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% add words to TeX's hyphenation exception list

\hyphenation{author another created financial paper re-commend-ed
colla-bo-ration dege-ne-racy}

% declarations for front matter

\title{Relic neutrino asymmetry, CMB and large-scale structure}
\author{Sergio Pastor\thanks{Speaker, e-mail: pastor@sissa.it}$~^{\rm a}$
and 
Julien Lesgourgues\thanks{E-mail: lesgour@sissa.it.
Work supported by INFN and by the TMR network grant
ERBFMRXCT960090.}
\address{SISSA--ISAS and INFN, Sezione di Trieste\\
Via Beirut 2-4, I-34013 Trieste, Italy}}

\begin{document}

\begin{abstract}
We consider some consequences of the presence of a cosmological lepton
asymmetry in the form of neutrinos. A relic neutrino degeneracy enhances
the contribution of massive neutrinos to the present energy density of
the Universe, and modifies the power spectrum of radiation and matter.
Comparing with current observations of cosmic microwave background
anisotropies and large scale structure, we derive some constraints on
the relic neutrino degeneracy and on the spectral index in the case of a
flat Universe with a cosmological constant.
\end{abstract}

% typeset front matter (including abstract)

\maketitle

\section{Introduction}

It is generally assumed that our Universe contains an approximately
equal amount of leptons and antileptons. The lepton asymmetry would be
of the same order as the baryon asymmetry, which is very small as
required by Big Bang Nucleosynthesis (BBN) considerations.  The
existence of a large lepton asymmetry is restricted to be in the form
of neutrinos from the requirement of universal electric neutrality,
and the possibility of a large neutrino asymmetry is still open. {}From
a particle physics point of view, a lepton asymmetry can be generated
by an Affleck-Dine mechanism \cite{AF} without producing a large
baryon asymmetry (see ref.~\cite{Casas} for a recent model), or even
by active-sterile neutrino oscillations after the electroweak phase
transition \cite{Foot}. 

We have studied some cosmological implications of relic degenerate
neutrinos \cite{Paper} (here degenerate refers to
neutrino-antineutrino asymmetry, not to mass degeneracy). We do not
consider any specific model for generating such an asymmetry, and just
assume that it was created well before neutrinos decouple from the
rest of the plasma.  An asymmetry of order one or larger can have
crucial effects on the global evolution of the Universe. Among other
effects, it changes the decoupling temperature of neutrinos, the
primordial production of light elements at BBN, the time of equality
between radiation and matter, or the contribution of relic neutrinos
to the present energy density of the Universe. The latter changes
affect the evolution of perturbations in the Universe. We focus on the
anisotropies of the Cosmic Microwave Background (CMB), and on the
distribution of Large Scale Structure (LSS). We calculate the power
spectrum of both quantities, in the case of massless degenerate
neutrinos, and also for neutrinos with a mass of $0.07$ eV, as
suggested to explain the experimental evidence of atmospheric neutrino
oscillations at Super-Kamiokande \cite{SK}.

The effect of neutrino degeneracy on the LSS power spectrum was
studied in ref.~\cite{Larsen}, as a way of improving the agreement
with observations of mixed dark matter models with eV neutrinos, in
the case of high values of the Hubble parameter.  Adams \& Sarkar
\cite{Sarkar} calculated the CMB anisotropies and the matter power
spectrum, and compared them with observations in the
$\Omega_\Lambda=0$ case for massless degenerate neutrinos. More
recently, Kinney \& Riotto \cite{Kinney} also calculated the CMB
anisotropies for massless degenerate neutrinos in the
$\Omega_\Lambda=0.7$ case.

\vspace{-0.25cm}
\section{Energy density of massive degenerate neutrinos}
\label{energy}

The energy density of one species of massive degenerate neutrinos and
antineutrinos, described by the distribution functions $f_\nu$ and
$f_{\bar{\nu}}$, is (we use $\hbar=c=k_B=1$ units)
%
\beq
\rho_\nu \! + \! \rho_{\bar{\nu}}= \!\!
\int_0^\infty \!\!\! \frac{dp}{2\pi^2} ~p^2 \sqrt{p^2 \! + \! m_\nu^2}
(f_\nu(p) \! + \! f_{\bar{\nu}}(p))
\label{defrhonu}
\eeq
%
valid at any moment. Here $p$ is the magnitude of the 3-momentum and
$m_\nu$ is the neutrino mass.

When the early Universe was hot enough, the neutrinos were in
equilibrium with the rest of the plasma via the weak interactions. In
that case the distribution functions $f_\nu$ and $f_{\bar{\nu}}$
changed with the Universe expansion, keeping the form of a Fermi-Dirac
distribution,
%
\beq 
f_{\nu,\bar{\nu}}(p)=\Frac{1}{\exp \left(\frac{p}{T_\nu} \mp
\frac{\mu}{T_\nu}\right)+1}
\label{FD}
\eeq
%
Here $\mu$ is the neutrino chemical potential, which is nonzero if a
neutrino-antineutrino asymmetry has been previously produced. Later
the neutrinos decoupled when they were still relativistic, and from
that moment the neutrino momenta just changed according to the
cosmological redshift.  If $a$ is the expansion factor, the neutrino
momentum decreases keeping $ap$ constant.  At the same time the
neutrino degeneracy parameter $\xi \equiv \mu/T_\nu$ is conserved,
with a value equal to that at the moment of decoupling. Therefore one
can still calculate the energy density of neutrinos now from
\eq{defrhonu} and \eq{FD}, replacing $\mu/T_\nu$ by $\xi$ and
$p/T_\nu$ by $p/(y_\nu T_0)$, where $T_0 \simeq 2.726$ K and $y_\nu$
is the present ratio of neutrino and photon temperatures, which is not
unity because once decoupled the neutrinos did not share the entropy
transfer to photons from the successive particle annihilations that
occurred in the early Universe. 
%In the standard case, the massless
%non-degenerate neutrinos decoupled just before the electron-positron
%pairs annihilated to photons, from which one gets the standard factor
%$y_\nu=(4/11)^{1/3}$.

In the presence of a significant neutrino degeneracy $\xi$ the
decoupling temperature $T(\xi)$ is higher than in the standard case,
\cite{Freese,Kang}. The reaction rate $\Gamma$ of the weak processes,
that keep the neutrinos in equilibrium with the other species, is
reduced because some of the initial or final neutrino states will be
occupied. The authors of ref.~\cite{Kang} found that the neutrino
decoupling temperature is $T_{dec}(\xi) \approx
0.2\xi^{2/3}\exp(\xi/3)$ MeV (for $\nu_\mu$ or $\nu_\tau$). Therefore
if $\xi$ is large enough, the degenerate neutrinos decouple before the
temperature of the Universe drops below the different mass thresholds,
and are not heated by the particle-antiparticle annihilations,
reducing the ratio of neutrino and photon temperatures with respect
to the standard value $y_\nu=(4/11)^{1/3}$.

The present contribution of these degenerate neutrinos to the energy
density of the Universe can be parametrized as $\rho_\nu = 10^4 h^2
\Omega_\nu$ eV cm$^{-3}$, where $\Omega_\nu$ is the neutrino energy
density in units of the critical density $\rho_c=3H^2M_P^2/8\pi$,
$M_P=1.22 \times 10^{19}$ GeV is the Planck mass and $H=100h$ Km
s$^{-1}$ Mpc$^{-1}$ is the Hubble parameter.
%
The value of $\rho_\nu$ can be calculated as a function of the
neutrino mass and the neutrino degeneracy $\xi$, or equivalently the
present neutrino asymmetry $L_\nu$ defined as the following ratio of
number densities
%
\beq 
L_\nu \equiv \frac{n_\nu-n_{\bar{\nu}}}{n_\gamma} =
\frac{1}{12\zeta (3)} y^3_\nu [\xi^3 + \pi^2 \xi]
\label{Lnu}
\eeq
%
We show\footnote{Here we assume $\xi>0$, but the results are also
valid for $\xi<0$ provided that $\xi$ and $L_\nu$ are understood as
moduli.} in figure \ref{lnumass} the contours in the $(m_\nu,L_\nu)$
plane that correspond to some particular values of $h^2
\Omega_\nu$. In the limit of small degeneracy (vertical lines) one
recovers the well-known bound on the neutrino mass $m_\nu \lsim 46$ eV
for $h^2 \Omega_\nu=0.5$. On the other hand, for very light
neutrinos the horizontal lines set a maximum value on the neutrino
degeneracy, that would correspond to a present neutrino chemical
potential $\mu_0 \lsim 7.4 \times 10^{-3}$ eV, also for $h^2
\Omega_\nu=0.5$. In the intermediate region of the figures the
neutrino energy density is $\rho_\nu \simeq m_\nu n_\nu (\xi)$ and the
contours follow roughly the relation
$L_\nu (m_\nu/\mbox{eV})\simeq 24.2 h^2\Omega_\nu$.

A similar calculation has been recently performed in reference
\cite{PalKar}. Note however that the ratio of neutrino and
photon temperatures was not properly taken into account for large
$\xi$.

The presence of a neutrino degeneracy can modify the outcome of BBN
(for a review see \cite{Sarkar96}). First a larger neutrino energy
density increases the expansion rate of the Universe, thus enhancing
the primordial abundance of $^4$He. This is valid for a nonzero $\xi$
of any neutrino flavor.  In addition if the degenerate neutrinos are
of electron type, they have a direct influence over the weak processes
that interconvert neutrons and protons. This last effect depends on
the sign of $\xi_{\nu_e}$, and one gets $-0.06 \lsim
\xi_{\nu_e} \lsim 1.1$ \cite{Kang},
%Both effects may be simultaneously important and it could be possible in
%principle to explain the observed primordial abundances with a large
%baryon density, $\Omega_B h^2 \approx 1$ \cite{Freese,Kang}.  However
%this possibility is ruled out by the fact that in that case our
%Universe would have been radiation dominated during a longer period
%and the observed large-scale structure would be difficult to
%explain.
%
%\beq
%-0.06 \lsim \xi_{\nu_e} \lsim 1.1
%\label{bbn}
%\eeq
%
while a sufficiently long matter dominated epoch requires
$|\xi_{\nu_\mu,\nu_\tau}| \lsim 6.9$ \cite{Kang}.  This estimate
agrees with our analysis in section \ref{comparison} and places a
limit shown by the horizontal line in figure \ref{lnumass} in
the case of degenerate $\nu_\mu$ or $\nu_\tau$.
%
\begin{figure}[htb]
\vspace{-0.5cm}
\centerline{\psfig{file=lnumass.ps,angle=-90,width=0.49\textwidth}}
\vspace{-0.75cm}
\caption{Present energy density of massive degenerate neutrinos as a
function of the neutrino asymmetry.}
%The curves correspond to
%different values of $h^2 \Omega_\nu$ and the horizontal line is the upper 
%bound from \eq{lsf}.}
\label{lnumass}
\end{figure}
%
%\section{Power spectra calculation}
%\label{power}
%
%We compute the power spectra of CMB anisotropies and large-scale
%structure using the Boltzmann code {\tt cmbfast} by Seljak \&
%Zaldarriaga \cite{SelZal}, adapted to the case of one family of
%degenerate neutrinos ($\nu$, $\bar{\nu}$), with mass $m_\nu$ and
%degeneracy parameter $\xi$.  Let us first review the required
%modifications. We use the notations of Ma \& Bertschinger
%\cite{MaBer}, and for all issues not specific to our case, we refer
%the reader to this review.
%
%Background quantities can be rewritten in terms of the dimensionless
%parameters\footnote{We assume $y_\nu = (4/11)^{1/3}$, and
%therefore $\xi \leq 12$ \cite{Kang}.} $M=m_\nu/T_{\nu 0}$ and $Q =
%ap/T_{\nu 0}$, where the scale factor is defined so that $a = 1$
%today.  For neutrinos with $m_\nu=0.07$ eV, $M \simeq 417$.  The mean
%density, pressure and phase-space distributions can be writen as
%functions of $(M,Q,\xi)$, and in the case of massive degenerate
%neutrinos, they must be calculated for each value of the scale factor,
%and also at the beginning of the code in order to find $\Omega_{\nu}$
%today. On the other hand, for massless neutrinos, there is an exact
%analytic solution
%
%\begin{equation}
%\bar{\rho}_{\nu} + \bar{\rho}_{\bar{\nu}}
%= 3 (\bar{P}_{\nu} + \bar{P}_{\bar{\nu}} )
%= \frac{7}{8} 
%\frac{\pi^2}{15} 
%T^4_{\nu} N_{eff} (\xi)
%\end{equation}
%
%where we have defined an effective number of massless neutrino families
%$N_{eff} \equiv 3 + 30/7 (\xi / \pi)^2 + 15/7 ( \xi / \pi )^4$. Thus
%the mean density and pressure of degenerate neutrinos are given by the
%ones for one massless non-degenerate family, multiplied by $N_{eff}$.
%Let us now consider perturbed quantities.  We define $\Psi_{\nu}$ and
%$\Psi_{\bar{\nu}}$, the perturbations of the phase space distribution for
%$\nu$ and $\bar{\nu}$, through
%%
%\begin{eqnarray}
%\delta f_{\nu} (\vec{x}, Q, \hat{n}, \tau) &=& f_{\nu} (Q) 
%\Psi_{\nu} (\vec{x}, Q, \hat{%n}, \tau)~, \
%\nonumber \\
%
%\delta f_{\bar{\nu}} (\vec{x}, Q, \hat{n}, \tau) 
%&=& f_{\bar{\nu}} (Q) \Psi_{\bar{\nu}} %(\vec{x}, Q, 
%\hat{n}, \tau)
%\end{eqnarray}
%
%($\hat{n}$ is the momentum direction: $\vec{p} \equiv p \hat{n}$).
%For our purpose, which is to integrate the linearized Einstein equations, 
%it can be shown that only the following linear combination is relevant
%
%\begin{equation}
%\Psi \equiv \frac{ f_{\nu} \Psi_{\nu} + f_{\bar{\nu}} \Psi_{\bar{\nu}}}
%{ f_{\nu} + f_{\bar{\nu}}}.
%\end{equation}
%
%Using the Boltzmann equations for $\Psi_{\nu}$ and $\Psi_{\bar{\nu}}$,
%it is straightforward to show that the evolution of $\Psi$ (in Fourier
%space and in the synchronous gauge, see \cite{MaBer}, eq. (40)) 
%depends on $\xi$ only through the gravitational source term, which
%is proportional to the quantity $(d \ln ( f_{\nu} )/d \ln Q)$.
%In the case $\xi=0$, this quantity has a
%simple interpretation: it is the $Q$-dependence of a planckian
%perturbation of the phase space distribution. In other words, a shift
%of the blackbody temperature $\Delta T / T(\vec{x}, \hat{n}, \tau)$
%corresponds to a perturbation
%
%\begin{equation}
%\Psi (\vec{x}, Q, \hat{n}, \tau) 
%= - \frac{\Delta T}{T}(\vec{x}, \hat{n}, \tau) 
%\frac{d \ln ( f_{\nu} )}{d \ln Q}. 
%\end{equation}
%
%Thus the planckian shape is unaltered for massless neutrinos or
%for massive neutrinos when they are still relativistic. When
%$\xi\neq0$, one gets
%
%\begin{equation} \label{dlnfdlnq}
%\frac
%{d \ln ( f_{\nu} + f_{\bar{\nu}} )}
%{d \ln Q} =
%- 
%\frac
%{Q (1 + \ch \xi~\ch Q)}
%{(\ch \xi + e^{-Q})(\ch \xi + \ch Q)}.
%\end{equation} 
%
%When neutrinos are still relativistic, $\Psi$ is proportional to
%this quantity, even if it cannot be simply interpreted in
%terms of blackbody temperature perturbations.
%
%We can now specify all the changes required in {\tt cmbfast}, first in
%the case of massive degenerate neutrinos. As usual, $\Psi$ can be
%expanded in a Legendre series: $\Psi = \sum_{l=0}^{\infty} (-i)^l (2l
%+1) \Psi_l P_l$.  It is easy to show that for each multipole $\Psi_l$,
%the evolution equation and the initial condition are both identical to
%those of the non-degenerate case, provided that we replace $(d \ln (
%f_{\nu} )/d \ln Q)$ by eq. (\ref{dlnfdlnq}).  So, in summary, one only
%needs to modify the homogeneous phase-space distribution, its
%logarithmic derivative with respect to $Q$, and the initial
%calculation of $\Omega_{\nu}$.  
%Also, in order to obtain a good
%precision in the CMB anisotropy spectra, one must set $l=5$ for the
%number of multipoles $\Psi_l$ to be time-integrated.  For transfer
%functions, the value $l=25$ proposed by the code is sufficient.
%
%In the case of massless degenerate neutrinos, 
%the $Q$-dependence of the Boltzmann equation can be
%integrated away, just like in the non-degenerate case. For this
%purpose, we must introduce the $Q$-independent variable $F_{\nu}$
%
%\begin{equation}
%F_{\nu} (\vec{k}, \hat{n}, \tau) \equiv
%\frac{\int Q^3 dQ (f_{\nu} + f_{\bar{\nu}}) \Psi} 
%{\int Q^3 dQ (f_{\nu} + f_{\bar{\nu}})}
%\equiv \sum_{l=0}^{\infty} (-i)^l (2l+1) F_{\nu l} P_l,
%\end{equation}
%
%The multipoles $F_{\nu l}$ are exactly identical for degenerate and
%non-degenerate massless neutrinos, because they share the same
%evolution equations and initial conditions. So, the effect of $\xi$
%arises only through the background quantities and is completely
%described by introducing an effective number of massless neutrinos.

\section{Effects on the power spectra}
\label{results}
%We have checked that the results for the CMB anisotropies 
%and transfer functions for different values of $\xi$, in the case
%of a very small mass $m_{\nu} \leq 0.001$ eV, match
%exactly those obtained with the unmodified version of {\tt cmbfast},
%when the appropriate effective neutrino number $N_{eff}$ is specified.

\begin{figure*}[t]
\vspace{-0.5cm}
\begin{eqnarray}
\psfig{file=figCMB.ps,width=0.48\textwidth}~~~~~
\psfig{file=figPK.ps,width=0.48\textwidth}
\nonumber
\end{eqnarray}
\vspace{-1.5cm}
\caption{CMB anisotropy and matter power spectra
for different models with one family of massless (solid lines) and
$m_{\nu} = 0.07$ eV (dashed lines) degenerate neutrinos. From bottom
to top (from top to bottom for $P(k)$), $\xi=0,3,5$.  
Cosmological parameters are fixed as described in the text. 
%We neglect reionization and tensor contribution.
}
\label{fig.CMB}
\end{figure*}
%
%\begin{figure}[htb]
%\begin{eqnarray}
%\centerline{\psfig{file=figCMB.ps,width=0.5\textwidth}}
%\nonumber \\
%\centerline{\psfig{file=figPK.ps,width=0.5\textwidth}}
%\nonumber
%\end{eqnarray}
%\caption{CMB anisotropy spectrum and present power spectrum of matter density
%for different models with one family of massless (solid lines) and
%$m_{\nu} = 0.07$ eV (dashed lines) degenerate neutrinos. From bottom
%to top (from top to bottom for $P(k)$), $\xi=0,3,5$.  
%Cosmological parameters are fixed as described in the text. 
%We neglect reionization and tensor contribution.
%}
%\label{fig.CMB}
%\end{figure}
%

We compute the power spectra of CMB anisotropies and LSS
using the Boltzmann code {\tt cmbfast} by Seljak \&
Zaldarriaga \cite{SelZal}, adapted to the case of one family of
degenerate neutrinos ($\nu$, $\bar{\nu}$), with mass $m_\nu$ and
degeneracy parameter $\xi$. Our modifications to the code
are reviewed and explained in \cite{Paper}.  

The effect of $\xi$ and $m_{\nu}$ on the CMB anisotropy spectrum can
be seen in figure \ref{fig.CMB}. We choose a set of cosmological
parameters ($h=0.65$, $\Omega_b=0.05$, $\Omega_{\Lambda}=0.70$,
$\Omega_{CDM}=1-\Omega_b-\Omega_{\nu}-\Omega_{\Lambda}$,
$Q_{rms-ps}=18~\mu$K, flat primordial spectrum, no reionization, no
tensor contribution), and we vary $\xi$ from 0 to 5, both in the case
of massless degenerate neutrinos and degenerate
neutrinos with $m_{\nu}=0.07$ eV.

Let us first comment the massless case.  The main effect of $\xi$ is
to boost the amplitude of the first peak\footnote{In fact, this is not
true for very large values of $\xi$, where recombination can take
place still at the end of radiation domination, and anisotropies are
suppressed.  However in such a case the location of the first peak is
$l \gsim 450$, and the matter power spectrum is strongly
suppressed.}. Indeed, increasing the energy density of radiation
delays matter-radiation equality, which is known to boost the acoustic
peaks, and to shift them to higher multipoles, by a factor $( (1 +
a_{eq}/a_*)^{1/2} - (a_{eq}/a_*)^{1/2})^{-1}$ ($a_{eq}$ increases with
$\xi$, while the recombination scale factor $a_*$ is almost
independent of the radiation energy density). Secondary peaks are then
more affected by diffusion damping at large $l$, and their amplitude
can decrease with $\xi$.

In the case of degenerate neutrinos with $m_{\nu}=0.07$ eV, the
results are quite similar in first approximation. Indeed, the effects
described previously depend on the energy density of neutrinos at
equality. At that time, they are still relativistic, and identical to
massless neutrinos with equal degeneracy parameter.  However, with a
large degeneracy, $\Omega_{\nu}$ today becomes significant: for
$\xi=5$, one has $\Omega_{\nu}=0.028$, i.e. the same order of
magnitude as $\Omega_b$.  Since we are studying flat models,
$\Omega_{\nu}$ must be compensated by less baryons, cold dark matter
(CDM) or $\Omega_{\Lambda}$. In our example, $\Omega_b$ and
$\Omega_{\Lambda}$ are fixed, while $\Omega_{CDM}$ slightly
decreases. This explains the small enhancement of the first peak
compared to the massless case (3.4\% for $\xi=5$).  Even if this
effect is indirect, it is nevertheless detectable in principle,
possibly by future satellite missions {\it MAP} and {\it Planck} (even
if one does not impose the flatness condition, the effect of
$\Omega_{\nu}$ will be visible through a modification of the
curvature).

%In figure \ref{fig.CMB}, for
%$\xi=0$, the first peak maximum is enhanced by only 0.37\%, while for
%$\xi=5$, the first peak amplitude increases by 3.4\%, 
%possibly detectable by the
%future satellite missions {\it MAP} and {\it Planck}.
%, unless there are
%large parameter degeneracies.  It is well-known that such degeneracies
%are generally removed when CMB and LSS data are combined for parameter
%extraction \cite{Tegmark}.

We also plot in figure \ref{fig.CMB} the power spectrum $P(k)$,
normalized on large scales to COBE. The effect of both parameters
$\xi$ and $m_{\nu}$ is now to suppress the power on small scales.
Indeed, increasing $\xi$ postpones matter-radiation equality, allowing
less growth for fluctuations crossing the Hubble radius during
radiation domination. Adding a small mass affects the recent evolution
of fluctuations, and has now a direct effect: when the degenerate
neutrinos become non-relativistic, their free-streaming suppresses
the growth of fluctuations for scales within the Hubble radius.  This
effect, already known for non-degenerate neutrinos \cite{Huetal}, is
enhanced in the presence of a neutrino degeneracy, since the average
neutrino momentum is shifted to larger values.

%For non-degenerate neutrinos, this effect is known to reduce power on
%those scales by a relative amount $\Delta P/P \sim 8 \Omega_{\nu} /
%\Omega_0$ \cite{Huetal} (we introduced $\Omega_0 = 1 -
%\Omega_{\Lambda}$). So, even with $m_{\nu}=0.07$ eV and $\xi=0$, it is
%significant, especially at low $\Omega_0$. In the models of figure
%\ref{fig.CMB}, $P(k)$ decreases by $\sim 5$ \%, in agreement with the
%theoretical prediction ($\Omega_{\nu}=1.8 \times 10^{-3}$,
%$\Omega_0=0.3$). However, at $\xi=5$ (i.e.  $\Omega_{\nu}=0.028$),
%this effect is even larger: $P(k)$ decreases by a factor 2.2, instead
%of an expected 1.7. This effect is likely related to the phase-space
%distribution of neutrinos with a chemical potential: their average
%momentum is shifted to larger values, making the free-streaming
%suppression mechanism even more efficient.

Our results for massless degenerate neutrinos can be compared with
those of previous works. We found the same effect of $\xi$ on the CMB
for $\Omega_{\Lambda}=0$ as in \cite{Sarkar}, while the revised
results in \cite{Kinney} also agree 
with our calculations for $\Omega_{\Lambda}=0.7$.

\section{Comparison with observations}

\label{comparison}

Since the degeneracy increases dramatically the amplitude of the first
CMB peak, we expect large $\xi$ values to be favored in the case of
cosmological models known to predict systematically a low peak (unless
a large blue tilt is invoked, which puts severe constraints on
inflation).
%On the other
%hand, a high $\xi$ is likely to be allowed (or even favored) for
%models that predict systematically a low peak, unless a large scalar
%spectral index $n \geq 1.2$ ({\it blue tilt}) is invoked. For
%instance, the degeneracy is likely to be favored by: (i) a large
%contribution of tensor perturbations; (ii) a significant effect from
%reionization; (iii) a low baryon density; (iv) a large $h$ ($h \geq
%0.7$); (v) flat models with $\Omega_{\Lambda} \leq 0.6$; etc.  For
%such models, the peak amplitude can be boosted by $\xi$, keeping $n$
%close to one, which is more natural from the point of view of
%inflation. However, a careful case-by-case analysis is required, since
%the effects of $\xi$ and $n$ on CMB and LSS spectra are far from being
%equivalent.  
Our goal here is not to explore systematically all possibilities, but
to briefly illustrate how $\xi$ can be constrained by current
observations for flat models with different values of
$\Omega_{\Lambda}$. Recent results from supernovae, combined with CMB
constraints, favor flat models with $\Omega_{\Lambda} \sim 0.6-0.7$.

We choose a model with $h=0.65$, $\Omega_b=0.05$,
$Q_{rms-ps}=18~\mu$K, no reionization and no tensor contribution, and
look for the allowed window in the space of free parameters
($\Omega_{\Lambda},\xi,n$).  The allowed window is defined as the
intersection of regions preferred at the 95\% confidence level by four
independent experimental tests, based on $\sigma_8$ estimation,
Stromlo-APM redshift survey, bulk velocity reconstruction, and CMB
anisotropy measurements. Details concerning these tests can be found
in \cite{Paper}.

%
%For simplicity, we take into account only a few constraints on the
%matter power spectrum, known to be representative of the large amount
%of available data: the value of $\sigma_8$ (the variance of mass
%fluctuations in a sphere of radius $R=8 h^{-1} $Mpc) given for flat
%models in \cite{ViaLid}, at 95 \% confidence level (CL);
%a $\chi^2$ comparison with the STROMLO-APM redshift survey
%\cite{Loveday}, at scales well within the linear regime, also with 95
%\% CL; and finally, the constraint on bulk velocity at $R=
%50 h^{-1}$ Mpc \cite{KolDek}, taking into account the cosmic variance.
%Except for the updated $\sigma_8$ constraint, we use exactly the same
%experimental tests as in \cite{LPS}, and refer the reader to this
%paper for details. For CMB data, we perform a $\chi^2$ analysis based
%on 19 experimental points and window functions, taking into account
%the Saskatoon calibration uncertainty, in the way suggested by
%\cite{LineBar}. The list of data that we use is given in \cite{LPS},
%and again allowed regions correspond to 95 \% CL.  We do not
%take into account the most recent experiments, for which window
%functions are still unpublished; they are anyway in good agreement
%with the data considered here.


\begin{figure*}[t]
\vspace{-0.5cm}
\begin{eqnarray}
\psfig{file=figWIN00.ps,width=0.47\textwidth}~~~~
\psfig{file=figWIN60.ps,width=0.47\textwidth}
\nonumber
\end{eqnarray}
\vspace{-1.5cm}
\caption{LSS and CMB constraints in ($\xi$, $n$) space
for $\Omega_{\Lambda}=0$ (left) and $\Omega_{\Lambda}=0.6$ (right).
The underlying cosmological model is flat, with $h=0.65$,
$\Omega_b=0.05$, $Q_{rms-ps}=18~\mu$K, no reionization, no tensor
contribution. The allowed regions are those where the labels are. For
LSS constraints, we can distinguish between degenerate neutrinos with
$m_{\nu} =0$ (solid lines) and $m_{\nu} =0.07$ eV (dotted lines).}
\label{fig.WIN}
\end{figure*}
%
%\begin{figure}[htb]
%\begin{eqnarray}
%\centerline{\psfig{file=figWIN60.ps,width=0.47\textwidth}}
%\nonumber \\
%\centerline{\psfig{file=figWIN00.ps,width=0.47\textwidth}}
%\nonumber
%\end{eqnarray}
%\caption{LSS and CMB constraints in ($\xi$, $n$) space
%for $\Omega_{\Lambda}=0.6$ (top) and $\Omega_{\Lambda}=0$ (bottom).
%The underlying cosmological model is flat, with $h=0.65$,
%$\Omega_b=0.05$, $Q_{rms-ps}=18~\mu$K, no reionization, no tensor
%contribution. The allowed regions are those where the labels are. For
%LSS constraints, we can distinguish between degenerate neutrinos with
%$m_{\nu} =0$ (solid lines) and $m_{\nu} =0.07$ eV (dotted lines).}
%\label{fig.WIN}
%\end{figure}

We plot in figure \ref{fig.WIN} the LSS and CMB allowed regions in
($\xi$, $n$) parameter space, for $\Omega_{\Lambda}=0$
and $0.6$.  
%For $\Omega_{\Lambda}=0.6$, the LSS window just comes out
%of $\sigma_8$ limits.  For $\Omega_{\Lambda}=0$, the lower LSS
%constraint is from $\sigma_8$, and the upper one from APM data. 
In the case of degenerate neutrinos with $m_{\nu} = 0.07$ eV, the LSS
regions are slightly shifted at large $\xi$, since, as we saw, the
effect of $\xi$ is enhanced (dotted lines on the figure). The CMB
regions do not show this distinction, given the smallness of the
effect and the imprecision of the data.  One can immediately see that
LSS and CMB constraints on $n$ are shifted in opposite direction with
$\xi$: indeed, the effects of $\xi$ and $n$ both produce a higher CMB
peak, while to a certain extent they compensate each other in $P(k)$.
So, for $\Omega_{\Lambda}\geq0.7$, a case in which a power spectrum
normalized to both COBE and $\sigma_8$ yields a too high
peak\footnote{At least, for the values of the other cosmological
parameters considered here.  This situation can be easily improved,
for instance, with $h=0.7$.}, a neutrino degeneracy can only make
things worst, and we find no allowed window at all.  In the other
extreme case $\Omega_{\Lambda}=0$, it is well known that the amplitude
required by $\sigma_8$ and the shape probed by redshift surveys favor
different values of $n$. We find that the neutrino degeneracy can
solve this problem with $\xi \gsim 3.5$, but the allowed window
is cut at $\xi \simeq 6$ by CMB data, and we are left with an
interesting region in which $\Omega_0=1$ models are viable. This
result is consistent with \cite{Sarkar}. However, current evidences
for a low $\Omega_0$ Universe are independent of the constraints used
here, so there are not many motivations at the moment to consider this
window seriously.  Finally, for $\Omega_{\Lambda}=0.5-0.6$, a good
agreement is found up to $\xi \simeq 3$. This upper bound could
marginally explain the generation of ultra-high energy cosmic rays by
the annihilation of high-energetic neutrinos on relic neutrinos with
mass $m_{\nu}= 0.07$ eV \cite{Gelmini}.

%\section{Conclusions}
%
%We have considered some cosmological implications of a large relic
%neutrino degeneracy. We have shown that this degeneracy enhances the
%contribution of massive neutrinos to the present energy density of the
%Universe. For instance, neutrinos with a small mass $m_{\nu} \sim
%10^{-2}$ eV can contribute significantly to $\Omega_0$, provided that
%there is a large neutrino-antineutrino asymmetry.
%
%Our main result is the computation of the power spectra of CMB
%anisotropies and matter density in presence of a neutrino
%degeneracy. We found, in agreement with \cite{Sarkar}, that it boosts
%the amplitude of the first CMB peak, shifts the peaks to larger
%multipoles, and supresses small scale matter fluctuations.  These
%effects follow the increase of neutrino energy density, that delays
%matter-radiation equality.
%
%We extended the calculation to the case of massive degenerate
%neutrinos, and showed the results for a mass of $0.07$ eV, as
%suggested by the Super-Kamiokande experiment. This mass has a small
%effect on CMB anisotropies. Indeed, such light neutrinos are still
%relativistic at recombination, but in presence of a degeneracy, they
%can account for a substantial part of the density today, of order
%$\Omega_{\nu} \sim 10^{-2}$.  Also, we showed that small scale matter
%fluctuations are much more suppressed when the degenerate neutrinos
%are massive, because free-streaming of non-relativistic neutrinos is
%more efficient when their average momentum is boosted by the chemical
%potential.
%
%We compared our results with observations, in the restricted case of a
%flat universe with arbitrary ($\Omega_{\Lambda}, \xi, n$) and fixed
%values of other cosmological parameters. We found that for
%$\Omega_{\Lambda} \simeq 0.5 - 0.6$, a large degeneracy is allowed, up
%to $\xi \simeq 3$. This upper bound could marginally explain the
%generation of ultra-high energy cosmic rays by the annihilation of
%high-energetic neutrinos on relic neutrinos with mass $m_{\nu}= 0.07$
%eV \cite{Gelmini}.  We also tried smaller values of
%$\Omega_{\Lambda}$, even if they are not favored by combined CMB and
%supernovae data. It turns out that a large degeneracy can account for
%both CMB and LSS constraints even with $\Omega_0 =1$, provided that
%$3.5 \leq \xi \leq 6$.
%
%Finally, it turns out that the degeneracy parameter and the mass of
%degenerate neutrinos have effects within the level of detectability of
%future CMB observations and redshift surveys, even with $m_{\nu} \sim
%0.07$ eV.  However, a careful analysis should be performed in order to
%detect possible parameter degeneracy between $\xi$, $m_{\nu}$ and
%other cosmological parameters.

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%
%\bibitem{Riotto}
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S.~Sarkar, \rpp{59}{96}{1493}

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U.~Seljak and M.~Zaldarriaga, \apj{469}{96}{437}

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%W.T.~Hu, Ph.D. Thesis (1995), E-mail preprint archive .

%\bibitem{Tegmark}
%D.J.~Eisenstein, W.~Hu and M.~Tegmark, preprint 

\bibitem{Huetal}
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%\bibitem{Perlmutter}
%S.~Perlmutter {\it et al.}, \nat{391}{98}{51}
%P.M.~Garnavich {\it et al.}, \apj{493}{98}{L53};
%A.G.~Riess {\it et al.}, \aj{116}{98}{1009}

%\bibitem{Lineweaver}
%C.~Lineweaver, \apj{505}{98}{L69}

%\bibitem{Wang}
%L.~Wang et al., preprint 
%R.R.~Caldwell, J.P.~Ostriker, and P.J.~Steinhardt, preprint 

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%P.~Viana and A.R.~Liddle, \mn{303}{99}{535}

%\bibitem{Borgani}
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%T.~Kolatt and A.~Dekel, \apj{479}{97}{592}

%\bibitem{LPS}
%J.~Lesgourgues, D.~Polarski and A.A.~Starobinsky, preprint 

%\bibitem{LineBar}
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%\bibitem{Bahcall}
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\bibitem{Gelmini}
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%preprint 

\end{thebibliography}

\end{document}
