diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzhepz" "b/data_all_eng_slimpj/shuffled/split2/finalzzhepz" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzhepz" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nUnderstanding the evolution of the carbon monoxide (CO) molecular abundance across redshifts is important from the point of view of galaxy formation, and the star formation history of the universe \\citep[for a recent review, see e.g.,][]{madau2014}. CO is strongly associated with star-forming galaxies \\citep{carilli2013}, and is the second most abundant molecular species in the interstellar medium, next to molecular hydrogen (H$_2$). {{The CO luminosity and gas mass of local galaxies are well-correlated with the far-infrared luminosity ($L_{\\rm FIR}$) which, in turn, is an indicator of star formation rate \\citep{kennicutt1998}. The CO molecule (unlike H$_2$) has a permanent dipole moment and a `ladder' of states for the rotational transitions, making it an ideal probe of the cold neutral phase of the interstellar medium (ISM). The two brightest CO lines are the (1-0) and (2-1) transitions with frequencies 115 and 230 GHz respectively, which also have overlaps with the frequency bands in CMB observations. CO has been studied both with surveys of local galaxies, e.g. \\citet{young1995, helfer2003, leroy2009} as well as in individual systems, e.g. \\citet{aravena2012, walter2014}.}}\n\nIntensity mapping, in which it is attempted to image the aggregate emission from several sources over very large volumes, does not require the resolution of individual galaxies. This technique has been successfully used to constrain the abundance and clustering of neutral hydrogen (HI) systems around redshift $z \\sim 1$ \\citep{chang10, masui13, switzer13}, and is a promising probe of cosmology, large scale structure in the universe and galaxy evolution. {The CO molecule offers interesting prospects for intensity mapping, both at intermediate and high redshifts. Intensity mapping with the CO line provides information about the spatial distribution of star formation.} At high redshifts ($z > 6$), performing a CO intensity mapping survey is expected to lead to valuable insights into the physics of galaxies that reionized the universe. It is also a useful probe of the earliest epochs of star formation activity \\citep{carilli2011} and large scale structure \\citep{righi2008, mashian2015}. \n\nAt intermediate redshifts ($1 < z < 4$), there are good prospects for detecting CO in intensity mapping surveys, especially around the peak of star formation activity at around $z \\sim 2-3$ \\citep[e.g.,][]{lilly1996, madau1998}. Recently, \\citet{keating2016} have provided the latest constraints on the CO (1-0) intensity power spectrum from the CO Power Spectrum Survey (COPSS) at $z \\sim 3$. Future surveys such as the CO Mapping Array Pathfinder (COMAP)\\footnote{http:\/\/www.astro.caltech.edu\/CRAL\/projects.html\\#comap} will aim to detect CO in emission over redshifts 2-3. There are also prospects for intensity mapping cross-correlations with the results from other surveys such as with neutral hydrogen (HI) and the redshifted \\textsc{c ii} (158 $\\mu$m) {transition} \\citep[e.g.,][]{switzer2017}. With the advent of facilities like the ALMA,\\footnote{http:\/\/www.almaobservatory.org\/} (Atacama Large Millimetre Array) and instruments on the LMT (Large Millimetre Telescope)\\footnote{http:\/\/www.lmtgtm.org\/}, a large number of CO detections in emission from galaxies will be possible. It will be also possible to observe higher transitions of the CO ladder of states, and cross-correlations of multiple spectral lines originating from the same redshift are expected to be useful in statistically isolating the intensity fluctuations \\citep{visbal2010}. \n\nOn the theoretical front, a number of approaches have focused on modelling the intensity mapping signal from various CO transitions at different redshifts, to be detected with current and future facilities. These include simulations and semi-analytical methods (SAMs) of galaxy formation \\citep{obreschkow2009, fu2012, li2015} which model the metallicity, atomic and molecular gas evolution in galaxies, as well as more empirical techniques starting from the far infrared (FIR) luminosity function \\citep{vallini2016}. {These have been studied both around the reionization epoch \\citep[$z \\gtrsim 6$;][]{righi2008, visbal2010,lidz2011, gong2011}, and at the peak of the star formation history \\citep[$z \\sim 2$;][]{pullen2013, li2015}.} One of the chief astrophysical uncertainties in the modeling of the CO power spectrum comes from the knowledge of the CO luminosity ($L_{\\rm CO}$) - host halo mass ($M$) relation. Various functional forms for this relationship have been suggested in the literature, based on the results of forward modelling such as SAMs and hydrodynamical simulations (a recent summary is provided in \\citet{li2015}). The theoretical models are found to lead to predictions spanning over an order of magnitude in the CO power spectrum \\citep[e.g.,][]{breysse2014, li2015}.\n\nIn this paper, we adopt a complementary, empirical approach, anchored to the observational data, towards understanding the evolution of the $L_{\\rm CO} - M$ relationship. We begin by reviewing (Sec. \\ref{sec:formalism}) the standard formalism and ingredients for calculating the CO power spectrum from intensity mapping. We also provide an overview of the theoretical models of CO from the literature. Using the empirically determined relations between the CO luminosity, star formation rate and host halo mass, we connect the low redshift CO galaxy measurements and the higher-redshift constraints from intensity mapping into an analytical halo model in Sec. \\ref{sec:model}. We provide convenient fitting forms and estimates on the errors in the derived parameters, that agree well with the observations of the CO luminosity function at intermediate redshifts. The form of the halo model enables ease of comparison to the empirically determined stellar mass-halo mass relation. In Sec. \\ref{sec:compare}, we explore the consistency of the approach with the results of previous literature, and estimate the mean and uncertainties in the CO power spectrum to be observed with current and future facilities. We summarize the results and discuss the future outlook in a brief concluding section (Sec. \\ref{sec:conc}). Throughout the paper, we use the $\\Lambda CDM$ cosmology with the cosmological parameters $h = 0.71, \\Omega_m = 0.281, \\Omega_b = 0.046, \\Omega_{\\Lambda} = 0.719, \\sigma_8 = 0.8, n_s = 0.963$.\n\n\n\n\\section{Formalism}\n\\label{sec:formalism}\nIn this section, we briefly review the standard equations involved in calculating the CO power spectrum observed in intensity mapping, similar studies are outlined in \\citet{breysse2014}, \\citet{mashian2015}, \\citet{pullen2013}.\n\nThe specific intensity of a CO line observed at a frequency, $\\nu_{\\rm obs}$ is given by:\n\\begin{equation}\n I(\\nu_{\\rm obs}) = \\frac{c}{4 \\pi} \\int_0^{\\infty} dz' \\frac{\\epsilon[\\nu_{\\rm obs} (1 + z')]}{H(z') (1 + z')^4}\n\\end{equation} \nin which $H(z)$ is the Hubble parameter at redshift $z$, and $\\epsilon[\\nu_{\\rm obs} (1 + z')]$ is the proper volume emissivity of the emitted line.\nWith the assumption that the profile of each CO line is a delta function at the frequency $\\nu_J$, we can express the emissivity as an integral of the host halo mass $M$:\n\\begin{equation}\n \\epsilon(\\nu, z) = \\delta_D(\\nu - \\nu_J) (1 + z)^3 f_{\\rm duty} \\int_{M_{\\rm min, CO}}^{\\infty} dM \\frac{dn}{dM} L_{\\rm CO}(M,z)\n\\end{equation} \nHere, $L_{\\rm CO} (M,z)$ is the specific luminosity of the CO line and it is assumed that a fraction $f_{\\rm duty}$ of all haloes above a mass $M_{\\rm min, CO}$ contribute to the CO emission.\n\nWith this, the specific intensity can be rewritten as:\n\\begin{equation}\nI(\\nu_{\\rm obs}) = \\frac{c}{4 \\pi} \\frac{1}{\\nu_{\\rm J} H(z_{\\rm J})} f_{\\rm duty} \\int_{M_{\\rm min, CO}}^{\\infty} dM \\frac{dn}{dM} L_{\\rm CO}(M,z)\n\\label{COspint}\n\\end{equation} \nThe brightness temperature, $T_{\\rm CO}$ can be derived from the specific intensity through the relation $I(\\nu_{\\rm obs}) = 2 k_B \\nu_{\\rm obs}^2 T_{\\rm CO} \/c^2$. \nThus the expression for the brightness temperature becomes:\n\\begin{equation}\n\\langle T_{\\rm CO} \\rangle = \\frac{c^3}{8 \\pi}\\frac{(1 + z_J)^2}{k_B \\nu_J^3 H(z_J)} f_{\\rm duty} \\int_{M_{\\rm min, CO}}^{\\infty} dM \\frac{dn}{dM} L_{\\rm CO}(M,z)\n\\label{tco}\n\\end{equation} \nIn order to derive the power spectrum to be observed in typical CO intensity mapping experiments, one also needs to model the clustering of the CO sources. In analogy with the methods for other species, e.g., neutral hydrogen intensity mapping, this can be done by weighting the dark matter halo bias by the CO luminosity-halo mass relation.\nWe thus have the expression for the clustering of CO sources:\n\\begin{equation}\n b_{\\rm CO}(z) = \\frac{\\int_{M_{\\rm min, CO}}^{\\infty} dM (dn\/dM) L_{\\rm CO} (M,z) b(M,z)}{\\int_{M_{\\rm min, CO}}^{\\infty} dM (dn\/dM) L_{\\rm CO} (M,z)}\n\\end{equation} \n{ where the $b(z)$ is the dark matter halo bias, e.g., given by \\citet{scoccimarro2001}}.\nThe shot noise contribution to the power, due to the number of haloes, can now be expressed as:\n\\begin{equation}\n P_{\\rm shot}(z) = \\frac{1}{f_{\\rm duty}}\\frac{\\int_{M_{\\rm min, CO}}^{\\infty} dM (dn\/dM) L_{\\rm CO} (M,z)^2}{\\left(\\int_{M_{\\rm min, CO}}^{\\infty} dM (dn\/dM) L_{\\rm CO} (M,z)\\right)^2}\n\\end{equation} \nGiven the above two expressions, we can express the signal (the power spectrum of the CO intensity fluctuations) as:\n\\begin{equation}\n P_{\\rm CO}(k,z) = \\langle T_{\\rm CO} \\rangle (z)^2 [b_{\\rm CO}(z)^2 P_{\\rm lin}(k,z) + P_{\\rm shot}(z)]\n\\end{equation} \nas a function of $k$ at every redshift, from which we also have the power spectrum in logarithmic $k$-bins:\n\\begin{equation}\n \\Delta_{k}^2(z) = \\frac{k^3 P_{\\rm CO}(k,z)^2}{2 \\pi^2}\n \\label{COpowspeclog}\n\\end{equation} \n\nIn some studies, e.g. \\citet{lidz2011}, $L_{\\rm CO}(M)$ is simply modelled as a linear relation: $L_{\\rm CO}(M) = A_{\\rm CO} M$, and $A_{\\rm CO}$ is a proportionality constant. This reduces the expressions in Eqs. \\ref{COspint} - \\ref{COpowspeclog} to integrals over the dark matter halo mass alone.\n\n\n\\subsection{Models in the literature}\n{{We thus see that one of the main astrophysical uncertainties in the measurement of the CO intensity power spectrum comes from the CO luminosity to host halo mass relation. \nSeveral approaches in the literature have been used to model this relation, some of which are briefly summarized in \\citet{li2015}. The astrophysical modelling typically requires (a) an SFR$-M$ relation and (b) an $L_{\\rm CO} -$ SFR relation. The various approaches towards modeling these are briefly described below (unless otherwise specified, the halo mass $M$ is in units of $M_{\\odot}$, $L_{\\rm CO}$ is in units of $L_{\\odot}$ and the SFR is in units of $M_{\\odot}$\/yr):}}\n\\begin{enumerate}\n\\item In \\citet{visbal2010}, the star formation rate is calculated as a function of halo mass as \n\\begin{equation}\n\\mathrm{SFR} = 6.2 \\times 10^{-11} \\left(\\frac{1+z}{3.5}\\right)^{3\/2} M\n\\end{equation}\nand the CO luminosity is calculated from the SFR as:\n \\begin{equation}\nL_{\\rm CO} = 3.7 \\times 10^3 \\ \\mathrm{SFR}\n\\end{equation}\nbased on the observations of M82 in \\citet{weiss2005}.\n\\item In \\citet{pullen2013}, two models are described, Model A and Model B. In Model A, the star formation rate is calculated as:\n\\begin{equation}\n\\mathrm{SFR} = 1.2 \\times 10^{-11} M^{5\/3}\n\\end{equation}\nand the CO luminosity as\n\\begin{equation}\n{L_{\\rm CO}} = 3.2 \\times 10^4 \\ \\mathrm{SFR}^{3\/5}\n\\end{equation}\nwhich is derived from the relations for $L_{\\rm CO} - L_{\\rm IR}$ \\citep{daddi2010} and the $L_{\\rm IR} - \\rm{SFR}$ \\citet{kennicutt1998}.\nIn Model B, empirical fits to the SFR are used, and the power spectra are multiplied by a rescaling factor (which leads to about a factor 5 higher predicted brightness temperature at $z \\sim 3$.)\n\\item In \\citet{lidz2011}, the SFR is assumed proportional to the halo mass: \n\\begin{equation}\n\\mathrm{SFR} = 1.7 \\times 10^{-10} M\n\\end{equation} \nand it is also assumed proportional to the CO luminosity:\n\\begin{equation}\nL_\\mathrm{CO} = 3.2 \\times 10^4 \\ \\rm{SFR}\n\\end{equation}\nand the 5\/3 power (assumed in the previous models) is replaced by unity for simplicity. \n\\item In \\citet{carilli2011}, the SFR assumed is that required to reionize the universe and keep it ionized, this is converted into an FIR luminosity by the relation \\citep{kennicutt1998}:\n\\begin{equation}\nL_{\\rm FIR} = 1.1 \\times 10^{10} \\ \\rm{SFR} \n\\end{equation}\n{{The FIR luminosity is, in turn}}, related to the specific luminosity of the CO line, measured in units of K km\/s pc$^2$ \\citep[the median relation derived by][]{daddi2010}: \n\\begin{equation}\n L'_{\\rm CO} = 0.02 \\ L_{\\rm FIR} \n\\end{equation} \nwhich can then be connected to the CO luminosity using \n\\begin{equation}\nL_{\\rm CO} = 3.11 \\times 10^{-11} \\nu_r^3 L'_{\\rm CO} \n\\label{lsunlprime} \n \\end{equation} \n { {where $\\nu_r$ is the rest frequency of the transition under consideration.}}.\n\n\\item In \\citet{righi2008}, the SFR- halo mass relation is derived following a merger history calculation with the extended Press-Schechter formalism of dark matter haloes \\citep{lacey1993}. The SFR is then converted to CO luminosity using the scaling:\n\\begin{equation}\n L_{\\rm CO} = 3.7 \\times 10^3 \\ \\rm{SFR}\n\\end{equation}\nfrom \\citet{weiss2005}.\n\n\\item In \\citet{gong2011}, the $L_{\\rm CO}$ is modelled as a function of the halo mass at the reionization epoch ($z \\sim 6-8$). It is fit using a function from the results of the semi-analytic modelling of \\citet{obreschkow2009}):\n\\begin{equation}\nL_{\\rm CO} = L_0 \\left(1 + \\frac{M}{M_c}\\right)^{-d} \\left(\\frac{M}{M_c}\\right)^b\n\\end{equation}\nwith the values $L_0 = 4.3 \\times 10^6, 6.2 \\times 10^6, 4 \\times 10^6 L_{\\odot}$, $b = 2.4, 2.6, 2.8$ and $M_c = 3.5 \\times 10^{11}, 3.0 \\times 10^{11}, 2.0 \\times 10^{11} M_{\\odot}$ at redshifts 6, 7 and 8 respectively.\n\n\\item \\citet{breysse2015} assume an SFR - CO relation derived from the results of \\citet{carilli2013, pullen2013, lidz2011} based on the FIR - CO luminosity connection, which can be expressed as:\n\\begin{equation}\n\\mathrm{SFR} = 9.8 \\times 10^{-18} \\left(\\frac{A_{\\rm CO}}{2 \\times 10^{-6}}\\right) M^{5 b_{\\rm CO}\/3}\n\\end{equation}\nwhere $L_{\\rm CO} (M) = A_{\\rm CO} M^{b_{\\rm CO}}$, and the fiducial values are $A_{\\rm CO} = 2 \\times 10^{-6}, b_{\\rm CO} = 1$.\n\n\\item \\citet{mashian2015} use a large velocity gradient (LVG) modelling and an empirically determined star formation rate evolution to predict the power spectra corresponding to several CO transitions in the reionization era ($z \\sim 6-10$). The star formation rate is modelled as a function of halo mass in the double power-law form:\n\\begin{eqnarray}\n\\rm{SFR} &=& a_1 M^{b_1}, M\\leq M_c; \\nonumber \\\\\n\\rm{SFR} & = & a_2 M^{b_2}, M \\geq M_c\t\t\n\\end{eqnarray}\n{{where $\\{a_1, a_2, b_1, b_2\\} = \\{2.4 \\times 10^{-17}, 1.1 \\times 10^{-5}, 1.6, 0.6\\}$ are the fitted parameters and the turnover occurs at the characteristic mass scale $M_c \\approx 10^{11.6} M_{\\odot}$.}}\n\n\\item \\citet{li2015} use simulations of the galaxy-halo connection at redshifts 2.4-2.8 to model the intensity map and power spectrum of the CO (1-0) line at these redshifts.\n\n\\item \\citet{fu2012} use different star formation prescriptions applied to semi-analytic models of galaxy formation to study the evolution of metals, atomic and molecular gas in galaxies including CO.\n\n\\end{enumerate}\n\nThese different approaches outlined above are found to lead roughly to an order of magnitude variation in the predicted CO luminosity - halo mass relation \\citep[e.g.,][]{breysse2014, li2015}.\n\n\n\n\n\\section{Modelling the CO observables}\n\\label{sec:model}\n\n{{\nIn this section, we begin by compiling the data available so far\\footnote{We assume that the data and the errors quoted are representative. The method outlined, however, is sufficiently general as to be adapted to modifications and extensions to this data.} in the context of the CO luminosity function, {{which is used in the subsequent analyses.}}\n\n\\begin{enumerate}\n\n\\item \\citet{keres2003} use a sample of $\\sim$ 300 galaxies from the FCRAO Extragalactic CO Survey \\citep{young1995} at $z = 0$ to derive a CO Luminosity Function (LF); and show that it is well fit by a Schechter function.\n\n\\item \\citet{keating2016} provide constraints on the CO luminosity function at $z \\sim 2.8$ by the measurement of the CO power spectrum in the COPSS (CO Power Spectrum Survey), this finds \n\\begin{equation}\nP_{\\rm CO} = 3.0 \\pm 1.3 \\times 10^3 \\mu {\\rm{K}}^2 (h^{-1} {\\rm{Mpc}}^3)\n\\end{equation}\nat $z \\sim 2.8$. This is combined with the data from direct detection efforts to place constraints on the CO LF at $z \\sim 3$, again assuming a Schechter form.\n\n\n\\item \\citet{aravena2012} detect CO in (1-0) emission from a sample of four results from the Jansky Very Large Array (JVLA) survey at $z \\sim 1.55$.\n\n\\item \\citet{walter2014} use the results of a blind search in the Hubble Deep Field North (HDF-N) to place constraints on the CO luminosity function for the (1-0), (2-1) and (3-2) transitions at median redshifts 0.33, 1.52 and 2.75.\n\\end{enumerate}\n\nThe galaxy emission data \\citep{keres2003} suggest that the CO luminosity function, $\\phi(L_{\\rm CO})$ at $z\\sim 0-3$ closely follows a Schechter form. Although the intensity mapping measurement \\citep{keating2016} does not contain enough information to imply that the CO luminosity function at high redshifts is well fit by the Schechter function, the analysis suggests that it may a reasonable assumption in the light of the data available so far. Previous research in HI \\citep{hpar2017, hpgk2016, hparaa2016}, suggests that this form of the luminosity function, reminiscent of a similar form for the HI (or stellar) mass function, leads to a distinct $L_{\\rm CO}$-halo mass (or, equivalently, HI-halo mass) relation, when either derived directly \\citep[e.g.,][]{hpar2017} or by abundance matching \\citep[e.g.,][]{behroozi2013, moster2013, hpgk2016}. This assumes a monotonic relationship between the CO galaxies and the host haloes.\n}}\n\n{The data \\citep[e.g., from COLD GASS,][]{saintonge2011, saintonge2011a, catinella2013} support a power law relation between the CO luminosity $L_{\\rm CO}$ and the star formation rate, with an index $0.557 \\sim 0.6$, which is consistent with theoretical predictions.} The existence of the star-forming main sequence \\citep[SFMS, e.g.,][]{brinchmann2004, salim2007} which connects the star formation rate and stelllar mass of star-forming galaxies, supports a power-law relation between the SFR and the stellar mass ($M_{*}$) across a range of wavelengths and redshifts \\citep[e.g.,][]{daddi2007}, such that we have $\\mathrm{SFR} \\propto M_{*}^{\\beta}$ for both the $z > 1$ and $z < 1$ regimes. The relation is fairly tight and its normalization changes with redshift.\n\n\nThese findings can thus be combined to a power law form for the CO luminosity as a function of stellar mass. \nFurther, using abundance matching of galaxies to dark matter haloes in simulations, the stellar mass - halo mass relation has been effectively modeled with a double power law behaviour \\citep{moster2010, moster2013, behroozi2013}, and the evolution in the free parameters is fixed by matching to higher redshifts.\n\nThe above discussion offers support for a double power law behaviour for the $L_{\\rm CO}- M$ relation (at redshift $z$) of the form:\n\\begin{equation}\nL_{\\rm CO} (M, z) = 2N(z) M [(M\/M_1(z))^{-b(z)} + (M\/M_1(z))^{y(z)}]^{-1}\n\\label{mosterco}\n\\end{equation}\nwith free parameters $M_1(z)$, $N(z)$, $b(z)$ and $y(z)$. These parameters are themselves composed of two terms, a constant term for $z \\sim 0$, and an evolutionary component:\n\\begin{eqnarray}\n\\log M_1(z) &=& \\log M_{10} + M_{11}z\/(z + 1) \\nonumber \\\\\nN(z) &=& N_{10} + N_{11}z\/(z + 1) \\nonumber \\\\\nb(z) &=& b_{10} + b_{11}z\/(z + 1) \\nonumber \\\\\ny(z) &=& y_{10} + y_{11}z\/(z + 1)\n\\label{comoster}\n\\end{eqnarray}\n\nThe CO luminosity function from\n\\citet{keres2003} with a sample of $\\sim$ 300 galaxies (from the FCRAO Extragalactic CO Survey at $z = 0$) is well fit by a Schechter function of the form: \n\\begin{equation}\n\\phi (L_{\\rm CO}) = \\frac{dn}{d \\log L_{\\rm CO}} = (\\ln 10) \\ \\rho ^* \\left(\\frac{L_{\\rm CO}}{L*} \\right)^{\\alpha + 1} \\exp-\\left(\\frac{L_{\\rm CO}}{L*}\\right)\n\\end{equation}\nwith the best-fit parameters: $\\rho* = 0.00072 \\pm 0.00035 \\ \\mathrm{Mpc}^{-3} \\mathrm{mag}^{-1}, \\alpha = -1.30 \\pm 0.16$ and $L* = (1.0 \\pm 0.2) \\times 10^7$ Jy km\/s Mpc$^{-2}$. The data points are shown in Fig. \\ref{fig:lumfunc} in red along with the associated error bars. \n\nWe use the Sheth-Tormen \\citep[][]{sheth2002} prescription of for the dark matter halo mass function $dn\/dM$. To recover the $L_{\\rm CO} - M$ relation, we use the matching of the abundances of the halo mass function and the fitted CO\nluminosity function, which can be expressed as \\citep[e.g.,][]{vale2004}:\n\\begin{equation}\n \\int_{M (L_{\\rm CO})}^{\\infty} \\frac{dn}{ d \\log_{10} M'} \\ d \\log_{10} M' = \\int_{L_{\\rm CO}}^{\\infty} \\phi(L_{\\rm CO}) \\ d \\log_{10} L_{\\rm CO}\n \\label{eqn:abmatchco}\n\\end{equation}\nIn the above equation, $dn \/ d \\log_{10} M$ is the number density of dark matter haloes with logarithmic\nmasses between $\\log_{10} M$ and $\\log_{10} (M$ + $dM)$, and $\\phi(L_{\\rm CO})$ is the\ncorresponding number density of CO-luminous galaxies in logarithmic luminosity bins. Solving\nEquation~(\\ref{eqn:abmatchco}) gives a relation between the CO luminosity\n$L_{\\rm CO}$ and the halo mass $M$. This approach assumes\nthat there is a monotonic relationship between $L_{\\rm CO}$ and $M$, which is reasonable in the light of the current data.\n\n\n{{Abundance matching of the CO luminosity function obtained from \\citet{keres2003}, to the dark matter halo mass function gives:\n$M_{10} = (4.17 \\pm 2.03) \\times 10^{12} M_{\\odot}, N_{10} = 0.0033 \\pm 0.0016 \\ \\mathrm{K \\ km\/s \\ pc}^2 M_{\\odot}^{-1}, b_{10} = 0.95 \\pm 0.46, y_{10} = 0.66 \\pm 0.32$. The data are binned into equally spaced logarithmic bins in luminosity, between $\\log L_{\\rm CO} = 6$ and $\\log L_{\\rm CO} = 11$ in units of K km\/s pc$^2$, with a bin width = 0.1 dex. {\\footnote{ {It can be checked (by increasing the bin width to 0.5 dex) that these results are not sensitive to reasonable changes in the number of bins within the quoted uncertainties.}}} Errors on the parameters are estimated by a combination of the errors on the data and the fitting uncertainties. Plots of the (i) luminosity function data from the results of \\citet{keres2003}, and (ii) the derived luminosity function from the abundance matched $L_{\\rm CO} - M$ relation are shown in Fig. \\ref{fig:lumfunc}. Also shown is the upper limit on the luminosity function measured by \\citet{walter2014} at $z \\sim 0.34$, for $L_{\\rm CO} \\sim 10^9$ K km\/s pc$^2$.}}\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[scale=0.4, width = \\columnwidth]{lcoredzero.pdf}\n \\end{center}\n \\caption{The luminosity function of CO galaxies at $z \\sim 0$. The red data points show the results from the FCRAO survey \\citep{young1995, keres2003} at $z \\sim 0$. The blue curve shows the derived luminosity function from the abundance matched best fitting parameters. The errors are indicated by the shaded regions. The black downward arrow shows the upper limit derived by \\citet{walter2014} at $z \\sim 0.34$.}\n\\label{fig:lumfunc}\n\\end{figure}\n\n{{The \\citet{keating2016} measurement may chiefly sample the shot noise portion of the CO power spectrum. The second moment of the CO luminosity function thus measured, is combined with the data from direct detection studies \\citep{decarli2014} and the absence of individual emitters within the COPSS dataset of $\\geq 5 \\sigma$ significance. This allows constraints on the CO luminosity function parameters at $z = 2.8$, assuming a Schechter functional form. These constraints are given by $\\rho* = 1.3^{+0.6}_{-0.7} \\times 10^{-3} \\ L_{\\odot}^{-1} \\rm{Mpc^{-3}} \\ \\rm{mag}^{-1}$ and $L_* = 4.5^{+1.4}_{-1.9} \\times 10^{10} \\ \\rm{K \\ km\/s \\ pc}^{\u22122}$, with the prior $\\alpha = -1.5 \\pm 0.75$ which is based on the SFR function parameters from \\citet{smit2012}.} \n\n{{The mean values and errors on the above Schechter function parameters can now be used to fit the $L_{\\rm CO} - M$ from \\eq{comoster} at $z \\sim 2.8$. This is done by matching the abundances of the halo mass function and the fitted CO luminosity function (with the associated errors) using \\eq{eqn:abmatchco}. \\footnote{For overall consistency, we assume an identical bin range and number of bins as for the case of the $z = 0$ analysis.}}} With this, we obtain the redshift evolution parameters to be:\n$M_{11} = -1.17 \\pm 0.85, N_{11} = 0.04 \\pm 0.03, b_{11} = 0.48 \\pm 0.35, y_{11} = -0.33 \\pm 0.24$. As in the previous case, the resultant errors are a combination of the fitting uncertainties and those from the data. \n\nThe resultant $L_{\\rm CO} - M$ relations at redshifts 0, 1 , 2 and 3, along with their associated errors are shown in Fig. \\ref{fig:hrlco}. Plotted for comparison at redshift $z \\sim 2$ are the model predictions (which assume a linear $L_{\\rm CO} - M$ relationship) from \\citet{pullen2013} Model A and \\citet{righi2008}. {{At redshift 3, the Model A prediction from \\citet{pullen2013} is shown, as well as the constraint derived by COPSS II on $A_{\\rm CO}(M)$, the coefficient of proportionality (assumed constant) between the CO luminosity and the host halo mass.}}\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[scale=0.3, width = \\columnwidth]{allz.pdf} \n \\end{center}\n \\caption{Best-fitting $L_{\\rm CO} - M$ relation (with $L_{\\rm CO}$ in units of K km\/s pc$^2$) at $z = 0$, $z = 1$, $z = 2$ and $z = 3$, from the combined results of the low redshift \\citet{keres2003} results and the higher redshift constraints from \\citet{keating2016}. The associated errors are shown by the grey bands. Also shown in the panels are the estimates of \\citet{pullen2013} Model A and \\citet{righi2008} at $z \\sim 2-3$. At $z \\sim3$, the {derived estimate from the COPSS II data \\citep{keating2016} which assumes a constant $A_{\\rm CO}(M)$ is also shown.}}\n\\label{fig:hrlco}\n\\end{figure}\n\n\n\n\n\n\\section{Comparison to data}\n\\label{sec:compare}\n\n\nWe have seen (Fig. \\ref{fig:lumfunc}) that the predicted luminosity function at $z \\sim 0$ is consistent by construction with the \\citet{keres2003} data, and is also consistent with the upper limit from \\citet{walter2014}. \n\n\\citet{aravena2012} use the results from a Jansky Very Large Array (JVLA) survey for CO 1-0 line emission from a candidate cluster at $z \\sim 1.55$, targeting four galaxies in the redshift range 1.47 to 1.59. Previous simulations were found to somewhat underestimate the number of CO galaxies detected at this redshift.\nIn Fig. \\ref{fig:red1_5} is plotted the model luminosity function with its associated error at $z \\sim 1.5$, compared to the findings of \\citet{aravena2012}. {{The data point shows the result for all the four galaxies in the sample, and is consistent with the model predictions.}} This is a consequence of the fact that the present model is also anchored to the high-redshift data [the $z \\sim 2.75$ measurements from \\citet{keating2016}]. Also plotted in Fig. \\ref{fig:red1_5} are the observational results from \\citet{walter2014}, in the redshift range $1.01 < z < 1.89$ (median redshift 1.52) from a blind search in the Hubble Deep Field North (HDF-N). \n\nIn Fig. \\ref{fig:red2_75} are plotted the results from this work at $z \\sim 2.75$ with the associated error bars, and for comparison, the results from COPSS \\citep{keating2016} which is consistent by construction. Also shown are the results from \\citet{walter2014} with the median redshift $z = 2.75$.\nOur findings are consistent with the fact that the \\citet{walter2014} and the COPSS \\citep{keating2016} results are in agreement, as also noted by \\citet{keating2016}. The present model predictions at these redshifts are also somewhat higher than those estimated by previous simulations. \n\nThe molecular hydrogen abundance can be constrained using estimates for the CO-to H$_2$ conversion factor and the total luminosity of the CO galaxies. With a typical value of $\\alpha = 4.3$, the cosmic hydrogen abundance is found to be $\\rho_{\\rm{H}_2} \\approx 10^8 M_{\\odot} \\ \\mathrm{Mpc}^{-3}$ at $z \\sim 3$, in good agreement with the results from data and theoretical models \\citep{obreschkow2009, lagos2011, sargent2014, popping2015, walter2014}.\n\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[scale=0.4, width = \\columnwidth]{phicoz1_5.pdf}\n \\end{center}\n \\caption{The CO luminosity function at $z \\sim 1.5$ with the associated error shown by the shaded bands. Also plotted for comparison are the results from \\citet{aravena2012} and \\citet{walter2014} at this redshift.}\n\\label{fig:red1_5}\n\\end{figure}\n\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[scale=0.4, width = \\columnwidth]{phicoz2_75.pdf}\n \\end{center}\n \\caption{The CO luminosity function at $z \\sim 2.75$. The associated errors are shown by the shaded bands. Also plotted for comparison are the results from \\citet[COPSS;][]{keating2016} and \\citet{walter2014} at this redshift.}\n\\label{fig:red2_75}\n\\end{figure}\n\nUsing the predicted $L_{\\rm CO} - M$ relation, we can estimate the magnitude and uncertainties of the CO temperature evolution and the intensity mapping power spectrum. We focus here on the 1-0 transition; analogous methods can be applied to the higher transitions as well. \n\nThe predicted $T_{\\rm CO}$ at various redshifts from the present model, using \\eq{tco} is shown in Fig. \\ref{fig:tcored3} by the blue dashed line.\\footnote{This is calculated following the conventions in \\citet{breysse2014}, Eq. (2.5) to enable ease of comparison with the compiled results in that work.} We assume fiducial values of $f_{\\rm duty} = 0.1$ and $M_{\\rm min, CO} = 10^9 h^{-1} M_{\\odot}$ in this plot. The shaded area indicates the model uncertainty.\\footnote{Note that the uncertainties in $f_{\\rm duty}$ and $M_{\\rm min, CO}$ are not included in the error band, which therefore represents a lower limit.} The figure also shows the predictions from various other models in the literature, compiled in \\citet{breysse2014} at $z \\sim 3$. It can be seen that the model predictions are consistent with the results of \\citet{righi2008} and \\citet{pullen2013} Model A in the previous literature, but below the Model B in \\citet{pullen2013}. {{This is as expected since the present model is matched to the results of \\citet{keating2016}, whose data are also found to be below Model B of \\citet{pullen2013}.}} The model is marginally consistent with the results of \\citet{visbal2010}.\n\\begin{figure}\n \\begin{center}\n \\includegraphics[scale=0.4, width = \\columnwidth]{tcored3.pdf}\n \\end{center}\n \\caption{The best-fitting evolution of the mean $T_{\\rm CO}$ (dashed blue curve) with orange error band. Results from the theoretical predictions of \\citet{visbal2010, pullen2013, righi2008} at $z \\sim 3$ are also shown for comparison (compiled in \\citet{breysse2014}).}\n\\label{fig:tcored3}\n\\end{figure}\n\n\nFinally, we can use the model predictions to compute the CO intensity mapping power spectrum using \\eq{COpowspeclog}. {{This calculation depends on the values of the minimum host halo mass, $M_{\\rm min}$, and also the duty cycle factor ($f_{\\rm duty}$). The minimum mass is assumed to be $M_{\\rm min, CO} = 10^9 h^{-1} M_{\\odot}$ throughout. The power spectra (in units of $\\mu K^2$) computed with two fiducial values of $f_{\\rm duty}$: 0.1 and 1, are shown in the top panel of Fig. \\ref{fig:powspec2}. These are compared with the model of \\citet{li2015} at the midpoint of the redshift range probed by the COMAP experiment ($z \\sim 2.4 - 2.8$) . The COMAP experiment sensitivity is also indicated on this panel by the red curve. \n\nAlthough tight constraints on $f_{\\rm duty}$ are difficult with the current data, most of the observational evidence suggests (and uses) a value of $f_{\\rm duty}$ close to unity \\citep{keating2016}. The bottom panel plots this along with the COPSS II data above the noise limit (black points).}}\n\n\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[scale=0.4, width = \\columnwidth]{powerspec2_6fdutyboth.pdf} \\includegraphics[scale=0.4, width = \\columnwidth]{powerspec3_0fduty1.pdf}\n \\end{center}\n \\caption{{{ The predicted best-fitting CO power spectrum, $\\Delta^2_{k}$ at redshifts $z = 2.6$ (top panel) and $z = 3.0$ (bottom panel). The associated errors are shown by the shaded bands. In the top panel, two values of $f_{\\rm duty}$ are considered: 0.1 and 1, and the simulation from \\citet{li2015} at the COMAP redshifts 2.4-2.8 is plotted as the dashed green curve. The COMAP sensitivity is shown in this panel as the red line. The bottom panel shows the power spectrum with an assumed 100\\% duty cycle, along with the results of COPSS II above the noise limit (shown by the black data points).}}}\n\\label{fig:powspec2}\n\\end{figure}\n\n\\section{Discussion and outlook}\n\\label{sec:conc}\nIn this paper, we have compiled the recent data in the field of carbon monoxide (CO) 1-0 emission line observations at low and intermediate redshifts. Here, we briefly summarize our results, discuss the scope of the technique and outline the possibilities for future work.\n\n\\begin{enumerate}\n\n\\item We have used the data at low redshifts to constrain the evolution of a parametric $L_{\\rm CO}$ - halo mass relation derived empirically. Given that the CO luminosity functions are well fit by the Schechter form, it is reasonable to expect the derived CO - halo mass relation to be modeled analogously to the stellar mass - halo mass relation \\citep[SHM;][]{behroozi2013, moster2013}. \n\n\\item This assumes a one-to-one-relationship between the host haloes and the CO-luminous galaxies, and also the completeness of the sample(s) under consideration. A caveat to the technique is that the halo mass function assumed is theoretical, and the assumption of matching the most massive haloes is involved. However, being completely empirical, this approach is free from the modelling uncertainties present in simulations, and at the same time is complementary to those studies. Extensions to this framework may be possible with the help of future data and comparison to high-resolution hydrodynamical simulations.\n\n\\item The evolution of the free parameters is determined from matching the available constraints at higher redshift ($z \\sim 3$) from intensity mapping. The resulting CO - halo mass relation is found to be consistent with most predictions from previous literature. It is also consistent with the results of surveys at intermediate redshifts \\citep{aravena2012,walter2014}. The associated errors not only encompass the uncertainties in the data, but also a range of uncertainties in the theoretical modelling. Thus, the fitting forms and errors contain the available theoretical and observational constraints on the intensity mapping power spectrum. \n\n\\item Using the empirically determined $L_{\\rm CO} - M$ relation with fiducial values of the minimum mass $M_{\\rm min}$ and duty fraction $f_{\\rm duty}$, one can predict the evolution of the integrated brightness temperature of the CO emission, $T_{\\rm CO} (z)$ and the power spectrum $P_{\\rm CO} (k,z)$ as a function of scale and redshift. These predictions are in, turn, consistent with the results of simulations in the literature. Table \\ref{table:final} summarizes the fitting functions for the $L_{\\rm CO} - M$ relation derived using the present approach.\n\n\\end{enumerate}\n\n\n\n\n\\begin{table}\n\\centering\n\\caption{Summary of the best-fitting $L_{\\rm CO} - M$ relation across $z \\sim 0-3$, and the free parameters involved. The $L_{\\rm CO}$ is in units of K km\/s pc$^2$ and all masses are in units of $M_{\\odot}$.}\n\\label{table:final}\n\\begin{tabular}{|c|}\n\\hline\n\\\\\n$L_{\\rm CO} (M, z) = 2N(z) M [(M\/M_1(z))^{-b(z)} + (M\/M_1(z))^{y(z)}]^{-1}$;\\\\ \n\\\\\n$\\log M_1(z) = \\log M_{10} + M_{11}z\/(z + 1)$\n \\\\\n \\\\\n$N(z) = N_{10} + N_{11}z\/(z + 1)$\n \\\\\n \\\\\n$b(z) = b_{10} + b_{11}z\/(z + 1)$\n \\\\\n \\\\\n$y(z) = y_{10} + y_{11}z\/(z + 1)$\n \\\\\n\\\\\n\\hline \n\\\\\n$M_{10} = (4.17 \\pm 2.03) \\times 10^{12} \\ M_{\\odot}$ ;\\ $M_{11} = -1.17 \\pm 0.85 $ \\\\\n\\\\ $N_{10} = 0.0033 \\pm 0.0016$ ; \\ $N_{11} = 0.04 \\pm 0.03$ \n \\\\\n\\\\$b_{10} = 0.95 \\pm 0.46$ ;\\ $b_{11} = 0.48 \\pm 0.35$ \n \\\\\n\\\\$y_{10} = 0.66 \\pm 0.32$ ;\\ $y_{11} = -0.33 \\pm 0.24$ \n \\\\\n\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\nTighter constraints on the power spectrum might be possible with new measurements from future detections (at low and intermediate redshifts) from a large sample of galaxies, {e.g., with the ALMA Spectroscopic Survey in the Hubble Ultra Deep Field (ASPECS) survey \\citep{walter2016}} and intensity mapping with facilities like the COMAP and the Y. T. Lee Array \\citep[YTLA;][]{ho2009}. Likewise, with the availability of new data, the model can be extended by, e.g., introducing merger histories and more accurate treatments of star formation \\citep[as done for the stellar-halo mass in, e.g.][]{moster2010}, and also to account for the turnover in the star-formation rate density beyond $z \\sim 3$. \n\nIt would be interesting to investigate the possibility of empirically constraining the $f_{\\rm duty}$ factor and connecting it to physically motivated duty cycles used in models of the UV luminosity function \\citep[e.g.,][]{tacchella2013}. With high-redshift data, the approach may be connected to the existing frameworks for modelling CO at close to the reionization epoch \\citep[$z \\sim 6-10$; as done in, e.g.,][]{mashian2015, gong2011}. Recently, a large sample of local CO-emitting galaxies has been compiled by \\citet{boselli2014}, which may be useful to constrain the CO density profiles and enable a more detailed characterization of the 1-halo term involved in the clustering \\citep[as done for HI in, e.g.,][]{hparaa2016}. Similarly, it would be useful to extend this approach towards the abundances of other molecules like \\textsc{c ii} \\citep[which has been modelled for the reionization epoch in, e.g.,][]{gong2012} and thereby facilitate the study of intensity mapping cross-correlations. \n\n\\section*{Acknowledgements}\n{I thank Kieran Cleary, Anthony Readhead, Adam Amara, Benny Trakhtenbrot, Tony Li and members of the COMAP (CO Mapping Array Pathfinder) collaboration for insightful discussions. I thank Jasjeet Bagla, Patrick Breysse, Kieran Cleary, Robert Feldmann, Girish Kulkarni, Anthony Pullen, Nirupam Roy, Sandro Tacchella and Livia Vallini for useful comments on the manuscript, and the referee for a detailed and helpful report.} My research is supported by the Tomalla Foundation. \n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and Main Results}\n\nThe Coxeter generators of the symmetric group $S_n$ are the transpositions $(1,2)$, $(2,3)$,\n\\dots, $(n-1,n)$.\nThe height of a permutation is defined distance to the identity element $e$,\n$$\nd(\\pi,e)\\, =\\, \\min\\{k \\geq 0\\, :\\, \\exists\\, \\tau_1,\\dots,\\tau_k \\in \\{(1,2),\\dots,(n-1,n)\\}\\ \\text{ such that }\\ \\pi = \\tau_1 \\cdots \\tau_k\\}\\, .\n$$\nMore generally, $d(\\pi_1,\\pi_2) = d(\\pi_2^{-1} \\pi_1,e)$.\nIt is easy to see that $d(\\pi,e) = d(\\pi^{-1},e)$.\nIn fact, another formula is\n$$\nd(\\pi,e)\\, =\\, |\\{(i,j) \\in \\mathbb{Z}^2\\, :\\, 1\\leq i\\pi_j\\}|\\, .\n$$\nThe pairs $(i,j)$ are called inversions of $\\pi$.\nIn \\cite{DiaconisRam}, Diaconis and Ram studied the Mallows measure, which is a probability measure on $S_n$ given by\n$$\n\\mathbbm{P}_{n}^{q}(\\pi)\\, =\\, \\frac{q^{d(\\pi,e)}}{P_n(q)}\\, ,\n$$\nwith $P_n(q)$ being a normalization constant.\nActually, Diaconis and Ram studied a Markov chain on $S_n$ for which the Mallows model gives the limiting\ndistribution.\nThis was followed up by another paper on a related topic by Benjamini, Berger, Hoffman\nand Mossel (BBHM) \\cite{BBHM} who related the biased shuffle and the Mallows model to the asymmetric\nexclusion process and the ``blocking'' measures (of Liggett, see \\cite{Liggett}, Chapter VIII, especially \nExample 2.8 and the end of Section 3).\nThey did this using Wilson's height functions \\cite{Wilson}.\nWe will discuss this more in Section \\ref{sec:Applications}.\nFor now, let it suffice that Diaconis and Ram identified the explicit formula for the normalization\nwhich they remarked is the ``Poincar\\'e polynomial'':\n$$\nP_n(q)\\, =\\, \\prod_{i=1}^{n} \\left(\\frac{q^{i}-1}{q-1}\\right)\\, =\\, [n]_q!\\, =\\, [n]_q \\cdots [1]_q\\, ,\\quad \\text{where}\\quad\n[n]_q\\, =\\, \\frac{q^n-1}{q-1}\\, .\n$$\nFurther references for the statistical applications\\footnote{\nIndependently, a similar $q$-deformed combinatorial formula was explained for a problem in quantum statistical\nmechanics, the ground state of the ferromagnetic $\\mathcal{U}_q(\\mathfrak{sl}_2)$-symmetric XXZ \nquantum spin chain, by Bolina, Contucci and Nachtergaele \\cite{BCN}.\nWe will comment more on this in Section 8.}\nof the Mallows model can be found in their paper.\n\nNote that, physically speaking, one would define a Hamiltonian energy function $H_n : S_n \\to \\mathbb{R}$ as\n$$\nH_n(\\pi)\\, =\\, \\frac{1}{n-1}\\, \\sum_{1\\leq i \\epsilon\\right\\}\\, =\\, 0\\, ,\n$$\nfor every continuous function $f : [0,1] \\times [0,1] \\to \\mathbb{R}$, where\n$$\nu(x,y)\\, =\\, \\frac{(\\beta\/2) \\sinh(\\beta\/2)}{\\big(e^{\\beta\/4} \\cosh(\\beta[x-y]\/2) - e^{-\\beta\/4} \\cosh(\\beta[x+y-1]\/2)\\big)^2}\\, .\n$$\n\\end{thm}\n\nNote that (one can show) the limit $\\beta \\to 0$ gives $1$.\nThe proof of Theorem \\ref{thm:main} uses a rigorous version of mean-field theory,\nas in the solution of the Curie-Weiss model.\nAn interesting feature is that the self-consistent mean-field equation leads us to the characterization of $u$ as\nthe solution of an integrable PDE\n$$\n\\frac{\\partial^2}{\\partial x \\partial y} \\ln u(x,y)\\, =\\, 2 \\beta u(x,y)\\, .\n$$\nIt is not unusual for mean-field problems to lead to integrable PDE's.\nWe demonstrate this briefly in the next section with the ubiquitous toy model,\nthe Curie-Weiss ferromagnet.\n\n\\section{Toy Model: The Curie-Weiss Ferromagnet}\n\nWe include this section merely to point out that mean-field problems often do lead to integrable PDE.\nHowever the issue is serious: in fact there is a recent paper by Genovese and Barra which we recommend for more\ndetails \\cite{GenoveseBarra}.\nOur approach merely summarizes their results (in our own words) as well as the earlier paper by Barra, himself \\cite{Barra}.\nThe configuration space of the CW model is $\\Omega_N = \\{+1,-1\\}^N = \\{\\sigma = (\\sigma_1,\\dots,\\sigma_n)\\, :\\, \\sigma_1,\\dots,\\sigma_n = \\pm 1\\}$.\nFor technical reasons, we choose the Hamiltonian as\n$$\nH_N(\\sigma,t,x)\\, =\\, - \\frac{t}{2N}\\, \\sum_{i,j=1}^{N} \\sigma_i \\sigma_j - x \\sum_{i=1}^{N} \\sigma_i\\, ,\n$$\nwe assume $t\\geq 0$ and $x \\in \\mathbb{R}$.\nDefining $m_N(\\sigma) = N^{-1} \\sum_{i=1}^{N} \\sigma_i$, which takes values in $[-1,1]$, we see that\n$$\nH_N\\, =\\, - N \\left(\\frac{t m_N^2}{2} + x m_N\\right)\\, .\n$$\nTherefore, defining\n$$\np_N(t,x)\\, =\\, \\frac{1}{N} \\ln \\sum_{\\sigma \\in \\Omega_N} e^{-H_N(\\sigma,t,x)}\\, ,\n$$\nwe easily see that\n$$\n\\frac{\\partial}{\\partial t} p_N(t,x)\\, =\\, \\frac{1}{2} \\langle m_N^2 \\rangle_{N,t,x}\\, ,\n$$\nand\n$$\n\\frac{\\partial^2}{\\partial x^2} p_N(t,x)\\, =\\, N \\left( \\langle m_N^2 \\rangle - \\langle m_N\\rangle^2\\right)\\, ,\n$$\nwhere\n$$\n\\langle f \\rangle\\, =\\, \\langle f\\rangle_{N,t,x}\\, =\\, \\frac{\\sum_{\\sigma \\in \\Omega_N} f(\\sigma) e^{-H_N(\\sigma,t,x)}}{\\sum_{\\sigma \\in \\Omega_N} e^{-H_N(\\sigma,t,x)}}\\, .\n$$\nActually it is easier to consider the ``order parameter,''\n$$\nu_N(t,x)\\, =\\, \\langle m_N \\rangle_{N,t,x}\\, =\\, \\frac{\\partial}{\\partial x} p_N(t,x)\\, ,\n$$\nfrom which $p_N(t,x)$ can be calculated by solving the ODE:\n$$\n\\begin{cases}\n\\frac{\\partial}{\\partial x} p_N(t,x)\\, =\\, u_N(t,x) & \\text{ for $x \\in \\mathbb{R}$,}\\\\\np_N(t,x) - |x| \\to \\frac{1}{2} t^2 & \\text{ as $x \\to \\pm \\infty$.}\n\\end{cases}\n$$\nThen we see that $u_N(t,x)$ satisfies the viscous Burgers equation (with velocity equal to the negative amplitude):\n$$\n\\begin{cases}\n\\frac{\\partial}{\\partial t}\\, u_N(t,x)\\, =\\, u_N(t,x)\\, \\frac{\\partial}{\\partial x}\\, u_N(t,x) + \\frac{1}{2N} \\cdot \\frac{\\partial^2}{\\partial x^2} u_N(t,x)\n& \\text{ for $t>0$ and $x \\in \\mathbb{R}$,}\\\\\nu_N(0,x)\\, =\\, \\tanh(x) \n& \\text{ for $x \\in \\mathbb{R}$.}\n\\end{cases}\n$$\nThis is an integrable PDE, using the Cole-Hopf transform.\nSee, for instance, Chapter 4 of Whitham, \\cite{Whitham}.\nActually, this leads to a solution in terms of Gaussian integrals.\nThe analogous transform in spin-configuration notation is the Hubbard-Stratonovich transform:\n$$\ne^{N t m^2\/2}\\, =\\, \\int_{-\\infty}^{\\infty} \\frac{e^{Nt (mx - x^2\/2)}}{\\sqrt{2\\pi\/t}}\\, dx\\, ,\n$$\nwhich ``linearizes'' the dependence of the Hamiltonian on $m_N$, in the exponential.\nThis trick is used to solve the Curie-Weiss model.\nSee, for example, Thompson \\cite{Thompson}.\n\nNote that in the $N \\to \\infty$ limit, one obtains $u(t,x) = \\lim_{N \\to \\infty} u_N(t,x)$ being the vanishing-viscosity\nsolution of the inviscid Burgers equation.\nShocks correspond to phase transitions.\nThe Lax-Oleinik variational formula for solutions of hyperbolic conservation laws applies.\nSee for example, Section 3.4.2 of Evans \\cite{Evans}.\nIn this context we claim that this is equivalent to the Gibbs variational formula, \nin the mean-field limit.\nWe review this next.\n\n\n\\section{Gibbs Variational Formula}\n\\label{sec:SCMFE}\n\nLet us begin by considering a general problem in classical statistical mechanics.\nSuppose that $\\mathcal{X}$ is a compact metric space, and suppose that there\nis a two-body interaction\n$$\nh : \\mathcal{X} \\times \\mathcal{X} \\to \\mathbb{R} \\cup \\{+\\infty\\}\\, .\n$$\nWe assume that $h$ is bounded below.\nThen for each $N \\geq 2$, one can consider the mean-field Hamiltonian $H_N : \\mathcal{X}^N \\to \\mathbb{R} \\cup \\{+\\infty\\}$\n$$\nH_N(x_1,\\dots,x_N) = \\frac{1}{N-1}\\, \\sum_{1\\leq i0$,}\\\\\n0 & \\text{ if $x<0$.}\n\\end{cases}\n$$\nSince $\\mu_0$ is absolutely continuous with respect to Lebesgue measure, all $x_1,\\dots,x_N$ and $y_1,\\dots,y_N$ are distinct,\nwith probability 1.\n(This is why we do not bother to specify $\\theta$ at the discontinuity point $0$.)\n\nLet $X_1<\\dots 0$ and $\\phi : [0,L_1] \\to \\mathbb{R}$, $\\psi : [0,L_2] \\to \\mathbb{R}$ both positive and continuous.\n\nNote that $\\frac{\\partial^2}{\\partial x \\partial y}$ is a wave operator, with characteristics directed along $x$ and $y$.\nSpecifically, defining $\\xi = (x+y)\/\\sqrt{2}$ and $\\zeta = (x-y)\/\\sqrt{2}$, we have\n$\\frac{\\partial^2}{\\partial x \\partial y} = \\frac{1}{2} (\\frac{\\partial^2}{\\partial \\xi^2} - \\frac{\\partial^2}{\\partial \\zeta^2})$,\nthe usual wave operator.\nTherefore, D'Alembert's formula for solutions of the wave equation allow us to reformulate (\\ref{eq:Liouville})\nas an integral equation, \n\\begin{equation}\n\\label{eq:CauchyIntegral}\n\\ln u(x,y)\\, =\\, \\ln \\phi(x) + \\ln \\psi(y) - \\ln \\alpha + 2 \\beta \\int_{[0,x] \\times [0,y]} u(x',y')\\, dx'\\, dy'\\, ,\n\\end{equation}\nwhich we prefer.\nThis equation is supposed to be solved for all $(x,y) \\in [0,L_1] \\times [0,L_2]$.\nWe have introduced the number $\\alpha = \\phi(0)$, which we also assumed\nequals $\\psi(0)$, for consistency since both are supposed to give $u(0,0)$.\n(Note that the initial surface, $([0,L_1] \\times \\{0\\}) \\cup (\\{0\\} \\times [0,L_2])$, is {\\it not}\na non-characteristic surface. This is the reason that our Cauchy problem\ndoes not require initial data for the tangential derivative of $u$ even though the wave equation\nis second order.)\nWe refer to Evans textbook for PDE's, (especially Section 2.4 on the wave equation and Section 4.6 on the Cauchy-Kovalevskaya theorem).\n\nAs we will see, the symmetry (\\ref{eq:symmetry}) is the key to solving both the Euler-Lagrange equation (\\ref{eq:EL}) and \nthe Cauchy problem (\\ref{eq:CauchyIntegral}).\n\n\n\\section{The Cauchy Problem}\n\n\nWe start with uniqueness for the Cauchy problem.\n\n\\begin{lemma}\n\\label{lem:IVP}\nFor any $L_1,L_2>0$, the Cauchy problem (\\ref{eq:CauchyIntegral}) having $\\phi=\\psi=\\alpha=1$\nhas at most one solution in the class of nonnegative integrable functions.\n\\end{lemma}\n\n\\begin{proof}\nSince $\\phi=\\psi=\\alpha=1$, equation (\\ref{eq:CauchyIntegral}) simplifies to\n$$\n\\ln u(x,y)\\, =\\, 2 \\beta \\int_{[0,x] \\times [0,y]} u(x',y')\\, dx'\\, dy'\\, .\n$$\nAssuming that $u$ is nonnegative and integrable, this implies that $\\ln u$ is bounded and continuous.\nThen, using these properties in the right-hand-side of the equation again (similarly as one does to prove elliptic regularity)\nwe deduce that $\\ln u$ is continuously differentiable and globally Lipschitz.\nIn particular, it is continuous up to the boundary.\n\n\nNow suppose that there are two solutions $u$ and $v$.\nLetting $z = \\ln u - \\ln v$, we have\n$$\nz(x,y)\\, =\\, 2 \\beta \\int_0^x \\int_0^y [1-e^{-z(x',y')}] u(x',y')\\, dx'\\, dy'\\, .\n$$\nSince both $\\ln u$ and $\\ln v$ are bounded, we see that $z$ is as well.\nTherefore, there exists a constant $K<\\infty$ such that $|1 - e^{-z}| \\leq K |z|$\nfor all values of $z$ in the range.\nSo we have\n$$\n|z(x,y)|\\, \\leq\\, \\beta K \\|u\\|_{\\infty} \\int_0^x \\int_0^y |z(x',y')|\\, dx'\\, dy'\\, .\n$$\nA version of Gronwall's lemma then implies that $z \\equiv 0$.\nWe outline this now, although our argument can probably be improved.\n\nLet $Z(t) = \\sup\\{ |z(x,y)|\\, :\\, (x,y) \\in (0,L_1) \\times (0,L_2)\\, ,\\ xy\\leq t\\}$.\nThen we obtain, after making the change of variables $(x,y) \\mapsto (x,t)$ where $t= xy$, and \nusing Fubini-Tonelli to integrate over $x$ first,\n$$\nZ(t)\\, \\leq\\,\\beta K \\|u\\|_{\\infty} \\int_0^t \\ln(t\/t') Z(t')\\, dt'\\, .\n$$\nWe rewrite this as \n$$\nZ(t)\\, \\leq\\, \\beta K \\|u\\|_{\\infty} \\int_0^t [\\ln(t\/L_1L_2) - \\ln(t'\/L_1L_2)] Z(t')\\, dt'\\, .\n$$\nSince $\\ln(t\/L_1L_2)\\leq 0$ for $t\\leq L_1L_2$, and since $Z\\geq 0$, we can drop\nthe term $\\ln(t\/L_1L_2) Z(t')$ in the integrand to obtain\n$$\nZ(t)\\, \\leq\\, \\beta K \\|u\\|_{\\infty} \\int_0^t |\\ln(t'\/L_1L_2)| Z(t')\\, dt'\\, .\n$$\nFinally, setting $\\zeta(t) = \\int_0^t |\\ln(t'\/L_1L_2)| Z(t')\\, dt'$, this leads to\n$$\n\\zeta'(t)\\, \\leq\\, \\beta K \\|u\\|_{\\infty} |\\ln(t\/L_1L_2)| \\zeta(t)\\, .\n$$\nBy Gronwall's inequality (see for example Appendix B of Evans \\cite{Evans}),\nwe obtain\n$$\n\\zeta(t)\\, =\\, e^{\\beta K \\|u\\|_{\\infty} \\int_0^t |\\ln(t'\/L_1L_2)|\\, dt'} \\zeta(0)\\, =\\, e^{\\beta K \\|u\\|_{\\infty} (1+|\\ln(t\/L_1L_2)|) t\/L_1L_2} \\zeta(0)\\, .\n$$\nBut $\\zeta(0) = 0$. Hence $\\zeta(t)=0$ for all $t$.\nThis implies $Z(t)=0$ for all $t$ which implies $z(x,y) = 0$ for all $x,y$.\n\\end{proof}\n\nNext we derive the explicit solution of (\\ref{eq:CauchyIntegral}), for the case $\\phi=\\psi=\\alpha=1$.\n\n\\begin{corollary}\n\\label{cor:Cauchy}\nSuppose that $L_1,L_2>0$ and either $\\beta\\leq 0$ or $L_1L_2 < 1\/\\beta$.\nThen the unique solution of the Cauchy problem (\\ref{eq:CauchyIntegral}) with $\\phi=\\psi=\\alpha=1$\nis\n$$\nu(x,y)\\, =\\, (1 - \\beta xy)^{-2}\\, .\n$$\n\\end{corollary}\n\n\n\\begin{proof}\nUniqueness was proved in Lemma \\ref{lem:IVP}, and it is trivial to check that this solves the PDE (\\ref{eq:LiouvillePDE}).\nTherefore, assuming that $u$ is integrable on $[0,L_1] \\times [0,L_2]$, we may derive D'Alembert's formula\nby standard calculus:\n\\begin{align*}\n\\ln u(x,y)\\, \n&=\\, \\int_0^x \\frac{\\partial}{\\partial x} \\ln u(x',y)\\, dx' + \\ln \\psi(y)\\\\\n&=\\, \\int_{(0,x) \\times (0,y)} \\frac{\\partial^2}{\\partial x \\partial y} \\ln u(x',y')\\, dx'\\, dy'\n+ \\int_0^x \\frac{\\partial}{\\partial x} \\ln \\phi(x')\\, dx' + \\ln \\psi(y)\\\\\n&=\\, 2 \\beta \\int_{(0,x) \\times (0,y)} u(x',y')\\, dx'\\, dy' + \\ln \\psi(y) + \\ln \\phi(x) - \\ln \\alpha\\, .\n\\end{align*}\nThe only issue is to check integrability, which amounts to checking $\\inf_{(x,y) \\in [0,L_1] \\times [0,L_2]} 1-\\beta xy>0$.\nThis holds if and only if \n$\\beta\\leq 0$ or $L_1 L_2 < 1\/\\beta$.\n\\end{proof}\n\nLet us briefly explain one approach to deriving this formula.\nFor nonlinear PDE's one always first guesses a scaling solution, in hopes of finding an explicit formula.\nBecause of the hyperbolic nature it makes sense to look for a solution $u(x,y) = U(xy)$\nfor some $U(z)$.\nThis leads to the ODE\n$$\n\\frac{d}{dz} \\ln U(z) + z \\frac{d^2}{dz^2} \\ln U(z)\\, =\\, 2 \\beta U(z)\\, ,\n$$\nwhich can also be expressed as\n$$\n\\frac{d}{dz} \\left(z\\, \\frac{d}{dz} \\ln U(z)\\right)\\, =\\, 2 \\beta U(z)\\, .\n$$\nThe idea of using a power law solution is natural because the derivative of the logarithm results in a power law, itself.\nTrying $U(z) = (1+cz)^p$ leads to \n$$\n\\ln U(z)\\, =\\, p \\ln(1+cz)\\quad \\Rightarrow\\quad\nz \\frac{d}{dz} \\ln \\phi(z)\\, =\\, \\frac{cpz}{1+cz}\\, =\\, p - \\frac{p}{1+cz}\\quad \n\\Rightarrow \\quad \n\\frac{d}{dz} \\left(z\\, \\frac{d}{dz} \\ln U(z)\\right)\\, =\\, \\frac{cp}{(1+cz)^2}\\, .\n$$\nSo, taking $p=-2$ and $c=-\\beta$, this solves the equation, and gives\n$U(z) = (1-\\beta z)^{-2} \\Rightarrow u(x,y) = (1-\\beta x y)^{-2}$.\nFinally, we are led to the solution of the general Cauchy problem.\n\n\\begin{corollary}\n\\label{cor:GenCauchy}\nSuppose that $\\phi, \\psi : [0,1] \\to \\mathbb{R}$ are continuous and satisfy\n$c\\leq \\phi,\\psi\\leq C$, for some constants $0 \\epsilon\\right\\}\\, =\\, 0\\, ,\n$$\nfor all $y \\in [0,1]$, where\n$$\n\\rho(x;y)\\, =\\, \\int_0^y u(x,y')\\, dy'\\, .\n$$\nThe scaling $p_N = \\frac{1}{2} + \\beta\/4N$ is the regime typically called ``weakly asymmetric.''\nSee, for example, Enaud and Derrida's paper \\cite{EnaudDerrida}, following the matrix method used, for example by Derrida,\nLebowitz and Speer \\cite{DerridaLebowitzSpeer}.\nNote that while they considered the nonequilibrium case, we consider the particle conserving, equilibrium case.\nOn the other hand, we are sure that the formula above is known.\n\nThe integral for $\\rho(x;y)$ is readily evaluated.\nSetting $\\phi$, $\\psi$, $\\Phi$, $\\Psi$ and $\\alpha$ as in Lemma \\ref{lem:marginal},\n\\begin{align*}\n\\rho(x;y)\\, \n&=\\, \\int_0^{y} \\frac{\\alpha \\phi(x) \\psi(y')}{(\\alpha - \\beta \\Phi(x) \\Psi(y'))^2}\\, dy'\\\\\n&=\\, \\frac{\\alpha \\phi(x)}{\\beta \\Phi(x) (\\alpha - \\beta \\Phi(x) \\Psi(y'))} \\bigg|_0^y\\\\\n&=\\, \\frac{\\phi(x) \\Psi(y)}{\\alpha - \\beta \\Phi(x) \\Psi(y)}\\, .\n\\end{align*}\nSubstituting in, and doing minor algebraic simplifications, we obtain\n$$\n\\rho(x;y)\\, =\\, \\frac{(1-e^{-\\beta y}) e^{-\\beta x}}{(1-e^{-\\beta}) - (1-e^{-\\beta x})(1-e^{-\\beta y})}\\, .\n$$\nFrom this formula it is obvious that the $\\beta \\to 0$ limit recovers $\\rho(x;y) \\equiv y$, as it should (for the symmetric case).\nAlso, after further ``simplifications,'' we obtain\n$$\n\\rho(x;y)\\, =\\, \\frac{e^{\\beta(\\frac{1}{2}-x)\/2} \\sinh(\\beta y\/2)}{e^{\\beta\/4} \\cosh(\\beta [x-y]\/2) - e^{-\\beta\/4} \\cosh(\\beta[x+y-1]\/2)}\\, .\n$$\nIn particular, one can observe that the particle-hole\/reflection symmetry is manifest in this formula due to the invariance under the transformation \n$(\\beta,x) \\mapsto (-\\beta,1-x)$.\n\n\n\nFinally, we note that we can partially undo the scaling limit by taking $\\beta \\to \\infty$ with $x=y+t\/\\beta$ (assuming $01$, correspond to the selection time scale\nbeing slower than the interaction time scale, and the limit value of\n$s\\to\\infty$ recovers the round-robin procedure. In fact, the\nequivalence of the limit $s\\to\\infty$ to the round-robin scheme\npoints to the latter being a form of 'mean-field' theory, in which\nindividuals reproduce so slowly that it makes sense to replace\npairwise interactions by the interaction with the 'average player'.\n\nAs for the games, we will consider the important case of symmetric\n$2 \\times 2$ games, in which the payoffs are given by the following matrix\n\\begin{equation} \\label{eq:2by2game}\n\\begin{array}{ccc}\n & \\mbox{ }\\, 1 & \\!\\!\\!\\!\\!\\! 2 \\\\\n\\begin{array}{c} 1 \\\\ 2 \\end{array} & \\left(\\begin{array}{c} a \\\\ c\n\\end{array}\\right. & \\left.\\begin{array}{c} b \\\\ d\n\\end{array}\\right), \\end{array} \\end{equation} whose rows give the\npayoff obtained by each strategy when confronted with the other or\nitself, and $a,b,c,d>0$. Let $n$ be the number of individuals using\nstrategy 1, also referred as type 1 individuals. After each\nreproduction event $n$ may stay the same, increase by one, or\ndecrease by one. Considering the definition of the dynamics, the\ncorresponding transition probabilities will depend on the fitness\nearned by each type during the interaction step and on their\nfrequencies. As both quantities will depend ultimately on $n$, we\nhave a Markov process with a tridiagonal transition matrix (i.e., a\nbirth-death process \\cite{Karlin:1975}) whose non-zero coefficients\nare \\begin{equation}\\begin{split}\nP_{n,n-1} &= \\frac{n}{N} E\\left(\\frac{F_2}{F_1+F_2} \\:\\bigg|\\: n \\right) \\\\\nP_{n,n+1} &= \\frac{N-n}{N} E\\left(\\frac{F_1}{F_1+F_2} \\:\\bigg|\\: n\n\\right) \\end{split} \\label{eq:tranprob} \\end{equation} and $P_{n,n} =\n1 - P_{n,n-1} - P_{n,n+1}$. $F_i$ is the payoff obtained by all\nplayers of type $i$, and $E( \\cdot | n )$ denotes the expected value\nconditioned to a population of $n$ individuals of type 1.\n\nWe stress that the parameter $s$ enters through the expected values\nof the relative fitness of each type (\\ref{eq:tranprob}). Indeed, if\nwe restrict ourselves to the limit $s\\to\\infty$,\nthese expected values are given directly by the pairing probabilities\nand the payoffs corresponding to each pair\n\\begin{eqnarray}\n && E \\left(\\left.\\frac{F_1}{F_1+F_2} \\:\\right|\\: n \\right) = \\\\\n &&\\frac{n(n-1)a + n(N-n)b}{n(n-1)a + n(N-n)(b+c) + (N-n)(N-n-1)d} \\nonumber\n\\end{eqnarray} as would be obtained by the round-robin scheme.\nHowever, as we will see below, finite values of $s$ often lead to\nresults completely different from this special case.\n\nThe solution to the birth-death process we have just described\ncan be obtained in a\nstandard manner \\cite{Karlin:1975}. Denoting by $p_n$ the\nfixation probability of type 1 (i.e. the probability of ending up in a\npopulation with all individuals of type 1) when starting from a\npopulation with $n$ players of this type, we have\n\\begin{equation}\np_n = P_{n,n-1}p_{n-1} + P_{n,n}p_n + P_{n,n+1}p_{n+1},\n\\end{equation}\nwith $p_0=0$ and $p_N=1$. The solution to this\nequation is given by\n\\begin{equation}\n\\label{eq:pn}\np_n = Q_n\/Q_N, \\quad Q_n =\n\\displaystyle 1 + \\sum_{j=1}^{n-1}\\prod_{i=1}^j\n\\frac{P_{i,i-1}}{P_{i,i+1}}, \\quad n>1\n\\end{equation}\nwith $Q_1=1$.\nAs stated above, the interesting case arises for\nfinite values of the parameter $s$. For general $s$,\na straightforward combinatorial analysis of\nall the possible sequences of $s$ pairings leads to\n\\begin{widetext}\n\\begin{equation}\n\\label{eq:expvalue}\nE\\left(\\frac{F_1}{F_1+F_2} \\:\\Big|\\: n \\right) = \\sum_{i=0}^s \\sum_{j=0}^{s-i} \\left[ 2^{s-i-j}\n\\frac {s!n^{s-j}(n-i)^i(N-n)^{s-i}(N-n-1)^j} {i!j!(s-i-j)!(N(N-1))^s }\n \\frac {2ai + b(s-i-j)} {2ai + 2dj + (b+c)(s-i-j)} \\right].\n\\end{equation}\n\\end{widetext}\nThis lengthy combinatorial expression reduces, in the limit case $s=1$\nof extremely rapid selection, to\n\\begin{equation}\\begin{split}\nP_{n,n-1} &= \\frac{n(N-n)}{N(N-1)} \\left( 1\n+ \\frac{c-b}{c+b}\\frac{n}{N} - \\frac{1}{N} \\right) \\\\\nP_{n,n+1} &= \\frac{n(N-n)}{N(N-1)} \\left( \\frac{2b}{b+c} +\n\\frac{c-b}{c+b}\\frac{n}{N} - \\frac{1}{N} \\right). \\end{split}\n\\label{eq:tranprob-s1} \\end{equation} The above equations are the\nfirst hint of the effect of time scales. Indeed, by noting that, for\nthis extreme case, only the coefficients of the skew diagonal of\n(\\ref{eq:2by2game}) appear in (\\ref{eq:tranprob-s1}) we reach the\nsurprising conclusion that if the time scale of selection equals that\nof interaction, the evolutionary outcome of any game will be\ndetermined solely by the performance of each strategy when confronted\nwith the other, and independently of the results when dealing with\nitself. However, as we will now see, there are another non-trivial,\nimportant differences.\n\nTo make our study as general as possible, we have analyzed all twelve\nnon-equivalent symmetric $2 \\times 2$ games \\cite{Rapoport:1966}.\nThese games can be further classified into three categories,\naccording to their Nash equilibria and their dynamical behavior\nunder the replicator dynamics with round-robin interaction:\n\nI.\\ There are six games with $a>c$ and $b>d$, or $ac$ and $bc$. They have\nseveral Nash equilibria, one of them with a mixed strategy. This\nmixed strategy equilibrium is the global attractor of the round-robin\nreplicator dynamics. The two pure strategies are unstable in this\ncase.\n\nLet us first consider an example of class I, namely the Harmony game\n\\cite{Licht:1999} ($a=1$, $b=0.25$, $c=0.75$, $d=0.01$). This is a\nno-conflict game, in which all players obtain the maximum payoff by\nfollowing strategy 1. As Fig.\\ \\ref{fig:1}(a) shows, this is the\nresult for large values of $s$, with a fixation probability $p_n\n\\approx 1$ for almost all $n$. On the other hand, Fig.\\\n\\ref{fig:1}(a) also shows that, for small $s$, strategy 2, i.e., the\ninefficient (in the sense of lowest payoff) one, is selected by the\ndynamics, unless starting from initial conditions with almost all\nindividuals of type 1.\n\n\\begin{figure} \\includegraphics[width=7.2cm]{Roca1.eps}\n\\caption{\\label{fig:1}Fixation probabilities in the games (a) Harmony\n($a=1$, $b=0.25$, $c=0.75$, $d=0.01$) and (b) Stag-Hunt ($a=1$,\n$b=0.01$, $c=0.8$, $d=0.2$), for $s$ equal to 1 ($\\circ$), 5\n($\\Box$), 10 ($\\triangle$), 100 ($+$), or $\\to\\infty$ ($\\times$).\nNote that, in figure (a), curves overlap for $s=10, 100$ and\n$\\to\\infty$. Population size $N=100$.} \\end{figure}\n\nFor class II, a good paradigm is the Stag-Hunt game\n\\cite{Skirms:2003} ($a=1$, $b=0.01$, $c=0.8$, $d=0.2$),\nwhich is a coordination game: Strategy 1 maximizes the\nmutual benefit, whereas strategy 2\nminimizes the risk of loss, and the conflict results from\nhaving to choose between these two options. As Fig.\\\n\\ref{fig:1}(b) reveals, the round-robin result is obtained for large\n$s$: both strategies are attractors, with the basin boundary located\nat the frequency corresponding to the mixed strategy equilibrium,\ni.e. $x = (d-b)\/(a-c+d-b) \\approx 0.49$. However, for small values of\n$s$ this boundary shifts to greater frequency values, thus reflecting\nan advantage of strategy 2. In the extreme $s = 1$ case this strategy\nbecomes the unique attractor.\n\nIt is interesting to note that Fig.\\ \\ref{fig:1} shows that\nthere is not a general crossover at $s \\approx N$. In the Harmony\ngame, the round-robin regime is mostly reached for $s \\simeq 10\\ll N$,\nwhereas in the Stag-Hunt game this does not happen until $s \\simeq\n100 = N$.\n\nFinally, let us consider the Snowdrift game \\cite{Sudgen:2004}\n($a=1$, $b=0.2$, $c=1.8$, $d=0.01$) as an example of class III. This\nis also a dilemma game, as each player has to choose between strategy\n1, which maximizes the population gain, and strategy 2, which gives\nindividuals the maximum payoff by exploiting the opponent. With\nround-robin dynamics both strategies coexists in the long run, with\nfrequencies corresponding to the mixed strategy equilibrium. However,\nour dynamics can never maintain coexistence indefinitely, because by\nconstruction one of the absorbing states (all players of type 1 or\nall of type 2) will be reached sooner or later with probability 1.\nNonetheless, it is possible to study the duration of metastable\nstates by using the mean time in each population state before\nabsorbtion, $t_n$ \\cite{Karlin:1975}. Figure \\ref{fig:2} shows the\nresults for two values of $s$ and a broad range of initial\nconditions. For $s$ large ($s=100$), the population stays for a long\ntime near the value corresponding to the mixed strategy equilibrium\n$x = (d-b)\/(a-c+d-b) \\approx 0.19$, independently of the initial\nnumber of type 1 individuals. A smaller value of $s=10$ (not shown)\ninduces a shift of the metastable equilibrium to smaller values of\n$n$, again almost independently of the initial conditions. Finally,\nfor an even smaller value of $s$ ($s=5$), there is no metastable\nequilibrium, but a fluctuation towards the $x=0$ absorbing state,\nwhich clearly depends on the initial conditions.\n\n\\begin{figure} \\includegraphics[width=7.64cm]{Roca2.eps}\n\\caption{\\label{fig:2}Mean time before fixation in the Snowdrift game\n($a=1$, $b=0.2$, $c=1.8$, $d=0.01$), for $s$ equal to 5 (a) and 100\n(b). Initial values of $n$ equal to 20 ($\\circ$), 50 ($\\triangle$)\nand 80 ($+$). Note that curves in (b) overlap. Population size\n$N=100$.} \\end{figure}\n\nHaving given examples of all three classes, we will summarize the\nrest of our study by saying that the remaining $2 \\times 2$ games\nbehave in a similar way, with rapid selection (small $s$) favoring in\nall cases the type that has the greatest coefficient in the skew\ndiagonal of the payoff matrix. For the remaining five games of class\nI this results in a reinforcement of the dominant strategy (the\nPrisoner's Dilemma \\cite{Axelrod:1981} being a prominent example).\nThe other two games of class II exhibit once again a displacement of\nthe basins of attraction, whereas the other two class III games\ndisplay the suppression of the coexistence state in favor of one of\nthe strategies. We thus see that rapid selection leads very generally\nto outcomes entirely different from those of round-robin dynamics.\n\nIt is important to realize that our results do not change\nqualitatively with the system size. Considering for instance the\nStag-Hunt game, the change in the basins of attraction is practically\nindependent of the population size. The main effect of working with\nlarger sizes is a steeper transition between the basins of\nattraction. Indeed, due to the inherent stochasticity of finite\npopulation sizes, smaller populations have a more blurred basin\nboundary, with points in each basin having an increasing non-zero\nprobability of reaching the other basin \\cite{Cabrales:2000}. Our\nresults for all other symmetric $2 \\times 2$ games are equally\nrobust. In fact, for very rapid selection, $s=1$, the limit $N \\to\n\\infty$ of the transition probabilities, Eq. (\\ref{eq:tranprob-s1}),\nshows that they depend only on the frequencies of both types.\n\nIt could be argued that in our model only $s$ pairs of individuals\nplay in each round, resulting in a very small effective population,\nthis being the fundamental cause of the reported results. To probe\ninto this issue, we have introduced a background of fitness\n\\cite{Claussen:2005,Nowak:2004a}, so that every player has an\nintrinsic probability of being selected, regardless of the outcome of\nthe game, and thus guaranteeing a population of $N$ players. Indeed,\nin most applications, agents interact through more than one type of\ngame and there are external contributions to fitness(environmental factors,\nfashions or media influence in a social context, etc.). Let $f_b$ \nbe the normalized\nfitness background, so that each individual has a background of\nfitness $s f_b \/ N$ before selection takes place; $f_b=1$ means that\nthe overall fitness coming from the game and from the background are\napproximately equal, for every value of $s$ and $N$. Figure\n\\ref{fig:3} shows the results for the Stag-Hunt game. A small fitness\nbackground of $f_b=0.1$ gives fixation probabilities very similar to\nthose with $f_b=0$ (Fig. \\ref{fig:2}(b)). For larger values, $f_b=1$,\nthe displacement of the basin boundary is smaller, but still\nperfectly noticeable. And a very large fitness background, $f_b\n\\gtrsim 10$ (not shown), drives the dynamics to random selection for\nevery value of $s$, because in this case the influence of the game is\nalmost negligible. Again, for the remaining symmetric $2 \\times 2$\ngames, our conclusions remain valid as well in the presence of a\nbackground of fitness. Consequently, our results are not merely due\nto a finite size effect of a small effective population of players.\n\n\\begin{figure} \\includegraphics[width=7.2cm]{Roca3.eps}\n\\caption{\\label{fig:3} Fixation probability in the Stag-Hunt game\n($a=1$, $b=0.01$, $c=0.8$, $d=0.2$) with a background of fitness\n$f_b$ equal to 0.1 (a) and 1 (b). Values of $s$: 5 ($\\Box$), 10\n($\\triangle$) 100 ($+$). Population size $N=100$.} \\end{figure}\n\nIn summary, we have proven that considering independent interaction\nand selection time scales leads to highly non-trivial,\ncounter-intuitive results. We have demonstrated the generality of\nthis conclusion by considering all symmetric $2 \\times 2$ games and\nshowing that rapid selection may lead to changes of the\nasymptotically selected equilibria, to changes of the basins of\nattraction of equilibria, or to suppression of long-lived metastable\nequilibria. This result has major implications for applying\nevolutionary game theory to model a specific problem, as the\nassumption of slow selection and consequently of round-robin dynamics\nmay or may not be correct. Indeed, as the example in\n\\cite{Sanchez:2005} shows, rapid selection may lead to the\nunderstanding of problems where Darwinian, individual evolution was\nthought not to play a role because round-robin dynamics was used. We\nenvisage that successful modelling in rapidly changing environments,\nsuch as social or (sub-)culture dynamics, will need a careful\nconsideration of the involved time scales along the lines discussed\nhere.\n\nWe thank R.\\ Toral for a critical reading of the manuscript. This\nwork is supported by MEC (Spain) under grants\nBFM2003-0180, BFM2003-07749-C05-01,\nFIS2004-1001 and NAN2004-9087-C03-03 and by Comunidad de Madrid\n(Spain) under grants\nUC3M-FI-05-007, SIMUMAT-CM and MOSSNOHO-CM.\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nModified theories of gravity have drawn the attention of the scientific\ncommunity because of the geometric mechanics that they provide to describe\nvarious phenomena in nature. In this concept, new geometrodynamical degrees of\nfreedom are introduced in the gravitational field equations in such a way so\nas to modify Einstein's General Relativity (GR). These additional terms can\nhave either theoretical or phenomenological origin \\cite{clifton}. Among the\nvarious\\ proposed modified theories of gravity, $f\\left( R\\right) $-gravity\n\\cite{Buda} has been the main subject of study in various works over different\nareas of gravitational physics.\n\n$f\\left( R\\right) $-gravity is a fourth-order theory, where the the action\nintegral involves a function of the spacetime scalar curvature $R$. General\nRelativity, with or without cosmological constant, is the special limit of\n$f\\left( R\\right) $-gravity when the theory becomes of second-order; that\nis, when $f$ is a linear function of the Ricci scalar. It belongs to a more\ngeneral family of theories that take into account curvature terms in order to\nmodify the Einstein-Hiblert action \\cite{cur1,cur4,cur2,cur3}. The\ngravitational field equations of $f\\left( R\\right) $-gravity are dynamically\nequivalent to that of O'Hanlon theory \\cite{Hanlon}, where a Lagrange\nmultiplier is introduced so as to reduce the order of the theory by increasing\nthe number of degrees of freedom through the introduction of a scalar field\n\\cite{Sotiriou,defelice}. This scalar field is nonminimally coupled to gravity\nand recovers Brans-Dicke theory \\cite{Brans} with a zero Brans-Dicke\nparameter.\\ Hence, $f\\left( R\\right) $-gravity is also related to families\nof Horndeski theories \\cite{hor}, which means that it is free of\nOstrogradsky's instabilities \\cite{wood1,wood2}. For a recent discussion on\nthe correspondence among $f\\left( R\\right) $-gravity and other theories\nthrough the various frames, together with relevant implications on\nconservation laws, see \\cite{san1, san2}.\n\nAs we mentioned before, the applications of $f\\left( R\\right) $-gravity in\ngravitational physics cover various subjects. As far as cosmology is\nconcerned, the theory is used both to model the inflationary phase of the\nuniverse \\cite{star,inf1,inf2,inf3,inf4,inf5,inf6} and also as a dark energy\ncandidate to describe the late-time acceleration phase\n\\cite{de00,de01,de0,de1,de2,de3,de4,de5,de6}. In \\cite{bian1} it was found\nthat new Kasner-like solutions exist, while some cosmological solutions in\nlocally rotational spacetimes were derived in \\cite{bian2,bian3}. Some effects\non the Mixmaster universe can be found in \\cite{bian4,bian5}. In general, the various implications of $f(R)$-gravity - and of other theories of gravitation as well - from a cosmological perspective can be seen in \\cite{Odnew1,Odnew2}. Other studies on\ngravitational collapse can be encountered in \\cite{col1,col2}, while static\nspherically solutions were derived in \\cite{staa0,staa1,staa2} with or without\na constant Ricci scalar. Black hole solutions have been also investigated in\nthe literature, for instance see \\cite{bh00,bh03,bh00a,bh01,bh02,bh04,bh05,bh06} and\nreferences therein, as also physical phenomena like, binary black hole merge\n\\cite{bhaa1}, anti-evaporation \\cite{bhaa2}, black hole thermodynamics\n\\cite{akbar,farr} and many other.\n\nIt is well known that black string solutions can be easily constructed in\nEinstein's gravity by trivially embedding black hole solutions in higher\ndimensions. The same is also true for certain classes of modified theories of\ngravitation (for example it holds for a certain type of Lovelock Lagrangians\n\\cite{Kastor,Giribet}). In many cases, numerical solutions have been presented in the\nliterature \\cite{Wiseman,Kobayashi,Kudoh}. However, exact solutions are always\nof special interest and there exists an extended bibliography over the subject\ncovering a great number of gravitational configurations: from three\ndimensional charged black strings \\cite{Horne}, to cosmological constant\nsolutions in an arbitrary number of dimensions \\cite{Adolfo}, even in the\npresence of axionic scalar fields \\cite{Adolfo2}. Some exact solutions which\ndescribe rotating black strings in $f\\left( R\\right) $-gravity in the\npresence of a electromagnetic field were derived in \\cite{stri2}, while some\nasymptotic black strings solutions can be found in \\cite{stri1} and a cosmic\nsting solution in four dimensions in the context of scalar tensor theory has\nbeen given in \\cite{stri3}. The stability of black string solutions is always\nan issue, since in general these geometries are unstable, see for example\n\\cite{Gregory,Alex1,Alex2}. However, counterexamples of this general rule\nexist and stable solutions may also arise \\cite{stable1,stable2}. In what regards other interesting gravitational solutions, a toroidal black hole has also emerged in the Einstein nonlinear sigma model \\cite{AlexFab}. What is more, in the literature one can also find brane solutions in higher dimensional $f(R)$-gravity \\cite{Chakra1,Chakra2}\n\nIn this work we start by investigating analytical solutions of power law\n$f\\left( R\\right) $-gravity in a four-dimensional static spacetime. In this\ncontext we derive the general analytical solution and try to see under which\nconditions interesting gravitational objects may be described by it. We find\nthat a black brane\/string solution can be distinguished for certain values of\nthe involved parameters in the case where the metric is axisymmetric. The\noutline of the paper is as follows: In Section \\ref{themodel}, we briefly\ndiscuss $f\\left( R\\right) $-gravity and derive the field equations for the\nspacetime of our consideration. In Section \\ref{gensolution}, we obtain the\ngeneral solution for the induced system of equations. Section \\ref{solution}\nincludes the main results of our analysis which is the black brane\/string\nsolution of the field equations for the power-law theory $f(R)=R^{k}$. In\nSection \\ref{thermo}, we calculate the surface gravity and thus the\ntemperature of the system as well as the entropy on the horizon. Finally, in\nSection \\ref{discus}, we discuss our results and draw our conclusions.\n\n\\section{Preliminaries}\n\n\\label{themodel}\n\nThe action integral of $f\\left( R\\right) $-gravity constitutes a\nmodification of the Einstein-Hilbert action and is given by the following\nexpressio\n\\begin{equation}\n\\mathcal{S}=\\int dx^{4}\\sqrt{-g}f\\left( R\\right) , \\label{ac.01\n\\end{equation}\nwhere $R$ is the Ricci scalar constructed from the spacetime metric $g_{\\mu\n\\nu}$. It follows from (\\ref{ac.01}) that the field equations of Einstein's GR\nare recovered when $f\\left( R\\right) $ is a linear function of $R$.\n\nVariation with respect to the metric tensor gives the field equation\n\\begin{equation}\nf_{,R}R_{\\mu\\nu}-\\frac{1}{2}fg_{\\mu\\nu}-\\left( \\nabla_{\\mu}\\nabla_{\\nu\n}-g_{\\mu\\nu}\\nabla_{\\sigma}\\nabla^{\\sigma}\\right) f_{,R}=0, \\label{ac.02\n\\end{equation}\nwhere we have assumed that we are in vacuum, i. e. there does not exist any\nmatter source. The Ricci scalar contains second order derivatives of the\ncoefficients of the metric $g_{\\mu\\nu}$, hence, relation (\\ref{ac.02})\nprovides a fourth-order system of differential equations.\n\nAn alternative way to write the latter is by the use of Einstein's tensor\ntogether with the definition of an energy-momentum tensor of geometric origin.\nIn particular, we can rewrite (\\ref{ac.02}) as\n\\begin{equation}\nR_{\\mu\\nu}-\\frac{1}{2}Rg_{\\mu\\nu}=k_{eff}T_{\\mu\\nu}^{eff}, \\label{ac.03\n\\end{equation}\nwhere $T_{\\mu\\nu}^{eff}$ is the effective energy momentum tensor that includes\nthe terms which make the theory deviate from General Relativity,\n\\begin{equation}\nT_{\\mu\\nu}^{eff}=\\left( \\nabla_{\\mu}\\nabla_{\\nu}-g_{\\mu\\nu}\\nabla_{\\sigma\n}\\nabla^{\\sigma}\\right) f_{,R}+\\frac{1}{2}\\left( f-Rf_{,R}\\right) g_{\\mu\n\\nu} \\label{ac.04\n\\end{equation}\nwhile $k_{eff}=\\left( f_{,R}\\left( R\\right) \\right) ^{-1}$ is a varying\ngravitational constant. From the latter it follows that the theory is defined\nin the Jordan frame.\n\nApart from the limit in which $f_{,RR}\\left( R\\right) =0$, where General\nRelativity is recovered, it can be observed that any constant Ricci curvature\n($R=R_{0}$) solution of General Relativity also satisfies field equations\n(\\ref{ac.03}) if the following algebraic condition relating the free\nparameters of the $f\\left( R\\right) $ function to $R_{0}$ \\cite{barot}\nholds\n\\begin{equation}\n2 f\\left( R_{0}\\right) -R_{0}f_{,R}\\left( R_{0}\\right) =0. \\label{ac.05\n\\end{equation}\n\n\nIn our work we assume the following static metric with line elemen\n\\begin{equation}\nds^{2}=-a(r)^{2}dt^{2}+N(r)^{2}dr^{2}+b(r)^{2} d\\phi^{2}+ c(r)^{2} d\\zeta^{2}\n\\label{genline\n\\end{equation}\nIt is interesting to note that, when $b(r)=c(r)$ and the line element is\ninvariant under rotations in the $\\zeta-\\phi$ plane, the general solution of a\ngeometry characterized by a constant Ricci scalar $R_{0}\\neq0$ is\n\\begin{equation}\n\\label{metRconst}ds^{2} = \\mp\\left( r^{2} -\\frac{1}{r}\\right) dt^{2} +\n\\frac{12}{R_{0}} \\left( \\frac{1}{r}-r^{2}\\right) ^{-1} dr^{2} + r^{2}\n\\left( d\\phi^{2} + d\\zeta^{2}\\right) ,\n\\end{equation}\nwhich - as we noted earlier - is also a solution of GR in the presence of a\ncosmological constant. Line element \\eqref{metRconst} can be also seen to be a\nspecial case of a more general solution presented in \\cite{Lemos} including\nalso an electromagnetic field. Clearly, the minus branch of \\eqref{metRconst}\ncharacterizes a black brane or string (depending on the topology of the ($\\phi,\n\\zeta$) surface) when $R_{0}<0$. For $f(R)$-gravity the constant scalar\ncurvature is related to the parameters of the relative model through algebraic\nequation \\eqref{ac.05}.\n\nThe existence of a solution like \\eqref{metRconst} is our motive to start\ninvestigating a more general setting given by line element \\eqref{genline}.\nThus, we begin by considering $b(r)\\neq c(r)$ and turn to the general case\nwhere $R$ is not a constant. The Ricci scalar is calculated to be\\footnote{The\nprime \\textquotedblleft$^{\\prime}$\\textquotedblright\\ denotes a total\nderivative with respect to the variable $r$, that is $a^{\\prime}\\left(\nr\\right) =\\frac{da\\left( r\\right) }{dr}$.}\n\\begin{equation}\nR=-\\frac{2}{N^{2}}\\left( \\frac{a^{\\prime\\prime}}{a}+\\frac{b^{\\prime\\prime\n}{b}+ \\frac{c^{\\prime\\prime}}{c}+ \\frac{a^{\\prime}b^{\\prime}}{ab\n+\\frac{a^{\\prime}c^{\\prime}}{ac}+ \\frac{b^{\\prime}c^{\\prime}}{bc}+\\right) +2\n\\frac{N^{\\prime}}{N^{3}}\\left( \\frac{a^{\\prime}}{a}+\\frac{b^{\\prime}}{b} +\n\\frac{c^{\\prime}}{c}\\right) . \\label{ac.06a\n\\end{equation}\nand the gravitational field equations (\\ref{ac.02}) are\n\\begin{equation}\nR^{\\prime2} f_{,RRR} + \\left[ \\left( \\frac{b^{\\prime}}{b}+ \\frac{c^{\\prime\n}{c}-\\frac{N^{\\prime}}{N}\\right) R^{\\prime}+ R^{\\prime\\prime}\\right] f_{,RR}\n- \\left( \\frac{a^{\\prime\\prime}}{a} + \\frac{a^{\\prime} b^{\\prime}}{a b} +\n\\frac{a^{\\prime} c^{\\prime}}{a c} - \\frac{a^{\\prime} N^{\\prime}}{a N}\\right)\nf_{,R} -\\frac{1}{2} N^{2} f = 0 \\label{fe.01\n\\end{equation\n\\begin{equation}\nR^{\\prime} \\left( \\frac{a^{\\prime}}{a} +\\frac{b^{\\prime} }{b} +\n\\frac{c^{\\prime} }{c}\\right) f_{,RR} - \\left( \\frac{a^{\\prime\\prime}}{a} +\n\\frac{b^{\\prime\\prime}}{b} + \\frac{c^{\\prime\\prime}}{c} - \\frac{a^{\\prime}\nN^{\\prime}}{a N} - \\frac{b^{\\prime} N^{\\prime}}{b N} - \\frac{c^{\\prime}\nN^{\\prime}}{c N} \\right) f_{,R} -\\frac{1}{2} N^{2} f =0 \\label{fe.02\n\\end{equation\n\\begin{equation}\nR^{\\prime2} f_{,RRR} + \\left[ \\left( \\frac{a^{\\prime}}{a}+ \\frac{c^{\\prime\n}{c}-\\frac{N^{\\prime}}{N}\\right) R^{\\prime}+ R^{\\prime\\prime}\\right] f_{,RR}\n- \\left( \\frac{b^{\\prime\\prime}}{b} + \\frac{a^{\\prime} b^{\\prime}}{a b} +\n\\frac{b^{\\prime} c^{\\prime}}{b c} - \\frac{b^{\\prime} N^{\\prime}}{b N}\\right)\nf_{,R} -\\frac{1}{2} N^{2} f = 0 \\label{fe.03\n\\end{equation}\n\\begin{equation}\nR^{\\prime2} f_{,RRR} + \\left[ \\left( \\frac{a^{\\prime}}{a}+ \\frac{b^{\\prime\n}{b}-\\frac{N^{\\prime}}{N}\\right) R^{\\prime}+ R^{\\prime\\prime}\\right] f_{,RR}\n- \\left( \\frac{c^{\\prime\\prime}}{c} + \\frac{a^{\\prime} c^{\\prime}}{a c} +\n\\frac{b^{\\prime} c^{\\prime}}{b c} - \\frac{c^{\\prime} N^{\\prime}}{c N}\\right)\nf_{,R} -\\frac{1}{2} N^{2} f = 0 \\label{fe.04\n\\end{equation}\n\n\nAn alternative way to derive the field equations (\\ref{fe.01})-(\\ref{fe.03}),\ntogether with \\eqref{ac.06a} for the scalar curvature, is with the use of a\npoint-like Lagrangian in the minisuperspace approach. In particular, the\naction of the Euler-Lagrange vector over the Lagrangian\n\\begin{equation}\n\\label{Lanc}L = \\frac{2}{N} \\left[ f_{R} \\left( c a^{\\prime}b^{\\prime}+a\nb^{\\prime}c^{\\prime}+ b a^{\\prime}c^{\\prime}\\right) +f_{RR}\\left( b c\na^{\\prime}R^{\\prime}+ a c b^{\\prime}R^{\\prime}+ a b c^{\\prime}R^{\\prime\n}\\right) \\right] + N a b c \\left( f - R f_{R}\\right)\n\\end{equation}\nwith respect to the variables~$\\left\\{ N,a,b,c,R\\right\\} $, produces a\nsystem of equations equivalent to (\\ref{ac.06a})-(\\ref{fe.04}). It is\nimportant to mention that (\\ref{Lanc}) is a singular Lagrangian, due to the\npresence of the degree of freedom $N$ for which no corresponding velocity\nappears. In this case $N$ plays a role similar to that of the lapse function\nin cosmological Lagrangians, the difference being that the evolution of the\npresent system is in the $r$ variable.\n\n\\section{The analytic solution}\n\n\\label{gensolution}\n\nThe method of selecting some special form for the metric so as to determine\nthe function $f\\left( R\\right) $ afterwards by construction is generally\nproblematic and lacks physical information. So, in this work, we prefer to\nwork in the inverse direction by first proving the integrability of the field\nequations; that is, the existence of an actual solution for the theory, and\nafterwards to try and derive a closed-form expression for it. In order to\nprove the integrability of the system we need to determine conservation laws\nthat will help us reduce the order of the dynamical equations (\\ref{ac.06a\n)-(\\ref{fe.04}). Among the different ways for the determination of\nconservation laws the symmetry method and the singularity analysis have been\nextensively applied in $f\\left( R\\right) $-gravity in several cosmological\nstudies \\cite{sym1,sym2,sym3,sym4}.\n\nAs we discussed above a well-known solution admitted by the system\n(\\ref{ac.06a})-(\\ref{fe.04}) is that of General Relativity with a cosmological\nconstant. However, that is only a particular solution and it is in general\nunstable, since the stability conditions depend on the form of $f\\left(\nR\\right) $. By the term particular solution we mean that there are present\nfewer integration constants due to the fact that there is a difference on the\nphysical degrees of freedom between General Relativity and $f\\left( R\\right)\n$ (with $f_{RR}\\neq0$) gravity.\n\nWe choose to work with the symmetry method and in order to simplify the\nproblem we redefine the lapse function\n\\begin{equation}\n\\label{lapNc}N = \\frac{n}{a b c \\left( R f_{R}- f\\right) },\n\\end{equation}\nwhich leads to the equivalent Lagrangian that is of the form\n\\begin{equation}\nL\\left( n,q^{\\alpha},q^{\\prime\\alpha}\\right) =\\frac{1}{n}G_{\\alpha\\beta\n}q^{\\prime\\alpha}q^{\\prime\\beta}-n, \\label{Lag2\n\\end{equation}\nwhere $q=(a,b,c,R)$ and $G_{\\mu\\nu}$ is\n\\begin{equation}\n\\label{minmetc}G_{\\mu\\nu} = \\left( 2 a b c \\left( R f_{R}-f\\right) f_{R}\n\\right)\n\\begin{pmatrix}\n0 & c & b & b c \\frac{f_{RR}}{f_{R}}\\\\\nc & 0 & a & a c \\frac{f_{RR}}{f_{R}}\\\\\nb & a & 0 & a b \\frac{f_{RR}}{f_{R}}\\\\\nb c \\frac{f_{RR}}{f_{R}} & a c \\frac{f_{RR}}{f_{R}} & a b \\frac{f_{RR}}{f_{R}}\n& 0\n\\end{pmatrix}\n.\n\\end{equation}\n\n\nIt has been shown in \\cite{tchris1} that linear in the momenta conserved\nquantities of the original constrained Lagrangian can be constructed by\nKilling vector fields of $G_{\\alpha\\beta}$. The latter is the singular system\nrealization of the Jacobi-Eisenhart metric used in Newtonian mechanics. The\nidea behind its utilization is to map solutions of a given energy to geodesic\nflows on Riemannian spaces \\cite{Arnold} (for applications and examples in\npseudo-Riemannian spaces see \\cite{ein1,ein2}).\n\nThe mini-superspace is four dimensional and its metric $G_{\\mu\\nu}$ is\nconformally flat. The simple transformation $(a,b,c,R)\\rightarrow(x,y,z,w)$\nwith\n\\begin{equation}\na=e^{\\frac{(3-6k)x+\\sqrt{2}(1-2k)y+(k+1)z}{12k-6}},\\;b=e^{\\frac{1}{12}\\left(\n\\frac{9(k-1)w}{k-2}+\\frac{2(4k-5)z}{2k-1}+\\sqrt{2}y\\right) },\\;c=e^{\\frac\n{(6k-3)x+\\sqrt{2}(1-2k)y+(k+1)z}{12k-6}},\\;R=e^{\\frac{\\sqrt{2}(k-2)y-(k+1)w\n{4k^{2}-10k+4}},\n\\end{equation}\nleads $G_{\\mu\\nu}$, as given by \\eqref{minmetc}, to become\n\\begin{equation}\nG_{\\mu\\nu}=(k-1)ke^{w+z}\\mathrm{diag}\\{-1,\\frac{1-k}{2k-1},\\frac{k^{2\n-1}{(1-2k)^{2}},-\\frac{3(k-1)^{2}(k+1)}{2(k-2)^{2}(2k-1)}\\}. \\label{minmetc2\n\\end{equation}\nThe minisuperspace admits six Killing fields, which in these coordinates are\n\\begin{equation\n\\begin{split}\n& \\xi_{1}=\\partial_{x},\\quad\\xi_{2}=\\partial_{y},\\quad\\xi_{3}=\\partial\n_{z}-\\partial_{w},\\quad\\xi_{4}=y\\partial_{x}+\\frac{(1-2k)x}{k-1}\\partial_{y}\\\\\n& \\xi_{5}=\\frac{3\\left( 2k^{2}-3k+1\\right) w}{2(k-2)^{2}}+z\\partial\n_{x}+\\frac{(1-2k)^{2}x}{k^{2}-1}\\partial_{z}-\\frac{(1-2k)^{2}x}{k^{2\n-1}\\partial_{w},\\\\\n& \\xi_{6}=-\\frac{(2k-1)\\left( \\left( 6k^{2}-9k+3\\right) w+2(k-2)^{2\nz\\right) }{2(k-2)^{2}}\\partial_{y}-\\frac{(1-2k)^{2}y}{k+1}\\partial_{z\n+\\frac{(1-2k)^{2}y}{k+1}\\partial_{w}.\n\\end{split}\n\\end{equation}\nAs is well known, they can be used to construct the conserved quantities of\nthe form $Q_{i}=\\xi^{\\alpha}\\frac{\\partial L}{\\partial q^{\\prime\\alpha}}$.\nWith the help of first order relations like $Q_{i}=$const. it is easy to\nderive the general solution for a generic $k\\neq1$. Of course, as we see from\n\\eqref{minmetc2}, the cases $k=1\/2$, $k=5\/4$ and $k=2$ have to be treated\nseparately. The general solutions of these particular cases are given later in\nthe appendix. By having excluded the aforementioned values, the final result\nin the original coordinates (after absorbing some unnecessary constants of\nintegration reads):\n\\begin{subequations}\n\\label{valuec\n\\begin{align}\na(r) & =e^{(\\alpha+\\gamma)r}(\\cosh r)^{\\frac{k-1}{5-4k}}\\label{ref1}\\\\\nN(r) & =C\\,e^{\\frac{\\beta+2\\alpha}{2k-1}r}(\\cosh r)^{\\frac{(8k-7)(k-1)\n{(2k-1)(5-4k)}}\\label{ref2}\\\\\nb(r) & =e^{\\beta r}(\\cosh r)^{\\frac{k-1}{5-4k}}\\label{ref3}\\\\\nc(r) & =e^{(\\alpha-\\gamma)r}(\\cosh r)^{\\frac{k-1}{5-4k}} \\label{ref4\n\\end{align}\nwith $C$, $\\alpha$, $\\beta$ and $\\gamma$ being the remaining constants of\nintegration. Of the latter three only two are actually independent, say\n$\\alpha$ and $\\gamma$, then it can be seen that (\\ref{ref1})-(\\ref{ref4}) is a\nsolution under the condition\n\\end{subequations}\n\\begin{equation}\n\\beta=-\\frac{\\left\\vert k-2\\right\\vert \\sqrt{\\frac{\\alpha^{2}(2k-1)(5-4k)^{2\n+2(k-1)\\left( \\gamma^{2}\\left( 8k^{2}-14k+5\\right) -3(k-1)^{2}\\right)\n}{5-4k}}+\\alpha(k-2)(2k-3)}{2(k-2)(k-1)} \\label{betaconst\n\\end{equation}\nA first important remark that we can make by studying (\\ref{ref1\n)-(\\ref{ref4}) is that, a typical uniform black string solution (i.e. one with\n$c(r)=$const.) is not possible in this setting. For the latter to happen we\nshould require $k=1$, which corresponds to GR and it is a value excluded from\nthis analysis. The scalar curvature that corresponds to this solution is\n\\begin{equation}\nR=\\frac{6(k-1)ke^{-\\frac{2(\\beta+2\\alpha)}{2k-1}r}}{C^{2}(2k-1)(4k-5)}\\left[\n\\cosh(r)\\right] ^{\\frac{2(2-k)}{(2k-1)(4k-5)}}.\n\\end{equation}\n\n\nAt this point we can formulate the line element in such a way so that we can\nidentify $b(r)$ as a radial type of variable. If we, for example, choose the\nparameter $\\beta$ to be\n\\begin{equation}\n\\beta=\\pm\\frac{k-1}{5-4k}, \\label{betavalue\n\\end{equation}\nthen $b(r)$ can become a radial variable $\\tilde{r}$ in the following manner:\nBy performing the transformation\n\\begin{equation}\nr=\\pm\\frac{1}{2}\\ln\\left( \\tilde{r}^{\\frac{5-4k}{k-1}}-1\\right) ,\n\\label{transfrad\n\\end{equation}\nwe get $b(\\tilde{r})=2^{\\frac{k-1}{4k-5}}\\tilde{r}$, with the constant value\n$2^{\\frac{k-1}{4k-5}}$ being irrelevant since it can be absorbed in the metric\nwith a scaling transformation. Under a transformation like \\eqref{transfrad},\nthe quantities $a$, $b$, $c$ and $Ndr$ transform as scalars. After a few\nreparametrizations and scalings over all of the variables, we can re-write the\nensuing line-element as (for simplicity instead of $\\tilde{r}$ we write $r$\nunderstanding that it is a different variable than the one used in expressions\n(\\ref{ref1})-(\\ref{ref4})):\n\\begin{equation} \\label{gensolr}\n\\begin{split}\nds^{2}= & -r^{2}\\left( \\frac{r^{\\frac{5-4k}{k-1}}}{m}-1\\right)\n^{\\frac{k-1}{4k-5}\\pm\\left( \\alpha+\\gamma\\right) }dt^{2}+r^{\\frac{2\\left(\n2k^{2}-2k-1\\right) }{(1-k)(2k-1)}}\\left( \\frac{r^{-\\frac{4k-5}{k-1}}\n{m}-1\\right) ^{-\\frac{8k^{2}-12k+2}{(1-2k)(5-4k)}\\mp\\frac{2\\alpha}{1-2k\n}dr^{2}\\\\\n& +r^{2}d\\phi^{2}+r^{2}\\left( \\frac{r^{\\frac{5-4k}{k-1}}}{m}-1\\right)\n^{\\frac{k-1}{4k-5}\\pm\\left( \\alpha-\\gamma\\right) }d\\zeta^{2},\n\\end{split}\n\\end{equation}\nwhere the constant $m$ appearing in \\eqref{gensolr} is associated with the initial $C$ that appears in \\eqref{ref2}. Wherever a double sign appears, the upper corresponds to a solution for $k>2$,\nwhile the lower for $k<2$. Of course, we always have to remember that not both\nof $\\alpha$ and $\\gamma$ are free parameters, but bound through the condition\n$\\beta=\\pm\\frac{k-1}{5-4k}$, with $\\beta$ being given by \\eqref{betaconst}.\nThe properties of the resulting spacetime are related to the type of theory\nthat we study and which is characterized by the number $k$. The scalar\ncurvature for the metric at hand is\n\\begin{equation}\nR=\\frac{6k(4k-5)}{(k-1)(2k-1)}r^{\\frac{2(k-2)}{(2k-1)(k-1)}}\\left(\n\\frac{r^{\\frac{5-4k}{k-1}}}{m}-1\\right) ^{\\frac{2k-3}{(2k-1)(4k-5)}\\mp\n\\frac{2\\alpha}{2k-1}} \\label{Ricciabc\n\\end{equation}\nwhich indicates that, depending on the value of $k$, we can have curvature\nsingularities at $r=0$, $r=m^{\\frac{1-k}{4k-5}}$ and at $r\\rightarrow+\\infty$.\n\nAs we can see from \\eqref{Ricciabc}, the only way to ensure that the term\n$\\frac{r^{\\frac{5-4 k}{k-1}}}{m}-1$ does not appear in the scalar curvature -\nso that it does not lead to its divergence either at $r=m^{\\frac{1-k}{4 k-5}}$\nor at infinity - is by setting\n\\begin{equation}\n\\alpha= \\pm\\frac{3-2 k}{2(5-4k)}.\n\\end{equation}\nThis value inevitably results, through \\eqref{betaconst}, in $\\gamma= \\pm\n\\frac{1}{2}$ that in its turn implies $b(r)=c(r)$. Thus, we are led in this\nway to start considering the case where the metric admits a toroidal type of symmetry.\n\n\\section{The black brane solution}\n\n\\label{solution}\n\nWe proceed by considering the aforementioned special case where $b(r)=c(r)$.\nThis is of special interest since, as we are about to see, it produces a black\nbrane\/string solution. The latter depends on how you interpret the $\\phi$,\n$\\zeta$ variables and the type of topology with which you endow the\ntwo-surface that they construct. We can either consider a topology\n$S^{1}\\times\\mathbb{R}$ (cylinder), $S^{1}\\times S^{1}$ (torus) or $\\mathbb{R} \\times \\mathbb{R}$ (plane) which imply\nthat one, both or none (respectively) of $\\phi$ and $\\zeta$ are bounded and periodic variables.\n\nTwo procedures can be followed to extract the solution: we can either start\nfrom the beginning, following a similar procedure like that of a previous\nsection and consider the resulting mini-superspace using ansatz\n\\begin{equation}\nds^{2}=-a(r)^{2}dt^{2}+N(r)^{2}dr^{2}+b(r)^{2}\\left( d\\phi^{2}+d\\zeta\n^{2}\\right) \\label{torline\n\\end{equation}\nfor the line element, or simply arrange the parameters in the general solution\n(\\ref{ref1})-(\\ref{ref4}) so that $b(r)=c(r)$. Obviously, the latter happens\nwhen $\\gamma=\\alpha-\\beta$, which means that the resulting solution depends on\nthe parameters $C$, $\\alpha$ and $\\beta$, with the latter two related through\n\\eqref{betaconst}, when $\\gamma=\\alpha-\\beta$ has been substituted in it.\nUnder these conditions it is easy to see that the general solution for the\n$b(r)=c(r)$ case is\n\\begin{subequations}\n\\label{valuec2\n\\begin{align}\na(r) & =e^{(2\\alpha-\\beta)r}(\\cosh r)^{\\frac{k-1}{5-4k}}\\label{tor1}\\\\\nN(r) & =C\\,e^{\\frac{2\\alpha+\\beta}{2k-1}r}(\\cosh r)^{\\frac{(8k-7)(k-1)\n{(2k-1)(5-4k)}}\\label{tor2}\\\\\nb(r) & =e^{\\beta r}(\\cosh r)^{\\frac{k-1}{5-4k}}\\label{tor3}\\\\\nc(r) & =e^{\\beta r}(\\cosh r)^{\\frac{k-1}{5-4k}} \\label{tor4\n\\end{align}\nwith $\\alpha$ and $\\beta$ bound to satisfy the algebraic relation\n\\end{subequations}\n\\begin{equation}\n40\\alpha^{2}+8\\alpha^{2}k(4k-9)+4\\alpha\\beta(5-4k)+\\beta^{2\n(4k-5)(4k-3)-3(k-2)k-3=0. \\label{torbeta\n\\end{equation}\n\n\nOnce more we can choose to study a particular solution belonging in this\nconfiguration. As in the previous section, we choose the parameter $\\beta$ to\nbe given by \\eqref{betavalue}. A value that helps us, with simple\ntransformation like \\eqref{transfrad}, to associate $b(r)$ to a radial\ndistance, only this time we additionally have $c(r)=b(r)$. With the choice\n\\eqref{betavalue}, the constraint \\eqref{torbeta} implies that\n\\begin{equation}\n\\alpha=\\pm\\frac{1-k}{4k-5}\\quad\\text{or}\\quad\\alpha=\\pm\\frac{3-2k}{2(5-4k)}.\n\\end{equation}\nThe first set of values leads to a solution of the form\\footnote{Again, we understand that the $r$ variable appearing in the following line elements is different from the one in \\eqref{valuec2}.}\n\\begin{equation}\nds^{2}=-r^{2}dt^{2}+r^{\\frac{2(-2k^{2}+2k+1)}{(k-1)(2k-1)}}\\left(\n1-\\frac{r^{\\frac{5-4k}{k-1}}}{m}\\right) ^{\\frac{2k}{1-2k}}dr^{2}+r^{2}\\left(\nd\\theta^{2}+d\\phi^{2}\\right) ,\n\\end{equation}\nwhich describes a spacetime that, depending on $k$, can have its singularity\npoints at the origin $r=0$, at infinity and at $r=m^{\\frac{1-k}{4k-5}}$. It is\ninteresting to note that just for the value $k=\\frac{3}{2}$ the corresponding\ngeometry has a curvature singularity only at $r=0$, while $r=m^{\\frac\n{1-k}{4k-5}}$ is just a coordinate singularity. In all the other cases where\n$r=m^{\\frac{1-k}{4k-5}}$ is just a coordinate singularity, a curvature\nsingularity at $r\\rightarrow\\infty$ necessarily occurs.\n\nThe most interesting situation appears when $\\alpha= \\pm\\frac{3-2 k}{2(5-4\nk)}$. With this set of values, we can write the two following line elements\ndepending on the value of $k$:\n\n\\begin{enumerate}\n\\item For $1\\frac{5}{4}$\n\\begin{equation}\n\\label{linesol2}ds^{2} = -r^{2} \\left( 1- \\frac{r^{\\frac{5-4 k}{k-1}}\n{m}\\right) dt^{2} + r^{\\frac{-4 k^{2}+4 k+2}{(k-1) (2 k-1)}} \\left( 1-\n\\frac{r^{\\frac{5-4 k}{k-1}}}{m}\\right) ^{-1} dr^{2} + r^{2} \\left( d\\phi\n^{2}+ d\\zeta^{2}\\right) .\n\\end{equation}\n\n\\end{enumerate}\n\nWe make this distinction so that the sign of $g_{tt}$ (the coefficient of\n$dt^{2}$ in the line element) is negative in each case when $r>m^{\\frac{1-k}{4\nk-5}}$. The two line elements are associated with constant, complex scalings\namong the coordinates and the essential constant $m$ that appears in them.\nBoth of them are of course solutions to the field equations\n\\eqref{fe.01}-\\eqref{fe.04} for an $f(R)=R^{k}$ theory.\n\nThe scalar curvature reads\n\\begin{equation}\n\\label{Ricsol}R = \\pm\\frac{6 k (4 k-5) r^{\\frac{2 (k-2)}{2 k^{2}-3 k+1}\n}{(k-1) (2 k-1)\n\\end{equation}\nwith the plus corresponding to the first line element, while the minus to the\nsecond. In both situations the Kretschmann scalar $K=R^{\\kappa\\lambda\\mu\\nu\n}R_{\\kappa\\lambda\\mu\\nu}$ is\n\\begin{equation\n\\begin{split}\nK = & \\frac{4}{(1-2 k)^{2} (k-1)^{2} m ^{2}} \\Big[\\left( 56 k^{4}-264\nk^{3}+468 k^{2}-370 k+111\\right) r^{-\\frac{2 \\left( 8 k^{2}-16 k+9\\right)\n}{(2 k-1) (k-1)}}\\\\\n& -2 m (k-2)^{2} (2 k-1) r^{\\frac{-8 k^{2}+18 k-13}{(2 k-1) (k-1)}} +3 m^{2}\n\\left( 8 k^{4}-20 k^{3}+13 k^{2}-2 k+2\\right) r^{\\frac{4 (k-2)}{(k-1) (2\nk-1)}}\\Big] .\n\\end{split}\n\\end{equation}\nThus, it can be seen that for the range of values $k<1\/2$ and $10$). We have to note here that, although we had initially excluded $k=2$\nfrom the general solution, its consideration separately leads to the same\nresulting spacetimes \\eqref{linesol1} and \\eqref{linesol2}, so we can include\nit at this point as an admissible value for $k$.\n\nLine element \\eqref{linesol1} is appropriate for describing a (compactified) black\nbrane or string in vacuum $f(R)=R^{k}$ gravity for those theories for which\n$12$, the curvature\nsingularity goes form the origin to infinity. Hence, we can state that, in the\ntheories corresponding to those values, such an object does not occur. We may\nnow proceed with studying the basic thermodynamic quantities for the\nstring and the compactified brane that appears in this context.\n\n\\section{Thermodynamic properties}\n\n\\label{thermo}\n\nWe continue by calculating basic thermodynamic quantities like the temperature\nand the entropy for line elements \\eqref{linesol1} and \\eqref{linesol2} for\nthose values of $k$ for which they describe a black object. That is\n$1