Astron. Astrophys. 363, 825-836 (2000)

1. Introduction

The atmospheric neutrino results from Super-Kamiokande (Fukuda et al. 1998) and MACRO (Ambrosio et al. 1998) experiments indicate that neutrinos oscillate. Those data are consistent with oscillations, but do not exclude oscillations to a sterile neutrino (see e.g. Foot et al. 1996; Foot & Volkas 1997, 1999). The small value of the difference of the squared masses () and the strong mixing angle () suggest that these neutrinos are nearly equal in mass as predicted by many models of particle physics beyond the standard model (see e.g. Primack & Gross 1998 and the references therein). Also, the LSND experiment (Athanassopoulos et al. 1998) support oscillations with eV² and other different types of solar neutrino experiments (Bahcall et al. 1998) suggest that could oscillate to a sterile neutrino with eV².

The direct implication of neutrino oscillations is the existence of non-zero neutrino masses in the eV range, and consequently a not negligible hot dark matter contribution to the total mass density of the universe (i.e. a density parameter ).

In view of the uncertainty in the direct neutrino experiments, a possible significant contribution can be obtained from the study of the cosmological implications of neutrino oscillations. In particular, the Cosmic Microwave Background (CMB) anisotropy pattern reflects the conditions in the universe at the time of the last scattering between photons and electrons, occurring at a redshift for standard recombination models. This means that all the physical processes occurring before this epoch could have left imprints on the CMB angular power spectra.

The standard Cold Dark Matter (CDM) model normalized to COBE/DMR data (Smoot et al. 1992; Wright et al. 1994; Górski et al. 1994; Bennett et al. 1996a) predicts both the amplitude and the shape of the CMB power spectrum at small scales inconsistent with the observations of the Large Scale Structure (LLS) of the universe as derived by galaxy surveys (e.g. Scott & White 1994; White et al. 1995; Primack et al. 1995). Recent works (Primack 1998; Gawiser & Silk 1998) show that the Cold + Hot Dark Matter (CHDM) model is the single model with whose predictions agree with both LSS observations and current CMB anisotropies data. The CDM model (Primack et al. 1995) with two 2.4eV neutrinos contributing with to the total density of the universe and with another contribution from cold dark matter and a small baryon fraction, agrees remarkably with all available observations only if the Hubble parameter is km s^-1Mpc^-1 ( km s^-1Mpc).

However, increasing evidence for larger value of the Hubble parameter, , (see e.g. Fukugita et al. 1999 and the references therein) are inconsistent with standard CHDM models. Such high values of the Hubble parameter make an universe suspiciously young unless the cosmological constant is non-zero. A way to improve the agreement of the standard CHDM models with eV neutrino mass with high values of Hubble parameter is to consider the relic neutrino degeneracy (the degeneracy parameter is defined as: , where is the neutrino chemical potential and is the neutrino temperature), that enhances the contribution of neutrinos to the total energy density of the universe (Larsen & Madsen 1995). The cosmological implications of the neutrino degeneracy has been often considered in the literature: it can change the neutrino decoupling temperature (Freese et al. 1983; Kang & Steigman 1992), the abundances of light elements at the big bang nucleosynthesis (BBN) (Steigman et al. 1977; Kang & Steigman 1992), the CMB anisotropies and the matter power spectrum (Kinney & Riotto 1999). For the CHDM models with degenerated neutrinos, the increase of the neutrino chemical potential increases the effective number of neutrino species in the relativistic era (Larsen & Madsen 1995; Hannestad 2000), and can bring the power spectral shape parameter in the range required by observations [for CDM models with low baryonic content the power spectral shape parameter (which basically measures the horizon scale at matter-radiation equality) is defined as (Dodelson et al. 1994): , where counts the relativistic degrees of freedom and corresponds to the standard model with photons and three massless neutrino species; observations require ]. The free-streaming properties of degenerated neutrinos also differ: when the neutrino chemical potential is increased, the neutrino number density is higher, although the neutrino energy density and the mean momentum change a little. The effect is the increase of the radiation energy density and the delay of the matter-radiation equality. It is shown (Larsen & Madsen 1995) that an CHDM model with one 2.4eV neutrino and is in good agreement with the observed LSS power spectrum of density fluctuations if the neutrino degeneracy parameter is . Constraints on neutrino degeneracy coming from BBN (Kang & Steigman 1992) indicate and . Recent works (Pal & Kar 1999; Lesgourgues & Pastor 1999; Kinney & Riotto 1999; Lesgourgues et al. 1999) derive bounds on neutrino degeneracy parameter in agreement with BBN predictions, by combining the current observations of the CMB anisotropies and LSS data. The likelihood analysis of the CMB anisotropy data obtained by the Boomerang experiment indicates bounds on massless neutrino degeneracy parameter (Hannestad 2000) of if only one massless neutrino species is degenerated and if the asymmetry is equally shared among three massless species.

On the other hand, evidences have been accumulated that we live in a low matter density universe (see e.g. Fukugita et al. 1999 and the references therein). Indications like Hubble diagram of Type 1a supernovae (Riess et al. 1998; Perlmutter et al. 1997) and the acoustic peak distribution in the CMB anisotropy power spectra (Hancock et al. 1998; Efstathiou et al. 1999) point to a universe dominated by vacuum energy (cosmological constant ) that keeps the universe close to flat. The combined analysis of the latest CMB anisotropy data and Type 1a supernovae data (Efstathiou et al. 1999) indicates and (95% confidence errors) for matter and vacuum energy densities respectively. These values are close to those favoured by other arguments (see e.g. Efstathiou et al. 1999 and the references therein) like the ages of globular clusters, observations of large scale structure, baryon abundance in clusters.

Adding a Hot Dark Matter component to the CDM model (CHDM) leads to a worse fit to LSS and CMB data, resulting in a limit on the total neutrino mass (Gawiser 2000) of eV for a primordial scale invariant power spectrum and eV for a primordial scale free power spectrum. A stronger upper limit is obtained (Fukugita et al. 1999) from the matching condition of the LSS power spectrum normalization (defined as the rms amplitude of the galaxy power spectrum in a sphere of radius 8Mpc) at the COBE scale and at the cluster scale. For the case of a CHDM model having , , and a primordial scale invariant power spectrum, it is found an upper limit of the total non-degenerated neutrino mass of if the Hubble constant Km s^-1 Mpc^-1.

In this paper we study the signature of relic degenerated neutrino oscillations on the CMB anisotropy and polarization power spectra and address its detectability with the future CMB anisotropy space missions, MAP (Microwave Anisotropy Probe) (see Bennett et al. 1996b) and PLANCK (Mandolesi et al. 1998a; Puget et al. 1998). We consider the impact of neutrino oscillations in the epoch after nucleosynthesis. Neutrino oscillations are mediated by weak interactions: this fact implies that the order of magnitude of the interaction rate is less than the Hubble expansion rate after nucleosynthesis. In order to get an appreciable effect we consider a large neutrino asymmetry as left by processes occurred before and during nucleosynthesis. An example of such large asymmetries is given by considering a relic neutrino degeneracy (Larsen & Madsen 1995), which has been recently subject of renewed interest (Kinney & Riotto 1999; Lesgourges & Pastor 1999; Pal & Kar 1999). We analyze the oscillations of relic degenerated neutrinos under the simple assumption of oscillations occurring between two active neutrino flavors

which have the following mass hierarchy:

We assume the third neutrino as massless and non-degenerate. We consider a total neutrino mass contribution of eV and a total neutrino degeneracy parameter accordingly to the present observational data summarized above. We assume a scale invariant primordial power spectrum, the presence of the scalar modes with spectral index and ignore the contribution of the tensorial modes and the reionisation effects. This model is consistent with the large scale structure and CMB anisotropy data, allowing in the same time a pattern of neutrino masses consistent with the results from atmospheric neutrino oscillations experiments.

In Sect. 2 we draw the basic formalism of the neutrino oscillation model necessary for understanding its impact on CMB angular power spectra. The features induced by the degenerated neutrino oscillations on the CMB anisotropy and polarization power spectra are presented in Sect. 3. In Sect. 4 we discuss the MAP and PLANCK capability of detecting neutrino oscillations. Finally, we summarize our main conclusions in Sect. 5.

Throughout the paper we employ the system of units in which .

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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