SUPERMASSIVE FERMION BALLS AND CONSTRAINTS FROM STELLAR DYNAMICS NEAR Sgr A$^{*}$
NEVEN BILI\'{C}$^{a,b}$
GARY B.\ TUPPER$^a$
and RAOUL D.\ VIOLLIER$^a$
All presently known stellar-dynamical constraints on the size and mass of the supermassive compact dark object at the Galactic center are consistent with a ball of self-gravitating, nearly non-interacting, degenerate fermions with mass between 76 and 491 keV/$c^{2}$, for a degeneracy factor $g$ = 2. Sterile neutrinos of 76 keV/$c^{2}$ mass, which are mixed with at least one of the active neutrinos with a mixing angle $\theta \sim 10^{-7}$, are produced in about the right amount in the early Universe and may be responsible for the formation of the supermassive degenerate fermion balls and black holes at the galactic centers via gravitational cooling.
viollier.tex
%====================================================================%
%                  BLOIS.TEX     2003                              %
% This latex file rewritten from various sources for use in the      %
% preparation of the standard proceedings Volume, latest version     %
% for the Neutrino'96 Helsinki conference proceedings                %
% by Susan Hezlet with acknowledgments to Lukas Nellen.              %
% Some changes are due to David Cassel.                              %
%====================================================================%

%\documentstyle[11pt,blois,epsfig]{article}
\documentclass[11pt]{article}
\usepackage{blois,epsfig}


\bibliographystyle{unsrt}
% for BibTeX - sorted numerical labels by order of
% first citation.

% A useful Journal macro
\def\Journal#1#2#3#4{{#1} {\bf #2}, #3 (#4)}

% Some useful journal names
\def\NCA{\em Nuovo Cimento}
\def\NIM{\em Nucl. Instrum. Methods}
\def\NIMA{{\em Nucl. Instrum. Methods} A}
\def\NPB{{\em Nucl. Phys.} B}
\def\PLB{{\em Phys. Lett.}  B}
\def\PRL{\em Phys. Rev. Lett.}
\def\PRD{{\em Phys. Rev.} D}
\def\ZPC{{\em Z. Phys.} C}

% Some other macros used in the sample text
\def\st{\scriptstyle}
\def\sst{\scriptscriptstyle}
\def\mco{\multicolumn}
\def\epp{\epsilon^{\prime}}
\def\vep{\varepsilon}
\def\ra{\rightarrow}
\def\ppg{\pi^+\pi^-\gamma}
\def\vp{{\bf p}}
\def\ko{K^0}
\def\kb{\bar{K^0}}
\def\al{\alpha}
\def\ab{\bar{\alpha}}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\CPbar{\hbox{{\rm CP}\hskip-1.80em{/}}}

%AUTHOR_STYLES_AND_DEFINITIONS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcommand{\simlt}
{\mbox{\raisebox{-0.5ex}{$\textstyle \; \sim$}
\raisebox{ 0.8ex}{$\textstyle  \!\!\!\!\!\!\! <$  }}}
%
%\parindent 0cm
%\parskip .5cm
%
\newcommand{\simgt}
{\mbox{\raisebox{-0.5ex}{$\textstyle \; \sim$}
\raisebox{ 0.8ex}{$\textstyle  \!\!\!\!\!\!\! >$  }}}
%
%temp replacement due to no font
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                %
%    BEGINNING OF TEXT                           %
%                                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\vspace*{4cm}
\title{SUPERMASSIVE FERMION BALLS AND CONSTRAINTS
FROM STELLAR DYNAMICS NEAR Sgr A$^{*}$}
\author{NEVEN BILI\'{C}$^{a,b}$,
GARY B.\ TUPPER$^a$, and RAOUL D.\ VIOLLIER$^a$
}
\address{$^a$Institute of Theoretical Physics and Astrophysics,
 Department of Physics, University of Cape Town,
 Private Bag, Rondebosch 7701, South Africa \\
Email: viollier@physci.uct.ac.za \\
$^b$Rudjer Bo\v skovi\'c Institute,
P.O. Box 180, 10002 Zagreb, Croatia \\
Email: bilic@thphys.irb.hr}



\maketitle

\abstracts{
All presently known stellar-dynamical constraints
on the
size and mass of the supermassive compact dark object
at the Galactic center are consistent with a ball of
self-gravitating, nearly non-interacting, degenerate
fermions with mass between 76 and 491 keV/$c^{2}$,
for a degeneracy factor $g$ = 2.
Sterile neutrinos of 76 keV/$c^{2}$ mass, which are
mixed with at least one of the active neutrinos with
a mixing angle $\theta \sim 10^{-7}$, are produced in
about the right amount in the early Universe and
may be responsible for the formation of the
supermassive degenerate fermion balls and black
holes at the galactic centers via gravitational
cooling.}
%
\section{Introduction}
In a recent paper Sch\"{o}del {\it et al} reported
a new set of
data \cite{schod1}
%\footnote{R. Sch\"{o}del {\it et al}, {\em Nature} {\bf 419} (2002) 694.},
including the corrected old
measurements \cite{eck2}
%\footnote{A. Eckart {\it et al}, {\em Mon. Not. R.
%Astron. Soc.} {\bf 331} (2002) 917.}
on the projected positions of the star S2(S0-2)
that was observed during the last decade with the
ESO telescopes in La Silla (Chile). The combined
data suggest that S2 (S0-2)is moving on a
Keplerian orbit with a period of 15.2 yr around
the enigmatic strong radiosource Sgr A$^{*}$ that
is widely believed to be a black hole with a mass
of about 2.6 $\times$ 10$^{6} M_{\odot}$
\cite{eck2,ghez3}.
%%%%%%%%%%%%%%%%\end{document}
%\footnote{A. Ghez {\it et al}, {\em Astrophys. J.} {\bf 509} (1998) 678.}
The salient feature of the new adaptive optics
data is that, between April and May 2002,
S2(S0-2) apparently sped past the point of
closest approach with a velocity $v$ $\sim$
6000 km/s at a distance of about 17 light-hours
\cite{schod1} or 123 AU from Sgr A$^{*}$.
Another star, S0-16 (S14), which was observed
during the last few years by Ghez {\it et al}
\cite{ghez4}
%\footnote{A. Ghez {\it et al}, invited talk at the
%Galactic Center conference, 4-8 October, 2002, Kalua-Kona (Hawaii).}
with the Keck telescope in Hawaii, made
recently a spectacular U-turn, crossing the
point of closest approach at an even smaller
distance of 8.32 light-hours or 60 AU from
Sgr A$^{*}$ with a velocity $v$ $\sim$
9000 km/s. Ghez {\it et al} \cite{ghez4} thus conclude
that the gravitational potential around
Sgr A$^{*}$ has approximately $r^{-1}$ form,
for radii larger than 60 AU, corresponding to
1169 Schwarzschild radii of 26 light-seconds
or 0.051 AU for a 2.6
$\times$ 10$^{6} M_{\odot}$ black hole.
Although the baryonic alternatives are
presumably ruled out, this still leaves
some room for the interpretation of the
supermassive compact dark object at the
Galactic center in terms of a finite-size
non-baryonic dark matter object rather
than a black hole. In fact, the supermassive
black hole
paradigm may eventually only be proven
or ruled out by comparing it with
credible alternatives in terms of
finite-size non-baryonic objects
\cite{mun9}.

The purpose of this paper is to explore,
using the example of a sterile neutrino
as the dark matter particle candidate,
the implications of the recent observations
for the degenerate fermion ball scenario of
the supermassive compact dark objects which
was developed during the last decade
\cite{mun9,viol5,viol6,bil7,bil8,mun10,bil11}.

\section{Stellar-dynamical constraints for fermion balls}
%\footnote{R.D. Viollier, D. Trautmann and G.B. Tupper, {\em Phys. Lett.} {\bf B306} (1993) 79.}\footnote{R.D. Viollier, %{\em Prog. Part. Nucl. Phys.} {\bf 32} (1994) 51.}\footnote{N. Bili\'{c}, F. Munyaneza and R.D. Viollier,
%{\em Phys. Rev.} {\bf D59} (1999) 024003.}\footnote{N. Bili\'{c}, R.L. Lindebaum, G.B. Tupper and R.D. Viollier,
%{\em Phys. Lett.} {\bf B515} (2001) 105.}\footnote{F.Munyaneza, D. Tsiklauri and R.D. Viollier, {\em Astrophys. J.}
%{\bf 509} (1998) L105; ibid. {\bf 526} (1999) 744.}
%As the bulk part of the dark matter in the halo of our Galaxy is presumably in the form of massive non-baryonic %particles, it is conceivable that the cold and degenerate dark matter components could sink towards the center, %eventually forming a degenerate supermassive object at the Galactic center \cite{bil8}. In fact, in order to prove or %rule out the black hole scenario, it is essential that we develop credible alternatives to the black hole scenario in %terms of supermassive, finite-size, non-baryonic objects.
In a self-gravitating ball of degenerate
fermionic matter, the gravitational
pressure is balanced by the degeneracy
pressure of the fermions due to the
Pauli exclusion principle.
Nonrelativistically, this scenario is
described by the Lane-Emden equation
with polytropic index $p = 3/2$. Thus
the radius $R$ and mass $M$ of a ball
of self-gravitating, nearly non-interacting
degenerate fermions of mass $m$ and
degeneracy $g$ scale as \cite{viol6}
\begin{eqnarray}
%\begin{array}{lcl}
R=\left[\frac{91.869
\hbar^{6}}{m^{8} G^{3}}\left( \frac{2}{g}
\right)^{2} \frac{1}{M} \right]^{1/3}
\hspace*{- 0.02cm}=3610.66\;\mbox{ld}
\left(\frac{15\;\mbox{keV}}{mc^{2}}
\right)^{8/3} \left(\frac{2}{g} \right)^{2/3}
\left( \frac{M_{\odot}}{M} \right)^{1/3}.
%\end{array}
\end{eqnarray}
Here 1.19129 ld = 1 mpc = 206.265 AU.
% and $m$
%is the fermion mass. The degeneracy factor
%$g$ = 2 describes either spin 1/2
%fermions (without antifermions) or spin
%1/2 Majorana fermions. ($\equiv$
%antifermions). For Dirac fermions and
%antifermions, or spin 3/2 fermions
%(without antifermions), we have $g$ = 4.
Using the canonical value
$M = 2.6 \times 10^{6} M_{\odot}$
and $R$ $\leq$ 60 AU for the supermassive
compact dark object at the Galactic center,
we obtain a minimal fermion mass of
$m_{\rm min}$ = 76.0 keV/$c^{2}$ for
$g$ = 2, or $m_{\rm min}$ = 63.9 keV/$c^{2}$
for $g$ = 4.

The maximal mass for a degenerate
fermion ball, calculated in a general
relativistic framework based on the
Tolman-Oppenheimer-Volkoff equations,
is the Oppenheimer-Volkoff (OV) limit
\cite{bil7}
\begin{equation}
M_{\rm OV} = 0.38322 \;
\frac{M_{\rm Pl}^{3}}{m^{2}} \,
\left( \frac{2}{g} \right)^{1/2}
=  2.7821 \times 10^{9} M_{\odot} \,
\left( \frac{15 \; \mbox{keV}}{mc^{2}}
\right)^{2} \, \left( \frac{2}{g}
\right)^{1/2}  ,
\end{equation}
where
$M_{\rm Pl} = (\hbar c/G)^{1/2} = 1.2210
\times 10^{19}$ GeV is the Planck mass.
Thus, for $m_{\rm min}$ = 76.0 keV/$c^{2}$
and $g$ = 2, or $m_{\rm min}$ = 63.9 keV/$c^{2}$
and $g$ = 4, we obtain
%\begin{equation}
$M_{\rm OV}^{\rm max} = 1.083 \times 10^{8}
M_{\odot}$.
%\end{equation}
In this scenario all supermassive compact
dark objects with mass $M > M_{\rm OV}^{\rm max}$
must be black holes, while those with
$M \leq M_{\rm OV}^{\rm max}$ are fermion balls.
Choosing as the OV limit the canonical mass
of the compact dark object at the center of
the Galaxy,
$M_{\rm OV}^{\rm min}$ = 2.6 $\times$ 10$^{6}$
$M_{\odot}$,
yields a maximal fermion mass of
$m_{\rm max}$ = 491 keV/$c^{2}$
for $g$ = 2, or $m_{\rm max}$ =
413 keV/$c^{2}$ for $g$ = 4.
%In this ultrarelativistic limit, there is
%little difference between the black hole
%and degenerate fermion ball scenarios, as the
%radius of the fermion ball is 4.45 compared
%to 3 Schwarzschild radii for the radius of
%the event horizon of a non-rotating black
%hole of the same mass. In fact, varying the
%fermion mass between $m_{\rm min}$ and $m_{\rm max}$,
%one can smoothly interpolate between a fermion
%ball of the largest acceptable size and a fermion
%ball of the smallest possible size, at the limit
%between fermion balls and black holes.

The masses of the supermassive compact
dark objects
discovered so far at the centers of both
active and
inactive galaxies are all in the range
\cite{korm12}
%\footnote{J. Kormendy and L.G. Ho, astro-ph/0003268; L.C. Ho and J. Kormendy, astro-ph/0003267.}
%\begin{equation}
$10^{6} M_{\odot} \; \simlt \; M \; \simlt \;
3 \times 10^{9} M_{\odot}$.
%\end{equation}
Thus, as $M_{\rm OV}^{\rm max}$ falls into
this range as well, we need both supermassive
fermion balls
($M \leq M_{\rm OV}^{\rm max}$)
and black holes ($M > M_{\rm OV}^{\rm max}$)
to describe the observed phenomenology.
At first sight, such a hybrid scenario
does not
seem to be particularly attractive.
However, it
is important to note that a similar
scenario is
actually realized in Nature, with
the co-existence
of neutron stars
%which have masses
%$M \leq M_{\rm OV}^{n}$,
and stellar-mass black
holes
%with mass $M > M_{\rm OV}^{n}$,
as observed
in stellar binary systems in the Galaxy
\cite{bland13}.
%\footnote{R. Blandford and N. Gehrels, {\em Physics Today}, June 1999, page 40.}
%Here the OV limit
%$M_{\rm OV}^{n}$, which includes the nuclear
%interaction of the neutrons, is  somewhat uncertain
%due to the unknown equation of state.
%But the consensus of the experts \cite{bland13}
%is that it must be in the range
%\begin{equation}
%1.4 \; M_{\odot} \; \leq \; M_{\rm OV}^{n} \;
%\simlt \; 3 \; M_{\odot} \; \; .
%\end{equation}
%None of the observed neutron stars have masses
%larger than 1.4 $M_{\odot}$, while there are at
%least nine candidates for stellar-mass black holes
%larger than 3 $M_{\odot}$ \cite{bland13}.
It is thus conceivable that Nature allows for the
co-existence of supermassive fermion balls and
black holes as well.
%Of course, we would expect
%characteristic differences in the properties of
%supermassive fermion balls and black holes.
%Similarly, pulsars and stellar-mass black holes
%are quite different, as pulsars have a strong
%magnetic field and a hard baryonic surface,
%while black holes are surrounded by an immaterial
%event horizon instead. However, one may also argue
%that the astrophysical differences between
%supermassive black holes and fermion balls close
%to the OV limit are not so easy to detect because
%both objects are of non-baryonic nature.
%%%%%%%%%%%%%%%%%%%%%%
\section{Cosmological constraints for
sterile neutrino dark matter}
%%%%%%%%%%%%%%%%%%%%%%%%
If the supermassive compact dark object at the
Galactic center is indeed a degenerate fermion
ball of mass
$M$ = 2.6 $\times$ 10$^{6} M_{\odot}$ and radius
$R$ $\leq$ 60 AU, the fermion mass must be in the
range
%\begin{eqnarray}
%\begin{array}{c}
$76.0 \; \mbox{keV}/c^{2} \; \leq \; m \; \leq \;
491 \; \mbox{keV}/c^{2} \; \; \mbox{for}
\; \; g = 2$, or
$63.9 \; \mbox{keV}/c^{2} \; \leq \; m \;
\leq \; 413 \; \mbox{keV}/c^{2} \; \;
\mbox{for} \; \; g = 4$.
%\end{array}
%\end{eqnarray}
It would be most economical if this particle
could represent the dark matter particle of
the Universe, as well.
%%%%%MOVED LOWER DOWN TO JUST BEFORE CH.4
%In fact, it has been recently shown \cite{bil8}
%that an extended cloud of degenerate fermionic
%matter will eventually undergo gravitational
%collapse and form a degenerate supermassive
%fermion ball in a few free-fall times, if the
%collapsed mass is below the OV limit. During
%the formation, the binding energy of the nascent
%fermion ball is released in the form of high-energy
%ejecta at every bounce of the degenerate fermionic
%matter through a mechanism similar to gravitational
%cooling that is taking place in the formation of
%degenerate boson stars  \cite{bil8}.
%If the mass of the collapsed object is above
%the OV limit, the collapse inevitably results
%in a supermassive black hole.
%%%%%%
The conjectured fermion could be a sterile
neutrino $\nu_{s}$ which does not participate
in the weak interactions.
We will now assume that its mass and degeneracy
factor is
$m_{s}$ = 76.0 keV/$c^{2}$ and $g_{s}$ = 2,
corresponding
to the largest fermion ball that is consistent
with the stellar-dynamical constraints. In order
to make sure that this fermion is actually
produced in the early Universe it must be mixed
with at least one active neutrino, e.g., the
$\nu_{e}$. Indeed, for an electron neutrino
asymmetry
$L_{\nu_{e}} =
\frac{n_{\nu_{e}} - n_{\overline{\nu}_{e}}}
{n_{\gamma}} \sim 10^{-2}$
and a mixing angle $\theta_{es} \sim 10^{-7}$
\cite{abza14},
incoherent resonant and non-resonant active
neutrino scattering in the early Universe
produces sterile neutrino matter amounting
to the required fraction
$\Omega_{m} h^{2} = \left( 0.135_{- 0.009}^{+0.008}
\right)$ \cite{wmap15} of the critical density
of the Universe today.
% Here $n_{\nu_{e}}$,
%$n_{\bar{\nu}_{e}}$ and $n_{\gamma}$ are the
%electron neutrino, electron antineutrino and
%photon number densities, respectively.
An electron
neutrino asymmetry of $L_{\nu_{e}} \sim 10^{-2}$
is compatible with the observational limits
\cite{kang16,espo17}
which is constrained by
$- 4.1 \times 10^{-2} \; \leq \; L_{\nu_{e}}
\; \leq 0.79$.

It is interesting to note that
incoherent resonant scattering of active
neutrinos produces
quasi-degenerate sterile neutrino matter,
while incoherent non-resonant active neutrino
scattering yields sterile neutrino matter that
has approximately a thermal spectrum
\cite{abza14}. Quasi-degenerate sterile neutrino
matter may contribute towards the formation of
the supermassive compact dark objects at the
galactic centers, while thermal sterile
neutrino matter is mainly contributing to the
dark matter of the galactic halos.
In fact, it has been recently shown \cite{bil8}
that an extended cloud of degenerate fermionic
matter will eventually undergo gravitational
collapse and form a degenerate supermassive
fermion ball in a few free-fall times, if the
collapsed mass is below the OV limit. During
the formation, the binding energy of the nascent
fermion ball is released in the form of high-energy
ejecta at every bounce of the degenerate fermionic
matter through a mechanism similar to gravitational
cooling that is taking place in the formation of
degenerate boson stars  \cite{bil8}.
If the mass of the collapsed object is above
the OV limit, the collapse inevitably results
in a supermassive black hole.
\section{Observability of
degenerate sterile neutrino balls}
The mixing of the sterile neutrino with at
least one of the active neutrinos necessarily
causes
radiative decay of the sterile into an active
neutrino and a photon. The lifetime
is \cite{boeh18}
%il the main decay mode of the $\nu_{s}$
%into three active neutrinos \cite{fie18} with a
%lifetime of
%\begin{equation}
%\tau \left( \nu_{s} \rightarrow 3 \nu \right)
%= \frac{192 \; \pi^{3}}{G_{F}^{2} \; m_{s}^{5}
%\; \sin^{2} \;
%\theta_{es}} = \frac{\tau (\mu^{-} \rightarrow
%e^{-} + \bar{\nu}_{e} + \nu_{\mu})}{\sin^{2}
%\theta_{es}} \;
%\left( \frac{m_{\mu}}{m_{s}} \right)^{5} \; \; ,
%\end{equation}
%which is presumably unobservable as the available
%neutrino energy is too small. Here
%$\tau (\mu^{-} \rightarrow e^{-} + \bar{\nu}_{e}
%+ \nu_{\mu})$ and $m_{\mu}$ are the lifetime and
%mass of the muon. However, there is a subdominant
%radiative decay mode of the sterile into an active
%neutrino and a photon with a branching ratio
%\cite{pal20}
%\begin{equation}
%B (\nu_{s} \rightarrow \nu \gamma) =
%\frac{\tau (\nu_{s} \rightarrow 3 \nu)}{\tau
%(\nu_{s} \rightarrow \nu \gamma)}
%= \frac{27 \; \alpha}{8 \pi} = 0.7840 \times 10^{-2} \; \; ,
%\end{equation}
%where $\alpha = e^{2}/\hbar c$ is the fine
%structure constant. The lifetime of this potentially
%observable decay mode is
%thus
\begin{equation}
\tau \left( \nu_{s} \rightarrow \nu \gamma \right)
= \frac{8 \pi}{27 \; \alpha} \;
\frac{1}{\sin^{2} \theta_{es}} \;
\left( \frac{m_{\mu}}{m_{s}} \right)^{5} \;
\tau \; (\mu^{-} \rightarrow e^{-} +
\bar{\nu}_{e} + \nu_{\mu})
\end{equation}
yielding, for $\theta_{es} = 10^{-7}$ and
$m_{s}$ = 76.0 keV/$c^{2}$, a lifetime of
$\tau (\nu_{s} \rightarrow \nu \gamma) =
0.46 \times 10^{19}$ yr.
%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%
Although the X-ray luminosity due to the
radiative decay of diffuse sterile neutrino dark
matter in the Universe is presumably not observable,
because it is well below the X-ray background
radiation at this energy \cite{abza14}, it is
perhaps possible to detect such hard X-rays in
the case of sufficiently concentrated dark matter
objects.
In fact, this could be the smoking gun for both
the existence of the sterile neutrino and the
fermion balls. For instance, a ball of $M = 2.6
\times 10^{6} M_{\odot}$ consisting of degenerate
sterile neutrinos of mass
$m_{s}$ = 76.0 keV/$c^{2}$ \cite{mun10},
degeneracy factor $g_{s}$ = 2, and mixing angle
$\theta_{es}$ = 10$^{-7}$
would emit 38 keV photons with a luminosity
$L_{X} = 1.6 \times 10^{34} \mbox{erg/s}$
%\frac{M c^{2}}{2 \tau (\nu_{s}
%\rightarrow \nu \gamma)} = 1.6 \times 10^{34} \;
%\mbox{erg/s} \; \; ,
%\end{equation}
within a radius of 60 AU, 8.32 light hours or
7.6 $\times$ 10$^{-3}$ arcsec of Sgr A$^{*}$,
assumed to be at a distance of 8 kpc. The
current upper limit for X-ray emission from the
vicinity of Sgr A$^{*}$ is
$\nu L_{\nu} \sim 3 \times 10^{35}$ erg/s,
for an X-ray energy of $E_{X} \sim 60$ keV
\cite{maha19},
where $L_{\nu} = dL/d \nu$ is
the spectral luminosity. Thus the X-ray line at
38 keV could presumably only be detected if
either the angular or the energy resolution
or both, of the present X-ray detectors are
increased.
\section*{References}
\begin{thebibliography}{99}
\bibitem{schod1} R.\ Sch\"{o}del {\it et al},
{\em Nature}   {\bf 419}, 694 (2002).
\bibitem{eck2} A.\ Eckart {\it et al},
 {\em Mon.\ Not.\ R.\ Astron.\ Soc.}
{\bf 331}, 917 (2002).
\bibitem{ghez3} A.\ Ghez {\it et al},
 {\em Astrophys.\ J.} {\bf 509}, 678 (1998).
\bibitem{ghez4} A.\ Ghez {\it et al},
invited talk at the Galactic Center
conference, 4-8 October, 2002,
Kalua-Kona (Hawaii).
\bibitem{mun9} F.\ Munyaneza, D.\ Tsiklauri
and R.D.\ Viollier, {\em Astrophys.\ J.}
{\bf 509}, L105 (1998); ibid.\
{\bf 526}, 744 (1999).
\bibitem{viol5} R.D.\ Viollier,
D.\ Trautmann and G.B.\ Tupper,
 {\em Phys.\ Lett.} B {\bf 306}, 79 (1993).
\bibitem{viol6} R.D.\ Viollier,
{\em Prog.\ Part.\ Nucl.\ Phys.}
{\bf 32}, 51 (1994).
\bibitem{bil7} N.\ Bili\'{c},
F.\ Munyaneza and R.D.\ Viollier,
 {\em Phys.\ Rev.} D {\bf 59}, 024003 (1999).
\bibitem{bil8}N.\ Bili\'{c},
R.L.\ Lindebaum, G.B.\ Tupper and R.D.\ Viollier,
 {\em Phys.\ Lett.} B {\bf 515}, 105 (2001).
\bibitem{mun10} F.\ Munyaneza and
R.D.\ Viollier, {\em Astrophys.\ J.}
{\bf 564}, 274 (2002).
\bibitem{bil11} N.\ Bili\'{c},
F.\ Munyaneza, G.B.\ Tupper and R.D.\ Viollier,
 {\em Prog.\ Part.\ Nucl.\ Phys.}
{\bf 48}, 291 (2002).
\bibitem{korm12} J.\ Kormendy and L.G.\ Ho,
astro-ph/0003268; L.C.\ Ho and J.\ Kormendy,
astro-ph/0003267.
Published in ``Encyclopedia of Astronomy
and Astrophysics'' (Institute of Physics Publishing),
January 2001.
\bibitem{bland13} R.\ Blandford and N.\ Gehrels,
 {\em Physics Today} , June 1999, p.40.
\bibitem{abza14} K.\ Abazajian, G.M.\ Fuller and
M.\ Patel, {\em Phys.\ Rev.} D {\bf 64}, 023501
(2001).
\bibitem{wmap15} WMAP Collaboration,
D.N.\ Spergel {\it et al}, astro-ph/0302209,
to appear in  {\em Astrophys.\ J.}
\bibitem{kang16} K.\ Kang and G.S.\ Steigman,
 {\em Nucl.\ Phys.} B {\bf 372}, 494 (1992).
\bibitem{espo17} S.\ Esposito, G.\ Mangano,
G.\ Miele and O.\ Pisanti, {\em JHEP}
{\bf 09}, 38 (2000);
M.\ Orito, T.\ Kajino, G.J.\ Matthews and
R.N.\ Boyd, astro-ph/0005446.
\bibitem{boeh18} V.\ Barger, R.J.N.\ Phillips
and S.\ Sarkar, {\em Phys.\ Lett.} B {\bf 352},
365 (1995);
B {\bf 356}, 617(E) (1995).
\bibitem{maha19} R.\ Mahadevan, {\em Nature}
{\bf 394}, 651 (1998); R.\ Narayan {\it et al},
 {\em Astrophys.\ J.} {\bf 492}, 554 (1998).
\end{thebibliography}
\end{document}
%


%%%%%%%%%%%%%%%%%%%%%%
% End of blois.tex  %
%%%%%%%%%%%%%%%%%%%%%%

astro-ph0310172.extracted_bib
\begin{thebibliography}{99}
\bibitem{schod1} R.\ Sch\"{o}del {\it et al},
{\em Nature}   {\bf 419}, 694 (2002).
\bibitem{eck2} A.\ Eckart {\it et al},
 {\em Mon.\ Not.\ R.\ Astron.\ Soc.}
{\bf 331}, 917 (2002).
\bibitem{ghez3} A.\ Ghez {\it et al},
 {\em Astrophys.\ J.} {\bf 509}, 678 (1998).
\bibitem{ghez4} A.\ Ghez {\it et al},
invited talk at the Galactic Center
conference, 4-8 October, 2002,
Kalua-Kona (Hawaii).
\bibitem{mun9} F.\ Munyaneza, D.\ Tsiklauri
and R.D.\ Viollier, {\em Astrophys.\ J.}
{\bf 509}, L105 (1998); ibid.\
{\bf 526}, 744 (1999).
\bibitem{viol5} R.D.\ Viollier,
D.\ Trautmann and G.B.\ Tupper,
 {\em Phys.\ Lett.} B {\bf 306}, 79 (1993).
\bibitem{viol6} R.D.\ Viollier,
{\em Prog.\ Part.\ Nucl.\ Phys.}
{\bf 32}, 51 (1994).
\bibitem{bil7} N.\ Bili\'{c},
F.\ Munyaneza and R.D.\ Viollier,
 {\em Phys.\ Rev.} D {\bf 59}, 024003 (1999).
\bibitem{bil8}N.\ Bili\'{c},
R.L.\ Lindebaum, G.B.\ Tupper and R.D.\ Viollier,
 {\em Phys.\ Lett.} B {\bf 515}, 105 (2001).
\bibitem{mun10} F.\ Munyaneza and
R.D.\ Viollier, {\em Astrophys.\ J.}
{\bf 564}, 274 (2002).
\bibitem{bil11} N.\ Bili\'{c},
F.\ Munyaneza, G.B.\ Tupper and R.D.\ Viollier,
 {\em Prog.\ Part.\ Nucl.\ Phys.}
{\bf 48}, 291 (2002).
\bibitem{korm12} J.\ Kormendy and L.G.\ Ho,
astro-ph/0003268; L.C.\ Ho and J.\ Kormendy,
astro-ph/0003267.
Published in ``Encyclopedia of Astronomy
and Astrophysics'' (Institute of Physics Publishing),
January 2001.
\bibitem{bland13} R.\ Blandford and N.\ Gehrels,
 {\em Physics Today} , June 1999, p.40.
\bibitem{abza14} K.\ Abazajian, G.M.\ Fuller and
M.\ Patel, {\em Phys.\ Rev.} D {\bf 64}, 023501
(2001).
\bibitem{wmap15} WMAP Collaboration,
D.N.\ Spergel {\it et al}, astro-ph/0302209,
to appear in  {\em Astrophys.\ J.}
\bibitem{kang16} K.\ Kang and G.S.\ Steigman,
 {\em Nucl.\ Phys.} B {\bf 372}, 494 (1992).
\bibitem{espo17} S.\ Esposito, G.\ Mangano,
G.\ Miele and O.\ Pisanti, {\em JHEP}
{\bf 09}, 38 (2000);
M.\ Orito, T.\ Kajino, G.J.\ Matthews and
R.N.\ Boyd, astro-ph/0005446.
\bibitem{boeh18} V.\ Barger, R.J.N.\ Phillips
and S.\ Sarkar, {\em Phys.\ Lett.} B {\bf 352},
365 (1995);
B {\bf 356}, 617(E) (1995).
\bibitem{maha19} R.\ Mahadevan, {\em Nature}
{\bf 394}, 651 (1998); R.\ Narayan {\it et al},
 {\em Astrophys.\ J.} {\bf 492}, 554 (1998).
\end{thebibliography}