diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzarjz" "b/data_all_eng_slimpj/shuffled/split2/finalzzarjz" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzarjz" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nConsider the linear transport equation in $1$D \n\\begin{equation}\\label{eq:transport}\n\\partial_t f + v \\partial_x f - \\partial_x \\Phi \\partial_v f = 0,\n\\end{equation}\nfor an unknown function $f: [0,\\infty) \\times \\mathbb R_x \\times \\mathbb R_v \\to \\mathbb R_{\\geq 0}$\nwith a smooth external confining potential $\\Phi:\\mathbb R \\to \\mathbb R$. \n\nThe following is the main result of this note:\n\\begin{theorem}\\label{thm:main}\nLet $\\epsilon>0$ and $\\Phi(x) = \\frac {x^2}2 + \\frac{\\epsilon x^4}2$. Consider the unique solution $f$ to \\eqref{eq:transport} with initial data $f \\restriction_{t=0} = f_0$ such that \n\\begin{itemize}\n\\item $f_0: \\mathbb R_x \\times \\mathbb R_v \\to \\mathbb R_{\\geq 0}$ is smooth, and\n\\item there exists $c_s >0$ such that $\\mathrm{supp}(f_0) \\subseteq \\{ (x,v): c_s \\leq \\frac{v^2}2 + \\Phi(x) \\leq c_s^{-1} \\}$.\n\\end{itemize}\n\nThen, for $\\epsilon$ sufficiently small, there exists $C>0$ depending on $\\epsilon$ and $c_s$ such that the following estimate holds:\n$$\\sup_{x\\in \\mathbb{R}} |\\partial_t \\varphi|(t,x) \\leq C \\langle t\\rangle^{-2} \\sup_{(x,v) \\in \\mathbb R\\times \\mathbb R} \\sum_{|\\alpha|+ |\\beta| \\leq 2} |\\partial_x^\\alpha \\partial_v^\\beta f_0|(x,v),$$\nwhere $\\varphi$ is defined by\n\\begin{equation}\\label{eq:phi}\n\\partial_{xx}^2\\varphi(t,x) = \\int_{\\mathbb{R}} f(t,x,v)\\, \\mathrm{d} v,\\quad \\varphi(t,0) = \\partial_x \\varphi(t,0) = 0.\n\\end{equation}\n\\end{theorem}\n\nA few remarks of the theorem are in order.\n\n\\begin{remark}[Nonlinear Vlasov--Poisson system]\nThe reason that we are particularly concerned with $\\partial_t \\varphi$ is that it appears to be the quantity relevant for the stability of the\nzero solution for the nonlinear Vlasov--Poisson system in $1$D; see Section~\\ref{sec:VP}.\n\nIt should be noted that $\\varphi$ itself is not expect to decay to $0$ (since $\\int_{\\mathbb{R}} f\\, \\mathrm{d} v \\geq 0$). Thus the decay for $\\partial_t \\varphi$ can be viewed as a measure of the rate that $\\varphi$ approaches the limit $\\lim_{t\\to +\\infty} \\varphi(t,x)$.\n\n\\end{remark}\n\n\\begin{remark}[Derivatives of $\\partial_t\\varphi$]\nFor the applications on the Vlasov--Poisson system, one may also wish to obtain estimates for the derivatives of $\\partial_t \\varphi$. It is easy to extend our methods to obtain\n$$|\\partial_x \\partial_t\\varphi |\\lesssim \\langle t\\rangle^{-1},\\quad |\\partial_x^2 \\partial_t \\varphi|\\lesssim 1.$$\nNotice that these decay rates, at least by themselves, do not seem sufficient for a global nonlinear result.\n\\end{remark}\n\n\\begin{remark}[Phase mixing and the choice of $\\Phi$]\nThe result in Theorem~\\ref{thm:main} can be interpreted as a quantitative phase-mixing statement. It is well-known that for \n$$\\Phi(x) = \\frac {x^2}2,$$\nthe solution to \\eqref{eq:transport} does \\underline{not} undergo phase mixing (see chapter 3 in \\cite{BinTre_2011}). It is therefore important that we added the $\\frac{\\epsilon x^4}2$ term in the definition of the potential.\n\nOn the other hand, there are other choices of $\\Phi$ for which analogues of Theorem~\\ref{thm:main} hold. We expect that as long as $\\Phi$ is even and satisfies the non-degeneracy condition of \\cite{pRoS2020}, then a similar decay estimate holds. The particular example we used is only chosen for concreteness.\n\\end{remark}\n\n\\begin{remark}[Method of proof]\nIt is well-known that the linear transport equation \\eqref{eq:transport} can be written in action-angle variables, say $(Q, K)$, in which case \\eqref{eq:transport} takes the form\n\\begin{equation}\\label{eq:trivial}\n\\part_t f - c({K})\\part_{{Q}} f=0.\n\\end{equation}\n\nWhen $c'(K)$ is bounded away from $0$, phase mixing in the sense that $f$ converges \\emph{weakly} to a limit can be obtained after solving \\eqref{eq:trivial} with a Fourier series in $Q$; see \\cite{pRoS2020}. The point here is that $\\varphi$ is a (weighted) integral of $f$ over a region of phase space that is most conveniently defined with respect to the $(x,v)$ (as opposed to the action-angle) variables.\n\nWe quantify the strong convergence of $\\varphi_t\\to 0$ by finding an appropriate commuting vector field $Y$ that is adapted to the action-angle variables. The fact that $\\varphi$ is naturally defined as an integral over $v$ in $(x,v)$ coordinates makes it tricky to prove decay using this vector field. Furthermore, we are only able to prove $1\/\\jap{t}^2$ decay; this is for instance in contrast to the decay of the density for the free transport equation on a torus.\n\\end{remark}\n\n\\medskip\n\n\n\\subsection{Related result}\n\n\\subsection*{Linear phase mixing results}\nIn the particular context of Theorem~\\ref{thm:main}, decay of $\\partial_t\\varphi$, but without a quantitative rate, can be inferred from the work \\cite{pRoS2020}.\n\nThere are many linear phase mixing result, the simplest setting for this is the linear free transport equation. This is well-known; see for instance notes \\cite{villani2010landau} by Villani.\n\nOne of the most influential work on phase mixing is the groundbreaking paper \\cite{Landau1946} of Landau wherein he proposes a linear mechanism for damping for plasmas that does not involve dispersion or change in entropy. In the case of $\\mathbb T^d$, this is even understood in a nonlinear setting; see the section on nonlinear results below. The situation is more subtle in $\\mathbb R^d$, see \\cite{jBnMcM2020}, \\cite{rGjS1994}, \\cite{rGjS1995} and \\cite{dHKttNfR2020}.\n\nSee also \\cite{jBfW2020}, \\cite{faou2021linear} and \\cite{iT2017} for linear results on related models. In particular, we note that \\cite{faou2021linear} also rely on action-angle variables in their analysis.\n\n\\subsection*{Relation with other phase-mixing problems with integrable underlying dynamics}\nAs pointed out in \\cite{pRoS2020}, phase space mixing is relevant for the dynamics of kinetic models in many physical phenomena from stellar systems and dark matter halos to mixing of relativistic gas surrounding a black hole. See \\cite{dominguez2017description} for related discussions on dark matter halos. We also refer the interested reader to \\cite{BinTre_2011} for further background and discussions of phase mixing in other models, including the stability of galaxies.\n\nWe hope that the present work would also be a model problem and aid in understanding more complicated systems such as those described in \\cite{pRoS2020}. One particularly interesting problem is the stability of the Schwarzschild solution to the Einstein--Vlasov system in spherical symmetry. \n\n\\subsection*{Nonlinear phase mixing results}\nNonlinear Landau damping for Vlasov--Poisson on $\\mathbb{T}^d$ was first proven in analytic regularity by Mouhot--Villani in their landmark paper \\cite{cMcV2011}. Since then their work has been extended and simplified in \\cite{jBnMcM2016} and \\cite{eGtNiR2020a}.\n\nSee also other nonlinear results, e.g.~in \\cite{jB2017}, \\cite{jBnMcM2018}, \\cite{chaturvedi2021vlasov}, \\cite{faou2016landau}, \\cite{dHKttNfR2019}, \\cite{bY2016}.\n\n\n\n\\subsection*{Collisional problems with confining potentials}\n\nConfining potentials for kinetic equations have been well-studied, particularly for collisional models. Linear stability results can be found in \\cite{carrapatoso2021special}, \\cite{dolbeault2009hypocoercivity}, \\cite{dolbeault2015hypocoercivity}, \\cite{duan2011hypocoercivity} and \\cite{duan2012hypocoercivity}.\n\nIn this connection, it would also be of interest to understand how phase mixing effects (studied in the present paper) interact with collisional effects (cf.~\\cite{jB2017}, \\cite{chaturvedi2021vlasov}, \\cite{iT2017}.)\n\n\n\\section{The Vlasov--Poisson system}\\label{sec:VP}\n\nThe motivation of our result is the Vlasov--Poisson system:\n\\begin{equation}\\label{eq:VP}\n\\begin{cases}\n\\part_t f+v\\part_x f-(\\part_x \\Phi+\\part_x \\varphi)\\part_v f=0, \\\\\n-\\part^2_x \\varphi=\\int_{\\mathbb{R}} f\\d v.\n\\end{cases}\n\\end{equation}\n\nNote that \\eqref{eq:VP} can be rewritten as \n\\begin{equation}\\label{e.VP}\n\\part_t f+\\{H,f\\}=0,\n\\end{equation}\nwhere $H$ is the Hamiltonian given by \n\\begin{equation}\\label{eq:nonlin-hamiltonian}\nH(x,v)=\\frac{v^2}{2}+\\Phi(x)+\\varphi(t,x).\n\\end{equation}\n\nNotice that $f \\equiv 0$ is a solution to \\eqref{eq:VP}, and the transport equation \\eqref{eq:transport} is \nthe linearization of \\eqref{eq:VP} near the zero solution.\n\nOne cannot hope that the term $\\partial_x \\varphi$ in the nonlinear term decays as $t\\to +\\infty$. \n(This can be seen by noting that $\\int_{\\mathbb{R}} f\\d v \\geq 0$ pointwise.) At best one can hope \nthat $\\partial_x \\varphi$ converges to some (non-trivial) limiting profile as $t\\to +\\infty$. For $f$ satisfying \nthe linear equation \\eqref{eq:transport}, such convergence (without a quantitative rate) has been shown\nin \\cite{pRoS2020}.\n\nIn anticipation of the nonlinear problem, it is important to understand the quantitative convergence. Since\n$\\partial_x\\varphi$ does not converge to $0$, it is natural to understand the decay rate of $\\partial_t \\partial_x \\varphi$.\n\nAs a first step to understand \\eqref{eq:VP}, we look at the linearized problem \\eqref{eq:transport} around the zero solution and prove that we get integrable decay for $\\varphi_t$ in the linearized dynamics.\n\n\\begin{remark}\nNote that the Poisson's equation above reads\n$$-\\part^2_x \\varphi=\\rho.$$\nIn particular, $\\varphi$ is only defined up to a harmonic function, i.e.~a linear function a $x$. \nIn Theorem~\\ref{thm:main}, we remove this ambiguity by setting $\\varphi(0) = (\\partial_x \\varphi)(0) = 0$. \nNotice that other normalization, e.g., $\\varphi(-\\infty) = (\\partial_x \\varphi)(-\\infty)=0$ would not change the function $\\varphi_t = \\partial_t \\varphi$.\n\\end{remark}\n\n\\section{The action-angle variables}\\label{sec:action}\n\n\\subsection{First change of variables}\nFrom now on we will consider the Hamiltonian $$H=\\frac{v^2}{2}+\\Phi(x).$$ This is the Hamiltonian for the equations \\eqref{eq:transport}, which is also \\eqref{eq:nonlin-hamiltonian} without the $\\varphi$ (the self-interaction term). As an intermediate step to getting the action-angle variables we use the change of coordinates \n\\begin{align*}\n(t,x,v)\\mapsto (t,\\chi,H) &\\hspace{5 em}\\text{when $x>0$}\\\\\n(t,x,v)\\mapsto (t,\\pi-\\chi,H) &\\hspace{5 em}\\text{when $x\\leq 0$},\n\\end{align*}\nwhere $\\chi:=\\arcsin\\left(\\frac{v}{\\sqrt{2H}}\\right).$\\\\\nFirst we check if the change of variables is well defined by calculating the Jacobian for $x>0$,\n\\begin{equation*}\nJ=\\begin{pmatrix}\n\\part_t t& \\part_x t&\\part_v t\\\\\n\\part_t H& \\part_x H&\\part_v H\\\\\n\\part_t \\chi& \\part_x \\chi&\\part_v \\chi\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n1&0&0\\\\\n0&\\Phi_x&v\\\\\n0&-\\frac{v}{2H}\\cdot\\frac{\\Phi_x}{\\sqrt{\\Phi}}&\\frac{\\sqrt{\\Phi}}{H}\n\\end{pmatrix}\n\\end{equation*}\nNow \n\\begin{align*}\n\\det(J)&=\\frac{\\Phi_x}{\\sqrt{\\Phi}}\\frac{(\\Phi+v^2\/2)}{H}\\\\\n&=\\frac{\\Phi_x}{\\sqrt{\\Phi}}.\n\\end{align*}\nSimilarly for $x\\leq 0$, \\begin{align*}\n\\det(J)&= - \\frac{\\Phi_x}{\\sqrt{\\Phi}}.\n\\end{align*}\nHence, \\begin{align*}\n\\det(J)\n&=\\sign{x}\\frac{\\Phi_x}{\\sqrt{\\Phi}}.\n\\end{align*}\nNext by chain rule and using that $H$ is independent of $t$, we get\n\n\\begin{align*}\n\\part_x&=\\sign{x}\\part_x \\chi\\part_{\\chi}+\\part_x H\\part_H\\\\\n&=-\\frac{v}{2H}\\cdot\\frac{\\Phi_x}{\\sqrt{\\Phi}}\\part_{\\chi}+\\Phi_x\\part_H,\n\\end{align*}\n\\begin{align*}\n\\part_v&=\\sign{x}\\part_v \\chi\\part_{\\chi}+\\part_v H\\part_H\\\\\n&=\\frac{\\sqrt{\\Phi}}{H}\\part_{\\chi}+v\\part_H.\n\\end{align*}\nPluggin this in \\eref{VP}, we get the equation\n\\begin{equation}\\label{e.VP_chi_H_lin}\n\\part_t f-\\sign{x}\\frac{\\Phi_x}{\\sqrt{\\Phi}}\\part_\\chi f=0.\n\\end{equation}\n\n\n\\subsection{Second change of variables} The coefficient in front of $\\part_\\chi f$ in \\eqref{e.VP_chi_H_lin} depends on both $\\chi$ and $H$. To take care of this, we reparametrize $\\chi$ (in a manner depending on $H$). More precisely, for a fixed $H$, we define $Q(\\chi,H)$ such that \n$$\\frac{\\d Q}{\\d \\chi}=\\frac{c(H)}{a(\\chi,H)},\\quad Q(0,H)=0,$$\nwhere $a(\\chi,H)=\\sign{x}\\frac{\\Phi_x(x)}{\\sqrt{\\Phi(x)}}$ such that $x=x(\\chi,H).$ To fix $c(H)$, we require that for every $H$, \n\\begin{equation}\\label{eq:C.def}\n2\\pi=\\int_0^{2\\pi}\\d Q=c(H)\\int_0^{2\\pi}\\frac{1}{a(\\chi,H)}\\d \\chi.\n\\end{equation}\nNow we define the change of variables, $(\\chi,H)\\mapsto (Q,K)$ where $K=H$. Then note,\n$$a(\\chi,H)\\partial_\\chi=c(H)\\part_Q$$\nand $$\\part_{H}=\\part_{K}+\\frac{\\part Q}{\\part H}\\part_Q.$$\nThus in these coordinates, we can rewrite \\eref{VP_chi_H_lin} as \n\\begin{equation}\\label{e.VP_Q_H_lin}\n\\part_t f-c({K})\\part_Q f=0.\n\\end{equation}\n\n\nFurther, the Jacobian is \n\\begin{equation*}\n\\begin{pmatrix}\n\\part_H K&\\part_\\chi K\\\\\n\\part_H Q& \\part_\\chi Q\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n1&0\\\\\n\\part_H Q&\\frac{c(H)}{a(\\chi,H)}\n\\end{pmatrix}.\n\\end{equation*}\nNote that the determinant is $\\frac{c(H)}{a(\\chi,H)}$. Further, since \n\\begin{align*}\na(\\chi,H)&=\\sign{x}\\frac{\\Phi_x(x)}{\\sqrt{\\Phi(x)}}\\\\\n&=\\sqrt{2}\\frac{1+2\\varepsilon x^2}{\\sqrt{1+\\varepsilon x^2}},\n\\end{align*}\n\twe have that $a(\\chi,H) \\approx 1$ when $x$ is in a compact subset of $\\mathbb R$. As a result the determinant is bounded away from zero. For more details see Lemma~\\ref{lem:c-prime-bound}. \n\n\n\n\\section{The commuting vector field}\nWe first define the vector field $$Y=tc'(H)\\partial_Q-\\part_K.$$ In this section we prove that this vector field commutes with the transport operator as in \\eqref{e.VP_Q_H_lin} and that $|c'(H)|>0.$\n\\subsection{Commutation property}\n\nThe following commutation formula is an easy computation and thus we leave out the details.\n\\begin{lemma}\\label{lem:commutation}\nLet $Y = t c'(H) \\partial_Q - \\partial_{K}$. Then\n$$[\\partial_t - c(H) \\partial_Q, Y] = 0.$$\n\\end{lemma}\n\nThe following is an easy consequence of Lemma~\\ref{lem:commutation}:\n\\begin{lemma}\\label{lem:infinity-est}\nLet $f$ be a solution to \\eqref{e.VP_Q_H_lin} with initial data satisfying assumptions of Theorem~\\ref{thm:main}. Then\n$$\\sup_{(t,Q,K) \\in [0,\\infty)\\times \\mathbb T^1 \\times [c_s, c_s^{-1}]}\\hspace{.1em}\\sum_{\\ell\\leq 2} |Y^{\\ell} f|(t,Q,K) \\lesssim \\sup_{(x,v) \\in \\mathbb R\\times \\mathbb R} \\hspace{.1em}\\sum_{|\\alpha|+ |\\beta| \\leq 2} |\\partial_x^\\alpha \\partial_v^\\beta f_0|(x,v).$$\n\\end{lemma}\n\\begin{proof}\nBy Lemma~\\ref{lem:commutation}, we have that $Y^\\ell f$ satisfies the transport equation \\eqref{e.VP_Q_H_lin} for any $\\ell \\in \\mathbb N \\cap \\{0\\}$. Hence we get the estimate\n$$\\sup_{(t,Q,K) \\in [0,\\infty)\\times \\mathbb T^1 \\times [c_s, c_s^{-1}]}\\hspace{.1em}\\sum_{\\ell\\leq 2} |Y^{\\ell} f|(t,Q,K)\\lesssim \\sup_{(Q,K) \\in \\mathbb T^1 \\times [c_s, c_s^{-1}]}\\hspace{.1em}\\sum_{\\ell \\leq 2} |\\part_K^{\\ell} f_0|(Q,K).$$\nSince the change of variables $(Q,K)\\to (x,v)$ is well-defined and bounded away from zero, we get the required result.\n\\end{proof}\n\\subsection{Positivity of $|c'(K)|$}\nWe prove that $|c'(K)|$ is uniformly bounded below on the support of $f$. This plays a key role in the next section ensuring phase mixing.\n\\begin{lemma}\\label{lem:c-prime-bound}\nFor every $c_s <+\\infty$, there exists $\\epsilon_0>0$ such that whenever $\\epsilon \\in (0,\\epsilon_0]$, there is a small constant $\\delta >0$ (depending on $c_s$ and $\\epsilon$) such that \n$$\\inf_{K \\in [c_s, c_s^{-1}]} |c'(K)| = \\inf_{H \\in [c_s, c_s^{-1}]} |c'(H)| \\geq \\delta.$$\n\\end{lemma}\n\\begin{proof}\nBy definition of $c(H)$, we have\n$$\\frac{2\\pi}{c(H)}=\\int_0^{2\\pi}\\left|\\frac{\\sqrt{\\Phi}}{\\Phi'}\\right|\\d \\chi,$$\nso that using $\\Phi=\\frac{x^2}{2}+\\frac{\\epsilon}{2} x^4$, we obtain\n$$\\frac{2\\pi}{c(H)}=\\int_0^{2\\pi}\\frac{\\sqrt{1+\\epsilon x^2}}{\\sqrt{2}(1+2\\epsilon x^2)}\\d \\chi.$$\n\nNotice that for $H \\in [c_s, c_s^{-1}]$, $|x|$ is bounded. It follows that $c(H) \\approx 1$. Therefore, to prove strict positivity of $|c'(H)|$, it suffices to prove positivity of $\\left|\\frac{c'(H)}{c^2(H)}\\right|.$ Note that \n\\begin{equation}\\label{eq:c'\/c}\n\\begin{split}\n\\frac{-2\\pi c'(H)}{c^2(H)}&=\\frac 1{\\sqrt 2}\\int_0^{2\\pi}\\part_H\\left(\\frac{\\sqrt{1+\\epsilon x^2}}{1+2\\epsilon x^2}\\right)\\d \\chi\\\\\n&=\\frac 1{\\sqrt 2}\\int_0^{2\\pi}\\part_H x\\left[\\frac{\\epsilon x}{\\sqrt{1+\\epsilon x^2}(1+2\\epsilon x^2)}-\\frac{4\\epsilon x\\sqrt{1+\\epsilon x^2}}{(1+2\\epsilon x^2)^2}\\right]\\d \\chi\\\\\n&=\\frac 1{\\sqrt 2}\\int_0^{2\\pi}\\part_H x\\left[\\frac{-3\\epsilon x-2\\epsilon^2x^3}{\\sqrt{1+\\epsilon x^2}(1+2\\epsilon x^2)^2}\\right]\\d \\chi.\n\\end{split}\n\\end{equation}\n\nNow we calculate $\\part_H x.$ First we use the equation, $H=\\frac{v^2}{2}+\\Phi(x).$ Precisely, we have \n$$1=v\\part_H v+\\Phi'(x) \\part_H x.$$ Thus \n\\begin{equation}\\label{eq:dxH}\n\\part_H x=\\frac{1-v\\part_H v}{\\Phi'}.\n\\end{equation}\nNext we use that $\\frac{v}{\\sqrt{2H}}=\\sin\\chi,$\n$$0=\\frac{\\part_H v}{\\sqrt{2H}}-\\frac{v}{(2H)^{\\frac{3}{2}}}.$$\nThus $\\part_H v=\\frac{v}{2H}.$ Plugging this into \\eqref{eq:dxH}, we get that \n\\begin{equation}\\label{eq:dxH.final}\n\\part_H x=\\frac{1-\\frac{v^2}{2H}}{\\Phi'}=\\frac{\\cos^2 \\chi}{\\Phi'(x)} = \\frac{\\cos^2\\chi}{x+2\\epsilon x^3},\n\\end{equation}\nwhere in the last equality we used $\\Phi=\\frac{x^2+\\epsilon x^4}{2}$.\n\nPlugging \\eqref{eq:dxH.final} back into \\eqref{eq:c'\/c}, we get that\n\\begin{align*}\n\\frac{-2\\pi c'(H)}{c^2(H)}&=\\int_0^{2\\pi}\\frac{\\cos^2\\chi}{\\sqrt{2} x(1+2\\epsilon x^2)}\\left[\\frac{-x(3\\epsilon +2\\epsilon^2x^2)}{\\sqrt{1+\\epsilon x^2}(1+2\\epsilon x^2)^2}\\right]\\d \\chi\\\\\n&=-\\int_0^{2\\pi}{\\cos^2\\chi}\\left[\\frac{(3\\epsilon +2\\epsilon^2x^2)}{\\sqrt{2(1+\\epsilon x^2)}(1+2\\epsilon x^2)^3}\\right]\\d \\chi.\n\\end{align*}\nFinally note that since $|x|$ is bounded on the region of interest, after choosing $\\epsilon_0$ sufficiently small, we have\n$$\\frac{(3\\epsilon +2\\epsilon^2x^2)}{\\sqrt{2(1+\\epsilon x^2)}(1+2\\epsilon x^2)^3}\\approx \\epsilon,$$ \nand thus $|c'(H)|>\\delta$. \\qedhere\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Decay for $\\varphi_t$}\nIn this section we finally prove the decay for $\\varphi_t$ (recall Theorem~\\ref{thm:main}). \n\nTo keep the notation lean, we will often suppress the explicit dependence on $t$.\n\\begin{lemma}\\label{lem:phi_t}\nFor $f$ satisfying the assumptions of Theorem~\\ref{thm:main}, and $\\varphi$ defined as in \\eqref{eq:phi}, we have the following formula\n$$\\varphi_t(x')=\\int_0^{x'}\\int_{\\mathbb{R}}v[f(y,v)-f(0,v)]\\d v\\d y.$$\n\\end{lemma}\n\\begin{proof}\nBy the continuity equation (following directly from \\eqref{eq:transport}), we have that $$\\rho_t=\\int_{\\mathbb{R}}v \\part_x f\\d v.$$\nThus $$-\\part^2_x \\varphi_t=\\rho_t=\\int_{\\mathbb{R}}v \\partial_x f\\d v.$$\nSolving the Laplace's equation (with boundary conditions \\eqref{eq:phi}), we get\n$$\\varphi_t(x')=\\int_0^{x'}\\int_0^y \\int_{\\mathbb{R}}v\\part_x f(z,v)\\d v\\d z\\d y.$$\nIntegrating by parts in $z$, we get\n$$\\varphi_t(x')=\\int_0^{x'}\\int_{\\mathbb{R}}v[f(y,v)-f(0,v)]\\d v\\d y.$$\n\\end{proof}\n\nIn view of Lemma~\\ref{lem:phi_t}, it suffices to bound $\\int_0^{x'}\\int_{\\mathbb{R}} v f(0,v) \\d v\\d y$ and $\\int_0^{x'}\\int_{\\mathbb{R}} v f(y,v) \\d v\\d y$, which will be achieved in the next two subsections respectively.\n\n\\subsection{Decay for the term involving $f(0,v)$}\nWe first prove decay for $\\int_{\\mathbb{R}} v f(0,v)\\d v.$ {Before proving the main estimate in Proposition~\\ref{prop:term-x=0}, we first prove a lemma.}\n\n\\begin{lemma}\\label{lem:Q.at.pi.2}\nThe level set $\\{x = 0\\}$ corresponds to the level sets $\\{ Q = \\frac \\pi 2 \\} \\cap \\{ Q = -\\frac \\pi 2\\} \\cup \\{(x,v) = (0,0)\\}$. \n\\end{lemma}\n\\begin{proof}\nFirst note that level set $\\{x = 0\\}$ corresponds to the level sets $\\{ \\chi = \\frac \\pi 2 \\} \\cap \\{ \\chi = -\\frac \\pi 2\\} \\cup \\{(x,v) = (0,0)\\}$. This is because when $x = 0$, $\\Phi(x)= 0 $, and thus by definition (when $v \\neq 0$) $\\chi:=\\arcsin\\left(\\frac{v}{\\sqrt{2H}}\\right) = \\arcsin (\\pm 1) = \\pm \\frac \\pi 2$.\n\nIt thus remains to show that\n\\begin{equation}\\label{eq:chi.Q.pi.2}\n\\chi = \\pm \\frac \\pi 2 \\iff Q = \\pm \\frac \\pi 2.\n\\end{equation}\n\nFix $H$, then since $a(\\chi,H)=\\lvert\\frac{\\Phi_x}{\\sqrt{\\Phi}}\\rvert$ is independent of $v$, we have\n$$c(H)\\int_0^{\\pi} \\frac{1}{a(\\chi,H)}\\d \\chi=c(H)\\int_{\\pi}^{2\\pi} \\frac{1}{a(\\chi,H)}\\d \\chi.$$\nFurther, by the evenness of $\\Phi$, we have\n$$c(H)\\int_0^{\\pi\/2} \\frac{1}{a(\\chi,H)}\\d \\chi=c(H)\\int_{\\pi\/2}^{\\pi} \\frac{1}{a(\\chi,H)}\\d \\chi.$$\nFinally, since we have by construction, $$c(H)\\int_0^{2\\pi} \\frac{1}{a(\\chi,H)}\\d \\chi=2\\pi,$$ we have that $$Q(\\chi=\\pi\/2,H)=c(H)\\int_0^{\\pi\/2} \\frac{1}{a(\\chi,H)}\\d \\chi=\\pi\/2.$$\nSimilarly, $Q(\\chi=-\\pi\/2,H)=-\\pi\/2.$ Combining these, we obtain \\eqref{eq:chi.Q.pi.2}. \\qedhere \n\\end{proof}\n\n\\begin{proposition}\\label{prop:term-x=0}\nFor $f$ satisfying the assumptions of Theorem~\\ref{thm:main}, we have the following estimate:\n$$\\Big| \\int_{\\mathbb{R}} v f(0,v)\\d v \\Big| \\lesssim \\langle t \\rangle^{-2} \\sup_{(x,v) \\in \\mathbb R\\times \\mathbb R} \\hspace{.1em}\\sum_{|\\alpha|+ |\\beta| \\leq 2} |\\partial_x^\\alpha \\partial_v^\\beta f_0|(x,v).$$\n\\end{proposition}\n\\begin{proof}\nThe transport equation preserves $L^\\infty$ bounds so that by the support properties, we obviously have\n$$\\Big| \\int_{\\mathbb{R}} v f(0,v)\\d v \\Big| \\lesssim \\sup_{(x,v) \\in \\mathbb R\\times \\mathbb R} \\hspace{.1em} | f_0|(x,v).$$\nIn other words, it suffices to prove the desired bound with $t^{-2}$ instead of $\\langle t \\rangle^{-2}$.\n\nNow note that $$\\int_{\\mathbb{R}} v f(0,v)\\d v=\\int_0^\\infty v [f(0,v)-f(0,-v)]\\d v.$$\nFor clarity of notation, we let $$\\bar{f}(t,Q,K) = f(t,x,v).$$ Now writing in the $(K,Q)$ variables, and using Lemma~\\ref{lem:Q.at.pi.2} together with the fact that $K = H = \\frac {v^2}2$ when $x=0$, we have\n$$\\int_{\\mathbb{R}} v f(0,v)\\d v=\\int_0^\\infty v [f(0,v)-f(0,-v)]\\d v=\\int_0^\\infty [\\bar f(\\pi\/2,K)-\\bar f(-\\pi\/2,K)]\\d { K}.$$\nBy the fundamental theorem of calculus, we have\n$$\\int_0^\\infty [\\bar f(\\pi\/2,K)- \\bar f(-\\pi\/2,K)]\\d { K}=\\int_{-\\pi\/2}^{\\pi\/2}\\int_0^\\infty \\part_Q \\bar f(Q,K)\\d {K}\\d Q.$$\nNext, the Cauchy--Schwarz inequality implies\n\\begin{align*}\n\\int_{-\\pi\/2}^{\\pi\/2}\\int_0^\\infty \\part_Q \\bar f(Q,K)\\d {K}\\d Q&=\\sqrt{\\pi}\\left(\\int_{-\\pi\/2}^{\\pi\/2}\\left(\\int_0^\\infty \\part_Q \\bar f(Q,K)\\d {K}\\right)^2\\d Q\\right)^{\\frac{1}{2}}\\\\\n&\\lesssim \\left(\\int_{0}^{2\\pi}\\left(\\int_0^\\infty \\part_Q \\bar f(Q,K)\\d {K}\\right)^2\\d Q\\right)^{\\frac{1}{2}}.\n\\end{align*}\nNow using Poincare's inequality we get that for any $\\ell\\geq 2$\n\\begin{equation}\\label{eq:easy.term.after.Poincare}\n\\left(\\int_{0}^{2\\pi}\\left(\\int_0^\\infty \\part_Q \\bar f(Q,K)\\d {K}\\right)^2\\d Q\\right)^{\\frac{1}{2}}\\lesssim \\left(\\int_{0}^{2\\pi}\\left(\\int_0^\\infty \\part^{\\ell}_Q \\bar f(Q,K)\\d {K}\\right)^2\\d Q\\right)^{\\frac{1}{2}}.\n\\end{equation}\n\nNow take $\\ell = 2$. We write $\\part_Q=\\frac{1}{c'(K)t}(Y+\\part_{K})$ so that \n\\begin{equation*}\n\\begin{split}\n&\\: \\left(\\int_{0}^{2\\pi}\\left(\\int_0^\\infty \\part^2_Q \\bar f(Q,K)\\d {K}\\right)^2\\d Q\\right)^{\\frac{1}{2}} \\\\\n= &\\: \\left(\\int_{0}^{2\\pi}\\left(\\int_0^\\infty \\frac{1}{|c'(K)|^2 t^2} ( Y^2 \\bar f+ 2\\part_{K} Y \\bar f+ \\partial^2_K \\bar f) (Q,K)\\d {K}\\right)^2\\d Q\\right)^{\\frac{1}{2}} \\\\\n\\lesssim &\\: \\frac 1{t^2} \\left(\\int_{0}^{2\\pi}\\left(\\int_0^\\infty (\\sum_{k=0}^2 |Y^k \\bar f|) (Q,K)\\d {K}\\right)^2\\d Q\\right)^{\\frac{1}{2}}.\n\\end{split}\n\\end{equation*}\nwhere in the last step we have integrated by parts in $K$ and bounded $\\frac 1{|c'(K)|}$, $\\frac{|c''(K)|}{|c'(K)|}$, etc.~using Lemma~\\ref{lem:c-prime-bound} and the smoothness of $c$.\n\nFinally, since $f(t,Q,K)$ is non-zero for $c_s\\leq K\\leq c_s^{-1}$ and $Q\\in [0,2\\pi]$, we can take supremum in $K$ and $Q$ followed by Lemma~\\ref{lem:infinity-est} to get the required result.\\qedhere\n\\end{proof}\n\n\n\\begin{remark}\nNotice that since we can take any $\\ell \\geq 2$ in \\eqref{eq:easy.term.after.Poincare}, we can write each $\\part_Q=\\frac{1}{c'(H)t}(Y+\\part_{K})$ and integrate by parts in $K$ many times to show that the term in Proposition~\\ref{prop:term-x=0} in fact decays faster than any inverse polynomial (depending on smoothness of $f$)! \n\nIn other words, the decay rate that we obtain in Theorem~\\ref{thm:main} is instead limited by the term treated in Proposition~\\ref{prop:bulk-term} below.\n\\end{remark}\n\n\\subsection{Decay for the term involving $f(y,v)$} \n\n{We now turn to the other term in Lemma~\\ref{lem:phi_t}. Before we obtain the main estimate in Proposition~\\ref{prop:bulk-term}, we first prove two simple lemmas.}\n\\begin{lemma}\\label{lem:Jacobian}\nUnder the change of variables $(x,v)\\mapsto (Q,K)$ as in Section~\\ref{sec:action}, the volume form transforms as follows:\n$$\\d v \\d x = c(K) \\,\\d Q\\d{K}.$$\n\\end{lemma}\n\\begin{proof}\nThe Jacobian determinant for the change of variables $(x,v)\\mapsto (\\chi,H)$ is $a(\\chi,H)=\\lvert\\frac{\\Phi_x}{\\sqrt{\\Phi}}\\rvert$. Further the Jacobian determinant for the change of variables $(\\chi,H)\\to (Q,K)$ is $\\frac{c(H)}{a(\\chi,H)}$ and hence the Jacobian determinant for $(x,v)\\to (Q,K)$ is $c(H) = c(K)$. \\qedhere\n\\end{proof}\n\n\\begin{lemma}\\label{lem:f-to-g}\nLet $\\bar{f}(Q,K) = f(x,v)$ as above. There exists a function $\\bar{g}(Q,K)$ such that\n\\begin{equation}\\label{eq:f-to-g}\n\\partial_Q^2 \\bar{g} = \\partial_Q \\bar{f}\n\\end{equation}\nand\n\\begin{equation}\\label{eq:g-bound}\n\\max_{\\ell\\leq 2} \\sup_{K} \\norm{Y^{\\ell} \\bar{g}}_{L^2_Q} \\lesssim \\max_{\\ell\\leq 2} \\sup_{Q,K} |Y^{\\ell} \\bar{f}|.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWe use the Fourier series of $f$ in $Q$ to get that $$\\bar{f}(Q,K)=\\sum_{k=-\\infty}^{k=\\infty} \\widehat{\\bar{f}}_k(K)e^{ikQ}.$$\nNow we define $$\\bar{g}(Q,K):=\\sum_{k\\in\\mathbb{Z}\\backslash\\{0\\}} \\frac{1}{ik}\\widehat{\\bar{f}}_k(K)e^{ikQ}.$$ Then we see that $\\partial^2_Q \\bar{g}=\\partial_Q \\bar{f}.$\n\nUsing Plancheral's theorem we can easily see that\n$$\\max_{\\ell\\leq 2} \\sup_{K} \\norm{Y^{\\ell} \\bar{g}}_{L^2_Q} \\lesssim \\max_{\\ell\\leq 2} \\sup_{K} \\norm{Y^{\\ell} \\bar{f}}_{L^2_Q}.$$\nFinally, the result follows by taking supremum in $Q$. \\qedhere\n\\end{proof}\n\n\\begin{proposition}\\label{prop:bulk-term}\nFor $f$ satisfying the assumptions of Theorem~\\ref{thm:main}, we have the following estimate:\n$$\\int_0^{x'}\\int_{\\mathbb{R}}vf(t,y,v)\\d v\\d y\\lesssim \\jap{t}^{-2} {\\sup_{(x,v) \\in \\mathbb R\\times \\mathbb R} }\\hspace{.1em} \\sum_{|\\alpha|+ |\\beta| \\leq 2} |\\partial_x^\\alpha \\partial_v^\\beta f_0|(x,v).$$\n\\end{proposition}\n\\begin{proof}\n{As in the proof of Proposition~\\ref{prop:term-x=0}, boundedness is obvious and thus it suffices to prove an estimate with $\\langle t\\rangle^{-2}$ replaced by $t^{-2}$.}\n\nWe first note that\n$$\\int_0^{x'}\\int_{\\mathbb{R}}vf(y,v)\\d v\\d y=\\int_0^{x'}\\int_0^\\infty v[f(y,v)-f(y,-v)]\\d v\\d y.$$\nAgain let $$\\bar{f}(t,Q,K) = f(t,x,v).$$\nNext we use the change of variables $(x,v)\\mapsto (Q,K)$, that $v=\\sqrt{2H}\\sin \\chi$ and Lemma~\\ref{lem:Jacobian} followed by the fundamental theorem of calculus to obtain\n\\begin{align*}\n&\\: \\int_0^{x'}\\int_0^\\infty v[f(y,v)-f(y,-v)]\\d v\\d y\\\\\n= &\\: \\int_0^{\\Phi(x')}\\int_0^{\\pi\/2} c(K)\\sqrt{2K}S(Q,K)[\\bar f(Q,K)-\\bar f(-Q,K)]\\d Q\\d {K}\\\\\n&+\\int_{\\Phi(x')}^\\infty\\int_{\\mathfrak{Q}_K}^{\\pi\/2} c(K)\\sqrt{2K}S(Q,K)[\\bar f(Q,K)-\\bar f(-Q,K)]\\d Q\\d {K}\\\\\n=&\\:\\int_0^{\\Phi(x')}\\int_0^{\\pi\/2}\\int_{-Q}^{Q} c(K)\\sqrt{2K}S(Q,K)\\part_Q \\bar f(Q',K)\\d {Q'}\\d Q\\d {K}\\\\\n&+\\int_{\\Phi(x')}^\\infty\\int_{\\mathfrak{Q}_{K}}^{\\pi\/2}\\int_{-Q}^{Q} c(K)\\sqrt{2K}S(Q,K)\\part_Q \\bar f(Q',K)\\d {Q'}\\d Q\\d {K}\\\\\n=: &\\: T_1+T_2,\n\\end{align*}\nwhere we have define\n\\begin{itemize}\n\\item $S(Q,K):=\\sin\\chi$, and \n\\item $\\mathfrak{Q}_K$ to be the angle in $(Q,K)$ coordinates corresponding to angle $\\arccos\\left(\\frac{\\Phi(x')}{H}\\right)$ in $(\\chi,H)$ coordinates.\n\\end{itemize}\n\nNow using Fubini's theorem, we have \n$$T_1=\\int_{-\\pi\/2}^{\\pi\/2}\\int_0^{\\Phi(x')}\\left(\\int^{\\pi\/2}_{|Q'|} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}\\part_Q \\bar f(Q',K)\\d {K}\\d {Q'}$$\nand \n\\begin{align*}\nT_2&=\\int_{-\\pi\/2}^{\\pi\/2}\\int_{\\Phi(x')}^{\\mathfrak{H}_{Q'}}\\left(\\int_{|Q'|}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}\\part_Q \\bar f(Q',K)\\d {K}\\d {Q'}\\\\\n&+\\int_{-\\pi\/2}^{\\pi\/2}\\int_{\\mathfrak{H}_{Q'}}^\\infty\\left(\\int_{\\mathfrak{Q}_{K}}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}\\part_Q \\bar f(Q',K)\\d {K}\\d {Q'},\n\\end{align*}\nwhere $\\mathfrak{H}_{Q'}$ is such that $\\left(\\arccos\\left(\\frac{\\Phi(x')}{\\mathfrak{H}_{Q'}}\\right),\\mathfrak{H}_{Q'}\\right)$ in $(\\chi,H)$ coordinates gets mapped to $(|Q'|,\\mathfrak{H}_{Q'})$ in $(Q,K)$ coordinates \n(such an $\\mathfrak{H}_{Q'}$ exists because $\\chi=\\arccos\\left(\\frac{\\Phi(x')}{H}\\right)$ increases as $H$ does and $Q$ is monotone\\footnote{Since $a(\\chi,H)>0$, we have that $Q$ is monotonically increasing as a function of $\\chi$ and vice-versa.} in $\\chi$.)\n\nPutting the above together we get,\n\\begin{align*}\nT_1+T_2=&\\int_{-\\pi\/2}^{\\pi\/2}\\int_{0}^{\\mathfrak{H}_{Q'}}\\left(\\int_{|Q'|}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}\\part_Q \\bar f(Q',K)\\d {K}\\d {Q'}\\\\\n&+\\int_{-\\pi\/2}^{\\pi\/2}\\int_{\\mathfrak{H}_{Q'}}^\\infty\\left(\\int_{\\mathfrak{Q}_{K}}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}\\part_Q \\bar f(Q',K)\\d {K}\\d {Q'}.\n\\end{align*}\nNow we use \\eqref{eq:f-to-g} from Lemma~\\ref{lem:f-to-g} and that $\\part_Q=\\frac{1}{c'({K})t}(Y+\\part_{K})$ to get that\n\\begin{align*}\nT_1+T_2=&t^{-1}\\int_{-\\pi\/2}^{\\pi\/2}\\int_{0}^{\\mathfrak{H}_{Q'}}\\frac{1}{c'({K})}\\left(\\int_{|Q'|}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K} (Y+\\part_{K})\\part_Q \\bar{g}(Q',K)\\d {K}\\d {Q'}\\\\\n&+t^{-1}\\int_{-\\pi\/2}^{\\pi\/2}\\int_{\\mathfrak{H}_{Q'}}^\\infty\\frac{1}{c'({K})}\\left(\\int_{\\mathfrak{Q}_{K}}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}(Y+\\part_{K})\\part_Q \\bar{g}(Q',K)\\d {K}\\d {Q'}.\n\\end{align*}\n\nNext we integrate by parts in $K$. Since $\\mathfrak{Q}_{\\mathfrak{H}_{Q'}}=|Q'|$, we see that the boundary terms exactly cancel! Hence,\n\\begin{align*}\n&t^{-1}\\int_{-\\pi\/2}^{\\pi\/2}\\int_{0}^{\\mathfrak{H}_{Q'}}\\frac{1}{c'(K)}\\left(\\int_{|Q'|}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K} \\part_{K}\\part_Q \\bar{g}(Q',K)\\d {K}\\d {Q'}\\\\\n&+t^{-1}\\int_{-\\pi\/2}^{\\pi\/2}\\int_{\\mathfrak{H}_{Q'}}^\\infty\\frac{1}{c'(K)}\\left(\\int_{\\mathfrak{Q}_{K}}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}\\part_{K}\\part_Q \\bar{g}(Q',K)\\d {K}\\d {Q'}\\\\\n&\\quad=-t^{-1}\\int_{-\\pi\/2}^{\\pi\/2}\\int_{0}^{\\mathfrak{H}_{Q'}}\\part_K\\left(\\frac{1}{c'(K)}\\left(\\int_{|Q'|}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}\\right)\\part_Q \\bar{g}(Q',K)\\d {K}\\d {Q'}\\\\\n&\\quad -t^{-1}\\int_{-\\pi\/2}^{\\pi\/2}\\int_{\\mathfrak{H}_{Q'}}^\\infty\\part_K\\left(\\frac{1}{c'(K)}\\left(\\int_{\\mathfrak{Q}_{K}}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}\\right)\\part_Q \\bar{g}(Q',K)\\d {K}\\d {Q'}.\n\\end{align*}\n\nSince there is no boundary term we can integrate by parts after writing $\\part_Q=\\frac{1}{c'({K})t}(Y+\\part_{K})$ once more. Next note that that $\\bar{g}(Q,K)$ is nonzero only for $K \\in [c_s,c_{s}^{-1}]$ and that derivatives of $\\frac{c(K)}{c'(K)}$ is bounded as $|c'(K)|\\geq \\delta$ by Lemma~\\ref{lem:c-prime-bound}. Futher, $S(Q,K)=\\sin\\chi$ is smooth as a function of $K$. Thus $$\\sum_{\\ell\\leq 2} \\partial_K^\\ell \\left(\\frac{1}{c'(K)}\\left(\\int_{|Q'|}^{\\pi\/2} S(Q,K)\\d Q\\right)c(K)\\sqrt{2K}\\right)\\lesssim 1.$$\n\nBy Cauchy--Schwarz in $Q'$ and $K$, we get that\n$$T_1+T_2\\lesssim \\sum_{\\ell\\leq 2}\\sup_{K}\\norm{Y^\\ell \\bar{g}}_{L^2_Q }.$$\nFinally, an application of \\eqref{eq:g-bound} from Lemma~\\ref{lem:f-to-g} followed by Lemma~\\ref{lem:infinity-est} gives us the required bound. \\qedhere\n\n\\end{proof}\n\\begin{proof}[Proof of Theorem~\\ref{thm:main}]\nThe proof follows by using Lemma~\\ref{lem:phi_t} and combining the estimates from Proposition~\\ref{prop:term-x=0} and Proposition~\\ref{prop:bulk-term}. \n\\end{proof}\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and outline}\n\\label{sec:intro}\n\nThe goal of this paper is to understand the energy levels of near-extremal charged black holes in D dimensions from the perspective of the Euclidean gravitational path integral. The description of such black holes simplifies due to their $AdS_2 \\times X_{D-2}$ near-horizon geometry, where $X_{D-2}$ is a compact space, specifying the geometry of the horizon. \n\n\nThe spectrum of near-extremal black holes has been a source of confusion over the years.\\footnote{By near-extremal we mean large charge black holes with a small but non-zero temperature ($T\\ll r_0^{-1}$, where $r_0$ is the horizon radius at extremality). } The first manifestation of this was pointed out by Preskill et al \\cite{Preskill:1991tb} (see also \\cite{Maldacena:1998uz} and \\cite{Page:2000dk}). The thermodynamics derived from a semiclassical evaluation of the gravitational path integral gives a temperature-dependent mass above extremality $E = M-M_0$ as $E \\sim 2\\pi^2 \\Phi_r T^2 $, where $M_0$ is the extremal mass. A similar analysis gives the semiclassical entropy $S=S_0 + 4\\pi^2 \\Phi_r T$, where $S_0=A_0\/(4G_N)$ is proportional to the extremal area $A_0$. This behavior is universal, but the precise value of the parameter $\\Phi_r$ depends on the details of the model. Because of this scaling, it was noticed in \\cite{Preskill:1991tb} that the thermodynamic description of near-extremal black holes breaks down at low enough temperatures, $T\\lesssim 1\/\\Phi_r$. At such temperatures, the naive semiclassical analysis suggests that emitting even a single Hawking quanta is sufficient to change the black hole's temperature by a large amount.\\footnote{An equivalent issue was pointed out in \\cite{Maldacena:1998uz} due to the dependence of $\\Phi_r$ with Newton constant $G_N$ in particular models.} String theory microstate counting examples \\cite{Maldacena:1996ds} indicate the resolution of this issue is the presence of a mass gap of order $E_{gap}\\sim 1\/\\Phi_r$ in the black hole spectrum.\\footnote{In this paper, we will focus on scales of order $\\sim 1\/\\Phi_r$ also captured by the gravity description. Of course, at a completely different regime of even lower temperatures $T \\sim e^{-S_0}$, we expect to find a discrete spectrum with spacing of order $e^{-S_0}$. This scale requires a UV completion of the gravitational theory.} However, its origin from the gravitational description has, so far, been elusive.\n\nA second manifestation of the gap problem is the question of the extremal black hole degeneracy. If the gravity description somehow breaks down at the energy scale $E \\sim 1\/\\Phi_r$ then it is possible that the black hole entropy $S_0$, obtained by a semiclassical computation, does not correctly capture the degeneracy of the extremal black hole; rather, $e^{S_0}$ could instead capture the number of extremal and near-extremal states within a $\\sim 1\/\\Phi_r$ energy interval. Nevertheless, while some authors claimed the extremal entropy vanishes \\cite{Hawking:1994ii}, string theory examples that preserve sufficient supersymmetry showed that the degeneracy at extremality (assumed to be captured by an index) matches with the Bekenstein-Hawking area $S_0$ \\cite{Strominger:1996sh}. \n\nIn \\cite{Iliesiu:2020qvm} these questions have been revisited using lessons from SYK \\cite{kitaevTalks, Maldacena:2016hyu} and Jackiw-Teitelboim (JT) gravity \\cite{Teitelboim:1983ux, Jackiw:1984je, Almheiri:2014cka, Jensen:2016pah, Maldacena:2016upp, Engelsoy:2016xyb,Iliesiu:2019lfc, Kapec:2019ecr}. The procedure used to compute the higher dimensional Euclidean path integral for non-supersymmetric black holes is the following. First, we separate the full near extremal geometry into the near-horizon $AdS_2 \\times X_{D-2}$ throat (where the interesting physics takes place) and the far away region, which we take to be \nflat. The second step is to recast the $D$-dimensional theory on the throat as a 2D theory on $AdS_2$. It was argued in \\cite{Iliesiu:2020qvm} that the temperature-dependence of the Euclidean path integral near-extremality is solely captured by a JT gravity theory (composed of fluctuations around the $AdS_2$ metric and a dilaton which captures the size of $X_{D-2}$), a dimensional reduction of $D$-dimensional gauge fields to 2D, and a gauge field originating from isometries of the horizon $X_{D-2}$. This reduction was considered classically in \\cite{Sachdev:2015efa, Almheiri:2016fws, Anninos:2017cnw, Turiaci:2017zwd, Nayak:2018qej, Moitra:2018jqs, Hadar:2018izi, Castro:2018ffi, Larsen:2018cts, Moitra:2019bub, Sachdev:2019bjn, Hong:2019tsx, Castro:2019crn, Charles:2019tiu,Larsen:2020lhg} but the new feature of \\cite{Iliesiu:2020qvm} is the explanation that quantum effects at low temperatures are also captured by this simple theory. It can be shown that JT gravity exactly reduces to a boundary mode, the Schwarzian theory, which lives in the boundary of the throat (shown in figure \\ref{fig:4DnearBH}), and describes the breaking of the emergent $SL(2,\\mathbb{R})$ symmetry in the throat. In this theory, one can compute the path integral exactly and extract the near-extremal spectrum.\\footnote{Integrating out the massive KK modes and other fields have the only possible effect of introducing temperature-independent logarithmic corrections to $S_0$ previously computed in \\cite{Banerjee:2010qc, Banerjee:2011jp, Sen:2011ba, Sen:2012cj, Iliesiu:2020qvm} and thus such modes will not affect the computation of the density of states.}\n\n\\begin{figure}[t!]\n\\begin{center}\n \\begin{tikzpicture}[scale=1]\n \\pgftext{\\includegraphics[scale=0.6]{nearextBHWiggleCropped.png}} at (0,0);\n \\draw (-5.5,3.5) node {\\small Asymptotic flat region};\n \\draw (-3.9,-1) node[text width=5cm,align=center] {\\small Near-horizon boundary \\\\ $\\tau \\rightarrow f(\\tau)$ JT mode};\n \\draw (-2.3,-3.5) node[text width=10cm,align=center] {\\small Horizon};\n \\draw (3,-2.3) node[text width=5cm,align=center] {\\small Near AdS$_2$ region\\\\ Supported by a constant $U(1)$ flux};\n \n \\draw (4.5,1.5) node[text width=5cm,align=center] {\\small Near extremal black hole\\\\ Charge $Q \\sim $ Mass};\n \\end{tikzpicture}\n \\caption{\\label{fig:4DnearBH} The near extremal 4D black hole. A similar picture applies to near-extremal rotating BTZ black hole in AdS$_3$.}\n \\end{center}\n \\end{figure}\n \nWith this new perspective, the authors of \\cite{Iliesiu:2020qvm} addressed the puzzling questions about the thermodynamics of non-supersymmetric near-extremal Reissner-Nordstr\\\"om black holes. Regarding the first confusion pointed out above, the scale $T\\sim 1\/\\Phi_r$ identified by \\cite{Preskill:1991tb} was found to be the scale at which quantum effects in the gravitational path integral become large, and the Schwarzian description becomes strongly coupled. This effect signals that the breaking of the $SL(2, \\mathbb R)$ conformal symmetry in the throat becomes important \\cite{Almheiri:2016fws}. However, in contrast with string theory proposals \\cite{Maldacena:1996ds}, it was found that for non-supersymmetric black holes, there is no gap at the scale $E\\sim 1\/\\Phi_r$ and the gravitational path integral predicts a density of states that goes smoothly to zero as $E\\to0$. For convenience, this density of states is reproduced here in figure \\ref{fig:plot-of-rho} in the left column. Regarding the second confusion about the ground state degeneracy of the extremal black hole, \\cite{Iliesiu:2020qvm} also predicts the entropy of the extremal black hole to be much smaller than the naive prediction given by the area of the extremal horizon. A similar conclusion was obtained for near-extremal rotating black holes in 3D gravity \\cite{Ghosh:2019rcj,Maxfield:2020ale}.\n \n \n \nThe results of \\cite{Iliesiu:2020qvm} thus show a very different spectrum of near-extremal black holes compared to results obtained from microscopic counting: string theory predicts that the ground state degeneracy of extremal black holes should agree with Bekenstein-Hawking extremal area \\cite{Strominger:1996sh} and that the value of the mass gap $E_{gap}\\sim 1\/\\Phi_r$ can be extrapolated from the weak coupling regime \\cite{Maldacena:1996ds}. The purpose of this paper is to resolve this puzzle. The solution is that most examples in string theory involve supergravity and the extremal black hole preserves some supersymmetries. In the cases studied in this paper, the temperature dependence of the Euclidean path integral is captured by a supersymmetric generalization of JT gravity, which describes the breaking of an emergent superconformal symmetry.\n\n\n In this paper, we will discuss the case of near-BPS near-extremal charged black holes in two setups. The first is in four dimensions ($D=4$), where such objects are described classically by the Reissner-Nordstr\\\"om solution with a $AdS_2 \\times S^2$ throat. We will consider such black holes in 4D ungauged $\\cN=2$ supergravity in asymptotically flat space \\cite{Freedman:1976aw}. The situation is depicted in Figure~\\ref{fig:4DnearBH}. In our conventions $M_0=Q\/\\sqrt{G_N}$, proportional to the charge of the black hole, and $\\Phi_r= \\sqrt{G_N}\\hspace{0.1cm} Q^3$. Such a 4D theory has the right ingredients for SUSY preserving extremal black holes \\cite{Gibbons:1982fy,Tod:1983pm,Ferrara:1995ih}. The BPS nature of such gravitational solutions allows one to identify them with corresponding string theory constructions, for example \\cite{Maldacena:1997de, Ooguri:2004zv,Simons:2004nm,Denef:2007vg}. The second setup we will consider is a near-extremal rotating black hole in $(4,4)$ supergravity in $AdS_3$, with a near-horizon geometry $AdS_2 \\times S^1$. This system is relevant for comparison with the D1\/D5 system \\cite{Strominger:1996sh}. Even though some questions about the spectrum can be addressed using solely the $(4,4)$ super-Virasoro algebra in this particular case thanks to the presence of an $AdS_3$ throat, we will be able to show in section \\ref{sec:ads3} that the spectrum matches with the one derived from JT gravity. The Virasoro analysis obviously does not generalize to other cases, for example the Reissner-Nordstr\\\"om black hole mentioned above, but the JT analysis does and is universal as we explain below.\n\n \nThe main feature in 4D $\\cN=2$ supergravity and $(4,4)$ supergravity in $AdS_3$, compared to Einstein gravity, is the fact that the emerging symmetry in the throat becomes the superconformal group $PSU(1,1|2) \\supset SL(2,\\mathbb{R})\\times SU(2)$. In 4D, this includes the $AdS_2$ conformal symmetry and the $S^2$ isometries as bosonic subgroups \\cite{Kallosh:1997qw, Boonstra:1998yu,Claus:1998yw,Michelson:1999kn}. In $AdS_3$ the $SU(2)$ arises from a 3D gauge field. In the examples from string theory there is a second $SU(2)$ gauge field coming from the isometries of an extra $S^3$ factor. This second $SU(2)$ charge can be turned on without breaking supersymmetry, and corresponds to the black hole of \\cite{Breckenridge:1996is} which has rotation along the $S^3$. We find in both cases the relevant 2D mode in the throat controlling the temperature dependence of the partition function is given by $\\cN=4$ super-JT gravity, which describes the breaking of $PSU(1,1|2)$ and becomes strongly coupled at low temperatures. We will first define this theory and solve it exactly to extract the temperature dependence of the partition function, and from it the black hole spectrum. In order to do this, we will show that $\\cN=4$ super-JT gravity is equivalent to a $\\cN=4$ super-Schwarzian theory, which we introduce in this paper. We solve this theory using either path integral localization or canonical quantization.\\footnote{Previous attempts of defining the $\\mathcal{N}=4$ super-Schwarzian are \\cite{Aoyama:2018lfc, Aoyama:2018voj, Galajinsky:2020hsy} however, explicit formulae for all the components of the $\\mathcal{N}=4$ super-Schwarzian were not presented. Furthermore, to our knowledge, no previous quantization attempt has been made. }\n \n\\begin{figure}[t!]\n\\begin{center}\n \\begin{tikzpicture}[scale=0.65]\n \\pgftext{\\includegraphics[scale=0.46]{dosem.pdf}} at (0,0);\n \\draw (-5.5,-0.4) node {$e^{S_0}$};\n \\draw (-5.5,2.75) node {\\small $\\rho(E)$};\n \\draw (-1.7,-3.2) node {\\small $\\sim 1\/\\Phi_r$};\n \\draw (4.55,-3) node {\\small $E$};\n \\draw (-1,-4.5) node {\\small (a) Einstein gravity, $J=0$};\n \\end{tikzpicture}\n \\hspace{0.5cm}\n \\begin{tikzpicture}[scale=0.65]\n \\pgftext{\\includegraphics[scale=0.51]{dossugra.pdf}} at (0,0);\n \\draw (-5.7,-0.4) node {$e^{S_0}$};\n \\draw (-5.8,2.75) node {\\small $\\rho(E)$};\n \\draw (-1.7,-3.2) node {\\small $E_{gap}= \\frac{1}{8 \\Phi_r}$};\n \\draw (5,-3) node {\\small $E$};\n \\draw (-0.5,-4.5) node {\\small (b) SUGRA, $J=0$};\n \\end{tikzpicture}\n \\\\\n \\vspace{0.8cm}\n \\begin{tikzpicture}[scale=0.65]\n \\pgftext{\\includegraphics[scale=0.46]{dosj1.pdf}} at (0,0);\n \\draw (-5.5,-0.4) node {$e^{S_0}$};\n \\draw (-5.5,2.75) node {\\small $\\rho(E)$};\n \\draw (-1.7,-3.2) node {\\small $J(J+1)\/2\\Phi_r$};\n \\draw (4.55,-3) node {\\small $E$};\n \\draw (-1,-4.5) node {\\small (c) Einstein gravity, $J\\neq0$};\n \\end{tikzpicture}\n \\hspace{0.5cm}\n \\begin{tikzpicture}[scale=0.65]\n \\pgftext{\\includegraphics[scale=0.51]{dosj1susy.pdf}} at (0,0);\n \\draw (-5.7,-0.4) node {$e^{S_0}$};\n \\draw (-5.8,2.75) node {\\small $\\rho(E)$};\n \\draw (-1.7,-3.2) node {\\small $J^2\/2\\Phi_r$};\n \\draw (5,-3) node {\\small $E$};\n \\draw (-0.5,-4.5) node {\\small (d) SUGRA, $J\\neq0$};\n \\end{tikzpicture}\n \\caption{\\label{fig:plot-of-rho} Schematic shape of the black hole spectrum at fixed $SU(2)$ charge $J$ as a function of energy above extremality $E$. In 4D $J$ is angular momentum while in $AdS_3$ it is the $SU(2)$ charge (one of the two rotations along extra $S^3$ in string theory) that breaks supersymmetry. We show the semiclassical answer (red dashed) and the solution including quantum effects (purple). (a) Answer for Einstein gravity. We see there is no gap at scale $E\\sim 1\/\\Phi_r$ and the extremal entropy goes to zero. (b) Answer for supergravity (either $\\cN=2$ in 4D or $\\cN=(4,4)$ in 3D). We find a gap at the scale $E_{gap}=\\frac{1}{8\\Phi_r}$ and a number $e^{S_0}$ of extremal states, consistent with string theory expectations. (c) Einstein gravity spectrum for $J\\neq 0$. (d) Supergravity spectrum for $J\\neq 0$, the jumps indicate contributions from different supermultiplets $\\mathbf{J}, \\mathbf{J+1\/2}$ and $\\mathbf{J+1}$.}\n \\end{center}\n \\end{figure}\n\nTo contrast the results found in this paper, we show in Figure~\\ref{fig:plot-of-rho} the spectrum of near-extremal black holes derived from Einstein gravity in 4D coupled to a Maxwell field in the left panel. As previously mentioned, we see a small extremal entropy and a lack of a gap. In the right column, we show the main result of this paper, the density of states for near-extremal black holes in 4D $\\cN=2$ supergravity and $(4,4)$ supergravity in asymptotically $AdS_3$. Through a computation of the Euclidean path integral, independent of the UV completion of the theory, our analysis verifies and predicts the following results:\n\\begin{itemize}\n\\item We find that extremal BPS states exhibit an exact degeneracy. This degeneracy is given by the Bekenstein-Hawking horizon area, which is consistent with extremal microstate countings in string theory \\cite{Strominger:1996sh}. This is not true for extremal non-BPS states with $J\\neq 0$.\n\n\\item We observe the presence of a mass gap $E_{gap}=1\/(8\\Phi_r)$. In the context of $(4,4)$ supergravity in asymptotically $AdS_3$ and the D1\/D5 system, we verify the mass gap estimated at weak coupling from long strings \\cite{Maldacena:1996ds}. Here, this prediction is also expanded to black holes in 4D $\\cN=2$ supergravity. \n\n\\item We find that the extremal states are solely bosonic, implying the Witten index from a microscopic model coincides with the black hole degeneracy (For previous arguments see for example \\cite{Sen:2009vz, Mandal:2010cj, Benini:2015eyy}). This implies that the typical counting of microstates in string theory using an index, such as \\cite{Strominger:1996sh}, is correct.\n\n\\item A previous attempt to obtain the gap from a gravitational perspective was studied by Maldacena and Strominger \\cite{Maldacena:1997ih}. Their argument is only correct for black holes in AdS$_3$, and not for Reissner-Nordstr\\\"om for example. Our analysis that takes into account the breaking of $PSU(1,1|2)$ can in principle be applied to any system with this pattern of symmetry breaking and applies to situations without an $AdS_3$ factor in the throat.\n\n\\item In combination with the prior non-supersymmetric results from \\cite{Iliesiu:2020qvm}, we conclude that these features found in string theory examples heavily rely on supersymmetry and are special to supergravity. \n\\end{itemize}\n\n\n\nWithout jumping into all the details, we will give a summary of the technical results derived below. The $\\cN=4$ super-Schwarzian theory describing the spectrum of supersymmetric black holes, is given by the following action\n\\beq\n\\label{eq:intro-N=4-Schw-action}\nI_{\\cN=4} =-\\Phi_r \\int d\\tau \\left[ \\text{Sch}(f,\\tau) + {\\rm Tr} \\left( g^{-1} \\partial_\\tau g\\right)^2 + ({\\rm fermions}) \\right],\n\\eeq\nwhere the Schwarzian derivative is defined as \n\\beq\n\\text{Sch}(f,\\tau) \\equiv \\frac{\\partial_\\tau^3 f}{\\partial_\\tau f} - \\frac{3}2 \\left(\\frac{\\partial_\\tau^2 f}{\\partial_\\tau f} \\right)^2.\n\\eeq\nIn the action \\eqref{eq:intro-N=4-Schw-action}, $f(\\tau)$ is a reparametrization mode, $g(\\tau)$ is a $SU(2)$ element, and we ignore the terms involving fermionic fields $\\eta(\\tau)$ and $\\bar{\\eta}(\\tau)$ in the doublet of $SU(2)$. The field $g(\\tau)$ describes fluctuations in the angular momentum $J$ of the black hole in the 4D setup comes from the isometries of $S^2$. This theory has $\\cN=4$ Poincar\\'e supersymmetry but \\textbf{breaks} the emergent superconformal $PSU(1,1|2)$ space-time symmetry.\\footnote{Nonetheless, $PSU(1,1|2)$ is an important global symmetry for the action \\eqref{eq:intro-N=4-Schw-action}. We will clarify this in section \\ref{sec:space-time-and-global-symm}.} According to the unbroken $\\cN=4$ Poincar\\'e supersymmetry, supermultiplets organize into $\\mathbf{J}=(J)\\oplus2(J-\\frac{1}{2})\\oplus(J-1)$ for $E\\neq0$ and states $(J)$ for $E=0$. Solving this theory exactly gives the density of supermultiplet states labeled by their highest angular momentum $J$ as\n\\beq\n\\rho(J,E) = e^{S_0} \\delta_{J,0}\\delta(E)+ \n\\frac{e^{S_0}J}{4\\pi^2 \\Phi_r E^2}\\sinh \\left(2 \\pi \\sqrt{2\\Phi_r(E-E_0(J))} \\right)\\hspace{0.1cM} \\Theta\\Big(E-E_0(J)\\Big)\\,, \n\\eeq\nwhere we define $E_0(J)\\equiv J^2\/(2\\Phi_r)$. The details of this formula can be found in section \\ref{sec:N=4-super-Schw}. For example, states with zero angular momentum come from the contributions with $J=0$, $J=1\/2$ since $\\mathbf{1\/2}=(1\/2)\\oplus 2 (0)$ and $J=1$ since $\\mathbf{1}=(1)\\oplus 2(1\/2)\\oplus (0)$. This is the origin of the plot in the right panel of figure \\ref{fig:plot-of-rho}. In the continuous part states with angular momentum $J$ start at an energy $E_0(J)$. This is not necessarily surprising since the same feature appears in the non-supersymmetric case \\cite{Iliesiu:2020qvm}. From the gravity perspective this is the correction to the extremality bound of non-BPS extremal charged slightly-rotating black holes. The surprising feature in the supersymmetric theory is that there are no states with zero angular momentum in the range $0$ is a quadratic bilinear form invariant under adjoint transformations. Such a form can be obtained from the quadratic Casimir of $\\mathfrak{psu}(1,1|2)$.\\footnote{The quadratic Casimir of the superalgebra can be written as, \\be\nC_2 = L_0^2 - \\frac{1}2 \\{L_1, L_{-1}\\} - T_i T^i +\\frac{1}4 (i \\sigma^2)^\\b{}_\\a [G_p{}^\\a ,\\bar G^{\\,p}{}_\\b],.\\nn\n\\ee\nRewriting $C_2 = g_{AB} X^A X^B$ with $X = \\{L_0, L_\\pm,\\,T_i,\\,G_p{}^\\a,\\, \\bar G^{\\,p}{}_\\a\\}$, we can define the invariant quadratic form by using the metric, $\n \\eta_{AB}^{\\mathfrak{psu(1,1|2)}} = (-1)^{[X_A]} (g_{AB}^{-1})\n$, \nwhere $[X_A] = 0,1$ for bosonic, and respectively, fermionic generators. In such a case, we can define the supertrace as, $\n\\text{Str} B F = \\< B, \\, F\\> = \\eta_{AB} B^A F^B$.\n} We define the gauge field in terms of the supermultiplet of the frame $e^a$ and spin connection $\\omega$, also consisting of the SU(2) gauge field $B^i$ and the gravitinos $\\psi_{p}{}^{ \\a}$ as:\\footnote{We choose such conventions for the $\\mathfrak{sl}(2, \\mR)$ components of the gauge field in order to agree with \\cite{Saad:2019lba}. }\n\\be \n\\label{eq:gauge-field-ansatz}\nA(x) &= \\sqrt{\\frac{\\Lambda}{2}} \\left[e^1(x) L_0 + \\frac{e^2(x)}2 \\left(L_{1} - L_{-1}\\right) \\right]- \\frac{\\omega(x)}2 \\left(L_1+L_{-1}\\right) + B^i(x) T_i + \\nonumber \\\\ &+ \\left(\\frac{\\Lambda}{2}\\right)^{\\frac{1}{4}}\\left(\\bar{\\psi}^{p}{}_{ \\a}(x) G_{p}{}^{ \\a} + \\psi_{p}{}^{ \\a}(x) \\bar{G}^p{}_{ \\a}\\right)\\,.\n\\ee\nThe zero-form field $\\phi(x)$ is fixed in terms of the supermultiplet of the $SL(2, \\mR)$ Lagrange multipliers $\\phi^{1, 2}$ and $\\phi^0$, \n\\be\n\\label{eq:Lagr-multiplier}\n\\phi(x) &= 2\\phi^1(x) L_0 + {\\phi^2(x)} \\left(L_1 - L_{-1} \\right)- {\\phi^0(x)} \\left(L_1+L_{-1}\\right) + b^i (x) T_i + \\nn \\\\ &+ \\left(\\frac{\\Lambda}{2}\\right)^{-\\frac{1}{4}}\\left(\\bar{\\lambda}^{p}{}_{ \\a }(x) G_{p}{}^{ \\a}+ \\lambda_{p}{}^{ \\a }(x) \\bar G^{p}{}_{ \\a}\\right)\\,.\n\\ee\nHere, $\\lambda$ and $\\bar \\lambda$, and $\\psi$ and $\\bar \\psi$ should be understood as independent components of $A$ and are not related to the Hermitian conjugates of each other. In such a case, the action can be written as,\n\\be\n\\label{eq:JT-N=4-action}\nI_{JT}^{\\cN=4}& = {i} \\int \\bigg( \n\\underbrace{\\phi^0}_{\\sim\\text{ Dilaton}}\\bigg[\\underbrace{d\\omega + \\frac{\\Lambda}4 \\epsilon_{ab} e^a \\wedge e^b}_{\\frac{d^2 x}2 \\sqrt{g} (R+\\Lambda)} -\\underbrace{\\sqrt{2\\Lambda}\\bar{\\psi}^p{}_\\alpha \\wedge \\psi_p {}^\\alpha }_{\\substack{\\text{Gravitino $\\psi_p{}^\\a$ contribution}\\\\\\text{multiplying the dilaton}}}\\bigg] \\nn \\\\ &- \\sqrt{\\frac{\\Lambda}{2}}\\underbrace{\\phi^a}_{\\substack{\\text{Super-torsion}\\\\ \\text{Lagr.~multiplier}}}\\underbrace{\\left[de^a - \\epsilon_{ab}\\,\\omega\\wedge e^b +2\\left(\\bar \\gamma_a\\right)^{\\alpha}{}_{\\beta}\\bar{\\psi}^{p}{}_\\alpha\\wedge \\psi_p{}^{\\beta} \\right]}_{\\text{super-torsion component }\\tau^a} \\\\ \n&- \\tr_{SU(2)} \\underbrace{\\left[b \\left(dB + B \\wedge B \\right)\\right]}_{SU(2)\\text{ BF theory}} \\,\\,+ \\,\\, \\sqrt{\\frac{\\Lambda}{2}}\\underbrace{b^q{}_p \\bar{\\psi}^p{} \\wedge \\bar{\\gamma}_3\\psi_q{} }_{SU(2)\\text{ BF super-partner}}\\nn \n+ \\, \\underbrace{2\\lambda\\mathcal{D} \\bar{\\psi} +2\\mathcal{D}^*\\psi \\bar{\\lambda}}_{\\substack{\\text{Gravitino $\\psi^p{}_\\a$ and }\\\\\\text{dilatino $\\lambda_p{}^\\a$ contribution}}}\\bigg)\\,, \n\\ee\nwhere we have outlined the important terms in the action which will ease the comparison with the near-horizon action which we shall obtain in section \\ref{ssec:4dSugra}.\\footnote{Above, our normalization of the $SU(2)$ trace is given such that $\\tr_{SU(2)}(T^i T^j) = -\\frac{1}2 \\delta^{ij}$. The covariant derivative is given by $\\mathcal{D}=\\bar{\\gamma}_3d+ \\sqrt{\\frac{\\Lambda}{2}}\\bar{\\gamma}_ae^a+\\frac{1}{2}\\omega+\\frac{\\bar{\\gamma}_3}{2}B^i (\\sigma^i)$, and $\\cD^*$ is the conjugate of $\\cD$. We choose $\\bar{\\gamma}_1=\\sigma_1,\\,\\, \\bar{\\gamma}_2=-\\sigma_3$, and $\\bar \\gamma_3 = \\bar \\gamma_1 \\bar \\gamma_2 = i\\sigma_2$. }\n\n\nThe equations of motion for $\\phi^{1,2}$ act as Lagrange multipliers and imposes that the super-torsion vanishes. After integrating out $\\phi^{1,2}$, one can replace $e^1$, $e^2$ and $\\omega$ in terms of the metric $g_{\\mu \\nu}$ to obtain the supergravity action \\eqref{eq:JT-N=4-action} in the second order formalism. The JT gravity dilaton is given by $-i \\phi^0 \\equiv \\Phi$. When comparing the gravitational theory obtained from the dimensional reduction of the near-horizon region of near-BPS black holes to the $\\cN=4$ super-JT action we will use this latter form. For simplicity, in the remainder of this section we will fix the cosmological constant, $\\Lambda = 2$. Later, when discussing the effective action for the near-horizon region of the near-BP black holes we will revert to a general cosmological constant, determined by the radius of the black hole.\n\n\nWe can complete the dictionary between the BF theory \\eqref{eq:JT-N=4-action} and the second-order super-JT gravity by noting that infinitesimal $PSU(1,1|2)$ gauge transformations in the former, map to infinitesimal diffeomorphisms in the latter. In particular, the infinitesimal supersymmetric transformations on all the fields in \\eqref{eq:JT-N=4-action} can be obtained by considering the infintesimal gauge transformation whose gauge parameter takes the form $\\Lambda = \\epsilon^{p\\a} G_{p \\a} + \\bar \\epsilon^{p \\a} \\bar G_{p \\a}$. \n\n\nWith this mapping between the BF theory \\eqref{eq:JT-N=4-action} and $\\cN=4$ super-JT gravity in mind, we can now analyze the supersymmetric boundary conditions applied to the theory \\eqref{eq:JT-N=4-action} which will be necessary in section \\ref{sec:higherD} to understand the gluing of the near-horizon region of near-BPS black holes to asymptotic flatspace. As we will see, these boundary conditions will reduce the gravitational path integral to that of the $\\cN=4$ super-Schwarzian. \n\n\n\n\\subsection{The super-JT boundary conditions}\n\\label{sec:super-JT-bdy-cond}\n\nWe begin, just like in the non-supersymmetric case, by imposing that the boundary metric is fixed and proper boundary length is large, $L = \\beta\/\\varepsilon$, with $\\varepsilon$ small. Working in Fefferman-Graham gauge, where the metric can be written as\n\\be \n\\label{eq:FG-gauge-metric}\nds^2 = dr^2 + \\left(\\frac{1}4 e^{2r} - \\tilde \\cL(\\tau) + \\dots \\right) d\\tau^2 \\,,\n\\ee\nwe consider the boundary to be at fixed, but large, $r$. To satisfy the boundary conditions we identify $\\tau \\sim \\tau+\\beta$ and cut-off the geometry at $e^{-r} = \\varepsilon\/2$. We fix the leading component of the boundary metric and allow the sub-leading component, $\\tilde \\cL(\\tau)$, to vary. The $\\dots$ represent terms that are further sub-leading in $r$ which we do not need to fix.\n\nSimilarly, we require that the dilaton takes an asymptotically large and constant value at the boundary, $\\Phi|_{\\partial \\cM} \\equiv -i\\phi_0|_{\\partial \\cM} = \\Phi_r\/\\varepsilon$. Next, we discuss the boundary conditions for the super-partners of the frame and spin-connection. If working in the Fefferman-Graham gauge \\eqref{eq:FG-gauge-metric}, then, in order to preserve supersymmetry, we impose that the leading component of the gravitino is fixed and vanishes $\\psi^{p \\alpha} = O(e^{-r\/2})$ and $\\bar \\psi^{q \\alpha} = O(e^{-r\/2})$. Similarly, to again preserve supersymmetry, we impose that the leading contribution of the dilatino vanishes at asymptotically large values $\\lambda = O(e^{-r\/2})$. Finally, we describe the boundary conditions for the $SU(2)$ gauge field and its Lagrange multiplier. From the perspective of the higher dimensional black holes which we will study in section \\ref{sec:higherD}, the $SU(2)$ gauge field appears from the isometry of $S^2$ along which we perform the dimensional reduction. We would therefore, like to fix Dirichlet boundary conditions at the boundary of the asymptotically flat region. As we will describe in detail in section \\ref{sec:higherD}, imposing these boundary conditions in the asymptotically flat region translates to fixing a linear combination of the zero-form field $b$ and the $SU(2)$ gauge field $B$ at the boundary of the near-horizon region. \n\n\nWe would now like to translate the boundary conditions discussed above in the second-order formalism to boundary conditions in the BF theory \\eqref{eq:JT-N=4-action}. We will follow the steps outlined in \\cite{Grumiller:2017qao} (explained also recently in \\cite{Saad:2019lba}) in non-supersymmetric JT gravity and will obtain results similar to \\cite{Cardenas:2018krd}, which studied boundary conditions in JT supergravity with an $OSp(2, \\cN)$ isometry group. Fixing the Fefferman-Graham gauge for the metric \\eqref{eq:FG-gauge-metric} yields the value of the frame $e^{1, \\, 2}$ and spin connection $\\omega$ \\cite{Saad:2019lba}\n\\be \n\\label{eq:FG-1st-order-formalism}\ne^1 = dr \\,, \\qquad e^2= \\left(\\frac{1}2 e^r - \\tilde \\cL(\\tau) e^{-r}\\right)d\\tau \\,, \\qquad \\omega = -\\left(\\frac{1}2 e^r + \\tilde \\cL(\\tau) e^{-r}\\right)d\\tau \\,.\n\\ee \nFixing the decaying piece of the gravitino we can gauge fix all other components of the $PSU(1,1|2)$ gauge field along the boundary to be\n\\be\n\\label{eq:PSU(1,1|2)-gauge-field-form}\nA_\\tau(\\tau) &= L_1 \\frac{e^r}2 + L_{-1} \\tilde \\cL(\\tau) e^{-r} + B^i_\\tau(\\tau) T_i \\nn \\\\ &+ e^{-\\frac{r}2} \\frac{\\bar\\psi^p(\\tau)}2 G_{p, \\, -\\frac{1}2} + e^{-\\frac{r}2} \\frac{\\psi^p(\\tau)}2 \\bar G_{p,\\, -\\frac{1}2} + O(e^{-2r})\\,,\n\\ee\nwhere we have used the shorthand notation $\\psi^{p}(\\tau) \\equiv \\psi^{p,\\, \\frac{1}2} $ and $\\bar \\psi^{p}(\\tau) \\equiv \\bar \\psi^{p,\\, \\frac{1}2} $ on the boundary $\\partial \\cM$. \nWe can now impose the equations of motion of $A_\\tau$ in the proximity of the boundary $D_\\tau \\phi = 0$, or rather due to only fixing the leading orders of $A_\\tau$ and $\\phi$, $D_\\tau \\phi = O(e^{-r})$, where $D$ denotes the $PSU(1,1|2)$ covariant derivative. This implies that the zero-form field $\\phi$ should be constrained to take the form \n\\be\n\\label{eq:b.c.-for-dilaton}\n\\phi(\\tau) &= -{i\\Phi_r} L_1 e^r + {2 i\\Phi_r'} L_0 +(\\bar \\psi \\lambda + \\psi \\bar \\lambda - 2i \\tilde\\cL \\Phi_r - 2i\\Phi_r'')L_{-1} e^{-r} + b^i(x) T_i \\nn \\\\&+ \\bar \\lambda^p G_{p,\\frac{1}2} e^{r\/2} - \\bigg(2(\\bar \\lambda^p)' - i\\Phi_r \\bar \\lambda^p - B_\\tau^i (\\sigma^i{}^*)^p{}_q\\, \\bar \\lambda^q\\bigg) G_{p,-\\frac{1}2}e^{-r\/2} \\nn \\\\ &+ \\lambda^p \\bar G_{p,\\frac{1}2}e^{r\/2} - \\bigg(2 (\\lambda^p)' - i \\Phi_r \\lambda^p - B_\\tau^i (\\sigma^i\\,)^p{}_q\\, \\lambda^q\\bigg) \\bar G_{p,-\\frac{1}2} e^{-r\/2} + O(e^{-2r})\\,,\n\\ee\nwhere, again, we have used the short-hand notation $\\lambda^{p}(\\tau) = \\lambda^{p, \\frac{1}2}$ and $\\bar \\lambda^{p}(\\tau) =\\bar \\lambda^{p, \\frac{1}2}$ on the boundary $\\partial \\cM$ and where $\\Phi'_r \\equiv \\partial_\\tau \\Phi_r(\\tau)$ denotes the derivative with respect to the boundary time $\\tau$.\\footnote{We will motivate why the parameter $\\Phi_r$ in \\eqref{eq:b.c.-for-dilaton} can be identified with the renormalized dilaton shortly.}\n\nIf we want to impose the boundary conditions for the dilaton ($\\Phi_r = \\text{const}$), the dilatinos ($\\lambda^{p, \\,\\frac{1}2} = 0$ and $\\bar \\lambda^{p,\\,\\frac{1}2} =0$) and the zero-form field $b^i(x)$ we can then relate the gauge field in \\eqref{eq:PSU(1,1|2)-gauge-field-form} to the zero-form field $\\phi(\\tau)$ as $-2i \\Phi_r \\hspace{0.1cm}A_\\tau(\\tau) = \\phi(\\tau)$, where $\\Phi_r$ is the renormalized value of the dilaton. Thus, in the first-order formalism and in the gauge in which $A_\\tau$ takes the form \\eqref{eq:PSU(1,1|2)-gauge-field-form}, the boundary condition that we want to impose is $\\delta(2i\\Phi_r A_\\tau(\\tau) + \\phi(\\tau))|_{\\partial \\cM}=0$. The boundary term necessary for such a boundary condition to have a well defined variational principle is \\cite{Cardenas:2018krd}:\n\\be \n\\label{eq:bdy-term-JT-grav-first-order}\nI_{BF,\\,\\text{bdy.}} = \\frac{i}2 \\int_{\\partial \\cM} \\text{Str}\\, \\phi A = { \\Phi_r} \\int_{\\partial \\cM} d\\tau\\, \\text{Str}\\, A_\\tau^2 \\, . \n\\ee\nIntegrating out the field $\\Phi$ in the bulk we find that the bulk term yields no contribution. \nReplacing $A_\\tau$ in \\eqref{eq:bdy-term-JT-grav-first-order} we find that the action can then be rewritten as\n\\be \nI_{BF,\\,\\text{bdy.}} = -{\\Phi_r} \\int_{\\partial \\cM} d\\tau \\left[\\tilde \\cL(\\tau) + \\frac{1}2 \\left((B^1_\\tau)^2+(B^2_\\tau)^2+ (B^3_\\tau)^2\\right)\\right]\n\\ee\nThus, it is convenient to define \n\\be\n\\label{eq:def-cL(u)-and-bdy-Lagr}\n\\cL(\\tau) = \\tilde \\cL(\\tau) + \\frac{1}2 \\left((B^1_\\tau)^2+(B^2_\\tau)^2+ (B^3_\\tau)^2\\right)\\,, \\,\\, \\text{ from which } \\,\\, I_{BF,\\,\\text{bdy.}} = -{\\Phi_r} \\int_{\\partial \\cM} d\\tau \\, \\cL(\\tau)\\,.\n\\ee\nWe will not determine $\\cL(\\tau)$ explicitly. Rather we will see how $\\cL(\\tau)$ (and all other variables in \\eqref{eq:PSU(1,1|2)-gauge-field-form}) transform under the gauge transformations that preserve the asymptotic form of \\eqref{eq:PSU(1,1|2)-gauge-field-form}. In general in a BF theory with gauge group $G$, boundary gauge transformations are parametrized by functions $g$ in $\\text{loop}(G)\/G$. However, since we are preserving the asymptotic form \\eqref{eq:PSU(1,1|2)-gauge-field-form} which comes from working in the Fefferman-Graham gauge \\eqref{eq:FG-1st-order-formalism} the space of gauge transformations is instead parametrized by $\\text{Diff}(S^{1|4})\/PSU(1, 1|2)$. Therefore, the way in which $\\cL(\\tau)$ transforms under this special class of gauge transformations will yield how the boundary Lagrangian transforms under $\\text{Diff}(S^{1|4})\/PSU(1, 1|2)$. From here, we will show in section \\ref{sec:N=4-transf-law} that the boundary Lagrangian can be identified with the $\\cN=4$ super-Schwarzian derivative. Thus, to do this identification, we note that $\\cL(\\tau)$, $B_\\tau(\\tau)$, and $\\psi^p(\\tau)$ transform as \n\\bea \n\\label{eq:gauge-transf-preserving-asymp}\n\\delta_\\L \\cL &=&\\xi \\cL'+2 \\cL \\xi'+ \\xi'''- B^i_\\tau (t^i)'+\\frac{1}{2}\\left(3\\bar{\\psi} \\epsilon'+\\bar \\psi'\\epsilon -3\\bar{\\epsilon}' \\psi -\\bar{\\epsilon} \\psi'\\right), \\nn \\\\ \n\\delta_\\L \\psi^p &=& \\xi ( \\psi^p)'+\\frac{3}{2}\\psi^p \\xi'-\\epsilon^p \\cL -2(\\epsilon^p)''-\\frac{1}{2}t^i(\\sigma^i){}^p{}_q\\, \\psi^q + (B^i_\\tau)' (\\sigma^i){}^p_q \\epsilon^q +2 B^i_\\tau (\\sigma^i{})^p{}_q\\,(\\epsilon^q)',\\nn\\\\ \n\\delta_\\L B^i_\\tau&=& \\left( \\xi B^i_\\tau\\right)'-(t^i)'+ \\frac{1}2 \\bar \\psi \\sigma^i\\epsilon+ \\frac{1}2 \\bar{\\epsilon} \\sigma^i \\psi + i \\epsilon^{ijk} \\,t^j\\,B^k_\\tau\\,,\n\\eea\nunder the gauge transformation which preserves the form of $A_\\tau$ in \\eqref{eq:PSU(1,1|2)-gauge-field-form}:\n\\be\n\\Lambda(\\tau) &= \\frac{\\xi}2 L_1 - \\xi' L_{0} + \\left(\\frac{1}2 \\bar \\psi \\epsilon +\\frac{1}2 \\psi \\bar \\epsilon - \\frac{1}2 \\xi (B^a)^2 +\\cL \\xi + \\xi'' \\right) L_{-1} - (t^i -\\xi B^i_\\tau) T_i \\nn \\\\ &+ \\frac{1}2 \\bar \\epsilon^p G_{p,\\frac{1}2} - \\left((\\bar \\epsilon^p)' - \\frac{1}2 \\xi \\bar \\psi^p - \\frac{1}2 B_\\tau^i (\\sigma^i{}^*){}^p_q\\, \\bar \\epsilon^q\\right) G_{p,-\\frac{1}2} + \\frac{1}2 \\epsilon^p \\bar G_{p,\\frac{1}2} \\nn \\\\ &- \\left( \\epsilon^p - \\frac{1}2 \\xi \\psi^p - \\frac{1}2 B_\\tau^i (\\sigma^i)^p_q\\, \\epsilon^q\\right) \\bar G_{p,-\\frac{1}2}\n\\ee\nIt is not a coincidence that $\\Lambda(\\tau)$ takes the same form (up to the redefinition \\eqref{eq:def-cL(u)-and-bdy-Lagr}) as $\\phi(\\tau)$ in \\eqref{eq:b.c.-for-dilaton}. This follows from requiring that the leading result in both $D_\\tau \\Lambda$ (since we require that the gauge transformation $\\Lambda$ not change the asymptotic form of $A_\\tau$ \\eqref{eq:PSU(1,1|2)-gauge-field-form}) and $D_\\tau \\phi$ (since we impose the equation of motion for $A$ by also using \\eqref{eq:PSU(1,1|2)-gauge-field-form}) both vanish. With these transformations in mind, we now proceed by introducing the necessary technology to define the $\\cN=4$ Schwarzian derivative. Following that, we will finally show that this Schwarzian derivative can be identified with the boundary Lagrangian $\\cL(\\tau)$ from \\eqref{eq:def-cL(u)-and-bdy-Lagr}. \n\n\nFinally, we can extend this analysis to the case when the $SU(2)$ chemical potential, $\\a$, is turned on. In order to do this we can generalize the previous boundary condition from $ A_\\tau - \\frac{i \\phi}{2\\Phi_r} = 0$ to $ A_\\tau - \\frac{i\\phi}{2\\Phi_r} = \\frac{2\\pi i}{\\beta} \\alpha T^3 $, where $\\alpha\\sim \\alpha +1$. This does not modify the boundary conditions of gravity and the fermions but now the $SU(2)$ gauge field boundary condition is $B_\\tau - \\Phi^{-1}_r b = \\frac{2\\pi i}{\\beta} \\alpha T^3$, supporting the identification of $\\alpha$ with a chemical potential. In section \\ref{sec:subtleties} we explain how from the 4D near-extremal black hole perspective this boundary condition is equivalent to fixing the holonomy of the gauge field in the asymptotically flat region, which is related to fixing the boundary 4D angular velocity. \n\n\n\\subsection{The $\\cN=4$ supersymmetric boundary mode} \n\nSo far we have seen that $\\cN=4$ super-JT gravity can be reduced to a boundary theory. In this section, we will see this theory can be recasted as a $\\cN=4$ super-reparametrization mode with a super-Schwarzian action. We will first review in \\ref{sec:N=4-superDiffeos} the definition of super-diffeomorphisms. In \\ref{sec:super-Schw-action} we will define the super-Schwarzian derivative that will be the action and in \\ref{sec:N=4-transf-law} we will put everything together to write down the final boundary action. \n\n\\subsubsection{Super-diffeomorphisms}\n\\label{sec:N=4-superDiffeos}\n\nTo match with the $\\cN=4$ super-JT theory defined previously we will be interested in $SU(2)$ extended $\\cN=4$ supersymmetry.\\footnote{Other choices with also $\\cN=4$ are the $O(4)$ extended algebra studied for example in \\cite{Schoutens:1988ig}, we will not be interested in those for the purposes of this paper.} We will begin by giving a super-space description of $\\cN=4$ super-diffeomorphisms. \n\nConsider an $\\cN=4$ super-line parametrized by a bosonic variable $\\tau$ and fermionic variables $\\theta^p$ and $\\bar{\\theta}_q$, where $p,q=1,2$. The coordinates $\\theta^p$ and $\\bar{\\theta}_q$ transform respectively in the fundamental and antifundamental representations of a local $SU(2)$ symmetry. We will omit the indices and simply write $\\theta$ and $\\bar{\\theta}$ when it is clear by context. We define the four super-derivatives as \n\\beq\nD_p = \\frac{\\partial}{\\partial \\theta^p} + \\frac{1}{2} \\bar{\\theta}_p \\partial_\\tau ,~~~~~\\bar{D}^q = \\frac{\\partial}{\\partial \\bar{\\theta}_q} + \\frac{1}{2} \\theta^q \\partial_\\tau.\n\\eeq\nThe $SU(2)$ indices will be raised and lowered by the antisymmetric tensors $\\varepsilon_{pq}$ and $\\varepsilon^{pq}$ with $\\varepsilon_{12}=\\varepsilon^{21}=1$. We will denote by $\\sigma^i$ with $i=1,2,3$ the Pauli matrices and indices will be contracted from `left-bottom' to `right-up', e.g. $\\bar{\\theta} \\sigma^i \\theta = \\bar{\\theta}_p (\\sigma^i)^p_q \\theta^q$. We will sometimes group the coordinates of the $\\cN=4$ super-line as $Z=(\\tau, \\theta^p, \\bar{\\theta}_p)$. \n\nWe will study general reparametrizations of the super-line, which will become the degrees of freedom in the path integral that defines the Schwarzian theory. These have the following form \n\\beq\\label{reparametrizations}\n\\tau \\to \\tau'(\\tau, \\theta, \\bar{\\theta}) ,~~~~\\theta^p \\to \\theta'{}^p(\\tau, \\theta, \\bar{\\theta}), ~~~~\\bar{\\theta}_q \\to \\bar{\\theta}'_q(\\tau, \\theta, \\bar{\\theta}),\n\\eeq\nand satisfy a set of constrains given by \n\\bea\\label{repa_constraints}\nD_p \\bar{\\theta}'_q =0&,&~~~\\bar{D}^p \\theta'{}^b=0,\\\\\nD_p \\tau' - \\frac{1}{2} (D_p \\theta'{}^q)\\bar{\\theta}'_q=0&,&~~~~\\bar{D}^p \\tau' - \\frac{1}{2} (\\bar{D}^p \\bar{\\theta}'_q)\\theta'{}^q=0\n\\eea\nanalyzed in \\cite{Matsuda:1988qf} and \\cite{Matsuda:1989kp}. They guarantee that the superderivatives transform homogeneously and preserve the global $SU(2)$ symmetry. We will refer to the space of solutions of these constrains as $\\cN=4$ super-diffeomorphisms and denote it as ${\\rm Diff}(S^{1|4})$, indicating one bosonic and four fermionic directions. \n\nNext we will look at some examples. The simplest super-reparametrizations are purely bosonic. They are given in terms of an arbitrary function $f(\\tau)$ and an arbitrary $SU(2)$ matrix $g(\\tau)$. The solutions of the constraints have the form \n\\bea\\label{eq:repbos}\n\\tau &\\to& f(\\tau) + \\frac{1}{8} f''(\\tau) (\\bar{\\theta}\\theta)^2,\\\\\n\\theta^p &\\to& g\\Big( \\tau -\\frac{1}{2}\\bar{\\theta}\\theta\\Big)^p_q ~\\theta^q~ \\sqrt{f'\\Big(\\tau-\\frac{1}{2}\\bar{\\theta}\\theta\\Big)} , \\\\\n\\bar{\\theta}_p &\\to& \\bar{\\theta}_q~ \\bar{g}\\Big(\\tau + \\frac{1}{2}\\bar{\\theta}\\theta\\Big)^q_p ~\\sqrt{f'\\Big(\\tau+\\frac{1}{2}\\bar{\\theta}\\theta\\Big)}.\n\\eea\nAnother simple example are the `chiral' and `anti-chiral' fermionic transformations. The chiral ones are parametrized in terms of two fermionic functions $\\eta^p(\\tau)$ in the fundamental of $SU(2)$ and the reparametrization is given by\n\\bea\\label{eq:repfer}\n\\tau &\\to&\\tau + \\frac{1}{2} \\bar{\\theta} \\eta\\Big(\\tau+\\frac{1}{2}\\bar{\\theta}\\theta\\Big) + \\frac{1}{4} \\partial_\\tau (\\bar{\\theta} \\eta(\\tau))^2,\\\\\n\\theta^p &\\to& \\theta^p + \\eta^p\\Big(\\tau-\\frac{1}{2}\\bar{\\theta}\\theta\\Big) , \\\\\n\\bar{\\theta}_p &\\to& \\bar{\\theta}_p\\Big( 1 + \\bar{\\theta} \\eta'\\Big(\\tau-\\frac{1}{2}\\bar{\\theta}\\theta\\Big) \\Big).\n\\eea\nThe anti-chiral are parametrized by $\\bar{\\eta}_p(\\tau)$ in the antifundamental, and the reparametrization is given by\n\\bea\\label{eq:repfer2}\n\\tau &\\to&\\tau + \\frac{1}{2} \\theta \\bar{\\eta}\\Big(\\tau-\\frac{1}{2}\\bar{\\theta}\\theta\\Big) + \\frac{1}{4} \\partial_\\tau (\\theta \\bar{\\eta}(\\tau))^2,\\\\\n\\theta^p &\\to&\\theta^p\\Big( 1 +\\theta \\bar{\\eta}'\\Big(\\tau+\\frac{1}{2}\\bar{\\theta}\\theta\\Big) \\Big) , \\\\\n\\bar{\\theta}_p &\\to& \\bar{\\theta}_p +\\bar{\\eta}_p\\Big(\\tau+\\frac{1}{2}\\bar{\\theta}\\theta\\Big).\n\\eea\nThe most general element of ${\\rm Diff}(S^{1|4})$ is obtained by a general fermionic transformation followed by a bosonic one.\\footnote{ In writing the Schwarzian derivative, the order of the composition is important. In particular, to compute the $\\cN=4$ super-Schwarzian we compose the bosonic transformation with a fermionic one in this order. The order of compositions will however be unimportant when localizing the path integral.} We parametrize ${\\rm Diff}(S^{1|4})$ in terms of the degrees of freedom\n$\nf(\\tau),~~g(\\tau)\\in SU(2),~~\\eta^p(\\tau),~~\\bar{\\eta}_p(\\tau),\n$\nwhere $p=1,2$. It is hard to determine the finite form that has all parameters turned on since such answer contains a large number of terms. Therefore, we will not write it down explicitly although its straightforward to get from the results presented so far. Instead, it is useful to analyze the most general infinitesimal transformation. Up to $\\mathcal{O}(\\eta)$ is given by \n\\bea\n\\tau &\\to&f(\\tau-\\frac{1}{2}\\bar{\\eta}(\\tau-\\frac{1}{2}\\bar{\\theta}\\theta)\\theta+\\frac{1}{2}\\bar{\\theta}\\eta(\\tau+\\frac{1}{2}\\bar{\\theta}\\theta)) + \\frac{1}{8}f''(\\tau)\\left((\\bar{\\theta}+\\bar{\\eta})(\\theta+\\eta)\\right)^2\\\\\n\\theta^p&\\to& F^p{}_q\\left(\\tau-\\frac{1}{2}\\bar{\\theta}\\theta\\right)\\left(\\theta^q+\\eta^q\\left(\\tau-\\frac{1}{2}\\bar{\\theta}\\theta\\right)\\right)+\\partial_{\\tau}\\left(F^p{}_q\\left(\\tau\\right)\\theta^q \\theta \\bar{\\eta}\\left(\\tau\\right)\\right)\\\\ \n\\bar{\\theta}_p &\\to& \\left(\\bar{\\theta}_q+\\bar{\\eta}_q\\left(\\tau+\\frac{1}{2}\\bar{\\theta}\\theta\\right)\\right)F^q{}_p\\left(\\tau+\\frac{1}{2}\\bar{\\theta}\\theta\\right)+ \\partial_{\\tau}\\left(\\eta\\left(\\tau\\right)\\bar{\\theta}\\bar{\\theta}_qF^q{}_p\\left(\\tau\\right)\\right)\n\\eea\nwhere we use $F^p{}_q(\\tau)=\\sqrt{f'(\\tau)}g^p{}_q(\\tau) $.\n\nWe introduce now an important sub-group of the super-reparametrizations, the global superconformal group $PSU(1,1|2)$. It is generated by six bosonic variables (three from $SL(2)$ and three from $SU(2)$) and eight fermionic. The general case can be found in \\cite{Matsuda:1989kp}. Instead we will write down the bosonic generators \n\\bea\\label{infintesimalparameterb}\n\\tau &\\to& \\frac{a\\tau+b}{c\\tau+d} - \\frac{c}{4(c\\tau+d)^3}(\\bar{\\theta}\\theta)^2 ,\\\\\n\\theta^p &\\to&[e^{i \\vec{t} \\cdot \\vec{\\sigma}}]^p_q\\theta^q \\frac{1}{c(\\tau - \\frac{1}{2} \\bar{\\theta}\\theta)+d} ,\\\\\n\\bar{\\theta}_p &\\to& \\bar{\\theta}_q [e^{-i \\vec{t} \\cdot \\vec{\\sigma}}]^q_p \\frac{1}{c(\\tau + \\frac{1}{2} \\bar{\\theta}\\theta)+d}\\label{infintesimalparameterb-last} ,\n\\eea\nwith $a,b,c,d\\in \\mathbb{R}$ such that $a d- bc=1$ and $c>0$. This is precisely of the form \\eqref{eq:repbos} where $f(\\tau)$ is a global conformal transformation $PSL(2,\\mathbb{R})$ while $(t^1, t^2, t^3)$ parametrize a global $SU(2)$ transformation. The fermionic generators can be parametrized by two constant spinor doublets $\\eta$ and $\\tilde{\\eta}$ and act as \n\\bea\n\\label{infintesimalparameterf-first}\n\\tau &\\to& \\tau + \\frac{1}{2} (\\bar{\\theta}-\\bar{\\tilde{\\eta}}) \\eta - \\frac{1}{2} \\bar{\\eta} (\\theta-\\tilde{\\eta}) ,\\\\\n\\theta^p &\\to& \\theta^p+ \\eta^p - \\tilde{\\eta}^p,\\\\\n\\bar{\\theta}_p &\\to&\\bar{\\theta}_p + \\bar{\\eta}_p - \\bar{\\tilde{\\eta}}_p,\n\\label{infintesimalparameterf}\n\\eea\n As anticipated there are eight fermionic generators. As explained above, the most general case is a composition of bosonic and fermionic. \n\n\\subsubsection{Super-Schwarzian action}\n\\label{sec:super-Schw-action}\n\nThe Schwarzian derivative associated to reparametrizations of the $\\cN=4$ super-circle was defined by Matsuda and Uematsu in \\cite{Matsuda:1989kp} (see also \\cite{Matsuda:1988qf}). The Schwarzian derivative is given in terms of superspace variables as \n\\beq\\label{Sderivative}\nS^i (Z; Z') = - 2 D \\sigma^i \\bar{D} \\log \\left(\\frac{1}{2} (D_p \\theta'{}^q)(\\bar{D}^p \\bar{\\theta}'_q)\\right).\n\\eeq\nIt satisfies the following chain rule \n\\beq\n\\label{chainrule}\nS^i (Z; Z') = \\frac{1}{ 2} (\\sigma^i)^p_q(\\sigma^j)^{q'}_{p'} (D_p \\theta''{}^{p'})(\\bar{D}^q \\bar{\\theta}''_{q'}) S^j (Z'';Z')+S^i (Z;Z'').\n\\eeq\nAnother defining property is the fact that $S^i (Z;Z')=0$ whenever the super-reparametrization is an element of the global $PSU(1,1|2)$ as in \\eqref{infintesimalparameterb}. This will prove consequential in section \\ref{sec:space-time-and-global-symm} when studying the global symmetries of the $\\cN=4$ super-Schwarzian action which we shall define shortly. \n\nThe derivative $S^i$ is a superfield with several components. We can extract the bosonic piece we want to associate to the Schwarzian action. In the notation of \\cite{Matsuda:1989kp} it is \n\\beq\nS^i (Z; Z') \\supset - \\bar{\\theta} \\sigma^i \\theta \\hspace{0.1cm}S_b(\\tau, \\theta, \\bar{\\theta}; \\tau',\\theta',\\bar{\\theta}')\n\\eeq\nOne motivation of this choice is to look at purely bosonic transformations defined in equation \\eqref{eq:repbos}. For simplicity let's briefly consider $Z'=(\\tau',\\theta',\\bar{\\theta}')$ where $\\tau'$, $\\theta'$ and $\\bar \\theta'$ take the special form \\eqref{eq:repbos}, in terms of an arbitrary $f(\\tau)$ and set $g(\\tau)=1$. Then the definition above gives the super-Schwarzian \n\\beq\n\\label{eq:just-in-terms-of-Schw}\nS^i (Z,Z') = -{\\rm Sch}(f,\\tau) \\bar{\\theta} \\sigma^i \\theta.\n\\eeq\nAnother motivation is that when we interpret $S^i$ as a superconformal generator, that component generates bosonic translations along the circle and we want to identify this as the action of the Schwarzian theory. \n\nThe super-Schwarzian satisfies the constrains $D_p D_q S^i = \\bar{D}^p \\bar{D}^q S^i=0$. Therefore, as a superfield it can be expanded in the following components \n\\bea\\label{sschwarzian}\nS^i &=& 2S_{T}^i + \\bar{\\theta} \\sigma^i S_f + \\bar{S}_f \\sigma^i \\theta - \\bar{\\theta} \\sigma^i \\theta S_b + i \\epsilon^{ijk} \\bar{\\theta} \\sigma^j \\theta \\partial_\\tau S_{T}^k \\nonumber\\\\\n&&- \\frac{1}{2}(\\bar{\\theta}\\theta) \\bar{\\theta}\\sigma^i \\partial_\\tau S_f + \\frac{1}{2} (\\bar{\\theta} \\theta) \\partial_\\tau \\bar{S}_f \\sigma^i \\theta + \\frac{1}{4} (\\bar{\\theta} \\theta)^2 \\partial^2_\\tau S_T^i \n\\ea\nThen the super-Schwarzian action is \n\\beq\n\\label{Saction}\nI_{\\cN=4} = - \\Phi_r\\int d\\tau S_b[f(\\tau),g(\\tau),\\eta(\\tau)]\n\\eeq\nwhere $\\Phi_r$ can be viewed as a coupling constant whose role we shall soon discuss and the factor of $12$ above is chosen such that it simplifies the factor in \\eqref{eq:just-in-terms-of-Schw}. \n\nWe can rewrite the action in super-field notation by defining $S = \\bar{\\theta} \\sigma^i \\theta S^i,$ where we sum over $i=1,2,3$. Then, using the expansion of $S^i$ gives,\\footnote{This is a simpler expression of $S^i$ for which more components are present. We can construct $S$ from an $S^i$ as long as $S^i$ satisfies $D_p D_q S^i = \\bar{D}^p \\bar{D}^q S^i=0$. To derive this we use that \n\\bea\n&&(\\bar{\\theta} \\sigma^i \\theta)(\\bar{\\theta} \\sigma^i \\theta)=-3(\\bar{\\theta} \\theta)^2,~~~~~~\\varepsilon^{ijk}(\\bar{\\theta} \\sigma^i \\theta)(\\bar{\\theta} \\sigma^j \\theta)=0,\\nn\\\\\n&& (\\bar{\\theta} \\sigma^i \\theta) (\\bar{\\theta} \\sigma^i G) = -3 (\\bar{\\theta} \\theta) (\\bar{\\theta} G),~~~~(\\bar{\\theta} \\sigma^i \\theta) (\\bar{G} \\sigma^i \\theta) = 3 (\\bar{\\theta} \\theta) (\\bar{G} \\theta) \\nn\\,.\n\\eea\n}\n\\beq\nS =2 \\bar{\\theta} \\sigma^i S_T^i \\theta - 3 (\\bar{\\theta} \\theta) \\bar{\\theta} S_f + 3 (\\bar{\\theta} \\theta) \\bar{S}_f \\theta + 3 S_b (\\bar{\\theta}\\theta)^2\\,.\n\\eeq\nThen the action \\eqref{Saction} can be rewritten as $I_{\\cN=4}\\sim \\Phi_r \\int d\\tau d^4\\theta S$.\nNote that in terms of the $ S[f(\\tau),g(\\tau),\\eta(\\tau)]$ there is no obvious chain rule analogous to \\eqref{chainrule}. For this reason it will oftentimes be easier to work with $S^i$ instead of the super-field $S$. To make things concrete, it is informative to write the Schwarzian action when focusing on purely bosonic components, when setting $\\eta(\\tau) = 0$ in \\eqref{Saction}:\n\\be\n\\label{eq:bosonic-compoents-of-the-action}\nI_{\\cN=4,\\text{ bosonic}}[f(\\tau), g(\\tau), \\eta(\\tau) = 0] =- \\int_0^\\b d\\tau \\Phi_r\\left[\\text{Sch}( f, \\tau ) + \\Tr(g^{-1} \\partial_\\tau g)^2\\right] \\,.\n\\ee\nSince $\\eta(\\tau)=0$ is a solution to the equations of motion we will soon use the above action to extract the classical saddle point when quantizing the super-Schwarzian action at the level of the path integral.\n\n\n\\subsubsection{Transformation law and a match with the JT boundary term } \n\\label{sec:N=4-transf-law}\n\n\nIn this section, we shall derive explicitly the infinitesimal transformation rules of the $\\cN=4$ super-Schwarzian (\\ref{sschwarzian}). We will expand $f(\\tau)\\approx\\tau+\\xi(\\tau)$, $g \\approx 1 + i t^{i}(\\tau) \\sigma^i$, $\\eta \\approx \\epsilon(\\tau)$ and $\\bar{\\eta}\\approx\\bar{\\epsilon}(\\tau)$ and work to linear order in $\\xi$, $t^i$, $\\epsilon$ and $\\bar{\\epsilon}$. A convenient way to encode the infinitesimal reparametrizations of the super line (\\ref{reparametrizations}) that automatically satisfies the constraints (\\ref{repa_constraints}) is to use a super-field as in \\cite{Matsuda:1988qf,Matsuda:1989kp}: \n\\bea\\label{repara_superfield}\nE(Z)=\\xi(\\tau+\\frac{1}{2}\\bar{\\theta}\\theta)+\\xi(\\tau-\\frac{1}{2}\\bar{\\theta}\\theta)+\\bar{\\theta} \\epsilon(\\tau-\\frac{1}{2}\\bar{\\theta}\\theta)-\\bar{\\epsilon}(\\tau+\\frac{1}{2}\\bar{\\theta}\\theta)\\theta+\\frac{1}{2}\\bar{\\theta}\\sigma^i\\theta t^i(\\tau),\n\\eea\nUnder such a reparametrization (\\ref{repara_superfield}), the super-Schwarzian (\\ref{sschwarzian}) transforms in the same way as the super-energy momentum tensor, which is given in \\cite{Matsuda:1989kp}: \n\\bea\\label{SStransformation}\n\\delta_E S^i= \\partial_{\\tau}\\left(E(Z)S^i\\right)+D E(Z)\\bar{D} S^i+ \\bar{D}E(Z)D S^i - i \\epsilon^{ijk}\\left(D\\sigma^j\\bar{D}E\\right) S^k-2D\\sigma^i\\bar{D}\\partial_{\\tau}E(z),\n\\eea\n Now substitute (\\ref{sschwarzian}) and (\\ref{repara_superfield}) into (\\ref{SStransformation}), and collect in components: \n\\bea\n\\delta_E S^i= 2\\delta_E S_T^i + \\bar{\\theta} \\sigma^i \\delta_ES_f + \\delta_E\\bar{S}_f \\sigma^i \\theta - \\bar{\\theta} \\sigma^i \\theta \\delta_E S_b + i \\epsilon^{ijk} \\bar{\\theta} \\sigma^j \\theta \\partial_\\tau \\delta_E S_T^k+\\dots,\n\\eea\nwe obtain the infinitesimal transformation of $S_T^i, S_f, \\bar{S}_f, S_b$. Note that the terms in $\\dots$ are purely determined by lower components, and thus it is enough to focus on the terms up to $\\mathcal{O}(\\bar{\\theta}\\theta).$ As a result, the transformations of $S_T^i, S_f, \\bar{S}_f, S_b$ are given by:\n\\bea \n\\label{eq:Sb-transformation}\n\\delta_E S_b&=&\\xi S_b'+2S_b \\xi'+ \\xi'''- S_T^i (t^i)'+\\frac{1}{2}\\left(3\\bar{S}_f\\epsilon'+\\bar{S}_f'\\epsilon-3\\bar{\\epsilon}'S_f-\\bar{\\epsilon} S_f'\\right), \\nonumber\\\\ \n\\delta_E S_f^p&=& \\xi (S_f^{p})'+\\frac{3}{2}S_f^p \\xi'-\\epsilon^p S_b-2(\\epsilon^p)''-\\frac{1}{2}t^i (\\sigma^i){}^p_qS_f^q+(S_T^i)' (\\sigma^i{})^p_q\\epsilon^q+2 S_T^i (\\sigma^i{})^p_q(\\epsilon^q)',\\nonumber\\\\ \n\\delta_E S_T^i&=& \\left( \\xi S_T^i\\right)'-(t^i)'+\\frac{1}2 \\bar{S}_f\\sigma^i\\epsilon+ \\frac{1}2\\bar{\\epsilon}\\sigma^iS_f+i \\epsilon^{ijk}t^j S_T^k.\n\\eea\n\nWe note that they exactly agree with the infinitesimal transformation deduced from the boundary action of the BF theory \\eqref{eq:gauge-transf-preserving-asymp} with the field identification:\n\\be \n\\cL &\\leftrightarrow S_b\\,,\\nn \\\\ \n\\psi^p &\\leftrightarrow S_f^p\\,,\\nn \\\\ \nB^i_\\tau &\\leftrightarrow S_T^i\\,.\n\\ee\nSince the infinitesimal transformations and ${PSU}(1,1|2)$ invariance suffices to determine the global form of the action as the $\\cN=4$ super-Schwarzian, it then follows that the boundary action \\eqref{eq:def-cL(u)-and-bdy-Lagr} from BF theory agrees with our definition of the Schwarzian action \\eqref{Saction}. Therefore, the path integral in the $\\cN=4$ super-JT gravity with the boundary conditions discussed in section \\ref{sec:super-JT-bdy-cond} can be reduced to that for the $\\cN=4$ super-Schwarzian action defined by \\eqref{Saction}. \n\n\\subsection{Spacetime and global symmetries}\n\\label{sec:space-time-and-global-symm}\nBefore analyzing the quantization of the super-Schwarzian theory, it is useful to study the space-time and global symmetries present in this action. \n\nA useful way to discuss symmetries of a quantum field theory is to cast it in terms of \\emph{internal symmetries} and \\emph{spacetime symmetries}. The continuous internal symmetries of (\\ref{Saction}) form PSU$(1,1|2)$,\\footnote{It is sometimes confusing whether the group is SU$(1,1|2)$ or PSU$(1,1|2)$. We note that SU$(1,1|2)$ contains an extra $U(1)$ factor generated by identity, due to the cancellation of SL$(2)$ part and $SU(2)$ part of the super-trace in the algebra \\cite{BrittoPacumio:1999ax}. We do not have such an $U(1)$ factor in \\ref{infintesimalparameterb}, and, therefore, our symmetry group is PSU$(1,1|2).$ } generated by (\\ref{infintesimalparameterb})--(\\ref{infintesimalparameterf}), directly acting on $\\left(f(\\tau), g(\\tau), \\bar{\\eta}_p(\\tau), \\eta^p(\\tau)\\right).$ They are zero modes of the $\\cN=4$ super-Schwarzian derivative, and the action (\\ref{Saction}) is invariant due to the chain rule (\\ref{chainrule}). Specifically,\n\\be\n\\label{eq:invariance-under-PSU(1,1|2)}\nS^i(Z, h \\circ Z'(Z)) = S^i(Z, Z'(Z)) \\,,\n\\ee\nwhere $h$ is a composition of the bosonic and fermionic transformations in (\\ref{infintesimalparameterb})--(\\ref{infintesimalparameterf}) which we apply to the super-reparametrization $Z'(Z)$.\nThese transformations are the supersymmetric generalization of the $SL(2,\\,\\mR):f(\\tau) \\to \\frac{a f(\\tau)+b}{c f(\\tau) +d}$, and we will have to quotient out such transformations as we proceed to compute the partition function \\cite{Jensen:2016pah,Maldacena:2016upp,Engelsoy:2016xyb}, in order to obtain a well-defined partition function. Furthermore, aside from the $PSU(1, 1|2)$ that has an $SU(2)$ subgroup which acts on the left of the field $g(\\tau)$ there is an additional $SU(2)$ symmetry which acts on the right of the field $g(\\tau)$. This additional symmetry does not act on $f(\\tau)$ or on the fermionic fields $\\bar \\eta$ and $\\eta$. Thus, to summarize the continuous internal symmetry is, up to discrete factors, $PSU(1,1|2) \\times SU(2)$.\\footnote{There is also an outer $SU(2)$ inherited from the $PSU(1,1|2)$ algebra. It acts on $(\\eta^1\\left(\\tau\\right), -\\bar{\\eta}_2\\left(\\tau\\right))$ as a doublet. In the $4d$ setup we later consider, this is the $R$ symmetry of the $\\mathcal{N}=2$ supergravity that is broken by stringy effects. }\n\nThe $\\cN=4$ super-Schwarzian theory also admits \\emph{spacetime} symmetries. To see this consider the transformation \\eqref{eq:Sb-transformation}; for the transformation $\\xi(\\tau) = \\xi$ corresponding to $\\tau \\to \\tau + \\xi$ (time translations), $t^i(\\tau) =t^i$ corresponding to constant infinitesimal rotation of $\\theta^p$ ($R$-symmetry rotations), and $\\epsilon(\\tau) = \\epsilon$ corresponding to $\\theta \\to \\theta+\\epsilon$ (super-translations), the action $S_b$ is invariant up to a total derivative. Together, all such infinitesimal transformations generate the $\\cN = 4$ super-Poincar\\'e spacetime symmetry. They satisfy the algebra\n\\bea\n\\{\\bar{Q}_p, Q^q\\}=2\\delta^q_p H, ~~~\\text{with }~~\\bar{Q}_p=i\\frac{\\partial}{\\partial \\theta^p}+\\bar{\\theta}_p\\partial_{\\tau},\\,\\,\\, Q^p=-i\\frac{\\partial}{\\partial \\bar{\\theta}_p}-\\theta^p \\partial_{\\tau},\\,\\,\\,H=\\partial_\\tau\n\\eea\nwhile all other commutators vanish. \nIt is straightforward to identify these $\\cN=4$ supercharges from the Noether procedure using the transformation \\eqref{eq:Sb-transformation}: $S_b$ is the charge generating time translations (and is thus the Hamiltonian of the theory), $S_f$ is the generator of super-translations, and $S_T^i$ is the generator of R-symmetry rotations. Note that the Hamiltonian here is \\emph{not} one of the generators of PSU$(1,1|2),$ since it is in fact the super-Schwarzian itself, and thus proportional to the quadratic Casimir of the PSU$(1,1|2).$ This is quite analogous to the non-supersymmetric case \\cite{kitaevTalks, Mertens:2017mtv, Lin:2019qwu}. \n\nWe would like to stress that even though $PSU(1,1|2)$ plays a big role in constructing the Schwarzian action, the theory is not invariant under spacetime superconformal $PSU(1,1|2)$ symmetries. The spectrum is not organized according to $PSU(1,1|2)$ representations. Only the $\\cN=4$ super-Poincar\\'e sub-group is a spacetime symmetry. \n\nIt is also useful to briefly discuss some of the discrete internal and spacetime symmetries of the theory. The $\\cN=4$ super-Schwarzian has a time reversal symmetry $\\cT$ which acts on the fermionic fields as $\\cT \\eta(\\tau) = i \\eta(-\\tau)$ and $\\cT\\bar \\eta(\\tau) = i \\bar \\eta(-\\tau)$ and on the bosonic fields as $\\cT f(\\tau) = f(-\\tau)$ and $\\cT g(\\tau) = g(-\\tau)$; in such a case, $\\cT^2 = 1$ and the symmetry is $\\mathbb Z_2^\\cT$. There is also a $\\mathbb Z_2^F$ fermionic symmetry $(-1)^F$ which solely acts on the fermionic fields.\\footnote{The same time-reversal properties are also true in the $\\cN=1$ theory \\cite{Stanford:2019vob}. } Thus, the symmetry is $\\mZ_2^F \\times \\mZ_2^\\cT$. We can now address discrete factors for the internal symmetry group of the theory. In (\\ref{infintesimalparameterb})--(\\ref{infintesimalparameterf}) we note that the transformation given by the center of $SL(2, \\mR)$ with $a=d=-1$ and $b=c=0$ (acting as $\\eta \\to -\\eta$ and $\\bar \\eta\\to -\\bar \\eta$) is redundant with the composition of the two center transformations for the two $SU(2)$, the first of which acts as $g \\to -g$, $\\eta \\to -\\eta$, $\\bar \\eta\\to -\\bar \\eta$, and the second of which acts solely on $g \\to -g$. Furthermore, this transformation is again redundant with the $(-1)^F$ symmetry mentioned previously. Thus, the bosonic subgroup of the symmetry group for the theory is given by \n\\be\n\\frac{SL(2, \\mR) \\times SU(2) \\times SU(2) \\times \\mZ_2^F}{\\mZ_2} \\times \\mZ_2^\\cT\\,.\n\\ee \nConsequently, we note that even-dimensional representations of $SU(2)$ (half-integer spins) need to be fermionic and odd-dimensional representations of $SU(2)$ (integer spins) need to be bosonic. Since we will be able to decompose our partition function as a sum over $SU(2)$ characters, this fact will play an important role in easily obtaining the supersymmetric index from the theory by using the result for the partition function. \n\n\\section{Quantizing the $\\cN=4$ Schwarzian theory}\n\\label{sec:N=4-super-Schw}\n\nIn this section, we will study the $\\cN=4$ super-Schwarzian theory in more detail. In particular we will compute the exact partition function and density of states. We will do the calculation in two ways, first using the localization method of Stanford and Witten \\cite{Stanford:2017thb} and second using the 2D CFT approach of \\cite{Mertens:2017mtv}, and find agreement. Then we will analyze the spectrum that we derive and point out some salient features like the large zero temperature degeneracy and the presence of a gap.\n\n\\subsection{The action}\n\n\nThe $\\cN=4$ super-line can be parametrized in superspace by $(\\tau, \\theta^p, \\bar{\\theta}_p)$, $p=1,2$, where $\\theta$ and $\\bar{\\theta}$ are Grassman variables transforming as fundamental and anti-fundamental of an $SU(2)$ symmetry. $\\cN=4$ super-reparametrizations are parametrized by a bosonic field $f(\\tau)\\in {\\rm Diff}(S^1)$, a local transformation $g(\\tau) \\in SU(2)$ (or more precisely the loop group) and fermionic fields $\\eta^p(\\tau)$ and $\\bar{\\eta}_p(\\tau)$. In terms of a super-reparametrization these fields can be roughly written as \n\\bea\n\\tau &\\to& f(\\tau) + \\ldots,\\\\\n\\theta^p &\\to& g^p_{~q}(\\tau) \\theta^q \\sqrt{f'(\\tau)} + \\eta^p(\\tau) + \\ldots, \\\\\n\\bar{\\theta}_p &\\to& \\bar{\\theta}_q \\hspace{0.1cm}\\bar{g}^q_{~p}(\\tau) \\sqrt{f'(\\tau)} + \\bar{\\eta}_p(\\tau) + \\ldots.\n\\ea\nThe dots correspond to terms that are fixed by the super-reparametrization constrains and can be found in the previous section. We also defined a Schwarzian action $I_{\\cN=4} =-\\Phi_r \\int d\\tau S_b [ f, g, \\eta, \\bar{\\eta}]$ invariant under $PSU(1,1|2)$ transformations acting on the fields $(f,g,\\eta,\\bar{\\eta})$. The bosonic component of this action is \n\\beq\nI_{\\cN=4} =- \\Phi_r \\int_0^\\b d\\tau \\left[\\text{Sch}(f,\\tau) + \\Tr(g^{-1} \\partial_\\tau g)^2 + ({\\rm fermions})\\right]\n\\eeq\nwhich gives the usual Schwarzian action and a particle moving on $SU(2)$. The extra terms involve the fermions $\\eta$ and $\\bar{\\eta}$. \n\nIn this section, we will compute the Euclidean path integral giving the partition function,\\footnote{We leave the measure implicit in this formula. We take the measure to be the Pfaffian of the symplectic form over the integration space ${\\rm Diff}(S^{1|4})\/PSU(1,1|2)$, studied in \\cite{Aoyama:2018lfc}.}\n\\beq\nZ (\\beta,\\alpha) = \\int \\frac{\\mathcal{D} f \\mathcal{D} g \\mathcal{D} \\eta \\mathcal{D} \\bar{\\eta}}{PSU(1,1|2)} ~\\exp\\left( \\Phi_r \\int d\\tau S_b[f,g,\\eta,\\bar{\\eta}] \\right)\\,, \n\\eeq\nwhere $\\Phi_r$ is a dimensionful coupling constant of the theory. The inverse temperature $\\beta$ and chemical potentials $\\alpha$ appears in the path integral through the boundary conditions of the fields:\n\\beq\nf(\\tau+\\beta)=f(\\tau),~~~~g(\\tau+\\beta) = e^{2 \\pi i \\alpha \\sigma^3} g(\\tau),~~~ \\eta(\\tau+ \\beta) =- e^{2\\pi i \\alpha \\sigma^3} \\eta(\\tau),\n\\eeq\nand similarly for $\\bar{\\eta} $. In the rest of this section we will evaluate this path integral as a function of $\\beta$, $\\alpha$ and the coupling $\\Phi_r$, and rewrite it as a trace over a Hilbert space with a possibly continuous spectrum. \n \n \\subsection{The partition function}\n \n\\subsubsection{Method 1: Fermionic localization}\n\\label{sec:fermionic-localization}\n\nIn this section, we will solve the theory following \\cite{Stanford:2017thb}. The integration space is a coadjoint orbit and the super-Schwarzian action generates a $U(1)$ symmetry. Even though we will not work out the measure and symplectic form in detail, we will assume it is chosen such that we can apply the Duistermaat-Heckman theorem. Therefore, we will compute the classical saddles, the one-loop determinants, and put everything together into the final answer \\eqref{eq:localization-part-function}.\n\n\\paragraph{The $\\cN=4$ saddle-point:} As previously mentioned, the bosonic part of the $\\cN=4$ Schwarzian action is given by\n\\be\n\\label{eq:bosonic-compoents-of-the-action-2}\nI_{\\cN=4,\\text{ bosonic}} =- \\int_0^\\b d\\tau \\Phi_r\\left[\\text{Sch}(f,\\tau) + \\Tr(g^{-1} \\partial_\\tau g)^2\\right] \\,.\n\\ee\nThe equations of motion for $f(\\tau)$ and $g(\\tau)$ imply that:\n\\be\n\\partial_\\tau \\text{Sch}(f,\\tau) = 0\\,, \\qquad \\partial_\\tau \\Tr (g^{-1} \\partial_\\tau g)^2 = 0\n\\ee\nThe solution for the Schwarzian is well-known and is given by $f(\\tau) = \\tan(\\pi \\tau\/\\b)$. The solution for the $SU(2)$ adjoint field takes the form $g = \\exp(i t_i \\e^i \\tau)$, where $\\e^i$ is a constant that needs is set by the boundary conditions for the field $g(\\tau)$. To make the computation easier we note that all solutions can be transformed to the diagonal form ($g = \\exp(i \\sigma_3 \\e^3 \\tau)$) using an $SU(2)$ transformation. If require that the field $g$ be periodic, than we have that $\\e^{3} = 2\\pi n\/\\beta$, with $n \\in \\mZ$. More generally, the $SU(2)$ symmetry could have a fugacity which would imply that the field $g$ is no longer periodic; rather, it has $g(\\beta) = z\\, g(0)$ where $z \\in SU(2)$ is the fugacity. Once again, since the partition function only depends on the conjugacy class of $z$, we can perform an $SU(2)$ transformation to diagonalize $z = \\exp(2\\pi i \\a \\sigma_3 )$. The solution for $h$ is then given by $g = \\exp\\left(2\\pi i \\sigma_3 (n+\\a) \\frac{\\tau}\\beta\\right)$. \n\nIn such a case, the on-shell value of the action $I_{\\cN=4,\\text{bosonic}}$ is given by: \n\\be \n\\label{eq:on-shell-action-N=4}\nI_{\\cN=4,\\text{bosonic}}^{\\,\\text{on-shell}} = -\\frac{2\\pi^2 \\Phi_r} {\\b} \\left(1 -4 (n+\\a) ^2 \\right)\\,,\n\\ee\nNow that we have the on-shell action we can proceed by computing the one-loop determinant which is sufficient for fully computing the partition function. \n \n\n\\paragraph{The one-loop determinant:} To compute the one-loop determinant, we have to account for all quadratic fluctuations in the theory. The quadratic fluctuations of the Schwarzian field have been analyzed in great detail \\cite{Maldacena:2016upp, Stanford:2017thb}, and its contributions to the one-loop determinant is given, up to an overall proportionality constant, by\n\\be \n\\label{eq:Schwarzian-one-loop-determinant}\n\\det_{\\text{Schw., one-loop}} = \\left(\\frac{\\Phi_r}{\\beta}\\right)^\\frac{3}2\\,.\n\\ee\nThe quadratic fluctuation around the saddle-point of the $SU(2)$-group element can be parametrized as $g(\\tau) = \\exp\\left( \\sigma_3 \\left[2\\pi i(n+\\a) \\frac{\\tau}\\beta + \\epsilon^3(\\tau) \\right]\\right)\\exp\\left( \\epsilon^2(\\tau) \\sigma_2\\right) \\exp\\left( \\epsilon^1(\\tau) \\sigma_1\\right)$, and yields a contribution to the action \\cite{Picken:190160}\n\\be \nI_{SU(2), \\text{ quad}} &= \\frac{8\\pi^2 \\Phi_r} {\\b} (n+\\a) ^2\\nn \\\\ &-2 \\Phi_r \\int_0^\\b d\\tau \\left((\\epsilon^1(\\tau)')^2 + (\\epsilon^2(\\tau)')^2 - (\\epsilon^3(\\tau)')^2 + \\frac{8\\pi (n+\\a)}{\\b} \\epsilon^2(\\tau) \\epsilon^1(\\tau)' \\right)\\,,\n\\ee\nand the one-loop determinant obtained from integrating out these modes for each value of $n$ is given by \\cite{Picken:190160}:\n\\be\n\\label{eq:SO3-one-loop-determinant}\n \\det_{SU(2), \\text{ one-loop}} = \\frac{\\Phi_r^{3\/2} (n+\\a) }{ \\beta^{3\/2} \\sin(2\\pi \\a)} \\,.\n\\ee\nFinally, we discuss the quadratic contribution of the fermionic fields. By using the saddle-point solution for $f(\\tau)$ and $g(\\tau)$ and by quadratically expanding the super-Schwarzian action:\n\\be\n I_{\\text{ferm., quad.}} = \\Phi_r\\int_0^{\\beta} d\\tau\\, &\\bigg(\\eta^p \\left[\\frac{2\\pi^2}{\\beta^2}\\left(1+2(n+\\a)^2 \\right) \\partial_\\tau - \\partial_\\tau^3\\right]\\bar \\eta_p + \\nn \\\\&+ \\partial_\\tau \\eta^p \\left[\\frac{12\\pi^2(n+\\a)^2}{\\b^2} + \\frac{8i \\pi (n+\\a)\\partial_\\tau}{\\beta} - 3\\partial_\\tau^2 \\right] \\bar \\eta_p\\bigg) \n \\ee\nExpanding the fermionic fields in Fourier modes, \n\\bea \\eta^1(\\tau) &=&e^{i\\frac{2\\pi (n+\\a)\\tau}{\\b} }\\sum_{m_1 \\in \\dots, -\\frac{1}2, \\, \\frac{1}2, \\dots} \\sqrt{\\frac{\\b}{2\\pi}} \\,\\eta^1_{m_1} e^{-i\\frac{2\\pi m_1\\tau}{\\b} }\\\\\n\\eta^2(\\tau) &=&e^{-i\\frac{2\\pi (n+\\a)\\tau}{\\b} }\\sum_{m_2 \\in \\dots, -\\frac{1}2, \\, \\frac{1}2, \\dots} \\sqrt{\\frac{\\b}{2\\pi}}\\, \\eta^2_{m_2} e^{i\\frac{2\\pi m_2\\tau}{\\b}}\\,,\n\\ea\nwhere we impose anti-periodic boundary conditions for the fermionic fields when $\\alpha= 0$ and we impose boundary conditions consistent with the introduction of the fugacity $z$ when $\\alpha\\neq 0$. We can then rewrite the action as, \n\\be\n I_{\\text{ferm., quad.}} = \\frac{2\\pi^2 i \\Phi_r}{\\beta} \\left[\\sum_{p=1, 2}\\,\\sum_{m_p \\in \\dots, -\\frac{1}2, \\, \\frac{1}2, \\dots} (m_p - n - \t\\a) (4m_p^2-1) \\eta^p_{m_p} \\bar \\eta^p_{-m_p}\\right]\\,.\n\\ee\nWe are interested in computing the dependence of the one-loop determinant on $n$, $\\alpha$, $\\beta$ and $\\Phi_r$. The $\\beta$ and $\\Phi_r$ dependence is captured by the existence of the four-fermionic zero modes with $m_p = \\pm 1\/2$. As in \\cite{Stanford:2017thb}, to compute the rest of the one-loop determinant, we will regularize this result by dividing the result by the one-loop determinant with $n=0$ and $\\alpha = 0$. We thus find that the regularized one-loop determinant is given, again up to a proportionality constant, by \n\\be\n\\label{eq:ferm-one-loop}\n\\det_{\\text{ferm., one-loop}} = \\frac{\\beta^4}{\\Phi_r^4} \\prod_{p=1, 2}\\,\\,\\prod_{m_p \\in \\dots, -\\frac{5}2, -\\frac{3}2, \\frac{3}2, \\frac{5}2, \\dots} \\frac{m-n -\\a}{m} = \\frac{\\beta^4}{\\Phi_r^4} \\frac{\\cos(\\pi\\a)^2}{(1 - 4(n+\\a)^2)^2 }\\,. \n\\ee\n\n\n\\paragraph{Final answer:} Thus, accounting for the saddle-point value of the action \\eqref{eq:on-shell-action-N=4} together with the one-loop determinants \\eqref{eq:Schwarzian-one-loop-determinant}, \\eqref{eq:SO3-one-loop-determinant}, and \\eqref{eq:ferm-one-loop}, we find that the partition function of the $\\cN=4$ Schwarzian theory is given, up to an overall proportionality constant, by: \n\\be \n\\label{eq:localization-part-function}\nZ_{\\cN=4\\,\\,\\,\\text{Schw.}} &= \\sum_{n \\in \\mathbb Z} \\det_{\\text{Schw., one-loop}} \\,\\,\\det_{SU(2), \\text{ one-loop}}\\, \\det_{\\text{ferm., one-loop}} e^{-I_{\\cN=4,\\text{bosonic}}^{\\,\\text{on-shell}} } \\nn \\\\ &=\\sum_{n \\in \\mathbb Z} \\frac{ \\beta \\cot(\\pi \\a) (\\a+n)}{\\Phi_r (1- 4(n+\\a)^2)^2} e^{\\frac{2\\pi^2 \\Phi_r} {\\b} \\left(1 -4 (n+\\a) ^2 \\right)}\n\\ee \nWe will thus continue by matching this result using the completely distinct method of canonical quantization, after which we will come back to a detailed analysis of the spectrum associated to \\eqref{eq:localization-part-function} in section \\ref{sec:density-of-states}. \n\n\\subsubsection{Method 2: Canonical quantization} \\label{sec:meth2cq}\nIn this section, we will compute the partition function of the $\\cN=4$ super-Schwarzian theory using the canonical quantization approach of \\cite{Mertens:2017mtv} (for a very recent discussion explaining the connection to the localization approach see also \\cite{Alekseev:2020jja}). \n\nWe will illustrate briefly the idea first. The localization formula we used above can be applied to integrals that generally have the following form \n\\beq\nZ = \\int dqdp ~e^{-H(p,q)},\n\\eeq\nwhere the integral is over a symplectic space (a classical phase space) with coordinates $(q,p)$ and $H(q,p)$ generates via the Poisson brackets a $U(1)$ symmetry. In the case of the bosonic Schwarzian theory the integration manifold is ${\\rm Diff}(S^1)\/SL(2,\\mathbb{R})$ which a coadjoint orbit of the Virasoro group (and, therefore, symplectic), and $H(p,q)$ is the Schwarzian action. Instead of using localization, we can obtain this integral using the following identity \n\\beq\\label{eq:SKJDKD}\n\\lim_{\\hbar\\to 0} {\\rm Tr}\\left( e^{-H(p,q)}\\right) = \\int dqdp ~e^{-H(p,q)},\n\\eeq\nwhere the left-hand side is $\\hbar\\to 0$ limit of the trace evaluated over the Hilbert space obtained by quantizing the phase space. In the case of the Schwarzian, the quantization of ${\\rm Diff}(S^1)\/SL(2,\\mathbb{R})$ is the identity representation of the Virasoro algebra with central charge $c \\sim 1\/\\hbar$. The left hand side of \\eqref{eq:SKJDKD} can by very easily computed at finite $c$ as a Virasoro vacuum character by counting descendants, and a very simple calculation gives the Schwarzian path integral \\cite{Mertens:2017mtv}. In the bosonic case, the main advantage of this method is the possibility to compute correlation functions which are not available using localization. In this case, we will use it as a double check on our previous result.\n\n\nFor the case of the $\\cN=4$ super-Schwarzian the integration space is ${\\rm Diff}(S^{1|4})\/PSU(1,1|2)$ which is a coadjoint orbit of super-Virasoro and, therefore, also symplectic. We will assume that the quantization of this phase space, the Hilbert space in \\eqref{eq:SKJDKD}, is equivalent to the identity representation of the small $\\cN=4$ Virasoro algebra with central charge $c\\sim 1\/\\hbar$. The $\\cN=4$ super-Schwarzian partition function is then the semiclassical limit of the vacuum character. \n\nLets begin then by recalling the super-Virasoro algebra involved in this problem. The bosonic generators are $L_n$ and $T_n^i$ where $n$ is an integer and $i=1,2,3$ label the generators of a Kac-Moody $SU(2)$ at level $k$. Their algebra is \n\\bea\n[ L_m,L_n ]&=& (m-n)L_{m+n}+\\frac{k}{2} m(m^2-1)\\delta_{n+m,0} \\label{eq:smalln4gen1}\\\\\n\\text{[} T^i_m,T^j_n \\text{]} &=& i \\epsilon^{ijk}T^k_{m+n} + \\frac{k}{2} m \\delta_{m+n,0} \\delta_{i,j} \\label{eq:smalln4gen2}\\\\\n\\text{[} L_m , T_n^i\\text{]} & = & -n T_{m+n}^i.\\label{eq:smalln4gen3}\n\\ea\nThe central charge of the bosonic Virasoro sector is $c=6k$, which is fixed by a Jacobi identity. The fermionic generators are $G_r^p$ and $\\bar{G}_s^p$, $p=1,2$. They transform in the fundamental and antifundamental of the $SU(2)$. The Fourier mode parameter $r,s$ are integer in the Ramond sector or half-integer in the Neveu-Schwarz sector. The rest of the algebra, involving the fermionic generators, can be found for example in \\cite{Eguchi:1987sm}, and is given by\n\\bea\n&&\\{ G_r^p, \\bar{G}_s^q\\} = 2 \\delta^{pq} L_{r+s}-2(r-s)\\sigma^i_{pq} T^{i}_{r+s}+\\frac{k}{2}(4r^2-1)\\delta_{r+s,0},\\nn \\\\\n&& [T_m^i,G_r^p]=-\\frac{1}{2}\\sigma^i_{pq}G^q_{m+r},~~[T^i_m,\\bar{G}_r^p]=\\frac{1}{2}\\sigma^i_{pq}{}^\\star \\bar{G}_{m+r}^q,~~ \\{ G_r^p, G_s^q\\}=\\{\\bar{G}_r^p,\\bar{G}_s^q\\}=0\\nn \\\\\n&&[L_m,G_r^p]=\\left(\\frac{m}{2}-r\\right)G_{m+r}^p,~~[L_m,\\bar{G}_r^p]=\\left(\\frac{m}{2}-r\\right)\\bar{G}_{m+r}^p,\\label{eq:smalln4gen4}\n\\ea\nwhere $\\sigma^i_{pq}$ are the Pauli matrices. In that reference, Eguchi and Taormina also construct the unitary representations of the algebra. \n\nFor the application we have in mind in this paper, the Schwarzian path integral, we will only need the massless representations in the NS sector, due to the fact that we want the Schwarzian fermions to be antiperiodic as explained in \\cite{Mertens:2017mtv}.\\footnote{If we wanted the Schwarzian theory Witten index we would use the characters in the Ramond sector.} General representations are labeled by $h$, the eigenvalue of $L_0$, and $\\ell$, the spin of the $SU(2)$ representation, and the massless sector has $(h=\\ell, \\ell)$ with half-integer $\\ell=0, \\frac{1}{2}, \\ldots, \\frac{k}{2}$. For the Schwarzian path integral we will need the $\\ell=0$ representation. The characters are defined by\n\\beq\n\\chi_{\\ell} (k;q,z) \\equiv {\\rm Tr}_{NS}\\left[ (-1)^F q^{L_0-\\frac{c}{24}} z^{T^3_0}\\right],\n\\eeq\nover a representation $\\ell$ of the algebra. We need to insert $(-1)^F$ such that the fermions along the quantization direction are periodic and survive the semiclassical limit (see discussion in \\cite{Mertens:2017mtv}). \n\nThese characters were computed by Eguchi and Taormina \\cite{Eguchi:1987wf} by simply counting states. They are given by the following expression\\footnote{The origin of the first factor in the right hand side is explained in section 5 of \\cite{Kraus:2006nb}.}\n\\beq\n\\chi_{\\ell} (k;q,z=e^{2\\pi i y}) = e^{2\\pi i k \\frac{y^2}{\\tau}} q^{-\\frac{k}{4}} q^{\\ell+\\frac{1}{4}} \\frac{i \\theta_3(q,-z) \\theta_3(q,-z^{-1})}{\\eta(q)^3 \\theta_1(q,z^2)} \\left[ \\mu(z,q) - \\mu(z^{-1},q) \\right],\n\\eeq\nwhere $\\theta_3(q,z)$ is the Jacobi theta function and we defined the function \n\\beq\n\\mu(z,q) \\equiv \\sum_{n\\in\\mathbb{Z}} \\frac{q^{(k+1)n^2+(2\\ell+1)n}z^{2(k+1)n + 2 \\ell + 1}}{(1- z q^{n+\\frac{1}{2}})(1- z q^{n+\\frac{1}{2}})}.\n\\eeq\n\nNow we have all the ingredients to extract the Schwarzian partition function from the $\\hbar \\sim 1\/k \\to 0$ ($c\\to\\infty$) limit applied to the above expression for the identity $\\ell=0$ representation. As explained in \\cite{Mertens:2017mtv} we need to consider the following scaling \n\\beq\nz=e^{2 \\pi i \\alpha \\tau},~q=e^{2\\pi i \\tau}~~~~~{\\rm with}~~~~\\tau = \\frac{i}{k} \\frac{4\\pi \\Phi_r}{\\beta}.\n\\eeq\nWe then take $\\tau \\cdot k$ fixed in the limit and this constant is related to the ratio $\\Phi_r\/\\beta$ in the Schwarzian theory. This choice of $z$ and $q$ is written directly in terms of $\\alpha$ and $\\beta$ which will become the chemical potential and inverse temperature in the Schwarzian limit. \n\nWhen taking the Schwarzian limit we will only keep track of the dependence on $\\alpha$ and $\\beta$ since any prefactor can be absorbed in a redefinition of the zero-point entropy and energy. We will not go over all the details but some useful intermediate steps are \n\\beq\n\\mu(z,q)-\\mu(z^{-1},q) \\sim \\frac{8 e^{\\frac{2\\pi^2 \\Phi_r}{\\beta}4\\alpha^2}}{\\pi^2 |\\tau|^2} \\sum_{n\\in \\mathbb{Z}} \\frac{(\\alpha+n)}{(1-4(\\alpha+n)^2)^2} e^{-\\frac{2\\pi^2 \\Phi_r}{\\beta} 4(n+\\alpha)^2}.\n\\eeq\nUsing eq (3.15) of \\cite{Ahn:2003tt} gives the following limit\n\\beq\ni \\left( \\frac{\\theta_4(q,z)}{\\eta(q)^3}\\right)^2 \\frac{\\eta(q)^3}{\\theta_4(q,z^2q^{\\frac{1}{2}})} \\sim \\frac{\\tau}{\\tan \\pi \\alpha},\n\\eeq\nwhich is related in a simple way to the Jacobi theta functions appearing in the character. Including the rest of the terms the semiclassical $k\\to\\infty$ limit of the vacuum character is \n\\beq\n\\chi_{\\ell=0} (k\\to\\infty;q,z) \\sim \\sum_{n\\in \\mathbb{Z}} \\frac{\\beta \\cot(\\pi \\alpha)(\\alpha+n)}{\\Phi_r(1-4(n+\\alpha)^2)^2} e^{\\frac{2 \\pi^2 \\Phi_r}{\\beta}(1-4(n+\\alpha)^2)}.\n\\eeq\nThis precisely reproduces the partition function computed by localization, given in equation \\eqref{eq:localization-part-function} (An analogous match was checked in \\cite{Mertens:2017mtv} for the case of $\\cN=1$ and $\\cN=2$ super-Schwarzian).\n\nWe can mention some interesting features of this expression. First of all the factor of $\\cot \\pi \\alpha$ is important for the formula to make sense. When $\\alpha \\to 0$ or $1$ it is crucial to include this factor for the final answer to be finite. The same happens when $\\alpha \\to 1\/2$ since otherwise the sum would be divergent. \n\nFrom the 2D CFT perspective the identity representation is invariant under the generators of the global $PSU(1,1|2)$ algebra. In terms of the Virasoro algebra those generators are \n\\beq\n\\text{Bosonic:}~~~~L_{-1},L_0,L_1,~~~~T_0^1,T_0^2,T_0^3,~~~~\\text{Fermionic:}~~~~G_{\\pm \\frac{1}{2}}^a,~~~\\bar{G}^a_{\\pm \\frac{1}{2}},\n\\eeq\nwhich satisfy the same superalgebra as \\eqref{eq:psu(1,1|2)-superalgebra}. \nIt is important to take the fermionic generators in the NS sector. These produce the pre-factor of $\\beta^1$ in the character. In the localization calculation this factor basically counts the number of bosonic and fermionic zero modes $Z \\sim \\beta^{(\\#{\\rm fermion})\/2 - (\\#{\\rm bosons})\/2}$. In the case of the small $\\cN=4$ algebra there are $8$ fermionic zero modes and $6$ bosonic zero modes, giving a factor of $\\beta$. \n\n\\subsection{$\\cN=4$ supermultiplets}\n\nBefore extracting the spectrum from the exact partition function we first explain what properties we expect it to have. The super-Schwarzian theory we are studying captures the explicit breaking of the superconformal symmetry group $PSU(1,1|2)$. Still, as we have seen in section \\ref{sec:space-time-and-global-symm}, translations, super-translations and rigid $SU(2)$ rotations are symmetries. We can write the fermionic generators as $Q_p$ and $\\bar{Q}^p$ with $p=1,2$. Then a part of the algebra that we will use here is \n\\beq\n\\{ Q_p, \\bar{Q}^q\\} = 2 \\delta_a^b H,~~~\\{Q_p,Q_q\\}=\\{ \\bar{Q}_p, \\bar{Q}_q\\} =0\n\\eeq\nThese generators can be written in terms of the Schwarzian fields but we will not need it for the manipulations here. Imagine we first diagonalize $H$ and look at some states with energy $E$. Then as long as $E\\neq 0$ the operators $Q \\sim \\hat{a}$ act as a $SU(2)$ doublet of lowering fermionic operators and $\\bar{Q}\\sim \\hat{a}^\\dagger$ as a $SU(2)$ doublet of rising fermionic operators. To construct a representation we can begin with a state $|J\\rangle$ which transforms as a spin $J$ representation of $SU(2)$, constructed such that $Q_p|J\\rangle=0$. The supermultiplet will have states acting with a single charge $\\bar{Q}^q|J\\rangle$, which can be expanded into $(1\/2)\\otimes J = (J-1\/2) \\oplus (J+1\/2)$; and acting with two charges $\\bar{Q}_1 \\bar{Q}_2 |J\\rangle$ of spin $J$. Therefore, the supermultiplet with $E\\neq 0$, starting with $J\\neq 0$ is made of $(J-1\/2)\\oplus 2 (J) \\oplus (J+1\/2)$. When we construct a supermultiplet starting with a singlet $|0\\rangle$, the $\\bar{Q}^q|0\\rangle$ transforms as a doublet and $\\bar{Q}_1\\bar{Q}_2|0\\rangle$ as another singlet, giving $2(0)\\oplus(1\/2)$. Labeling the supermultiplet by the state with highest $SU(2)$ spin, the $E\\neq 0 $ part of the spectrum should organize as \n\\bea\n\\mathbf{J}&=&(J) \\oplus 2 (J-1\/2) \\oplus (J-1),~~~~J\\geq 1\\\\\n\\mathbf{1\/2}&=&(1\/2) \\oplus 2 (0).\n\\ea\nFinally we might also have states with $E=0$. Starting with a spin-$J$ representation $|J\\rangle$, having $H|J\\rangle =0$ implies that all supercharges annihilate the state and, therefore, that's the whole supermultiplet.\n\nTaking these considerations into account, we can expect the partition function of the $\\cN=4$ super-Schwarzian theory to be expanded in the following way\n\\bea\nZ(\\beta,\\alpha) &=& \\sum_J \\chi_J(\\alpha) \\rho_{\\rm ext}(J) + \\int dE ~e^{-\\beta E} \\left( \\chi_{1\/2}(\\alpha) +2\\chi_{0}(\\alpha)\\right) \\rho_{\\rm cont}(1\/2,E) \\nonumber\\\\\n&&+\\sum_{J\\geq 1} \\int dE ~e^{-\\beta E} \\left( \\chi_{J}(\\alpha) +2\\chi_{J-\\frac{1}{2}}(\\alpha)+ \\chi_{J-1}(\\alpha)\\right) \\rho_{\\rm cont}(J,E),\\label{sqwewq}\n\\ea\nwhere the sums are over half-integer $J$ and $\\chi_J(\\alpha)\\equiv \\sum_{m=-J}^J e^{4\\pi i \\alpha m} = \\frac{\\sin (2J+1)2\\pi \\alpha}{\\sin 2\\pi \\alpha}$ is the character of a spin-$J$ representation of $SU(2)$. In the first line, the first term corresponds to states with $E=0$ while the second term to the $E\\neq 0$ multiplet $\\mathbf{1\/2}$. The second line corresponds to the sum over all other $E\\neq 0$ supermultiplets. Therefore, $\\rho(J,E)$ is the density of supermultiplets with energy $E\\neq 0$ and highest spin $J$, while $\\rho_{\\rm ext}(J)$ is the density of $E=0$ states of spin $J$. \n\nWe will see in the next section that the spectrum of the $\\cN=4$ super-Schwarzian derived from the exact partition function we computed above has precisely this form (although with only singlet $J=0$ zero energy states). \n\n\n\n\n\\subsection{Exact density of states}\n\\label{sec:density-of-states}\nThe final answer for the exact $\\mathcal{N}=4$ super-Schwarzian theory partition function is given by the following function of inverse temperature $\\beta$ and $SU(2)$ chemical potential $\\alpha$ as\n\\beq\\label{exactPFN4d}\nZ(\\beta,\\alpha) =e^{S_0} \\sum_{n\\in \\mathbb{Z}} \\frac{\\beta}{\\Phi_r} \\frac{2 \\cot(\\pi \\alpha)(\\alpha+n)}{\\pi^3(1-4(n+\\alpha)^2)^2} e^{\\frac{2 \\pi^2 \\Phi_r}{\\beta}(1-4(n+\\alpha)^2)}\\,.\n\\eeq\nWe have fixed the overall normalization in a way that will be convenient later. We will write this answer as a trace over a Hilbert space (albeit with continuous spectrum) realizing it has precisely the form \\eqref{sqwewq}.\n\nTo understand the physics of this partition function we want to extract the density of states as a function of energy at fixed $SU(2)$ charge, which we will refer to as angular momentum (anticipating the application to near extremal black holes in 4D). To do that we begin by performing an inverse Laplace transform and define the fixed-chemical-potential density of states \n\\beq\nZ(\\beta,\\alpha) = \\int dE e^{-\\beta E} D(\\alpha, E).\n\\eeq\nApplying this to our result \\eqref{exactPFN4d} gives\n\\beq\nD(\\alpha,E) = D_{E=0}(\\alpha) \\delta(E) + D_{\\rm cont}(\\alpha,E),\n\\eeq\nwhere we separate the BPS and continuous part of the spectrum,\\footnote{The sum in $D_{E=0}(\\alpha)$ is at face value divergent. To regulate it we used the following prescription $\\lim_{N\\to\\infty} \\sum_{n=-N}^N\\frac{4(\\alpha+n)}{\\pi \\tan \\pi \\alpha (1-4(\\alpha+n)^2)} = 1$. We can verify that this is the correct prescription by checking that after integrating over energies, this gives back the original partition function.}\n\\bea\n\\label{toto}D_{E=0}(\\alpha) &=&e^{S_0} \\sum_{n\\in\\mathbb{Z}} \\frac{4(\\alpha+n)}{\\pi \\tan \\pi \\alpha (1-4(\\alpha+n)^2)} = e^{S_0}\\label{extD} \\\\\\label{contD}\nD_{\\rm cont}(\\alpha,E)&=&e^{S_0}\\sum_{n\\in \\mathbb{Z}} \\frac{4(\\alpha+n)}{\\pi \\tan \\pi \\alpha} \\frac{I_2\\left(2\\pi \\sqrt{2 \\Phi_r E(1-4(\\alpha+n)^2)}\\right)}{E(1-4(\\alpha+n)^2)}\n\\ea\nWe see the first line corresponding only to states with zero energy is independent of $\\alpha$. This means it only gets contributions from zero charge (angular momentum) states. We chose the normalization of the partition function such that this gives $\\exp \\left( S_0 \\right)$ and can be interpreted as the degeneracy of ground states.\n\nTo find the density of states we use the following identity to rewrite \\eqref{contD} as \n\\beq\\label{ble}\nD_{\\rm cont}(\\alpha,E) =e^{S_0} \\sum_{m=1}^\\infty \\frac{m \\sin 2\\pi m \\alpha}{ \\tan \\pi \\alpha} \\frac{\\sinh \\left(2 \\pi \\sqrt{2\\Phi_rE-\\frac{1}{4}m^2}\\right)}{2 \\pi^2 \\Phi_r E^2} \\Theta\\Big(E-\\frac{m^2}{8\\Phi_r}\\Big).\n\\eeq\nWe defined the Heaviside function $\\Theta(x)$ such that $\\Theta(x>0)=1$ and $\\Theta(x<0)=0$. The dependence with the chemical potential can be expanded in $SU(2)$ characters in the following simple way \n\\bea\\label{charactedID4}\n\\frac{2 \\sin 2\\pi m \\alpha}{ \\tan \\pi \\alpha} &=& \\chi_{J}(\\alpha) +2\\chi_{J-\\frac{1}{2}}(\\alpha)+ \\chi_{J-1}(\\alpha),~~~J\\equiv m\/2,~{\\rm with}~m>1\\\\\n\\frac{2 \\sin 2\\pi \\alpha}{ \\tan \\pi \\alpha} &=& \\chi_{1\/2}(\\alpha) +2\\chi_{0}(\\alpha),~~\\hspace{2.3cm}J\\equiv m\/2,~{\\rm with}~m=1\n\\eea\nwhere in the right hand side we defined the angular momentum $J$ in terms of the integer $m$. In principle we can use this formula to extract the density of states for each $SU(2)$ representation. Instead we will notice this is precisely the combination in equation \\eqref{sqwewq}\nwhere $J$ now labels the supermultiplet $\\mathbf{J}$. The second line with $m=1$ involves the special case $\\mathbf{1\/2}$. Comparing \\eqref{sqwewq} with \\eqref{ble} we can extract the density of supermultiplets $\\rho_{\\rm cont}(J,E)$ for $E\\neq0$ and using \\eqref{extD} we can write the density of $E=0$ states $\\rho_{\\rm ext}(J)$. The final answer is given by\n\\bea\n\\rho_{\\rm ext}(J) &=& e^{S_0} \\delta_{J,0}.\\label{sksks}\\\\\n\\rho_{\\rm cont}(J,E) &=& \n\\frac{e^{S_0}J}{4\\pi^2 \\Phi_r E^2}\\sinh \\left(2 \\pi \\sqrt{2\\Phi_r(E-E_0(J))} \\right)\\hspace{0.1cM} \\Theta\\Big(E-E_0(J)\\Big), \\hspace{0.1cm}\\text{for }J\\geq\\frac{1}{2},\\label{sksks2}\n\\ea\n where the gap for each supermultiplet labeled by $J$ is given by $E_0(J) \\equiv J^2\/(2\\Phi_r)$.\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=0.65]\n \\pgftext{\\includegraphics[scale=0.5]{superdos.pdf}} at (0,0);\n \\draw (-5.7,-0.4) node {$e^{S_0}$};\n \\draw (-5.8,2.75) node {\\small $\\rho(E)$};\n \\draw (-2,-3) node {\\small $E_{\\rm gap}$};\n \\draw (-0,-3) node {\\small $E_0(1)$};\n \\draw (-0,-1.5) node {\\small $\\mathbf{1}$};\n \\draw (-2.3,-1.5) node {\\small $\\mathbf{1\/2}$};\n \\draw (-4.6,-1.5) node {\\small $\\mathbf{0}$};\n \\draw (5,-3) node {\\small $E$};\n \n \\end{tikzpicture}\n \\hspace{0.6cm}\n \\begin{tikzpicture}[scale=0.65]\n \\pgftext{\\includegraphics[scale=0.47]{dosj0.pdf}} at (0,0);\n \\draw (-5.4,-0.4) node {$e^{S_0}$};\n \\draw (-5.4,2.75) node {\\small $\\rho(E)$};\n \\draw (-2,-3) node {\\small $E_{\\rm gap}$};\n \\draw (0.2,-3) node {\\small $E_0(1)$};\n \\draw (5,-3) node {\\small $E$};\n \n \\end{tikzpicture}\n \\caption{\\textbf{Left:} Density of supermultiplets labeled by the highest spin $\\mathbf{J}$. We show $\\mathbf{0}$, which is simply a delta function at $E=0$; $\\mathbf{1\/2}$ which is continuous but starts at $E_{\\rm gap}\\equiv E_0(1\/2)$; and $\\mathbf{1}$ which is also continuous starting at $E_0(1)$. \\textbf{Right:} Degeneracy for all states with $J=0$. These come from $\\mathbf{0}$, the delta function at $E=0$, $\\mathbf{1\/2}$, starting at $E_{\\rm gap}$, and $\\mathbf{1}$, starting at $E_0(1)$. All other supermultiplets do not have a $J=0$ component.}\n \\label{fig:my_label}\n\\end{figure}\n\n\n\nUsing this result we can get a simple picture of the shape of the spectrum. First we have a number $e^{S_0}$ of states at exactly $E=0$ which are all in the supermultiplet $\\mathbf{0}$, an $SU(2)$ singlet. These are the extremal BPS states of the black hole as we will see in the next section. For small energies there are no states until we reach the gap in the spectrum given by the threshold energy for the supermultiplet $\\mathbf{\\frac{1}{2}}=(\\frac{1}{2})\\oplus 2(0)$, given by \n\\beq\nE_{\\rm gap} = \\frac{1}{8 \\Phi_r},\n\\eeq\nand for $E>E_{\\rm gap}$ we have a continuum of states. Something similar is true for higher multiplets $J > 1\/2$, but now the continuum starts at a supermultiplet-dependent gap\n\\beq\nE_{0}(J) = \\frac{1}{2\\Phi_r} J^2.\n\\eeq\nIt is perhaps not surprising that states with spin $J$ start at $E_{0}(J)$. The surprising feature is that there are no states with $J=0$ at energies $01\/2$ contribute in the following way \n\\beq\ne^{2\\pi i J} \\left[\\chi_J(\\alpha) - 2 \\chi_{J-1\/2}(\\alpha)+\\chi_{J-1}(\\alpha)\\right].\n\\eeq\nIt is easy to see that when $\\alpha=0$ this combination exactly vanishes since there is the same number of bosonic and fermionic states in the supermultiplet. The same is true for $\\mathbf{1\/2}$ which gives $\\chi_{1\/2}(\\alpha)-2\\chi_0(\\alpha)$ and also vanishes for $\\alpha=0$. Therefore, the Witten index of the $\\cN=4$ super-Schwarzian theory is given by $e^{S_0}$ and counts the number of ground states.\n\nAs a final comment, there are two different definitions of the $\\cN=2$ super-Schwarzian theory that differ on the presence of a 't Hooft anomaly, as we review in Appendix \\ref{app:N2}. Since the gauge group of the $\\cN=4$ super-Schwarzian is $SU(2)$ we think there cannot be such anomaly \\cite{Kapec:2019ecr} and the theory is unique, but this deserves further investigation.\n\n\n\\subsection{Comparison with a pure bosonic theory} \n\\label{sec:compbosth}\nTo finish this section we would like to compare this solution to a non-supersymmetric version of the theory, such as the one in \\cite{Iliesiu:2020qvm}. Imagine we have a bosonic Schwarzian theory coupled to an $SU(2)$ mode. The action is \n\\beq\\label{ejksjw}\nI =-\\Phi_r \\int d\\tau \\hspace{0.1cm} \\text{Sch}(f,\\tau) + K \\int d\\tau {\\rm Tr} \\left( g^{-1} \\partial_\\tau g\\right)^2 ,\n\\eeq\nwhere $K$ and $\\Phi_r$ are independent parameters. This theory can be solved exactly \\cite{Mertens:2019tcm}. The density of states as a function of energy and angular momentum $J$ is given by\n\\beq\n\\rho_{bos.}(J,E) = e^{S_0} \\sinh \\left( 2\\pi \\sqrt{2\\Phi_r\\left( E- E_{bos.}(J)\\right)}\\right) \\Theta (E-E_{bos.}(J)),~~~E_{bos.}(J)\\equiv\\frac{J(J+1)}{2K}.\n\\eeq\nThe bosonic sector of the supersymmetric theory is special in two ways. First of all it necessarily has $K=\\Phi_r$. Therefore, at least semiclassically one can compute the gap scale by measuring the following quantity \n\\beq\n\\left( \\frac{\\partial J}{\\partial \\Omega} \\right)_{T=0,\\Omega=0}= K = \\Phi_r,\n\\eeq\nwhere $\\Omega=i\\alpha\/\\beta$ is the potential conjugated to $J$ (from 4D perspective, angular velocity). The second, and more important, feature is the fact that for the bosonic theory \\eqref{ejksjw}, even if $K=\\Phi_r$, there are states with $J=0$ for any energy $E>0$, namely $\\rho(J=0,E>0)\\neq 0$ and $\\rho(J=0,E=0)=0$. The $\\cN=4$ supersymmetric Schwarzian theory is completely different. We find a delta function at $E=0$ describing $e^{S_0}$ states. Moreover, even for $J=0$, there are no states in the range $00$.\\footnote{For the opposite orientation, we can take $\\beta_L\\to0$ then we have large $P<0$.} Second, we take large $k\\gg 1$ so that the gravity description in the bulk is accurate. Finally, we take $\\beta_L \\sim k \\gg 1$, which implies that we are looking at very low temperatures or states with $E \\sim P$. Since the state without left-moving excitations preserves supersymmetry, this is also a near-BPS limit \\cite{Coussaert:1993jp}. \n\nWe will follow the calculation first in the fixed $\\beta_L,\\beta_R$ ensemble. As explained in \\cite{Ghosh:2019rcj} when taking this near-extremal limit we can inverse Fourier transform to obtain fixed $P$ ensemble by basically replacing $\\beta_R \\to 2\\pi \\sqrt{c\/(24 P)}$ and $\\beta_L \\to 2 \\beta$ at the end of the calculation. We also keep the left-moving $SU(2)$ chemical potential $\\alpha$ fixed in this limit, and consider zero right-moving charge. Either taking the limit of the character or doing the reduction, the near-extremal near-BPS limit of the partition function is \n\\beq\\label{eq:RReqnschwarzian}\nZ_{R-R}(\\beta,\\alpha) \\sim e^{2\\pi \\sqrt{k P}} \\sum_{n\\in \\mathbb{Z}} \\frac{\\beta}{k} \\frac{\\cot(\\pi \\alpha)(\\alpha+n)}{(1-4(n+\\alpha)^2)^2} e^{\\frac{ \\pi^2 k}{2\\beta}(1-4(n+\\alpha)^2)}\n\\eeq\nWe will not repeat the calculation here since it is completely analogous to section \\ref{sec:meth2cq}. The first term $e^{2\\pi \\sqrt{k P}}$ comes from the right-moving identity character which is basically evaluated in the usual Cardy limit, since $\\bar{q}' \\to 0$. The rest comes from the evaluation of the left-moving identity character, in the limit $q'\\to 1$. \n\nThe parameters describing the near-extremal near-BPS effective theory analyzed in section \\ref{sec:N=4-super-Schw} are given in terms of the level $k$ and the angular momentum $P$, by \n\\begin{equation}\n S_0 = 2 \\pi \\sqrt{k P} ,~~~~\\Phi_r = \\frac{k}{4}.\n\\end{equation}\nTaking the inverse Laplace transform of this, we obtain the same density of states as the $\\cN=4$ super-Schwarzian theory with these parameters. In particular, we find a large degeneracy of BPS states given by $e^{2\\pi \\sqrt{ k P}}$, we find a gap to the first excited black hole state $E_{gap}=1\/(2k)$, and this predicts the index matches with the black hole degeneracy. \n\nThis can be easily generalized to cases with non-zero $SU(2)_R$ charge $\\bar{T}_0^3=J_R$. In this cases states with $T_0^3=0$ are still BPS and the contribution from the right-moving sector replaces $S_0 \\to 2\\pi \\sqrt{k P - J_R^2}$. With this modification, the spectrum as a function of temperature and $SU(2)_L$ chemical potential is still given by \\eqref{eq:RReqnschwarzian}, controlled by the $\\mathcal{N}=4$ super-Schwarzian.\n\n\\paragraph{Alternative construction -- preserving $\\mathbf{(4,0)}$ SUSY:} In this case we obtain a similar conclusion for $\\beta_R \\to 0$ and large $\\beta_L$. The difference now is that we can take instead $\\beta_L \\to 0 $ and $\\beta_R$ large. This extremal limit breaks supersymmetry since the right-moving sector of the theory is purely bosonic. Therefore, we expect that in this case, the black hole spectrum has, to leading order, no extremal states and no gap, similar to \\cite{Ghosh:2019rcj} (or \\cite{Iliesiu:2020qvm}).\n\n\n\\subsection{Comparison to string theory constructions}\nIn this section, we will very briefly mention some string theory constructions using D-branes that can be dimensionally reduced to the $AdS_3$ supergravity theories studied above.\n\nFor example, take type IIB string theory compactified on $M^4$, being either $T^4$ or $K3$. We consider the D1-D5 system, which at low energies can be described by either a $(4,4)$ 2D superconformal field theory or supergravity on $AdS_3 \\times S^3$. Compactifying down to $AdS_3$ gives the $(4,4)$ supergravity theory we studied above. The bosonic part of the spectrum is a 3D metric on $AdS_3$, and a gauge symmetry coming from the $S^3$ factor in the metric $SO(4)\\sim SU(2)_L \\times SU(2)_R$ separated into left- and right-movers. For the reasons explained in the previous section, we expect the presence of other fields to leave the conclusions below unchanged. For this theory, we can derive the level of the $SU(2)$ current algebra by matching the chiral anomaly\n\\begin{equation}\n k = Q_1 Q_5 ~~{\\rm for}~T^4,~~~~k=Q_1Q_5+1~~{\\rm for}~K3.\n\\end{equation}\nFor concreteness, we look at the $T^4$ case below. \n\nSo far we have vacuum $AdS_3$. We can add some momentum $P$ along the D1-string direction, which is identified in the BTZ gravitational description with the angular momentum $P$ defined above \\cite{Cvetic:1998xh}. The BPS extremal states correspond to no left-moving excitations of the string. Looking at low temperatures we have a near-extremal near-BPS black hole string with $AdS_2 \\times S^1 \\times S^3$ horizon. The parameters of the effective low-energy $AdS_2$ theory from the microscopic model is\n\\begin{equation}\n S_0 = 2 \\pi \\sqrt{Q_1 Q_5 P},~~~~~\\Phi_r = \\frac{Q_1Q_5}{4}\n\\end{equation}\nUsing our solution we see we have $e^{S_0}$ states at extremality, consistent with \\cite{Strominger:1996sh}. Our analysis also explains why the index matches with the black hole degeneracy. This can be easily generalized to near BPS states with non-zero $SU(2)_R$ charges corresponding at zero temperature to BPS black holes with angular momentum in $S^3$ \\cite{Breckenridge:1996is}.\n\nFrom our gravitational analysis giving the low-temperature dependence of the partition function, we have also derived the gap to the first excited black hole, and it is $E_{gap}=1\/(8\\Phi_r)$. In terms of the microscopic model parameters, it is \n\\begin{equation}\n E_{gap} = \\frac{1}{2 Q_1 Q_5}.\n\\end{equation}\nThis answer matches with the string theory approach from \\cite{Maldacena:1996ds}. From this perspective, the extremal black hole states come from counting string configurations at the brane system with only left movers. The lowest energy excitation comes from the first excitation of long strings wound $Q_1Q_5$ times along the branes.\n\n\\paragraph{Alternative construction -- Black holes in type I string theory:} We can also analyze a similar model in type I string theory instead of type II. We consider a D1-D5 brane system but now the supergravity theory emerging in $AdS_3$ has $(4,0)$ supersymmetry \\cite{Johnson:1998vd, Oz:1999it}. When we have an extremal black hole made out of right-movers, we expect the spectrum near-extremality to be analogous to the $\\cN=4$ super-Schwarzian. On the other hand, when the extremal black hole is made out of left-movers, supersymmetry is broken, and the near-extremal spectrum will look like the non-supersymmetric cases studied in \\cite{Ghosh:2019rcj} or \\cite{Iliesiu:2020qvm}.\n\nFinally, there are other interesting compactifications which we do not analyze in this paper, but whose role we briefly mention in the discussion section.\n\n\\section{Discussion}\n\\label{sec:conclusion}\n\nIn this paper, we have defined and solved $\\cN=4$ super-JT gravity. We show it reduces to a $\\cN=4$ generalization of the Schwarzian theory, which can be exactly solved. We argue that this theory captures the temperature-dependence of the gravitational path integral evaluated around near-extremal black holes in higher dimensions. We showed that both $\\cN=2$ ungauged supergravity in 4D flat space and $(4,4)$ supergravity in $AdS_3$ reduce to $\\cN=4$ super-JT in the near-horizon region of near-extremal black hole backgrounds. We found a gravitational explanation of the large extremal black hole degeneracy and for the presence of a gap in the spectrum. Thus our work addresses the strong tension between the non-supersymmetric results of \\cite{Iliesiu:2020qvm} and past micro-state countings in string theory \\cite{Strominger:1996sh}.\n\nWe finish here with some open questions and future directions:\n\n\\subsection*{Generalization to other black holes in AdS}\n\nWhile in this paper, we have focused on near-BPS black holes in 4D $\\cN=2$ supergravity in flatspace or in $(4,4)$ supergravity in $AdS_3$, there are numerous other near-extremal black hole solutions which are of interest in the AdS\/CFT correspondence.\\footnote{Similar considerations are also useful in computing quantum corrections to the Hartle-Hawking wavefunction in dS \\cite{Maldacena:2019cbz}.}\n\nThe first near-BPS solutions which we have not fully analyzed are those on $AdS_3 \\times S^3 \\times S^3 \\times S^1$ \\cite{Elitzur:1998mm}. The special feature about this compactification is that the extremal solution exhibits a large $\\cN=4$ symmetry, with $SU(2)_k\\times SU(2)_{k'}\\times U(1)$ current algebra. It would be interesting to analyze the 2D theory emerging in the throat for near-extremal near-BPS black holes. In this case, the symmetry of the boundary mode is now $D(2,1,\\alpha)$. We do not yet know how to study this version of the $\\cN=4$ Schwarzian theory, and we hope to address such a construction in future work.\n\nWe would also like to briefly mention the existence of near-BPS solutions in higher dimensional AdS. Extremal black holes in such theories typically preserve a smaller amount of supersymmetry; for instance, in AdS$_4$, such black holes exhibit an $OSp(2|2)$ isometry in the near-horizon region. The effective theory capturing the breaking of $OSp(2|2)$ was found to be $\\cN=2$ super-JT gravity \\cite{Forste:2020xwx} and the boundary dynamics is analogously given by the $\\cN=2$ super-Schwarzian (whose properties we have reviewed in appendix \\ref{app:N2}). As we explain in appendix \\ref{app:N2}, the $\\cN=2$ super-Schwarzian has a gap depending on the value of $\\hat q$ (which gives the periodicity of the identification of the $U(1)$ field $\\sigma$) and on whether or not the theory exhibits an anomaly (related to how we weigh the different saddles in the path integral). Thus, to conclude whether near-BPS black holes in such a theory exhibit a mass gap, we need to perform a rigorous analysis to account for all possible massless Kaluza-Klein modes that can appear in the near-horizon region, determine the analog of $\\hat q$ in supergravity and understand the situations in which the action of the boundary mode can exhibit an anomaly. \n\n\nOne purpose for studying the partition function of near-BPS black holes from the bulk perspective is to understand the gap in scaling dimensions between BPS and the near-BPS states in the dual CFT. If we find that the effective theory which captures the near-horizon dynamics exhibits a gap (as it did for the black holes in flatspace studied in this paper), then this translates to a scaling dimension gap, $\\Delta_{\\text{gap}}\\sim 1\/N^2$. It would be interesting to understand whether this gap in scaling dimensions is consistent with predictions from the large charge bootstrap \\cite{Jafferis:2017zna} in SCFTs. Finally, in comparing the partition function on the CFT side to the black hole partition function within a fixed large charge sector, there may be a mismatch coming from configurations with multiple black holes. Thus, it would be interesting to understand such corrections coming from multi-centered black hole solutions \\cite{Denef:2007vg, Denef:2000nb, Bates:2003vx, Sen:2007pg}.\\footnote{We thank G.~Moore for pointing out past works on this issue.} \n\n\n\n\n\n\n\\subsection*{On a possible $\\cN=4$ SYK model}\n\nAnother interesting possibility is whether there is a UV completion of the $\\mathcal{N}=4$ Super-Schwarzian theory in some quantum mechanical models, along the lines of \\cite{Fu:2016vas} for $\\mathcal{N}=1,2.$ To be more specific, we would like a random quantum mechanical model, with a stable unitary nearly conformal fixed point at low energies, with unbroken $\\mathcal{N}=4$ supersymmetry. A model involving dynamical bosons typically causes instability and exhibits supersymmetry breaking in the infrared (shown either as an operator with complex scaling dimension in the spectrum as in \\cite{Giombi:2017dtl,Klebanov:2018fzb}, or the absence of supersymmetric Dyson-Schwinger solutions as in \\cite{Anninos:2016szt,Chang:2018sve} ), and is rather inconvenient to study at finite $N$. In fact, in such a theory with a supermultiplet with $(b, \\psi, \\dots),$ and in a supersymmetric nearly conformal fixed point, $\\Delta_{\\psi}=\\Delta_b+\\frac{1}{2}.$ The Dyson-Schwinger equation of the dominant interaction in the infrared would constrain the dimensions of various fields so that the sum of scaling dimensions of the fields in the interaction is one, i.e.~$ n \\Delta_b+ 2m \\Delta_{\\psi}+ \\dots=1$, with $n,m\\in \\mathbb{Z}_{\\geq 0}$. For the nearly conformal fixed point to be unitary, we require $\\Delta_{b},\\Delta_{\\psi}\\geq 0.$ Together with the supersymmetric constraint, we conclude the only non-trivial solutions possible is that \n\\begin{equation}\\label{bosondim}\n \\Delta_b=0, \\Delta_{\\psi}=\\frac{1}{2}\\,.\n\\end{equation}\nHowever, such a solution indicates that the two-point function of $b$ must be logarithmic and typically causes a divergence in the Dyson-Schwinger equations (or \\eqref{bosondim} ceases to be a solution as in \\cite{Popov:2019nja}). \n\nOn the other hand, we may consider a theory with a fermionic super-multiplet. This scenario brings about yet another complication. Unlike the case of $\\mathcal{N}=1,2$, there is no relevant deformation of the $\\mathcal{N}=4$ that exists in the UV free theory. To see this, we can work in the $\\mathcal{N}=4$ superspace,\\footnote{Here, we note that this argument does not rule out possible theories without any kind of superspace realization. In particular, in one dimension one can consider a first derivative action in bosons as in \\cite{Tikhanovskaya:2020elb}, and this modifies the supersymmetry constraint to $\\Delta_b=\\Delta_{\\psi}$. Thus it allows a greater number of relevant interactions. } and the allowed action is \n\\begin{equation} \n \\mathcal{L} \\sim \\int d^2\\theta W(\\Psi)+ h.c. + \\int d^4\\theta K(\\Psi, \\bar{\\Psi}),\n\\end{equation}\nwhere schematically \n\\begin{equation}\n \\Psi= \\psi+ \\theta b+ \\dots,\n\\end{equation}\n where $\\Psi$ can sit in any representation of $SU(2)$ with half-integer $J$, and $\\psi$ is the lowest component. As a result, any local interaction must have dimension at least one due to $\\int d^2\\theta \\dots $, which implies that it cannot be relevant. This contrasts with the constructions of $\\mathcal{N}=1,2$ SYK-like models, where $\\int d\\theta W(\\Psi)$ is allowed and can produce relevant interactions. We can still ask if it's possible to have a marginally relevant deformation. Even if this were the case, in the infrared, a similar argument to (\\ref{bosondim}) would suggest \n \\begin{equation}\n \\Delta_{\\psi}=0, \\Delta_b=\\frac{1}{2}, \\dots,\n \\end{equation}\nwhich coincides with the dimensions of the free theory. This analysis suggests that constructing an interacting IR fixed point is difficult when starting from a UV theory with the same amount of supersymmetry. Therefore, one might be tempted to consider a scenario in which the $\\cN=4$ supersymmetry solely emerges in the IR and is not present in the UV. We leave a more thorough investigation into these issues for future work. \n \n \n\\subsection*{Higher genus corrections to super-JT}\n\nMotivated by the existence of the gap in the leading density of states for the $\\cN=2$ and $\\cN=4$ super-Schwarzian, it would be interesting to understand whether the gap survives when accounting for corrections coming from higher genus geometries contributing to the 2D theory. Relatedly, due to the existence of the gap, it is interesting to note that the contribution from disk topologies to the spectral form factor $\\< Z(\\beta-i t) Z(\\beta + it)\\>$ dominates even at very late times. This result contrasts with non-supersymmetric or $\\cN=1$ JT gravity, where at late times, the cylindrical topology starts dominating, leading to a ``ramp'' in the spectral form factor, followed by a plateau at even later times. It would also be interesting to understand whether the genus expansion of the $\\cN=2$ and $\\cN=4$ super-JT gravity has an interpretation in terms of a matrix integral; this interpretation needs to go beyond the three Dyson\nensembles \\cite{Dyson:1962es} and the seven Altland-Zirnbauer ensembles \\cite{altland1997nonstandard}, whose gravitational interpretation was studied in \\cite{Stanford:2019vob}. \n\nThese non-perturbative corrections are relevant from the perspective of solving 2D $\\cN=4$ gravity exactly. It is not clear whether these corrections could be reliable from the higher dimensional picture, but it would be interesting if the presence of supersymmetry could help better understand issues of factorization in the D1\/D5 system, as one example. We leave this for future work.\n\n\n\n\n\n\\subsection*{Acknowledgements} \n\n\nWe thank R. Campos Delgado, A.~Castro, G.~Horowitz, I.~Klebanov, J.~Maldacena, S.~Pufu, D.~Stanford, H.~Verlinde and E.~Witten for valuable discussions and comments on the draft. MTH is supported in part by Department of Energy Grants DE-SC0007968, DE-SC0009988, and the Princeton Gravity Initiative. LVI was supported in part by the Simons Collaboration on the Conformal Bootstrap, a Simons Foundation Grant with No. 488653, and by the Simons Collaboration on Ultra-Quantum Matter, a Simons Foundation Grant with No. 651440. GJT is supported by a Fundamental Physics Fellowship. WZ was supported in part by the US NSF under Grants No. PHY-1620059 and PHY-1914860.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn order to tame the increasing data demands, small cell deployments are of key importance. To date, primary focus of small cell networks is to enhance the overall capacity by bringing users closer to serving base stations~\\cite{li2014throughput}. Small cells can be deployed as standalone self organizing networks, or operating in conjunction with the existing macro cellular network. Due to the ever-increasing data traffic demands, an ultra-dense deployment of small cells is not a cost-effective strategy due to CAPEX\/OPEX issues. At the same time, recent developments in unmanned aerial vehicles (UAVs), driven by Google and Facebook bring forward the idea of using UAVs for coverage extension and capacity enhancements. UAVs can be used as aerial access points acting as \\emph{pivot} between macro and small cell tiers. UAVs can autonomously provide a reliable multi-connectivity in areas prone to high demand or link failures. UAVs can be used as aerial access points, or relays between disconnected networks and enhanced connectivity~\\cite{guo2014performance}. A smart combination of all these networks can provide a vast range of applications in civilian networks~\\cite{merwaday2015uav}. One of the major issues faced by these networks is on-demand\/on-the-fly capacity provisioning. Capacity refers to data rate transmission towards ground users, whereas delays refer to latency of data transmission~\\cite{mozaffari2015unmanned}. Using drone small cells as aerial support to existing cellular network can handle these high traffic situations more cost-efficiently. While deploying a single UAV is relatively easy using a maximum coverage point over the demand area, deploying more UAVs operating in coordination is more challenging due to interference from other aerial nodes~\\cite{mozaffari2015drone}. Thus, an efficient approach is required that not only provides efficient topology for UAVs based on user demands, but also improved connectivity and enhanced coverage. Using multiple UAVs as relays between the existing macro cells and small cell networks is the primary focus of this letter, in which the goal is to increase capacity, and lower transmission delays.\n\nIn this letter, a cost function based multiple UAVs deployment model is presented. The proposed model uses user demand patterns to assign a cost and density function to each area and UAVs. These cost and density functions are then used to match each UAV to a particular demand zone via a reverse neural model based on user demand patterns~\\cite{chandrashekarappa2014forward}.\n\\begin{figure}[!hb]\n \\centering\n \n\\includegraphics[width=200px]{fig1}\\\\\n \\caption{Illustration of an UAV-overlaid network deployment, with UAVs acting as relays between MBS and UEs.}\\label{fig1}\n\\end{figure}\n\\section{System Model}\nThe proposed model aims at provisioning continuous data between macro and small cell user equipments (UEs). UAVs further enhance load balancing by forming multiple intermediate links between the macro cell and the small cell UEs. The proposed model focuses on UAV-to-UE links rather the MBS-to-UAV backhaul capacity links. A traditional small and macro cell network is shown in Fig.~\\ref{fig1}. This network provides a direct link between the UEs and the macro cell base station (MBS). However, with a continuous increase in number of users within the coverage of a particular small cell, it becomes almost impossible to sustain connectivity without any loss of data. This is a problematic issue for future generation 5G networks, whereby demand is set to be greater than the available capacity. This is the focus of this work harnessing the formation of UAVs for a reliable and load balanced network. The proposed approach leverages a cost based framework which uses neural demand patterns to identify areas with high demands. A predictive chart is formed at the MBS acting as a launch point for UAVs. This chart helps in finding appropriate positions and topology for UAVs. Network stability is attained by minimizing a cost function associated with both demand areas and deployed UAVs. A delay threshold is defined to analyze the network performance, defined as the network state with minimum delay in providing capacity coverage in high demand areas, minimum errors in mapping UAVs to particular demand area, and high reliability in terms of data rate delivery. The model consists of deploying $n$ UAVs each one capable of handling $S_{n}$ service requests in a zone governed by an MBS. If $S_{r}$ requests are made by active users in a small cell zone that the MBS is unable to handle, then the minimum number of UAVs required to handle this traffic is $\\frac{S_{r}}{S_{n}}$. Moreover, the optimal placement of multiple-UAVs in the required zone is a major issue. For this, the concept of \\emph{zone guider lines} is considered where \\emph{zone guider lines} divide a particular area into a set of small regular areas acting independently. For a low-complexity solution, the number of UAV is kept equal to the number of guider lines. The traditional hexagonal cell is divided into standard guider lines, then, the area with high user requests are marked. Next, the existing guider lines form the maximum and minimum limit for introducing new guider lines which will embark the area to be governed by a UAV. This procedure is presented in Fig~\\ref{fig3}.\n\\begin{figure}[!hb]\n \\centering\n \n\\includegraphics[width=200px,height=180px]{fig3}\\\\\n \\caption{Area distribution in the macro cell with an illustration of referral guider lines.}\\label{fig3}\n\\end{figure}\n\\subsection{Assumptions}\nIn order to validate the effectiveness of the proposed approach, some generic assumptions are as follows:\n\\begin{itemize}\n \\item UAVs operate on same frequency spectrum.\n \\item Each UAV is of the same make, whose configuration does not affect its positioning.\n\\end{itemize}\n\\section{Proposed Approach}\nUAVs are used as high altitude base stations to cover certain geographical areas. The proposed approach uses a cost based neural model to find the appropriate user demand zones where UAVs is placed. The cost function includes the cost of operation which has to be minimum, and the cost of handling UAVs. Thus, the proposed approach focuses on finding the appropriate cost function for demand areas and UAVs, allocating them to MBS, and defining a neural model to minimize the cost function. In the proposed model, UAVs operate at an altitude $h$ over an area $A$, with $x$ number of users. Users service request $S_{r}$ come with an arrival rate of $\\gamma$, and a mean packet size of 1\/$\\mu$. The load\/delay, denoted by $L$ (in seconds) for a user at location $y$, is computed as~\\cite{samarakoon2014opportunistic}:\n\\begin{equation}\\label{eq:d1}\nL (y) =\\frac{\\gamma}{W \\;\\log(1+SINR(y))\\times \\mu}.\n\\end{equation}\nThe channel modeling includes radio range and pathloss. Moreover, round robin approach is applied for scheduling. Further, the system model is defined with respect to UAVs rather than the MBS, assumed to operate on orthogonal band. The area load $L_{a}$ is given by~\\cite{samarakoon2014opportunistic}:\n\\begin{equation}\\label{eq:d2}\nL_{a}=\\int_{y \\in A} L(y) dy.\n\\end{equation}\nHere, $W$ is the system bandwidth, and assuming that UAVs operate on the same frequency spectrum, the signal-to-interference-plus-noise ratio $\\left(SINR\\right)$ from the $i^{th}$ UAV to a given UE at location $y$, considering UAV-to-UAV interference, is:\n\\begin{equation}\nSINR(y)= \\frac{\\frac{P \\; K }{R_{iy}^{\\alpha}}}{\\sum_{j=1, j\\neq i}^{n} \\frac{P \\;K}{ R_{jy}^{\\alpha}} + N_{0}},\n\\end{equation}\nwhere $P$ is the UAV transmission power, $K$ is a factor that accounts for the geometrical parameters such as transmitter and receiver antenna heights, $R_{iy}$ is the distance between the $i^{th}$ UAV and the UE at location $y$, $\\alpha$ is the path loss exponent, and $N_{0}$ is the noise power spectral density. The spectral efficiency $N_{S}$ for a user at location $y$ following round robin scheduling is given by:\n\\begin{equation}\nN_{S}=W.\\;\\frac{\\log_{2} \\left( 1+SINR(y) \\right)}{x}\n\\end{equation}\nThe cost function is a function of capacity, delay, availability of line of sight (LOS), and coverage. Further, let $D_{f}$ denote the density function that quantifies the population of active\/non-active users based on users' request patterns. $D_{f}$ accounts for the number of active users $x$, packet loss (call drops) $C_{d}$, service requests $S_{r}$, and the total amount of users a cell can handle is $T_{r}$. For the considered network, two variants of the density function, one for a given area $D_{f}^{A}$, and the other for UAVs $D_{f}^{U}$ are computed as\n\\begin{equation}\\label{eq:1}\nD_{f}^{A}=\\min\\left(\\frac{\\left(\\frac{x}{T_{r}}\\right)^{S_{r}}\\;e^{- \\left(\\frac{x}{T_{r}}\\right)}}{S_{r}!}\\right),\n\\end{equation}\nand\n\\begin{equation}\\label{eq:2}\nD_{f}^{U}=\\min\\left(\\frac{\\left(\\frac{L_{a}}{n}\\right)^{S_{n}}\\;e^{- \\left(\\frac{L_{a}}{n}\\right)}}{S_{n}!}\\right),\n\\end{equation}\nrespectively. $D_{f}^{A}$ accounts for user distribution over the area, in which a higher value requires more UAVs, and a minimum value shows the efficient connectivity with no further requirement of intermediate relays. $D_{f}^{U}$ accounts for pending user requests in area A with respect to the number of UAVs deployed. A higher value denotes the requirement of more UAVs, and a minimum denotes efficient service handling using current deployment. Here, $\\frac{x}{T_{r}}$ denotes the ratio of active users to the total users a cell can handle. For 100\\% accuracy in mapping, $\\frac{x}{T_{r}}=1$. (\\ref{eq:1})-(\\ref{eq:2}) account for the cost function provided constraint (\\ref{eq:3}) holds.\n\\begin{equation}\\label{eq:3}\n\\sqrt{\\frac{1}{S_{r}} \\sum_{i=1}^{S_{r}} \\left( T_{r}^{i}-C_{d}^{i}\\right)} \\leq \\frac{x}{T_{r}}.\n\\end{equation}\nFor an efficient operation, the deviation in (\\ref{eq:3}), the number of users with unhandled service requests, should be kept minimum. Furthermore, the per area and UAV cost function $C_{f}^{A}$ and $C_{f}^{U}$ is given as:\n\\begin{equation}\\label{eq:4}\nC_{f}^{A}=\\min\\left(D_{f}^{A}\\; L_{a}\\; \\left(\\eta_{1}S_{r} + \\eta_{2}T_{r} \\right)\\right),\n\\end{equation}\nand\n\\begin{equation}\\label{eq:5}\nC_{f}^{U}=\\min\\left(D_{f}^{U}\\; R^{\\alpha}_{iy}\\; \\left( \\eta_{1}S_{r}+\\eta_{2} x\\right)\\right), LOS=true,\n\\end{equation}\nrespectively. Here, $\\eta_{1}$ and $\\eta_{2}$ are network balancing constants such that $\\left(\\eta_{1}, \\eta_{2} \\right) \\in \\left(0,1\\right)$. In general, $\\eta_{1}$ is driven by the network bandwidth and link speed, whereas $\\eta_{2}$ is driven by the number of active connections. For ideal state, $\\eta_{1}$ and $\\eta_{2}$ equals 1. In general, $0.5 \\leq \\eta_{1} \\leq 1$ and $\\eta_{1} \\leq \\eta_{2} \\leq 1$ which denotes that the network transfer rate must be higher than half of the initial configured rate. Both cost functions $C_{f}^{A}$ and $C_{f}^{U}$ are governed by the constraints of $D_{f}^{A}$ and $D_{f}^{U}$. The complete availability of LOS is one of the key driving factor for continuous connectivity. The overall cost function $C_{f}^{O}$ is computed at the MBS to maintain the overall network connectivity such that\n\\begin{equation}\\label{eq:6}\nC_{f}^{O}=\\min\\left( \\frac{1}{n} \\sum_{i=1}^{n} \\left(C_{f}^{U}\\right)_{i} + \\sum_{j=1}^{A_{T}} \\left(\\frac{C_{f}^{A}}{U_{T}} \\right)_{j}\\right).\n\\end{equation}\nHere, $U_{T}$ is the number of UAVs allocated to a particular area, $A_{T}$ is the number of total demand areas. With more UAVs, more resources are available in terms of transmission power, yielding high throughput and reduced delay. Further, the delay ($L_{d}$) at each node is computed as:\n\\begin{equation}\nL_{d}=L_{transmission}+L_{propagation}+L_{queue}+L_{processing}.\n\\end{equation}\nHere, $L_{transmission}$ is the transmission delay defined as the load\/delay of a particular user and is equal to $L$ (\\ref{eq:d1}), $L_{propagation}$ is the ratio of the distance between nodes to the propagation speed, $L_{queue}$ is the waiting time of the packet, and $L_{processing}$ is the network operational time.\n\\section{Neural Demand Patterns and Network Capacity}\nThe goal is to optimize the density functions and minimize the cost functions defined in (\\ref{eq:1}), (\\ref{eq:2}), (\\ref{eq:4}), (\\ref{eq:5}), and (\\ref{eq:6}). By controlling $D_{f}^{A}$ and $D_{f}^{U}$, the user distribution with pending requests are controlled, which in turn, minimizes $C_{f}^{A}$, $C_{f}^{U}$, and $C_{f}^{O}$. This minimization provides guaranteed service to UEs. Thus, the aim of the neural model is to accurately map UAVs to demand areas so as to minimize these cost functions.\n\\begin{figure}[!ht]\n \\centering\n \n\\includegraphics[width=180px,height=170px]{fig4}\\\\\n \\caption{Reverse neural model modeling the user demand pattern: $C_{f}^{A}$ refers to area cost function, $C_{f}^{U}$ refers to UAV cost function, $C_{f}^{O}$ is the overall cost function.}\\label{fig4}\n\\end{figure}\nDemand patterns are used to minimize the demand and cost functions by efficiently deploying aerial nodes as intermediate nodes between high demand areas and the MBS. These demand patterns are driven by a reverse multi-hierarchical neural model which is a combination of input, hidden, and output layer. Although neural models are slow and complex in operation, the reverse neural model accounts for accurate mapping of UAVs to demand areas with lesser iterations. This provides a low-complexity approach for UAV-to-Area mapping. The output for the proposed model is computed in forward direction, i.e. from Layer 3 to Layer 1, but nodes are deployed in reverse direction i.e. from Layer 1 to Layer 3, as shown in Fig.~\\ref{fig4}. Input involves demand areas that require services of UAVs. MBS is the driving factor of this reverse neural model, and it act as an output layer. UAVs are the intermediates, and are not having permanent connections with the demand areas, thus, acts as a hidden layer. These neural patterns are then topologically rearranged to form a stable network with minimized cost function. Layer 3 defines the cost function for underlying demand areas, layer 2 is for aerial nodes, and output layer 1 is mapped to the MBS. At first, a mesh like interface is initialized for the neural model (Fig.~\\ref{fig4}), then, the whole model is rearranged to allocate links that will lower the cost functions of all the zones. This procedure is governed by series of steps given in Algorithm~\\ref{algo1}.\n\\begin{algorithm}[!ht]\n\\fontsize{7}{9}\\selectfont\n\\caption{UAV to Area Mapping}\n\\label{algo1}\n\\begin{algorithmic}[1]\n\\State \\textbf{Input}: $U \\longleftarrow UAVs$, $A \\longleftarrow Demand \\;areas$\n\\State Initialize Network\n\\State Divide A into subdivisions $D_{M}$\n\\While{(U are not mapped to subarea $D_{M}$ \\&\\& ($C_{f}^{O}$=minimum))}\n\\State Store $D_{M}$ in area[] following descending order for $C_{f}^{A}$\n\\State i=1\n\\While{($i \\leq count(area[])$)}\n\\State allocate U to $D_{M}$, such that min($C_{f}^{U}$) is mapped to max($C_{f}^{A}$)\n\\State Compute $C_{f}^{A}$, $C_{f}^{U}$, $C_{f}^{O}$\n\\If {((U is mapped to $area[i]$) \\&\\& ($C_{f}^{A}(i)$=minimum))}\n\\State continue\n\\Else\n\\State re-initialize $U$, $C_{f}^{A}$\n\\State reset\n\\EndIf\n\\State \\textbf{end if}\n\\State $i=i+1$\n\\EndWhile\n\\State \\textbf{end while}\n\\EndWhile\n\\State \\textbf{end while}\n\\end{algorithmic}\n\\end{algorithm}\nInitially, the demand area $A$ is subdivided into small segments $D_{M}$, and the cost function is computed for each of the subdivided area. Each of the cost function is ranked in descending order, in which the UAV with a minimum cost function is mapped to the area with maximum cost function. This helps in balancing the overall load of the network. After initial allocation, all the cost functions are recomputed, and UAVs are mapped to the next high demand area based on the cost function. This procedure continues till a minimum is attained. With an efficient deployment, the cost function is minimized by sub-dividing the density function based on the area so that active users becomes equal to the total registered users i.e. $\\frac{x}{T_{r}}=1$ such that equation (\\ref{eq:1}) reduces to\n\\begin{equation}\nD_{f,avg}^{A}=\\frac{1}{e\\times S_{r}!}\n\\end{equation}\nwhere e=2.71828 approx. For verification at any stage, the average density function of each area must be less than equal to $D_{f,avg}^{A}$.\n\\begin{table}[!ht]\n\\fontsize{7}{9}\\selectfont\n\\centering\n\\caption{Parameter Configurations}\\label{self_conf\n\\begin{tabular}{l l l}\n\\hline\\\\\n\\textbf{Parameter} & \\textbf{Value} & \\textbf{Description}\\\\\n\\hline\n\\hline\\\\\n$A$ & 10000x10000 sq. m. & Simulation Area\\\\\nMBS & 10& Number of Macro Cell Base Station\\\\\n$T_{r}$ & 1200 (per MBS) & Max Users in a Cell\\\\\n$n$& 6 (per MBS)& Number of UAVs\\\\\n$S_{n}$ & 200 & Service Requests handled by each UAV\\\\\n$N_{0}$ & -170 dBm\/Hz & Noise Power Spectral Density\\\\\n$\\frac{1}{\\mu}$ & 1024 B&Packet Size \\\\\n$h$ &200-500 Feet& UAV altitude\\\\\n$\\frac{\\gamma}{\\mu}$ & 256 kbps& Offered Traffic\\\\\n$\\alpha$& 4&Path loss Exponent\\\\\n$K$ & -11 dB & Transmission Constant\\\\\n$P$ & 35 dBm & UAV Transmission Power \\\\\n$S_{r}$& 30-50 per zone& Service Requests\\\\\n$W$ & 10 MHz& System Bandwdith\\\\\n$x$ & 400 & Active Users\\\\\n\\hline\n\\end{tabular}\n\\end{table\n\\begin{figure}[!ht]\n\\begin{minipage}[t]{0.20\\textwidth}\n\\centering\n\\includegraphics[width=120px,height=80px]{g1}\n\\caption{\\fontsize{6}{6}\\selectfont Networks Delays vs. Extra Users }\n\\label{g1}\n\\end{minipage}\n\\hspace{\\fill}\n\\begin{minipage}[t]{0.26\\textwidth}\n\\centering\n\\includegraphics[width=120px,height=80px]{g7}\n\\caption{\\fontsize{6}{6}\\selectfont Throughput Coverage vs. Path loss Exponent}\n\\label{g3}\n\\end{minipage}\n\\end{figure}\n\\begin{figure}[!ht]\n\\begin{minipage}[t]{0.20\\textwidth}\n\\centering\n\\includegraphics[width=120px,height=80px]{g8}\n\\caption{\\fontsize{6}{6}\\selectfont Throughput Coverage vs. Extra Users}\n\\label{g4}\n\\end{minipage}\n\\hspace{\\fill}\n\\begin{minipage}[t]{0.26\\textwidth}\n\\centering\n\\includegraphics[width=120px,height=80px]{g9}\n\\caption{\\fontsize{6}{6}\\selectfont 5th Percentile Spectral Efficiency vs. Extra Users }\n\\label{g5}\n\\end{minipage}\n\\end{figure}\n\\begin{figure}[!ht]\n\\begin{minipage}[t]{0.20\\textwidth}\n\\centering\n\\includegraphics[width=120px,height=80px]{g12}\n\\caption{\\fontsize{6}{6}\\selectfont 5th Percentile Spectral Efficiency vs. Path loss Exponent }\n\\label{g6}\n\\end{minipage}\n\\hspace{\\fill}\n\\begin{minipage}[t]{0.26\\textwidth}\n\\centering\n\\includegraphics[width=120px,height=80px]{g5}\n\\caption{\\fontsize{6}{6}\\selectfont Probability of Guaranteed SINR vs. Extra Users }\n\\label{g7}\n\\end{minipage}\n\\end{figure}\n\\section{Performance Evaluation}\nThe proposed model is analyzed using network simulations, to efficiently allocates areas with higher demand patterns to UAVs based on their cost function. All parameters and configurations are presented in Table~\\ref{self_conf}. Active users refer to the number of users in a cell requesting services. Thus, at any time instance, more than the maximum supported user requests can be present in the cell. Results were recorded for delays, network capacity, reliability, and value of cost function with and without use of UAVs. Delays were traced for complete data sharing with 1000 iterations of user requests over 1000 seconds. Availability of LOS is the primary condition for UAVs to form a link with the UE. Finding appropriate position in the user demand areas causes UAVs to adjust their altitude to guarantee LOS towards the UE. High altitude provides less interference and appropriate LOS, but also induces more delays. Thus, an optimum altitude with availability of LOS is required for better coverage. For analysis, the altitude was varied between 200 ft. and 500 ft. with a multi-antenna relay support for communication and backhaul link capacity of 1.2 Gbps. Delay threshold is fixed at 200 ms, which defines the upper limit above which the packet drop increases abruptly. Results show that the proposed approach leveraging UAVs yields 37.7\\% lesser delays in comparison with a network comprising of small cell and macro cell. The performance delay for various UAV altitudes is given in Fig.~\\ref{g1}. Throughput coverage defined as the percentage of users whose SINR is above the threshold (0.03 bps\/Hz) is shown in Fig.~\\ref{g3}. Therein, the use of UAVs increases the overall 5th percentile throughput coverage by 15.5\\%, as shown in Fig.~\\ref{g3} and Fig.~\\ref{g4}. Moreover the optimal placement of UAVs according to user demand leveraging the reverse neural deployment model enhances the 5th percentile spectral efficiency. The accurate mapping optimally places UAVs according to demand patterns, thus, improving the 5th percentile spectral efficiency by 38\\% approx., as shown in Fig.~\\ref{g5} and Fig.~\\ref{g6}. Finally, Fig.~\\ref{g7} plots the probability of guaranteed SINR for a particular user in a given macro cell. Clearly, it can be noticed that the use of UAVs provides much guaranteed SINR above the threshold defined by $\\eta_{1}$ in (\\ref{eq:5}). Here, the SINR threshold is kept at 0.55 defining the value below which the network is unable to provide efficient connectivity to users. Results show that the proposed user demand based network model is capable of providing better capacity and prolonged connectivity than the existing cellular network.\n\\section{Conclusion}\nIn this letter, user demand based network model is proposed using multiple UAVs. The proposed model uses density and cost functions to compute areas with higher demands, whereby UAVs are deployed based on these cost functions. Analysis proved that the proposed model is capable of providing better capacity, reliability, and prolonged connectivity in comparison to existing ground-based wireless networks.\n\\bibliographystyle{ieeetr}\n\\nocite{*}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\\section{Introduction}\nEvergrowing concerns about user-privacy, censorship and central authority in popular social media have motivated both the development of federated social networks such as Mastodon and Diaspora~\\cite{rochko2018mastodon,Bielenberg:2012:TGD}, as well as research in academia~\\cite{Anderson:2009:PSN, Shakimov:2009:PCA}.\nThese networks aim to promote user control by decentralizing authority and relying on open-source software and open standards.\nAt the time of this writing, Mastodon has over 1 million users and 3500 instances which demonstrates the increasing acceptance of distributed social networks.\nAs with traditional social media, one key success factor of such a network is an active and engaged community.\n\nAs a community grows, overwhelming amounts of content make it increasingly difficult for a user to find interesting topics and other users to interact with.\nFor that reason, popular platforms such as Twitter, LinkedIn and Facebook introduce recommender systems that set out to solve a particular recommendation task.\nOne prominent example is the ``Who to Follow'' service by Twitter~\\cite{Gupta:2013:WFS}.\nDue to their recent emergence, those recommender systems do not exist for federated social networks, yet.\nHowever, they are needed to make distributed social media attractive to large user groups as well as competitive to centralized networks.\nAt the same time, recommender systems will contribute to develop, grow and sustain an active community.\n\nTo make federated social networks more attractive and feature complete, we implement and evaluate a topology-based user recommender based on personalized PageRank~\\cite{Page:1999:PCR}, a commonly used algorithm for link-prediction in social networks.\nWe compare this method against collaborative filtering based on link intersections~\\cite{Hannon:2010:RTU} and a random link predictor baseline~\\cite{Liben-Nowell:2003:LPP}.\nThe experiments are carried out on Mastodon, a federated social network for which user relations do not require reciprocation, and the network forms a directed graph.\nWe expect that the method and results are transferable to any other federated social network with similar characteristics.\n\nWe evaluate the systems in an offline and online scenario. For the offline evaluation, we collect an unbiased sample of the Mastodon user graph.\nThis sample is created by performing a Metropolis-Hastings Random Walk (MHRW) adapted for directed graphs~\\cite{Wang:2011:UGS,Wang:2010:USD}.\nThe collected data contains about 25\\% of the entire userbase of Mastodon.\nWe then evaluate the recommender systems according to standard performance metrics used in ranked retrieval systems, and deploy the two best performing methods to an online setting.\nBoth algorithms generate a list of personalized recommendations for 19 Mastodon users participating in the online trial and performance is measured with the balanced interleaving approach~\\cite{Joachims:2003:ERP}.\n\nThis paper is structured as follows.~\\cref{sec:dataset} explains how data are collected for the offline experiments and discusses the recommendation algorithms and their evaluation.\nIn~\\cref{sec:results} we present and discuss experimental results.~\\cref{sec:conclusion} concludes this paper and provides directions for future work.\n\n\\section{Dataset and Methods}\n\\label{sec:dataset}\n\n\\subsection{Recommendation Algorithms}\n\\label{sec:recommendation-algorithms}\nThe user recommendation problem for social networks can be formalized as follows.\nGiven a graph $G = (V,E)$ where $V$ and $E$ are vertices and edges, we seek to predict an interaction between a user $u \\in V$ and $v \\in V$ denoted by edge $(u,v)$.\nIn networks such as Mastodon and Twitter, a user interaction does not require reciprocation.\nThus, the graph is directed.\nWe consider two broad approaches to generate recommendations: (1) collaborative filtering-based recommendation and (2) topology-based recommendation.\n\nWith respect to the collaborative filtering, we use an approach inspired by~\\cite{Hannon:2010:RTU}. Each user $u \\in V$ is represented by a profile and recommendations are generated based on the similarity of profiles.\nWe distinguish between the three best performing strategies in~\\cite{Hannon:2010:RTU}:\n\\begin{description}[leftmargin=!,labelwidth=\\widthof{$following(u)$}]\n\t\\item[$\\mathit{following}(u)$] The set of user ID's $u$ follows\n\t\\item[$\\mathit{followers}(u)$] The set of user ID's that follow $u$\n\t\\item[$\\mathit{combined}(u)$] The combined set of following and follower ID's\n\\end{description}\nWe consider these profiles as documents to be indexed in a general purpose search engine. In order to generate recommendations for a user, the corresponding profile is extracted first.\nAfterwards, the retrieval system is queried with the profile and it ranks the indexed documents by their relevance to the query.\nEach ID in the user profile is a token of the query. If a query consists of more than 10,000 tokens, we create a random subset of 10,000 tokens.\nUnlike~\\citet{Hannon:2010:RTU}, we use BM25 instead of TF-IDF to estimate the relevance score of each document and set parameters to common defaults ($k_1 = 1.2$, $b = 0.75$)~\\cite{Manning:2008:IIR:1394399}.\nThe final recommendation list contains the top-$k$ documents with highest retrieval score.\n\nThe collaborative filtering recommendations are compared to topology-based recommendations.\nSeveral methods have been proposed in literature which make use of link-based ranking algorithms such as HITS, PageRank and SALSA\\@.\nDue to the novelty of generating recommendations for federated social networks, we restrict our experiments to the personalized PageRank algorithm~\\cite{Page:1999:PCR} whose efficient computation is well-understood and which is used in the Twitter recommender system~\\citep{Gupta:2013:WFS}.\nWe apply the personalized PageRank for a seed node which is the user we want to generate recommendations for.\nAfter convergence, the list of user recommendations is constructed by taking the top-$k$ nodes with highest PageRank. Following~\\cite{Liben-Nowell:2003:LPP}, we set the damping factor $\\lambda = 0.85$.\n\n\\subsection{Data Collection}\nAcquiring the complete graph of a social network is always infeasible due to API limits and time constraints~\\cite{Wang:2011:UGS}.\nAn additional concern arises in a distributed social network.\nAs data is not stored at a central authority, there is no single API that provides access to all parts of the network.\nInstead, data is scattered around different sub-networks.\nBoth issues are addressed within this section.\n\nTo overcome the time constraint, we apply the Metropolis-Hastings Random Walk (MHRW) to acquire an unbiased sample that is still representative of the complete graph.\nMHRW is a Markov-Chain Monte Carlo algorithm that can be used to obtain node samples with a uniform probability distribution~\\cite{Wang:2011:UGS}.\nAs the MHRW is only applicable to undirected graphs, we apply a generalization that considers all directed edges as bidirectional edges~\\citep{Wang:2010:USD}.\nWe do not consider graph sampling methods such as Random Walk and Breadth-First Sampling as it has been shown that these methods yield samples biased towards high degree nodes~\\cite{Gjoka:2010:WFC}.\n\nDue to the fact that a distributed social network has no central API, one has to query the API of each individual sub-network referred to as \\textit{instance}.\nIn case of Mastodon, there are two public endpoints to acquire incoming and outgoing links: \\texttt{\/following} and \\texttt{\/followers}\\footnote{The following API URL pattern applies to any Mastodon instance:\\\\ \\texttt{https:\/\/\/users\/\/.json}}.\nWhenever the MHRW visits an unexplored node, followers and followings of that node are fetched and stored in a document-oriented database.\nThis database is also used as a cache: if the random walk transitions to a node which it has already visited, we use the cached result rather than querying the API again.\nDuring the data collection, we apply fair crawling policies. Only instances that allow crawling as defined by the \\texttt{robots.txt} are considered.\nFurthermore, concurrent requests are throttled such that no more than 10 requests per second are issued (a rate which we believe any web server can sustain).\n\n\\subsection{Dataset Statistics}\n\\label{sec:dataset-statistics}\n\\cref{tab:dataset-statistics} summarizes the properties of the collected graph. The initial graph ($t_1$) has been crawled from the 16\/05\/18 until 17\/05\/18.\nThe MHRW was executed for 5500 iterations.\nDuring the crawl, 138 instances were disregarded either because of their \\texttt{robots.txt} or because they were no longer available.\nIn order to acquire a newer version of that graph ($t_2$), we visited the same users five days later and recorded new relationships.\nThe number of visited users in $t_2$ is slightly lower than in $t_1$, as some profiles were deleted or their instances became unavailable.\nThe updated graph is used as the ground-truth when evaluating our recommender systems.\n\nIt can be observed that the Network Average Clustering Coefficient (NCC) and the fraction of nodes in the largest Strongly Connected Component (SCC) is almost equal for the two given graphs.\nFurthermore, the graph is mildly disassortative.\nIt is important to mention that although the total number of nodes found $|V|$ is high (253,000), accounting for about 25\\% of the total Mastodon users, the number of visited nodes $|V^*|$ is much smaller (about 3400).\nIncoming and outgoing edges are only known for visited nodes.\n\n\\begin{table}[t]\n\\centering\n\\caption{Statistics of crawled graphs. The initial crawl at $t_1$ and the newer crawl of the same users at $t_2$.}\n\\label{tab:dataset-statistics}\n\\begin{tabular}{@{}llllllll@{}}\n\\toprule\nGraph & $|V|$ & $|V^*|$ & $|E|$ & Assort. & Deg. & NCC & SCC \\\\ \\midrule\n$t_1$ & 253,822 & 3437 & 754,037 & -0.015 & 5.94 & 0.31 & 0.175 \\\\\n$t_2$ & 255,638 & 3383 & 754,667 & -0.016 & 5.9 & 0.31 & 0.173 \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\subsection{Evaluation}\n\\label{sec:evaluation}\nThe algorithms presented in~\\cref{sec:recommendation-algorithms} are evaluated in two phases: an offline evaluation and an online evaluation.\nFor the offline evaluation we measure precision at rank $k$ (p@k), Mean Average Precision (MAP) and success at rank $k$ (s@k), which are popular metrics for the evaluation of ranked retrieval systems~\\cite{Manning:2008:IIR:1394399}.\nThe newer graph at time $t_2$ serves as the ground-truth, whereas the graph at time $t_1$ can be seen as the training graph.\nIn information retrieval terms, the generated list of recommendations are the retrieved documents and the list of users a target user follows at time $t_2$ are the relevant documents. Significance is tested using a two-tailed paired t-test. We denote improvements with $^{\\blacktriangle}$ ($p<0.01$), deteriorations with $^{\\blacktriangledown}$ ($p<0.01$), and no significance by $^{\\circ}$.\n\nDuring the offline evaluation, all systems generate a list of 100 recommendations based on the training graph at time $t_1$.\nThis list is then compared with the actual links added to the graph in between time $t_1$ and $t_2$ (see~\\cref{sec:dataset-statistics}).\nIn case of the collected dataset, 329 of 3437 visited users started to follow another individual, and thus added a link to the graph.\nOnly for this set of users, recommendations are generated and evaluated.\n\nThe online evaluation is performed as follows.\nA recommendation bot is created on the Mastodon instance associated with the institute of the authors\\footnote{See \\url{https:\/\/mastodon.utwente.nl\/@Followdon}}.\nAfterwards, we ask users to follow this bot if they wish to receive personalized recommendations. For each participant, we generate a static web page consisting of a list of $N$ recommendations with the option to start to follow a suggested user.\nA link to this web page is then send to the user and we track the user interactions. A recommendation is considered relevant if the participant starts to follow a suggested user.\nThe recommendations of two algorithms are presented using balanced-interleaving, which is a relatively inexpensive evaluation method for online experiments compared to conventional A\/B testing.\nWe refer the reader to~\\cite{Joachims:2003:ERP} for a thorough discussion of this evaluation method.\n\nOne complication arises in the online evaluation. As an up-to-date graph is unavailable at recommendation time, such a graph has to be created.\nFor this, we explore the vicinity of a recommendation target $u \\in V$ by applying an egocentric random walk for a fixed amount of iterations.\nThis strategy resembles the ``circle-of-trust'' used in the Twitter recommender system~\\cite{Gupta:2013:WFS}.\nThe random walk is performed as follows. At each iteration, the algorithm either transitions to a random neighbor of the current user with probability $\\gamma$, or jumps back to $u$ with probability $1 - \\gamma$. In our experiments, we execute the random walk for 200 iterations and set $\\gamma = 0.8$.\nHere, we do not claim that this is the most efficient way of generating recommendations in an online setup. It is merely a way to deal with incomplete data in federated social networks.\n\n\\section{Results}\n\\label{sec:results}\n\n\\subsection{Offline Evaluation}\nThe collaborative filtering approach shows a consistently higher performance than a topology-based system using PageRank (see~\\cref{tab:results-offline}).\nWith respect to the success at rank $k$ metric, profile-based approaches (R2--R4) have up to two times higher retrieval scores than PageRank (R5).\nThe individual profiling strategies perform all rather similarly, which aligns with the findings in~\\cite{Hannon:2010:RTU}.\nAlso, a baseline system (R1) which generates recommendations by selecting 100 random users from the network topology is outperformed by a large margin.\nIn \\cref{fig:precision-at-k}, it can be observed that shorter recommendation lists have a higher precision for all recommendation strategies.\nPrecision at rank $k$ remains stable starting from a list length of $k = 50$ items.\nThis suggests that shorter lists are to be preferred in an online scenario.\n\n\\begin{table}[t]\n\\centering\n\\caption{Experimental results of offline evaluation. Significance for model in line $i > 1$ is tested against line $i - 1$.}\n\\label{tab:results-offline}\n\\begin{tabular}{@{}clllll@{}}\n\\toprule\n\\textbf{ID} & \\textbf{System} & \\textbf{MAP} & \\textbf{s@1} & \\textbf{s@5} & \\textbf{s@10} \\\\ \\midrule\nR1 & Random & 0.001 & 0.000 & 0.000 & 0.055 \\\\\nR2 & Profile (following) & \\textbf{0.019}$^{\\blacktriangle}$ & \\textbf{0.033}$^{\\blacktriangle}$ & 0.085$^{\\blacktriangle}$ & 0.152$^{\\blacktriangle}$ \\\\\nR3 & Profile (followers) & \\textbf{0.019}$^{\\circ}$ & 0.030$^{\\circ}$ & 0.100$^{\\circ}$ & 0.167$^{\\circ}$ \\\\\nR4 & Profile (combined) & 0.018$^{\\circ}$ & \\textbf{0.033}$^{\\circ}$ & \\textbf{0.106}$^{\\circ}$ & \\textbf{0.173}$^{\\circ}$ \\\\\nR5 & Pers.\\ PageRank & 0.014$^{\\circ}$ & 0.018$^{\\circ}$ & 0.061$^{\\blacktriangledown}$ & 0.082$^{\\blacktriangledown}$ \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.8\\columnwidth]{figures\/precisionatk-curve}\n\\caption{Precision for different recommendation list lengths ($k$) in offline evaluation.}\n\\label{fig:precision-at-k}\n\\end{figure}\n\nIt is important to mention that the list of possible suggestions from the profile-based recommender is smaller than the list from the PageRank recommender, which complicates the discussion.\nOnly visited nodes (see~\\cref{sec:dataset-statistics}) have been indexed in the document retrieval system.\nThis significantly reduces the pool size of possible users ($\\approx$3k).\nIn contrast, the PageRank recommender can suggest any user in the topology ($\\approx$255k).\nOne could overcome this issue as follows.\nEach user, regardless of whether or not it has been visited during the data collection, could be added to the search index.\nThen, incoming relationships can be inferred by inspecting the outgoing links of visited users.\nBy adding these relations as \\textit{followers} to the documents of unexplored nodes, the \\textit{following} strategy of the collaborative filtering can be applied.\nHowever, as no outgoing links are known for unexplored users, the \\textit{following} and \\textit{combined} strategies are not fully applicable.\nDue to time constraints, we did not further investigate this issue.\n\nFurthermore, it is worth to note that the chosen window of five days between $t_1$ and $t_2$ might not have been long enough to capture sufficient user activity.\nIn between the training snapshot at $t_1$ and the testing snapshot at $t_2$, six new connections were added to each user on average.\nThis gives rise to an interesting trade-off.\nFor a longer time span, one can capture larger amounts of activity within the network.\nIntuitively, more links will be added as users start to follow other users.\nHowever, the farther two snapshots are apart, the larger is the risk that the network deviates too much from the original structure.\nUsers might stop following other users or profiles could be deleted.\nMore severely, entire instances could become unavailable due to a temporary downtime, or they could even be discontinued.\nThis is a unique concern related to the distributed nature of federated social networks.\n\nFinally, we want to motivate which recommendation systems are evaluated in the online trial based on the results presented above.\nFrom~\\cref{tab:results-offline} it can be observed that the profile-based recommendation strategies perform rather similarly.\nHowever, the combined strategy (R4) performs best with respect to the success at rank 10 metric, which one seeks to maximize in an online system where 10 recommendations are presented to the user.\nTherefore, we pick R4 as the first recommendation system.\nAlthough the personalized PageRank recommender (R5) has a lower performance than the other profiling strategies, we expect that it produces valuable recommendations which are significantly different from the profile-based strategies.\nThis is due to the fact that it considers the network topology when generating recommendations.\nTherefore, we apply the balanced-interleaving evaluation to systems R4 and R5.\n\n\\subsection{Online Evaluation}\nThe online evaluation shows that neither the profile-based nor the topology-based system is superior (see~\\cref{tab:summary-online-evaluation}). Nineteen users participated in our online study. On average, they started to follow 1.8 users from our recommendations. For 5 users the profile-based approach performed best. For another 5 users, the topology-based approach performed best. For the remaining 9 users both system performed equally well, or no recommendation was followed.\nThe fact that valuable recommendations were generated that resulted in new followings shows that the two systems can be useful in practice.\nHowever, a larger group of participants is required to draw final conclusions on the recommender system performance.\n\n\\begin{table}[t]\n\\centering\n\\caption{Summary of online evaluation.}\n\\label{tab:summary-online-evaluation}\n\\begin{tabular}{@{}p{.7\\columnwidth}c@{}}\n\\toprule\n\\textbf{Characteristic} & \\textbf{Value} \\\\ \\midrule\nNumber of participants & 19 \\\\\nProfile-based recommender (R4) superior & 5\\\\\nPageRank recommender (R5) superior & 5\\\\\nDraw & 2 \\\\\nNo user interaction & 7 \\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\subsection{Practical Considerations}\nThe generation of online recommendations turned out to be costly because the complete network data is not available.\nIn contrast to centralized social media, federated social networks do not have a single authority which stores data about the entire network graph.\nThe proposed method of crawling the vicinity of a target user at recommendation time (see~\\cref{sec:evaluation}) comes with a high overhead in network traffic and is not suitable for real-time systems that have to support large amounts of users.\nIn addition to that, the method is sensitive to the size of the vicinity.\nWe expect that a larger number of iterations yields a better picture of a user's vicinity, which in turn increases the quality of recommendations. However, an exploration of different parameter settings has been out of scope of this study.\nThe data collection issue is even more severe in the offline evaluation which requires large and representative samples of the entire network.\n\nTo reduce the overhead associated with crawling in an online setting, one might attempt to gradually construct a cached representation of the entire network graph.\nWhenever a recommendation is generated for a user, the vicinity is added to that graph. On subsequent recommendations, one might reuse parts of this network to avoid additional crawling.\nThis approach has two important issues that have to be considered. First, one has to address the question when parts of the network are considered to be out of date (i.e., when the cache expires). Second, and more importantly, such an approach seems to be in conflict with the intentions behind decentralization.\nBy constructing a database that aims to capture the entire network graph, one starts to centralize the data of a federated social network.\\looseness=-1\n\n\\section{Conclusion}\\label{sec:conclusion}\nUser recommendation algorithms commonly applied to centralized social media can be applied to incomplete data from federated social networks with the goal of developing an engaged community.\nWe showed that collaborative filtering-based recommenders outperform a topology-based recommender on a large unbiased sample of the federated social network Mastodon. The two recommenders outperform a random recommender by a large margin.\nA subsequent live user experiment on Mastodon using balanced interleaving shows that the two recommender approaches perform on par.\nAcquiring a sufficiently large snapshot of the network topology for offline recommendation proofed to be difficult and costly. Keeping the snapshot up-to-date needs constant re-sampling. Online recommendation was done by sampling the graph neighborhood for the current user.\n\nThere are several directions for future work. First, studying the extent to which incomplete data impacts the recommender performance may derive methods that are tailored towards federated social networks which operate with limited amounts of data.\nSecond, user recommendation algorithms in popular social media increasingly utilize user context information such as location data and interests.\nIt remains unclear how such data can be effectively acquired and utilized in federated social networks while preserving privacy.\nThird, BM25 might not be the best ranking function for the presented recommender approach, and it should be compared to functions that also use popularity-based scoring.\nFinally, one may investigate how decentralized communication protocols such as ActivityPub can be extended to support community building algorithms while maintaining the notion of decentralized network data.\\looseness=-1\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{s1:introduction}\n\nThe Advanced LIGO (aLIGO) observatories completed their first observing run ``O1''\nearly-2016, operating at a factor of $3-4$ higher gravitational-wave\n(GW) strain sensitivity than their first-generation \ncounterparts~\\cite{Shoemaker2009}.\nDuring O1, they made the first direct observation of gravitational \nwaves~\\cite{LIGOVirgo2016a}. Emitted by a pair of coalescing black holes, these\nwaves heralded an era of observational GW astrophysics as they traveled \nthrough Earth.\nTowards the end of this decade, we expect aLIGO to reach its design sensitivity.\nIn addition to the US-based efforts, we also expect the French-Italian detector Advanced\nVirgo~\\cite{aVIRGO,aVirgo2}, Japanese detctor KAGRA~\\cite{kagra,Somiya:2011np},\nand LIGO-India~\\cite{2013IJMPD..2241010U} to begin observing at comparable\nsensitivities within a few years. With a global network of sensitive GW\n observatories, we can expect GW astronomy to face significant developments over the\ncoming years.\n\n\nCoalescing compact binaries of stellar-mass black holes (BH) and\/or\nneutron stars (NS) are the primary targets for the second\ngeneration GW detectors~\\cite{Timmes:1995kp,Fryer:1999mi,RevModPhys.74.1015,\n2010ApJ...714.1217B,2010ApJ...715L.138B,Dominik:2014yma,Belczynski:2006zi,\n2012ApJ...749...91F,\nWex:1998wt,1991ApJ...379L..17N,Mandel:2015spa,Abbott:2016nhf}.\nA binary system of black holes was recently observed by \naLIGO~\\cite{LIGOVirgo2016a}. Previously, stellar-mass black holes had only \nbeen observed by inference in mixed binaries with stellar companion (through\nelectromagnetic observations of the companion)~\\cite{Lewin2010,\nRemillard:2006fc,Fragos:2010tm}.\nNeutron stars, on the other hand, have had numerous sightings. Thousands of\nelectromagnetically emitting neutron stars, or pulsars, have been \ndocumented~\\cite{Manchester:2004bp},\nin varied situations: as radio pulsars~\\cite{Lattimer:2012nd,Manchester:2004bp},\nin binary systems with a stellar companion~\\cite{1971ApJ...169L..23M,\nBond:2002eh,Lattimer:2012nd,Manchester:2004bp},\nand in binary neutron stars (BNS)~\\cite{Hulse:1975uf,Taylor:1982wi,\nWeisberg:2010zz,Lattimer:2012nd,Manchester:2004bp}.\nMixed binaries of black holes and neutron stars, is an astrophysically\ninteresting class of systems~\\cite{Wex:1998wt,\n1991ApJ...379L..17N,Janka1999,Fryer:2015jpa}, that has not yet been detected.\nWe expect to observe $\\mathcal{O}(10)$ mixed binaries per year with\naLIGO~\\cite{Abadie:2010cf}.\n\n\n\n\n\nNSBH binaries are of interest for multiple reasons. For instance,\nthey have been long associated with (as possible progenitors of) short\nGamma-ray Bursts (SGRBs)~\\cite{eichler:89,1992ApJ...395L..83N,moch:93,Barthelmy:2005bx,\n2005Natur.437..845F,2005Natur.437..851G,Shibata:2005mz,Paschalidis2014,\nTanvir:2013}. Depending on their equation of state (EoS), NSs can get disrupted by\nthe tidal field of their companion BHs. Once disrupted, most of the NS\nmaterial falls into the hole over an $\\mathcal{O}(1$ms$)$ time-scale,\nwith the rest partly getting ejected as unbound material\nand partly forming an accretion disk around the BH.\nThis short lived ($0.1-1s$) disk-BH system is hypothesized to drive SGRBs\nthrough the production of relativistic jets~\\cite{Foucart:2015a,\nLovelace:2013vma,Deaton2013,Foucart2012,Shibata:2005mz,Paschalidis2014}.\nHowever, whether or not such a system forms depends also on the nature of\nthe BH. Massive BHs (with $m_\\mathrm{BH}\\gtrsim 12M_\\odot$), as well as BHs with\nlarge retrograde spins, tend to swallow the NS whole without forming a\ndisk~\\cite{Foucart:2013psa}.\nOn the other hand, {\\it low-mass} BHs with $m_\\mathrm{BH}\\in[3M_\\odot, 12M_\\odot]$\\footnote{\nThe upper limit on BH mass that allows for NS disruption may very well be\nhigher, depending strongly on the magnitude of BH spin~\\cite{Foucart:2014nda}.},\ncan disrupt their companion NSs much before merger, forming long-sustained disks\nthat are required to sustain SGRBs~\\cite{Shibata:2007zm,2010PhRvD..81f4026F,\nLovelace:2013vma,Foucart:2014nda,Kawaguchi:2015}.\nA {\\it coincident} detection of both GWs and gamma-rays from an NSBH merger,\nwill provide us with a unique opportunity to confirm this hypothesized link\nbetween NSBH mergers and GRBs~\\cite{Abbott:2016wya}.\n\n\n\n\n\nAnother question that compact object mergers can help answer is `what is the \nnature of matter at nuclear densities supported by NSs'?\nA large fraction of past work aimed at measuring NS matter effects from GW\nsignals has consisted of inquiries about BNSs~\\cite{Lee1999a,Lee1999b,Lee2000,\noechslin:07,Read:2008iy,Markakis:2010mp,Markakis:2011vd,stergioulas:11,\nEast:2011xa,Lackey2014,Wade:2014vqa,Bauswein:2014qla}. In this paper, we will\ninstead focus on NSBHs.\nDuring the course of early inspiral, the tidal field of the BH produces a\ndeformation in its companion NS. The quadrupolar moment of the star associated\nwith this deformation also depends on its material properties, through an EoS-dependent\ntidal deformability parameter $\\Lambda_\\mathrm{NS}$. This induced quadrupolar moment\nchanges over the orbital time-scale, resulting in the emission of GWs in {\\it\ncoherence} with the orbital waves.\nThese waves draw more energy from the orbit and increase the inspiral rate (as\ncompared to an equivalent BBH)~\\cite{Flanagan2008}.\nCloser to merger, the strong tidal field of the BH can disrupt the NS. The\nquadrupolar moment of the disrupted binary system falls monotonically over a\nmillisecond time-scale~\\cite{Kyutoku:2010zd,Lackey:2013axa,Lovelace:2013vma,\nFoucart:2015a,Pannarale:2015jia}, resulting in the damping of GW amplitude.\nThis penultimate stage also depends strongly on the internal structure and energy\ntransport mechanism of the NS, and carries the strongest tidal signature in the\nGW spectrum~\\cite{Foucart:2014nda,Deaton2013}.\n\n\n\n\nGravitational waves emitted by coalescing NSBH binaries carry subtle hints of\nthe NS EoS from inspiral through to merger. During early inspiral, the tidal\ndephasing is relatively weak and has a frequency dependence equivalent to a\n$5^{th}$ Post-Newtonian (PN) order effect~\\cite{Vines2011}. Closer to merger,\na disruptive fate of the NS\ncan result in a strong suppression of GW emission above a cut-off \nfrequency~\\cite{Pannarale:2015jia}. Some past studies of tidal measurements\nwith NSBH binaries have used PN inspiral-only waveforms~\\cite{Maselli:2013rza}.\nIn doing so, however, they ignore (i) the merger signal which could contain significant\ninformation for NSBHs, and (ii) the errors due to unknown vaccum terms in PN \nwaveforms, which could dominate over the tidal terms themselves~\\cite{Barkett2015,\nYagi:2014}.\nSome other studies that account for merger effects via the use of complete\nnumerical simulations~\\cite{Foucart:2013psa}, are limited in the binary\nparameter space they sample.\nOthers, that do the same through the use of phenomenological waveform\nmodels~\\cite{Lackey2011,Lackey:2013axa} use the Fisher matrix to estimate\n$\\Lambda_\\mathrm{NS}$ measurement errors. Fisher matrix estimates may become\nunreliable at realistic signal-to-noise ratios (SNR)~\\cite{Vallisneri:2007ev},\nsuch as those as we might expect in the upcoming observing runs of GW\ndetectors~\\cite{Abadie:2010cf}, and we improve such studies with a\nfully Bayesian treatment of the problem here.\n\n\n\n\n\nIn this paper we study the measurability of neutron star's tidal deformability\nfrom realistic binaries of {\\it low}-mass BHs and NSs by aLIGO. We also probe\nhow tidal effects affect the estimation of other binary parameters for the same\nclass of systems. This study improves upon previous work in the following ways.\nFirst, we include tidal effects during inspiral and merger in a consistent\nway, by using the waveform model of Lackey {\\it et al.}~\\cite{Lackey:2013axa}\n(abbreviated henceforth to ``LEA'').\nSecond, we include the effect of black hole spin on tidal GW signals, in\naddition to the effect of BH mass, tidal deformability of the NS, and the SNR.\nThird, we perform a complete Bayesian analysis, instead of using the Fisher matrix\napproximation.\nAnd fourth, we explore how our measurement errors decrease as we gain information\nfrom multiple (realistic) events.\n\n\nWe now outline the main questions and results discussed in this paper.\nFirst, we probe the effect of ignoring tidal effects in\nthe recovery of non-tidal binary parameters, such as \ncomponent masses and spins. This is the case for current and planned aLIGO\nefforts.\nTo do so, we first use the enhanced-LEA (or ``LEA+'', see Sec.~\\ref{s2:waveforms})\nmodel to generate a set of realistic signals;\nand then use non-tidal (BBH) waveform filters to estimate the underlying\nbinary masses and spins with a Markov-chain Monte Carlo. Here and throughout,\nwe use the zero-detuning high-power design sensitivity curve~\\cite{Shoemaker2009}\nto characterize the expected detector noise.\nWe find that, for individual events, ignoring tidal effects will affect mass\nand spin-estimation only marginally; only for very loud signals (SNRs $\\gtrsim 30$)\nwill the systematic biases be large enough to exceed the underlying\nstatistical uncertainty. Furthermore, detection searches can ignore tidal\neffects without loss of sensitivity.\n\n\n\nSecond, we study the ability of aLIGO to constrain neutron star tidal \ndeformability with a single observation of an NSBH merger. For this, we\nuse the same setup for signal waveforms as before, but replace the filter\ntemplate model with one that includes tidal effects from inspiral\nthrough to merger (i.e. LEA+)~\\cite{Lackey:2013axa}. For most binaries with\nBH masses outside of the mass-gap $(2-5M_\\odot)$~\\cite{Bailyn:1997xt,\nKalogera:1996ci,Kreidberg:2012,Littenberg:2015tpa} and\/or realistic signal-to-noise\nratios (SNR), we find it difficult to put better than a factor of $2$ bound\non $\\Lambda_\\mathrm{NS}$ with a single observation. As we can see from\nFig.~\\ref{fig:TT_LambdaCIWidths90_0_Lambda_SNR}, it is only at SNRs \n$\\rho\\gtrsim 20-30$ (under otherwise favorable circumstances, such as a stiff\nequation of state) that we are able\nto bring this down to a $\\pm 75\\%$ bound on $\\Lambda_\\mathrm{NS}$. For signals louder\nthan $\\rho =30$, we can constrain $\\Lambda_\\mathrm{NS}$ to a much more meaningful degree\n(within $\\pm 50\\%$ of its true value).\nWhile this is discouraging at first, we turn to ask: what if we combine\ninformation from a population of low-SNR observations?\n\n\n\n\nThe EoS of matter at nuclear densities is believed to be universal among all\nneutron stars. The Tolman-Oppenheimer-Volkoff equation~\\cite{Tolman:1939jz,\nOppenheimer:1939ne,1934PNAS...20..169T}\nwould then predict that NS properties satisfy a universal relationship\nbetween $\\Lambda_\\mathrm{NS}$ and $m_\\mathrm{NS}$. As the final part of\nthis paper, we combine information from multiple observations of realistic NSBH\nsystems and perform a fully-Bayesian analysis of how our estimation of\n$\\Lambda_\\mathrm{NS}$ changes as we accumulate detections. This is similar to an earlier\nstudy~\\cite{DelPozzo:13} aimed at binary neutron stars.\nWe restrict ourselves to a population of NSs with masses clustered very tightly\naround $1.35M_\\odot$ (with a negligible variance), and negligible spins. We\nsample different nuclear EoSs by sampling entire populations fixing different\nvalues for the NS tidal deformability.\nFor all populations, we take source locations to be uniformly distributed in\nspatial volume, and source orientations to be uniform on the $2-$sphere. To\nsummarize, we find the following:\n(a) Our median estimate for $\\Lambda_\\mathrm{NS}$ starts out prior dominated, but \nconverges to within $10\\%$ of the true value within $10-20$ detections.\n(b) Measurement uncertainties for $\\Lambda_\\mathrm{NS}$, on the other hand, depend on\n$\\Lambda_\\mathrm{NS}$ itself. We find that for hard equations of state (with \n$\\Lambda_\\mathrm{NS}\\geq 1000$), $10-20$ observations are sufficient to constrain\n$\\Lambda_\\mathrm{NS}$ within $\\pm 50\\%$. For softer equations of state, the same level\nof certainty would require substantially more ($25-40$) observations.\n(c) Further, if the astrophysical ``mass-gap''~\\cite{Bailyn:1997xt,\nKalogera:1996ci,Kreidberg:2012,Littenberg:2015tpa} is real, we find that $20-50\\%$\nadditional observations would be required to attain the same measurement\naccuracy as above. (d) Putting tighter constraints on the $\\Lambda_\\mathrm{NS}$ of a\npopulation would require $50+$ NSBH observations, in any scenario.\nAnd, (e) it is the loudest $5-10$ events that will furnish the bulk\nof tidal information, and not the combination of a large number of \nlow-SNR events.\nAll of the above is possible within a few years of design\naLIGO operation~\\cite{Abadie:2010cfa}.\n\n\n\n\nIn this paper, we restrict our parameter space to span mass-ratios\n$q:=m_\\mathrm{BH}\/m_\\mathrm{NS}\\in[2,5]$, dimensionless BH spin (aligned with orbit)\n$\\chi_\\mathrm{BH}\\in[-0.5, +0.75]$, and dimensionless NS tidal deformability \n$\\Lambda_\\mathrm{NS}:= G\\left(\\frac{c^2}{G m_\\mathrm{NS}}\\right)^5\\lambda \\in[500, 2000]$.\nThese ranges are governed by the calibration of the LEA+ model which we use as\nfilters. \nMost of the disruptive NSBH simulations that LEA+ has been calibrated to\ninvolve $1.35M_\\odot$ NSs, and it is unclear how reliable the model is \nfor different NS masses~\\cite{Lackey:2013axa,Pannarale:2015jka}. This\nmotivates us to conservatively fix NS masses to $1.35M_\\odot$ in our \nsimulated signals (not templates). But, since the domain of calibration of\nLEA+ excludes NS spin completely, we fix $\\chi_\\mathrm{NS}=0$ in both signals as well as\nfilter templates. We expect the effect of ignoring NS mass and spin variations\nin our NSBH populations to be less severe than for\nBNSs~\\cite{Agathos:2015a}, considering the higher mass-ratios of NSBHs.\nThe accuracy of our quantitative results depends on the reliability of LEA+,\nwhich is the only model of its kind in current literature. A more recent \nwork~\\cite{Pannarale:2015jka} improves upon the amplitude description of LEA+,\nbut needs to be augmented with a compatible phase model. Overall, we expect\nour broad conclusions here to hold despite modeling inaccuracies (with errors\nnot exceeding $\\mathcal{O}(10\\%)$~\\cite{Pannarale:2015jka}).\nFinally, our results apply to LIGO instruments at design sensitivity,\nwhich they are projected to attain by $2019$~\\cite{Shoemaker2009,\nAbbott:2016wya}.\n\n\nThe remainder of the paper is organized as follows. \nSec.~\\ref{s1:techniques} discusses data analysis techniques and resources \nused in this paper, such as the waveform model, and parameter estimation \nalgorithm.\nSec.~\\ref{s1:PEwithnoNS} discusses the consequences of ignoring tidal \neffects in parameter estimation waveform models.\nSec.~\\ref{s1:PEwithNS} discusses the measurability for the leading order\ntidal parameter $\\Lambda_\\mathrm{NS}$ at plausible SNR values.\nSec.~\\ref{s1:multiple_observations} discusses the improvement in our\nmeasurement of $\\Lambda_\\mathrm{NS}$ with successive (multiple) observations of\nNSBH mergers.\nFinally, in Sec.~\\ref{s1:discussion} we summarize our results and discuss\nfuture prospects with Advanced LIGO.\n\n\n\n\n\n\\section{Techniques}\\label{s1:techniques}\n\\begin{figure*}\n\\centering \n\\includegraphics[trim=20 18 18 18 0,clip=true,width=1.8\\columnwidth]{SingleSystemEta_q4_0_mc2_25_chi0_50}\\\\\n\\caption{{\\bf Illustrative posterior probability distributions for mass-ratio $\\eta$ at different SNR values:}\nWe show here probability distributions for mass ratio $\\eta$ as measured\nfor the same signal at different SNRs. The intrinsic parameters of the source\nare: $q = m_\\mathrm{BH}\/m_\\mathrm{NS} = 5.4M_\\odot\/1.35M_\\odot = 4$, $\\chi_\\mathrm{BH}=+0.5$, and $\\Lambda_\\mathrm{NS}=2000$;\nand the signal is injected at SNRs $\\rho=\\{20,30,50\\}$ (left to right). The templates\n{\\it ignore} tidal effects.\nIn each panel: the dashed red line marks the median value\n$\\eta^\\mathrm{Median}$, while the dashed green line show the true value\n$\\eta^\\mathrm{Injected}$. The darker shading shows\nthe recovered $90\\%$ credible interval for $\\eta$, $(\\Delta\\eta)^{90\\%}$.\nComparing systematic and statistical errors, we find that:\nat $\\rho=20$, $\\eta$ measurement is dominated by statistical\nerrors; at $\\rho=30$, the two become comparable; and \nfor louder signals ($\\rho\\simeq50$), the systematic errors dominate.\n}\n\\label{fig:SingleSystemEtaPDFvsSNR}\n\\end{figure*}\n\n\n\\subsection{Waveform Models}\\label{s2:waveforms}\n\n\n\nLackey {\\it et al.}~(LEA)~\\cite{Lackey:2013axa} developed a complete inspiral-merger\nwaveform model for disrupting NSBHs. Theirs is a frequency-domain\nphenomenological model that includes the effect of BH and NS masses and spins\n$\\{m_\\mathrm{BH}, \\chi_\\mathrm{BH}, m_\\mathrm{NS}\\}\\equiv\\vec{\\theta}$ and NS tidal deformability\n$\\Lambda_\\mathrm{NS}$. It was calibrated to a suite of $134$ numerical relativity (NR)\nsimulations of NSs inspiraling into spinning BHs, with\nNS masses ranging between $1.2M_\\odot\\leqm_\\mathrm{NS}\\leq 1.45M_\\odot$,\nmass-ratios $2\\leq q\\leq 5$, and BH spins $-0.5\\leq\\chi_\\mathrm{BH}\\leq+0.75$.\nThey also sample a total of $21$ two-parameter nuclear EoSs to cover the\nspectrum of NS deformability.\nThe GW strain $\\tilde{h}(f)$ per the LEA model can be written as\n\\begin{equation}\n \\tilde{h}_\\mathrm{NSBH}(f, \\vec{\\theta}, \\Lambda_\\mathrm{NS}) = \\tilde{h}_\\mathrm{BBH}(f, \\vec{\\theta})\\,A(f, \\vec{\\theta}, \\Lambda_\\mathrm{NS})\\,e^{\\mathrm{i} \\Delta\\Phi(f, \\vec{\\theta}, \\Lambda_\\mathrm{NS})},\n\\end{equation}\nwith NS spin $\\chi_\\mathrm{NS}=0$ identically. Here, $\\tilde{h}_\\mathrm{BBH}$ is\nan underlying BBH waveform model. In the original LEA model,\nthis was taken to be the SEOBNRv1\nmodel~\\cite{Taracchini:2012} of the Effective-one-body (EOB)\nfamily~\\cite{Buonanno99}. The factor $A(\\cdot)$ adjusts\nthe amplitude of the BBH model to match that of an NSBH merger of otherwise\nidentical parameters, with NS-matter effects parametrized by $\\Lambda_\\mathrm{NS}$.\nDuring early inspiral this term is set to\nunity, but is a sensitive function of $\\Lambda_\\mathrm{NS}$ close to merger. The term with\n$\\Delta\\Phi$ corrects the waveform phasing. During inspiral,\n$\\Delta\\Phi$ is set to the PN tidal phasing corrections,\nat the leading and next-to-leading orders~\\cite{Vines2011}; close to merger,\nadditional phenomenological terms are needed. Both $A$ and $\\Delta\\Phi$ are\ncalibrated to all $134$ available NR simulations.\n\n\nIn this paper we use LEA for our signal and template modeling, but switch the \nunderlying BBH model to SEOBNRv2 (and refer to it as enhanced-LEA or\n``LEA+'')~\\cite{Taracchini:2013rva}. We using the reduced-order\nfrequency-domain version of SEOBNRv2, which has the additional benefit of\nreducing computational cost~\\cite{Purrer:2015tud}. We expect this enhancement\nfrom LEA$\\rightarrow$LEA+ to make our conclusions more robust because: (a) the \nSEOBNRv2 model is more accurate~\\cite{Kumar:2015tha,Kumar:2016dhh}, and (b)\nthe differences between the two EOB models are caused by the\ninaccuracies of SEOBNRv1 during the {\\it inspiral} phase, many orbits before \nmerger~\\cite{Kumar:2015tha}.\nSince LEA only augments inspiral phasing with PN tidal terms, our\nchange in the underlying BBH model does not change LEA's construction, {\\it and}\nincreases the overall model accuracy during inspiral.\nFinally, we note that we approximate the full GW signal with its dominant\n$l=|m|=2$ modes, that are modeled by LEA+. For use in future LIGO science\nefforts, we have implemented the LEA+ model in the LIGO Algorithms\nLibrary~\\cite{LAL}.\n\n\\subsection{Bayesian methods}\\label{s2:bayesian}\n\n\n\nThe process of measuring systematic and statistical measurement errors\ninvolves simulating many artificial GW signals, and inferring source binary\nparameters from them using Bayesian statistics.\nWe start with generating a signal waveform, using the model LEA+, and injecting\nit in zero noise to obtain a stretch of data $d_n$. Source intrinsic parameters\n$\\vec{\\Theta}:=\\{m_\\mathrm{BH},m_\\mathrm{NS},\\chi_\\mathrm{BH},\\Lambda_\\mathrm{NS}\\}$ are reconstructed from this\ninjected signal. Extrinsic parameters $\\vec{\\theta}:=\\{t_c,\\phi_c\\}$ \nrepresenting the time of and phase at the arrival of signal are marginalized\nover numerically and analytically (respectively), while source location and\norientation parameters such as its luminosity distance, sky location, inclination\nand polarization angles are absorbed into a normalization, as describe later,\nand subsequently maximized over. This is justified because in this paper we\nconsider the {\\it single-detector case}. Using Bayes' theorem, the joint inferred\nprobability distribution of $\\vec{\\Theta}$ can be evaluated as\n\\begin{equation}\\label{eq:postprob}\n p(\\vec{\\Theta} | d_n, H) = \\dfrac{p(d_n|\\vec{\\Theta}, H)\\,p(\\vec{\\Theta} | H)}{p(d_n|H)}.\n\\end{equation}\nHere, $p(\\vec{\\Theta} | H)$ is the {\\it a priori} probability of binary parameters\n$\\vec{\\Theta}$\ntaking particular values, given $H$ - which denotes all our collective knowledge,\nexcept for expectations on binary parameters that enter\nour calculations explicitly. Throughout this paper, we impose priors that are\nuniform in individual component masses, BH spin, and the tidal deformability of\nthe NS. In addition, we restrict mass-ratios to $q\\geq 2$, as LEA+ is not\ncalibrated for $1\\leq q\\leq 2$. $p(d_n|\\vec{\\Theta}, H)$ is the {\\it likelihood}\nof obtaining the given stretch of data $d_n$ if we assume that a\nsignal parameterized by $\\vec{\\Theta}$ is buried in it, and is given by\n\\begin{equation}\\label{eq:likelihood}\n p(d_n| \\vec{\\Theta}, H) \\equiv \\mathcal{L}(\\vec{\\Theta}) = \\mathcal{N}\\, \\mathrm{exp}[-\\frac{1}{2} \\langle d_n - h | d_n - h\\rangle ],\n\\end{equation}\nwhere $h\\equiv h(\\vec{\\Theta})$ is a filter template with parameters \n$\\vec{\\Theta}$, $\\langle\\cdot|\\cdot\\rangle$ is a suitably defined\ndetector-noise weighted inner-product\\footnote{The inner product\n$\\langle\\cdot|\\cdot\\rangle$ is defined as\n\\begin{equation}\n\\langle a|b\\rangle \\equiv 4\\,\\mathrm{Re}\\left[\\int_0^\\infty \\dfrac{\\tilde{a}(f) \\tilde{b}(f)^*}{S_n(|f|)}\\,\\mathrm{d} f\\right],\n\\end{equation}\nwhere $\\tilde{a}(f)$ is the Fourier transform of the finite time series $a(t)$,\nand $S_n(|f|)$ is the one-sided amplitude spectrum of detector noise. In this\nwork, we use the zero-detuning high-power design sensitivity curve~\\cite{\nShoemaker2009} for Advanced LIGO, with $15$Hz as the lower frequency cutoff.},\nand $\\mathcal{N}$ is the normalization constant that absorbs source distance,\norientation and sky location parameters. As in Ref.~\\cite{Purrer:2015nkh} we\nuse a likelihood that is maximized over the template\nnorm, allowing us to ignore the extrinsic parameters that only enter in the\ntemplate norm through $\\mathcal{N}$. As a result, we only need to sample over\n$\\vec{\\Theta}$ (or $\\vec{\\Theta} - \\{\\Lambda_\\mathrm{NS}\\}$ in the case of non-tidal\ntemplates).\nThe denominator in Eq.~\\ref{eq:postprob} is the {\\it a priori} probability of finding\nthe particular signal in $d_n$ and we assume that each injected signal is as\nlikely as any other. From the joint probability distribution\n$p(\\vec{\\Theta} | d_n, H)$ so constructed, extracting the measured probability distribution\nfor a single parameter (say $\\alpha$) involves integrating\n\\begin{equation}\\label{eq:marginalize}\n p(\\alpha | d_n, H) = \\int\\mathrm{d} \\vec{\\Theta}_\\alpha\\, p(\\vec{\\Theta} | d_n, H),\n\\end{equation}\nwhere $\\vec{\\Theta}_\\alpha$ is the set of remaining parameters, i.e.\n$\\vec{\\Theta}_\\alpha:=\\vec{\\Theta} - \\{\\alpha\\}$.\n\nWe use the ensemble sampler Markov-chain Monte-Carlo algorithm implemented in\nthe {\\tt emcee} package~\\cite{emcee}, to sample the probability distribution \n$p(\\vec{\\Theta} | d_n, H)$. We run 100 independent chains, each of which is\nallowed to collect 100, 000 samples and combine samples from chains that have\na Gelman-Rubin statistic~\\cite{gelman1992} close to unity. This procedure yields\nabout 10,000 independent samples.\nOne simplification we make to mitigate computational cost is to set the\nfrequency sampling interval to $\\Delta f=0.4$~Hz, which we find to be\nsufficient for robust likelihoods calculations in zero noise~\\cite{Purrer:2015nkh}.\nWe integrate\nEq.~\\ref{eq:marginalize} to obtain marginalized probability distributions\nfor the NS tidal deformability parameter: $p(\\Lambda_\\mathrm{NS}|d_n,H)$. We will quote\nthe median value of this distribution as our {\\it measured} value for\n$\\Lambda_\\mathrm{NS}$, and the $90\\%$ credible intervals associated with the distribution\nas the statistical error-bars.\n\n\n\n\n\\section{How is PE affected if we ignore NS matter effects?}\\label{s1:PEwithnoNS}\n\\begin{figure*}\n\\centering \n\\includegraphics[trim=20 20 18 22 0,clip=true,width=1.95\\columnwidth]{TNMchirpBiasesOverCIWidths_CI90_0_Lambda_SNR30_70_linear}\n\\caption{{\\bf Ratio of systematic to statistical errors in measuring $\\mathcal{M}_c$, ignoring tidal effects:}\nWe show here the ratio of systematic and statistical\nmeasurement uncertainties for the binary chirp mass over the NSBH parameter \nspace. Each panel shows the same as a function of BH mass and spin. NS mass\nis fixed at $m_\\mathrm{NS}=1.35M_\\odot$, and its spin is set to zero. Down each column,\nwe can see the effect of the increasing tidal deformability of the NS at fixed\nSNR. Across each row, we can see the effect of increasing the signal strength\n(SNR), with the tidal deformability of the NS fixed. We show dashed contours\nfor $\\mathtt{R}_{\\mathcal{M}_c}=10\\%, 25\\%, 50\\%\\cdots$, with interleaving filled color\nlevels separated by $5\\%$.\nFor BBHs, the statistical errors dominate systematic ones for contemporary\nwaveform models~\\cite{Inprerp-LVC-WaveModels:2016,Kumar:2016dhh}. We find that\nits not much different for NSBH binaries, until we get to very high SNRs\n$\\rho\\gtrsim 70$.\n}\n\\label{fig:TN_chirpMassBias_vs_Lambda_SNR}\n\\end{figure*}\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[trim=20 20 18 21 0,clip=true,width=1.95\\columnwidth]{TNEtaBiasesOverCIWidths_CI90_0_Lambda_SNR20_50_linear}\n\\caption{{\\bf Ratio of systematic to statistical errors in measuring $\\eta$, ignoring tidal effects:}\nThis figure is similar to Fig.~\\ref{fig:TN_chirpMassBias_vs_Lambda_SNR}\nwith the difference that here we show the ratio of systematic and statistical\nerror sources for the symmetric mass-ratio $\\eta$ and not chirp mass. We find\nthat for fairly loud GW signals, at $\\rho\\simeq 50$, not including the\neffects of tidal deformation of the NS on GW emission can become the dominant\nsource of error for astrophysical searches with Advanced LIGO. However, for\nquieter signals with $\\rho\\leq 30$, it will have a negligible effect on the\nmeasurement of $\\eta$. We remind the reader that the SNRs here are always\nsingle detector values.\n}\n\\label{fig:TN_EtaBias_vs_Lambda_SNR}\n\\end{figure*}\n\\begin{figure*}\n\\centering\n\\includegraphics[trim=20 20 18 20 0,clip=true,width=1.95\\columnwidth]{TNChiBHBiasesOverCIWidths_CI90_0_Lambda_SNR_linear}\n\\caption{{\\bf Ratio of systematic to statistical errors in measuring $\\chi_\\mathrm{BH}$, ignoring tidal effects:}\nThis figure shows the ratio of the systematic and statistical\nmeasurement errors for BH spins $\\mathtt{R}_{\\chi_\\mathrm{BH}}$. Information is arranged identically\nto Fig.~\\ref{fig:TN_chirpMassBias_vs_Lambda_SNR}, \nand~\\ref{fig:TN_EtaBias_vs_Lambda_SNR}, with the level spacing of filled contours\nincreased to $15\\%$.\nSimilar to the case of mass parameters, we find that below $\\rho\\approx 30$,\nignoring tidal effects in templates introduces minor systematic effects,\nwhich remain sub-dominant to the statistical measurement uncertainties.\n}\n\\label{fig:TN_BHspinBias_vs_Lambda_SNR}\n\\end{figure*}\n\\begin{figure*}\n\\centering \n\\includegraphics[trim=10 20 8 18 0,clip=true,width=1.9\\columnwidth]{SingleSystemLambdaVary_q4_0_mc2_25_chi0_50_snr50}\n\\caption{{\\bf Illustrative posterior probability distributions for NS tidal\ndeformability $\\Lambda_\\mathrm{NS}$:}\nWe show here probability distributions recovered for the NS tidal\ndeformability parameter $\\Lambda_\\mathrm{NS}$ from three GW injections, with parameters:\n$q = m_\\mathrm{BH}\/m_\\mathrm{NS} = 5.4M_\\odot\/1.35M_\\odot = 4$, $\\chi_\\mathrm{BH}=+0.5$, and \n$\\Lambda_\\mathrm{NS}=\\{1000,1500,2000\\}$ from left to right. The injection SNR is fixed at\n$\\rho=50$. The templates {\\it include} tidal effects, with a prior $0\\leq\\Lambda_\\mathrm{NS}\\leq 4000$.\nIn each panel- the dashed red line marks the median value for\n$\\Lambda_\\mathrm{NS}$, and the dashed green line marks its {\\it true} value.\nThe darker shading shows the $90\\%$ credible interval, whose width\n$(\\Delta\\Lambda_\\mathrm{NS})^{90\\%}$ is a direct measure of our statistical uncertainty.\nBy comparing the measurement uncertainty for these three injections, we see\nthat $(\\Delta\\Lambda_\\mathrm{NS})^{90\\%}$ grows very slowly with $\\Lambda_\\mathrm{NS}$. Therefore,\nthe fractional measurement error - $(\\Delta\\Lambda_\\mathrm{NS})^{90\\%}\/\\Lambda_\\mathrm{NS}$ -\ndecreases monotonically as $\\Lambda_\\mathrm{NS}$ increases (with signal strength fixed).\n}\n\\label{fig:SingleSystemLambdaPDFvsSNR}\n\\end{figure*}\nPast (and future) efforts with Advanced LIGO have used (or plan to use) BBH\nwaveform templates to search for and characterize NSBH sources. In doing so,\nthey ignore the signature of NS tidal effects on the emitted GWs. In this\nsection we present a fully Bayesian analysis of the effect of this\nsimplification on the recovery of non-tidal parameters from NSBH signals.\n\n\nWe inject LEA+ NSBH signals into zero noise, and run an MCMC sampler on\nthem using equivalent BBH templates (same model, tidal terms $\\rightarrow 0$).\nWe fix $m_\\mathrm{NS}=1.35M_\\odot$ and $\\chi_\\mathrm{NS}=0$, and explore a range of NS\nequations of state via the single tidal deformability parameter\n$\\Lambda_\\mathrm{NS}\\in\\{500, 800, 1000, 1500, 2000\\}$. Our injections also span a\nrectangular grid in the BH parameter space, with vertices at\n$q\\in\\{2,3,4,5\\}$, i.e.\n$m_\\mathrm{BH}\\in\\{2.7M_\\odot, 4.05M_\\odot, 5.4M_\\odot, 6.75M_\\odot\\}$, and BH spins\n$\\chi_\\mathrm{BH}\\in\\{-0.5, 0, +0.5, +0.75\\}$. \nFinally, we sample all other source-related parameters, that determine the\nsignal strength but not character\\footnote{For aligned-spin signals and\naligned-spin templates both, we only consider the contribution of the dominant\n$l=|m|=2$ waveform multipoles. This approximation has the additional benefit\nof combining the dependence of the waveforms on inclination, polarization\nand sky location angles, as well as on distance, into the luminosity\nor {\\it effective} distance. This quantity only appears as an overall scaling\nfactor, and therefore only affects signal strength~\\cite{Sathyaprakash:2009xs}.\n}, by sampling the SNR $\\rho\\in\\{20,30,50,70\\}$. Our choice of injection\nparameters here\nis motivated by two factors: (i) previous studies of the signatures of NS tidal\neffects on gravitational waves~\\cite{FoucartEtAl:2011,Foucart:2013psa,\nFoucart:2014nda} (which suggest that necessary conditions for the observation\nof tidal effects with aLIGO include high SNRs and a low-mass spinning companion\nBH); and (ii) technical constraints of our chosen LEA+ model~\\cite{Lackey:2013axa}.\nAt design sensitivity, if we expect $0.2-300$ NSBH detections a \nyear~\\cite{Abadie:2010cfa}, we can expect to see $0.02-25$ \n{\\it disruptive}\\footnote{We assume here that BH mass values are {\\it uniformly}\nlikely from $2M_\\odot$ to $\\sim 35M_\\odot$~\\cite{LIGOVirgo2016a}, but NSs are\ndisrupted in NSBH mergers only if $q\\leq 6$ and $\\chi_\\mathrm{BH}\\geq 0$~\\cite{Foucart:2014nda,\nFoucart:2013psa}.\n} NSBH mergers a year, of which we will have $0.005-7$ observations with \n$\\rho\\geq 20$, and $0.002-3$ a year with $\\rho\\geq 30$.\nTherefore, our injection parameters span a physically interesting subset of NSBH\nbinaries, that is {\\it also} likely observable in the near future. \nFor our Bayesian priors, we choose uniform distributions for both component\nmasses and black hole spin: $m_\\mathrm{BH}\\in[1.2,25]M_\\odot$; $m_\\mathrm{NS}\\in[1.2,3]M_\\odot$;\nand $-0.75\\leq \\chi_\\mathrm{BH}\\leq +0.75$.\n\n\n\nThe effect of ignoring\ntidal corrections in templates will manifest as a systematic shift\nof recovered median parameter values away from what they would be if we had used\ntidal templates with identical priors. \nIn zero noise, we expect the probability distributions recovered using tidal\ntemplates to be multi-dimensional Gaussians with the maximum likelihood\nparameter values approaching their true values. If the priors are not\nrestrictive, we expect the recovered median to also converge to the true value.\nHowever,\nthe LEA model imposes significantly more restrictive priors (both mass-ratio\nand spin) than SEOBNRv2~\\cite{Taracchini:2013rva,Lackey:2013axa}, which shifts\nthe median value of parameters recovered using {\\it our} tidal templates away\nfrom their true value. If we use LEA+ priors for our non-tidal templates, it\nwould add a caveat to our original question 'can we estimate non-tidal NSBH\nparameters with equivalent BBH templates'. Instead, we approximate the median\ntidally recovered parameters by their true injected values, as one would expect\nto recover with an ideal model for tidally disruptive NSBH mergers. With this\ncaveat, we estimate {\\it systematic} measurement bias\/errors as the\ndifferences between median and {\\it injected} parameter values. \nAs an illustration, in Fig.~\\ref{fig:SingleSystemEtaPDFvsSNR} we show the\nrecovered probability distributions for binary mass ratio $\\eta$ for\nthree NSBH injections, with $\\rho=20$ (left), $30$ (middle), and $\\rho=50$\n(right), and other parameters held fixed ($m_\\mathrm{NS}=1.35M_\\odot$, $\\chi_\\mathrm{NS}=0$,\n$m_\\mathrm{BH}=5.4M_\\odot$, $\\chi_\\mathrm{BH}=+0.5$ and $\\Lambda_\\mathrm{NS}=2000$). In each panel, both\nthe {\\it true} and median values of $\\eta$ are marked, and we use\nthe shift between the red and green vertical lines as our estimate of systematic\nmeasurement errors. Darker shading in all panels marks $90\\%$ credible intervals,\nwhose width $(\\Delta\\eta)^{90.0\\%}$ we use as a direct measure of our\n{\\it statistical} measurement uncertainty\/error\\footnote{We generalize the\nnotation $(\\Delta X)^{90.0\\%}$ to mean the $90\\%$ credible interval width\nfor any measured source parameter $X$.}.\nFor the illustrated binary,\nwe see clearly that even when the signal is moderately loud, with $\\rho=20$,\nstatistical errors dominate over systematics for $\\eta$. As we turn up the SNR\nfurther, the two error sources become comparable at $\\rho\\sim 30$,\nand systematic errors dominate finally when $\\rho\\simeq 50$.\n\n\n\n\n\nCredible intervals $(\\Delta X)^{90\\%}$ showing the precision with which\n$X=\\{\\mathcal{M}_c,\\eta,\\chi_\\mathrm{BH}\\}$ can be measured, are presented in\nAppendix~\\ref{as1:nontidalerrors}.\nWe remind ourselves that this {\\it precision} is only meaningful so long as the\nmeasurement is {\\it accurate} to begin with. Therefore, we define $\\mathtt{R}_X$ as\nthe ratio between systematic and statistical errors associated with the\nmeasurement of parameter $X$,\n\\begin{equation}\\label{eq:arr}\n\\mathtt{R}_X = \\dfrac{(X^\\mathrm{Median} - X^\\mathrm{Injected})}{(\\Delta X)^{90\\%}},\n\\end{equation}\nin order to compare the relative magnitude of both. Only when\n$|\\mathtt{R}_X| \\ll 1$ can we ignore tidal effects in our templates\nwithout hampering the measurement of non-tidal parameters from NSBH signals.\nWhen $\\mathtt{R}_X$ approaches a few tens of percent of unity, we can begin to favor\ntidal templates for NSBH studies.\n\n\nWe start with calculating $\\mathtt{R}_{\\mathcal{M}_c}$ as a function of various source\nparameters and show it in Fig.~\\ref{fig:TN_chirpMassBias_vs_Lambda_SNR}.\n$\\mathcal{M}_c$ is the leading order\nmass combination that affects the GW strain emitted by compact binaries as they\nspiral in, and is therefore determined the most precisely. We notice\nimmediately that for $\\rho\\leq 30$ the systematics are well under control\nand we can obtain reliable chirp mass estimates for NSBH signals using BBH\ntemplates.\nFor louder and less likely SNRs ($\\rho\\simeq 50$), we find that\n$\\mathtt{R}_{\\mathcal{M}_c}$ can become comparable to unity, but only if the BH has\nprograde spin $\\chi_\\mathrm{BH}\\gtrsim 0.4$, {\\it and} the true NS tidal deformability\nis large enough, s.t. $\\Lambda_\\mathrm{NS} \\gtrsim 1000$.\nWe therefore conclude that only for very loud signals, with\n$\\rho\\gtrsim 50-70$, will the inclusion of tidal terms in template\nmodels improve $\\mathcal{M}_c$ estimation. For lower SNRs, inclusion of new\nphysical content in templates will instead get washed out by detector noise.\nIn addition, we also note that $\\mathtt{R}_{\\mathcal{M}_c}\\geq 0$ always,\ni.e. $\\mathcal{M}_c$ is always being over-estimated. This is to be expected since\nthe tidal deformation of the NS drains energy faster from the orbit during\ninspiral (as compared to the BBH case), and its disruption close to merger\nreduces GW signal power at high frequencies. Both of these effects make the\nresulting signal resemble a BBH signal of higher chirp (or total) mass, although\nwe expect the latter effect to be dominant~\\cite{Pannarale:2011pk}.\n\n\n\nNext, in Fig.~\\ref{fig:TN_EtaBias_vs_Lambda_SNR}, we show the ratio of\nmeasurement errors $\\mathtt{R}_\\eta$ for the symmetric mass-ratio.\nGoing through the figure from left to right, we find that for realistic\nSNRs ($\\rho\\leq 30$) the systematics remain below statistical errors for\n$\\eta$ measurement. The worst case is of the most deformable NSs\n($\\Lambda_\\mathrm{NS} = 2000$), but even for them systematics in $\\eta$ are $2\\times$\nsmaller than the statistical measurement errors.\nMoving to louder signals with $\\rho\\simeq 50$, we find that for binaries of\nfairly deformable NSs ($\\Lambda_\\mathrm{NS}\\gtrsim 1500$) and low-mass BHs\n($m_\\mathrm{BH}\\leq 5M_\\odot$) that have prograde spins ($\\chi_\\mathrm{BH}\\gtrsim +0.4$), our\nmeasurement of mass-ratio can be seriously compromised by ignoring tidal\nphysics in template models. \nThis pattern is continued at even higher SNRs, as we can see from\nFig.~\\ref{fig:TN_EtaBias_vs_Lambda_SNR}. We therefore conclude that, even if\nunder moderate restrictions on BH and NS parameters, $\\rho=30-50$ is loud enough\nto motivate the use of tidal templates in aLIGO data analyses.\nIn addition, we also notice that, unlike for $\\mathcal{M}_c$, the median value of\n$\\eta$ is always {\\it lower} than its true value, which is what we expect if we\nwant BBH templates to fit NSBHs that disrupt and merge at lower frequencies.\n\n\n\nMoving on from mass to spin parameters, we now consider the measurement of BH\nspin angular momentum $\\chi_\\mathrm{BH}$. The ratio of systematic and statistical errors\nfor $\\chi_\\mathrm{BH}$ are shown in\nFig.~\\ref{fig:TN_BHspinBias_vs_Lambda_SNR}. The presentation of information in this \nfigure is identical to that of Fig.~\\ref{fig:TN_chirpMassBias_vs_Lambda_SNR}\nand~\\ref{fig:TN_EtaBias_vs_Lambda_SNR}. A diverging colormap is used so that both \nextremes of the colorbar range point to large systematic biases, while its zero (or\nsmall) value lies in the middle.\nFor the lowest SNR considered ($\\rho=30$), $\\chi_\\mathrm{BH}$ bias is about $2\\times$\nsmaller than its statistical measurement uncertainty, and is therefore mostly\nnegligible. Both do become somewhat comparable, but only when we have the most\ndeformable NSs in orbit around low-mass BHs. \nAt higher SNRs $(\\rho\\simeq50-70)$, we find that the systematics in $\\chi_\\mathrm{BH}$\nmeasurement can dominate completely, especially for binaries containing\nmass-gap violating BHs and\/or deformable NSs with $\\Lambda_\\mathrm{NS}\\geq1000$.\nFrom Fig.~\\ref{fig:TN_BHspinBias_vs_Lambda_SNR} we additionally note that when\nthe source spin magnitudes approach the highest allowed, i.e. at both extremes\nof the $x$-axes, $\\chi_\\mathrm{BH}\\times\\mathtt{R}_{\\chi_\\mathrm{BH}}<0$. This is to be expected because\nthe median of the recovered posterior distributions for $\\chi_\\mathrm{BH}$ can only get\npushed inwards from the boundaries.\n\n\nSummarizing these results, we find that irrespective of system parameters,\nbelow a signal-to-noise ratio of $30$, our measurements of mass and spin\nparameters of astrophysical NSBH binaries will remain limited by the intrinsic\nuncertainty due to instrument noise, and do not depend on whether we include\ntidal effects in template models. However, when the signal-to-noise ratio\nexceeds $30$ the systematic bias in binary mass and spin measurements become\ncomparable to and can exceed the uncertainty due to noise. Of the different\nnon-tidal parameters considered, we find that the measurement of $\\eta$\ndegrades worst (in a relative-error sense) due to the use of BBH templates in\ndeciphering an NSBH signal. Of all the sub-categories, we find that tidal\ntemplates could especially help with the parameter estimation of astrophysical\n{\\it mass-gap violating} NSBH binaries,\n\n\n\n\n\n\n\n\\section{What do we gain by using templates that include NS matter effects?}\\label{s1:PEwithNS}\n\\begin{figure*}\n\\centering \n\\includegraphics[trim=35 21 15 21 0,clip=true,width=2.2\\columnwidth]{TTLambdaRawCIWidths90_0_Lambda_SNR}\n\\caption{{\\bf Statistical uncertainty in $\\Lambda_\\mathrm{NS}$ measurement:}\nHere we show the statistical uncertainty in the measurement of\n$\\Lambda_\\mathrm{NS}$. In each panel, the\nsame is shown as a function of the BH mass and spin, keeping $\\Lambda_\\mathrm{NS}$ and\ninjection's SNR $\\rho$ fixed (noted in the panel). Rows contain panels\nwith the same value of $\\Lambda_\\mathrm{NS}$, with $\\rho$ increasing from left to right.\nColumns contain panels with the same value of $\\rho$, with $\\Lambda_\\mathrm{NS}$ \nincreasing from top to bottom.\nContours at $(\\Delta\\Lambda_\\mathrm{NS})^{90\\%}=\\{50\\%, 75\\%, 100\\%, 150\\%, 200\\%\\}\\times\\Lambda_\\mathrm{NS}^\\mathrm{Injected}$ demarcate regions where we can constrain the\n$\\Lambda_\\mathrm{NS}$ parameter well (within a factor of two of the injected value).\nWe note that, as expected, the measurement accuracy for $\\Lambda_\\mathrm{NS}$ improves\nwith (i) increasing SNR, (ii) increasing $\\Lambda_\\mathrm{NS}$, (iii) increasing BH spin,\nand (iv) decreasing BH mass.\n}\n\\label{fig:TT_LambdaCIWidths90_0_Lambda_SNR}\n\\end{figure*}\n\\begin{figure}\n\\centering \n\\includegraphics[trim=10 10 0 10 0,clip=true,width=1.05\\columnwidth]{TTSNRThresholdFor100LambdaMeasurement_BHspin_BHmass_Lambda2000_0_CI90_0}\n\\caption{\nWe show here, as a function of BH mass and spin, the {\\it minimum} signal\nstrength (SNR) required to constrain $\\Lambda_\\mathrm{NS}$ within an interval of width\nequal to $100\\%$ of its true value, i.e. with $\\pm 50\\%$ error-bars. The NS mass\nis fixed at $1.35M_\\odot$, spin at zero, and $\\Lambda_\\mathrm{NS}=2000$.\nWe can see that, even in the most conducive circumstances with large aligned \n$\\chi_\\mathrm{BH}$ and a comparable mass BH, we can only constrain $\\Lambda_\\mathrm{NS}$ to better\nthan $\\pm 50\\%$ {\\it if} the SNR is $\\gtrsim 29$. In the era of design\nsensitivity LIGO instruments, we expect this to happen approximately once in a\nyear of observation~\\cite{Abadie:2010cfa}.\n}\n\\label{fig:TT_SNRThresholds_BHspin_BHmass_CI90_0}\n\\end{figure}\n\n\nIn the previous section, we showed that the effects of the tidal deformation of\nNSs by their companion BHs become discernible in\nthe GW spectrum under certain favorable conditions, including (a) BH mass is\nsufficiently small, (b) BH spin is positive aligned, i.e. $\\chi_\\mathrm{BH}\\gtrsim +0.4$,\n(c) the NS is not very compact, with $\\Lambda_\\mathrm{NS}\\gtrsim 1000$, and (d) the\nsource location and orientation are such that its GW SNR $\\gtrsim 30$.\nBoth condition (a) and (b) enhance the tidal distortion of the star and increase\nthe number of orbits the system goes through at small separation, where the\ndifferences between NSBH and BBH signals are maximal.\nConditions (a)-(c) also reduce the onset frequency of the disruption of the NS,\nallowing for it to happen earlier in the orbit. \nWe expect that these conditions are also the ones which should maximize the\nlikelihood of {\\it measuring} tidal effects in NSBH signals. Here,\nwe turn the question around to ask: under similarly favorable circumstances,\ncan we gain insights about the internal structure of neutron stars from GW\nobservations?\n\n\n\nIn this section, we calculate the accuracy with which we measure $\\Lambda_\\mathrm{NS}$ from\n{\\it single} GW observations. We sample the same set of disruptive NSBH mergers\nas in the previous section, i.e. those with $q=\\{2,3,4,5\\}$,\n$\\chi_\\mathrm{BH}=\\{-0.5,0,+0.5,+0.75\\}$, and $\\Lambda_\\mathrm{NS}=\\{500, 800, 1000, 1500, 2000\\}$;\nfixing the NS mass $m_\\mathrm{NS}=1.35 M_\\odot$ and $\\chi_\\mathrm{NS}=0$. For each unique\ncombination of these parameters, we inject LEA+ signals into zero noise and\nperform a fully Bayesian parameter estimation analysis of each with LEA+\ntemplates. Our priors on component masses and spins remain as in the\nprevious section, with mass-ratio additionally restricted to $2\\leq q\\leq 6$,\nand $\\Lambda_\\mathrm{NS}$ sampled uniformly from $[0,4000]$.\nAs an illustration of individual injections, we show the recovered probability\ndistribution for $\\Lambda_\\mathrm{NS}$ for three specific configurations in \nFig.~\\ref{fig:SingleSystemLambdaPDFvsSNR}. We fix\n$q = m_\\mathrm{BH} \/m_\\mathrm{NS} = 5.4M_\\odot\/1.35M_\\odot = 4$, with $\\chi_\\mathrm{BH}=+0.5$, and\nvary $\\Lambda_\\mathrm{NS}$ over $\\{1000, 1500, 2000\\}$ between the three panels.\nThe SNR is fixed at $\\rho=50$. The darker shaded regions mark the $90\\%$ credible\ninterval on $\\Lambda_\\mathrm{NS}$. We note that $\\Lambda_\\mathrm{NS}$ is estimated to within\n$\\pm 2000$ of its true value at this SNR. Another interesting thing to note \nis that while $(\\Delta\\Lambda_\\mathrm{NS})^{90\\%}$ slowly grows with $\\Lambda_\\mathrm{NS}$, the\nfractional uncertainty\n\\begin{equation}\n\\delta\\Lambda_\\mathrm{NS}^{90\\%}:= (\\Delta\\Lambda_\\mathrm{NS})^{90\\%}\/\\Lambda_\\mathrm{NS}\n\\end{equation}\ndecreases instead.\nFurther illustrations, showing the correlation between tidal and non-tidal\nparameters, are presented in Appendix~\\ref{as1:illustrations}.\nWe will continue here to focus on the measurement of $\\Lambda_\\mathrm{NS}$ itself.\n\n\n\n\nIn Fig.~\\ref{fig:TT_LambdaCIWidths90_0_Lambda_SNR} we show the main results of\nthis section. In each panel, as a function of black hole mass and spin, we show\nthe measured $90\\%$ credible interval widths $(\\Delta\\Lambda_\\mathrm{NS})^{90\\%}$. These\ncorrespond to the full width of the dark shaded regions in the illustrative\nFig.~\\ref{fig:SingleSystemLambdaPDFvsSNR}. The effect of increasing signal\nstrength can be seen as we go from left to right in each row. The effect of the\nNS tidal deformability parameter $\\Lambda_\\mathrm{NS}$ on its own measurability can be\nseen by comparing panels within each column, with the NS becoming more\ndeformable from top to bottom. \nA uniform pattern emerges in the left-most column, which corresponds to $\\rho=20$.\nWe find that at this signal strength, our measurement of $\\Lambda_\\mathrm{NS}$ is\ndominated by the width of our prior on it. The $90\\%$ credible intervals span\nthe entire allowed range for $\\Lambda_\\mathrm{NS}$, making a reasonable estimation of\n$\\Lambda_\\mathrm{NS}$ at $\\rho\\simeq20$ difficult.\nIncreasing the signal strength to $\\rho=30$ gives marginally better results,\nbringing down the statistical uncertainties to within $\\pm 75-100\\%$ of the\ntrue $\\Lambda_\\mathrm{NS}$ value~\\footnote{The symmetric error-bars of $\\pm\\mathrm{X}\\%$\ncorrespond to $\\delta\\lambdans^{90\\%} = 2\\mathrm{X}\\%$.}.\nIt is not until we reach an SNR as high as $\\rho\\gtrsim 50$, can we put\nmeaningful (i.e. $\\mathcal{O}(10\\%)$) constraints on $\\Lambda_\\mathrm{NS}$. For e.g.,\nwith a {\\it single} observation of a $q=4$ binary with $\\chi_\\mathrm{BH}\\geq 0.6$ and\n$\\rho = 50$~\\footnote{For an optimally oriented source with\n$q=4, m_\\mathrm{NS}=1.35M_\\odot, \\chi_\\mathrm{BH}=0.6$, an SNR of $\\rho = 50$ corresponds to\na luminosity distance of $\\approx 113$Mpc.}, we would be able to estimate \n$\\Lambda_\\mathrm{NS}$ to within $\\pm 40\\%$ of its true value (which is equivalent to\nmeasuring the ratio of NS radius to mass with an uncertainty of about\n$\\pm 10\\%$).\nThese results agree well with Sec.~\\ref{s1:PEwithnoNS}, and are consistent with\nFisher matrix estimates at high SNRs~\\cite{Lackey:2013axa}.\n\n\n\n\nAmongst other source parameters, BH mass and spin play a dominant role. A smaller\nBH with a larger spin always allows for a more precise measurement on $\\Lambda_\\mathrm{NS}$.\nWe can see this in the bottom right corner of each panel in\nFig.~\\ref{fig:TT_LambdaCIWidths90_0_Lambda_SNR}, which corresponds to low-mass BHs\nwith large spins, and is simultaneously the region of smallest measurement errors on $\\Lambda_\\mathrm{NS}$.\nThe actual deformability of the NS also plays an important role on its own\nmeasurability. For e.g., when $\\Lambda_\\mathrm{NS}\\leq 1000$, it is fairly difficult\nto meaningfully constrain $\\Lambda_\\mathrm{NS}$ without requiring the source to be\nclose ($\\approx 100$Mpc) with a GW SNR $\\rho\\gtrsim 50$. Quantifying this further,\nin Fig.~\\ref{fig:TT_SNRThresholds_BHspin_BHmass_CI90_0} we show the minimum\nsignal strength required to attain a certain level of credibility in our\n$\\Lambda_\\mathrm{NS}$ measurement, as a function of BH properties. The NS is allowed\nthe most favorable (hardest) EoS considered, with $\\Lambda_\\mathrm{NS}^\\mathrm{true}=2000$.\nWe first note that, even with the most favorable BH and NS properties, achieving\na $\\pm 50\\%$ measurement certainty on $\\Lambda_\\mathrm{NS}$ will require a GW SNR\n$\\rho\\gtrsim 30$. If we additionally restrict BH masses to lie outside of the so-called\nastrophysical mass-gap~\\cite{Bailyn:1997xt,Kalogera:1996ci,Kreidberg:2012,\nLittenberg:2015tpa}, we will simultaneously need to restrict BH spins\nto $\\chi_\\mathrm{BH}\\gtrsim +0.5$ to obtain the same measurement credibility at the same\nsource location.\n\n\nIt is interesting to note that the parameter ranges most favorable to\nthe measurability of $\\Lambda_\\mathrm{NS}$ are also those which produce\nrelatively more massive post-merger disks~\\cite{Foucart2012}. That is, \nthe subset of NSBHs that potentially produce SGRBs (using a sufficiently-large\ndisk mass as an indicator) would be the same subset most favorable\nfor measurement of tidal effects. Therefore the rate of SGRBs in the \nlocal universe (allowing for the fraction that are produced by NSBHs versus\nBNSs) would be an indicator of the rate of events most favorable for nuclear\nequation of state measurements.\n\n\nIn summary, with a single moderately loud ($\\rho\\lesssim 30$) GW signal from\na disruptive BHNS coalescence, we can constrain\nthe NS compactness parameter $\\Lambda_\\mathrm{NS}$ within $\\pm 100\\%$ of its true value.\nTo measure better with one observation, we will need a more fine-tuned source, with\n$\\rho\\geq 30$ and high BH spins, or $\\rho\\geq 50$.\nFinally, we note that these results are {\\it conservative}, and \nBHs with spins $\\chi_\\mathrm{BH} > 0.75$ will prove to be even more favorable laboratories\nfor $\\Lambda_\\mathrm{NS}$ measurement. However, we are presently unable to explore this case\nin quantitative detail due to waveform model restrictions~\\cite{Lackey:2013axa},\nwhich will also restrict our analyses of GW signals during the upcoming LIGO\nobserving runs.\n\n\n\n\\section{Combining observations: looking forward with Advanced LIGO}\\label{s1:multiple_observations}\n\\begin{figure}\n\\centering \n\\includegraphics[trim=18 18 18 10 0,clip=true,width=\\columnwidth]{pdfLambda_vs_N_L800.pdf}\\\\\n\\includegraphics[trim=18 18 18 10 0,clip=true,width=\\columnwidth]{FillBetweenErrorBarsLambda_vs_N_L800.pdf}\n\\caption{{\\bf Recovery of $\\Lambda_\\mathrm{NS}$ for an increasingly large population of BH-NS signals.}\n{\\it Top}: Posterior probability distributions for $\\Lambda_\\mathrm{NS}$ (colored curves), and\nassociated $90\\%$ credible intervals (grey vertical lines), shown for different number\nof accumulated observations N. Distributions are normalized to unit area. \n{\\it Bottom}: Measured median value of $\\Lambda_\\mathrm{NS}$ (as solid circles) and the\nassociated $90\\%$ credible intervals (as the vertical extent of filled region), shown as\na function of number of observations N. Solid horizontal line indicates the true value of\n$\\Lambda_\\mathrm{NS}=800$. Dashed and dotted horizontal lines (a pair for each line-style) demarcate\n$\\pm 25\\%$ and $\\pm 50\\%$ error bounds.\n}\n\\label{fig:TT_Lambda_vs_N_L800_CI90_0}\n\\end{figure}\n\\begin{figure}\n\\centering \n\\includegraphics[width=1.05\\columnwidth,trim=1cm 0 0 0]{FillBetweenRelErrorBarsLambda_vs_NShifted_AllLambda.pdf}\n\\caption{{\\bf Improvement in $\\Lambda_\\mathrm{NS}$ measurement accuracy for different NS EoS:}\nIn this figure, the filled regions show how our measurement of $\\Lambda_\\mathrm{NS}$\nimproves as the number of observed events ($N$, shown on $x$-axis) increases.\nEach color corresponds to an independent population with its true value of\n$\\Lambda_\\mathrm{NS}$ given in the legend. For each population, we show the median \n$\\Lambda_\\mathrm{NS}$ value (as filled circles), as well as the associated\n$90\\%$ credible intervals for the measurement (as the vertical extent of the\nfilled region about the median), as functions of $N$.\n}\n\\label{fig:TT_Lambda_vs_N_CI90_0}\n\\end{figure}\n\\begin{figure*}\n\\centering\n\\includegraphics[trim=1cm 0 0 0, width=1.025\\columnwidth]{LambdaMedian_vs_N_AllPopulation}\n\\includegraphics[trim=0 0 1cm 0, width=1.025\\columnwidth]{LambdaMedian_vs_N_AstroPopulation}\\\\\n\\includegraphics[trim=1cm 0 0 0, width=1.025\\columnwidth]{LambdaMedian90pc_vs_N_AllPopulation}\n\\includegraphics[trim=0 0 1cm 0, width=1.025\\columnwidth]{LambdaMedian90pc_vs_N_AstroPopulation}\\\\\n\\caption{\n{\\bf No Mass-Gap}, {\\it top left}: The top figure shows the median value of the recovered\nprobability distribution for $\\Lambda_\\mathrm{NS}$, as a function of the number of observed \nevents $N$. There are four ensembles of curves,\ncorresponding to $\\Lambda_\\mathrm{NS}=\\{500,1000,1500,2000\\}$, with a hundred\nindependent population draws within each ensemble. One curve in each ensemble\nis highlighted in color, representing the realizations already plotted in\nFig.~\\ref{fig:TT_Lambda_vs_N_CI90_0}.\nIn the same color we show $\\pm 10\\%$ error-bounds on $\\Lambda_\\mathrm{NS}$ with\nhorizontal dash-dotted lines.\n{\\bf No Mass-Gap}, {\\it bottom left}: Here we show the interval of $\\Lambda_\\mathrm{NS}$ values within\nwhich the median $\\Lambda_\\mathrm{NS}$ lies for $90\\%$ of the populations in\neach ensemble shown in the top left panel.\nWe observe that within $10-25$ observations, the median of the measured \ncumulative probability distribution for $\\Lambda_\\mathrm{NS}$ converges to within $10\\%$\nof its true value.\n{\\bf Mass-Gap}, {\\it right column}: These panels are identical to their counterparts on the left,\nwith the only difference that the BH masses in each population are restricted\nto lie {\\it outside} the astrophysical mass-gap (i.e. paradigm B). The\ndifference that\nwe observe under this paradigm is that we need more ($30+$) events to achieve \nthe same ($10\\%$) measurement accuracy for populations with $\\Lambda_\\mathrm{NS}<1000$.\nFor more deformable neutron stars, $10-25$ events would suffice.\n}\n\\label{fig:TT_LambdaMedian_vs_N_AllInOne}\n\\end{figure*} \n\\begin{figure*}\n\\centering \n\\includegraphics[width=1.025\\columnwidth,trim=1cm 0 0 0]{LambdaCIWidths90pc_vs_N_AllPopulation}\n\\includegraphics[width=1.025\\columnwidth,trim=0 0 1cm 0]{LambdaCIWidths90pc_vs_N_AstroPopulation}\n\\caption{{\\bf No Mass-Gap} ({\\it left}): This panel shows the width of $\\Lambda_\\mathrm{NS}$ interval within\nwhich the $90\\%$ credible intervals for $\\Lambda_\\mathrm{NS}$ lie, for $90\\%$ of \nthe populations in each ensemble, as a function of the number of observed events\n$N$. Details of how this is calculated are given in the text.\nThe populations are sampled under paradigm A, which allows BH masses to\nfall within the astrophysical mass-gap.\nEach panel corresponds to a unique value of populations' $\\Lambda_\\mathrm{NS}$,\ndecreasing from $2000\\rightarrow 500$ as we go from top to bottom.\nOne curve in each ensemble is highlighted in color (thin lines), representing the \nrealizations already plotted in Fig.~\\ref{fig:TT_Lambda_vs_N_CI90_0}.\n{\\bf Mass-Gap} ({\\it right}): This panel shows populations drawn under paradigm B, which\nrespects the mass-gap.\nWe find that with approximately $25$ or so events, we begin to put\nstatistically meaningful constraints on $\\Lambda_\\mathrm{NS}$, restricting it to within\n$\\pm 50\\%$ of the true value. We can expect to achieve this with a few years\nof design aLIGO operation~\\cite{Abadie:2010cfa}. Further tightening of \n$\\Lambda_\\mathrm{NS}$ credible intervals will require $40+$ events.\n}\n\\label{fig:TT_LambdaError_vs_N_L500_2000_CI90_0_AllInOne}\n\\end{figure*}\n\\begin{figure*}\n\\centering \n\\includegraphics[width=1.025\\columnwidth,trim=1cm 0 0 0]{LambdaMedianCIWidths90pc_vs_N_AllPopulation_SNRSorted}\n\\includegraphics[width=1.025\\columnwidth,trim=0 0 1cm 0]{LambdaCIWidths90pc_vs_N_AstroPopulation_SNRSorted}\n\\caption{This figure is similar to Fig.~\\ref{fig:TT_LambdaError_vs_N_L500_2000_CI90_0_AllInOne},\nwith the only difference being that events in each population\nhave been sorted according to their signal strength (SNR), instead of\ntheir simulated chronology. We note that information about\nthe tidal deformability of neutron stars comes primarily \nfrom the loudest $5-10$ events, whether we allow BH masses\nin the mass gap (left panel) or restrict them to\n$m_\\mathrm{BH}\\geq 5M_\\odot$ (right panel). Left inset zooms\nin on the main figure for the first $15$ events. Right inset\nshows the actual (ensemble mean) SNR value for each event. We find that\nevents with $\\rho\\gtrsim 20-30$ provide the bulk of tidal information\nin our analysis.\n}\n\\label{fig:TT_LambdaError_vs_N_L500_2000_CI90_0_AllInOne_SNRSorted}\n\\end{figure*}\n\n\nIn the previous section, we showed that single observations of NSBH\ncoalescences at moderate SNRs have little information about the internal\nstructure of neutron stars that will be accessible to Advanced LIGO at its\ndesign sensitivity. We expect all neutron stars to share the same equation of\nstate, and hence the same $\\Lambda_\\mathrm{NS}(m_\\mathrm{NS})$. In addition, we know that the mass\ndistribution of (most) NSs that have not been spun up to millisecond periods\n(which are the ones we focus on in this paper, by setting $\\chi_\\mathrm{NS}\\approx 0$) is\nnarrowly peaked around $\\sim 1.35M_\\odot$~\\cite{Kiziltan2013}. Therefore,\ninformation from multiple NSBH observations can be combined to improve our\nestimation of $\\Lambda_\\mathrm{NS}$. We explore the same in this section within a fully\nBayesian framework. We refer the reader to Ref.~\\cite{Mandel:2009pc,Lackey2014,\nWade:2014vqa} for similar analyses of BNS inspirals.\n\n\n\n\nAn intuitive understanding of the problem is gained by considering first\nmultiple {\\it identical} sources with realistic but different SNRs. Let us consider the case\nof a population of optimally oriented binaries~\\footnote{An optimally oriented\nbinary is one which is located directly overhead the detector, with the \norbital angular momentum parallel to the line joining the detector to the\nsource. Such a configuration maximizes the observed GW signal strength in \nthe detector.}, distributed uniformly in spatial volume out\nto a maximum {\\it effective} distance~\\footnote{{\\it effective} distance $D$ \nis a combination of distance to the source, its orientation, and its sky\nlocation angles; and has a one-to-one correspondence with SNR for non-precessing\nsources. This is so because for such sources, their location and orientation\nremain constant over the timescales within which they sweep through\naLIGO's sensitive frequency band.}.\n$D^\\mathrm{max}$. $D^\\mathrm{max}$ is set by the minimum SNR \nthreshold $\\rho_\\mathrm{min}$ at which a source is considered\ndetectable~\\footnote{which\nwe take as $\\rho_\\mathrm{min}=10$ throughout.}. Next, we divide this volume into $I$\nconcentric shells, with radii $D_i$. If we have a measurement error\n$\\sigma_0$ for $\\Lambda_\\mathrm{NS}$, associated with a source located at $D=D_0$,\nthe same error for the same source located within the $i-$th shell would\nbe $\\sigma_i=\\sigma_0 \\dfrac{D_i}{D_0}$. Ref.~\\cite{Markakis:2010mp}\ncalculated that the combined error $\\sigma$ from $N$ independent\nmeasurements of $\\Lambda_\\mathrm{NS}$ in such a setting to be\n\\begin{align}\\label{eq:1oversigma}\n\\frac{1}{\\sigma^2} =& \\sum_{i=1}^I \\frac{N_i}{\\sigma_i^2} = \\left(\\frac{D_0}{\\sigma_0}\\right)^2 \\sum_{i=1}^I\\frac{N_i}{D_i^2}\\\\ \\nonumber =& \\left(\\frac{D_0}{\\sigma_0}\\right)^2 \\int_0^{D^\\mathrm{max}} \\dfrac{4\\pi D^2 n}{D^2}\\mathrm{d} D = \\left(\\frac{D_0}{\\sigma_0}\\right)^2 \\dfrac{3N}{(D^\\mathrm{max})^2},\n\\end{align}\nwhere $N_i$ is the number of sources within the $i-$th shell (s.t.\n$N:=\\sum N_i$), and $n$ is the number density of sources in volume.\nThe root-mean-square (RMS) averaged measurement error from $N$ sources is \nthen~\\cite{Markakis:2010mp}\n\\begin{equation}\\label{eq:rmsSigmaIdenticalSources}\n \\sigma_{avg} := \\frac{1}{\\sqrt{1\/\\sigma^{2}}} = \\frac{\\sigma_0}{D_0} D^\\mathrm{max} \\frac{1}{\\sqrt{3 N}},\n\\end{equation}\ngiven a fiducial pair $(\\sigma_0, D_0)$. It is straightforward to deduce from\nEq.~\\ref{eq:rmsSigmaIdenticalSources} that measurement uncertainty scales as \n$1\/\\sqrt{N}$, and the uncertainty afforded by a single observation with a high\nSNR $\\rho_c$ can be attained with $N = \\rho_c^2\/300$ realistic observations\nthat have $\\rho\\geq\\rho_\\mathrm{min}$. E.g., to get to the\nlevel of certainty afforded by a single observation with $\\rho=70$, we would\nneed $49\/3\\approx 16-17$ realistic (low SNR) detections.\n\nWhile we discussed Eq.~\\ref{eq:rmsSigmaIdenticalSources} for a population\nof optimally oriented sources, it is valid for a more general population\ndistributed uniformly in effective volume~\\cite{Markakis:2010mp}\n($\\propto D^3$).\nHowever, it \nstill only applies to sources with identical masses and spins, and we \novercome this limitation by performing a fully Bayesian analysis next.\n\n\n\\textbf{Astrophysical source population: }\\label{s2:astro_multiple}\nImagine that we have $N$ stretches of data, $d_1, d_2, \\cdots, d_N$, each \ncontaining a single signal emitted by an NSBH binary. Each of these signals can\nbe characterized by the non-tidal source parameters\n$\\vec{\\theta} := \\{m_\\mathrm{BH}, m_\\mathrm{NS}, \\chi_\\mathrm{BH}, \\chi_\\mathrm{NS}, \\vec{\\alpha}\\}$,\nand $\\{\\Lambda_\\mathrm{NS}\\}$, where $\\vec{\\alpha}$ contains extrinsic parameters,\nsuch as source distance, inclination, and sky location angles.\nAs before, let $H$ denote all of our collective prior knowledge; for instance,\n$H$ includes our assumption that all NSs in a single population have the same\ndeformability parameter $\\Lambda_\\mathrm{NS}$, and that its cumulative measurement is\ntherefore possible.\nThe probability distribution for $\\Lambda_\\mathrm{NS}$, given $N$ unique and\nindependent events, is\n\\begin{eqnarray}\n p(\\Lambda_\\mathrm{NS} |&& \\hspace{-4mm}d_1, d_2, \\cdots, d_N, H)\\hspace{50mm}\\nonumber\\\\ &=& \\dfrac{p(d_1,d_2,\\cdots,d_N |\\Lambda_\\mathrm{NS} , H)\\,p(\\Lambda_\\mathrm{NS}|H)}{\\int p(\\Lambda_\\mathrm{NS} |H) p(d_1,d_2,\\cdots,d_N |\\Lambda_\\mathrm{NS} , H)\\mathrm{d}\\Lambda_\\mathrm{NS}},\\label{eq:p11}\\\\\n &=& \\dfrac{p(\\Lambda_\\mathrm{NS}|H) \\prod_{i} p(d_i|\\Lambda_\\mathrm{NS}, H)}{\\int p(\\Lambda_\\mathrm{NS} ) p(d_1,d_2,\\cdots,d_N |\\Lambda_\\mathrm{NS} , H)\\mathrm{d}\\Lambda_\\mathrm{NS}},\\label{eq:p12} \\\\\n &=& \\dfrac{p(\\Lambda_\\mathrm{NS}|H) \\prod_{i} \\left( p(\\Lambda_\\mathrm{NS} |d_i, H)\\dfrac{p(d_i)}{p(\\Lambda_\\mathrm{NS}|H)} \\right)}{\\int\\, p(\\Lambda_\\mathrm{NS}|H )\\, p(d_1,d_2,\\cdots,d_N |\\Lambda_\\mathrm{NS} , H)\\mathrm{d}\\Lambda_\\mathrm{NS}}\\label{eq:p13};\n\\end{eqnarray}\nwhere Eq.~\\ref{eq:p11} and Eq.~\\ref{eq:p13} are application of Bayes' theorem,\nwhile Eq.~\\ref{eq:p12} comes from the mutual independence of all events.\nAssuming in addition that all events are {\\it equally likely}: \n$p(d_i) = p(d_j) = p(d)$, we get\n\\begin{eqnarray}\n p(&&\\hspace{-4mm}\\Lambda_\\mathrm{NS} | d_1, d_2, \\cdots, d_N, H)\\hspace{50mm}\\nonumber\\\\\n &=& p(\\Lambda_\\mathrm{NS})^{1-N}\\times \\dfrac{p(d)^N}{\\int p(\\Lambda_\\mathrm{NS}) p(d_1, d_2, \\cdots, d_N |\\Lambda_\\mathrm{NS}, H)\\mathrm{d}\\Lambda_\\mathrm{NS}}\\nonumber\\\\ &&\\hspace{3mm}\\times\\prod_i p(\\Lambda_\\mathrm{NS} |d_i, H)\\label{eq:p22},\n\\end{eqnarray}\nwhere the prior probability $p(\\Lambda_\\mathrm{NS}|H)$ is written $p(\\Lambda_\\mathrm{NS})$ for\nbrevity. {\\it A priori}, we assume that no particular value of $\\Lambda_\\mathrm{NS}$ is\npreferred over another within the range $[0, 4000]$, i.e.\n\\begin{equation}\\label{eq:lprior}\n p(\\Lambda_\\mathrm{NS} | H) = \\dfrac{1}{4000}\\,\\mathrm{Rect}\\left(\\frac{\\Lambda_\\mathrm{NS}-2000}{4000}\\right).\n\\end{equation}\nWith a uniform prior, the first two factors in Eq.~\\ref{eq:p22} can be\nabsorbed into a normalization factor $\\mathcal{N}$, simplifying it to\n\\begin{equation}\\label{eq:lambdaMultiple}\n p(\\Lambda_\\mathrm{NS} | d_1, d_2, \\cdots, d_N; H) = \\mathcal{N}\\prod_{i=1}^N p(\\Lambda_\\mathrm{NS} | d_i, H).\n\\end{equation}\n\nIn the second set of terms in Eq.~\\ref{eq:lambdaMultiple} (of the form \n$p(\\Lambda_\\mathrm{NS} | d_i, H)$), each is the probability distribution for $\\Lambda_\\mathrm{NS}$\ninferred {\\it a posteriori} from the \\textit{i}-th observation by marginalizing\n\\begin{equation}\\label{eq:margpost}\n p(\\Lambda_\\mathrm{NS} | d_i, H) = \\int\\, p(\\vec{\\theta}, \\Lambda_\\mathrm{NS} | d_i, H)\\, \\mathrm{d} \\vec{\\theta},\n\\end{equation}\nwhere $p(\\vec{\\theta}, \\Lambda_\\mathrm{NS} | d_i, H)$ is the inferred joint probability \ndistribution of all source parameters $\\vec{\\theta}\\cup\\{\\Lambda_\\mathrm{NS}\\}$ for the \n$i$-th event, as given by Eq.~\\ref{eq:postprob}. We note that \nFig.~\\ref{fig:SingleSystemLambdaPDFvsSNR} illustrates $p(\\Lambda_\\mathrm{NS} | d_i, H)$\nfor three individual events. By substituting\nEq.~\\ref{eq:lprior}-\\ref{eq:margpost} into Eq.~\\ref{eq:lambdaMultiple}, we\ncalculate the probability distribution for $\\Lambda_\\mathrm{NS}$ as measured using $N$\nindependent events.\n\n\n\n\nOur goal is to understand the improvement in our measurement of $\\Lambda_\\mathrm{NS}$\nwith the number of recorded events. To do so, we simulate a population~\\footnote{%\nA population here is an ordered set of events, and an event itself is the \nset of parameters describing one astrophysical NSBH binary.}\nof $N$ events, and quantify what we learn from each successive observation \nusing Eq.~\\ref{eq:lambdaMultiple}. This allows us to quantify how\nrapidly our median estimate for $\\Lambda_\\mathrm{NS}$ converges to the true value,\nand how rapidly our credible intervals for the same shrink, with increasing\n$N$. Finally, we generate and analyze an ensemble of populations in order to\naverage over the stochastic process of population generation itself.\n\n\nIn order to generate each population, the first step is to fix\nthe NS properties: (i) NS mass $m_\\mathrm{NS}=1.35M_\\odot$, (ii) NS spin $\\chi_\\mathrm{NS}=0$\nand (iii) NS tidal deformability $\\Lambda_\\mathrm{NS}=$ fixed value chosen from\n$\\{500,800,1000,1500,2000\\}$. Next, we generate events, by sampling BH mass\n(uniformly) from $m_\\mathrm{BH}\\in[3M_\\odot,6.75M_\\odot]$, BH spin (uniformly)\nfrom $\\chi_\\mathrm{BH}\\in[0, 1]$, orbital inclination from $\\iota\\in[0,\\pi]$, and \nsource location uniform in spatial volume\\footnote{with a minimum SNR \n$\\rho_\\mathrm{min}=10$}.\nWe restrict ourselves to positive aligned BH spins, since binaries with\nanti-aligned spins have very little information to add at realistic SNRs,\nas demonstrated in Fig.~\\ref{fig:TT_LambdaCIWidths90_0_Lambda_SNR}. This is\nto be taken into account when the number of observations is related to detector\noperation time. \nWe repeat this process till we have an ordered set of $N$ events.\nSince we want to analyze not just a single realization of an astrophysical\npopulation, but an ensemble of them, we make an additional approximation to\nmitigate computational cost. Complete Bayesian parameter estimation is\nperformed for a set of simulated signals whose parameters are the vertices\nof a regular hyper-cubic grid (henceforth ``G'') in the space of\n$\\{q\\}\\times\\{\\chi_\\mathrm{BH}\\}\\times\\{\\rho\\}$, with each sampled at $q=\\{2,3,4,5\\}$,\n$\\chi_\\mathrm{BH}=\\{-0.5,0,0.5,0.75\\}$, and $\\rho=\\{10,20,30,50,70\\}$.\nAll events in each population draw are substituted by their respective nearest\nneighbours on the grid G.\nOur chosen signal parameter distribution is different from some other studies\nin literature, which often sample from more astrophysically motivated population\ndistribution functions~\\cite{Mandel:2009pc}. We chose one that is sufficiently\nagnostic in absence of actual known NSBHs, and pragmatic enough for generating\npopulation ensembles.\n\n\n\n\nIn Fig.~\\ref{fig:TT_Lambda_vs_N_L800_CI90_0} we show illustrative results\nfor a single population with neutron star deformability $\\Lambda_\\mathrm{NS}=800$.\nIn the top panel, each curve shows the \nprobability distributions for $\\Lambda_\\mathrm{NS}$ as inferred from $N$ events, with $N$\nranging from $1-80$. We also mark the $90\\%$ credible intervals associated\nwith each of the probability distribution curves. The first few observations\ndo not have enough information to bound $\\Lambda_\\mathrm{NS}$ much more than\nour prior from Eq.~\\ref{eq:lprior} does. \nIn the bottom panel, we present information derived from the top panel.\nThe line-circle curve shows the measured median value from $N$ observations.\nThe pair of dashed (dotted) horizontal lines mark\n$\\pm25\\%$ ($\\pm50\\%$) error bars. At each $N$, the range spanned by the \nfilled region is the $90\\%$ credible interval deduced from the same \nevents. This figure somewhat quantifies the qualitative deductions we made\nfrom the left panel. We find that the median does track the true value quickly,\nreaching within its $10\\%$ with $10-15$ observations. This is as one expects of \ninjections in zero noise where random fluctuations are unable to shift the\nmedian away from the true value, so long as the measurement is not restricted\nby the prior. With the same information, our credible intervals also shrink to $\\pm 25\\%$.\nIn Fig.~\\ref{fig:TT_Lambda_vs_N_CI90_0} we show further results from four\nindependent populations for $\\Lambda_\\mathrm{NS}=\\{500,1000,1500,2000\\}$. As in the \nright panel of Fig.~\\ref{fig:TT_Lambda_vs_N_L800_CI90_0}, the line-circle curves\ntrack the median $\\Lambda_\\mathrm{NS}$, while the filled regions show\nthe associated $90\\%$ credible intervals. From the figure, we observe\nthe following: (i) the shrinkage of credible interval widths with increasing\n$N$ happens in a similar manner for each $\\Lambda_\\mathrm{NS}$, \nand (ii) it takes\napproximately $20$ events to distinguish definitively (with $90\\%$ credibility)\nbetween deformable NSs with $\\Lambda_\\mathrm{NS}=2000$ and compact NSs with \n$\\Lambda_\\mathrm{NS}=500$, or equivalently to distinguish between hard, moderate and soft\nnuclear equations of state. This is comparable to what has been found for\nbinary neutron stars~\\cite{DelPozzo:13,Chatziioannou:2015uea,Agathos:2015a}.\n\n\n\nSo far we have discussed individual realizations of NSBH populations. The \nunderlying stochasticity of the population generation process makes it\ndifficult to draw generalized inferences (from a single realization of an\nNSBH population) about the measurability of $\\Lambda_\\mathrm{NS}$. In order to mitigate\nthis, we discuss ensembles of population draws next. In\nFig.~\\ref{fig:TT_LambdaMedian_vs_N_AllInOne} we show the median\n$\\Lambda_\\mathrm{NS}$ as a function of the number of observed\nevents, for four population ensembles, with a hundred population draws\nin each ensemble. Lets focus on the {\\it top left} panel first. In it, we\nshow the median $\\Lambda_\\mathrm{NS}$ for all populations in four ensembles,\nwith true $\\Lambda_\\mathrm{NS}=\\{2000,1500,1000,500\\}$ from top to bottom.\nPopulations highlighted in color are simply those that we discussed in\nFig.~\\ref{fig:TT_Lambda_vs_N_CI90_0}. Dash-dotted horizontal lines\ndemarcate $\\pm10\\%$ error intervals around the true $\\Lambda_\\mathrm{NS}$ values.\nThe panel just below it shows the range of $\\Lambda_\\mathrm{NS}$ that encloses the\nmedian $\\Lambda_\\mathrm{NS}$ for $90\\%$ of the populations in {\\it each}\nensemble. In other words, this panel shows the range of $\\Lambda_\\mathrm{NS}$ within which\nthe median $\\Lambda_\\mathrm{NS}$ value for $90\\%$ of NSBH populations is\nexpected to lie. From these panels, we observe that our median $\\Lambda_\\mathrm{NS}$\nvalues will be within $10\\%$ of the {\\it true} value after $\\sim 25$\ndetections of less deformable neutron stars ($\\Lambda_\\mathrm{NS}\\leq 1000$), or\nafter as few as $15$ detections of more deformable neutron stars\n($\\Lambda_\\mathrm{NS}\\geq 1500$). This is not surprising because we inject simulated\nsignals in {\\it zero} noise, which ensures that the median not be shifted away\nfrom the true value. That it takes $15+$ events for the median to approach\nthe true value is a manifestation of the fact that the measurement is limited\nby the prior on $\\Lambda_\\mathrm{NS}$ when we have fewer than $15$ events.\nThe results discussed in Fig.~\\ref{fig:TT_Lambda_vs_N_L800_CI90_0},\n\\ref{fig:TT_Lambda_vs_N_CI90_0} and the left two panels\nof Fig.~\\ref{fig:TT_LambdaMedian_vs_N_AllInOne} apply to the parameter\ndistribution spanned by the grid G. This distribution allows for $m_\\mathrm{BH}$\nas low as $2.7M_\\odot$ (i.e. $q=2$).\nGiven that disruptive signatures are strongest for small $m_\\mathrm{BH}$, we now\ninvestigate an alternate paradigm in which no black hole masses fall within\nthe mass gap $2-5M_\\odot$ suggested by astronomical\nobservations~\\cite{Bailyn:1997xt,Kalogera:1996ci,Kreidberg:2012,\nLittenberg:2015tpa}. We will henceforth denote our standard paradigm, which\ndoes not respect the mass-gap, as paradigm A; with paradigm B being\nthis alternate scenario.\nBoth right panels of the figure are identical to their corresponding left\npanels, but drawn under population paradigm B. Under this paradigm, we\nexpectedly find that information accumulation is much slower. It would\ntake $25-40$ detections with $\\rho\\geq10$ under this paradigm, for our median\n$\\Lambda_\\mathrm{NS}$ to converge within $10\\%$ of its true value.\n\n\nFinally, we investigate the statistical uncertainties associated with\n$\\Lambda_\\mathrm{NS}$ measurements. We use $90\\%$ credible intervals as our measure of\nthe same. First, we draw an ensemble of a hundred populations each for\n$\\Lambda_\\mathrm{NS}=\\{500,1000,1500,2000\\}$. For each population $i$ in each ensemble,\nwe construct\nits $90\\%$ credible interval $[{\\Lambda_\\mathrm{NS}^{90\\%}}_{i-},{\\Lambda_\\mathrm{NS}^{90\\%}}_{i+}]$.\nNext, we construct the interval $[X^-,Y^-]$ that contains ${\\Lambda_\\mathrm{NS}^{90\\%}}_{i-}$\nfor $90\\%$ of the populations in each ensemble; and similarly $[X^+,Y^+]$\nfor ${\\Lambda_\\mathrm{NS}^{90\\%}}_{i+}$. Finally, in the left panel of \nFig.~\\ref{fig:TT_LambdaError_vs_N_L500_2000_CI90_0_AllInOne}, we show the\nconservative width $|Y^+ - X^-|$ that contains the $90\\%$ credible\nintervals for $90\\%$ of all populations in each ensemble~\\footnote{Drawn\nunder paradigm A (mass-gap not respected).}.\nFrom top to bottom, the population $\\Lambda_\\mathrm{NS}$\ndecreases from $\\Lambda_\\mathrm{NS}=2000\\rightarrow 500$, corresponding to decreasingly\ndeformable NSs with softer equations of state. We observe the following:\n(i) for moderately-hard to hard equations of state with $\\Lambda_\\mathrm{NS}\\geq 1000$,\nwe can constrain $\\Lambda_\\mathrm{NS}$ within $\\pm 50\\%$ using only $10-20$ events, and\nwithin $\\pm 25\\%$ (marked by black circles) with $25-40$ events; (ii) for softer \nequations of state with\n$\\Lambda_\\mathrm{NS}<1000$, we will achieve the same accuracy with $20-30$ and $50+$ \nevents, respectively; and (iii) for the first $5$ or so observations, our\nmeasurement spans the entire prior allowed range:\n$\\Lambda_\\mathrm{NS}\\in[0,4000]$, as shown by the plateauing of the $90\\%$ \ncredible intervals towards the left edge to $90\\%$ of $4000$, i.e. $3600$.\nThe right panel in Fig.~\\ref{fig:TT_LambdaError_vs_N_L500_2000_CI90_0_AllInOne}\nis identical to the left one, with the difference that populations are drawn\nunder paradigm B, which does {\\it not} allow for BH masses to fall within\nthe mass-gap. We find that\nfor NSs with $\\Lambda_\\mathrm{NS}\\leq 1000$, it would take $25-40$ events to\nconstrain $\\Lambda_\\mathrm{NS}$ within $\\pm 50\\%$ and $50+$ events to constrain it\nwithin $\\pm 25\\%$. This is somewhat slower than paradigm A, as is to be\nexpected since here we preclude the lowest mass-ratios, which correspond to\nsignals with largest tidal signatures. For $\\Lambda_\\mathrm{NS}>1000$ we find that we\ncan constrain $\\Lambda_\\mathrm{NS}$ within $\\pm 50\\%$ with a similar number of events as\nfor paradigm A, but will need more ($30-40$, as compared to $25-40$) events\nto further constrain it to within $\\pm 25\\%$ of the true value.\nUnder either paradigms, we find that measuring $\\Lambda_\\mathrm{NS}$ better than\n$25\\%$ will require $\\mathcal{O}(10^2)$ observations of disruptive NSBH\nmergers.\nThese results demonstrate that our probed set of disruptive NSBH mergers is as\ninformative of NS tidal properties as are BNS populations, if we assume a uniform mass\ndistribution for NSs and zero NS spins, and possibly more if NS masses are\nnot distributed uniformly~\\cite{Agathos:2015a}.\nHowever, being a subset of all NSBH binaries, the accumulation of information\nfrom NSBH signals in general may be slower than from binary neutron stars,\ndepending on the distribution of BH masses in coalescing NSBH binaries.\n\n\n{\\it Do all events matter?} In Ref.~\\cite{Lackey2014} the authors demonstrate\nthat the overwhelming majority of information about the NS equation of state\ncomes from the loudest $\\sim 5$ events in the case of binary neutron star\ndetections, and not from the combined effect of a large number of low-SNR\nsystems.\nThe question naturally arises if the same is true for NSBH sources as well.\nTherefore, in Fig.~\\ref{fig:TT_LambdaError_vs_N_L500_2000_CI90_0_AllInOne_SNRSorted}\nwe re-evaluate the accumulation of information with each successive NSBH \ndetection, sorting the events in each population by their SNR instead of \nsimulated chronology. We find the same qualitative behavior as in the case of \nBNSs~\\cite{Lackey2014}. \nWhether or not there is an astrophysical mass gap, we find\nthat the bulk of tidal information will be furnished by the loudest $5-10$\nNSBH detections of aLIGO detectors. {\\it This result is especially encouraging\nto NR follow-up efforts for GW detections, as we now know that only a handful\nof loudest NSBH events (with SNRs $\\rho\\gtrsim 20-30$) are the ones that may merit\nfull numerical-GR + magnetohydrodynamical follow-up simulations.}\n\n\n\nTo summarize, in this section we study the improvement in our measurement of \nNS deformability parameter $\\Lambda_\\mathrm{NS}$ with an increasing number of events. We\ndo so by simulating plausible populations of disrupting NSBH binaries (with\n$\\rho\\geq 10$). We find that:\n(i) for more deformable neutron stars (harder equation of states), the median\nvalue of $\\Lambda_\\mathrm{NS}$ comes within $10\\%$ of the true value with as \nfew as $10$ events, while achieving the same accuracy for softer equations of \nstate will take $15-20$ source detections; (ii) the statistical uncertainty\nassociated with $\\Lambda_\\mathrm{NS}$ measurement shrinks to within $\\pm50\\%$ with\n$10-20$ events, and to within $\\pm 25\\%$ with $50+$ events, when source \n$\\Lambda_\\mathrm{NS}\\geq 1000$; (iii) for softer equations of state, the same could take\n$25-40$ and $50+$ events, respectively for the two uncertainty thresholds;\nand (iv) if BHs really do observe the astrophysical mass-gap, the information\naccumulation is somewhat slower than if they do not. We conclude that within\n$20-30$ observations, aLIGO would begin to place very interesting bounds on \nthe NS deformability, which would allow us to rule out or rank different\nequations of state for neutron star matter. Within this population, we also\nfind that it will be the loudest $5-10$ events that will furnish most of the\ntidal information. Our key findings are summarized in \nFig.~\\ref{fig:TT_LambdaMedian_vs_N_AllInOne} -\n\\ref{fig:TT_LambdaError_vs_N_L500_2000_CI90_0_AllInOne_SNRSorted}.\n\n\n\n\n\n\\section{Discussion}\\label{s1:discussion}\n\n\nThe pioneering observation of gravitational waves by Advanced LIGO\nharbingers the dawn of an era of gravitational-wave astronomy where observations\nwould finally drive scientific discovery~\\cite{Abbott:2016blz}. As confirmed by\nthe first observations~\\cite{Abbott:2016blz,Abbott:2016nmj,Abbott:2016nhf},\nstellar-mass compact binary mergers emit GWs right in the sensitive frequency\nband of the LIGO observatories, and are their primary targets.\nNeutron star black hole binaries form a physically distinct sub-class of\ncompact binaries. We expect to detect the first of them in the upcoming\nobserving runs~\\cite{Abbott:2016ymx}, and subsequently at a healthy rate of\n$0.2-300$ mergers a year when aLIGO detectors reach design\nsensitivity~\\cite{Abadie:2010cf}.\n\nNSBH binaries are interesting for various reasons. Unlike BBHs, the presence of\nmatter allows for richer phenomena to occur alongside the strong-field\ngravitational dynamics. The quadrupolar moment of the NS changes during the\ncourse of inspiral, which increases the inspiral rate of the binary and alters the\nform of the emitted gravitational waves. Close to merger, under restricted but\nplausible conditions, the neutron star is disrupted by the tidal field of its \ncompanion black hole and forms an accretion disk around it. This disruption\nreduces the quadrupolar moment of the system, and decreases the amplitude of\nthe emitted GWs from the time of disruption through to the end of ringdown.\nBoth of these phenomena are discernible in their gravitational-wave signatures\nalone. In addition, if the neutron star matter is magnetized, the magnetic\nwinding above the remnant black hole poles can build up magnetic fields\nsufficiently to power short gamma-ray bursts (SGRB)~\\cite{Foucart:2015a,\nLovelace:2013vma,Deaton2013,Foucart2012,Shibata:2005mz,Paschalidis2014}.\nTherefore a coincident observation of gravitational waves from an NSBH merger\nand a SGRB can potentially confirm the hypothesis that the former is a\nprogenitor of the latter~\\cite{eichler:89,1992ApJ...395L..83N,moch:93,\nBarthelmy:2005bx,2005Natur.437..845F,2005Natur.437..851G,Shibata:2005mz,\nTanvir:2013,Paschalidis2014}.\n\n\nIn this paper we study the observability of tidal signatures in the\ngravitational-wave spectrum of NSBH binaries. More specifically, we investigate\nthree questions. First, what is the effect of not including tidal effects in \ntemplates while characterizing NSBH signals? Second, if we do include tidal \neffects, how well can we measure the tidal deformability of the NS\n(parameterized by $\\Lambda_\\mathrm{NS}$) from individual NSBH signals? And third, as we\nobserve more and more signals, how does our knowledge of $\\Lambda_\\mathrm{NS}$ improve?\nIn the following, we summarize our main findings.\n\n\n\n\n\nFirst, we study the effects of not including tidal terms in our search\ntemplates while characterizing NSBH signals. We expect that the waveform\ntemplate that best fits the signal would compensate for the reduced number of\ndegrees of freedom in the template model by moving away from the true\nparameters of the binary. This should result in a {\\it systematic} bias in \nthe recovered values of non-tidal source parameters, such as its masses \nand spins. In order to quantify it, we inject tidal signals into zero noise,\nand perform a Bayesian parameter estimation analysis on them using templates\n{\\it without} tidal terms.\nWe use the LEA+ model (c.f. Sec.~\\ref{s2:waveforms}) to produce tidal waveforms\nthat incorporate the effect\nof NS distortion during inspiral, and of its disruption close to merger. Our\ninjected signals sample the region of NSBH parameter space where NS disruption\nprior to binary merger is likely {\\it and} can be modeled using LEA+. Their\nparameters are given by combinations of $q=m_\\mathrm{BH}\/m_\\mathrm{NS}=\\{2,3,4,5\\},\n\\chi_\\mathrm{BH}=\\{-0.5, 0, 0.5, 0.75\\}$ and $\\Lambda_\\mathrm{NS}=\\{500,800,1000,1500,2000\\}$.\nOther parameters, such as source location and orientation, that factor out of\n$h(t)$ as amplitude scaling are co-sampled by varying $\\rho=\\{20,30,50,70\\}$.\n\n\n\nAt low to moderate SNRs ($\\rho\\lesssim 30$), we find that using BBH templates\ndoes not significantly hamper our estimation of non-tidal parameters for NSBH\nsignals. In the worst case, when the BH mass is within the astrophysical \nmass-gap~\\cite{Bailyn:1997xt,Kalogera:1996ci,Kreidberg:2012,Littenberg:2015tpa}\nand its spin is positive aligned, the systematic biases in $\\eta$ and $\\chi_\\mathrm{BH}$\nmeasurements do become somewhat comparable to statistical errors (ratio\n$\\sim 0.5-0.8$) under very restrictive conditions~\\footnote{requiring a\ncompanion BH with mass $m_\\mathrm{BH}\\lesssim 4.5M_\\odot$ (i.e. in the astrophysical\nmass-gap), and the\nhardest NS EoS considered (with $\\Lambda_\\mathrm{NS}\\simeq 2000$).}, but never exceed them.\nAt high SNRs ($\\rho\\gtrsim 50$), systematic biases in $\\mathcal{M}_c$ become larger\nthan the statistical uncertainties. For $\\eta$ and $\\chi_\\mathrm{BH}$ the difference\nis more drastic with the systematics reaching up to $4\\times$ the statistical\nerrors. We therefore conclude that $\\rho\\simeq 30-50$ is loud enough to\nmotivate the use of tidal templates for even the estimation of non-tidal\nparameters from NSBH signals.\nWe also conclude that low-latency parameter estimation algorithms, designed to\nclassify GW signals into electromagnetically active (NSBH and NSNS) and\ninactive (BBH) sources, can use BBH templates to trigger GRB \nalerts~\\cite{2012A&A...541A.155A,Singer:2014qca,Singer:2015ema,Pankow:2015cra,\nAbbott:2016wya,Abbott:2016gcq} for NSBH signals with low to moderate SNRs\n($\\rho\\lesssim 30$).\nThis is so because the primary requirement of identifying NS-X binaries (X =\n\\{NS, BH\\}) can be achieved just as easily with BBH templates, on the basis of\nthe smaller component's mass\\footnote{The smaller component mass is unlikely\nto be significantly biased by missing tidal effects in filter templates below\n$\\rho\\simeq 30$, as we show above.}.\nWe also speculate that NSBH detection searches are unlikely to be\naffected by the choice of ignoring tidal effects in matched-filtering\ntemplates, if these effects are too subtle to manifest in parameter estimation\nbelow $\\rho\\simeq 30$.\n\n\n\n\n\nSecond, we turn the question around to ask: can we measure the tidal effects if\nour template models did account for them? Tidal effects in our waveform model\nare parameterized using a single deformability parameter \n$\\Lambda_\\mathrm{NS}\\propto (R\/M)_\\mathrm{NS}^5$. In order to quantify the \nmeasurability of $\\Lambda_\\mathrm{NS}$, we inject the same tidal signals as before, and\nthis time perform a Bayesian analysis on them using {\\it tidal} templates. \nThe results are detailed in Sec.~\\ref{s1:PEwithNS}.\nAt low SNRs ($\\rho\\simeq 20$), we find that the best we can do is to constrain\n$\\Lambda_\\mathrm{NS}$ within $\\pm 75\\%$ of its true value at $90\\%$ credible level. This\ntoo only if the BH is spinning sufficiently rapidly, with $\\chi_\\mathrm{BH}\\gtrsim +0.7$,\nand the NS has $\\Lambda_\\mathrm{NS}\\gtrsim 1000$. At moderate SNRs ($\\rho\\simeq 30$), we\ncan constrain $\\Lambda_\\mathrm{NS}$ a little better, i.e. within $\\pm 50\\%$ of its true\nvalue. This level of accuracy, however, again requires that BH spin\n$\\chi_\\mathrm{BH}\\gtrsim+0.7$ and $\\Lambda_\\mathrm{NS}\\gtrsim 1000$. Binaries with smaller BH spins\nand\/or softer NS EoSs will furnish worse than $\\pm 75\\%-\\pm 100\\%$ errors for\n$\\Lambda_\\mathrm{NS}$. This trend continues as we increase the SNR from $\\rho=30-50$. It\nis not before we reach an SNRs as high as $\\rho\\simeq 70$ that we can shrink\n$\\Lambda_\\mathrm{NS}$ errors substantially with a single observation (i.e. within\n$\\pm 25\\%$ of its true value).\nIn summary, we find that with a single but moderately loud NSBH signal,\nAdvanced LIGO can begin to put a factor of $1-2\\times$ constraints on NS tidal\ndeformability parameter. These constraints can subsequently be used to assess\nthe likelihood of various candidate equations of state for nuclear matter, and\npossibly to narrow the range they span.\n\n\n\n\n\nThird, knowing that single observations can furnish only so much information\nabout the NS equation of state, we move on to investigate how well we do with\nmultiple signals. In order to quantify how $\\Lambda_\\mathrm{NS}$ measurement improves\nwith the number of observed events $N$, we generate populations of NSBH signals\nand combine the information extracted from each event.\nThe population generation procedure is as follows. The neutron star mass is\nheld fixed at $1.35M_\\odot$, its spin at $\\chi_\\mathrm{NS}=0$, and its tidal\ndeformability is fixed to each of $\\Lambda_\\mathrm{NS}=\\{500,1000,1500,2000\\}$. Black hole\nmass is sampled uniformly from the range $[2,5]\\times 1.35=[2.7, 6.75]M_\\odot$,\nand spin from $\\chi_\\mathrm{BH}\\in[0,1]$. As before, our parameter choice here is given\nby the intersection set of the mass range that allows for neutron star disruption\nand the range supported by LEA+~\\cite{Foucart2012,Foucart:2013a,Lackey:2013axa}.\nIn order to keep the computational cost reasonable, we make an additional\napproximation. For every population generated, we replace the parameters of each\nevent by their nearest neighbor on the uniform grid G, which has vertices\nat: $q=\\{2,3,4,5\\}\\times\\chi_\\mathrm{BH}=\\{-0.5,0,0.5,0.75\\}\\times\\Lambda_\\mathrm{NS}=\\{500,800,\n1000,1500,2000\\}\\times\\rho=\\{10,20,30,50,70\\}$.\nThis way, we only have to run full Bayesian parameter estimation analysis on\nthis fixed set of signals. \nThere are two sources of error that enter the deductions we make from\na single population generated in the manner described above. First, since the\ninjection parameters are pushed to their nearest neighbor on a grid, we\nfind discrete jumps in $\\Lambda_\\mathrm{NS}$ errors as a function of $N$. And second, an\nindividual population is one particular realization of a stochastic process and\ncould have excursions that may never be found in another population. To\naccount for both of these limitations, we generate an ensemble of populations,\nand conservatively combine information from all of them\\footnote{See \nSec.~\\ref{s1:multiple_observations} for further details.}.\n\n\n\n\nWe probe two astrophysical paradigms, one that allows for BH masses to lie\nwithin the astrophysical mass-gap (paradigm A), and one that does not (paradigm\nB).\n{\\it For paradigm A}, we find the following: (i) for the softer equations of\nstate that result in less deformable neutron stars, $15-20$ detections bring\nthe measured probability distribution for $\\Lambda_\\mathrm{NS}$ entirely within the prior,\nwhich ensures that the median $\\Lambda_\\mathrm{NS}$ tracks the true value to within $10\\%$.\n(ii) For NSBH populations with more deformable NSs ($\\Lambda_\\mathrm{NS}> 1000$),\nthe same is achievable within as few as $10$ (or $15$ at most) realistic\nobservations. (iii) The statistical uncertainty associated with $\\Lambda_\\mathrm{NS}$\nmeasurement can be restricted to be within $\\pm50\\%$ using $10-20$ observations\nwhen $\\Lambda_\\mathrm{NS}> 1000$), and using $25-40$ observations for softer equations\nof state. All of the above is possible within a few years of design\naLIGO operation~\\cite{Abadie:2010cfa}, if astrophysical BHs are allowed\nmasses $< 5M_\\odot$ (i.e. in the mass-gap). However, further\nrestricting $\\Lambda_\\mathrm{NS}$ will require $50+$ NSBH observations.\n{\\it For paradigm B}, we find the information accumulation to be somewhat slower.\nWhile the quantitative inferences for populations with $\\Lambda_\\mathrm{NS}>1000$ are\nnot affected significantly, we find that $\\Lambda_\\mathrm{NS}< 1000$ populations require \n$10-20\\%$ more events to attain the same measurement accuracy as under\nparadigm A. In either case, the accumulation of information from the general\nNSBH population will likely be slower than from BNS inspirals~\\cite{Mandel:2009pc,\nLackey2014,Wade:2014vqa,Agathos:2015a}, depending on the mass distribution of \nstellar-mass black holes. Though, template models for the latter may be more\nuncertain due to missing point-particle PN terms at orders comparable to\ntidal terms~\\cite{Lackey2014}.\nWe conclude that within as few as $20-30$ observations of disruptive NSBH\nmergers, aLIGO will begin to place interesting bounds on NS deformability.\nThis, amongst other things, will allow us to rank different equations of \nstate for neutron star matter from most to least likely, within a few years'\ndetector operation.\nWe also find that, within this population, the loudest $5-10$ events (with SNRs\n$\\rho\\gtrsim 20-30$) will provide us with most of the tidal information,\nand will therefore merit full NR follow-up.\nOur methods and results are detailed in Sec.~\\ref{s1:multiple_observations}.\n\n\n\n\n\n\n\n\n\nFinally, we note that the underlying numerical simulations used to calibrate\nthe waveform model used here have not been verified against\nindependent codes so far.\nIt is therefore difficult to assess the combined modeling error of LEA+ and its\neffect\non our results. Our results here are, therefore, limited by the limitations of\nour waveform model, and presented with this caveat. However, we do expect the\ncombined effect of modeling errors to {\\it not} affect our {\\it qualitative}\nconclusions, especially since the underlying point-particle component of LEA+\nincludes all high-order terms, unlike past BNS studies~\\cite{Lackey2014,\nWade:2014vqa}\nIn future, we plan to further the results presented here by using more recent\ntidal models~\\cite{Pannarale:2015jka,Hinderer:2016eia}, that\nmay improve upon LEA+\\footnote{One of them~\\cite{Pannarale:2015jka} is only an\namplitude model though, which has to be augmented with a compatible phase model\nfirst.}.\n\n\n\n\n\\begin{acknowledgments}\n We thank Benjamin Lackey, Francesco Pannarale, Francois Foucart, and Duncan Brown\n for helpful discussions. We gratefully acknowledge support\n for this research at CITA from NSERC of Canada, the Ontario Early \n Researcher Awards Program, the Canada Research\n Chairs Program, and the Canadian Institute for Advanced Research.\n Calculations were performed at the Vulcan\n supercomputer at the Albert Einstein Institute;\n H.P. and P.K. thank the Albert-Einstein Institute,\n Potsdam, for hospitality during part of the time where this research\n was completed. M.P. thanks CITA for hospitality where part of the work\n was carried out.\n\\end{acknowledgments}\n\n\\begin{appendix}\n\n\\section{Statistical uncertainty in measuring non-tidal parameters}\\label{as1:nontidalerrors}\n\\begin{figure*}\n\\centering \n\\includegraphics[trim = {2cm 0 0 0},width=2.\\columnwidth]{TNMchirpCIWidths90_0_Lambda_SNR}\\\\\n\\includegraphics[trim = {2cm 0 0 0},width=2.\\columnwidth]{TNEtaCIWidths90_0_Lambda_SNR}\\\\\n\\includegraphics[trim = {2cm 0 0 0},width=2.\\columnwidth]{TNChiBHCIWidths90_0_Lambda_SNR}\n\\caption{{\\bf Statistical measurement uncertainty for NSBH parameters,\nignoring tidal effects:}\nWe show here the statistical uncertainty associated with our measurement of\nnon-tidal parameters $\\mathcal{M}_c, \\eta,$ and $\\chi_\\mathrm{BH}$ (at $90\\%$ credibility),\nover the signal parameter space. Individual panels show the same as a function\nof BH mass and spin. Across each row, we see the effect of increasing signal\nstrength (i.e. SNR) with the tidal deformability of the NS $\\Lambda_\\mathrm{NS}$ fixed.\nDown each column, we see the effect of increasing $\\Lambda_\\mathrm{NS}$, at fixed SNR.\nTidal effects are ignored in templates.\n}\n\\label{fig:CIWidths90_Lambda_SNR}\n\\end{figure*}\nIn Fig.~\\ref{fig:CIWidths90_Lambda_SNR}, we show how {\\it precisely} can we\nmeasure non-tidal NSBH parameters $X=\\{\\mathcal{M}_c,\\eta,\\chi_\\mathrm{BH}\\}$ using BBH templates.\nThe three panels correspond to $\\mathcal{M}_c$ (top), $\\eta$ (middle), and $\\chi_\\mathrm{BH}$\n(bottom), and show the width of these credible intervals $(\\Delta X)^{90\\%}$\nas a function of BH mass\/spin (within each sub-panel), and NS properties, i.e.\n$\\Lambda_\\mathrm{NS}$ (downwards in each column)~\\footnote{We restrict NS mass to\n$1.35M_\\odot$ and its spin to zero. Varying its tidal deformability $\\Lambda_\\mathrm{NS}$\ndoes not significantly change the measurement uncertainties for non-tidal\nbinary parameters, as is evident from comparing the two rows in each panel of\nFig.~\\ref{fig:CIWidths90_Lambda_SNR}.}.\nFrom the left-most column, we find that: (i) at $\\rho=20$ the chirp mass is\nmeasured remarkably well - to a precision of $0.16\\%$ of its true value, and\n(ii) so is $\\chi_\\mathrm{BH}$. (iii) The dimensionless mass-ratio $\\eta$ is determined\nmore loosely, with $25+\\%$ uncertainty. If the signal is even louder\n($\\rho\\geq 30$), all three measurements gain further precision, especially\n$\\eta$, for which the relative errors shrink down to single-digit percents.\nWe remind ourselves that these results do not tell the full story since the\nprecision of a measurement is only meaningful if the measurement is accurate \nto begin with. In our case there are tidal effects that have not been\nincorporated into our search (BBH) templates, which can lead to a systematic\nbias in parameter recovery. We refer the reader to Sec.~\\ref{s1:PEwithnoNS} for\na comparative study of both systematic and statistical errors.\n\n\n\n\n\\section{Illustrations of Bayesian posteriors}\\label{as1:illustrations}\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1.05\\columnwidth,trim=2cm 0 0 0]{AllParamsMcEtPDF1D2D_q4_mc2_25_chi0_50_snr20\n\\includegraphics[width=1.05\\columnwidth,trim=0.7cm 0 1cm 0]{AllParamsMcEtPDF1D2D_q4_mc2_25_chi0_50_snr50\n\\caption{{\\bf Illustrative posterior probability distributions for NSBH parameters,\nfor signals at different SNRs:}\nWe illustrate here two sets of two-dimensional joint probability distributions,\ndiffering only in signal strength, with $\\rho=20$ in the left panel, and\n$\\rho=50$ in the right. The injected parameters are \n$q = m_\\mathrm{BH}\/m_\\mathrm{NS} = 5.4M_\\odot\/1.35M_\\odot = 4$, $\\chi_\\mathrm{BH}=+0.5$, and \n$\\Lambda_\\mathrm{NS}=2000$. Contours are shown for $\\{1-,2-,3-,\\cdots\\}\\sigma$ confidence levels.\nTemplates include tidal effects, as evident in the bottom rows\nof both panels which show the correlation of $\\Lambda_\\mathrm{NS}$ with non-tidal \nparameters. Contrasting the two panels illustrates the effect of increasing the\nSNR on various parameter measurements.\n}\n\\label{fig:SingleSystemLambda2DPDFs}\n\\end{figure*}\n\n\n\n\n\n\nIn Fig.~\\ref{fig:SingleSystemLambda2DPDFs} we show the correlation of\nmass, spin, and tidal parameter measurements. We keep the binary parameters\nas in Fig.~\\ref{fig:SingleSystemLambdaPDFvsSNR}, with $\\Lambda_\\mathrm{NS}=2000$, and\nset $\\rho=20$ (left panel) or $\\rho=50$ (right panel).\nWe find that the measurement of $\\Lambda_\\mathrm{NS}$ is weakly degenerate with\nother parameters, and at realistic SNRs it would improve by a few tens of \npercent if we knew non-tidal parameters to better accuracy. The predominant\nfactor that would enhance the measurement accuracy for $\\Lambda_\\mathrm{NS}$ is \nnevertheless the signal strength. Only when $\\rho\\gtrsim 50$ can we\nexpect $\\Lambda_\\mathrm{NS}$ measurement to be limited by its degeneracy with \nnon-tidal parameters (at a factor of few level), as also reported by previous\nstudies~\\cite{Lackey:2013axa}.\n\n\n\n\n\n\n\n\\section{Phenomenology of $\\Lambda_\\mathrm{NS}$ measurement errors}\n\\begin{figure}\n\\centering \n\\includegraphics[width=1.05\\columnwidth]{PowerLawCoefficient_LambdaErrorvsN_vs_N.pdf}\n\\caption{%\nAssuming a power-law dependence of the measurement error on the number of\nevents: $\\delta\\Lambda_\\mathrm{NS}\\propto 1\/N^\\alpha$, we show $\\alpha$ in this figure\nas a function of the number of observed events $N$. Shown are five families\nof $100$ population draws each, with each family corresponding to one of\n$\\Lambda_\\mathrm{NS}=\\{500,800,1000,1500,2000\\}$. Each grey curve corresponds to one\nof these $100\\times5 = 500$ populations. The thicker curves, one from each\nfamily, shows the population we discussed in\nFig.~\\ref{fig:TT_Lambda_vs_N_L800_CI90_0}-\\ref{fig:TT_Lambda_vs_N_CI90_0}.\nWe find that a power-law is a good approximation for the concerned dependence,\nand information accumulates {\\it faster} than $1\/\\sqrt{N}$. We estimate\n$\\alpha\\simeq 0.7^{+0.2}_{-0.2}$.\n}\n\\label{fig:TT_PowerLawLambdaErrorVsN}\n\\end{figure}\n\\begin{figure}\n\\centering \n\\includegraphics[width=\\columnwidth]{PowerLawCoefficient_LambdaErrorvsLambda_vs_N_AllPopulations.pdf}\n\\caption{%\nIn this figure, which is similar to Fig.~\\ref{fig:TT_PowerLawLambdaErrorVsN},\nwe quantify the dependence of $\\delta\\Lambda_\\mathrm{NS}$ on $\\Lambda_\\mathrm{NS}$ itself. Of \nthe five families of simulated NSBH populations, we construct $100$\nindependent sets taking one population from each family. With each of \nthese $100$ sets, and assuming a power-law dependence:\n$\\delta\\Lambda_\\mathrm{NS}\\propto\\Lambda_\\mathrm{NS}^\\beta$, we estimate $\\beta$ and show it in\nthis figure as a function of the number of observed events $N$. The thicker\ncurve corresponds to the populations discussed in\nFig.~\\ref{fig:TT_Lambda_vs_N_CI90_0}.\nWe find that $\\beta$ can be estimated to lie within $[1\/6,5\/6]$ with a\nlikely value close to $1\/2$. Since $0<\\beta<1$, the relative error\n$\\delta\\Lambda_\\mathrm{NS}\/\\Lambda_\\mathrm{NS}$ {\\it decreases} as the star gets more \ndeformable, while the absolute error $\\delta\\Lambda_\\mathrm{NS}$ {\\it increases}.\n}\n\\label{fig:TT_PowerLawLambdaErrorVsLambda}\n\\end{figure}\nHere, we quantitatively explore the dependence of our statistical\nuncertainties for $\\Lambda_\\mathrm{NS}$ on the number of events, as well as on the true\nNS deformability itself. First, we will focus on the dependence on $N$. We\nassume a power-law dependence of the form\n$\\delta\\Lambda_\\mathrm{NS}\\propto\\ 1\/N^\\alpha$. For each of the $100$ populations \nfor each of $\\Lambda_\\mathrm{NS}=500-2000$, we compute the exponent $\\alpha$ as a\nfunction of the number of observed events $N$, and show it in \nFig.~\\ref{fig:TT_PowerLawLambdaErrorVsN}. There are $100\\times5=500$ curves\non the figure, with one highlighted for each value of population's $\\Lambda_\\mathrm{NS}$.\nThese highlighted values are only special in the sense that they correspond to\npopulations discussed earlier in this section (c.f.\nFig.~\\ref{fig:TT_Lambda_vs_N_L800_CI90_0}-\\ref{fig:TT_Lambda_vs_N_CI90_0}).\nWe immediately observe two things, (i) there is a globally similar dependence\non $N$ for all populations, and (ii) information accumulates {\\it faster} than\n$1\/\\sqrt{N}$. In fact, we find that if\n$\\delta\\Lambda_\\mathrm{NS}\\propto\\frac{1}{N^\\alpha}$, $\\alpha$ lines in the range\n$0.7_{-0.2}^{+0.2}$.\nNext, we focus on the dependence of $\\delta\\Lambda_\\mathrm{NS}$ on $\\Lambda_\\mathrm{NS}$ of the\npopulation itself. As suggested by Fisher-matrix studies~\\cite{Lackey:2013axa},\nand as for $N$, we assume the form $\\delta\\Lambda_\\mathrm{NS}\\propto\\Lambda_\\mathrm{NS}^\\beta$.\nFrom each set of $100$ populations with a given $\\Lambda_\\mathrm{NS}$ value, we draw one\nat random, and form a set of $5$ similarly drawn populations, one for each of\n$\\Lambda_\\mathrm{NS}=\\{500,800,1000,1500,2000\\}$. With each set, we determine $\\beta$\nfor different number of observed events $N$. In all, we make $100$ independent\n$5-$population sets and show the value of $\\beta$ measured from each in \nFig.~\\ref{fig:TT_PowerLawLambdaErrorVsLambda}. We find that the assumed\nrelation $\\delta\\Lambda_\\mathrm{NS}\\propto\\Lambda_\\mathrm{NS}^\\beta$ gets fairly robust for \nlarger values of $N$, with $\\beta$ converging to $\\beta=0.5^{+0.33}_{-0.33}$.\nThe fact that $0<\\beta<1$ implies that the relative error\n$\\delta\\Lambda_\\mathrm{NS}\/\\Lambda_\\mathrm{NS}$ {\\it decreases} with increasing $\\Lambda_\\mathrm{NS}$, while\nthe absolute error {\\it increases}.\nFrom these results, we conclude that the measurement uncertainty for\n$\\Lambda_\\mathrm{NS}$ after $N$ observations is\n\\begin{equation}\n \\delta\\Lambda_\\mathrm{NS}\\propto \\dfrac{\\Lambda_\\mathrm{NS}^{0.5^{+0.33}_{-0.33}}}{N^{0.7_{-0.2}^{+0.2}}}.\n\\end{equation}\nWe also find that while these results are inferred from paradigm A populations,\nparadigm B gives very similar results.\n\n\n\n\n\\section{Choice of underlying BBH model in LEA}\n\\begin{figure}\n\\centering \n\\includegraphics[width=1.05\\columnwidth]{match-q-spin1-PhenomC-mf0_01.pdf}\n\\caption{%\nWe compare two alternatives of the tidal NSBH model from Ref.~\\cite{\nLackey:2013axa}, which differ in their underlying BBH prescriptions. One which\nwe use in this study uses SEOBNRv2, while the other uses IMRPhenomC as its\nbase. In this figure, we show the normalized overlap (match) between the two for\n$2,000,000$ points sampled uniformly in the NSBH parameter space. We find that\nfor $q\\gtrsim 4$ the discrepancies between IMRPhenomC and SEOBNRv2 as reported\nin~\\cite{Kumar:2015tha} dominate over tidal terms.\n\\label{fig:PhenomC_vs_SEOBNRv2_LEA}\n}\n\\end{figure}\nThe waveform model used in this paper is a variant of those calibrated\nin Ref~\\cite{Lackey:2013axa}. In that work, the authors also calibrate\na tidal prescription with the phenomenological model IMRPhenomC~\\cite{\nSantamaria:2010yb}\nas the base BBH model. Previous work~\\cite{Kumar:2015tha,Kumar:2016dhh} has \nshown that IMRPhenomC can exhibit pathological behavior for mass-ratios\n$q\\gtrsim 4$ and\/or non-zero black hole spins. We compute noise-weighted\ninner-products between the two variants for $2,000,000$ points sampled\nover the NSBH parameter space, and show the results in \nFig.~\\ref{fig:PhenomC_vs_SEOBNRv2_LEA}. We restrict the comparison to frequencies\nthat are affected by the tidal disruption of the NS, by integrating\nthe inner-products from $f = \\mathrm{max}(15, 0.01\/M)$~Hz\n(where $M$ is expressed in seconds ($1M_\\odot \\simeq 4.925\\mu$S, see\nEq.~(32-34) of~\\cite{Lackey:2013axa}).\nWe find that the differences between the two variants of LEA+ have mismatches\nof a few percent, while the tidal corrections contribute at a sub-percent\nlevel. We conclude that the differences of the underlying BBH model in LEA+\ndominate over its tidal calibration, and since SEOBNRv2 has been shown to\nbe more reliable than IMRPhenomC~\\cite{Kumar:2015tha,Kumar:2016dhh}, we \nrecommend the use of SEOBNRv2-based LEA+ in upcoming LIGO-Virgo analyses.\n\n\n\\end{appendix}\n\n\n\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}