\\ Institute of Theoretical Physics and Astrophysics, \\ Department of Physics, University of Cape Town, \\ Private Bag, Rondebosch 7701, South Africa \\ } \maketitle \begin{abstract} We consider a self-gravitating ideal fermion gas at nonzero temperature as a model for the Galactic halo. The Galactic halo of mass $\sim 2 \times 10^{12} M_{\odot}$ enclosed within a radius of $\sim 200$ kpc implies the existence of a supermassive compact dark object at the Galactic center that is in hydrostatic and thermal equilibrium with the halo. The central object has a maximal mass of $\sim 2.3 \times 10^{6} M_{\odot}$ within a minimal radius of $\sim 18$ mpc for fermion masses $\sim 15$ keV. \end{abstract} %\pacs{PACS Nos.: 98.35.Gi, 98.35.Jk, 95.35.+d} In the past, self-gravitating neutrino matter has been suggested as a model for quasars, with neutrino masses in the $0.2 {\rm keV} \simlt m \simlt 0.5 {\rm MeV}$ range \cite{1}. More recently, supermassive compact objects consisting of weakly interacting degenerate fermionic matter, with fermion masses in the $10 \simlt m/{\rm keV}\simlt 20$ range, have been proposed \cite{2,3,4,5,6} as an alternative to the supermassive black holes that are believed to reside at the centers of many galaxies. So far the masses of $\sim 20$ supermassive compact dark objects at the galactic centers have been measured \cite{kor}. The most massive compact dark object ever observed is located at the center of M87 in the Virgo cluster, and it has a mass of $\sim 3 \times 10^9 M_{\odot}$ \cite{7}. If we identify this object of maximal mass with a degenerate fermion star at the Oppenheimer-Volkoff (OV) limit \cite{opp}, i.e., $M_{\rm{OV}} = 0.54 M_{\rm Pl}^3\, m^{-2} g^{-1/2} \simeq 3 \times 10^9 M_{\odot}$ \cite{4}, where $M_{\rm{Pl}}=\sqrt{\hbar c/G}$, this allows us to fix the fermion mass to $m \simeq 15$ keV for a spin and particle-antiparticle degeneracy factor of $g=2$. Such a relativistic object would have a radius of $R_{\rm{OV}}= 4.45 R_{\rm{S}} \simeq$ 1.5 light-days, where $R_{\rm{S}}$ is the Schwarzschild radius of the mass $M_{\rm{OV}}$. It would thus be virtually indistinguishable from a black hole of the same mass, as the closest stable orbit around a black hole has a radius of 3 $R_{\rm{S}}$ anyway. Near the lower end of the observed mass range is the compact dark object located at the Galactic center \cite{eck} with a mass of $M_{\rm c} \simeq 2.6 \times 10^6 M_{\odot}$. Interpreting this object as a degenerate fermion star consisting of $m \simeq 15$ keV and $g= 2$ fermions, the radius is $R_{\rm c} \simeq 21$ light-days $\simeq 7 \times 10^4 R_{\rm S}$ \cite{2}, $R_{\rm S}$ being the Schwarzschild radius of the mass $M_{\rm c}$. Such a nonrelativistic object is far from being a black hole. The observed motion of stars within a projected distance of $\sim$ 6 to $\sim$ 50 light-days from Sgr A$^{*}$ \cite{eck} yields, apart from the mass, an upper limit for the radius of the fermion star $R_{\rm c} \simlt 22$ light-days. The required weakly interacting fermion of $\sim$ 15 keV mass cannot be an active neutrino, as it would overclose the Universe by orders of magnitude \cite{11}. However, the $\sim\! 15$ keV fermion could very well be a sterile neutrino, contributing $\Omega_{\rm d} \simeq$ 0.3 to the dark matter fraction of the critical density today. Indeed, as has been shown for an initial lepton asymmetry of $\sim 10^{-3}$, a sterile neutrino of mass $\sim$ 10 keV may be resonantly produced in the early Universe with near closure density, i.e., $\Omega_{\rm d} \sim$ 1 \cite{shi}. As an alternative possibility, the required $\sim$ 15 keV fermion could be the axino \cite{cov} or the gravitino \cite{lyt} in soft supersymmetry breaking scenarios. In the recent past, galactic halos have been successfully modeled as a self-gravitating isothermal gas of particles of arbitrary mass, the density of which scales asymptotically as $r^{-2}$, yielding flat rotation curves \cite{col}. As the supermassive compact dark objects at the galactic centers are well described by a gas of fermions of mass $m \sim 15$ keV at $T = 0$, it is tempting to explore the possibility that one could describe both the supermassive compact dark objects and their galactic halos in a unified way in terms of a fermion gas at finite temperature. We will show in this letter that this is indeed the case, and that the observed dark matter distribution in the Galactic halo is consistent with the existence of a supermassive compact dark object at the center of the Galaxy which has about the right mass and size. Degenerate fermion stars are well understood in terms of the Thomas-Fermi theory applied to self-gravitating fermionic matter at $T = 0$ \cite{2}. Extending this theory to nonzero temperature \cite{16,bil1,bil2,17}, it has been shown that at some critical temperature $T = T_{\rm c}$, a self-gravitating ideal fermion gas, having a mass below the OV limit enclosed in a sphere of radius $R$, may undergo a first-order gravitational phase transition from a diffuse state to a condensed state. However, this first-order phase transition can only take place if the Fermi gas is able to get rid of the large latent heat. As the short-range interactions of the fermions are negligible, the gas cannot release its latent heat; it will thus be trapped for temperatures $T < T_{\rm c}$ in a thermodynamic quasi-stable supercooled state close to the point of gravothermal collapse. The formation of a supercooled state close to the point of gravothermal collapse, may be understood as a process similar to that of violent relaxation \cite{lynd,bin}. Through the gravitational collapse of an overdense fluctuation, $\sim$ 1 Gyr after the Big Bang, part of gravitational energy transforms into the kinetic energy of random motion of small-scale density fluctuations. The resulting virialized cloud will thus be well approximated by a gravitationally stable thermalized halo. In order to estimate the particle mass-temperature ratio, we assume that a cold overdense cloud of the mass of the Galaxy $M$, stops expanding at the time $t_{\rm m}$, reaching its maximal radius $R_{\rm m}$ and minimal average density $\rho_{\rm{m}}= 3 M/(4 \pi R_{\rm{m}}^3)$. The total energy per particle is just the gravitational energy \begin{equation} E=-\frac{3}{5}\frac{GM}{R_{\rm{m}}} \, . \label{eq001} \end{equation} Assuming spherical collapse \cite{padma} one arrives at \begin{equation} \rho_{\rm{m}}=\frac{9\pi^2}{16} \bar{\rho}(t_{\rm{m}}) =\frac{9\pi^2}{16} \Omega_{\rm{d}} \rho_0 (1+z_{\rm{m}})^3, \label{eq002} \end{equation} where $\bar{\rho}(t_{\rm{m}})$ is the background density at the time $t_{\rm{m}}$ or cosmological redshift $z_{\rm{m}}$, and $\rho_0\equiv 3 H_0^2/(8\pi G)$ is the present critical density. We now approximate the virialized cloud by a singular isothermal sphere \cite{bin} of mass $M$ and radius $R$, characterized by a constant circular velocity $ \Theta=(2 T/m)^{1/2}$ and the density profile $ \rho(r)=\Theta^2/4\pi G r^2 .$ % \begin{equation} % \rho(r)=\frac{\Theta^2}{4\pi G r^2}\, . % \label{eq003} % \end{equation} Its total energy per particle is the sum of gravitational and thermal energies, i.e. \begin{equation} E=-\frac{1}{4}\frac{GM}{R} =-\frac{1}{4}\Theta^2 . \label{eq004} \end{equation} Combining Eqs. (\ref{eq001}), (\ref{eq002}), and (\ref{eq004}), we find \begin{equation} \Theta^2=\frac{6\pi}{5} G (6 \Omega_{\rm{d}} \rho_0 M^2)^{1/3}(1+z_{\rm{m}}) . \label{eq005} \end{equation} Taking $\Omega_{\rm{d}}=0.3$, $M=2\times 10^{12} M_{\odot}$, $z_{\rm{m}}=4$, and $H_0=65\,{\rm km\, s^{-1} Mpc^{-1}}$, we find $\Theta \simeq 220\, {\rm km\, s^{-1}}$, which corresponds to the mass-temperature ratio $m/T\simeq 4\times 10^6$. We now briefly discuss the general-relativistic Tho\-mas-Fer\-mi theory \cite{bil1,bil2} for a self-gravitating gas of $N$ fermions with mass $m$ at the temperature $T$ enclosed in a sphere of radius $R$. We denote by $p$, $\rho$, and $n$ the pressure, energy density, and particle number density of the gas, respectively. The metric is assumed to be static, spherically symmetric, and asymptotically flat, i.e., \begin{equation} ds^2=\xi^2 dt^2 -\frac{dr^2}{1-2{\cal{M}}/r} - r^2(d\theta^2+\sin \theta d\phi^2). \label{eq13} \end{equation} For numerical convenience, we introduce the parameter $\alpha=\mu/T$ and the substitution $\xi=(\varphi+1)^{-1/2} \mu/m$, where $\mu$ is the chemical potential associated with the conserved fermion number $N$. The equation of state for a self-gravitating gas may be represented in parametric form \cite{bil2} with appropriate mass and length scales $b$ and $a$, respectively, $b= (2/g)^{1/2} M^3_{\rm{Pl}}/ m^2$, $a= b\hbar / (c M_{\rm{Pl}}^2)$. Einstein's equations for the metric (\ref{eq13}) are given by \begin{equation} \frac{d\varphi}{dr} = -2(\varphi+1)\frac{{\cal M}+4\pi r^3 p}{r(r-2{\cal{M}})} \, , \label{eq88} \end{equation} \begin{equation} \label{eq89} \frac{d{\cal{M}}}{dr}=4\pi r^2 \rho. \end{equation} To these two equations we add \begin{equation} \frac{d{\cal N}}{dr}=4\pi r^2 (1-2{\cal{M}}/r)^{-1/2} n \label{eq93} \end{equation} imposing particle-number conservation as a condition at the boundary ${\cal N}(R)=N$. Equations (\ref{eq88})-(\ref{eq93}) should be integrated using the bo\-un\-da\-ry conditions at the origin, i.e., \begin{equation} \varphi(0)=\varphi_0 > -1 \, ; \;\;\;\;\; {\cal{M}}(0)=0 \, ; \;\;\;\;\; {\cal{N}}(0)=0. \label{eq90} \end{equation} It is useful to introduce the degeneracy parameter $\eta=\alpha \varphi/2$. As $\varphi$ is monotonously decreasing with increasing $r$, the strongest degeneracy is obtained at the center with $\eta_0=\alpha\varphi_0/2$. The parameter $\eta_0$, uniquely related to the central density and pressure, will eventually be fixed by ${\cal N}(R)=N$. For $r\geq R$, the function $\varphi$ yields the usual empty-space Schwarzschild solution \begin{equation} \varphi(r)=\frac{\mu^2}{m^2} \left(1-\frac{2 M}{r}\right)^{-1}-1\, , \label{eq91} \end{equation} with \begin{equation} M={\cal M}(R)=\int_0^R dr 4\pi r^2 \rho(r) . \label{eq92} \end{equation} The set of self-consistency equations (\ref{eq88})-(\ref{eq93}), with the boundary conditions (\ref{eq90})-(\ref{eq92}) defines the general-relativistic Thomas-Fermi equation. The numerical procedure is now straightforward. For a fixed, arbitrarily chosen $\alpha$, we first integrate Eqs. (\ref{eq88}) and (\ref{eq89}) numerically on the interval $[0,R]$ to find the solutions for various central values $\eta_0$. Integrating (\ref{eq93}) simultaneously yields ${\cal N}(R)$ as a function of $\eta_0$. We then select the value of $\eta_0$ for which ${\cal N}(R)=N$. The chemical potential $\mu$ corresponding to this particular solution is given by Eq. (\ref{eq91}) which in turn yields the parametric dependence on the temperature through $\alpha=\mu/T$. The quantities $N$, $T$,and $R$ are free parameters in our model and their values are dictated by physics. At $T=0$ the number of fermions $N$ is restricted by the OV limit $N_{\rm OV}=2.89 \times 10^{9}\, \sqrt{2/g} (15\,{\rm keV}/m)^2 M_{\odot}/m $. However, at nonzero temperature, stable solutions exist $N>N_{\rm OV}$, depending on temperature and radius. In the following $N$ is required to be of the order $2\times 10^{12} M_{\odot}/m$, so that for any $m$, the total mass is close to the estimated mass of the halo \cite{wilk}. As we have demonstrated, the expected temperature of the halo is given by $\alpha\simeq m/T=4\times 10^4$. Our choice of $R=200$ kpc is based on the estimated size of the Galactic halo. The only remaining free parameter is the fermion mass, which we shall fix at $m=15$ keV, and justify its choice {\em a posteriori}. For fixed $N$, there is a range of $\alpha$ where the Thomas-Fermi equation has multiple solutions. For example, for $N=2\times 10^{12}$ and $\alpha=4\times 10^6$ we find six solutions, which we denote by (1), (2), (3), (3'), (2'), and (1') corresponding to the values $\eta_0 =$ -30.53, -25.35, -22.39, 29.28, 33.38, and 40.48, respectively. In Fig.\ \ref{fig1} we plot the density profiles. For negative central value $\eta_0$, for which the degeneracy parameter is negative everywhere, the system behaves basically as a Maxwell-Boltzmann isothermal sphere. Positive values of the central degeneracy parameter $\eta_0$ are characterized by a pronounced central core of mass of about $2.5 \times 10^6 M_{\odot}$ within a radius of about 20 mpc. The presence of the core is obviously due to the degeneracy pressure. A similar structure was obtained in collisionless stellar systems modeled as a nonrelativistic Fermi gas \cite{chav}. \begin{figure}[p] \centering \epsfig{file=fig1.ps,width=10cm} \caption{ The density profile of the halo for $\eta_0=0$ (dotted line) and for the six $\eta_0$-values discussed in the text. Configurations with negative $\eta_0$ ((1)-(3)) are depicted by the dashed and those with positive $\eta_0$ ((1')-(3')) by the solid line. } \label{fig1} \end{figure} Fig.\ \ref{fig1} shows two important features. First, a galactic halo at a given temperature may or may not have a central core depending whether the central degeneracy papameter $\eta_0$ is positive or negative. Second, the closer to zero $\eta_0$ is, the smaller the radius at which the $r^{-2}$ asymptotic behavior of the density begins. The flattening of the Galactic rotation curve begins in the range $1 \simlt r/{\rm kpc} \simlt 10$, hence the solution $3'$ most likely describes the Galactic halo. This may be verified by calculating the rotation curves in our model. We know already from our estimate (\ref{eq005}) that our model yields the correct asymptotic circular velocity of 220 km/s. In order to make a more realistic comparison with the observed Galactic rotation curve, we must include two additional matter components: the bulge and the disk. The bulge is modeled as a spherically symmetric matter distribution of the form \cite{you} \begin{equation} \rho_{\rm b}(s)=\frac{e^{-hs}}{2s^3} \int_0^{\infty} du \frac{e^{-hsu}}{[(u+1)^8-1]^{1/2}} \, , \label{eq006} \end{equation} where $s=(r/r_0)^1/4$, $r_0$ is the effective radius of the bulge and $h$ is a parameter. We adopt $r_0=2.67$ kpc and $h$ yielding a bulge mass $M_{\rm b}= 1.5 \times 10^{10} M_{\odot}$ \cite{suc}. \begin{figure}[p] \centering \epsfig{file=fig2.ps,width=10cm} \caption{ Enclosed mass of the halo plus bulge versus radius for $\eta_0$ = 24 (dashed), 28 (solid), and 32 (dot-dashed line). } \label{fig2} \end{figure} In Fig.\ \ref{fig2} the mass of halo and bulge enclosed within a given radius is plotted for various $\eta_0$. The data points, indicated by squares, are the mass $M_{\rm c}=2.6 \times 10^6 M_{\odot}$ within 18 mpc, estimated from the motion of the stars near Sgr A$^*$ \cite{eck}, and the mass $M_{50}=5.4^{+0.2}_{-3.6}\times 10^{11}$ within 50 kpc, estimated from the motion of satellite galaxies and globular clusters \cite{wilk}. Variation of the central degeneracy parameter $\eta_0$ between 24 and 32 does not change the essential halo features. In Fig.\ \ref{fig3} we plot the circular velocity components: the halo, the bulge, and the disk. The contribution of the disk is modeled as \cite{per} \begin{equation} \Theta_{\rm d}(r)^2= \Theta_{\rm d}(r_{\rm o})^2 \frac{1.97 (r/r_{\rm o})^{1.22}}{ [(r/r_{\rm o})^2+0.78^2]^{1.43}} \, , \label{eq007} \end{equation} where we take $r_{\rm o}=13.5$ kpc and $\Theta_{\rm d}=100$ km/s. Here it is assumed for simplicity that the disc does not influence the mass distribution of bulge and halo. Choosing the central degeneracy $\eta_0=28$ for the halo, the data by Merrifield and Olling \cite{oll} are reasonably well fitted. We now turn to the discussion of our choice of the fermion mass $m=15$ keV for the degeneracy factor $g=2$. To that end we investigate how the mass of the central object, i.e., the mass $M_{\rm c}$ within 18 mpc, depends on $m$ in the interval 5 to 25 keV, for various $\eta_0$. We find that $m\simeq15$ gives always the maximal value of $M_{\rm c}$ ranging between 1.7 and 2.3 $\times 10^6 M_{\odot}$ for $\eta_0$ between 20 and 28. Hence, with $m\simeq 15$ keV we get the value closest to the mass of the central object $M_{\rm c}$ estimated from the motion of the stars near Sgr A$^*$ \cite{eck}. \begin{figure}[p] \centering %\epsfig{file=fig3.ps,width=10cm,rheight=8.0cm} \epsfig{file=fig3.ps,width=10cm} \caption{ Fit to the Galactic rotation curve. The data points are by Olling and Merrifield \protect\cite{oll}, for $R_0=8.5$ kpc and $\Theta_0=220$ km/s. } \label{fig3} \end{figure} In summary, using the Thomas-Fermi theory, we have shown that a weakly interacting self-gravitating fermionic gas at finite temperature yields a mass distribution that successfully describes both the center and the halo of the Galaxy. For a fermion mass $m \simeq 15$ keV, a reasonable fit to the rotation curve is achieved with the temperature $T = 3.75$ meV and the degeneracy parameter at the center $\eta_0=28$. With the same parameters, the masses enclosed within 50 and 200 kpc are $M_{50} = 5.04\times 10^{11} M_{\odot}$ and $M_{200} = 2.04\times 10^{12} M_{\odot}$, respectively. These values agree quite well with the mass estimates based on the motion of satellite galaxies and globular clusters \cite{wilk}. Moreover, the mass of $M_{\rm c} \simeq 2.27\times 10^6 M_{\odot}$, enclosed within 18 mpc, agrees reasonably well with the observations of the compact dark object at the center of the Galaxy. We thus conclude that both the galactic halo and center could be made of the same fermions. %________________________ %{\bf Acknowledgement} This research is in part supported by the Foundation of Fundamental Research (FFR) grant number PHY99-01241 and the Research Committee of the University of Cape Town. 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