As a summary of the last decade of neutrino experiments, two hierarchical neutrino mass splittings and three mixing angles have been measured. Furthermore, the standard model has three neutrinos: the motivation for considering deviations from the standard model in the form of extra sterile neutrinos has disappeared [655, 13]. Of course, deviations from the standard effective numbers of neutrino species could still indicate exotic physics which we will discuss below (Section 2.8.4).
New and future neutrino experiments aim to determine the remaining parameters of the neutrino mass matrix and the nature of the neutrino mass. Within three families of neutrinos, and given all neutrino oscillation data, there are three possible mass spectra: a) degenerate, with mass splitting smaller than the neutrino masses, and two non-degenerate cases, b) normal hierarchy (NH), with the larger mass splitting between the two more massive neutrinos and c) inverted hierarchy (IH), with the smaller spitting between the two higher mass neutrinos. Figure 36  illustrates the currently allowed regions in the plane of total neutrino mass, , vs. mass of the lightest neutrino, . Note that a determination of would indicate normal hierarchy and that there is an expected minimum mass . The cosmological constraint is from .
Cosmological constraints on neutrino properties are highly complementary to particle physics experiments for several reasons:
The hot big bang model predicts a background of relic neutrinos in the universe with an average number density of , where is the number of neutrino species. These neutrinos decouple from the CMB at redshift when the temperature was , but remain relativistic down to much lower redshifts depending on their mass. A detection of such a neutrino background would be an important confirmation of our understanding of the physics of the early universe.
Massive neutrinos affect cosmological observations in different ways. Primary CMB data alone can constrain the total neutrino mass , if it is above [526, finds at 95% confidence] because these neutrinos become non-relativistic before recombination leaving an imprint in the CMB. Neutrinos with masses become non-relativistic after recombination altering matter-radiation equality for fixed ; this effect is degenerate with other cosmological parameters from primary CMB data alone. After neutrinos become non-relativistic, their free streaming damps the small-scale power and modifies the shape of the matter power spectrum below the free-streaming length. The free-streaming length of each neutrino family depends on its mass.
Current cosmological observations do not detect any small-scale power suppression and break many of the degeneracies of the primary CMB, yielding constraints of  if we assume the neutrino mass to be a constant. A detection of such an effect, however, would provide a detection, although indirect, of the cosmic neutrino background. As shown in the next section, the fact that oscillations predict a minimum total mass implies that Euclid has the statistical power to detect the cosmic neutrino background. We finally remark that the neutrino mass may also very well vary in time ; this might be tested by comparing (and not combining) measurements from CMB at decoupling with low- measurements. An inconsistency would point out a direct measurement of a time varying neutrino mass .
Particle physics experiments are sensitive to neutrino flavours making a determination of the neutrino absolute-mass scales very model dependent. On the other hand, cosmology is not sensitive to neutrino flavour, but is sensitive to the total neutrino mass.
The small-scale power-suppression caused by neutrinos leaves imprints on CMB lensing: forecasts indicate that Planck should be able to constrain the sum of neutrino masses , with a error of 0.13 eV [491, 557, 289].
Euclid’s measurement of the galaxy power spectrum, combined with Planck (primary CMB only) priors should yield an error on of 0.04 eV [for details see 211] which is in qualitative agreement with previous work [e.g. 779]), assuming a minimal value for and constant neutrino mass. Euclid’s weak lensing should also yield an error on of 0.05 eV . While these two determinations are not fully independent (the cosmic variance part of the error is in common given that the lensing survey and the galaxy survey cover the same volume of the universe) the size of the error-bars implies more than detection of even the minimum allowed by oscillations. Moreover, the two independent techniques will offer cross-checks and robustness to systematics. The error on depends on the fiducial model assumed, decreasing for fiducial models with larger . Euclid will enable us not only to detect the effect of massive neutrinos on clustering but also to determine the absolute neutrino mass scale.
Since cosmology is insensitive to flavour, one might expect that cosmology may not help in determining the neutrino mass hierarchy. However, for , only normal hierarchy is allowed, thus a mass determination can help disentangle the hierarchy. There is however another effect: neutrinos of different masses become non-relativistic at slightly different epochs; the free streaming length is sightly different for the different species and thus the detailed shape of the small scale power suppression depends on the individual neutrino masses and not just on their sum. As discussed in , in cosmology one can safely neglect the impact of the solar mass splitting. Thus, two masses characterize the neutrino mass spectrum: the lightest , and the heaviest . The mass splitting can be parameterized by for normal hierarchy and for inverted hierarchy. The absolute value of determines the mass splitting, whilst the sign of gives the hierarchy. Cosmological data are very sensitive to ; the direction of the splitting – i.e., the sign of – introduces a sub-dominant correction to the main effect. Nonetheless,  show that weak gravitational lensing from Euclid data will be able to determine the hierarchy (i.e., the mass splitting and its sign) if far enough away from the degenerate hierarchy (i.e., if ).
A detection of neutrino-less double- decay from the next generation experiments would indicate that neutrinos are Majorana particles. A null result of such double- decay experiments would lead to a definitive result pointing to the Dirac nature of the neutrino only for degenerate or inverted mass spectrum. This information can be obtained from large-scale structure cosmological data, improved data on the tritium beta decay, or the long-baseline neutrino oscillation experiments. If the small mixing in the neutrino mixing matrix is negligible, cosmology might be the most promising arena to help in this puzzle.
Neutrinos decouple early in cosmic history and contribute to a relativistic energy density with an effective number of species . Cosmology is sensitive to the physical energy density in relativistic particles in the early universe, which in the standard cosmological model includes only photons and neutrinos: , where denotes the energy density in photons and is exquisitely constrained from the CMB, and is the energy density in one neutrino. Deviations from the standard value for would signal non-standard neutrino features or additional relativistic species. impacts the big bang nucleosynthesis epoch through its effect on the expansion rate; measurements of primordial light element abundances can constrain and rely on physics at . In several non-standard models – e.g., decay of dark matter particles, axions, quintessence – the energy density in relativistic species can change at some later time. The energy density of free-streaming relativistic particles alters the epoch of matter-radiation equality and leaves therefore a signature in the CMB and in the matter-transfer function. However, there is a degeneracy between and from CMB data alone (given by the combination of these two parameters that leave matter-radiation equality unchanged) and between and and/or . Large-scale structure surveys measuring the shape of the power spectrum at large scale can constrain independently the combination and , thus breaking the CMB degeneracy. Furthermore, anisotropies in the neutrino background affect the CMB anisotropy angular power spectrum at a level of through the gravitational feedback of their free streaming damping and anisotropic stress contributions. Detection of this effect is now possible by combining CMB and large-scale structure observations. This yields an indication at more than level that there exists a neutrino background with characteristics compatible with what is expected under the cosmological standard model [901, 285].
The forecasted errors on for Euclid (with a Planck prior) are at level , which is a factor better than current constraints from CMB and LSS and about a factor better than constraints from light element abundance and nucleosynthesis.
A recurring question is how much model dependent will the neutrino constraints be. It is important to recall that usually parameter-fitting is done within the context of a CDM model and that the neutrino effects are seen indirectly in the clustering. Considering more general cosmological models, might degrade neutrino constraints, and vice versa, including neutrinos in the model might degrade dark-energy constraints. Here below we discuss the two cases of varying the total neutrino mass and the number of relativistic species , separately.
In  it is shown that, for a general model which allows for a non-flat universe, and a redshift dependent dark-energy equation of state, the spectroscopic errors on the neutrino mass are in the range 0.036 – 0.056 eV, depending on the fiducial total neutrino mass , for the combination Euclid+Planck.
On the other hand, looking at the effect that massive neutrinos have on the dark-energy parameter constraints, it is shown that the total CMB+LSS dark-energy FoM decreases only by 15% – 25% with respect to the value obtained if neutrinos are supposed to be massless, when the forecasts are computed using the so-called “-method marginalized over growth-information” (see Methodology section), which therefore results to be quite robust in constraining the dark-energy equation of state.
For what concerns the parameter correlations, at the LSS level, the total neutrino mass is correlated with all the cosmological parameters affecting the galaxy power spectrum shape and BAO positions. When Planck priors are added to the Euclid constraints, all degeneracies are either resolved or reduced, and the remaining dominant correlations among and the other cosmological parameters are -, -, and -, with the - degeneracy being the largest one.
spectroscopic errors stay mostly unchanged whether growth-information are included or marginalised over, and decrease only by 10% – 20% when adding measurements. This result is expected, if we consider that, unlike dark-energy parameters, affects the shape of the power spectrum via a redshift-dependent transfer function , which is sampled on a very large range of scales including the turnover scale, therefore this effect dominates over the information extracted from measurements of . This quantity, in turn, generates new correlations with via the -term, which actually is anti-correlated with . On the other hand, if we suppose that early dark-energy is negligible, the dark-energy parameters , and do not enter the transfer function, and consequently growth information have relatively more weight when added to constraints from and alone. Therefore, the value of the dark-energy FoM does increase when growth-information are included, even if it decreases by a factor 50% – 60% with respect to cosmologies where neutrinos are assumed to be massless, due to the correlation among and the dark-energy parameters. As confirmation of this degeneracy, when growth-information are added and if the dark-energy parameters , , are held fixed to their fiducial values, the errors decrease from 0.056 eV to 0.028 eV, for Euclid combined with Planck.
We expect that dark-energy parameter errors are somewhat sensitive also to the effect of incoherent peculiar velocities, the so-called “Fingers of God” (FoG). This can be understood in terms of correlation functions in the redshift-space; the stretching effect due to random peculiar velocities contrasts the flattening effect due to large-scale bulk velocities. Consequently, these two competing effects act along opposite directions on the dark-energy parameter constraints (see methodology Section 5).
On the other hand, the neutrino mass errors are found to be stable again at , also when FoG effects are taken into account by marginalising over ; in fact, they increase only by 10% – 14% with respect to the case where FoG are not taken into account.
Finally, in Table 18 we summarize the dependence of the -errors on the model cosmology, for Euclid combined with Planck.15 We conclude that, if is 0.1 eV, spectroscopy with Euclid will be able to determine the neutrino mass scale independently of the model cosmology assumed. If is 0.1 eV, the sum of neutrino masses, and in particular the minimum neutrino mass required by neutrino oscillations, can be measured in the context of a CDM model.
Regarding the spectroscopic errors,  finds from Euclid, and , for Euclid+Planck. Concerning the effect of uncertainties on the dark-energy parameter errors, the CMB+LSS dark-energy FoM decreases only by with respect to the value obtained holding fixed at its fiducial value, meaning that also in this case the “-method marginalized over growth–information” is not too sensitive to assumptions about model cosmology when constraining the dark-energy equation of state.
About the degeneracies between and the other cosmological parameters, it is necessary to say that the number of relativistic species gives two opposite contributions to the observed power spectrum (see methodology Section 5), and the total sign of the correlation depends on the dominant one, for each single cosmological parameter. In fact, a larger value suppresses the transfer function on scales . On the other hand, a larger value also increases the Alcock–Paczynski prefactor in . For what concerns the dark-energy parameters , , , and the dark-matter density , the Alcock–Paczynski prefactor dominates, so that is positively correlated to and , and anti-correlated to and . In contrast, for the other parameters, the suppression produces the larger effect and results to be anti-correlated to , and positively correlated to and . The degree of the correlation is very large in the - case, being of the order with and without Planck priors. For the remaining cosmological parameters, all the correlations are reduced when CMB information are added, except for the covariance -, as happens also for the -correlations. To summarize, after the inclusion of Planck priors, the remaining dominant degeneracies among and the other cosmological parameters are -, -, and -, and the forecasted error is , from Euclid+Planck. Finally, if we fix to their fiducial values the dark-energy parameters , and , decreases from 0.086 to 0.048, for the combination Euclid+Planck.
for degenerate spectrum: ; for normal hierarchy: ,
for inverted hierarchy: , ; fiducial cosmology with massless neutrinos
In general, forecasted errors are obtained using techniques, like the Fisher-matrix approach, that are not particularly well suited to quantify systematic effects. These techniques forecast only statistical errors, which are meaningful as long as they dominate over systematic errors. Therefore, it is important to consider sources of systematics and their possible effects on the recovered parameters. Possible sources of systematic errors of major concern are the effect of nonlinearities and the effects of galaxy bias.
The description of nonlinearities in the matter power spectrum in the presence of massive neutrinos has been addressed in several different ways: [966, 779, 778, 780] have used perturbation theory,  the time-RG flow approach and [167, 166, 168, 928] different schemes of -body simulations. Another nonlinear scheme that has been examined in the literature is the halo model. This has been applied to massive neutrino cosmologies in [1, 421, 422].
On the other hand, galaxy/halo bias is known to be almost scale-independent only on large, linear scales, but to become nonlinear and scale-dependent for small scales and/or for very massive haloes. From the above discussion and references, it is clear that the effect of massive neutrinos on the galaxy power spectrum in the nonlinear regime must be explored via -body simulations to encompass all the relevant effects.
Here below we focus on the behavior of the DM-halo mass function (MF), the DM-halo bias, and the redshift-space distortions (RSD), in the presence of a cosmological background of massive neutrinos. To this aim,  and  have analysed a set of large -body hydrodynamical simulations, developed with an extended version of the code gadget-3 , which take into account the effect of massive free-streaming neutrinos on the evolution of cosmic structures.
The pressure produced by massive neutrino free-streaming contrasts the gravitational collapse which is the basis of cosmic structure formation, causing a significant suppression in the average number density of massive structures. This effect can be observed in the high mass tail of the halo MF in Figure 38, as compared with the analytic predictions of  (ST), where the variance in the density fluctuation field, , has been computed via camb , using the same cosmological parameters of the simulations. In particular, here the MF of sub-structures is shown, identified using the subfind package , while the normalization of the matter power spectrum is fixed by the dimensionless amplitude of the primordial curvature perturbations , evaluated at a pivot scale , which has been chosen to have the same value both in the CDM and in the CDM cosmologies.
In Figures 38 and 39, two fiducial neutrino masses have been considered, and . From the comparison of the corresponding MFs, we confirm the theoretical predictions, i.e., that the higher the neutrino mass is, the larger the suppression in the comoving number density of DM haloes becomes.
As is well known, massive neutrinos also strongly affect the spatial clustering of cosmic structures. A standard statistics generally used to quantify the degree of clustering of a population of sources is the two-point auto-correlation function. Although the free-streaming of massive neutrinos causes a suppression of the matter power spectrum on scales larger than the neutrino free-streaming scale, the halo bias is significantly enhanced. This effect can be physically explained thinking that, due to neutrino structure suppression, the same halo bias would correspond, in a CDM cosmology, to more massive haloes (than in a CDM cosmology), which as known are typically more clustered.
This effect is evident in Figure 39 which shows the two-point DM-halo correlation function measured with the Landy and Szalay  estimator, compared to the matter correlation function. In particular, the clustering difference between the CDM and CDM cosmologies increases at higher redshifts, as it can be observed from Figures 40 and 41 and the windows at redshifts of Figure 38. Note also the effect of nonlinearities on the bias, which clearly starts to become scale-dependent for separations .
As it happens for the MF and clustering, also RSD are strongly affected by massive neutrinos. Figure 42 shows the real and redshift space correlation functions of DM haloes as a function of the neutrino mass. The effect of massive neutrinos is particularly evident when the correlation function is measured as a function of the two directions perpendicular and parallel to the line of sight. As a consequence, the value of the linear growth rate that can be derived by modelling galaxy clustering anisotropies can be greatly suppressed with respect to the value expected in a CDM cosmology. Indeed, neglecting the cosmic relic massive neutrino background in data analysis might induce a bias in the inferred growth rate, from which a potentially fake signature of modified gravity might be inferred. Figure 43 demonstrates this point, showing the best-fit values of and , as a function of and redshift, where , being the halo effective linear bias factor, the linear growth rate and the pairwise velocity dispersion.
Living Rev. Relativity 16, (2013), 6
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