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%%%%%%%%% An Entropy Consistency Criterion for Neutralino Relic Gases  %%%%%%
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\begin{document}

\preprint{\large \bf \ v3}}

\title{An Entropy Consistency Criterion for Neutralino Relic Gases} 

\author{Luis G. Cabral-Rosetti$^\dagger$, Xavier Hern\'andez$^\ddagger$ 
and Roberto A. Sussman$^\dagger$}

%\email{luis@nuclecu.unam.mx, xavier@astroscu.unam.mx, sussman@nuclecu.unam.mx}

\affiliation{$^\dagger$Instituto de Ciencias Nucleares, Universidad Nacional 
Aut\'onoma de M\'exico (ICN-UNAM), A. P. 70--543, 04510 M\'exico D. F., 
M\'exico. \\ 
$^\ddagger$Instituto de Astronom\'{\i}a, Universidad Nacional Aut\'onoma de 
M\'exico (IA-UNAM), A. P. 70--264, 04510 M\'exico D. F., M\'exico.}

%\date{\today}

\begin{abstract}
We compare two independent estimates of the change in the entropy per particle
of a neutralino gas between two equilibrium states: the freeze-out era and 
present state virialized structures.  Demanding that both estimates yield the
same entropy change for neutralinos that are mostly B-inos or mostly
Higgsinos,  we find the B-ino to be favoured over the  Higgsino channel. These
results are consistent with the theoretical  analysis of cosmic ray data
reported by the HEAT collaboration.

\end{abstract}

\pacs{12.60.Jv, 14.80.Ly, 95.30.Cq, 95.30.Tg, 95.35.+d, 98.35.Gi}

\maketitle

%\section{Decoupling}
There are strong theoretical arguments favouring lightest supersymmetric 
particles (LSP) as making up the relic gas that forms the halos of actual 
galactic structures. Assuming that {\it R} parity is conserved and that 
the LSP is stable, it might be an ideal candidate for cold dark matter 
(CDM), provided it is neutral and has no strong interactions. The most 
favoured scenario \cite{Ellis,Report,Torrente} considers the LSP to be 
the lightest neutralino ($\tilde\chi_1^0$), a mixture of supersymmetric 
partners of the photon, $Z$ boson and neutral Higgs boson \cite{Report}. 
Since neutralinos must have decoupled once they were non-relativistic, 
it is reasonable to assume that they constituted originally a 
Maxwell-Boltzmann (MB) gas in thermal equilibrium with other components 
of the primordial cosmic plasma. In the present cosmic era, such a gas 
is either virialized in galactic and galactic cluster halos and in the 
process of virialization or still in the linear regime for superclusters 
and structures near the scale of homogeneity \cite{Peac}.

The equation of state of a non-relativistic MB neutralino gas is
\cite{Peac,RKT}
%
\bse
\label{MBNR}\ba \rho \ =&& \
m_{\chic{\tilde\chi_0^1}}\,n_{\chic{\tilde\chi_0^1}}\,
\left(1+\frac{3}{2\,x}\right),\qquad p \ = \
\frac{m_{\chic{\tilde\chi_0^1}}\,n_{\chic{\tilde\chi_0^1}}}{x}, 
\label{eqst}\\
 x \ \equiv&& \
\frac{m_{\chic{\tilde\chi_0^1}}}{T},
\label{beta_def}
\ea\ese
%
where $m_{\chic{\tilde\chi_0^1}}$ and $n_{\chic{\tilde\chi_0^1}}$ are the 
neutralino mass and number density.  Since we will deal exclusively with 
the lightest neutralino, we will omit henceforth the subscript 
$_{\chic{\tilde\chi_0^1}}$, understanding that all symbols of physical 
and observational variables (\textit{i.e.} $\Omega_0,\,m,\,\rho,\,n,$ etc.) 
will correspond to this specific particle. 
%
As long as the neutralino gas is in thermal equilibrium, the number density 
$n$ is 
%
\ba 
n \ \approx \ \neqq \ =&& \
\left[\frac{m}{\sqrt{2\,\pi}}\right]^3\,x^{-3/2}\,\exp\,
\left(-x\right),
\label{n_theq}
\ea        
% 
but, in general, $n$ satisfies the Boltzmann equation \cite{Report}
%
\ba 
\dot n + 3\,H\,n \ = \ -\la
\sigma|\textrm{v}|\ra\left[n^2-\left(\neqq\right)^2\right],
\label{boltz}
\ea
%
where $H$ is the Hubble expansion factor and $\la \sigma|\textrm{v}|\ra$ 
is the annihilation cross section. Since this gas is non-relativistic as
annihilation reactions ``freeze out'' and it decouples from the radiation 
dominated cosmic plasma, we can assume that $H=1.66\, \sqrt{g_*}\,T^2/m_p$ 
and
% 
\ba 
\la\sigma|\textrm{v}|\ra \ = \ a \ 
+ \ b\la \textrm{v}^2\ra,
\label{eq<sv>}
\ea 
%
where $m_p=1.22\times 10^{19}$ GeV is Planck's mass, $g_*=g_*(T)$ is the 
sum of relativistic degrees of freedom, $\la \textrm{v}^2\ra$ is the 
thermal averaging of the center of mass velocity (roughly 
$\textrm{v}^2\propto 1/x$ in non-relativistic conditions) and the constants 
$a$ and $b$ are determined by the parameters characterizing specific 
annihilation processes of the neutralino (s-wave or p-wave) \cite{Report}. 
The decoupling of the neutralino gas follows from the condition
%
\ba 
\Gamma \ \equiv \ n\,\la\sigma|\textrm{v}|\ra \ = \ H,
\label{fcond}
\ea
%
leading to the freeze out temperature $T_{\textrm{f}}$. Reasonable 
approximate  solutions of (\ref{fcond}) follow by solving for $x_f$ the 
implicit relation \cite{Report},
%
\ba 
\xf  = 
\ln\left[\frac{0.0764\,m_p\,c_0(2+c_0)\,(a+6\,b/\xf)\,m}{(g_{*{\textrm{f}}}
\,\xf)^{1/2}}\right],
\label{eqxf}
\ea  
%
where $g_{*{\textrm{f}}}=g_*(T_{\textrm{f}}) $ and $c_0\approx 1/2$ yields 
the best fit to the numerical solution of (\ref{boltz}) and (\ref{fcond}). 
>From the asymptotic solution of (\ref{boltz}) we obtain the present 
abundance of the relic neutralino gas \cite{Report}
%
\bse
\ba 
\Omega_0\,h^2 \ = &&\ Y_\infty\, \frac{\CS_0\, m} 
{\rho_{\textrm{crit}}/h^2}
\ \approx \ 2.82\times 10^8\,Y_\infty\,\frac{m}
{\textrm{GeV}},\label{eqOmega0}\\
 Y_\infty \
\equiv  &&\ \frac{n_0}{\CS_0} \nonumber\\ = && 
\left[0.264\,g_{*\textrm{f}}^{1/2}\,m_p\,m\left\{a/\xf+3(b-1/4\,a)
/\xf^2\right\}\right]^{-1},
\nonumber
\\
\label{eqYinf}
\ea
\ese
%
where $\CS_0\approx 4000\,\textrm{cm}^{-3}$ is the present radiation 
entropy density (CMB plus neutrinos),  
$\rho_{\textrm{crit}} = 1.05 \times 10^{-5}\,\textrm{GeV}\,\textrm{cm}^{-3}$. 
%
 
Since we have typical values $\xf\sim 20$ so that $T_{\textrm{f}} \agt$ GeV, 
we can use $g_{*{\textrm{f}}}\simeq 106.75$ \cite{Torrente} in equations 
(\ref{eqxf}) -- (\ref{eqYinf}). Equation (\ref{eqxf}) shows how $\xf$ has 
a logarithmic dependence on $m$, while theoretical considerations 
\cite{Ellis,Report,Torrente} related to the minimal supersymetric extensions 
of the Standard Model (MSSM) yield specific forms for $a$ and $b$ that also 
depend on $m$. Inserting into (\ref{eqOmega0})--(\ref{eqYinf}) the specific
forms of $a$ and $b$ for each annihilation channel leads to a specific range 
of $m$ that satisfies the ``abundance'' criterion based on current 
observational constraints that require $0.2 \alt \Omega_0 \alt 0.4$ and
$h\approx 0.65$ \cite{Peac}.
%
Suitable forms for $\la \sigma |v|\ra$ can be obtained for all types of
annihilation reactions \cite{Report}. If the neutralino is mainly pure 
B-ino, it will mostly annihilate into lepton pairs through t-channel 
exchange of right-handed sleptons. In this case the cross section is 
p-wave dominated and can be approximated by (\ref{eq<sv>}) with 
\cite{Torrente,Moroi,Olive}
%
\ba 
a \ \approx \ 0,\qquad b \ \approx
\ \frac{8\,\pi\,\alpha_1^2}{m^2\,\left[1+m_l^2/m^2\right]^2},
\label{sleptons}
\ea
%
where $m_{l}$ is the mass of the right-handed slepton ($m_{l} \sim m$ 
\cite{Torrente}) and $\alpha_1 = g_1^2/4 \pi \simeq 0.01$ is the fine 
structure coupling constant for the $U(1)_Y$ gauge interaction. If the 
neutralino is Higgsino-like, annihilating  into W-boson pairs, then the 
cross section is s-wave dominated and can be approximated by (\ref{eq<sv>}) 
with \cite{Torrente,Moroi,Olive}
%
\ba 
b \ \approx \ 0,\qquad a \ \approx \
\frac{\pi\,\alpha_2^2\,(1-m_{\chic W}^2/m^2)^{3/2}}
{2\,m^2\,(2-m_{_W}^2/m^2)^2},
\label{Wboson}
\ea
%
where $m_{_W}=80.44$ GeV is the mass of the W-boson and 
$\alpha_2 = g_2^2/4 \pi \simeq 0.03$ is the fine structure coupling 
constant for the $SU(2)_L$ gauge interaction.      
%
In the freeze out era the entropy per particle (in units of the Boltzmann 
constant $\kB$) for the neutralino gas is given by \cite{Peac,RKT}
%
\ba 
\sff \ = \ \left[\frac{\rho + p}{n\,T}\right]_{\textrm{f}} \ = \
\frac{5}{2} \ + \ \xf, 
\label{sf}
\ea
%
where we have assumed that chemical potential is negligible and have used 
the equation of state (\ref{eqst}). From (\ref{eqxf}) and (\ref{sf}), it 
is evident that the dependence of $\sff$ on $m$ will be determined by the 
specific details of the annihilation processes through the forms of $a$ 
and $b$. 

After the freeze out era, particle numbers are conserved and the neutralinos 
constitute a weakly interacting and practically collisionless 
self--gravitating gas. Their evolution between a spectrum of low contrast 
density 
perturbations at freeze out and the final virialized structures is 
extremely complex, involving a variety of dissipative effects characterized 
by collisional and collisionless relaxation processes \cite{Padma2,Padma3}, 
all of which result in extensive entropy production. 
However, the freeze out and present day virialized structures  
correspond (to a good approximation) to ``initial'' and ``final'' equilibrium
states of this gas.  Hence, the microcanonical ensamble in the ``mean field''
approximation  yields an adequate entropy definition that is well defined for
such a  self--gravitating gas at an intermediate scale, between the short
range  and long range regimes of the gravitational potential~\cite{Padma2}.
This  microcanonical entropy can be given in terms of the volume of phase
space 
\cite{Padma3} 
%
\ba 
s \ = \ \ln \,\left[\frac{\,(2mE)^{3/2}\,V\,}{(2\pi\hbar)^3}\right],
\label{mcsdef}
\ea
%
where $V$ and $E$ are local average values of volume and energy associated 
with the intermediate scale. For  non-relativistic velocities $v/c\ll 1$ 
prevailing in present day virialized halo gas, we have 
$V\propto 1/n\propto m/\rho$ and $E\propto m\,v^2/2\propto m/x$. In fact, 
under these assumptions definition (\ref{mcsdef}), evaluated at the 
freeze out, is consistent with (\ref{n_theq}) and (\ref{sf}), and so it is 
also valid immediately after the freeze out era (once particle numbers 
are conserved). Since (\ref{mcsdef}) is valid at both the initial and final 
states, respectively corresponding to the decoupling ($\sff,\,\xf,\,\nf$) 
and ($\sha,\,\xha,\,\nha$) values associated with a suitable halo 
structure, the change in entropy per particle that follows from 
(\ref{mcsdef}) between these two states is given by
%
\ba 
\Delta s = \ 
\ln\,\left[\frac{\nf}{\nha}\left(\frac{\xf}{\xha}\right)^{3/2}
\right],
\label{Delta_s}
\ea
%
The intermediate scale of the microcanonical description is an excellent 
approximation for gas particles near the symmetry center of the halo where 
the gas density enhancement is maximum but spacial gradients of all 
macroscopic quantities are negligible~\cite{Padma2,Padma3}. Hence, we will 
consider current halo macroscopic variables as evaluated at the center of 
the halo: $\shac,\,\xhac,\,\nhac$.      

In order to use equation (\ref{Delta_s}) we need to relate it to present 
day cosmological parameters $\Omega_0$ and $h$. Since density 
perturbations at the freeze out era were very small 
($\delta\,\nf/\nf \alt 10^{-4}$, \cite{Peac}), the density $\nf$ is  
practically homogeneous and so we can estimate it from the conservation 
of particle numbers: $\nf = n_0 \,(1+z_{\textrm{f}})^3$, and of photon 
entropy: 
$g_{*\textrm{f}}\CS_{\textrm{f}} = g_{*0}\,\CS_0\, \,(1+z_{\textrm{f}})^3$, 
valid from the freeze out era to the present for the unperturbed 
homogeneous background. Eliminating $ (1+z_{\textrm{f}})^3$ from these 
conservation laws yields
%
\bse
\label{eqnf}\ba \nf \ = \
n_0\,\frac{g_{*\textrm{f}}}{g_{*0}}\left[\frac{T_{\textrm{f}}}
{T_0^{\textrm{\tiny{CMB}}}}\right]^3
\ \simeq \ 27.3\,n_0\,
\left[\frac{x_0^{\textrm{\tiny{CMB}}}}{\xf}\right]^3,\\
\textrm{where}\quad x_0^{\textrm{\tiny{CMB}}} \ \equiv \
\frac{m}{T_0^{\textrm{\tiny{CMB}}}}
\ = \ 4.29\,\times\,10^{12}\,\frac{m}{\textrm{GeV}}
\ea\ese
%
where $g_{*0}=g_*(T_0^{\textrm{\tiny{CMB}}})\simeq 3.91$ and
$T_0^{\textrm{\tiny{CMB}}}=2.7\,\textrm{K}$. Since for present day conditions 
$n_0/\nhac=\rho_0/\rhohac$ and $\rho_0=\rho_{\textrm{crit}}\,\Omega_0\,h^2 $, 
we collect the results from (\ref{eqnf}) and write (\ref{Delta_s}) as 
%
\ba
\begin{array}{c}
\displaystyle 
\Delta s = 90.56 +
\ln\left[\left(\frac{m}{\textrm{GeV}}\right)^3\,\frac{h^2\,\Omega_0}
{(\xf\,\xhac)^{3/2}}\,\frac{\rho_{\textrm{crit}}}{\rhohac}\right]
\\[0.5cm]
\displaystyle
= \ 79.10 \ 
+ \ln\left[\left(\frac{m}{\textrm{GeV}}\right)^3\,\frac{h^2\,\Omega_0}
{(\xf\,\xhac)^{3/2}}\,\frac{\textrm{GeV/cm}^3}{\rhohac}\right]\, .
\label{shalo}
\end{array}
\ea 
%
Therefore, given $m$ and a specific form of $\la\sigma|\textrm{v}|\ra$ 
associated with $a$ and $b$, equation (\ref{shalo}) provides a theoretical 
estimate of the change in entropy per particle of the neutralino halo gas 
that resulted from the sum of all processes which acted between the original
equilibrium state at freeze out, and now,
to produce the relaxed, virialized galactic haloes we see today.
This is a function of 
the initial state given by $\xf$ in (\ref{eqxf}) and (\ref{sf}), on 
observable cosmological parameters $\Omega_0,\,h$ and on generic state 
variables associated to the halo structure. 

Realistic halo models follow from ``energy truncated'' (ET) distribution 
functions \cite{Padma3, BT, Katz2} that assume a maximal ``cut off'' velocity 
(an escape velocity). We can provide then a convenient empirical estimate 
of the halo entropy, $\shac$, from the microcanonical entropy definition 
(\ref{mcsdef}) in terms of phase space volume, but restricting this volume 
to the actual range of velocities (\textrm{i.e.} momenta) accessible to the 
central particles, that is up to a maximal escape velocity $v_e(0)$. From 
theoretical studies of dynamical and thermodynamical stability associated 
with ET distribution functions \cite{Katz2,cohn,HM} and from observational 
data for elliptic and LSB galaxies and clusters \cite{young,DBM,FDCHA2}, 
it is reasonable to assume
%
\ba 
v_e^2(0) \ = \ 2\,|\Phi(0)| \ \simeq \ \alpha \, \sigha^2(0),\quad 12\alt
\alpha\alt 18,\label{alphas}
\ea
%
where $\Phi(r)$ is the newtonian gravitational potential and 
$\sigha(0)=c^2/\xhac$ is the velocity dispersion at the center \cite{BT}. 
We  have then an observational estimate of the entropy per particle in present 
day galactic halos  
%
\ba &&\shac \ \simeq \
\ln\left[\frac{m^4\,v_{e}^3}
{(2\pi\hbar)^3\,\rhohac}\right]\nonumber\\ =&& 89.17
+\ln\left[\left(\frac{m}{\textrm{GeV}}\right)^4\,
\left(\frac{\alpha}{\xha_c}\right)^{3/2}
\,\frac{\textrm{GeV/cm}^3} {\rhohac}
\right],
\label{SHALO}
\ea    
%
Similar entropy expressions have been discussed in 
\cite{LGM}.
%
We can now evaluate the entropy change directly by subtracting (\ref{sf})
from (\ref{SHALO}), leading to
%
\ba
\Delta s^{\prime} && = \ 86.67 - \xf +\nonumber\\
&& \ \ln\left[ 
\left({\frac{\alpha}{\xha_c}}\right)^{3/2}\,
\left({\frac{m}{\textrm{GeV}}}\right)^{4}\,
\frac{\textrm{GeV/cm}^3} {\rhohac}
\right].
\label{constr}
\ea
%
Since we have evaluated the same quantity by means of two independent
estimates, we have a ``consistency criterion''  
%
\ba
\Delta s^{\prime} \ = \ \Delta s,
\label{Entropycons}
\ea
%
which, with the help of (\ref{shalo}) and (\ref{constr}), implies the
following non--trivial constraint 
%
\ba
\ln(h^{2} \Omega_0) \ = \ 7.57 - \xf + \ln\left[(\alpha\, \xf)^{3/2}
\frac{m}{\textrm{GeV}}
\right].
\label{Result}
\ea
%
linking the variables on which each of the two estimates depend. This relation
is a new estimate of the cosmological parameters $h^{2} \Omega_0$, as a
function of a structural quantity of galactic dark matter halos, $\alpha$,
the  mass of the dark matter particle, $m$ and the
temperature of the neutralino gas at freeze out, $\xf$. This last quantity
depends explicitly not only on the mass of the dark matter particle, but also
on its interaction cross section, and hence on its detailed phenomenological
physics via (\ref{eqxf}). 

In order to plot  $\Omega_0$ vs. $m$ by means of (\ref{Result}), we need to
find
$\xf=\xf\,(a,b,m)$ by solving (\ref{eqxf}) numerically and then to provide
adequate values for the constants $a$ and $b$. Taking $h=0.65$ and
given the uncertainty range of $\alpha$, we will obtain a specific region in
the $\Omega_0-m$ plane for each neutralino channel. Considering first
Higgsino-like neutralinos associated with (\ref{Wboson}), we obtain the shaded
region in figure 1a. In the same figure we have also plotted the relation
$\Omega_0=\Omega_0(m)$ provided by the abundance criterion (\ref{eqOmega0}).
Notice that within the observationally determined range of \,$0.2 < \Omega_0 <
0.4$ (horizontal dash lines), there is no intersection between the shaded
region and the abundance criterion curve. This implies that both criteria are
mutually inconsistent, and the option of Higgsino--like neutralinos making up
both cosmological and galactic dark matter, appears unlikely.  

Next we apply the same methodology for the mainly B-ino case associated with
(\ref{sleptons}). As shown in figure 1b, in this case the abundance criterion
curve falls  well within the shaded region defined by the entropy criterion.
Although we cannot improve on the mass estimate provided by the abundance
criterion alone, the consistency of both criteria reveals the B--ino channel
neutralino as a viable option for a dark matter particle candidate. 

\begin{figure*}[htb]
\centerline{
\epsfig{figure=fig1a_1b.eps,width=18.0cm}}
\caption{
Figures (a) and (b) respectively correspond to the Higgsino and B-ino 
channels. The shaded regions display $\Omega_0$ vs $m$ from our entropy
criterion  (\ref{Result}), the solid curve denoting $\Omega_0$ from the
cosmological abundance criterion (\ref{eqOmega0}). In all cases we take
$h=0.65$. The horizontal dashed lines give current estimates of
$\Omega_0=0.3\pm0.1$.  It is evident that the B-ino channels yields a better
fitting  of both the abundance and the entropy criteria, yielding a neutralino
mass range of $155-230$ GeV, consistent with HEAT results \cite{HEAT-TH}.}
\label{fig1}
\end{figure*}
%
The consistency test that we have derived for the neutralinos can be applied to
any given thermal dark matter candidate that can be proposed as the
constituent particle of the present galactic dark matter halos. The
specificity of each particle and/or annihilation channel follows from the
choice of forms for $a$ and $b$, obtained from adequate considerations of the
phenomenology of each case. The method is robust and is of a very general
applicability,  as it is largely insensitive to (i) the details of the
structure formation scenario that might be assumed and (ii) the complexities
of processes involved in the evolution of the relic gas from
its freeze--out to its actual state in virialized structures (entropy is a
function of state).   

Recent theoretical work by  E. A. Baltz \textit{et al.} \cite{HEAT-TH} 
confirmed that neutralino annihilation in the galactic halo can produce 
enough positrons to make up for the excess of cosmic ray positrons 
experimentally detected by the HEAT collaboration \cite{HEAT2}. Baltz 
\textit{et al.} concluded that for a boost factor $B_s \sim 30$ the 
neutralinos must be primarily B-inos with mass around 160 GeV. For 
$30 \alt B_s \alt 100$, the gaugino--dominated SUSY models complying 
with all constraints yield neutralino masses in the range of 
$150\,\textrm{GeV} \alt  m_{\chic{\tilde\chi^1_0}} \alt 400 \,\textrm{GeV}$. 
On the other hand, Higgsino dominated neutralinos are possible but only 
for $B_s \sim 1000$ with masses larger than 2 TeV. The results that we have 
presented in this letter are in agreement with these predictions, as we 
obtain roughly the same mass range for the B-ino dominated case 
(see figure 1b) and the Higgsino channel is shown to be less favoured in 
the mass range lower than TeV's.       
%
The current study of the parameter space of annihilation channels of LSP's 
in  MSSM is mostly based on finding out (by means of (\ref{eqxf}) and 
(\ref{eqOmega0})--(\ref{eqYinf})) the parameter ranges that yield relic gas
abundances compatible with observational 
constraints~\cite{Ellis,Report,Torrente}. However, equations 
(\ref{eqxf}) and (\ref{eqOmega0})--(\ref{eqYinf}) by themselves are 
insufficient to discriminate between annihilation channels. A more efficient 
study of the parameter space of MSSM can be achieved by the joint 
application of the abundance and entropy consistency criteria, but 
considering more general cross section terms (see for example \cite{Report}) 
than the simplified approximated forms (\ref{sleptons}) and (\ref{Wboson}).
More than providing final predictions, the aim of this letter is to introduce a
methodology, which in spite of the approximations involved, can be helpful
in the understanding of the physics of dark matter particles. This work is
currently in progress.
%
\begin{acknowledgments}
We acknowledge support from {\bf DGAPA-UNAM} grant {\tt IN109001} 
and {\bf CoNaCyT} grants {\tt I37307-E} and {\tt I39181-E}. 
\end{acknowledgments}
%
\begin{thebibliography}{99}

%Neutralino

\bibitem{Ellis}
John Ellis, Summary of DARK 2002: {\it 4th International Heidelberg 
Conference on Dark Matter in Astro and Particle Physics}, Cape Town, 
South Africa, 4-9 Feb. 2002. e-Print Archive: .

\bibitem{Report} G. Jungman, M. Kamionkowski and K. Griest,
{\it Phys. Rep.} {\bf 267}, 195 - 373 (1996).

\bibitem{Torrente} S. Khalil, C. Mu\~noz and E. Torrente-Lujan 
{\it New Jour. Phys.} {\bf 4}, 27 (2002). e-Print Archive: 
.

%libros de texto

\bibitem{Peac} \textit{Cosmological Physics}, J.A. Peacocok,
Cambridge University Press, 1999. See also \textit{Fundamentals of Cosmology},
J. Rich, Springer--Verlag, 2001.

\bibitem{RKT} \textit{Relativistic Kinetic Theory}, S.R. de Groot, W.A. van
Leeuwen and Ch.G. van Weert, North Holland Publishing Co., 1980.

%Secciones Eficases

\bibitem{Moroi} T. Moroi, M. Yamaguchi and T. Yanagida, 
{\it Phys. Lett.} {\bf B 342}, 105 - 110 (1995).

\bibitem{Olive} Keith A. Olive and Mark Srednicki, 
{\it Phys. Lett.} {\bf B 230}, 78 - 82 (1989).

%Termodinamica y Mecanica Estadistica

\bibitem{Padma2} T. Padmanabhan,
{\it Phys. Rep.} {\bf 188}, 285 - 362 (1990).

\bibitem{Padma3} T. Padmanabhan,\textit{Theoretical Astrophysics, Volume I:
Astrophysical Processes}, Cambridge University Press, 2000. 

%Dinamica galactica y FD

\bibitem{BT} \textit{Galactic Dynamics}, J. Binney and S. Tremaine, Priceton
University Press, 1987.

\bibitem{Katz2} J. Katz, \textit{MNRAS}, {\bf 190}, 497,
(1980).

\bibitem{cohn} H. Cohn, \textit{ApJ}, {\bf 242}, 765,
(1980).

\bibitem{HM} J. Hjorth and J. Madsen, \textit{MNRAS}, {\bf 253}, 703,
(1991).

\bibitem{young} P.J. Young, , \textit{ApJ}, {\bf 81}, 807,
(1976).

\bibitem{DBM} W.J.G. de Blok and S.S. McGaugh, \textit{MNRAS}, {\bf 290}, 533,
(1997).

\bibitem{FDCHA2} C. Firmani \textit{et al}, \textit{MNRAS}, {\bf 321}, 713,
(2001).

\bibitem{LGM} G.B. Lima Neto, D. Gerbal and I. M\'arquez, LANL preprint 
.

\bibitem{IS3} I.T. Iliev and P.R. Shapiro, \textit{ApJ}, {\bf 546}, L--5,
(2001).

\bibitem{HEAT-TH} E. A. Baltz, J. Edsj\"{o}, K. Freese and P. Gondolo, 
\textit{Phys. Rev. D},
\textbf{65}, 063511 (2002).

\bibitem{HEAT2} HEAT--pbar Collaboration, S. Coutu \textit{et al.}, in 
\textit{Proceedings of 27th
ICRC, 2001.}

\end{thebibliography}

\end{document}



