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

4. Detection of neutrino oscillations with CMB experiments

Different methods can be used to quantify the performance of a CMB experiment in measuring a given set of cosmological parameters. We estimate the errors on the oscillation parameters and and the lepton asymmetry by using the Fisher information matrix approximation, widely employed in the literature (e.g. Efstathiou & Bond 1999; Popa et al. 1999).

The Fisher information matrix elements measure the width and the shape of the likelihood function around its maximum (Efstathiou & Bond 1999). The minimum error that can be obtained on a parameter when we need to determine all parameters jointly, is given by:

depending on the experimental parameter data set and the target model and the class of considered cosmological models.

If only the temperature anisotropy power spectrum is used, the Fisher information matrix reads as (Efstathiou & Bond 1999):

If both anisotropy and polarization power spectra are used, the Fisher information matrix is given by (Zaldarriaga & Seljak 1997; Zaldarriaga 1997):

where: X and Y stands for T, E, C and B power spectra and is the inverse of the covariance matrix. For the purpose of this work we assume only scalar modes, then the relevant covariance matrix elements in Eqs. (21) (23) are:

Here we set for anisotropy and for polarization (with the sum performed over detector channels), and , where and are the relative noise per pixel for anisotropy and polarization channels (Knox 1995); and account for the beam smearing, is the Gaussian beam profile, and is the fraction of the sky used in the analysis.

Assessing PLANCK performances under realistic assumptions is a very delicate task: both instrumental systematic effects (e.g. Delabrouille 1998; De Maagt et al. 1998; Maino et al. 1999; Burigana et al. 1998 and references therein) and astrophysical contamination (e.g. Toffolatti et al. 1998, 1999; De Zotti et al. 1999a,b and references therein) have to be carefully understood, reduced by optimizing the telescope (Mandolesi et al. 1998b) and the instrumental design (Bersanelli & Mandolesi 1998; Puget et al. 1998) and accurately subtracted in the data analysis. For simplicity, we consider only the MAP and PLANCK "cosmological windows", i.e. those frequencies that are minimally affected by foregrounds contaminations, and neglect the degradation in CMB power spectrum recovering introduced by foreground contaminations and instrumental effects. As shown in Fig. 4, in the cosmological channels extragalactic source fluctuations are comparable to CMB ones only at , both for temperature (left panel) and polarization (right panel) anisotropies, whereas galactic fluctuations, which power spectrum overwhelms the extragalactic foreground one at , are below the CMB ones at least far from the galactic plane.

Fig. 4. Temperature (left panel) and polarization (right panel) power spectra for the neutrino oscillation target model considered here (see also the text) compared to astrophysical component power spectra: the galactic temperature fluctuations at moderate galactic latitudes, sum of synchrotron, free-free and dust fluctuations, and the extragalactic source temperature fluctuations modelled as in Toffolatti et al. (1998) and De Zotti et al. (1999a) and the polarization fluctuations from galactic synchrotron and dust emission and separately from extragalactic radiosources and infrared sources modelled according to De Zotti et al. (1999b) and references therein. We report also the white noise nominal power spectrum for LFI and MAP, taking into account the effect of resolution loss at high l due to the beam convolution.

Then, the possible imprinting of neutrino oscillations in the CMB anisotropy are not significantly masked from astrophysical contaminations at frequencies GHz. We exploit then separately different sets of PLANCK data: the "cosmological" LFI channels alone (70 and 100 GHz channels), the "cosmological" HFI channels alone (143 and 217 GHz channels and 100 GHz but for the temperature fluctuations only) and both together. For comparison, we consider also the set of data from the two MAP "cosmological" channels at 60 and 90 GHz. Table 1 lists the experimental parameters of the various experiments that we considered in our calculations.

Table 1. Summary of instrumental performances of future CMB space missions in their "cosmological channels". For MAP and PLANCK LFI we compute (Zaldarriaga & Seljak 1997).

The PLANCK and MAP data will be mainly used for determining the cosmological parameters; we exploit then the Fisher matrix method by evaluating together the uncertainties ( errors) of a simple set of cosmological parameters and of the parameters characterizing the neutrino oscillation model discussed here. As well known, the errors quoted for the considered parameters depend on the experiment sensitivity but also in part on the assumed target model and the set of considered parameters.

We assume as target model a spatially flat CHDM model without oscillations having: , , , , , eV, , (), eV², and . We also assume a scale invariant power spectrum, the presence only of the scalar modes having a spectral index , and ignore the contribution of the tensorial modes and reionisation effects. The tensorial modes have a small contribution only to the polarization signal at low order multipoles (Crittenden & Turok 1995; Seljak 1997; Kamionkowski & Kosowsky 1998) that will be difficult to detect against the polarized foregrounds. Also, the reionisation effects can help in breaking the degeneracy only in the presence of the polarization signal for a large value of the optical depth to Thomson scattering (Zaldarriaga et al. 1997).

We numerically compute the partial derivatives of the power spectra with respect to each relevant parameter by approximating them with the (possibly central) finite differences between the two power spectra obtained by varying with a quantity the parameter in consideration around its value assumed in the considered target model. Typically, is taken in the range between few percent, in order to have a step small enough for an accurate evaluation of the derivative and large enough for having quite sensitive and numerically stable variations of the power spectrum.

Table 2 presents 1- errors on the estimates of the relevant cosmological parameters and neutrino oscillation parameters obtained from the anisotropy alone and anisotropy plus polarization, for the experimental parameters listed in Table 1 and few values of . The errors presented in Table 2 arise because of the geometrical degeneracy (see Efstathiou & Bond 1999 and the references therein) that imposes limits on the determination of and , and the correlations existing among the energy density parameters and and .

Table 2. errors on the estimates of the cosmological parameters and neutrino oscillation parameters obtained from the power spectrum statistics.

These errors can be understood in terms of the ability of each experiment to determine the properties of Doppler peaks (Efstathiou & Bond 1999): the location of the first Doppler peak and the positions, heights and relative amplitudes of the first and subsidiary Doppler peaks. The Doppler peak positions [, where is the angular diameter distance to the last scattering, is the sound horizon distance and m is the Doppler peak number] suffer the degeneracy between and when is fixed (). From Eq. (19) value is given by the and values when the total neutrino mass is fixed, and therefore and are also correlated. The Doppler peak heights suffer the degeneracy between , , , and .

Clearly, the PLANCK LFI, probing the subsidiary Doppler peaks structure to high multipoles , will produce a significant improvement with respect to MAP (sensitive up to ) in the determination of all parameters reducing their uncertainties. A further improvement can be achieved by exploiting the better sensitivity and angular resolution (up to ) of PLANCK HFI, the full PLANCK performance at very high l being limitated essentially by the foreground contamination.

On the other hand, other kinds of astronomical observations provide information on the cosmological parameters, complementary to those derived from CMB anisotropies. Galaxy cluster observations limit the shape parameter and the normalization of the galaxy power spectrum and, combined with the galaxy peculiar velocity fields, provide informations on the matter density in the universe (Dekel 1994; Strauss & Willick 1995). Weak gravitational lensing provides also direct estimates of (see e.g Seljak 1998). The ratio between baryon and total mass in clusters (0.01 - 0.02) (see e.g. Dodelson et al. 1996 and the references therein), informations from the BBN [ (Burles & Tytler 1998)] and the Helium Lyman-Alpha Forest (Wadsley et al. 1999) constrain or if is known from other kinds of measurements. The complementarity between Type 1a supernovae luminosity distances and CMB anisotropy provides a powerful tool in breaking the degeneracy between and (see e.g. Perlmutter et al. 1997; Efstathiou & Bond 1999; Efstathiou et al. 1999 and the references therein). The ages of the oldest globular clusters set lower limits on the age of the universe (see e.g. Chaboyer 1998) and then on and if is well constrained. For a spatially flat universe this contributes to break the geometrical degeneracy in - plane. For CDM models with known baryonic content, measurements of from large scale structure, combined with independent determinations of , provide estimates on (Efstathiou & Bond 1999). Constraints on the scalar spectral index can be inferred also from limits on the primordial black hole abundance (Green et al. 1997) and on the CMB spectral distortions (Hu et al. 1994).

Recent CMB anisotropy measurements from BOOMERANG and MAXIMA-1 have been in fact used jointly to other astronomical observations to determine cosmological parameters through the so-called analysis with prior (Lange et al. 2000; Balbi et al. 2000). Of course, adding information from other kind of observations helps both in circumventing the degeneracy problem and in improving the accuracy in the determination of the cosmological parameters.

Such a kind of analysis is out of the aim of this work. We just estimate here lower limits on the MAP and PLANCK sensitivity in determining the neutrino oscillation parameters, by exploiting the Fisher matrix method, assuming that all the cosmological parameters are known and estimating only the joint errors on , , and that can be obtained from CMB anisotropy plus polarization data.

Table 3 presents these errors ( errors) for PLANCK (LFI and HFI together) and MAP experiments; together with the results shown in Table 2, it provides a range of PLANCK and MAP sensitivity to neutrino oscillation parameters. Of course, in this case we find a significant reduction of the errors on the neutrino oscillation parameters for both PLANCK and MAP. These values can be also considered as estimates of lower limits on the errors on the neutrino oscillation parameters, as they could be in principle stetted by PLANCK and MAP experiments.

Table 3. errors on the estimates of the neutrino oscillation parameters and lepton asymmetry obtained from anisotropy and polarization power spectrum statistics.

Fig. 5 presents few confidence regions of the neutrino oscillation parameters space that can be potentially detected by the PLANCK surveyor and few allowed regions of and at the same confidence level (CL) obtained by MACRO Collaboration (Ambrosio et al. 1998).

Fig. 5. Confidence regions of the neutrino oscillation parameter space that can be potentially detected by PLANCK surveyor by using CMB anisotropy measurements. Panel a : the allowed confidence regions at CL (A) and at CL (B) obtained for (continuous lines) and (dashed lines). The target model has eV2 and (see also the text). Panel b : the allowed confidence regions at CL (A) and at CL (B) for . The target model has: eV2 and (continuous lines) and eV2 and (dashed lines). In each panel we also present (dash-dotted lines) the allowed regions of and at 99 CL (A1) and 90 CL (B1) measured by the MACRO experiment.

According to the standard method (see e.g Particle Data Group 1998) an input model defined by the parameters (, ) is more likely to have occurred higher is the probability to obtain values larger than a specific value . The cumulative probability that defines a certain confidence region on the parameters is given by:

where , with , 9.21 at and CL respectively (for a distribution with two degrees of freedom). The likelihood function in the Eq. (25) is defined as (Efstathiou & Bond 1999):

where is the power spectrum for the target model, is the power spectrum for the input model having the same cosmological parameters as the target model and and are in the intervals ()eV² and () respectively.

Panel a) of Fig. 5 presents the confidence regions that could be obtained with the PLANCK surveyor for different fractions of sky. The target model is the model used in the previous section. Panel b) of the same figure presents the confidence regions obtained with PLANCK for and the same target model and for another target model.

In each panel we also present the allowed regions of and as derived from the MACRO experiment.

We conclude that exists a significant overlap between the region of the oscillation parameter space that can be potentially detected by PLANCK surveyor and that implied by the atmospheric neutrino oscillations data.

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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