WMAP 5year constraints on lepton asymmetry and radiation energy density: Implications for Planck
Abstract
In this paper we set bounds on the radiation content of the Universe and neutrino properties by using the WMAP5 year CMB measurements complemented with
most of the existing CMB and LSS data (WMAP5+All),
imposing also selfconsistent BBN constraints on the primordial helium abundance.
We consider lepton asymmetric cosmological models parametrized by the neutrino degeneracy parameter and the variation of the relativistic degrees of freedom, , due to possible other physical processes occurred between BBN and structure formation epochs.
We get a mean value of the effective number of relativistic neutrino species of (68% CL), bringing an important improvement over the similar result obtained from WMAP5+BAO+SN+HST data [57].
We also find a strong correlation between and , showing that we observe mainly via the effect of , rather than via neutrino anisotropic stress as claimed by the WMAP team [57].
WMAP5+All data provides a strong bound on helium mass fraction of (68% CL), that rivals the bound on obtained from the conservative analysis of the present data on helium abundance.
For neutrino degeneracy parameter we find a bound of (68%CL), that represent an important improvement over the similar result obtained by using the WMAP 3year data.
The inclusion in the analysis of LSS data reduces the upper limit of the neutrino mass to eV (95% CL) with respect to the values obtained from the analysis from WMAP5only data [56] and WMAP5+BAO+SN+HST data [57].
We forecast that the CMB temperature and polarization measurements observed with high angular resolutions and sensitivities by the future Planck satellite will reduces the errors on and down to (68% CL) and (68% CL) respectively, values fully consistent with the BBN bounds on these parameters.
This work has been done on behalf of PlanckLFI activities.
pacs:
CMBR theory, dark matter, cosmological neutrinos, big bang nucleosynthesis,
1 Introduction
The radiation budget of the Universe relies on a strong theoretical prejudice: apart from the Cosmic Microwave Background (CMB) photons, the relativistic background would consist of neutrinos and of possible contributions from other relativistic relicts. The main constraints on the radiation energy density come either from the very early Universe, where the radiation was the dominant source of energy, or from the observation of cosmological perturbations which carry the information about the time equality between matter and radiation.
In particular, the primordial light element abundance predictions in the standard theory of the Big Bang Nucleosynthesis (BBN) [1, 2, 3, 4] depend on the baryontophoton ratio, , and the radiation energy density at the BBN epoch (energy density of order MeV), usually parametrized by the effective number of relativistic neutrino species, .
Meanwhile, the number of active neutrino flavors have been fixed by boson decay
width to [4] and the combined study of the incomplete neutrino decoupling and the QED corrections indicate that the number of relativistic neutrino
species is [5]. Any departure of from this last value would be due to nonstandard neutrino features or to the contribution of other relativistic relics.
The solar and atmospheric neutrino oscillation experiments [6, 7] indicate the existence of nonzero neutrino masses in eV range.
There are also indications of neutrino oscillations with larger
masssquared difference,
coming from the short baseline oscillation experiments [8, 9],
that can be explained by adding one or two sterile neutrinos with eVscale mass to the standard scheme with three active neutrino flavors (see Ref.[10] for a recent analysis). Such results have impact on cosmology because sterile neutrinos can contribute to the number of relativistic degrees of freedom at the Big Bang Nucleosynthesis [11].
These models are subject to strong bounds on the sum of active neutrino masses from the combination of various cosmological data sets [12, 13],
ruling out a thermalized sterile neutrino component with eV mass [14, 15].
However, there is the possibility to accommodate the cosmological observations
with data from short baseline neutrino oscillation experiments by postulating
the existence of a sterile neutrino with the mass of few keV having a phasespace distribution significantly suppressed relative to the thermal distribution.
For both, nonresonant zero lepton number production and enhanced resonant production
with initial cosmological lepton number, the keV mass sterile neutrino produced via small mixing angle oscillation conversion of thermal active neutrinos [16] provides a
valuable Dark Matter (DM) candidate,
alleviating the accumulating contradiction between the CDM model
predictions on small scales and observations by smearing out the small scale structure
[17, 18, 19, 20].
A recent analysis of Xray and Lyman data indicate that keV sterile neutrinos
can be considered valuable DM candidates only if
they are produced via resonant oscillations with nonzero
lepton number or via other (nonoscillatory) production
mechanisms [21].
On the other hand, the possible existence of new particles such as axions and gravitons, the time variation of the physical constants and other nonstandard scenarios (see e.g. [22] and references therein) could contribute to the radiation energy density at BBN epoch.
At the same time, more phenomenological extensions to the standard neutrino sector
have been studied, the most natural being the consideration of the leptonic asymmetry
[23, 24, 25], parametrized by the neutrino degeneracy parameter
[ is the neutrino chemical potential and
is the present temperature of the neutrino background,
].
Although the standard model of particle physics predicts the value of leptonic asymmetry of the same order as the value of the baryonic asymmetry, , there are many particle physics scenarios in which a leptonic asymmetry much larger can be generated [26, 27].
One of the cosmological implications of a larger leptonic asymmetry is
the possibility to generate small baryonic asymmetry of the Universe through
the nonperturbative (sphaleron) processes [28, 29, 30].
Therefore, distinguishing between a vanishing and nonvanishing at the BBN epoch is a crucial test of the standard assumption that sphaleron effects equilibrate the
cosmic lepton and baryon asymmetries.
The measured neutrino mixing parameters implies that neutrinos
reach the chemical equilibrium before BBN [31, 32, 33], so that all
neutrino flavors are characterized by the same
degeneracy parameter, , at this epoch.
The most important impact of the leptonic asymmetry on BBN is
the shift of the beta equilibrium between protons and neutrons and the increase
of the radiation energy density parametrized by:
(1) 
The BBN constraints on N have been recently reanalyzed by comparing the theoretical predictions and experimental data on the primordial abundances of light elements, by using the baryon abundance derived from the WMAP 3year (WMAP3) CMB temperature and polarization measurements [34, 35, 36]: . In particular, the He abundance, , is quite sensitive to the value of . In the analysis of Ref. [37], the conservative error of helium abundance, [38], yielded to (95% CL) in good agreement with the standard value, but still leaving some room for nonstandard values, while more stringent error bars of helium abundance , [39], leaded to (95% CL) [40].
The stronger constraints on the degeneracy parameter obtained from BBN [41] gives (69% CL), adopting the conservative error analysis of of Ref. [38] and (68% CL), adopting the more stringent error bars of of Ref. [42].
The CMB anisotropies and LSS matter density fluctuations power spectra carry the signature
of the energy density of the Universe at the time of matterradiation equality
(energy density of order eV), making possible the measurement of through its effects on the growth of cosmological perturbations.
The number of relativistic neutrino species influences the CMB power spectrum by changing the time of matterradiation equality that enhances the integrated SachsWolfe effect, leading to a higher first acoustic Doppler peak amplitude. Also, the temperature anisotropy of the neutrino background
(the anisotropic stress) acts as an additional source term for the gravitational potential [43, 44], changing the CMB anisotropy power spectrum at the level of %.
The delay of the epoch of matterradiation equality shifts the LSS matter power spectrum turnover position toward larger angular scales, suppressing the power at small scales. In particular, the nonzero neutrino chemical potential leads to changes in neutrino freestreaming length and neutrino Jeans mass due to the increase of the neutrino velocity dispersion [45, 46].
After WMAP3 data release there are many works aiming to constrain from cosmological observations [14, 34, 37, 47, 48, 49].
Their results suggest large values for within 95% CL interval, some of them not including the standard value [14, 34, 37]. Recently Ref. [50] argues that the discrepancies are due to the treatment of the scaledependent biasing in the galaxy power spectrum inferred from the main galaxy sample of the Sloan Digital Sky Survey data release 2 (SDSSDR2) [51, 52]
and the large fluctuation amplitude reconstructed from the Lyman forest data
[53] relative to that inferred from WMAP3.
Discrepancies between BBN and cosmological data results on was interpreted as
evidence of the fact that further relativistic species are produced by particles decay between BBN and structure formation [48, 49].
Other theoretical scenarios include the violation of the spinstatistics in the neutrino sector [54], the possibility of an extra interaction between
the dark energy and radiation or dark matter, the existence of a BransDicke field which could mimic the effect of adding extra relativistic energy density between BBN and structure formation epochs [55].
A lower limit to (95% CL) was recently obtained from the analysis of the WMAP 5year (WMAP5) data alone [56], while the combination of the WMAP5 data with distance information from baryonic acoustic oscillations (BAO), supernovae (SN) and Hubble constant measured by Hubble Space Telescope (HST), leaded to (68% CL), fully consistent with the standard value [57].
The extra energy density can be splitted in two distinct uncorrelated contributions, first due to net lepton asymmetry of the neutrino background and second due to the extra contributions from other unknown processes:
(2) 
The aim of this paper is to obtain bounds on the neutrino lepton asymmetry and on the extra radiation energy density by using WMAP5 data in combination with most of the existing CMB and LSS measurements and selfconsistent BBN priors on . We also compute the sensitivity of the future Planck experiment [58] for these parameters testing the restrictions on cosmological models with extra relativistic degrees of freedom expected from high precision CMB temperature and polarization anisotropy measurements.
2 Leptonic asymmetric cosmological models
The density perturbations in leptonic asymmetric cosmological models
have been discussed in literature [59, 60, 45, 46, 63, 64].
We applied them to modify the Boltzmann Code for Anisotropies in the Microwave Background (CAMB) [65, 66, 67] to compute the CMB temperature and polarization anisotropies
power spectra and LSS matter density fluctuations power spectra for the case of
three degenerated neutrinos/antineutrinos with the total mass and degeneracy parameter . As neutrinos reach their approximate chemical potential
equilibrium before BBN epoch [31, 32, 33], we consider in our computation that all three flavors of neutrinos/antineutrinos have the same degeneracy parameter .
When the Universe was hot enough, neutrinos and antineutrinos of each flavor
behave like relativistic particles with FermiDirac phase space distributions:
(3) 
where is one flavor neutrino/antineutrino energy and is the comoving momentum. Hereafter, is the cosmological scale factor ( today). The mean energy density and pressure of one flavor of massive degenerated neutrinos and antineutrinos can be written as:
(4)  
(5) 
We modify in CAMB the expressions for the energy density and the pressure in the relativistic and nonrelativistic limits for the degenerate case [46] and follow the standard procedure to compute the perturbed quantities by expanding the phase space distribution function of neutrinos and antineutrinos into homogeneous and perturbed inhomogeneous components [67, 68, 69]. Since the gravitational source term in the Boltzmann equation is proportional to the logarithmic derivative of the neutrino distribution function with respect to comoving momentum, , we also modify this term to account for [46, 63].
As mentioned before, the BBN theory gives strong constraints on and
by comparing the measured light element abundance with the theoretical predictions. The only free parameter is the baryon to photon ratio, , that is obtained from the determination of from CMB measurements.
In particular, the He mass fraction, , affects the CMB angular power spectra through its impact on different evolution phases of the ionization/recombination history [70].
As previously demonstrated [71, 72, 73], the impact of selfconsistent BBN prior on has a net impact on parameter inference, improving the bounds on cosmological parameters compared to the analysis which treat as a constant or free parameter.
We use the public BBN code PArthENoPE [74] to compute the dependence of on
, as given in equation (2) and .
The accuracy on obtained by using PArthENoPE code is , being only limited by the experimental uncertainty on the neutron lifetime [76].
We also modify the recombination routine Recfast v1.4 [75] of the CAMB code to account for dependence on , and .
In our computation we implicitly assume that the value of is not changed between the epoch of BBN and matterradiation equality.
3 Analysis
We use the CosmoMC Monte Carlo Markov Chain (MCMC) public package [77] modified for our extended parameter space to sample from the posterior distribution giving the following experimental datasets:

The LSS power spectrum of the matter density fluctuations inferred from the galaxy clustering data of the Sloan digital Sky Survey (SDSS) [51, 52, 83, 84] and Twodegree Field Galaxy Redshift Survey (2dFGRS) [85]. In particular, the luminous red galaxies (LRG) sample from SDSS data release 5 (SDSSDR5) [83, 84] has more statistical significance than the spectrum retrieved from the SDSS main galaxy sample from data release 2 (SDSSDR2) [51, 52] eliminating the existing tension between the power spectra from SDSSDR2 and 2dFGRS. For this reason we include in our analysis the matter power spectra from SDSSLRG and 2dFGRS. We consider SDSSLRG data up to h Mpc and the 2dFGRS data up to h Mpc and apply the corrections due to the nonlinearity behavior and scale dependent bias [83], connecting the linear matter power spectrum, , and the galaxy power spectrum, , through:
(6) where the free parameters and are marginalized.

The BBN constraints on as obtained from PArthENoPE code, allowing and to span the ranges indicated in Table 1.
Hereafter, we will denote WMAP5+SDSSDR5+2dFGRS+SNIa+BBN data set as WMAP5+All.
We perform our analysis in the framework of the extended CDM cosmological model described by free parameters:
(7) 
Here and are the baryon and cold dark matter energy density parameters, is the Hubble expansion rate, is the redshift of reionization, is the scalar spectral index of the primordial density perturbation power spectrum and is its amplitude at the pivot scale hMpc. The additional three parameters denote the neutrino energy density , the neutrino degeneracy parameter and the contribution of extra relativistic degrees of freedom from other unknown processes . Table 1 presents the parameters of our model, their fiducial values used to generate the Plancklike simulated power spectra and the prior ranges adopted in the analysis.
Parameter  Symbol  Fiducial value  Prior range 

Baryon density  0.022  0.005 0.04  
Dark matter density  0.11  0.01 0.5  
Hubble constant  70  40 100  
Redshift of reionization  11  3 20  
Scalar spectral index  0.96  0.5 1.3  
Normalization  3.264  2.7 4  
Neutrino density  0.01  0 0.3  
Neutrino degeneracy parameter  0  1 1  
Number of extra rel. d.o.f.  0.046  3 3  
Helium mass fraction  0.248  0.6 
For the forecast from Plancklike simulated data we use the CMB temperature (TT), the
polarization (EE) and their crosscorelation (TE) power spectra obtained for our fiducial
cosmological model for multipoles up to and the expected experimental characteristics of the Planck frequency channels presented in Table 2 [58]. We assume a sky coverage of .
Following the method described in Ref.[61, 62], for each frequency channel we consider an homogeneous detector noise with the power spectrum given by:
(8) 
where is the frequency of the channel, is the FWHM of the beam and are the corresponding sensitivities per pixel of temperature (T) and polarization (P) maps. The global noise of the experiment is obtained as:
(9) 
FWHM  

(GHz)  (arcminutes)  ( K)  ( K) 
100  9.5  6.8  10.9 
143  7.1  6.0  11.4 
217  5.0  13.1  26.7 
We assume uniform prior probability on parameters (i.e. we assume that all values of parameters are equally probable) and compute the cumulative distribution function , quoting as upper and lower intervals at 68% CL the values at which is 0.84 and 0.16 respectively. For the case of neutrino mass, , we quote the upper limit at 95%CL to facilitate the comparison with other measurements.
4 Results
We start by making a consistency check, verifying that by using WMAP5+All data and imposing and priors we obtain results in agreement with the ones obtained by WMAP collaboration [56, 57].
In order to understand how the extra relativistic energy density and the leptonic asymmetry affect the determination of other cosmological parameters, we compute first the likelihood functions for WMAP5+All by imposing prior, extending then our computation over the whole parameter space for WMAP5+All and Plancklike simulated data. In Figure 1 we compare the marginalized likelihood probabilities obtained for the main cosmological parameters.
As neutrinos with eV mass decouple when they are still relativistic ( 2 MeV), the main effect of including is the change of relativistic energy density. This changes the redshift of matterradiation equality, , that affects the determination of from CMB measurements because of its linear dependence on [57]:
(10) 
Here =2.469 10 is the present photon energy density
parameter for
K.
As a consequence, and are linearly correlated,
with the width of degeneracy line given by the uncertainty in the determination of .
In Figure 2 we compare the marginalized posterior likelihood probabilities of neutrino parameters,
and obtained in our models.
The mean values of these parameters and the corresponding (68% CL) error bars are given in Table 3.
The LSS measurements provides an independent constraint on which helps to reduce the degeneracy between this parameter and . From WMAP5+All data with and priors we find a mean value of (68% CL). One should note that mean value of for the standard CDM model with is (68%CL) from WMAP5 data only [57].
Figure 3 presents the joint twodimensional marginalized distributions
(68% and 95% CL) in  plane (left panel) and  plane (right panel).
The thick solid lines in the left panel show the 68% and 95% CL limits calculated from the corresponding limits on obtained form the WMAP5+All with prior by using equation (10).
When we transform axis of the left panel to axis from right panel we observe a strong degeneracy between and .
This is valid for both WMAP5+All with =0 prior and
WMAP5+All with no neutrino priors.
For the last case, the nonzero neutrino chemical potential augments the value of by as given in equation (1). This imply a larger expansion rate of the Universe, an earlier weak process freeze out with a higher value for the neutron to proton density ratio, and thus a larger value of . On the other hand, a nonzero value of the electron neutrino chemical potential shifts the neutronproton beta equilibrium, leading to a large variation
of [72, 73].
Left panel from Figure 4 presents the twodimensional marginalized joint probability distributions (68% and 95% CL), showing the degeneracy between and . The total effect of is a noticeably increase of the projected error on .
As the anisotropic stress of neutrinos leaves distinct signatures in the CMB
power spectrum which are not degenerated with , we conclude from this analysis that we observe a nonzero value of from WMAP5+All data
mainly via the change of rather than by the effect of neutrino anisotropic stress.
Our conclusion is not in agreement with the claim of WMAP team concerning the strong evidence of neutrino anisotropic stress from a similar analysis of the WMAP5+BAO+SN+HST data [57].
We obtain from our analysis a mean value of (68% CL) from WMAP5+All with prior and (68% CL) form WMAP5+All with no neutrino priors.
These values bring an important improvement over the similar
result obtained from WMAP5+BAO+SN+HST data: (68% CL).
From Plancklike simulated data the 68% error is
.
The analysis of WMAP5+All data with prior provides a strong bound on helium mass fraction of (68% CL),
that rivals the bound on obtained from the conservative analysis of the present data on helium abundance [2].
Under the assumption of degenerated BBN this bound is weakened, leading to
(68% CL) from WMAP5+All with no neutrino priors, that reflects the strong dependence of on .
From Plancklike simulated data the 68% error on is
, fully consistent with bounds obtained
from the conservative analysis of the present data on helium abundance.
We get for neutrino degeneracy parameter a bound of (68%CL) that represent an important improvement over the similar result obtained by using the WMAP 3year data [45].
The CMB only is able to constrain through its contribution to the radiation energy density during radiation domination epoch and the BBN constraints
on .
We find that the sensitivity of CMB to from Plancklike simulated data
will reduce the error on down to (68% CL), value fully consistent with the BBN bounds.
In the right panel from Figure 4 we compare the twodimensional marginalized joint probability distributions (68% and 95% CL) between and ,
showing the potentiality of the future high sensitivity CMB measurements
to reduce the degeneracy between these parameters.
The inclusion in the analysis of LSS data reduces significantly the upper limit
of the neutrino mass to eV (95% CL) from WMAP5+All with
and priors.
One should note that the analysis of WMAP5+BAO+SN+HST leaded to eV (95% CL) [57].
Under the assumption of degenerated BBN this bound is weakened due to the
degeneracy between and . From WMAP5+All with no neutrino priors
we find eV (95%CL), while from Plancklike simulated data
with no neutrino priors we obtain eV.
5 Summary and conclusions
In this paper, we set bounds on the radiation content of the Universe and neutrino properties by using the WMAP5 year CMB measurements complemented with most of the existing CMB and LSS measurements (WMAP5+All), imposing also selfconsistent BBN constraints on the primordial helium abundance, which proved to be important in the estimation of cosmological parameters [72, 73].
We consider lepton asymmetric cosmological models parametrized by the neutrino degeneracy parameter and the variation of the relativistic degrees of freedom, , due to possible other physical processes occurred between BBN and structure formation epochs.
From our analysis we get a mean value of the effective number of relativistic neutrino
species form WMAP5+All with no neutrino priors, of (68% CL), bringing an important improvement over the similar result obtained from WMAP5+BAO+SN+HST data [57].
Although the LSS measurements provides an independent constraint on
which helps to reduce the degeneracy between this parameter and the redshift of matter radiation equality, , we find a strong correlation between and , showing that we observe a nonzero value from WMAP5+All data mainly due to the change of , rather than by the neutrino anisotropic stress.
The analysis of WMAP5+All data with prior provides a strong bound on helium mass fraction of (68% CL), that rivals the bound on obtained from the conservative analysis of the present data on helium abundance [2]. Under the assumption of degenerated BBN this bound is weakened to (68% CL), reflecting the strong dependence of on .
From the analysis of WMAP5+All data we get for neutrino degeneracy parameter a bound of (68%CL).
The inclusion in the analysis of LSS data reduces the upper limit of the neutrino mass to eV (95% CL), from WMAP5+All with no neutrino priors, with respect to the values obtained from the analysis from WMAP5only data [56] and WMAP5+BAO+SN+HST data [57].
The analysis of WMAP5+All measurements bring also an important improvement over the similar results obtained by using WMAP 1year measurements complemented with LSS data [60] and the WMAP 3year data alone [45, 88].
We forecast that the CMB temperature and polarization measurements observed
with high angular resolutions and sensitivity by the future Planck satellite will reduces the errors on and down
to (68% CL) and
(68% CL) respectively,
values fully consistent with the BBN bounds on these parameters [12].
Our forecasted errors on the cosmological
parameters from Plancklike simulated data are
also consistent those obtained in Ref.[72].
Priors: ; =0  Priors:  No neutrino priors  No neutrino priors  
WMAP5+All  WMAP5+All  WMAP5+All  Planck  
Parameter  
(eV)  
    
    
0.046  
Acknowledgements
We acknowledge the referee for useful comments and suggestions. L.P. and A.V. acknowledge the support by the ESA/PECS Contract ”Scientific exploitation of PlanckLFI data”
We also acknowledge the use of the GRID computing system facility at the Institute for Space Sciences Bucharest and would like to thank the staff working there.
References
q
Footnotes
 http://camb.info
 http://parthenope.na.infn.it
 http://cosmologist.info/cosmomc
References
 Wagoner R V, Fowler W A, Hoyle F, 1967 Astrophys. J. 148 3
 Olive K A, Steigman G, Walker, T P, 2000 Phys. Rep. 333 389 [astroph/9905320]
 Burles S, Nollett K M, Turner M S, 2001 Astrophys. J. 552 L1 [astroph/0010171]
 Eidelman S et al, 2004 Phys. Lett. B 592 1
 Mangano G, Miele G , Pastor S, Peloso M, 2002 Phys. Lett. B 534 8 [astroph/0111408]
 Fukuda Y et al (SuperKamiokande Collab.), 1998 Phys. Rev. Lett. 81 1562
 Ambrosio M et al (MACRO Collab.), 1998 Phys. Lett. B 434 451 [hepex/9807005]
 Athanassopoulos C et al, 1996 Phys. Rev. Lett. 77 3082 [nuclex/9605003]
 AguilarArevalo A A et al (MiniBooNE Collaboration) 2007 [hepex/0704.1500]
 Maltoni M. ans Schwets T, 2007 [hepex/0705.0107]
 Cirelli M , Marandella G, Strumia S, Vissani F, 2005 Nucl. Phys. B 708 215 [hepph/0403158]
 Hannestad S and Raffelt G G, 2006 J. Cosmol. Astropart. Phys. JCAP11(2006)016 [astroph/0607101]
 Kristiansen J, Eriksen H K and Elgar A, 2006 Phys. Rev. D 74 123005 [astroph/0608017]
 Seljak U, Solsar A and McDonald P, 2006 J. Cosmol. Astropart. Phys. JCAP10(2006)014 [astroph/0604335]
 Dodelson S, Melchiorri A and Slosar A, 2006 Phys. Rev. Lett. 97 041301 [astroph/0511500]
 Abazajian K N and Fuller G M, 2002 Phys. Rev. D 66 023526 [astroph/0204293]
 Dodelson S, Widrow L M, 1994 Phys. Rev. Lett. 72, 17 [hepph/9303287]
 Dolgov A D and Hansen S H, 2002 Astropart. Phys. 16 339 [hepph/0009083]
 Abazajian K, Fuller G M, Patel M, 2001 Phys. Rev. D 64 023501 [astroph/0101524]
 Asaka T, Sahaposhnikov M and Kusenko A, 2006 Phys. Lett. B 638 401 [hepph/0602150]
 Palazzo A, Cumberbatch D, Slosar A, Silk J, 2007 Phys. Rev. D 76, 103511 (2007) [arXiv:0707.1495]
 Sarkar S, 1996 Rept. Prog. Phys 59 1493 [hepph/9602260]
 Freese K, Kolb E W, Turner M S, 1983 Phys. Rev. D27 1689
 Ruffini R, Song D J, Stella L, 1983 Astron. Astrophys. 125 265
 Ruffini R, Song D J, Taraglio S, 1988 Astron. Astrophys. 190 1
 Smith C J, Fuller G M, Kishimoto C T, Abazajian K N, 2006 Phys. Rev. D 74 085008 [astroph/0608377]
 Chu Y Z, Cirelli M, 2006 Phys. Rev. D 74 085015
 Kuzmin V, Rubakov V, Shaposhnikov M, 1985 Phys. Lett. B 155 36
 Falcone D, Tramontano F, 2001 Phys. Rev. D 64 077302 [hepph/0102136]
 Buchmuller W, Di Bari P, Plumacher M, 2004 New Jour. Phys. 6 105 [hepph/0406014]
 Dolgov A D, Hansen S H, Pastor S, Petcov S T, Raffelt G, Semikoz D V, 2002 Nucl.Phys. B 632, 363 [hepph/0201287]
 Wong Y Y Y, 2002 Phys. Rev. D 66 025015 [hepph/0203180]
 Abazajian K N, Beacom J F, Bell N F, 2002 Phys. Rev. D 66 013008 [astroph/0203442]
 Spergel D N et al.(wmap cOLLABORATION), 2007 Astrophys. J. Suppl. 170 377 [astroph/0603449]
 Hinshaw et al. (WMAP Collaboration),2007 Astrophys. J. Suppl. 170 288 [astroph/0603451]
 Page L et al. (WMAP Collaboration), 2007 Astrophys. J.Suppl. 170 335 [astroph/0603450]
 Mangano G, Melchiorri A, Mena O, Miele G, Slosar A, 2007 J. Cosmol. Astropart. Phys. JCAP03(2007)006 [astroph/0612150]
 Olive K A, Skillman E D, 2004 Astrophys. J. 617 29 [astroph/0405588]
 Izotov Y I, Thuan T X, Stasinska G, 2007 Astrophys. J. 662 15 [astroph/0702072]
 Ichikawa K, Kawasaki M, Nakayama K, Senami M, Takahashi F, 2007 J. Cosmol. Astropart. Phys. JCAP05(2007)008 [hepph/0703034]
 Serpico P D, Raffelt G G, 2005 Phys.Rev. D 71 127301 [astroph/0506162]
 Izotov Y I, Thuan T X, 2004 Astrophys. J. 602 200 [astroph/0310421]
 Hu W,; Scott D, Sugiyama N, White M, 1995 Phys. Rev. D 52 5498 [astroph/9505043]
 Trotta R, Melchiorri A, 2005 Phys. Rev Lett. 95 011305 [astroph/0412066]
 Lattanzi M, Ruffini R, Vereshchagin G V, Phys. Rev. D 72 063003 [astroph/0509079]
 Ichiki k , Yamaguchi M, Yokoyama J, 2007 Phys.Rev D 75, 084017 [hepph/0611121]
 Hannestad S, Raffelt G G, 2006 J. Cosmol. Astropart. Phys. JCAP11(2006)016 [astroph/0607101]
 Cirelli M, Strumia A, 2006 J. Cosmol. Astropart. Phys. JCAP12(2006)013 [astroph/0607086]
 Ichikawa K, Kawasaki M, Takahashi F, 2007 J. Cosmol. Astropart. Phys. JCAP05(2007)007 [astroph/0611784]
 Hamann J, Hannestad S, Raffelt G G, Wong Y Y Y, 2007 [arXiv:0705.0440)]
 Tegmark M et al (SDSS Collaboration), 2004 Astrophys. J. 606 702 [astroph/0310725]
 Tegmark M et al (SDSS Collaboration), 2004 Phys. Rev. D 69 103501 [astroph/0310723]
 McDonald P et al, 2005 Astrophysical J. 635 761 [astroph/0407377]
 Dolgov A D, Hansen S H, Smirnov A Y, 2005 J. Cosmol. Astropart. Phys. JCAP06(2005)004 [astroph/0611784]
 de Felice A, Mangano G, Serpico P D, Trodden M, 2006 Phys. Rev. D 74 103005 [astroph/0510359]
 Dunkley J et al. (WMAP Collaboration), 2008 Astrophys. J. Supp. [arXiv:0803.0586]
 Komatsu et al. (WMAP Collaboration), 2008 Astrophys. J. Supp. [arXiv:0803.0547]
 The Planck Consortia, 2005 ”The Scientific Programme of Planck” ESASCI 1 [astroph/0604069 ]
 Kinney W H and Riotto A, 1999 Phys. Rev. Lett. 83 3366 [hepph/9903459]
 Crotty P, Lesgourgues J, Pastor S, 2004 Phys. Rev D 69 123007 [astroph/0302337]
 Perotto L, Lesgourgues J, Hannestad S Tu H, Wong Y Y Y, 2006 J. Cosmol. astropart. Phys. JCAP10(2006)013 [astroph/0606227]
 Popa L A, Vasile A, 2007 J. Cosmol. astropart. Phys. JCAP10(2007)017
 Lesgourgues L, Pastor S, 1999 Phys. Rev. D 60 103521 [hepph/9904411]
 Orito M, Kajino T, Mathews G J, Wang Y, 2002 Phys. Rev. D 65 123504 [astroph/0203352]
 Hu W, 2000 Phys.Rev. D 62 043007 [astroph/0002238]
 Challinor A, Lewis A, 2005 Phys. Rev., D 71 103010 [astroph/0502425]

Lewis A, Challinor A, Lasenby A, 2000
Astrophys. J. 538 473 [astroph/9911177]
^{1}  Ma C P, Bertschinger E, 1995 Astrophys. J. 455 7 [astroph/9401007]
 Seljak U, Zaldarriaga M, 1996 Astrophys. J. 469 437 [astroph/9603033]
 Trotta R, Hansen S H, 2004 Phys. Rev. D 69 023509 [astroph/0306588]
 Ichikawa K, Takahashi T, 2006 Phys. Rev. D 73 063528 [astroph/0601099]
 Hamann J, Lesgourgues J, Mangano G, 2007, 2008 J. Cosmol. astropart. Phys. JCAP(2008)004 [arXiv:0712.2826]
 Ichikawa K, Sekiguchi T, Takahashi T, 2007 [arXiv:0712.4327]

Pisanti O, Cirillo A, Esposito S, Iocco F, Mangano G, Miele G, Serpico P D, 2007
[arXiv:0705.0290v1]
^{2}  Seager S, Sasselov D D, Scott D, 1999 Astrophys. J. 523 L1L5 [astroph/9909275]
 Serpico P D, Esposito S, Iocco F, Mangano G, Miele G, Pisanti O, 2004 J. Cosmol. Astropart. Phys. JCAP12(2004)010 [astroph/0408076]

Lewis A and Bridle S, 2002
Phys. Rev. D 66 103511 [astroph/0205436]
^{3}  Nolta M et al.(WMAP Collaboration),2008 [arXiv:0803.0593v1]
 Netterfield C B et al.,2002 Astrophys. J. 571 604 [astroph/0104460]
 MacTavish C J et al., 2006 Astrophys. J. 647 799 [astroph/0507503]
 Reichardt, C. L et al., 2008 Astrophys. J. submitted [arXiv:0801.1491]
 Readhed A C S et al., 2004 Astrophys. J. 609 498 [astroph/0402359]
 Tegmark M et al.(SDSS Collaboration), 2006 Phys. Rev. 74 123507 [astroph/0608632]
 Percival W J et al., 2007 Astrophys. J. 657 645 [astroph/0608636]
 Cole S et al.(2dFGRS Collaboration), 2002 Mon. Not. R. Astron. Soc. 362 505 [astroph/0501174]
 Astier P et al., 2006 Astron. Astophys. 447 31 [astroph/0510447]
 Riess A G et al., 2004 Astrophys. J. 607 665 [astroph/0402512]
 Ichikawa K, Sekiguchi T, Takahashi T, 2008 [arXiv:0803.0889]