gamma_2}{2N}}\\cdots\\|n\\|_{L^2}^{1-\\f{3+2\\gamma_i}{2N}}\\|\\nabla^{N}n\\|_{L^2}^{\\f{3+2\\gamma_i}{2N}}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\\leq&C\\delta\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l-1,~i\\geq2}}\\|\\nabla^{{\\alpha}}n\\|_{L^2}^{\\theta}\\|\\nabla^{N}n\\|_{L^2}^{1-\\theta}\\|n\\|_{L^2}^{1-\\theta}\\|\\nabla^{N}n\\|_{L^2}^{\\theta}\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\\leq&C\\delta\\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2,\n\t\\end{split}\n\t\\end{equation*}\n\twhere\n\t$$\\theta=\\f{3(i-1)+2(m-\\gamma_1)}{2N},\\quad{\\alpha}=\\f{3(i-1)N}{3(i-1)+2(m-\\gamma_1)}\\leq \\f{3}{5}N\\leq N,$$\n\tprovided that $i\\geq2$ and $i-1\\leq m-\\gamma_1$.\n\tHence, the combination of estimates of terms $L_{441}$ and $L_{442}$ implies directly\n\t\\begin{equation*}\n\tL_{44}\\leq C\\delta\\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2.\n\t\\end{equation*}\n\tWe substitute estimates of terms from $L_{41}$ to $L_{44}$ into \\eqref{LL4} to find\\begin{equation}\\label{l4}\n\t\\begin{split}\n\t\t|L_4|\\leq C\\delta\\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2.\n\t\\end{split}\n\t\\end{equation}\n\tInserting \\eqref{l1}, \\eqref{l2}, \\eqref{l3} and \\eqref{l4} into \\eqref{ks22}, we thereby deduce that\\begin{equation}\\label{ws22}\n\t\\begin{split}\n\t\t\\Big|\\int \\nabla^{N}\\widetilde{S}_2\\cdot \\nabla^{N}v dx\\Big|\\leq C\\delta\\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2.\n\t\\end{split}\n\t\\end{equation}\n\tPlugging \\eqref{ws11} and \\eqref{ws22} into \\eqref{ehk1} gives \\eqref{en2} directly. Therefore, the proof of this lemma is completed.\n\\end{proof}\nFinally, we aim to recover the dissipation estimate for $n$.\n\\begin{lemm}\\label{ennjc}\n\tUnder the assumptions in Theorem \\ref{them3}, for $1\\leq k\\leq N-1$, we have\n\t\\begin{equation}\\label{en3}\n\t\\f{d}{dt}\\int \\nabla^k v\\cdot\\nabla^{k+1}ndx+\\|\\nabla^{k+1}n\\|_{L^2}^2\\leq C_2\\|(\\nabla^{k+1}v,\\nabla^{k+2}v)\\|_{L^2}^2,\n\t\\end{equation}\n\twhere $C_2$ is a positive constant independent of $t$.\n\\end{lemm}\n\\begin{proof}\n\tApplying differential operator $\\nabla^k$ to $\\eqref{ns5}_2$, multiplying the resulting equation by $\\nabla^{k+1}n$, and integrating over $\\mathbb{R}^3$, one arrives at\\begin{equation}\\label{vnjc}\n\t\\begin{split}\n\t\t\\int \\nabla^{k}v_t\\cdot\\nabla^{k+1}n dx+\\|\\nabla^{k+1}n\\|_{L^2}^2\\leq C \\|\\nabla^{k+2}v\\|_{L^2}^2+\\int \\nabla^k\\widetilde{S}_2\\cdot\\nabla^{k+1}n dx.\n\t\\end{split}\n\t\\end{equation}\n\tIn order to deal with $\\int \\nabla^{k}v_t\\cdot\\nabla^{k+1}n dx$, we turn\n\tthe time derivative of velocity to the density.\n\tThen, applying differential operator $\\nabla^k$ to the mass equation $\\eqref{ns5}_1$, we find\n\t\\[\\nabla^kn_t+\\gamma\\nabla^{k}\\mathop{\\rm div}\\nolimits v=\\nabla^k \\widetilde{S}_1.\\]\n\tHence, we can transform time derivative to the spatial derivative, i.e.,\\begin{equation*}\n\t\\begin{split}\n\t\t\\int \\nabla^{k}v_t\\cdot\\nabla^{k+1}n dx=&\\f{d}{dt}\\int \\nabla^{k}v\\cdot\\nabla^{k+1}n dx-\\int \\nabla^{k}v\\cdot\\nabla^{k+1}n_t dx\\\\\n\t\t=&\\f{d}{dt}\\int \\nabla^{k}v\\cdot\\nabla^{k+1}n dx+\\gamma\\int \\nabla^{k} v\\cdot\\nabla^{k+1}\\mathop{\\rm div}\\nolimits v dx-\\int \\nabla^{k} v\\cdot\\nabla^{k+1}\\widetilde{S}_1 dx\\\\\n\t\t=&\\f{d}{dt}\\int \\nabla^{k}v\\cdot\\nabla^{k+1}n dx-\\gamma\\|\\nabla^{k}\\mathop{\\rm div}\\nolimits v\\|_{L^2}^2-\\int \\nabla^{k+1}\\mathop{\\rm div}\\nolimits v\\cdot\\nabla^{k-1}\\widetilde{S}_1 dx\n\t\\end{split}\n\t\\end{equation*}\n\tSubstituting the identity above into \\eqref{vnjc} and integrating by parts yield\\begin{equation}\\label{nvjc2}\n\t\\begin{split}\n\t\t&\\f{d}{dt}\\int \\nabla^{k}v\\cdot\\nabla^{k+1}n dx+\\|\\nabla^{k+1}n\\|_{L^2}^2\\\\\n\t\t\\leq& C \\|(\\nabla^{k+1}v,\\nabla^{k+2}v)\\|_{L^2}^2+C\\int \\nabla^{k+1}\\mathop{\\rm div}\\nolimits v\\cdot\\nabla^{k-1}\\widetilde{S}_1dx-C\\int \\nabla^k\\widetilde{S}_2\\cdot\\nabla^{k+1}n dx.\n\t\\end{split}\n\t\\end{equation}\n\tAs for the term of $\\widetilde{S}_1$.\n\t\tIt then follows in a similar way to the estimates of term from $I_1$ to $I_4$ in Lemma \\ref{enn-1} that\n\n\n\n\n\t\\begin{equation}\\label{ss1}\n\t\\begin{split}\n\t\t\\Big|\\int \\nabla^{k+1}\\mathop{\\rm div}\\nolimits v\\cdot\\nabla^{k-1}\\widetilde{S}_1 dx\\Big|\\leq C\\|\\nabla^{k+2}v\\|_{L^2}\\|\\nabla^{k-1}\\widetilde{S}_1\\|_{L^2}\n\t\t\\leq C \\delta \\|\\nabla^{k+1}n\\|_{L^2}^2+C\\|(\\nabla^{k+1}v,\\nabla^{k+2}v)\\|_{L^2}^2.\n\t\\end{split}\n\t\\end{equation}\n\tTo deal with the term of $\\widetilde{S}_2$, we only need to follow the idea as the estimates of term from $L_1$ to $L_4$ in Lemma \\ref{enn}. Hence, we\n\tgive the estimates as follow\n\t\\begin{equation}\\label{ss2}\n\t\\begin{split}\n\t\t\\Big|\\int \\nabla^k\\widetilde{S}_2\\cdot\\nabla^{k+1}n dx\\Big|\\leq C \\|\\nabla^k\\widetilde{S}_2\\|_{L^2}\\|\\nabla^{k+1}n\\|_{L^2}\\leq C \\delta\\|(\\nabla^{k+1}n,\\nabla^{k+2}v)\\|_{L^2}^2.\n\t\\end{split}\n\t\\end{equation}\n\tWe then utilize \\eqref{ss1} and \\eqref{ss2} in \\eqref{nvjc2}, to deduce \\eqref{en3} directly.\n\\end{proof}\n\n\\underline{\\noindent\\textbf{The proof of Proposition \\ref{nenp}.}}\nWith the help of Lemmas \\ref{enn-1}-\\ref{ennjc},\nit is easy to establish the estimate \\eqref{eml}.\nTherefore, we complete the proof of Proposition \\ref{nenp}.\n\n\n\n\n\n\\subsection{Optimal decay of higher order derivative}\nIn this subsection, we will study the optimal decay rate for the\n$k-th$ $(2 \\leq k\\leq N-1)$ order spatial derivatives of global solution.\nIn order to achieve this target, the optimal decay rate of higher order spatial\nderivative will be established by the lower one.\nIn this aspect, the Fourier splitting method, developed by Schonbek(see \\cite{Schonbek1985}),\nis applied to establish the optimal decay rate for higher order derivative\nof global solution in \\cite{{Schonbek-Wiegner},{gao2016}}.\nHowever, We're going to use time weighted and mathematical induction to solve this problem.\n\n\\begin{lemm}\\label{N-1decay}\n\tUnder the assumption of Theorem \\ref{them3}, for $0\\leq k\\leq N-1$, we have\n\t\\begin{equation}\\label{n1h1}\n\t\\|\\nabla^k(n,v)\\|_{H^{N-k}}\\leq C (1+t)^{-\\f34-\\f k2},\n\t\\end{equation}\n\twhere $C$ is a positive constant independent of time.\n\\end{lemm}\n\\begin{proof}\nWe will take the strategy of induction to give the proof for\nthe decay rate \\eqref{n1h1}. In fact, the decay rate \\eqref{basic-decay}\nimplies \\eqref{n1h1} holds true for the the case $k=0, 1$.\nBy the general step of induction, assume that the decay rate \\eqref{n1h1}\nholds on for the case $k=m$, i.e.,\n\t\\begin{equation}\\label{inducass}\n\t\\|\\nabla^m (n,v)\\|_{H^{N-m}}\\leq C (1+t)^{-\\f34-\\f m2},\n\t\\end{equation}\nfor $m=1,...,N-2$.\nChoosing the integer $l=m$ in \\eqref{eml} and\nmultiplying it by $(1+t)^{\\f32+m+\\ep_0} (0<\\ep_0<1)$, we have\n\\begin{equation*}\n\\begin{split}\n\\f{d}{dt}\\Big\\{(1+t)^{\\f32+m+\\ep_0} \\mathcal{E}^N_m(t)\\Big\\}\n+(1+t)^{\\f32+m+\\ep_0}\\big(\\|\\nabla^{m+1}n\\|_{H^{N-m-1}}^2\n+\\|\\nabla^{m+1}v\\|_{H^{N-m}}^2\\big)\\leq C(1+t)^{\\f12+m+\\ep_0} \\mathcal{E}^N_m(t).\n\\end{split}\n\\end{equation*}\nIntegrating with respect to $t$, using the equivalent relation \\eqref{emleq} and the decay estimate \\eqref{inducass}, one obtains\n\\begin{equation}\\label{energy2}\n\\begin{split}\n&(1+t)^{\\f32+m+\\ep_0} \\mathcal{E}^N_m(t)\t\n    +\\int_0^t(1+\\tau)^{\\f32+m+\\ep_0}\\big(\\|\\nabla^{m+1}n\\|_{H^{N-m-1}}^2\n    +\\|\\nabla^{m+1}v\\|_{H^{N-m}}^2\\big)d\\tau\\\\\n\\leq& \\mathcal{E}^N_m(0)+C\\int_0^t(1+\\tau)^{\\f12+m+\\ep_0} \\mathcal{E}^N_m(\\tau)d\\tau\\\\\n\\leq&C\\|\\nabla^m(n_0,v_0)\\|_{H^{N-m}}^2\n     +C\\int_0^t(1+\\tau)^{\\f12+m+\\ep_0} \\|\\nabla^m(n,v)\\|_{H^{N-m}}^2d\\tau\\\\\n\\leq&C\\|\\nabla^m(n_0,v_0)\\|_{H^{N-m}}^2\n     +C\\int_0^t(1+\\tau)^{-1+\\ep_0}d\\tau\\leq C(1+t)^{\\ep_0}.\n\\end{split}\n\\end{equation}\nOn the other hand, taking $l=m+1$ in \\eqref{eml}, we have\n\\begin{equation}\\label{Ekk}\n\\begin{split}\t\\f{d}{dt}\\mathcal{E}^{N}_{m+1}(t)\n+\\|\\nabla^{m+2}n\\|_{H^{N-m-2}}^2+\\|\\nabla^{m+2}v\\|_{H^{N-m-1}}^2\\leq 0.\n\\end{split}\n\\end{equation}\nMultiplying \\eqref{Ekk} by $(1+t)^{\\f52+m+\\ep_0}$,\nintegrating over $[0, t]$ and using  estimate \\eqref{energy2}, we find\n\\begin{equation*}\n\\begin{split}\n&(1+t)^{\\f52+m+\\ep_0}\\mathcal{E}^{N}_{m+1}(t)\n+\\int_0^t(1+\\tau)^{\\f52+m+\\ep_0}\\big(\\|\\nabla^{m+2}n\\|_{H^{N-m-2}}^2\n+\\|\\nabla^{k+2}v\\|_{H^{N-m-1}}^2\\big)d\\tau\\\\\n\\leq& \\mathcal{E}^{N}_{m+1}(0)\n+\\int_0^t(1+\\tau)^{\\f32+m+\\ep_0}\\mathcal{E}^{N}_{m+1}(\\tau)d\\tau\\\\\n\\leq&C\\|\\nabla^{m+1}(n_0,v_0)\\|_{H^{N-m-1}}^2\n+\\int_0^t(1+\\tau)^{\\f32+m+\\ep_0}\\|\\nabla^{m+1}(n,v)\\|_{H^{N-m-1}}^2d\\tau\n    \t\\leq C(1+t)^{\\ep_0},\n\\end{split}\n\\end{equation*}\nwhich, togeter with the equivalent relation \\eqref{emleq}, yields immediately\n\\begin{equation*}\n\\|\\nabla^{m+1}(n,v)\\|_{H^{N-m-1}}\\leq C(1+t)^{-\\f34-\\f {m+1}2}.\n\\end{equation*}\nThen, the decay estimate \\eqref{n1h1} holds on for case of $k=m+1$.\nBy the general step of induction, we complete the proof of this lemma.\n\\end{proof}\n\n\n\\subsection{Optimal decay of critical derivative}\n\nIn this subsection, our target is to establish the optimal decay rate for the\n$N-th$ order spatial derivative of global solution $(n, v)$ as it tends to zero.\nThe decay rate of $N-th$ order derivative of global solution $(n, v)$ in Lemma \\ref{N-1decay}\nis not optimal since it only has the same decay rate as the lower one.\nThe loss of time decay estimate comes from the appearance of cross term\n$\\frac{d}{dt}\\int \\nabla^{N-1} v \\cdot \\nabla^N n dx$ in energy\nwhen we set up the dissipation estimate for the density.\nNow let us introduce some notations that will be used frequently in this subsection.\nLet $0\\leq\\varphi_0(\\xi)\\leq1$ be a function in $C_0^{\\infty}(\\mathbb{R}^3)$ such that\\begin{equation*}\n\\begin{split}\n\\varphi_0(\\xi)=\\left\\{\n\\begin{array}{ll}\n1,\\quad \\text{for}~~|\\xi|\\leq \\f{\\eta}{2},\\\\[1ex]\n0,\\quad\\text{for}~~|\\xi|\\geq \\eta,   \\\\[1ex]\n\\end{array}\n\\right.\n\\end{split}\n\\end{equation*}\nwhere $\\eta$ is a fixed positive constant. Based on the Fourier transform, we can define a low-medium-high-frequency decomposition $(f^l(x),f^h(x))$ for a function $f(x)$ as follows:\n\\begin{equation}\\label{def-h-l}\nf^l(x)\\overset{def}{=}\\mathcal{F}^{-1}(\\varphi_0(\\xi)\\widehat{f}(\\xi))~~\\text{and}~~f^h(x)\\overset{def}{=}f(x)-f^l(x).\n\\end{equation}\n\n\\begin{lemm}\\label{highfrequency}\nUnder the assumptions of Theorem \\ref{them3},\nthere exists a positive small constant $\\eta_3$, such that\n\\begin{equation}\\label{en6}\n\\begin{split}\t&\\f{d}{dt}\\Big\\{\\|\\nabla^{N}(n,v)\\|_{L^2}^2-\\eta_3\\int_{|\\xi|\\geq\\eta}\\widehat{\\nabla^{N-1}v}\\cdot \\overline{\\widehat{\\nabla^{N}n}}d\\xi \\Big\\}+\\|\\nabla^{N}v^h\\|_{L^2}^2+\\eta_3\\|\\nabla^{N}n^h\\|_{L^2}^2\\\\\n\t\t\\leq&  C_4\\|\\nabla^{N}(n^l, v^l)\\|_{L^2}^2+C(1+t)^{-3-N},\n\\end{split}\n\\end{equation}\n\twhere $C_4$ is a positive constant independent of time.\n\\end{lemm}\n\t\\begin{proof}\nTaking differential operating $\\nabla^{N-1}$ to the equation \\eqref{ns5}, it holds true\n\\begin{equation}\\label{ns6}\n\\left\\{\\begin{array}{lr}\n\\nabla^{N-1}n_t +\\gamma\\nabla^{N-1}\\mathop{\\rm div}\\nolimits v=\\nabla^{N-1}\\widetilde S_1,\\\\\n\\nabla^{N-1}v_t+\\gamma\\nabla^{N} n-\\mu_1\\nabla^{N-1}\\tri v-\\mu_2\\nabla^{N}\\mathop{\\rm div}\\nolimits v\n=\\nabla^{N-1}\\widetilde  S_2.\n\\end{array}\\right.\n\\end{equation}\nTaking the Fourier transform of $\\eqref{ns6}_2$, multiplying the resulting equation by $\\overline{\\widehat{\\nabla^{N}n}}$ and integrating on $\\{\\xi||\\xi|\\geq \\eta\\}$, it holds true\n\\begin{equation}\\label{f1}\n\\begin{split}\n&\\int_{|\\xi|\\geq\\eta}\\widehat{\\nabla^{N-1}v_t}\\cdot \\overline{\\widehat{\\nabla^{N}n}}d\\xi+\\gamma\\int_{|\\xi|\\geq\\eta}|\\widehat{\\nabla^Nn}|^2d\\xi\\\\\n=&\\int_{|\\xi|\\geq\\eta}\\big(\\mu_1\\widehat{\\nabla^{N-1}\\tri v}+\\mu_2\\widehat{\\nabla^N\\mathop{\\rm div}\\nolimits v}\\big)\\cdot \\overline{\\widehat{\\nabla^{N}n}}d\\xi+\\int_{|\\xi|\\geq\\eta}\\widehat{\\nabla^{N-1}\\widetilde  S_2}\\cdot \\overline{\\widehat{\\nabla^{N}n}}d\\xi.\n\\end{split}\n\\end{equation}\nIt is easy to deduce from $\\eqref{ns6}_1$ that\n\\begin{equation*}\n\\begin{split}\n\t\t\\widehat{\\nabla^{N-1}v_t}\\cdot \\overline{\\widehat{\\nabla^{N}n}}=&-i\\xi\\widehat{\\nabla^{N-1}v_t}\\cdot \\overline{\\widehat{\\nabla^{N-1}n}}=-\\widehat{\\nabla^Nv_t}\\cdot \\overline{\\widehat{\\nabla^{N-1}n}}\\\\\n\t\t=&-{\\partial}_t(\\widehat{\\nabla^Nv}\\cdot \\overline{\\widehat{\\nabla^{N-1}n}})+\\widehat{\\nabla^Nv}\\cdot \\overline{\\widehat{\\nabla^{N-1}n_t}}\\\\\n\t\t=&-{\\partial}_t(\\widehat{\\nabla^Nv}\\cdot \\overline{\\widehat{\\nabla^{N-1}n}})-\\gamma\\widehat{\\nabla^Nv}\\cdot\n\\overline{\\widehat{\\nabla^{N-1}\\mathop{\\rm div}\\nolimits v}}+\\widehat{\\nabla^Nv}\\cdot \\overline{\\widehat{\\nabla^{N-1}\\widetilde{S}_1}}.\n\t\\end{split}\n\t\\end{equation*}\nThen, substituting this identity into identity \\eqref{f1}, we have\n\\begin{equation}\\label{f2}\n\\begin{split}\n\t\t&-\\f{d}{dt}\\int_{|\\xi|\\geq\\eta}\\widehat{\\nabla^{N}v}\\cdot \\overline{\\widehat{\\nabla^{N-1}n}}d\\xi+\\gamma\\int_{|\\xi|\\geq\\eta}|\\widehat{\\nabla^Nn}|^2d\\xi\\\\\n\t\t=&\\int_{|\\xi|\\geq\\eta}\\big(\\mu_1\\widehat{\\nabla^{N-1}\\tri v}+\\mu_2\\widehat{\\nabla^N\\mathop{\\rm div}\\nolimits v}\\big)\\cdot \\overline{\\widehat{\\nabla^{N}n}}d\\xi+\\gamma\\int_{|\\xi|\\geq\\eta}\\widehat{\\nabla^{N}v}\\cdot \\overline{\\widehat{\\nabla^{N-1}\\mathop{\\rm div}\\nolimits v}}d\\xi \\\\\n\t\t&\\quad-\\int_{|\\xi|\\geq\\eta}\\widehat{\\nabla^{N}v}\\cdot \\overline{\\widehat{\\nabla^{N-1}\\widetilde  S_1}}d\\xi +\\int_{|\\xi|\\geq\\eta}\\widehat{\\nabla^{N-1}\\widetilde  S_2}\\cdot \\overline{\\widehat{\\nabla^{N}n}}d\\xi\\\\\n\t\t\\overset{def}{=}&M_1+M_2+M_3+M_4.\n\\end{split}\n\\end{equation}\nThe application of Cauchy inequality yields directly\n\\begin{equation}\\label{N1}\n\\begin{split}\n|M_1|\n\\leq& C\\int_{|\\xi|\\geq\\eta}|\\xi|^{2N+1}|\\widehat{v}||\\widehat{n}|d\\xi\n\\leq\\ep \\int_{|\\xi|\\geq\\eta}|\\xi|^{2N}|\\widehat{n}|^2d\\xi\n        +C_{\\ep}\\int_{|\\xi|\\geq\\eta}|\\xi|^{2(N+1)}|\\widehat{v}|^2d\\xi,\n\\end{split}\n\\end{equation}\nfor some small $\\ep$, which will be determined later.\nObviously, it holds true\n\\begin{equation}\\label{N2}\n|M_2| \\leq  C\\int_{|\\xi|\\geq\\eta}|\\xi|^{2N}|\\widehat{v}|^2d\\xi.\n\\end{equation}\nUsing the Cauchy inequality and definition of $\\widetilde{S}_1$, it is easy to check that\n\\begin{equation}\\label{m3}\n\\begin{split}\n|M_3|\n\\leq&C \\int_{|\\xi|\\geq\\eta}|\\xi|^{2(N+1)}|\\widehat{v}|^2d\\xi\n       +C\\int_{|\\xi|\\geq\\eta}|\\xi|^{2(N-2)}|\\widehat{\\widetilde  S_1}|^2 d\\xi \\\\\n\\leq&C \\int_{|\\xi|\\geq\\eta}|\\xi|^{2(N+1)}|\\widehat{v}|^2d\\xi\n       +C\\int_{|\\xi|\\geq\\eta}|\\xi|^{2(N-2)}|\\widehat{\\nabla nv+n\\nabla v}|^2d\\xi\\\\\n    & +C\\int_{|\\xi|\\geq\\eta}|\\xi|^{2(N-2)}|\\widehat{\\nabla\\bar\\rho v+\\bar\\rho\\nabla v}|^2d\\xi\\\\\n\\overset{def}{=}&\\int_{|\\xi|\\geq\\eta}|\\xi|^{2(N+1)}|\\widehat{v}|^2d\\xi+M_{31}+M_{32}.\n\\end{split}\n\\end{equation}\nUsing Plancherel Theorem  and Sobolev inequality, it is easy to check that\n\\begin{equation}\\label{m31}\n\\begin{split}\nM_{31}\n\\le  &C\\|\\nabla^{N-2}(\\nabla nv+n\\nabla v)\\|_{L^2}^2\\\\\n\\leq &C\\big(\\|\\nabla n\\|_{L^\\infty}^2\\|\\nabla^{N-2}v\\|_{L^2}^2\n      +\\|\\nabla^{N-1}n\\|_{L^2}^2\\|v\\|_{L^\\infty}^2\n      +\\|n\\|_{L^\\infty}^2\\|\\nabla^{N-1}v\\|_{L^2}^2\n      +\\|\\nabla^{N-2}n\\|_{L^2}^2\\|\\nabla v\\|_{L^\\infty}^2\\big)\\\\\t\t\n\\leq&C\\big(\\|\\nabla^{2}(n,v)\\|_{H^1}^2\\|\\nabla^{N-2}(n,v)\\|_{L^2}^2\n      +\\|\\nabla(n,v)\\|_{H^1}^2\\|\\nabla^{N-1}v\\|_{L^2}^2\\big)\\\\\n\\leq&C(1+t)^{-3-N},\n\\end{split}\n\\end{equation}\nwhere we have used the decay \\eqref{n1h1} in the last inequality.\nSimilarly, we also apply Hardy's inequality to obtain\n\\begin{equation}\\label{m32}\n\\begin{split}\nM_{32}\n\\leq &C\\int_{|\\xi|\\geq\\eta}|\\xi|^{2(N-1)}|\\widehat{\\nabla\\bar\\rho v+\\bar\\rho\\nabla v}|^2d\\xi\n\\le  C\\|\\nabla^{N-1}(\\nabla \\bar\\rho v+\\bar\\rho\\nabla v)\\|_{L^2}^2 \\\\\n\t\t\\leq&C\\sum_{0\\leq l\\leq N-1}\\Big(\\|(1+|x|)^{l+1}\\nabla^{l+1}\\bar\\rho\\|_{L^\\infty}\\|\\f{\\nabla^{N-1-l}v}{(1+|x|)^{l+1}}\\|_{L^2}+\\|(1+|x|)^{l}\\nabla^{l}\\bar\\rho\\|_{L^\\infty}\\|\\f{\\nabla^{N-l}v}{(1+|x|)^{l}}\\|_{L^2}\\Big)\\|\\nabla^{N+1}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\|(\\nabla^{N}v,\\nabla^{N+1}v)\\|_{L^2}^2,\n\\end{split}\n\\end{equation}\nwhere we have used the fact that for any suitable function $\\phi$,\nthere exists a positive constant $C$ dependent only on $\\eta$ such that\n\\begin{equation*}\n\\int_{|\\xi|\\geq\\eta}|\\xi|^{2(N-2)}|\\widehat{\\phi}|^2d\\xi\\leq C\\int_{|\\xi|\\geq\\eta}|\\xi|^{2(N-1)}|\\widehat{\\phi}|^2d\\xi.\n\\end{equation*}\nSubstituting the estimates \\eqref{m31} and \\eqref{m32} into \\eqref{m3},\none can get that\n\\begin{equation}\\label{N3}\n|M_3|\\leq C\\delta\\|(\\nabla^{N}v,\\nabla^{N+1}v)\\|_{L^2}^2 +C(1+t)^{-3-N}.\n\\end{equation}\nUsing Cauchy inequality and the definition of $\\widetilde  S_2$, we have\n\\begin{equation}\\label{m4}\n\\begin{split}\n|M_4|\n\\leq &C\\int_{|\\xi|\\geq\\eta}|\\xi|^{2N-1}|\\widehat{\\widetilde  S_2}|| {\\widehat{n}}|d\\xi\\\\\n\t\t\\leq&C\\int_{|\\xi|\\geq\\eta}|\\xi|^{2N-1}| {\\widehat{n}}||\\widehat{v\\cdot \\nabla v}|d\\xi\n     +C\\int_{|\\xi|\\geq\\eta}|\\xi|^{2N-1}| {\\widehat{n}}\n        |\\big(\\mu_1|\\widehat{\\wf(n+\\bar\\rho)\\tri v}|\n         +\\mu_2|\\widehat{\\wf(n+\\bar\\rho)\\nabla\\mathop{\\rm div}\\nolimits v}|\\big)d\\xi\\\\\n&\\quad+C\\int_{|\\xi|\\geq\\eta}|\\xi|^{2N-1}| {\\widehat{n}}|\n          |\\widehat{\\wg(n+\\bar\\rho)\\nabla n}|d\\xi\n      +C\\int_{|\\xi|\\geq\\eta}|\\xi|^{2N-1}| {\\widehat{n}}|\n          |\\widehat{\\h(n,\\bar\\rho)\\nabla \\bar\\rho}|d\\xi\\\\\n\\overset{def}{=}&M_{41}+M_{42}+M_{43}+M_{44}.\n\\end{split}\n\\end{equation}\nIn view of Plancherel Theorem, Sobolev inequality and\ncommutator estimate in Lemma \\ref{commutator}, we find\n\\begin{equation}\\label{M41}\n\\begin{split}\nM_{41}\n\\leq &\\ep \\|\\nabla^Nn\\|_{L^2}^2+C_{\\ep}\\|\\nabla^{N-1}(v\\cdot\\nabla v)\\|_{L^2}^2\\\\\n\\leq &\\ep \\|\\nabla^Nn\\|_{L^2}^2+C_{\\ep}\\|v\\|_{L^\\infty}^2\\|\\nabla^Nv\\|_{L^2}^2\n      +C_{\\ep}\\|[\\nabla^{N-1},v]\\cdot\\nabla v\\|_{L^2}^2\\\\\n\\leq &\\ep \\|\\nabla^Nn\\|_{L^2}^2+C_{\\ep}\\|\\nabla v\\|_{H^1}^2\\|\\nabla^Nv\\|_{L^2}^2\n      +C_{\\ep}\\|\\nabla v\\|_{L^\\infty}^2\\|\\nabla^{N-1} v\\|_{L^2}^2\\\\\n\\leq &\\ep \\|\\nabla^Nn\\|_{L^2}^2+C_{\\ep}(1+t)^{-3-N},\n\\end{split}\n\\end{equation}\nwhere we have used the estimate \\eqref{n1h1} in the last inequality.\nSimilarly, it is easy to check that\n\\begin{equation}\\label{m42}\n\\begin{split}\nM_{42}\n\\leq &\\ep \\|\\nabla^Nn\\|_{L^2}^2+C_{\\ep}\\|\\nabla^{N-1}\\big(\\wf(n+\\bar\\rho)\n        (\\mu_1\\tri v+\\mu_2\\nabla\\mathop{\\rm div}\\nolimits v)\\big)\\|_{L^2}^2\\\\\n\\leq &\\ep\\|\\nabla^Nn\\|_{L^2}^2+C_{\\ep}\\|(n,\\bar\\rho)\\|_{L^\\infty}^2\n      \\|\\nabla^{N+1}v\\|_{L^2}^2+C_{\\ep}\\|\\nabla n\\|_{L^3}^2\\|\\nabla^{N}v\\|_{L^6}^2\n      +C_{\\ep}\\|(1+|x|)\\nabla \\bar\\rho\\|_{L^\\infty}^2 \\|\\f{\\nabla^{N}v}{1+|x|}\\|_{L^2}^2\\\\\n     &\\quad+C_{\\ep}\\sum_{2\\leq l\\leq N-1}\\|\\nabla^l\\wf(n+\\bar\\rho)\\nabla^{N+1-l}v\\|_{L^2}^2\\\\\n\\leq &\\ep \\|\\nabla^Nn\\|_{L^2}^2+C_{\\ep}\\delta\\|\\nabla^{N+1}v\\|_{L^2}^2\n      +C_{\\ep}\\sum_{2\\leq l\\leq N-1}\\|\\nabla^l\\wf(n+\\bar\\rho)\\nabla^{N+1-l}v\\|_{L^2}^2.\n\\end{split}\n\\end{equation}\nNow let us deal with the last term on the right handside\nof \\eqref{m42}. Indeed, it is easy to deduce that\n\\begin{equation*}\n\\begin{split}\n&\\sum_{2\\leq l\\leq N-1}\\|\\nabla^l\\wf(n+\\bar\\rho)\\nabla^{N+1-l}v\\|_{L^2}\\\\\n\\leq& C\\sum_{2\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\0\\leq m\\leq l}}\\||\\nabla^{\\gamma_1}n|\\cdots|\\nabla^{\\gamma_{i}}n||\\nabla^{\\gamma_{i+1}}\\bar\\rho|\\cdots|\\nabla^{\\gamma_{i+j}}\\bar\\rho||\\nabla^{N+1-l}v|\\|_{L^2}.\n\\end{split}\n\\end{equation*}\nFor $m=0$, we apply the Hardy inequality to obtain\n\\begin{equation*}\n\\begin{split}\n\\sum_{2\\leq l\\leq N-1}\n\\sum_{\\gamma_1+\\cdots+\\gamma_{j}=l}\\|(1+|x|)^{\\gamma_{i+1}}\\nabla^{\\gamma_{i+1}}\\bar\\rho\\|_{L^\\infty}\n\\cdots\\|(1+|x|)^{\\gamma_{i+j}}\\nabla^{\\gamma_{i+j}}\\bar\\rho\\|_{L^\\infty}\n\\|\\f{\\nabla^{N+1-l}v}{(1+|x|)^{l}}\\|_{L^2}\n\\leq C\\delta\\|\\nabla^{N+1}v\\|_{L^2}.\n\\end{split}\n\\end{equation*}\nFor $1\\leq m\\leq l-1$,  the Sobolev inequality\nand decay estimate \\eqref{n1h1} imply\n\\begin{equation*}\n\\begin{split}\n&\\sum_{2\\leq l\\leq N-1}\n\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l-1}}\n  \\|\\nabla^{\\gamma_1}n\\|_{L^\\infty}\\cdots\\|\\nabla^{\\gamma_{i}}n\\|_{L^\\infty}\n  \\|(1+|x|)^{\\gamma_{i+1}}\\nabla^{\\gamma_{i+1}}\\bar\\rho\\|_{L^\\infty}\n  \\cdots\\|(1+|x|)^{\\gamma_{i+j}}\\nabla^{\\gamma_{i+j}}\\bar\\rho\\|_{L^\\infty}\n  \\|\\f{\\nabla^{N+1-l}v}{(1+|x|)^{l-m}}\\|_{L^2}\\\\\n\\leq&C\\sum_{2\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots\n                                    +\\gamma_{i}=m\\\\1\\leq m\\leq l-1}}\n\\|\\nabla^{\\gamma_1+1}n\\|_{H^1}\\cdots\\|\\nabla^{\\gamma_{i}+1}n\\|_{H^1}\\|\\nabla^{N+1-m}v\\|_{L^2}\n\\leq C(1+t)^{-\\f{4+N}{2}}.\n\\end{split}\n\\end{equation*}\nFor $m=l$, the Sobolev inequality\nand decay estimate \\eqref{n1h1} yield directly\n\\begin{equation*}\n\\begin{split}\n&\\sum_{2\\leq l\\leq N-1}\\sum_{\\gamma_1+\\cdots+\\gamma_{i}=l}\n\\|\\nabla^{\\gamma_1}n\\|_{L^\\infty}\n\\cdots\\|\\nabla^{\\gamma_{i-1}}n\\|_{L^\\infty}\n\\|\\nabla^{\\gamma_{i}}n\\|_{L^3}\\|\\nabla^{N+1-l}v\\|_{L^6}\\\\\n\\leq\n&\\sum_{2\\leq l\\leq N-1}\\sum_{\\gamma_1+\\cdots+\\gamma_{i}=l}\\|\\nabla^{\\gamma_1+1}n\\|_{H^1}\\cdots\n\\|\\nabla^{\\gamma_{i-1}+1}n\\|_{H^1}\\|\\nabla^{\\gamma_{i}}n\\|_{H^1}\\|\\nabla^{N+1-l}v\\|_{H^1}\n\\leq C(1+t)^{-\\f{4+N}{2}}.\n\\end{split}\n\\end{equation*}\nTherefore, we can obtain the following estimate\n\\begin{equation}\\label{m421}\n\\begin{split}\n\t\\sum_{2\\leq l\\leq N-1}\\|\\nabla^l\\wf(n+\\bar\\rho)\\nabla^{N+1-l}v\\|_{L^2}\n\t\\leq C\\delta\\|\\nabla^{N+1}v\\|_{L^2}+C(1+t)^{-\\f{4+N}{2}},\n\\end{split}\n\\end{equation}\nwhich, together with the estimate \\eqref{m42}, yields directly\n\\begin{equation}\\label{M42}\nM_{42}\n\t\\leq \\ep \\|\\nabla^Nn\\|_{L^2}^2+C_{\\ep}\\delta \\|\\nabla^{N+1}v\\|_{L^2}^2+C_{\\ep}(1+t)^{-4-N}.\n\\end{equation}\nOne can deal with the term $M_{43}$ in the manner of $M_{42}$. It holds true\n\\begin{equation}\\label{m43}\n\\begin{split}\n\tM_{43}\n\t\\leq&\\ep \\|\\nabla^Nn\\|_{L^2}^2+C_{\\ep}\\|(n,\\bar\\rho)\\|_{L^\\infty}^2\\|\\nabla^{N}n\\|_{L^2}^2+C_{\\ep}\\sum_{1\\leq l\\leq N-1}\\|\\nabla^l\\wg(n+\\bar\\rho)\\nabla^{N-l}n\\|_{L^2}^2\\\\\n\t\\leq&(\\ep+C_\\ep\\delta) \\|\\nabla^Nn\\|_{L^2}^2+C_{\\ep}\\sum_{2\\leq l\\leq N-1}\\|\\nabla^l\\wg(n+\\bar\\rho)\\nabla^{N-l}n\\|_{L^2}^2.\n\\end{split}\n\\end{equation}\nSimilar to the estimate \\eqref{m421}, it is easy to check that\n\\begin{equation*}\n\\begin{split}\n\t\\sum_{2\\leq l\\leq N-1}\\|\\nabla^l\\wg(n+\\bar\\rho)\\nabla^{N+1-l}v\\|_{L^2}\n\t\\leq C\\delta\\|\\nabla^{N}n\\|_{L^2}+C(1+t)^{-\\f{3+N}{2}},\n\\end{split}\n\\end{equation*}\nwhich, together with the inequality \\eqref{m43}, yields directly\n\\begin{equation}\\label{M43}\n\\begin{split}\n\tM_{43}\n\t\\leq&(\\ep+C_\\ep\\delta) \\|\\nabla^Nn\\|_{L^2}^2+C_{\\ep}(1+t)^{-3-N}.\n\\end{split}\n\\end{equation}\nFinally, let us deal with the term $M_{44}$.\nIndeed, the Hardy inequality and identity \\eqref{h} yield directly\n\\begin{equation}\\label{m44}\n\t\\begin{split}\n\t\tM_{44}\t\t\\leq&\\ep\\|\\nabla^{N}n\\|_{L^2}^2+C_{\\ep}\\|\\h(n,\\bar\\rho)\\nabla^{N}\\bar\\rho\\|_{L^2}^2+C_{\\ep}\\sum_{1\\leq l\\leq N-1}\\|\\nabla^l\\h(n,\\bar\\rho)\\nabla^{N-l}\\bar\\rho\\|_{L^2}^2\\\\\n\t\t\\leq&\\ep\\|\\nabla^{N}n\\|_{L^2}^2+C_{\\ep}\\|\\f{n}{(1+|x|)^{N}}\\|_{L^2}^2\\|(1+|x|)^{N}\\nabla^{N}\\bar\\rho\\|_{L^\\infty}^2+C_{\\ep}\\sum_{1\\leq l\\leq N-1}\\|\\nabla^l\\h(n,\\bar\\rho)\\nabla^{N-l}\\bar\\rho\\|_{L^2}^2\\\\\n\t\t\\leq&(\\ep+C\\delta)\\|\\nabla^{N}n\\|_{L^2}^2+C_{\\ep}\\sum_{1\\leq l\\leq N-1}\\|\\nabla^l\\h(n,\\bar\\rho)\\nabla^{N-l}\\bar\\rho\\|_{L^2}^2.\n\t\\end{split}\n\t\\end{equation}\n\tTo deal with the last term on the right side of \\eqref{m44}, we employ the estimate \\eqref{h2} of $\\h$, to find\\begin{equation*}\n\t\\begin{split}\n\t\t&\\|\\nabla^l\\h(n,\\bar\\rho)\\nabla^{N-l}\\bar\\rho\\|_{L^2}\\\\\n\t\t\\leq&C\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{1+j}=l\\\\\\gamma_1=m\\\\0\\leq m\\leq l}}\\|\\f{\\nabla^{\\gamma_1}n}{(1+|x|)^{N-m}}\\|_{L^2}\\|(1+|x|)^{\\gamma_{2}}\\nabla^{\\gamma_{2}}\\bar\\rho\\|_{L^\\infty}\\cdots\\|(1+|x|)^{\\gamma_{1+j}}\\nabla^{\\gamma_{1+j}}\\bar\\rho\\|_{L^\\infty}\\|(1+|x|)^{N-l}\\nabla^{N-l}\\bar\\rho\\|_{L^\\infty}\\\\\n\t\t&\\quad+C\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l,i\\geq2}}\\|\\f{\\nabla^{\\gamma_1}n}{(1+|x|)^{N-m}}\\|_{L^2}\\|\\nabla^{\\gamma_2}n\\|_{L^\\infty}\n\\cdots\\|\\nabla^{\\gamma_i}n\\|_{L^\\infty}\n\\|(1+|x|)^{\\gamma_{i+1}}\\nabla^{\\gamma_{i+1}}\\bar\\rho\\|_{L^\\infty}\\\\\n&\\quad  \\quad  \\quad  \\quad  \\quad  \\quad \\quad \\quad \\quad \\cdots\\|(1+|x|)^{\\gamma_{i+j}}\\nabla^{\\gamma_{i+j}}\\bar\\rho\\|_{L^\\infty}\\|(1+|x|)^{N-l}\\nabla^{N-l}\\bar\\rho\\|_{L^\\infty}\\\\\n\t\t\\overset{def}{=}&M_{441}+M_{442}.\n\t\\end{split}\n\t\\end{equation*}\nIn view of Hardy inequality, we can obtain the estimate\n\\begin{equation*}\n\\begin{split}\nM_{441}\n\t\\leq& C\\delta\\|\\nabla^N n\\|_{L^2}.\n\\end{split}\n\\end{equation*}\nTo deal with the term $M_{442}$, we can apply Hardy inequality and the decay estimate \\eqref{n1h1} to obtain\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\tM_{442}\n\t\t\t\\leq& C\\delta\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\0\\leq m\\leq l,~i\\geq2}}\\|\\nabla^{N-m+\\gamma_1}n\\|_{L^2}\\|\\nabla^{\\gamma_2+1}n\\|_{H^1}\\cdots\\|\\nabla^{\\gamma_i+1}n\\|_{H^1}\n\t\t\t\\leq C(1+t)^{-\\f{4+N}{2}}.\n\t\t\\end{split}\n\t\t\\end{equation*}\nThe combination of estimates from term $M_{441}$ to $M_{442}$ yields directly\n\\begin{equation*}\n\\sum_{1\\leq l\\leq N-1}\\|\\nabla^l\\h(n,\\bar\\rho)\\nabla^{N-l}\\bar\\rho\\|_{L^2}\n\\leq C\\delta\\|\\nabla^{N}n\\|_{L^2}+C(1+t)^{-\\f{4+N}{2}},\n\\end{equation*}\nwhich, together with the inequality \\eqref{m44}, yields directly\n\t\\begin{equation}\\label{M44}\n\t\\begin{split}\n\t\tM_{44}\n\t\t\\leq(\\ep+C_{\\ep}\\delta)\\|\\nabla^{N}n\\|_{L^2}^2+C_{\\ep}(1+t)^{-4-N}.\n\t\\end{split}\n\t\\end{equation}\nConsequently, by virtue of the estimates \\eqref{m4}, \\eqref{M41},\n\\eqref{M42}, \\eqref{M43} and \\eqref{M44}, it holds true\n\\begin{equation}\\label{M4}\nM_4 \\leq (\\ep+C_{\\ep}\\delta) \\|\\nabla^N n\\|_{L^2}^2\n    +C_{\\ep}\\delta \\|\\nabla^{N+1}v\\|_{L^2}^2+C_{\\ep}(1+t)^{-3-N}.\n\\end{equation}\nSubstituting the estimates \\eqref{N1}-\\eqref{N3} and \\eqref{M4} into \\eqref{f2}, we find\n\\begin{equation*}\n\\begin{split}\n&-\\f {d}{dt}\\int_{|\\xi|\\geq\\eta}\\widehat{\\nabla^{N}v}\\cdot \\overline{\\widehat{\\nabla^{N-1}n}}d\\xi+\\gamma\\int_{|\\xi|\\geq\\eta}|\\widehat{\\nabla^Nn}|^2d\\xi\\\\\n&\\leq(\\ep+C_{\\ep}\\delta)\\|\\nabla^N n\\|_{L^2}^2+C_{\\ep}\\|(\\nabla^N v,\\nabla^{N+1}v)\\|_{L^2}^2+C_{\\ep}(1+t)^{-3-N}.\n\t\\end{split}\n\t\\end{equation*}\nDue to the definition \\eqref{def-h-l}, there exists a positive constant $C$ such that\n\t\\begin{equation}\\label{vhvl}\n\t\\begin{split}\n\t\t\\|\\nabla^N v^h\\|_{L^2}^2\\leq C\\|\\nabla^{N+1}v^h\\|_{L^2}^2,\\quad \\|\\nabla^{N+1} v^l\\|_{L^2}^2\\leq C \\|\\nabla^{N}v^l\\|_{L^2}^2,\n\t\\end{split}\n\t\\end{equation}\n\tand choosing $\\ep$ and $\\delta$ suitably small, we deduce that\n\t\\begin{equation}\\label{en5}\n\t\\begin{split}\n\t\t&-\\f{d}{dt}\\int_{|\\xi|\\geq\\eta}\\widehat{\\nabla^{N}v}\\cdot \\overline{\\widehat{\\nabla^{N-1}n}}d\\xi+\\gamma\\int_{|\\xi|\\geq\\eta}|\\widehat{\\nabla^Nn}|^2d\\xi\n\t\t\\leq C\\|\\nabla^N (n,v)^l\\|_{L^2}^2+C_{3} \\|\\nabla^{N+1}v^h\\|_{L^2}^2+C(1+t)^{-3-N}.\n\t\\end{split}\n\t\\end{equation}\nRecalling the estimate \\eqref{en2} in Lemma \\ref{enn}, one has the following estimate\n\\begin{equation}\\label{en4}\n\\f{d}{dt}\\|\\nabla^{N}(n,v)\\|_{L^2}^2+\\|\\nabla^{N+1}v\\|_{L^2}^2\\leq C\\delta \\|(\\nabla^{N}n,\\nabla^{N+1}v)\\|_{L^2}^2.\n\\end{equation}\nMultiplying \\eqref{en5} by $\\eta_3$, then adding to \\eqref{en4}, and choosing $\\delta$ and $\\eta_3$ suitably small, then we have\n\\begin{equation*}\n\\begin{aligned}\t\t&\\f{d}{dt}\\Big\\{\\|\\nabla^{N}(n,v)\\|_{L^2}^2-\\eta_3\\int_{|\\xi|\\geq\\eta}\\widehat{\\nabla^{N-1}v}\\cdot \\overline{\\widehat{\\nabla^{N}n}}d\\xi \\Big\\}\n+\\|\\nabla^{N+1}v\\|_{L^2}^2+\\eta_3\\|\\nabla^{N}n^h\\|_{L^2}^2\\\\\n\\leq &C_4\\|\\nabla^{N}(n^l,v^l)\\|_{L^2}^2+C(1+t)^{-3-N}.\n\\end{aligned}\n\\end{equation*}\nUsing \\eqref{vhvl} once again, we obtain that\n\\begin{equation*}\n\\begin{aligned}\t\n&\\f{d}{dt}\\Big\\{\\|\\nabla^{N}(n,v)\\|_{L^2}^2\n  -\\eta_3\\int_{|\\xi|\\geq\\eta}\\widehat{\\nabla^{N-1}v}\\cdot\n  \\overline{\\widehat{\\nabla^{N}n}}d\\xi \\Big\\}\n  +\\|\\nabla^{N}v^h\\|_{L^2}^2+\\eta_3\\|\\nabla^{N}n^h\\|_{L^2}^2\\\\\n\\leq & C_4\\|\\nabla^{N}(n^l, v^l)\\|_{L^2}^2+C(1+t)^{-3-N}.\n\\end{aligned}\n\\end{equation*}\nTherefore, we complete the proof of this lemma.\n\\end{proof}\n\n\n    Observe the right handside of the estimate \\eqref{en6} in Lemma \\ref{highfrequency}, we need to estimate the low frequency of $\\nabla^N(n,v)$.\n    For this purpose, we need to analyze the initial value problem for linearized system of \\eqref{ns5}:\n    \\begin{equation}\\label{linear}\n    \\left\\{\\begin{array}{lr}\n\t\\widetilde n_t +\\gamma\\mathop{\\rm div}\\nolimits\\widetilde v=0,\\quad (t,x)\\in \\mathbb{R}^{+}\\times \\mathbb{R}^3,\\\\\n\t\\widetilde u_t+\\gamma\\nabla \\widetilde n-\\mu_1\\tri \\widetilde v-\\mu_2\\nabla\\mathop{\\rm div}\\nolimits \\widetilde v =0,\\quad (t,x)\\in \\mathbb{R}^{+}\\times \\mathbb{R}^3,\\\\\n\t(\\widetilde n,\\widetilde v)|_{t=0}=(n_0,v_0),\\quad x\\in \\mathbb{R}^3.\n    \\end{array}\\right.\n    \\end{equation}\n    In terms of the semigroup theory for evolutionary equations, the solution $(\\widetilde n,\\widetilde v)$ of the linearized system \\eqref{linear} can be expressed as\n    \\begin{equation}\\label{U}\n    \\left\\{\\begin{array}{lr}\n    \\widetilde U_t=A\\widetilde U,\\quad t\\geq0,\\\\\n    \\widetilde U(0)=U_0,\n\\end{array}\\right.\\end{equation}\n    where $\\widetilde U \\overset{def}{=}(\\widetilde{n},\\widetilde{v})^t$,\n    $U_0 \\overset{def}=(n_0,v_0)$\n    and the matrix-valued differential operator $A$ is given by\n    \\begin{equation*}\n    A = {\\left(\n\t\\begin{matrix}\n\t\t0 & -\\gamma \\mathop{\\rm div}\\nolimits \\\\\n\t\t-\\gamma \\nabla & \\mu_1\\tri+\\mu_2\\nabla\\mathop{\\rm div}\\nolimits\n\t\\end{matrix}\n\t\\right).}\n    \\end{equation*}\n    Denote $S(t)\\overset{def} =e^{tA}$, then the system \\eqref{U} gives rise to\n    \\begin{equation}\\label{uexpress}\\widetilde U(t)=S(t)U_0=e^{tA} U_0,\\quad t\\geq0. \\end{equation}\n    Then, it is easy to check that the following estimate holds\n    \\begin{equation}\\label{linearlow}\n    \\|\\nabla^N(S(t)U_0)\\|_{L^2}\\leq C(1+t)^{-\\f34-\\f N2}\\|U_0\\|_{L^1\\cap H^N},\n    \\end{equation}\n    where $C$ is a positive constant independent of time.\n    The estimate \\eqref{linearlow} can be found in \\cite{chen2021,duan2007}.\n    Finally, let us denote $F(t)=(\\widetilde{S_1}(t),\\widetilde{S_2}(t))^{tr}$, then\n    the system \\eqref{ns5} can be rewritten as follows:\n    \\begin{equation}\\label{nonlinear}\n    \\left\\{\\begin{array}{lr}\n    \tU_t=A U+F,\\\\\n    \tU(0)=U_0.\n    \\end{array}\\right.\n    \\end{equation}\n    Then we can use Duhamel's principle to represent the solution of system \\eqref{ns5} in term of the semigroup\n    \\begin{equation}\\label{Uexpress}\n    U(t)=S(t)U_0+\\int_0^tS(t-\\tau)F(\\tau)d\\tau.\n    \\end{equation}\n    Now, one can establish the estimate for the low frequency of $\\nabla^N(n,v)$ as follows:\n\\begin{lemm}\\label{lowfrequency}\nUnder the assumption of Theorem \\ref{them3}, we have\n\\begin{equation}\\label{lowfre}\n\\|\\nabla^N(n^l, v^l)(t)\\|_{L^2}\n\\leq C \\delta \\sup_{0\\leq s\\leq t}\\|\\nabla^N(n,v)(s)\\|_{L^2}+C(1+t)^{-\\f34-\\f N2},\n\\end{equation}\nwhere $C$ is a positive constant independent of time.\n\\end{lemm}\n\\begin{proof}\n\tIt follows from the formula \\eqref{Uexpress} that\n    \\begin{equation*}\n\t\\nabla^N(n, v)=\\nabla^N(S (t)U_0)+\\int_0^t\\nabla^{N}[S(t-\\tau)F(\\tau)]d\\tau,\n\t\\end{equation*}\n\twhich yields directly\n\t\\begin{equation}\\label{nvexpress}\n\t\\|\\nabla^N(n^l, v^l)\\|_{L^2}\n    \\leq \\|\\nabla^N(S (t)U_0)^l\\|_{L^2}+\\int_0^t\\|\\nabla^{N}[S(t-\\tau)F(\\tau)]^l\\|_{L^2}d\\tau.\n\t\\end{equation}\n\tSince the initial data $U_0=(n_0,v_0)\\in L^1\\cap H^N$, it follows from the\n    estimate \\eqref{linearlow} that\n    \\begin{equation}\\label{U0es}\n\t\\|\\nabla^N(S (t)U_0)^l\\|_{L^2}\\leq C(1+t)^{-\\f34-\\f N2}\\|U_0\\|_{L^1\\cap H^N}.\n\t\\end{equation}\n\tThen we compute by means of Sobolev inequality that\\begin{equation}\\label{nvl}\n\t\\begin{split}\n\t\t&\\int_0^t\\|\\nabla^{N}[S(t-\\tau)F(\\tau)]^l\\|_{L^2}d\\tau\n\t\t\\leq\\int_0^t\\||\\xi|^{N}|\\widehat{S}(t-\\tau)||\\widehat F(\\tau)|\\|_{L^2(|\\xi|\\leq \\eta)}d\\tau\\\\\n\t\t\\leq&\\int_0^{\\f t2}\\||\\xi|^{N}|\\widehat{S}(t-\\tau)|\\|_{L^2(|\\xi|\\leq \\eta)}\\|\\widehat F(\\tau)\\|_{L^\\infty(|\\xi|\\leq \\eta)}d\\tau+\\int_{\\f t2}^t\\||\\xi||\\widehat{S}(t-\\tau)|\\|_{L^2(|\\xi|\\leq \\eta)}\\||\\xi|^{N-1}\\widehat F(\\tau)\\|_{L^\\infty(|\\xi|\\leq \\eta)}d\\tau\\\\\n\t\t\\leq&\\int_0^{\\f t2}(1+t-\\tau)^{-\\f34-\\f N2}\\|\\widehat F(\\tau)\\|_{L^\\infty(|\\xi|\\leq \\eta)}d\\tau+\\int_{\\f t2}^t(1+t-\\tau)^{-\\f54}\\||\\xi|^{N-1}\\widehat F(\\tau)\\|_{L^\\infty(|\\xi|\\leq \\eta)}d\\tau\\\\\n\t\t\\overset{def}{=}&N_1+N_2.\n\t\\end{split}\n\t\\end{equation}\n\tNow we estimate the first term on the right handside of \\eqref{nvl} as follows:\n\\begin{equation}\\label{ss1N}\n\\begin{split}\nN_1=\n\\int_0^{\\f t2}(1+t-\\tau)^{-\\f34-\\f N2}\\|F\\|_{L^1}d\\tau\n\\leq C\\int_0^{\\f t2}(1+t-\\tau)^{-\\f34-\\f N2} \\big(\\|\\widetilde{S}_1\\|_{L^1}+\\|\\widetilde{S}_2\\|_{L^1}\\big)d\\tau.\n\\end{split}\n\\end{equation}\nIn view of the definitions of $\\widetilde{S}_i(i=1,2)$\nand decay estimate \\eqref{n1h1}, we have\n\\begin{equation}\\label{SS1}\n\\|\\widetilde{S}_1\\|_{L^1}\\leq C\\|\\nabla (n,v)\\|_{L^2}\\|(n,v)\\|_{L^2}+C\\|(1+|x|)\\nabla \\bar\\rho\\|_{L^2}\\|\\f{v}{1+|x|}\\|_{L^2}+C\\|\\bar\\rho\\|_{L^2}\\|\\nabla v\\|_{L^2}\\leq C\\delta (1+t)^{-\\f54},\n\\end{equation}\nand\n\\begin{equation}\\label{SS2}\n\\begin{split}\n\t\t\\|\\widetilde{S}_2\\|_{L^1}\\leq& C\\|v\\|_{L^2}\\|\\nabla v\\|_{L^2}+C\\|\\wf(n+\\bar\\rho)\\|_{L^2}\\|\\nabla^2 v\\|_{L^2}+C\\|\\wg(n+\\bar\\rho)\\|_{L^2}\\|\\nabla n\\|_{L^2}+C\\|\\h(n,\\bar\\rho)\\|_{L^2}\\|\\nabla \\bar\\rho\\|_{L^2}\\\\\n\t\t\\leq&C\\|v\\|_{L^2}\\|\\nabla v\\|_{L^2}+C\\|n\\|_{L^2}\\|\\nabla^2 v\\|_{L^2}+C\\|\\bar\\rho\\|_{L^2}\\|\\nabla^2 v\\|_{L^2}+C\\|n\\|_{L^2}\\|\\nabla n\\|_{L^2}\\\\\n\t\t&\\quad+C\\|\\bar\\rho\\|_{L^2}\\|\\nabla n\\|_{L^2}+C\\|\\f{n}{1+|x|}\\|_{L^2}\\|(1+|x|)\\nabla \\bar\\rho\\|_{L^2}\\\\\n\t\t\\leq&C\\delta (1+t)^{-\\f54}.\n\t\\end{split}\n\t\\end{equation}\nSubstituting the estimates \\eqref{SS1} and \\eqref{SS2} into \\eqref{ss1N},\nand using the estimate in Lemma \\ref{tt2}, it holds\n\\begin{equation}\\label{F1}\n\t\\begin{split}\n\t\tN_1\\leq C \\int_0^{\\f t2}(1+t-\\tau)^{-\\f34-\\f N2}(1+\\tau)^{-\\f54}d\\tau\\leq C (1+t)^{-\\f34-\\f N2}.\n\\end{split}\n\\end{equation}\nNext, let us deal with the $N_2$ term.\nFor any smooth function $\\phi$, there exists a positive constant $C$\ndependent only on $\\eta$, such that\n$$\\||\\xi|^{N-1}\\widehat \\phi\\|_{L^\\infty(|\\xi|\\leq \\eta)}\\leq C\\||\\xi|^{N-2}\\widehat \\phi\\|_{L^\\infty(|\\xi|\\leq \\eta)},$$\n\tthen we find that\n\\begin{equation}\\label{L}\n\t\\begin{split}\n\t\t\\||\\xi|^{N-1}\\widehat F\\|_{L^\\infty(|\\xi|\\leq \\eta)}\n\t\t\\leq& C \\|[\\nabla^{N-1}\\widetilde {S}_1]^l\\|_{L^1}+C\\|[\\nabla^{N-1}\\widetilde {S}_2]^l\\|_{L^1}\\\\\n\t\t\\leq&C\\|[\\nabla^{N-1}(\\nabla nv+n\\nabla v)]^l\\|_{L^1}+C\\|[\\nabla^{N-1}(\\nabla \\bar\\rho v+\\bar\\rho\\nabla v)]^l\\|_{L^1}+C\\|[\\nabla^{N-1}(v\\nabla v)]^l\\|_{L^1}\\\\\n\t\t&\\quad+C\\|[\\nabla^{N-2}\\big(\\wf(n+\\bar\\rho)(\\mu_1\\tri v+\\mu_2\\nabla \\mathop{\\rm div}\\nolimits v)\\big)]^l\\|_{L^1}+C\\|[\\nabla^{N-1}(\\wg(n+\\bar\\rho)\\nabla n)]^l\\|_{L^1}\\\\\n\t\t&\\quad+C\\|[\\nabla^{N-1}(\\h(n,\\bar\\rho)\\nabla \\bar\\rho)]^l\\|_{L^1}\\\\\n        \\overset{def}{=}&N_{21}+N_{22}+N_{23}+N_{24}+N_{25}+N_{26}.\n\t\\end{split}\n\t\\end{equation}\n\tFirst of all, applying the decay estimate \\eqref{n1h1}, then the term $N_{21}$ can be estimated as follows\n    \\begin{equation}\\label{L1}\n\t\\begin{split}\n\t\tN_{21}\\leq C\\sum_{0\\leq l\\leq N-1}\\big(\\|\\nabla^{l+1}n\\|_{L^2}\\|\\nabla^{N-l-1}v\\|_{L^2}+\\|\\nabla^{l}n\\|_{L^2}\\|\\nabla^{N-l}v\\|_{L^2}\\big)\n\t\t\\leq C (1+t)^{-1-\\f N2}.\n\t\\end{split}\n\t\\end{equation}\n\tBy virtue of Hardy inequality, we have\\begin{equation}\\label{L222}\n\t\\begin{split}\n\t\tN_{22}\\leq&C\\sum_{0\\leq l\\leq N-1}\\Big(\\|(1+|x|)^{l+1}\\nabla^{l+1}\\bar\\rho\\|_{L^2}\\|\\f{\\nabla^{N-l-1}v}{(1+|x|)^{l+1}}\\|_{L^2}+\\|(1+|x|)^{l}\\nabla^{l}\\bar\\rho\\|_{L^2}\\|\\f{\\nabla^{N-l}v}{(1+|x|)^{l}}\\|_{L^2} \\Big)\\\\\n\t\t\\leq&C \\delta\\|\\nabla^N v\\|_{L^2}.\n\t\\end{split}\n\t\\end{equation}\n\tIn view of the decay estimate \\eqref{n1h1}, it follows directly\n\t\\begin{equation}\\label{LL3}\n\t\\begin{split}\n\t\tN_{23}\\leq C\\sum_{0\\leq l\\leq N-1}\\|\\nabla^{l}v\\|_{L^2}\\|\\nabla^{N-l}v\\|_{L^2}\n\t\t\\leq C (1+t)^{-1-\\f N2}.\n\t\\end{split}\n\t\\end{equation}\n\tApplying the estimate \\eqref{widef} of function $\\wf$, we deduce that\\begin{equation}\\label{L4}\n\t\\begin{split}\n\t\tN_{24}\n\t\t\\leq&C\\int|\\wf(n+\\bar\\rho)||\\nabla^{N}v|dx+C\\sum_{1\\leq l\\leq N-2}\\sum_{\\gamma_1+\\cdots+\\gamma_{j}=l}\\int|\\nabla^{\\gamma_{1}}\\bar\\rho|\\cdots|\\nabla^{\\gamma_{j}}\\bar\\rho||{\\nabla^{N-l}v}|dx\\\\\n\t\t&\\quad+C\\sum_{1\\leq l\\leq N-2}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l}}\\int|\\nabla^{\\gamma_1}n|\\cdots|\\nabla^{\\gamma_{i}}n||\\nabla^{\\gamma_{i+1}}\\bar\\rho|\\cdots|\\nabla^{\\gamma_{i+j}}\\bar\\rho||{\\nabla^{N-l}v}|dx\\\\\n\t\t\\overset{def}{=}&N_{241}+N_{242}+N_{243}.\n\t\\end{split}\n    \\end{equation}\n\tIt follows from H{\\\"o}lder inequality that\\begin{equation*}\n\t\\begin{split}\n\t\tN_{241}\\leq&C\\|(n+\\bar\\rho)\\|_{L^2}\\|\\nabla^N v\\|_{L^2}\\leq C\\delta\\|\\nabla^N v\\|_{L^2}.\n\t\\end{split}\n    \\end{equation*}\n\tBy Hardy inequality, it is easy to deduce\\begin{equation*}\n\t\\begin{split}\n\t\tN_{242}\\leq& C\\sum_{1\\leq l\\leq N-2}\\sum_{\\gamma_1+\\cdots+\\gamma_{j}=l}\\|(1+|x|)^{\\gamma_{1}}\\nabla^{\\gamma_{1}}\\bar\\rho\\|_{L^2}\\|(1+|x|)^{\\gamma_{2}}\\nabla^{\\gamma_{2}}\\bar\\rho\\|_{L^\\infty}\\cdots\\|(1+|x|)^{\\gamma_{j}}\\nabla^{\\gamma_{j}}\\bar\\rho\\|_{L^\\infty}\\|\\f{\\nabla^{N-l}v}{(1+|x|)^{l}}\\|_{L^2}\\\\\n\t\t\\leq& C\\delta\\|\\nabla^N v\\|_{L^2}.\n\t\\end{split}\n\t\\end{equation*}\n\tWithout loss of generality, we assume that $1\\leq \\gamma_1\\leq\\cdots\\leq \\gamma_{i}\\leq N-2$. The fact $i\\geq2$ implies $\\gamma_{i-1}\\leq N-3\\leq N-2$. Thus, we can exploit Hardy inequality, Sobolev interpolation inequality \\eqref{Sobolev} in Lemma \\ref{inter} and the decay estimate \\eqref{n1h1} to obtain\\begin{equation*}\\begin{split}\n\t\tN_{243}\\leq&C\\sum_{1\\leq l\\leq N-2}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l}}\\|\\nabla^{\\gamma_1}n\\|_{L^\\infty}\\cdots\\|\\nabla^{\\gamma_{i-1}}n\\|_{L^\\infty}\\|\\nabla^{\\gamma_{i}}n\\|_{L^2}\\|(1+|x|)^{\\gamma_{i+1}}\\nabla^{\\gamma_{i+1}}\\bar\\rho\\|_{L^\\infty}\\\\\n\t\t&\\quad\\cdots\\|(1+|x|)^{\\gamma_{i+j}}\\nabla^{\\gamma_{i+j}}\\bar\\rho\\|_{L^\\infty}\\|\\f{\\nabla^{N-l}v}{(1+|x|)^{l-m}}\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\sum_{1\\leq l\\leq N-2}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l}}\\|\\nabla^{\\gamma_1+1}n\\|_{H^1}\\cdots\\|\\nabla^{\\gamma_{i-1}+1}n\\|_{H^1}\\|\\nabla^{\\gamma_i}n\\|_{L^2}\\|\\nabla^{N-m}v\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\sum_{1\\leq l\\leq N-2}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l}}(1+t)^{-\\f N2-\\f{5i}{4}-\\f74}\n\t\t\\leq C (1+t)^{-\\f32-\\f N2}.\n\t\\end{split}\n\t\\end{equation*}\n\tSubstituting the estimates of $N_{241}$, $N_{242}$ and $N_{243}$ into \\eqref{L4}, we arrive at\n\t\\begin{equation}\\label{L24}\n\tN_{24}\\leq C (1+t)^{-\\f32-\\f N2}+C\\delta\\|\\nabla^N v\\|_{L^2}.\n\t\\end{equation}\n\tOne can deal with the term $N_{25}$ in the same manner of $N_{24}$.\n    Then, it is easy to check that\n    \\begin{equation}\\label{L5}\n\tN_{25}\\leq C(1+t)^{-\\f32-\\f N2}+C\\delta \\|\\nabla^N n\\|_{L^2}.\n\t\\end{equation}\n\tFinally, let us deal with $N_{26}$. By virtue of the estimate \\eqref{h} and \\eqref{h2} of $\\h$, then we have\\begin{equation}\\label{l6}\\begin{split}\n\t\tN_{26}\\leq& C\\int|n||\\nabla^N \\bar\\rho|dx+ C\\sum_{1\\leq l\\leq N-1}\\sum_{\\gamma_1+\\cdots+\\gamma_{j}=l}\\int|n||\\nabla^{\\gamma_{1}}\\bar\\rho|\\cdots|\\nabla^{\\gamma_{j}}\\bar\\rho||\\nabla^{N-l}\\bar\\rho|dx\\\\\n\t\t&\\quad+C\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l}}\\int|\\nabla^{\\gamma_1}n|\\cdots|{\\nabla^{\\gamma_i}n}||\\nabla^{\\gamma_{i+1}}\\bar\\rho|\\cdots|\\nabla^{\\gamma_{i+j}}\\bar\\rho||\\nabla^{N-l} \\bar\\rho|dx\\\\\n\t\t\\overset{def}{=}& N_{261}+N_{262}+N_{263}.\n\t\\end{split}\n\t\\end{equation}\n\tIt follows from Hardy inequality that\\begin{equation}\\label{N261-2}\n\t\\begin{split}\n\t\tN_{261}+N_{262}\\leq& C\\|\\f{n}{(1+|x|)^N}\\|_{L^2}\\|(1+|x|)^N\\nabla^N \\bar\\rho\\|_{L^2}+C\\sum_{1\\leq l\\leq N-1}\\sum_{\\gamma_1+\\cdots+\\gamma_{j}=l}\\|\\f{n}{(1+|x|)^{N}}\\|_{L^2}\\\\\n\t\t&\\quad\\times\\|(1+|x|)^{\\gamma_{1}}\\nabla^{\\gamma_{1}}\\bar\\rho\\|_{L^\\infty}\\cdots\\|(1+|x|)^{\\gamma_{j}}\\nabla^{\\gamma_{j}}\\bar\\rho\\|_{L^\\infty}\\|(1+|x|)^{N-l}\\nabla^{N-l} \\bar\\rho\\|_{L^2}\\\\\n\t\t\\leq& C\\delta\\|\\nabla^Nn\\|_{L^2}.\n\t\\end{split}\n\t\\end{equation}\n\tFor the term $N_{263}$, it is easy to check that\\begin{equation}\\label{N263}\n\t\\begin{split}\n\tN_{263}\\leq&C\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1=m\\\\1\\leq m\\leq l}}\\|\\f{\\nabla^{\\gamma_1}n}{(1+|x|)^{N-\\gamma_1}}\\|_{L^2}\\|(1+|x|)^{\\gamma_{2}}\\nabla^{\\gamma_{2}}\\bar\\rho\\|_{L^\\infty}\\cdots\\|(1+|x|)^{\\gamma_{1+j}}\\nabla^{\\gamma_{1+j}}\\bar\\rho\\|_{L^\\infty}\\\\\n\t&\\quad\\|(1+|x|)^{N-l}\\nabla^{N-l} \\bar\\rho\\|_{L^2}+C\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l,i\\geq2}}\\|\\nabla^{\\gamma_1}n\\|_{L^\\infty}\\cdots\\|\\nabla^{\\gamma_{i-1}}n\\|_{L^\\infty}\\|\\f{\\nabla^{\\gamma_i}n}{(1+|x|)^{N-m}}\\|_{L^2}\\\\\n\t&\\quad\\times\\|(1+|x|)^{\\gamma_{i+1}}\\nabla^{\\gamma_{i+1}}\\bar\\rho\\|_{L^\\infty}\\cdots\\|(1+|x|)^{\\gamma_{i+j}}\\nabla^{\\gamma_{i+j}}\\bar\\rho\\|_{L^\\infty}\\|(1+|x|)^{N-l}\\nabla^{N-l} \\bar\\rho\\|_{L^2}\\\\\n\t\\overset{def}{=}&N_{2631}+N_{2632}.\n\t\\end{split}\n\t\\end{equation}\n\tWe employ Hardy inequality once again, to discover\n\t\\begin{equation*}\n\t\\begin{split}\n\t\tN_{2631}\n\t\t\\leq& C\\d\\|\\nabla^Nn\\|_{L^2}.\n\t\\end{split}\n\t\\end{equation*}\n\tWithout loss of generality, we assume that $1\\leq \\gamma_1\\leq\\cdots\\leq \\gamma_{i}\\leq N-1$. The fact $i\\geq2$ implies $\\gamma_{i}\\leq m-1\\leq N-2$. For the term $N_{2632}$, by virtue of Hardy inequality, Sobolev interpolation inequality \\ref{Sobolev} in Lemma \\ref{inter} and the decay estimate \\eqref{n1h1}, we deduce\n\t\\begin{equation*}\n\t\\begin{split}\n\t\tN_{2632}\n\t\t\\leq&C\\delta\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l,i\\geq2}}\\|\\nabla^{\\gamma_1+1}n\\|_{H^1}\\cdots\\|\\nabla^{\\gamma_{i-1}+1}n\\|_{H^1}\\|\\nabla^{N-m+\\gamma_i}n\\|_{L^2}\\\\\n\t\t\\leq&C\\delta\\sum_{1\\leq l\\leq N-1}\\sum_{\\substack{\\gamma_1+\\cdots+\\gamma_{i+j}=l\\\\\\gamma_1+\\cdots+\\gamma_{i}=m\\\\1\\leq m\\leq l,i\\geq2}} (1+t)^{-\\f N2+\\f12-\\f{5i}{4}}\n\t\t\\leq C (1+t)^{-2-\\f N2}.\n\t\\end{split}\n\t\\end{equation*}\n\tSubstituting estimates of terms $N_{2631}$ and $N_{2632}$ into \\eqref{N263}, we find\\begin{equation*}\n\tN_{263}\\leq C\\d\\|\\nabla^Nn\\|_{L^2}+C(1+t)^{-2-\\f N2}.\n\t\\end{equation*}\n\tInserting \\eqref{N261-2} and \\eqref{N263} into \\eqref{l6}, we get immediately\\begin{equation}\\label{L6}\n\tN_{26}\\leq C\\delta \\|\\nabla^N n\\|_{L^2}+C(1+t)^{-2-\\f N2}.\n\t\\end{equation}\n\tWe then conclude from \\eqref{L}-\\eqref{LL3}, \\eqref{L24}, \\eqref{L5} and \\eqref{L6} that\\begin{equation*}\n\t\\begin{split}\n\t\t\\||\\xi|^{N-1}\\widehat F\\|_{L^\\infty(|\\xi|\\leq \\eta)}\\leq&C\\delta\\|\\nabla^N (n,v)\\|_{L^2}+ (1+t)^{-1-\\f N2},\n\t\\end{split}\n\t\\end{equation*}\n\twhich, together with the definition of term $N_2$ and\n   the estimate in Lemma \\ref{tt2}, yields directly\n   \\begin{equation}\\label{kF1}\n\t\\begin{split}\n\t\tN_2\n\t\t\\leq&C \\int_{\\f t2}^t(1+t-\\tau)^{-\\f54}\\big(\\delta\\|\\nabla^N (n,v)\\|_{L^2}+ (1+\\tau)^{-1-\\f N2}\\big)d\\tau\\\\\n\t\t\\leq&C\\delta\\sup_{0\\leq\\tau\\leq t}\\|\\nabla^N (n,v)\\|_{L^2}\\int_{\\f t2}^t(1+t-\\tau)^{-\\f54}d\\tau+C(1+t)^{-1-\\f N2}\\\\\n\t\t\\leq&C\\delta\\sup_{0\\leq\\tau\\leq t}\\|\\nabla^N (n,v)\\|_{L^2}+C(1+t)^{-1-\\f N2}.\n\t\\end{split}\\end{equation}\n\tSubstituting \\eqref{F1} and \\eqref{kF1} into \\eqref{nvl}, it holds true\n    \\begin{equation}\\label{Unon}\n\t\\int_0^t\\|\\nabla^{N}[S(t-\\tau)F(U(\\tau))]^l\\|_{L^2}d\\tau\n    \\leq C\\delta\\sup_{0\\leq\\tau\\leq t}\\|\\nabla^N (n,v)\\|_{L^2}+C(1+t)^{-\\f34-\\f N2}.\n\t\\end{equation}\n\tInserting \\eqref{U0es} and \\eqref{Unon} into \\eqref{nvexpress}, one obtains immediately that\n\t\\begin{equation*}\n\t\\|\\nabla^N(n^l,v^l)\\|_{L^2}\\leq C \\delta \\sup_{0\\leq s\\leq t}\\|\\nabla^N(n,v)\\|_{L^2}+C(1+t)^{-\\f34-\\f N2}.\n\t\\end{equation*}\n\tThis completes the proof of this lemma.\n\\end{proof}\n\nFinally, we focus on establishing optimal decay rate for the $N-th$ order spatial derivative of solution.\n\\begin{lemm}\\label{optimaln}\n\tUnder the assumption of Theorem \\ref{them3}, we have\n\t\\begin{equation}\\label{n1h2}\n\t\\|\\nabla^N(n,v)(t)\\|_{L^2}\\leq C (1+t)^{-\\f34-\\f N2},\n\t\\end{equation}\n\twhere $C$ is a positive constant independent of $t$.\n\\end{lemm}\n\\begin{proof}\n\tWe may rewrite the estimate \\eqref{en6} in Lemma \\ref{highfrequency} as\\begin{equation}\\label{ddt}\n\t\\f{d}{dt}\\widetilde{\\mathcal E}^N(t)+\\|\\nabla^{N}v^h\\|_{L^2}^2+\\eta_3\\|\\nabla^{N}n^h\\|_{L^2}^2\n\t\\leq  C_4\\|\\nabla^{N}(n,v)^l\\|_{L^2}^2+C(1+t)^{-3-N}.\n\t\\end{equation}\n\twhere the energy $\\widetilde{\\mathcal E}^N(t)$ is defined by\n\t\\[\\widetilde{\\mathcal E}^N(t)\\overset{def}{=}\\|\\nabla^{N}(n,v)\\|_{L^2}^2-\\eta_3\\int_{|\\xi|\\geq\\eta}\\widehat{\\nabla^{N-1}v}\\cdot \\overline{\\widehat{\\nabla^{N}n}}d\\xi.\\]\n\tWith the help of Young inequality, by choosing $\\eta_3$ small enough, one obtains the equivalent relation\\begin{equation}\\label{endj}\n\tc_5\\|\\nabla^{N}(n,v)\\|_{L^2}^2\\leq\\widetilde{\\mathcal E}^N(t)\\leq c_6 \\|\\nabla^{N}(n,v)\\|_{L^2}^2,\n\t\\end{equation}\n\twhere the constants $c_5$ and $c_6$ are independent of time.\n\tThen adding on both sides of \\eqref{ddt} by $\\|\\nabla^{N}(n^l,v^l)\\|_{L^2}^2$\n     and applying the estimate \\eqref{lowfre} in Lemma \\ref{lowfrequency}, we find\n     \\begin{equation*}\n\t\\begin{split}\n\t\t\\f{d}{dt}\\widetilde{\\mathcal E}^N(t)+\\|\\nabla^{N}(n,v)\\|_{L^2}^2\n\t\t\\leq ( C_4+1)\\|\\nabla^{N}(n^l,v^l)\\|_{L^2}^2+C(1+t)^{-3-N}\\leq C\\delta \\sup_{0\\leq \\tau\\leq t}\\|\\nabla^N(n,v)\\|_{L^2}^2+C(1+t)^{-\\f32-N}.\n\t\\end{split}\n\t\\end{equation*}\n   By virtue of the equivalent relation \\eqref{endj}, we have\\begin{equation}\\label{en7}\n\t\\begin{split}\n\t\t\t\\f{d}{dt}\\widetilde{\\mathcal E}^N(t)+\\widetilde{\\mathcal E}^N(t)\n\t\t\\leq C\\delta \\sup_{0\\leq \\tau\\leq t}\\|\\nabla^N(n,v)\\|_{L^2}^2+C(1+t)^{-\\f32-N},\n\t\\end{split}\n\t\\end{equation}\n\twhich, using Gronwall inequality, gives immediately\\begin{equation}\\label{estimateE}\n\t\\begin{split}\n\t\t\\widetilde{\\mathcal E}^N(t)\\leq e^{-t} \\widetilde{\\mathcal E}^N(0)+C\\delta\\sup_{0\\leq \\tau\\leq t}\\|\\nabla^N(n,v)\\|_{L^2}^2\\int_0^te^{\\tau-t}d\\tau+C\\int_0^te^{\\tau-t}(1+\\tau)^{-\\f32-N}d\\tau.\n\t\\end{split}\n\t\\end{equation}\n\tIt is easy to deduce that\n    $$\n\t\\int_0^te^{\\tau-t}d\\tau\\leq C\n    $$\n    and\n    $$\\int_0^te^{\\tau-t}(1+\\tau)^{-\\f32-N}d\\tau\\leq C(1+t)^{-\\f32-N}.$$\n\tFrom the equivalent relation \\eqref{endj} and \\eqref{estimateE}, it holds\n\t\\begin{equation*}\n\t\\begin{split}\n\t\t\\sup_{0\\leq \\tau\\leq t}\\|\\nabla^N(n,v)(\\tau)\\|_{L^2}^2\\leq Ce^{-t}\\|\\nabla^N(n_0,v_0)\\|_{L^2}^2+C\\delta\\sup_{0\\leq \\tau\\leq t}\\|\\nabla^N(n,v)\\|_{L^2}^2+ C(1+t)^{-\\f32-N}.\n\t\\end{split}\n\t\\end{equation*}\n\tApplying the smallness of $\\delta$, one arrives at\n    \\begin{equation*}\n\t\t\\sup_{0\\leq \\tau\\leq t}\\|\\nabla^N(n,v)(\\tau)\\|_{L^2}^2\\leq C(1+t)^{-\\f32-N}.\n\t\\end{equation*}\n\tTherefore, we cpmlete the proof of this lemma.\n\\end{proof}\n\n\\underline{\\noindent\\textbf{The Proof of Theorem  \\ref{them3}.}}\nCombining the estimate \\eqref{n1h1} in Lemma \\ref{N-1decay} with\nestimate \\eqref{n1h2} in Lemma \\ref{optimaln}, then we can obtain the\ndecay rate \\eqref{kdecay} in Theorem \\ref{them3}.\n  Consequently, we finish the proof of Theorem \\ref{them3}.\n\n\n\\subsection{Lower bound of decay rate}\\label{lower}\nIn this subsection, the content of our analysis is to establish the lower bound of decay rate for the global solution and its spatial derivatives of the initial value problem \\eqref{ns5}.\nIn order to achieve this target, we need to analyze the linearized system \\eqref{linear}.\nWe obtain the following proposition immediately, whose proof is similar to \\cite{chen2021} and standard, so we omit here.\n\\begin{prop}\\label{lamma-lower}\nLet $U_0=(n_0,v_0)\\in L^1(\\R^3)\\cap H^l(\\R^3)$ with $l\\geq3$,\nassume that $M_n\\overset{def}{=} \\int_{\\R^3}n_0(x) d x$\nand $M_v\\overset{def}{=}\\int_{\\R^3}v_0(x) d x$\nare at least one nonzero.\nThen there exists a positive constant $\\widetilde c$ independent of time\nsuch that for any large enough $t$, the global solution $(\\widetilde n,\\widetilde v)$\nof the linearized system \\eqref{linear} satisfies\n\\begin{equation}\\label{linearnudecay}\n\\min\\{\\|{\\partial}_{x}^{k} \\widetilde n(t)\\|_{L^2(\\R^3)},\n      \\|{\\partial}_{x}^{k} \\widetilde v(t)\\|_{L^2(\\R^3)}\\}\n\\geq \\widetilde c(1+t)^{-\\f34-\\f{k}{2}},\\quad\\text{for}~~ 0\\leq k\\leq l.\n\\end{equation}\nHere $\\widetilde c$ is a positive constant depending only on $M_n$ and $M_v$.\n\\end{prop}\n\nDefine the difference $(n_{\\d},v_{\\d})\\overset{def}{=}(n-\\widetilde n,v-\\widetilde v)$,\nthen the quantity $(n_{\\d},v_{\\d})$ satisfies the following system:\n\\begin{equation}\\label{ns7}\n\\left\\{\\begin{array}{lr}\n\tn_{\\d t} +\\gamma\\mathop{\\rm div}\\nolimits v_{\\d}=\\widetilde S_1,\\quad (t,x)\\in \\mathbb{R}^{+}\\times \\mathbb{R}^3,\\\\\n\tv_{\\d t}+\\gamma\\nabla n_{\\d}-\\mu_1\\tri v_{\\d}-\\mu_2\\nabla\\mathop{\\rm div}\\nolimits v_{\\d} =\\widetilde  S_2,\\quad (t,x)\\in \\mathbb{R}^{+}\\times \\mathbb{R}^3,\\\\\n\t(n_\\d,v_\\d)|_{t=0}=(0,0).\n\\end{array}\\right.\n\\end{equation}\nBy virtue of the formula \\eqref{Uexpress}, the solution of system \\eqref{ns7}\ncan be represented as\\begin{equation*}\n(n_\\d,v_\\d)^{t}=\\int_0^tS(t-\\tau)F(U(\\tau))d\\tau.\n\\end{equation*}\nNext, we aim to establish the upper bounds of decay rates for the difference $(n_\\d,v_\\d)$ and its first order spatial derivative.\n\\begin{lemm}\n\tUnder the assumptions of Theorem \\ref{them4}, assume $(n_\\d,v_\\d)$ be the smooth solution of the initial value problem \\eqref{ns7}. Then, it holds on for $t\\geq0$,\n\t\\begin{equation}\\label{nuddecay}\n\t\\|\\nabla^k(n_\\d,v_\\d)(t)\\|_{L^2}\\leq \\widetilde C\\d(1+t)^{-\\f34-\\f k2},\\quad \\text{for}~~k=0,1,\n\t\\end{equation}\n\twhere $\\widetilde C$ is a constant independent of time.\n\\end{lemm}\n\\begin{proof}\n\tBy Duhamel's principle, it holds for $k\\geq0$,\\begin{equation}\\label{nvd}\n\t\\|\\nabla^k(n_\\d,v_\\d)(t)\\|_{L^2}\\leq \\int_0^t(1+t-\\tau)^{-\\f34-\\f k2}\\big(\\|(\\widetilde {S}_1,\\widetilde {S}_2)(\\tau)\\|_{L^1}+\\|\\nabla^k(\\widetilde {S}_1,\\widetilde {S}_2)(\\tau)\\|_{L^2}\\big)d\\tau.\n\t\\end{equation}\n\tIt then follows from decay estimates \\eqref{SS1} and \\eqref{SS2} that\\begin{equation}\\label{s1s2}\n\t\\|(\\widetilde {S}_1,\\widetilde {S}_2)\\|_{L^1}\\leq C\\d(1+t)^{-\\f54}.\n\t\\end{equation}\nIn view of Sobolev and Hardy inequality, it is easy to deduce that\n\\begin{equation}\\label{ds1}\n\\|\\widetilde {S}_1\\|_{L^2}\n\\leq C\\|\\bar\\rho\\|_{L^\\infty}\\|\\nabla n\\|_{L^2}\n     +C\\|(1+|x|)\\nabla\\bar\\rho\\|_{L^\\infty}\\|\\f{n}{1+|x|}\\|_{L^2}\n\\leq C\\delta\\|\\nabla n\\|_{L^2}\\leq C\\d(1+t)^{-\\f54},\n\\end{equation}\nand\n\\begin{equation}\\label{ds2}\n\\begin{aligned}\n\\|\\widetilde {S}_2\\|_{L^2}\n\\leq& C\\|v\\|_{L^\\infty}\\|\\nabla v\\|_{L^2}\n      +\\|(n+\\bar\\rho)\\|_{L^\\infty}\\|(\\nabla^2 v,\\nabla n)\\|_{L^2}\n      +\\|\\f{n}{1+|x|}\\|_{L^\\infty}\\|(1+|x|)\\nabla \\bar\\rho\\|_{L^2}\\\\\n\\leq& C\\delta\\|(\\nabla n,\\nabla v,\\nabla^2v)\\|_{L^2}\\leq C\\delta(1+t)^{-\\f54}.\n\\end{aligned}\n\\end{equation}\nThen, substituting the decay estimates \\eqref{s1s2}, \\eqref{ds1}\nand \\eqref{ds2} into \\eqref{nvd} with $k=0$, it holds\n\\begin{equation*}\n\\begin{split}\n\\|(n_\\d,v_\\d)(t)\\|_{L^2}\\leq& \\int_0^t(1+t-\\tau)^{-\\f34}\\big(\\|(\\widetilde {S}_1,\\widetilde {S}_2)(\\tau)\\|_{L^1}+\\|(\\widetilde {S}_1,\\widetilde {S}_2)(\\tau)\\|_{L^2}\\big)d\\tau\\\\\n\t\t\\leq& C\\d \\int_0^t(1+t-\\tau)^{-\\f34}(1+\\tau)^{-\\f54}d\\tau\\\\\n\t\t\\leq& C\\d(1+t)^{-\\f34},\n\\end{split}\n\\end{equation*}\nwhere we have used the Lemma \\ref{tt2} in the last inequality.\nSimilar to \\eqref{ds1} and \\eqref{ds2}, one arrives at\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\|\\nabla\\widetilde {S}_1\\|_{L^2}\\leq& C\\|\\bar\\rho\\|_{L^\\infty}\\|\\nabla^2 n\\|_{L^2}+C\\|(1+|x|)\\nabla\\bar\\rho\\|_{L^\\infty}\\|\\f{\\nabla n}{1+|x|}\\|_{L^2}+C\\|(1+|x|)^2\\nabla^2\\bar\\rho\\|_{L^\\infty}\\|\\f{n}{(1+|x|)^2}\\|_{L^2}\\\\\n\t\t\\leq& C\\delta\\|\\nabla^2 n\\|_{L^2}\\leq C\\d(1+t)^{-\\f74},\\\\\n\t\t\\|\\nabla\\widetilde {S}_2\\|_{L^2}\\leq& C\\|v\\|_{L^\\infty}\\|\\nabla^2 v\\|_{L^2}+\\|\\nabla v\\|_{L^3}\\|\\nabla v\\|_{L^6}+\\|n+\\bar\\rho\\|_{L^\\infty}\\|(\\nabla^3 v,\\nabla^2 n)\\|_{L^2}+\\|\\nabla n\\|_{L^3}\\|(\\nabla^2v,\\nabla n)\\|_{L^6}\\\\\n\t\t&+\\|(1+|x|)\\nabla\\bar\\rho\\|_{L^\\infty}\\|(\\f{\\nabla^2v}{1+|x|},\\f{\\nabla n}{1+|x|})\\|_{L^2}+\\|(1+|x|)\\nabla \\bar\\rho\\|_{L^\\infty}^2\\|\\f{ n}{(1+|x|)^2}\\|_{L^2}\\\\\n\t\t\\leq& C\\delta\\|(\\nabla^2 n,\\nabla^2 v,\\nabla^3v)\\|_{L^2}\\leq C\\delta(1+t)^{-\\f74},\n\t\\end{split}\n\t\\end{equation*}\nwhich, together with \\eqref{nvd}, \\eqref{s1s2} and Lemma \\ref{tt2},\nyields directly\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\|\\nabla(n_\\d,v_\\d)(t)\\|_{L^2}\\leq& \\int_0^t(1+t-\\tau)^{-\\f54}\\big(\\|(\\widetilde {S}_1,\\widetilde {S}_2)(\\tau)\\|_{L^1}+\\|\\nabla(\\widetilde {S}_1,\\widetilde {S}_2)(\\tau)\\|_{L^2}\\big)d\\tau\\\\\n\t\t\\leq&C\\d\\int_0^t(1+t-\\tau)^{-\\f54}(1+\\tau)^{-\\f54}d\\tau\\leq C\\d(1+t)^{-\\f54}.\n\t\\end{split}\n\t\\end{equation*}\n\tTherefore, the proof of this lemma is completed.\n\\end{proof}\nFinally, we establish the lower bound of decay rate for the global solution and its spatial derivatives of compressible Navier-Stokes equations with potential force.\nThen, the estimate \\eqref{lower-estimate} in Lemma \\ref{lemmalower} below will\nyield the optimal decay estimate in Theorem \\ref{them4}.\n\\begin{lemm}\\label{lemmalower}\nUnder the assumptions of Theorem \\ref{them4}, then for any large enough $t$, we have\n\\begin{equation}\\label{lower-estimate}\n\\min\\{\\|\\nabla^kn(t)\\|_{L^2},\\|\\nabla^kv(t)\\|_{L^2} \\}\n\\geq c_1(1+t)^{-\\f34-\\f k2},\\quad \\text{for}~~ 0\\leq k\\leq N,\n\\end{equation}\nwhere $c_1$ is a positive constant independent of time.\n\\end{lemm}\n\\begin{proof}\nBy virtue of the definition of $n_\\d$, it holds true\n\\begin{equation*}\n\\|\\nabla^k\\widetilde n\\|_{L^2}\\leq \\|\\nabla^kn\\|_{L^2}+\\|\\nabla^kn_\\d\\|_{L^2},\n\\end{equation*}\nwhich, together with the  lower bound decay \\eqref{linearnudecay} and upper bound decay \\eqref{nuddecay}, yields directly\n\\begin{equation}\\label{lowlow-01}\n\\|\\nabla^kn\\|_{L^2}\\geq \\|\\nabla^k\\widetilde n\\|_{L^2}-\\|\\nabla^kn_\\d\\|_{L^2}\n\\geq  \\widetilde c(1+t)^{-\\f34-\\f k2}-\\widetilde C\\d (1+t)^{-\\f34-\\f k2},\n\\end{equation}\nwhere $k=0,1$. It is worth noting that the small constant $\\delta$ is used\nto control the upper bound of initial data in $L^2-$norm instead of $L^1$ one\n(see \\eqref{phik} and \\eqref{initial-H2}).\nFrom the estimate \\eqref{linearnudecay} in Lemma \\ref{lamma-lower},\nthe constant $\\widetilde c$ in \\eqref{lowlow-01} only depends on the quantities $M_n$ and $M_v$.\nThen, we can choose $\\delta$ small enough such that\n$\\widetilde C \\d\\leq\\f{1}{2} \\widetilde c$, and hence,\nit follows from \\eqref{lowlow-01} that\n\\begin{equation}\\label{lowlow}\n\\|\\nabla^k n\\|_{L^2} \\geq  \\frac12\\widetilde c(1+t)^{-\\f34-\\f k2}, \\quad k=0,1.\n\\end{equation}\nBy virtue of the Sobolev interpolation inequality in Lemma \\ref{inter},\nit holds true for $k\\geq2$\n\\begin{equation*}\n\\|\\nabla n\\|_{L^2}\\leq C\\|n\\|_{L^2}^{1-\\f1k}\\|\\nabla^kn\\|_{L^2}^{\\f1k},\n\\end{equation*}\nwhich, together with the lower bound decay \\eqref{lowlow}\nand upper bound decay \\eqref{kdecay}, implies directly\n\\begin{equation}\\label{highlow}\n\\|\\nabla^kn\\|_{L^2}\n\\geq C\\|\\nabla n\\|_{L^2}^k\\|n\\|_{L^2}^{-(k-1)}\n\\geq C(1+t)^{-\\f{5k}{4}}(1+t)^{\\f{3(k-1)}{4}}\n\\geq c_1(1+t)^{-\\f34-\\f k2},\n\\end{equation}\nfor all $k \\geq 2$.\nIn the same manner, it is easy to deduce that\n\\begin{equation}\\label{vhighlow}\n\\|\\nabla^kv\\|_{L^2}\\geq c_1(1+t)^{-\\f34-\\f k2},\\quad \\text{for}~~k \\geq 0.\n\\end{equation}\nThen, the combination of estimates \\eqref{lowlow}, \\eqref{highlow}\nand \\eqref{vhighlow} yields the estimate \\eqref{lower-estimate}.\nTherefore, we complete the proof of this lemma.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\n\n\n\nThis research was partially supported by NNSF of China(11801586, 11971496, 12026244), Guangzhou Science and technology project of China(202102020769), Natural Science Foundation of Guangdong Province of China (2020A1515110942),\nNational Key Research and Development Program of China(2020YFA0712500).\n\n\n\\phantomsection\n\\addcontentsline{toc}{section}{\\refname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}