The first significant evidence for a finite neutrino mass [496] indicated the incompleteness of the standard model of particle physics. Subsequent experiments have further strengthened this evi- dence and improved the determination of the neutrino mass splitting required to explain observa- tions of neutrino oscillations.
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 [865, 18]. Of course, deviations from the standard effective numbers of neutrino species could still indicate exotic physics which we will discuss below (Section 2.7.4).
New and future neutrino experiments aim to determine the remaining parameters of the neu- trino mass matrix and the nature of the neutrino mass. Within three families of neutrinos, and
Figure 41: Constraints from neutrino oscillations and from cosmology in the m-Σ plane. Image reproduced by permission from [641]; copyright by IOP and SISSA.
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 hi- erarchy (IH), with the smaller spitting between the two higher mass neutrinos. Figure 41 [641] illustrates the currently allowed regions in the plane of total neutrino mass, Σ, vs. mass of the lightest neutrino,m. Note that a determination of Σ <0.1 eV would indicate normal hierarchy and that there is an expected minimum mass Σ>0.054 eV. The cosmological constraint is from [1007].
Cosmological constraints on neutrino properties are highly complementary to particle physics experiments for several reasons:
• Relic neutrinosproduced in the early universe are hardly detectable by weak interactions,
making it impossible with foreseeable technology to detect them directly. But new cosmolog- ical probes such as Euclid offer the opportunity to detect (albeit indirectly) relic neutrinos, through the effect of their mass on the growth of cosmological perturbations.
• Cosmology remains a key avenue to determine the absolute neutrino mass scale.
Particle physics experiments will be able to place lower limits on theeffective neutrino mass, which depends on the hierarchy, with no rigorous limit achievable in the case of normal hierarchy [896]. Contrarily, neutrino free streaming suppresses the small-scale clustering of large-scale cosmological structures by an amount that depends on neutrino mass.
• “What is the hierarchy (normal, inverted or degenerate)?” Neutrino oscillation data
are unable to resolve whether the mass spectrum consists in two light states with massmand a heavy one with massM – normal hierarchy – or two heavy states with massM and a light one with massm– inverted hierarchy – in a model-independent way. Cosmological observations, such as the data provided by Euclid, can determine the hierarchy, complementarily to data from particle physics experiments.
• “Are neutrinos their own anti-particle?” If the answer is yes, then neutrinos are Majorana fermions; if not, they are Dirac. If neutrinos and anti-neutrinos are identical, there could have been a process in the early universe that affected the balance between particles and anti-particles, leading to the matter anti-matter asymmetry we need to exist [497]. This question can, in principle, be resolved if neutrino-less double-β decay is observed [see 896, and references therein]. However, if such experiments [ongoing and planned, e.g., 355] lead to a negative result, the implications for the nature of neutrinos depend on the hierarchy. As shown in [641], in this case cosmology can offer complementary information by helping determine the hierarchy.
2.7.1
Evidence of relic neutrinos
The hot big bang model predicts a background of relic neutrinos in the universe with an average number density of∼100Nν cm−3, whereNν is the number of neutrino species. These neutrinos
decouple from the CMB at redshiftz∼1010 when the temperature wasT ∼o(MeV), 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 Σ, if it is above∼1 eV [704, finds Σ<1.3 eV at 95% confi- dence] because these neutrinos become non-relativistic before recombination leaving an imprint in the CMB. Neutrinos with masses Σ<1 eV become non-relativistic after recombination altering matter-radiation equality for fixed Ωmh2; this effect is degenerate with other cosmological param-
eters 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 Σ < 0.3 eV [1007] 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 Σ∼0.054 eV 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 [1245]; this might be tested by comparing (and not combining) measurements from CMB at decoupling with low-zmeasurements. An inconsistency would point out a direct measurement of a time varying neutrino mass [1247].
2.7.2
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 and prior to the experiment forecasts indicated that Planck should be able to constrain the sum of neutrino masses Σ, with a 1σerror of 0.13 eV [656, 739, 389]. In [962] Planck reported constraints on theNeff = 3.30+/−0.27 for the effective number of relativistic degrees of freedom, and an upper
limit of 0.23 eV for the summed neutrino mass. However the Planck cosmological constraints also reported a relatively low value of the Hubble parameter with respect to previous measurements, that resulted in several papers , for example [1259], that investigated the possibility that this tension could possibly be resolved by introducing an eV-scale (possibly sterile) neutrino. Combining the Planck results with large scale structure measurements or weak lensing measurements has resulted
in reported claims of even stronger constraints on the sum of neutrino masses, for example [1012] found an upper limit on the sum of neutrino masses of<0.18 eV (95% confidence) by combining with WiggleZ data. [145] and [565] combined Planck data with weak lensing data from CFHTLenS and found higher values for the sum of neutrino masses, as a result of tension in the measured and inferred values ofσ8between lensing and the CMB where the lensing prefers a lower value, however
[678] find that such a lower value ofσ8 is consistent with Baryon feedback models impacting the
small-scale distribution of dark matter.
Euclid’s measurement of the galaxy power spectrum, combined with Planck (primary CMB only) priors should yield an error on Σ of 0.04 eV [for details see 281] which is in qualitative agreement with previous work [e.g. 1028]), assuming a minimal value for Σ and constant neutrino mass. Euclid’s weak lensing should also yield an error on Σ of 0.05 eV [673]. 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 1σ detection of even the minimum Σ allowed by oscillations. Moreover, the two independent techniques will offer cross-checks and robustness to systematics. The error on Σ depends on the fiducial model assumed, decreasing for fiducial models with larger Σ. Euclid will enable us not only to detect the effect of massive neutrinos on clustering but also to determine the absolute neutrino mass scale. However, recent numerical investigations found severe observational degeneracies between the cosmological effects of massive neutrinos and of some modified gravity models [113]. This may indicate an intrinsic theoretical limit to the effective power of astronomical data in discriminating between alternative cosmological scenarios, and in constraining the neutrino mass as well. Further investigations with higher resolution simulations are needed to clarify this issue and to search for possible ways to break these cosmic degeneracies [see also 720, 711, 848].
2.7.3
Hierarchy and the nature of neutrinos
Since cosmology is insensitive to flavour, one might expect that cosmology may not help in deter- mining the neutrino mass hierarchy. However, for Σ<0.1 eV, 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 dis- cussed in [641], in cosmology one can safely neglect the impact of the solar mass splitting. Thus, two masses characterize the neutrino mass spectrum: the lightest m, and the heaviest M. The mass splitting can be parameterized by ∆ = (M−m)/Σ for normal hierarchy and ∆ = (m−M)/Σ 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, [641] 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 Σ<0.13).
A detection of neutrino-less double-β decay from the next generation experiments would in- dicate that neutrinos are Majorana particles. A null result of such double-β decay experiments would lead to a result pointing to the Dirac nature of the neutrino only for degenerate or inverted mass spectrum. Even in this case, however, there are ways to suppress the double-β decay signal, without the neutrinos being Dirac particles. For instance, the pseudo-Dirac scenario, which arises from the same Lagrangian that describes the see-saw mechanism [see e.g. 1020]. 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.
Figure 42: Left: region in the ∆-Σ parameter space allowed by oscillations data. Right: Weak lensing forecasts. The dashed and dotted vertical lines correspond to the central value for ∆ given by oscillations data. In this case Euclid could discriminate NI from IH with a ∆χ2 = 2. Image
reproduced by permission from [641]; copyright by IOP and SISSA.
2.7.4
Number of neutrino species
Neutrinos decouple early in cosmic history and contribute to a relativistic energy density with an effective number of species Nν,eff = 3.046. 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: ωrel =ωγ+Nν,effων, where ωγ denotes the energy density
in photons and is exquisitely constrained from the CMB, and ων is the energy density in one
neutrino. Deviations from the standard value forNν,eff would signal non-standard neutrino features
or additional relativistic species. Nν,eff impacts the big bang nucleosynthesis epoch through its
effect on the expansion rate; measurements of primordial light element abundances can constrain Nν,eff and rely on physics at T ∼ MeV [218]. 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 Nν,eff and Ωmh2 from CMB data alone (given
by the combination of these two parameters that leave matter-radiation equality unchanged) and betweenNν,eff andσ8 and/orns. Large-scale structure surveys measuring the shape of the power
spectrum at large scale can constrain independently the combination Ωmhand ns, thus breaking
the CMB degeneracy. Furthermore, anisotropies in the neutrino background affect the CMB anisotropy angular power spectrum at a level of ∼ 20% 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 2σlevel that there exists a neutrino background with characteristics compatible with what is expected under the cosmological standard model [1173, 384].
The forecasted errors onNν,eff for Euclid (with a Planck prior) are±0.1 at 1σlevel [673], which
is a factor∼5 better than current constraints from CMB and LSS and about a factor ∼2 better than constraints from light element abundance and nucleosynthesis.
2.7.5
Model dependence
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 ΛCDM 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 Σ and the number of relativistic speciesNeff, separately. Possible effects of modified gravity models
that could further degrade the neutrino mass constraints will not be discussed in this section.
2.7.6
Σ
forecasted error bars and degeneracies
In [281] 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 1σ spectroscopic errors on the neutrino mass Σ are in the range 0.036 – 0.056 eV, depending on the fiducial total neutrino mass Σ, for the combination Euclid+Planck.
On the other hand, looking at the effect that massive neutrinos have on the dark-energy pa- rameter 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 “P(k)-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 Σ 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 Σ and the other cosmological parameters are Σ-Ωde, Σ-Ωm, and Σ-wa, with the Σ-Ωdedegeneracy being the largest one.
2.7.6.1 Hierarchy dependence
In addition, the neutrino mass spectroscopic constraints depend also on the neutrino hierarchy. In fact, the 1σerrors on total neutrino mass for normal hierarchy are ∼ 17% – 20% larger than for the inverted one. It appears that the matter power spectrum is less able to give information on the total neutrino mass when the normal hierarchy is assumed as fiducial neutrino mass spectrum. This is similar to what found in [641] for the constraints on the neutrino mass hierarchy itself, when a normal hierarchy is assumed as the fiducial one. On the other hand, when CMB information are included, the Σ-errors decrease by∼35% in favor of the normal hierarchy, at a given fiducial value Σ|fid. This difference arises from the changes in the free-streaming effect due to the assumed mass
hierarchy, and is in agreement with the results of [738], which confirms that the expected errors on the neutrino masses depend not only on the sum of neutrino masses, but also on the order of the mass splitting between the neutrino mass states.
2.7.6.2 Growth and incoherent peculiar velocity dependence
Σ spectroscopic errors stay mostly unchanged whether growth-information are included or mar- ginalised over, and decrease only by 10% – 20% when adding fgσ8 measurements. This result
is expected, if we consider that, unlike dark-energy parameters, Σ affects the shape of the power spectrum via a redshift-dependent transfer functionT(k, z), which is sampled on a very large range of scales including the P(k) turnover scale, therefore this effect dominates over the information extracted from measurements offgσ8. This quantity, in turn, generates new correlations with Σ
that early dark-energy is negligible, the dark-energy parameters Ωde,w0 andwa do not enter the
transfer function, and consequently growth information have relatively more weight when added to constraints from H(z) and DA(z) 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 Σ and the dark-energy parameters. As confirmation of this degeneracy, when growth-information are added and if the dark-energy parameters Ωde, w0, wa are held fixed to their fiducial values,
the errorsσ(Σ) 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 inco- herent 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 atσ(Σ) = 0.056, also when FoG effects are taken into account by marginalising overσv(z); in fact, they increase only