Inferences from cosmological observations are made under the assumption that the Universe is homogeneous and isotropic, and as such it is well-described, in the context of general relativity, by a Friedmann-Lemaitre-Robertson-Walker (FLRW) metric. Small deviations from homogeneity and isotropy are modeled as perturbations over of a FLRW background.

In a FLRW Universe, expansion is described by the Friedmann equation² for the Hubble parameter H:

where G is the gravitational constant, K parameterizes the spatial curvature³, a is the cosmic scale factor and the ρ is the total energy density. This is given by the sum of the energy densities of the various components of the cosmic fluid.

Considering cold dark matter (c), baryons (b), photons (γ), dark energy (DE), and massive neutrinos (ν), and introducing the redshift 1 + z = a⁻¹, the Friedmann equation can be recast as:

where we have introduced the present value of the critical density required for flat spatial geometry ρcrit,0≡3H02/8πG (in general, we use a subscript 0 to denote quantities evaluated today), and the present-day density parameters Ω_i = ρ_i,0/ρ_crit,0 (since we will be always referring to the density parameters today, we omit the subscript 0 in this case). The scalings with (1 + z) come from the fact that the energy densities of non-relativistic matter and radiation scale with a⁻³ and a⁻⁴, respectively. For DE, in writing Equation (2) we have left open the possibility for an arbitrary (albeit constant) equation-of-state parameter w. In the case of neutrinos, since the parameter of their equation of state is not constant, we could not write a simple scaling with redshift, although this is possible in limiting regimes (see section 2.5). We use ρ_ν,tot to denote the total neutrino density, i.e., summed over all mass eigenstates. Finally, we have defined a “curvature density parameter” Ωk=-K/H02. From Equation (2) evaluated at z = 0 it is clear that the density parameters, including curvature, satisfy the constrain ∑iΩi=1.

Let us also introduce some extra notation and jargon that will be useful in the following. We will use Ω_m to refer to the total density of non-relativistic matter today. Thus, this in general includes dark matter, baryons, and those neutrinos species that are heavy enough to be non-relativistic today. In such a way we have that Ω_m + Ω_DE = 1 in a flat Universe (or Ω_m + Ω_DE = 1 − Ω_k in general), since the present density of photons and other relativistic species is negligible. Since many times we will have to consider the density of matter that is non-relativistic at all the redshifts that are probed by cosmological observables, i.e., dark matter and baryons but not neutrinos, we also introduce Ω_c+b, with obvious meaning. When we consider dark energy in the form of a cosmological constant (w = −1) we use Ω_Λ in place of Ω_DE to make this fact clear. Finally, we also use the physical density parameters ωi≡Ωih2, with h being the present value of the Hubble parameter in units of 100 km s⁻¹ Mpc⁻¹.

As we shall discuss in more detail in the following, cosmological observables often carry the imprint of particular length scales, related to specific physical effects. For this reason we recall some definitions that will be useful in the following. The causal horizon r_h at time t is defined as the distance traveled by a photon from the Big Bang (t = 0) until time t. This is given by:

Note that this is actually the comoving causal horizon; in the following, unless otherwise noted, we will always use comoving distances. We also note that the comoving horizon is equal to the conformal time η(t) (defined through dt = adη and η(t = 0) = 0). In a Friedmann Universe (i.e., one composed only by matter and radiation), the physical causal horizon is proportional, by a factor of order unity, to the Hubble length dH(t)≡H(t)-1. For this reason, we shall sometimes indulge in the habit of calling the latter the Hubble horizon, even though this is, technically, a misnomer.

A related quantity is the sound horizon r_s(t), i.e., the distance traveled in a certain time by an acoustic wave in the baryon-photon plasma. The expression for r_s is very similar to the one for the causal horizon, just with the speed of light (equal to 1 in our units) replaced by the speed of sound c_s in the plasma:

The speed of sound is given by cs=1/3(1+R), with R = (p_b + ρ_b)/(p_γ + ρ_γ) being the baryon-to-photon momentum density ratio. When the baryon density is negligible relative to the photons, cs≃1/3 and rs≃rh/3=η/3.

Imprints on the cosmological observables of several physical processes usually depend on the value of those scales at some particular time. For example, the spacing of acoustic peaks in the CMB spectrum is reminiscent of the sound horizon at the time of hydrogen recombination; the suppression of small-scale matter fluctuations due to neutrino free-streaming is set by the causal horizon at the time neutrinos become non-relativistic; and so on. Moreover, since today we see those scales through their projection on the sky, what we observe is actually a combination of the scale itself and the distance to the object that we are observing. We find then useful also to recall some notions related to cosmological distances. The comoving distance χ between us and an object at redshift z is

and this is also equal to η₀ − η(z). The comoving angular diameter distance d_A(z) is given by

so that

The angular size θ of an object is related to its comoving linear size λ through θ = λ/d_A(z). This justifies the definition of an object of known linear size as a standard ruler for cosmology. In fact, knowing λ, we can use a measure of θ to get d_A and make inferences on the cosmological parameters that determine its value through the integral in Equation (6).

Another measure of distance is given by the luminosity distance d_L(z), that relates the observed flux F to the intrinsic luminosity L of an object at redshift z:

Similarly to what happened for the angular diameter distance, this allows to use standard candles—objects of known intrinsic luminosity—as a mean to infer the values of cosmological parameters, after their flux has been measured.

According to the standard theory of neutrino oscillations, the observed neutrino flavors ν_α (α = e, μ, τ) are quantum superpositions of three mass eigenstates ν_i (i = 1, 2, 3):

where U is the Pontecorvo-Maki-Nakagawa-Sasaka (PMNS) mixing matrix. The PMNS matrix is parameterized by three mixing angles θ₁₂, θ₁₃, θ₂₃, and three CP-violating phases: one Dirac, δ, and two Majorana phases, α₂₁ and α₃₁. The Majorana phases are non-zero only if neutrinos are Majorana particles. They do not affect oscillation phenomena, but enter lepton number-violating processes like 0ν2β decay. The actual form of the PMNS matrix is:

where c_ij ≡ cos θ_ij and s_ij ≡ sin θ_ij.

In addition to the elements of the mixing matrix, the other parameters of the neutrino sector are the mass eigenvalues m_i (i = 1, 2, 3). Oscillation experiments have measured with unprecedented accuracy the three mixing angles and the two mass squared differences relevant for the solar and atmospheric transitions, namely the solar splitting Δmsol2=Δm212≡m22-m12≃7.6×10-5eV2, and the atmospheric splitting Δmatm2=|Δm312|≡|m32-m12|≃2.5×10-3eV2 (see e.g., [39–41] for a global fit of the neutrino mixing parameters and mass splittings). We know, because of matter effects in the Sun, that, of the two eigenstates involved, the one with the smaller mass has the largest electron fraction. By convention, we identify this with eigenstate “1,” so that the solar splitting is positive. On the other hand, we do not know the sign of the atmospheric mass splitting, so this leaves open two possibilities: the normal hierarchy (NH), where Δm312>0 and m₁ < m₂ < m₃, or the inverted hierarchy, where Δm312<0 and m₃ < m₁ < m₂.

Oscillation experiments are unfortunately insensitive to the absolute scale of neutrino masses. In this review, we will mainly focus on cosmological observations as a probe of the absolute neutrino mass scale. To a very good approximation, cosmological observables are mainly sensitive to the sum of neutrino masses Σm_ν, defined simply as

Absolute neutrino masses can also be probed by laboratory experiments. These will be reviewed in more detail in section 9, where their complementarity with cosmology will be also discussed. For the moment, we just recall the definition of the mass parameters probed by laboratory experiments. The effective (electron) neutrino mass m_β

can be constrained by kinematic measurements like those exploiting the β decay of nuclei. The effective Majorana mass of the electron neutrino m_ββ:

can instead be probed by searching for 0ν2β decay.

Our best description of the Universe is currently provided by the spatially flat ΛCDM model with adiabatic, nearly scale-invariant initial conditions for scalar perturbations. With the exception of some mild (at the ~2σ level) discrepancies that will be discussed in the part devoted to observational limits, all the available data can be fit in this model, that in its simplest (“base”) version is described by just six parameters. In the base ΛCDM model, the Universe is spatially flat (Ω_k = 0), and the matter and radiation content is provided by cold dark matter, baryons, photons, and neutrinos, while dark energy is in the form of a cosmological constant (w = −1). The energy density of photons is fixed by measurements of the CMB temperature, while neutrinos are assumed to be very light, usually fixing the sum of the masses to Σm_ν = 0.06 eV, the minimum value allowed by oscillation experiments. In this way, the energy density of neutrinos is also fixed at all stages of the cosmological evolution (see section 2.5). From Equation (2), and taking into account the flatness constraint, it is clear then that the background evolution in such a model is described by three parameters, for example⁴ h, ω_c, and ω_b, with Ω_Λ given by 1 − Ω_m. The initial scalar fluctuations are adiabatic and have a power-law, nearly scale invariant, spectrum, that is thus parameterized by two parameters, an amplitude A_s and a logarithimc slope n_s − 1 (with n_s = 1 thus corresponding to scale invariance). Finally, the optical depth to reionization τ parameterizes the ionization history of the Universe.

This simple, yet very successful, model can be extended in several ways. The extension that we will be most interested in, given the topic of this review, is a one-parameter extension in which the sum of neutrino masses is considered as a free parameter. We call this seven-parameter model ΛCDM + Σm_ν. This is also in some sense the best-motivated extension of ΛCDM, as we actually know from oscillation experiments that neutrinos have a mass, and from β-decay experiments that this can be as large as 2 eV. In addition to this minimal extension, we will also discuss how relaxing some of the assumptions of the ΛCDM model affects estimates of the neutrino mass. Among the possibilities that we will consider, there are those of varying the curvature (Ω_k), the equation-of-state parameter of dark energy (w), or the density of radiation in the early Universe (N_eff, defined in section 2.5).

There are many relevant extensions to the ΛCDM model that however we will not consider here (or just mention briefly). The most important one concerns the possibility of non-vanishing tensor perturbations, i.e., primordial gravitational waves, that, if detected, would provide a smoking gun for inflation. This scenario is parameterized through an additional parameter, the tensor-to-scalar ratio r. In the following, we will always assume r = 0. In any case, this assumption will not affect the estimates reported here, as the effect of finite neutrino mass and of tensor modes on the cosmological observables are quite distinct. Similarly, we will not consider the possibility of non-adiabatic initial perturbations, nor of more complicated initial spectra for the scalar perturbations, including those with a possible running of the scalar spectral index, although we report a compilation of relevant references in section 7 for the reader's convenience.

Given that cosmological observables carry the imprint of different epochs in the history of the Universe, we find it useful to shortly recall some relevant events taking place during the expansion history, and their relation to the cosmological parameters. For our purposes, it is enough to start when the temperature of the Universe was T ~ 1 MeV, i.e., around the time of Big Bang Nucleosynthesis (BBN) and neutrino decoupling. At these early times (z ~ 10¹⁰), since matter and radiation densities scale as (1 + z)³ and (1 + z)⁴, respectively, the Universe is radiation-dominated.

At T ~ 1 MeV (z ~ 10¹⁰), the active neutrinos decouple from the rest of the cosmological plasma. Before this time, neutrinos were kept in equilibrium by weak interactions with electrons and positrons, that were in turn coupled electromagnetically to the photon bath. After this time, the neutrino mean free path becomes much larger the the Hubble length, so they essentially move along geodesics, i.e., they free-stream. Shortly after neutrino decouple, electrons and positrons in the Universe annihilate, heating the photon-electron-baryon plasma, and, to a much lesser extent, the neutrino themselves (in section 2.5 we shall discuss in more detail the neutrino thermal history). After this time, the Universe can essentially be thought as composed of photons, electrons, protons and neutrons (either free, or, after BBN, bound together into the light nuclei), neutrinos, dark matter, and dark energy.

In this section, we discuss the thermal history of cosmic neutrinos.

As anticipated above, in the early Universe neutrinos are kept in equilibrium with the cosmological plasma by weak interactions. The two competing factors that determine if equilibrium holds are the expansion rate, given by the Hubble parameter H(z), and the interaction rate Γ(z) = n〈σv〉, where n is the number density of particles, σ is the interaction cross section, and v is the velocity of particles (brackets indicate a thermal average). In fact, neutrino interactions become too weak to keep them in equilibrium once Γ < H. The left-hand side of this inequality is set by the standard model of particle physics, as the interaction rate at a given temperature only depends on the cross-section for weak interactions, and thus, ultimately, on the value of the Fermi constant (σw~GF2T2). The right-hand side is instead set through Equation (2) by the total radiation density (the only relevant component at such early times): H2=(8πG/3)(ργ+ρν). In the framework of the minimal ΛCDM model, once the present CMB temperature is measured, the radiation density at any given temperature is fixed. Thus the temperature of neutrino decoupling, defined through Γ(T_ν,dec) = H(T_ν,dec) does not depend on any free parameter in the theory. A quite straightforward calculation shows that T_ν,dec ≃ 1 MeV [44].

While they are in equilibrium, the phase-space distribution f(p) of neutrinos is a Fermi-Dirac distribution⁵:

where it has been taken into account that at T ≳ 1 MeV, the active neutrinos are certainly ultrarelativistic (i.e., T_ν ≫ m_ν) and thus E(p) ≃ p. The distribution does not depend on the spatial coordinate x→, nor on the direction of momentum p^, due to the homogeneity and isotropy of the Universe. Before decoupling, the neutrino temperature T_ν is the common temperature of all the species in the cosmological plasma, that we denote generically with T, so that T_ν = T. We recall that the temperature of the plasma evolves according to g*s1/3aT=const., where g_∗s counts the effective number of relativistic degrees of freedom that are relevant for entropy [44].

Since decoupling happens while neutrinos are ultrarelativistic, it can be shown that, as a consequence of the Liouville theorem, the shape of the distribution function is preserved by the expansion. In other words, the distribution function still has the form Equation (15), with an effective temperature T_ν(z) (that for the sake of simplicity we will continue to refer to as the neutrino temperature) that scales like a⁻¹ (i.e., aT = const). We stress that this means that, when computing integrals over the distribution function, one still neglects the mass term in the exponential of the Fermi-Dirac function, even at times when neutrinos are actually non-relativistic.

Shortly after neutrino decouple, electrons and positrons annihilate and transfer their entropy to the rest of the plasma, but not to neutrinos. In other words, while the neutrino temperature scales like a⁻¹, the photon temperature scales like a-1g*s-1/3, and thus decreases slightly more slowly during e⁺e⁻ annihilation, when g_∗s is decreasing. In fact, applying entropy conservation one finds that the ratio between the neutrino and photon temperatures after electron-positron annihilation is Tν/T=(4/11)1/3. The photon temperature has been precisely determined by measuring the frequency spectrum of the CMB radiation: T₀ = (2.725 ± 0.002) K [45, 46], so that the present temperature of relic neutrinos should be T_ν,0 ≃ 1.95 K ≃ 1.68 × 10⁻⁴ eV.

The number density n_ν of a single neutrino species (including both neutrinos and their antiparticles) is thus given by:

where ζ(3) is the Riemann zeta function of 3, and in the last equality we have taken into account that g = 2 for neutrinos. This corresponds to a present-day density of roughly 113 particles/cm³.

The energy density of a single neutrino species is instead

This is the quantity that appears, among other things, in the right-hand side of the Friedmann equation (summed over all mass eigenstates). In the ultrarelativistic (T_ν ≫ m) and non-relativistic (T_ν ≪ m) limits, the energy density takes simple analytic forms:

These scalings are consistent with the fact that one expects neutrinos to behave as pressureless matter, ρν∝(1+z)3, in the non-relativistic regime, and as radiation, ρν∝(1+z)4, in the ultrarelativistic regime.

Given that the present-day neutrino temperature is fixed by measurements of the CMB temperature and by considerations of entropy conservation, it is clear from the above formulas how the present energy density of neutrinos depends only on one free parameter, namely the sum of neutrino masses Σm_ν defined in Equation (11). Introducing the total density parameter of massive neutrinos Ων≡∑iρνi,0/ρcrit,0, one easily finds from Equation (16):

where we have already included the effects of non-instantaneous neutrino decoupling, see below. In the instantaneous decoupling approximation, the quantity at denominator would be 94.2 eV.

On the other hand, the neutrino energy density in the early Universe only depends on the neutrino temperature, and thus it is completely fixed in the framework of the ΛCDM model. Using the fact that for photons ργ=(π2/15)T4, together with the relationship between the photon and neutrino temperatures, one can write for the total density in relativistic species in the early Universe, after e⁺e⁻ annihilation:

where N_ν is the number of neutrino families. In the framework of the standard model of particle physics, considering the active neutrinos, one has N_ν = 3. However, the above formula slightly underestimates the total density at early times; the main reason is that neutrinos are still weakly coupled to the plasma when e⁺e⁻ annihilation occurs, so that they share a small part of the entropy transfer. Moreover, finite temperature QED radiative corrections and flavor oscillations also play a role. This introduces non-thermal distortions at the subpercent level in the neutrino energy spectrum; the integrated effect is that at early times the combined energy densities of the three neutrino species are not exactly equal to 3ρ_ν, with ρ_ν given by the upper row of Equation (18), but instead are given by (3.046ρ_ν) [12, 47]. A recent improved calculation, including the full collision integrals for both the diagonal and off-diagonal elements of the neutrino density matrix, has refined this value to (3.045ρ_ν) [48]. It is then customary to introduce an effective number of neutrino families N_eff and rewrite the energy density at early times as:

In this review, we will consider N_eff = 3.046 as the “standard” value of this parameter in the ΛCDM model, and not the more precise value found in de Salas and Pastor [48], since most of the literature still makes use of the former value. This does not make any difference, however, from the practical point of view, given the sensitivity of present and next-generation instruments.

It is also customary to consider extensions of the minimal ΛCDM model in which one allows for the presence of additional light species in the early Universe (“dark radiation”). In this kind of extension, the total radiation density of the Universe is still given by the right-hand side of Equation (21), where now however N_eff has become a free parameter. In other words, Equation (21) becomes a definition for N_eff, that is, just a way to express the total energy density in radiation. The effect on the expansion history of this additional radiation component can be taken into account by the substitution

in the rhs of the Hubble equation (2), with ΔN_eff ≡ N_eff − 3.046. Note that this substitution fully captures the effect of the additional species only if this is exactly massless, and not just very light (as in the case of a light massive sterile neutrino, for example—see section 10).

It is often useful, to understand some of the effects that we will discuss in the following, to have a feeling for the time at which neutrinos of a given mass become non-relativistic, or, thinking the other way around, for the mass of a neutrino that becomes non-relativistic at a given redshift. The average momentum of neutrinos at a temperature T_ν is 〈p〉 = 3.15T_ν. We take as the moment of transition from the relativistic to the non-relativistic regime the time when 〈p〉 = m_ν. Then, using the fact that Tν(z)=(4/11)1/3T0(1+z)=1.68×10-4(1+z)eV, one has

This relation can be used to show e.g., that neutrinos with mass m_ν ≲ 0.6 eV turn non-relativistic around or after recombination. In the following, when discussing the effect of neutrino masses on the CMB anisotropies, we will assume that this is the case. Note however that the actual statistical analyses from which bounds on neutrino masses are derived do not make such an assumption. We also note that, given the current measurements of the neutrino mass differences, only the lightest mass eigenstate can still be relativistic today. Thus at least two out of the three active neutrinos become non-relativistic before the present time.

We conclude this section with a clarification on the role of neutrinos in determining the redshift of matter-radiation equality. Given the present bounds on neutrino masses, we know that equality likely takes place when neutrino are relativistic. In fact, observations of the CMB anisotropies constrain z_eq ≃ 3,400, so that neutrinos with mass m_ν ≃ 1.8 eV, just below the current bound from tritium beta-decay, turn non-relativistic at equality. Thus, for masses sufficiently below the tritium bound, the total density of matter at those times is proportional to Ω_c+b. The radiation density is instead provided by photons and by the relativistic neutrinos (and as such does not depend on the neutrino mass), plus any other light species present in the early Universe. So the redshift of equivalence is given by

where the last equality makes it clear that, in the framework of the minimal ΛCDM model, the redshift of equivalence only depends on the quantity ω_c+ω_b, since N_eff is fixed and ω_γ is determined through observations (it is basically the CMB energy density).

The CMB consists of polarized photons that, for the most part, have been free-streaming from the time of recombination to the present time. The pattern of anisotropies in both temperature (i.e., intensity) and polarization thus encodes a wealth of information about the early Universe, down to z = z_rec ≃ 1, 100. Moreover, given that the propagation of photons from decoupling to the present time is also affected by the cosmic environment, the CMB also has some sensitivity to physics at z < z_rec. Two relevant examples for the topic under consideration are the CMB sensitivity to the redshift of reionization (because the CMB photons are re-scattered by the new population of free electrons) and to the integrated matter distribution along the line of sight (because clustering at low redshifts modifies the geodesics with respect to an unperturbed FLRW Universe, resulting in a gravitational lensing of the CMB, see next section). However, the CMB sensitivity to these processes is limited due to the fact that these are integrated effects.

The information in the CMB anisotropies is encoded in the power spectrum coefficients CℓTT, i.e., the coefficients of the expansion in Legendre polynomials of the two-point correlation function. In the case of the temperature angular fluctuations ΔT(n^)/T¯:

For Gaussian fluctuations, all the information contained in the anisotropies can be compressed without loss in the two-point function, or equivalently in its harmonic counterpart, the power spectrum. A similar expression holds for the polarization field and for its cross-correlation with temperature. In detail, the polarization field can be decomposed into two independent components, known as E− (parity-even and curl-free) and B− (parity-odd and divergence-free) modes. Given that, it is clear that we can build a total of six spectra CℓXY with X, Y = T, E, B; however, if parity is not violated in the early Universe, the TB and EB correlations are bound to vanish. Let us also recall that, in linear perturbation theory, B modes are not sourced by scalar fluctuations. Thus, in the framework of the standard inflationary paradigm, primordial B modes can only be sourced in the presence of tensor modes, i.e., gravitational waves.

The shape of the observed power spectra is the result of the processes taking place in the primordial plasma around the time of recombination. In brief, in the early Universe, standing, temporally coherent acoustic waves set in the coupled baryon-photon fluid, as a result of the opposite action of gravity and radiation pressure [49]. Once the photons decouple after hydrogen recombination, the waves are “frozen” and thus we observe a series of peaks and throughs in the temperature power spectrum, corresponding to oscillation modes that were caught at an extreme of compression or rarefaction (the peaks), or exactly in phase with the background (the throughs). The typical scale of the oscillations is set by the sound horizon at recombination r_s(z_rec), i.e., the distance traveled by an acoustic wave from some very early time until recombination, see Equation (4). The position of the first peak in the CMB spectrum is set by the value of this quantity and corresponds to a perturbation wavenumber that had exactly the time to fully compress once. The second peak corresponds to the mode with half the wavelength, that had exactly the time to go through one full cycle of compression and rarefaction, and so on. Thus, smaller scales (larger multipoles) than the first peak correspond to scales that could go beyond one full compression, while larger scales (smaller multipoles) did not have the time to do so. In fact, scales much above the sound horizon are effectively frozen to their initial conditions, provided by inflation. This picture is complicated a little bit by the presence of baryons, that shift the zero of the oscillations, introducing an asymmetry between even and odd peaks. Finally, the peak structure is further modulated by an exponential suppression, due to the Silk damping of photon perturbations (further related to the fact that the tight coupling approximation breaks down at very small scales). This description also holds for polarization pertubations, with some differences, like the fact that the polarization perturbations have opposite phase with respect to temperature perturbations.

As noted above, the large-scale temperature fluctuations, that have entered the horizon very late and did not have time to evolve, trace the power spectrum of primordial fluctuations, supposedly generated during inflation. On the contrary, since there are no primordial polarization fluctuations, but those are instead generated at the time of recombination and then again at the time of reionization, the polarization spectra at large scales are expected to vanish, with the exception of the so-called reionization peak.

We can now understand how the CMB power spectra are shaped by the cosmological parameters, in a minimal model with fixed neutrino mass. The overall amplitude and slope of the spectra are determined by A_s and n_s, since these set the initial conditions for the evolution of perturbations. The height of the first peak strongly depends on the redshift of equivalence z_eq (that sets the enhancement in power due to the early ISW), while its position is determined by the angle θ_s subtended by the sound horizon at recombination. As we have discussed before, z_eq and θ_s are in turn set by the values of the background densities and of the Hubble constant. The baryon density further affects the relative heights of odd and even peaks, and also the amount of damping at small scales, through its effect on the Silk scale. The ratio of the densities of matter and dark energy fixes the redshift of dark energy domination and the amount of enhancement of large-scale power due to the late ISW. Finally, the optical depth at reionization τ induces an overall power suppression, proportional to e^−2τ, in all spectra, at all but the largest scales. This can be easily understood as the effect of the new scatterings effectively destroying the information about the fluctuation pattern at recombination, at the scales that are inside the horizon at reionization. Reionization also generates the large-scale peak in the polarization spectra, described above. Measuring the power spectra gives a precise determination of all these parameters: simplifying a little bit, the overall amplitude and slope give Ase-2τ and n_s (the latter especially if we can measure a large range of scales), the ratio of the peak heights and the amount of small-scale damping fix ω_b, while the position and height of the first peak fix θ_s and z_eq, and thus h and ω_b+c. The polarization spectra further help in that they are sensitive to τ directly, allowing to break the A_s − τ degeneracy, and that the peaks in polarization are sharper and thus allow, in principle, for a better determination of their position [50]. It is clear that adding one more degree of freedom to this picture, for example considering curvature, the equation of state parameter of dark energy, or the neutrino mass as a free parameter, will introduce parameter degeneracies and degrade the constraints.

Coming to massive neutrinos, as we have discussed in section 3, there is a combination of the following effects when Σm_ν, and consequently ω_ν, is increased, depending on how we are changing the other parameters to keep ∑iΩi=1: (i) an increase in θ_s; (ii) a smaller z_eq and thus a longer radiation-dominated era; (iii) a delay of the time of dark energy domination. These changes will in turn result in: (i) a shift towards the left of the position of the peaks; (ii) an increased height of the first peak, that is set by the amount of early ISW; (iii) less power at the largest scales, due to the smaller amount of late ISW. A more quantitative assessment of these effects can be obtained using a Boltzmann code, like CAMB [51] or CLASS [52], to get a theoretical prediction for the CMB power spectra in presence of massive neutrinos. These are shown in Figure 1. In the left panel we plot the unlensed CMB temperature power spectra for a reference model with Σm_ν = 0.06 eV (ων≃6.4×10-4) (the other parameters are fixed to their best-fit values from Planck 2015) and for three models with Σm_ν = 1.8 eV (ων≃1.9×10-2), where either h, ω_c, or Ω_Λ are changed to keep ∑iΩi=1. We consider three degenerate neutrinos with m_ν = 0.6 eV each, so that they become non-relativistic around recombination. We also show the ratio between these spectra and the reference spectrum in the right panel of the same figure.

These imprints are in principle detectable in the CMB, especially the first two, since the position and height of the first peak are very well measured; much less so the redshift of DE domination, due to the large cosmic variance at small ℓ's. However, following the above discussion, it is quite easy to convince oneself that these effects can be pretty much canceled due to parameter degeneracies. In fact, simplifying again a little bit, in standard ΛCDM we use the very precise determinations of the height and position of the first peak to determine θ_s and z_eq, and from them ω_c+b and h. In an extension with massive neutrinos, we still have the same determination of θ_s and z_eq, but we have to use them to fix three parameters, namely ω_c+b, h, and ω_ν, so that the system is underdetermined. One could argue that the amount of late ISW, as measured by the large-scale power, could be used to break this degeneracy, as it would provide a further constraint on the matter density (given that the DE density is fixed by the flatness condition). Unfortunately, measurements of the large-scale CMB power are plagued by large uncertainties, due to cosmic variance, so they are of little help in solving this degeneracy. Given the experimental uncertainties, then, it is clear that, when trying to fit a theory to the data, there will be a strong degeneracy direction corresponding to models having the same θ_s and z_eq, and thus with identical predictions for the first peak, and slightly different values of z_Λ, with very low statistical weight due to the large uncertainties in the corresponding region of the spectrum. In other words, the effects of neutrino masses will be effectively “buried” in the small-ℓ plateau, where experimental uncertainties are large. The situation is even worse in extended models, for example if we allow the spatial curvature or the equation of state of dark energy to vary [13]. In any case, the degeneracy between h and ω_c+b is not completely exact, so that the unlensed CMB still has some degree of sensitivity to neutrinos that were relativistic at recombination. For example, the Planck 2013 temperature data, in combination with high-resolution observations from ACT and SPT, were able to constrain Σm_ν < 1.1 eV after marginalizing over the effects of lensing [53].

Measurements of the distance-redshift relation of Supernovae Ia (SNIa) have provided the compelling evidence of the accelerated Universe [91, 92]. SNIa are produced in binary stellar systems in which one of the stars is a white dwarf. Accreting matter from its companion, the white dwarf explodes once it reaches the Chandrasekhar mass limit. Therefore, SNIa are standard candles, because their absolute magnitude can be theoretically inferred from models of stellar evolution. A comparison between the absolute magnitude and the apparent luminosity yields an estimate of their luminosity distance d_L(z). The expected value of d_L in turn depends on the underlying cosmological model. The constraints coming from SNIa in the Ω_m − Ω_Λ plane are orthogonal to those obtained from CMB. As a result, the combination of the two probes is extremely efficient in breaking the degeneracy between the two parameters. For this reason, SNIa are very useful for constraining models of dark energy and/or arbitrary curvature. Nonetheless, constraints on Σm_ν can benefit from the use of SNIa data, thanks to the improved bounds on Ω_m.

As already discussed, the effect of light massive neutrinos on the background evolution of the Universe can be also compensated by a change in the value of the Hubble constant H₀. Therefore, it is clear that any direct measurement of H₀ can be highly beneficial for putting bounds on Σm_ν. Direct measurements only rely on local distance indicators (i.e., redshift z ≪ 1), therefore they are little or not-at-all sensitive to changes in the underlying cosmological model. In contrast, indirect estimates from high-redshift probes, such as primary CMB, can suffer from model dependency.

Direct measurements of H₀ are based on the geometric distance calibration of nearby Cepheids luminosity-period relation and the subsequent calibration of SNIa over Cepheids observed in the same SNIa galaxy hosts (see e.g., [93] and references therein). The goal is to connect the precise geometric distances measured in the nearby Universe (usually referred to as “anchors”) with the distant SNIa magnitude-redshift relation in order to extract the estimate of H₀. The main systematics are of course related to the calibration procedure. Further improvements on the precision of direct measurements of H₀ are expected to come once the precise parallaxes measurements from the Gaia satellite will be available [94].

Local measurements of H₀ are not directly sensitive to Σm_ν. Besides, their results, in combination with cosmological probes, can break the degeneracy between cosmological parameters and improve constraints on Σm_ν. The main example is in fact the possibility to break the strong (inverse) degeneracy between H₀ and Σm_ν that affects CMB constraints.

Indirect estimates of H₀ can be obtained from CMB and BAO measurements. We have already seen in section 4 that the position and amplitude of the first acoustic peak in the CMB spectrum depends on H₀ in combination with other parameters. In addition, we shall mention that, once the BAO are calibrated with the precise determination of r_d from CMB, measurements of d_A/r_d and Hr_d (or d_V/r_d) yields bounds on H₀ that are competitive with CMB estimates and direct measurements.

We finally mention an additional independent measurement of H₀. The gravitational wave (GW) signal emitted by merging compact objects in combination with the observation of an electromagnetic counterpart has been proposed as a standard siren [95, 96]. The GW waveform reconstruction allows for a determination of the luminosity distance to the source. Precise determinations of the source localization can lead to percent accuracy in the luminosity distance estimation. The observation of the electromagnetic counterpart of the GW event is then essential to determine the redshift to the source. The full combination of distance-redshift pair can finally be employed to constrain H₀. In the absence of the detection of an electromagnetic counterpart, methods to infer the redshift of the source of the GW signal have been proposed (see e.g., [97]).

Before moving to report the current observational constraints, we find it useful to summarize the constraining power of different cosmological observables with respect to the neutrino mass. The discussion is somehow qualitative, also given the high-level complexity of the cosmological models. The purpose is also to underline the importance of combining different cosmological probes.

We start from the CMB. For the present discussion, it is useful to consider separately the information coming from the unlensed CMB (i.e., the primary CMB plus all the secondary effects with the exclusion of lensing) and that coming from the weak lensing of CMB photons. For what concerns the former, the sensitivity of the unlensed CMB to neutrino masses is somehow limited. This is mainly due to a geometrical degeneracy between h and ω_ν thanks to which one can simultaneously change the two parameters (decreasing h and increasing ω_ν) to keep θ_s constant, thus preserving the position of the first peak, with only limited changes to other parts of the spectrum (especially changes in the low-ℓ region, where the sensitivity is limited by cosmic variance, induced by variations in Ω_Λ). The height of the first peak is preserved by keeping ω_c fixed. Having access to the information contained in the CMB lensing, either through its effect on the temperature and polarization power spectra, or through a direct estimation of the lensing power spectrum, helps because Σm_ν also affects the matter distribution and then the amplitude of the lensing potential at small scales. This helps breaking the degeneracy described above.

To illustrate this point, in the upper panel of Figure 3 we show the parameter correlations derived by an analysis of the Planck observations of the temperature, over a wide range of scale, and large-scale polarization anisotropies. We remember that this dataset contains some information about lensing through the high-ℓ part of the temperature power spectrum. The negative degeneracy between Σm_ν and H₀ is particularly evident. Given that ω_c and ω_b are both measured quite well from the CMB, this also translates into a strong degeneracy with Ωm=(ωc+ωb)/h2 and Ω_Λ = 1 − Ω_m. Among the other parameters, one can notice mild correlations with A_s and τ. These are due to the small-scale effects related to the increased lensing in models with larger Σm_ν. The overall amplitude of the spectrum Ase-2τ is very precisely determined by CMB observations. On the other hand, the lensing amplitude depends on A_s but not on τ. So, the lensing amplitude can be kept constant by increasing both A_s and ω_ν. At this point τ has to be increased as well to preserve the scalar amplitude Ase-2τ.

Geometric measurements, like those coming from BAO, SNIa, or direct measurements of H₀, greatly help solving the geometrical degeneracy between H₀ and Σm_ν. This is evident in the lower panel of Figure 3, where we show parameter correlations from an analysis of the same dataset as above with the addition of BAO data, if one compares the (H₀, Σm_ν) square with the corresponding square in the upper panel. Measurements of large scale structures, and especially those that are directly sensitive to the total matter distribution at small scales, are very helpful, in that on the one hand they allow to further constrain Ω_m, A_s, and n_s and thus reduce degeneracies with these parameters; on the other hand, they allow to probe the regime in which neutrino free-streaming is important. Finally, it is also clear that a precise measurement of τ from a CMB experiment that is sensitive to the large-scale polarization (meaning that it can access a large fraction of the sky) will be highly beneficial.

We have focused our attention to the ΛCDM + Σm_ν model. In extended dark energy models (as well as modified gravity models), for example for arbitrary equations of state of the dark energy fluid, the degeneracy between Σm_ν and Ω_Λ is amplified. Both massive neutrinos and dark energy-modified gravity affect the late time evolution of the Universe, so that the individual effects on cosmological observables (mostly structures) can be reciprocally canceled.

CMB observations are probably the most mature cosmological measurements. The frequency spectrum is known with great accuracy [46]. Measurements of the power spectrum of CMB anisotropies in temperature are cosmic-variance limited down to very small scales (ℓ ~ 1, 500) and the quality of current CMB data in polarization is already good enough to tighten constraints on cosmological parameters [14, 16, 17, 19, 20]. The next generation of CMB experiments will further improve our knowledge of CMB polarization anisotropies [21, 30–33]. The main systematics involved in CMB measurements are due to foreground contamination (atmospheric, galactic, extragalactic), calibration uncertainties and spurious effects induced by an imprecise knowledge of the instrument (see e.g., [108–112] for a sample list of references).

The tightest constraints on Σm_ν from a single experiment come from the measurements of the Planck satellite [14]. In the context of a one-parameter extension of the ΛCDM cosmological model, the state of the art after the 2015 data release was as follows. The combination of the measurements of the CMB temperature anisotropies up to the multipole ℓ ≃ 2, 500 (hereafter, “Planck TT”) and the large scale (ℓ < 30) polarization anisotropies (hereafter “lowP”) leads to an upper bound of Σm_ν < 0.72 eV at 95% CL. The inclusion of the small scale (ℓ ≥ 30) polarization measurements (which we globally label as “Planck TE,EE”) provides a tighter upper bound of Σm_ν < 0.49 eV at 95% CL. This latter bound should be regarded as less conservative, as a small level of residual systematics could still affect the small scale polarization data.

The Planck collaboration also provides the most significant measurements of the CMB lensing potential power spectrum for the multipole range 40 < L < 400 (labeled as “lensing”) [113]. When this dataset is included in the analysis, the 95% CL constraints on Σm_ν become: Σm_ν < 0.68 eV for Planck TT+lowP+lensing and Σm_ν < 0.59 eV for Planck TT,TE,EE+lowP+lensing [14]. When combining the lensing reconstruction data from Planck with the measurements of the CMB power spectra, it should be kept in mind that CMB power spectra as measured by Planck prefer a slightly higher lensing amplitude than that estimated with the lensing reconstruction. As a result, the bounds on Σm_ν obtained by their combination have less weight for smaller values of Σm_ν than the corresponding bounds obtained from CMB power spectra only. Nevertheless, higher values of Σm_ν are still disfavored.

In 2016, new estimates of the reionization optical depth τ have been published by the Planck collaboration [43], obtained from the analysis of the high-frequency CMB maps, in 2015 still affected by unexplained systematics effects at large scales. The estimated 68% credible interval for τ coming from the EE−only low-ℓ data is τ = 0.055 ± 0.009. This estimate is lower than the corresponding interval obtained in 2015 from the analysis of the low-frequency maps (τ = 0.067 ± 0.023), though the two estimates are well in agreement with each other. The lower value of τ has an impact on the constraints on Σm_ν, due to the degeneracy between the optical depth and the amplitude of primordial perturbations A_s, as they together fix the normalization amplitude Ase-2τ. A lower τ implies a lower A_S and thus a lower lensing amplitude, leaving less room for large values of Σm_ν (that would further reduce lensing). If the “lowP” dataset is replaced by the new estimate of τ (labeled as “SimLow”), the 95% CL bounds improve as follows: Σm_ν < 0.59 eV for Planck TT+SimLow and Σm_ν < 0.34 eV for Planck TT,TE,EE+SimLow [43].

Although the CMB is an extremely powerful dataset, multiple degeneracies between cosmological parameters limit the constraining power on Σm_ν from CMB only, as seen in section 4.4. Measurements of the large scale structures (LSS) can help solving these degeneracies. LSS surveys map the distribution and clustering properties of matter at later times (or equivalently at lower redshift) than those accessible with CMB data and are directly sensitive to cosmological parameters that CMB data can only constrain indirectly, such as the total matter abundance at late times (see e.g., [114] for a review). In this section, we gather constraints on Σm_ν from different LSS probes alone and in combination with CMB data.

The most recent estimate of the Hubble constant has been reported in Riess et al. [93]. The authors improved over their previous measurement of H₀ from 3.3 to 2.4% thanks to an increased sample of reliable SNIa in nearby galaxies calibrated over Cepheids. Their final estimate, based on the combination of three different anchors, is H0=(73.24±1.74)kms-1Mpc-1, 3.2σ higher than the indirect estimate of H₀ from Planck TT+SimLow (3.4σ higher than Planck TT,TE,EE+SimLow) in the context of a ΛCDM cosmology with Σm_ν = 0.06 eV. Previous analyses from the same authors also pointed to a ~2σ tension between direct measurements of H₀ and indirect estimate from primary CMB anisotropies from Planck (although see [124] for a re-analysis of the same dataset which slightly reduces the discrepancy to within 1σ agreement). A discussion about the possible reasons behind this discrepancy and ways to alleviate it invoking non-standard cosmological scenarios are beyond the scope of this work. We refer the reader to the dedicated works [62, 125, 126] for further reading.

Since the Hubble constant and the sum of neutrino masses are anti-correlated, given the tension between the two probes it is clear that the combination of direct measurements of H₀ with CMB data leads to a preference for smaller values of Σm_ν with respect to CMB-only constraints. Indeed, several authors have pointed out the tight constraints on Σm_ν for such a combination. As an example, Vagnozzi et al. [98] showed that constraints on Σm_ν can be as tight as Σm_ν < 0.148 eV at 95% CL when Planck TT+lowP+BAO are complemented with a Gaussian prior on H₀ equal to the estimate of the Hubble constant in Riess et al. [93], to be compared with Σm_ν < 0.186 eV from Planck TT+lowP+BAO only. When lowP is replaced by a Gaussian prior on τ compatible with the new estimates from SimLow, these numbers change to Σm_ν < 0.115 eV (Σm_ν < 0.151 eV) with (without) the H₀ prior.

For the sake of completeness, we shall also mention that independent estimates of H₀ from BAO measurements conducted by the SDSS III-BOSS DR12 collaboration [23] are in agreement with CMB estimates (see also [62] for a recent discussion). See also Abbott et al. [127] for an additional independent estimate of H₀ with a combination of clustering and weak lensing measurements from DES-Y1 with BAO and BBN data. A discussion about the combination of different measurements of H₀ from cosmological probes and local measurements is also reported in Abbott et al. [127], Vega-Ferrero et al. [128].

Finally, we report that a standard siren measurement of H₀ has been performed after the detection of the neutron star-neutron star merger GW170817 [129–131]. The Hubble constant has been constrained as H0=70.0-8.0+12.0kms-1Mpc-1 at 68% CL. The accuracy of this determination is not comparable with the precise estimates of direct measurements and other cosmological constraints. However, the standard siren approach represents an additional independent estimate of H₀ and appears as a promising avenue as more GW events with electromagnetic counterparts are detected.

Concerning the inclusion of SNIa, the bounds from Planck TT+lowP improve from Σm_ν < 0.72 eV to Σm_ν < 0.33 eV at 95% CL when data from the Joint Lightcurve Analysis [105, 106] are included¹⁰. The most relevant systematics that affect SNIa measurements are related to the way in which SNIa light curves are standardized, with issues mostly arising from photometric calibrations and lightcurve fitting procedures.

The tightest bounds on Σm_ν from a single CMB experiment are those from the Planck satellite, reported in section 5.1. As already explained, this sensitivity mostly comes from the ability to (1) detect, at the level of CMB power spectrum, the smoothing effect of gravitational lensing of CMB photons, and, (2) directly reconstruct the lensing power spectrum itself. These effects arise at small angular scales (higher multipoles ℓ), therefore it is crucial to observe this region of the power spectrum with high accuracy in order to improve the sensitivity on Σm_ν. Improved measurements of the polarization power spectra at all scales are also important to break degeneracies between cosmological parameters. The main example is the effect that a better estimate of the reionization optical depth τ from the large scale polarization spectrum has on Σm_ν. Concerning the lensing power spectrum, this is internally reconstructed by the Planck collaboration with high statistical significance up to intermediate scales. However, the full power of this probe will be definitively unveiled when better measurements of polarization maps are available, enabling reconstruction from E-B estimators with lower variance and up to smaller scales [57].

A detailed summary of the expected sensitivity to cosmological parameters, including Σm_ν, of all pre-2020 and post-2020 CMB missions can be found in Errard et al. [134]. As relevant examples, in this section we focus on two classes of future (post 2020) CMB experiments: a space mission and a ground based telescope.

Recently, a proposal for a future CMB space mission has been submitted to the European Space Agency (ESA) in response to a call for medium-size mission proposals (M5). The mission, named Cosmic ORigin Explorer (CORE), is designed to have 19 frequency channels in the range 60−600 GHz for simultaneously solving for CMB and foreground signals, angular resolution in the range 2′−18′ depending on the frequency channel and aggregate sensitivity of 2 μK·arcmin [32] (for comparison, the Planck satellite has 9 frequency channels in the range 30−900 GHz, angular resolution in the range 5′−33′ and the most sensitive channel shows a temperature noise of 0.55 μK·deg at 143 GHz [135]). This experimental setup would enable to constrain Σmν=(0.072-0.051+0.037)eV at 68% CL assuming a ΛCDM model with a fiducial value of the sum of the neutrino masses Σm_ν = 0.06 eV, for the combination of CORE TT,TE,EE,PP (temperature and E-polarization auto and cross spectra and lensing power spectrum PP) [132]. This roughly corresponds to a sensitivity of σ(Σm_ν) ~ 0.044 eV (note that the target threshold for a 3σ detection in the minimal mass scenario is σ(Σm_ν) = 0.020 eV; for comparison, a simulated Planck-like experiment could only put an upper limit of Σm_ν < 0.315 eV at 68% CL for the same model). Other than to the capability of measuring with high precision the small scale polarization (also in order to reconstruct the lensing potential), part of this high sensitivity also comes from the improved limits that a science mission like CORE can put on τ: compared to Planck, CORE would achieve an almost cosmic-variance-limited (CVL) detection of the reionization optical depth [σ_CVL(τ) ≃ 0.002].

A roadmap towards a Stage-IV (S4) generation of CMB ground-based experiments¹¹ has been also developing [30]. The goal is to set a definitive CMB experiment with ~250,000 detectors surveying half of the sky, with angular resolution of 1′−2′ and a sensitivity of 1μK·arcmin at 150 GHz. The greatest contaminant for a ground-based experiment is the atmospheric noise, which highly reduces the accessible frequencies for CMB observations to a total of four windows, roughly 35, 90, 150, and 250 GHz. The main advantages with respect to a space-borne mission are a larger collecting area with an incredibly higher number of detectors (for a comparison, the CORE proposal accounts for a total of 2,100 detectors [32], the Planck satellite has 74 detectors [135]) and subsequent suppression of experimental noise. At large scales, the Stage-IV target is the recombination bump at ℓ > 20. The reduced sky fraction accessible from ground, foreground contaminations and atmospheric noise are the main issues that limit the possibility to target also the range ℓ < 20. Therefore, it is likely that S4 would be complemented by balloon-based and satellite-based measurements at the largest scales. As a result, forecasts for S4 relies on external measurements of τ. The sensitivity σ(Σm_ν) of S₄ TT,TE,EE,PP complemented with a Gaussian prior on the optical depth of τ = 0.06 ± 0.01 (roughly corresponding to the latest estimate from Planck-HFI [43]) is in the range [0.073 − 0.110] eV, depending on the angular resolution and noise level, for f_sky = 40% [30].

Neither of the two classes of future CMB mission proposals can achieve alone the necessary sensitivity to claim a detection of Σm_ν = 0.06 eV at the 3-σ level. Nevertheless, we will see in the next section that the combination of future CMB missions with future galaxy surveys could possibly lead to the first detection of neutrino masses from cosmology.

Improved performances from future galaxy surveys with respect to the current status can be achieved by mapping a larger volume of the sky, therefore increasing the number of samples observed and going deeper in redshift. In this section, we will briefly review the expected performances of the main Stage-IV LSS surveys.

The successor to SDSS III-BOSS survey will be the ground-based Dark Energy Spectroscopic Instrument¹² (DESI). It is designed to operate for 5 years and cover roughly a 14,000 deg² survey area. The extension in redshift is expected to be up to z = 1 for Luminous Red Galaxies (LRG), z = 1.7 for Emission Line Galaxies (ELG) and z = 3.5 for Lyα forests, for a total of over 20 million galaxy and quasar redshifts. With these numbers, DESI will improve over the BOSS survey by an order of magnitude in both volume covered and number of objects observed. It can achieve a 3.49% and 4.78% determination of the BAO signal across (d_A/r_d) and along (Hr_d) the line-of-sight, respectively, at z = 1.85, and 16% and 9% determination of the same quantities at the highest redshift achievable with Lyα forest z = 3.55 [35]. Even in the most conservative scenario when DESI BAO only (i.e., without including information from the broadband shape of the matter power spectrum and Lyα forests) are combined with future CMB experiments, the sensitivity on Σm_ν greatly improves. It goes down to σ(Σm_ν) = 0.021 eV for CORE TT,TE,EE,PP+DESI BAO, forecasting a ~3σ detection of Σm_ν in the minimal mass scenario [132]. In the case of S4+DESI BAO [30], σ(Σm_ν) is in the range [0.023 − 0.036] eV (or [0.020 − 0.032] eV) with a prior of τ = 0.06 ± 0.01 (or τ = 0.060 ± 0.006, the expected sensitivity from Planck-HFI [136]) and f_sky = 0.40, depending on the S4 angular resolution and noise level. For a 1′ resolution and a noise level lower than 2.5 μK · arcmin, σ(Σm_ν) could be further improved with a better measurement of τ down to the level of σ(Σm_ν) < 0.015 eV, that would guarantee a >4σ detection of Σm_ν in the minimal mass scenario.

The DESI mission will be complementary to the science goals of the Large Synoptic Survey Telescope¹³ (LSST), a Stage-IV ground-based optical telescope. The main science fields in which LSST will mostly operate are [36]: “Inventory of the Solar System, Mapping the Milky Way, Exploring the Transient Optical Sky, and Probing Dark Energy and Dark Matter”. These goals will be achieved by surveying a ~30,000 deg² area (2/3 of which in a “deep-wide-fast” survey mode) over 10 years, in six bands (ugrizy), with incredible angular resolution (~ 0.7″), producing measurements of roughly 10 billion stars and galaxies. Thanks to its peculiar observational strategy, LSST will provide multiple probes of the late-time evolution of the Universe with a single experiment, namely, weak lensing cosmic shear, BAO in the galaxy power spectrum, evolution of the mass function of galaxy clusters, and a compilation of SNIa redshift-distances. The expected sensitivity on Σm_ν [36] is in the range σ(Σm_ν) = [0.030 − 0.070] eV, depending on the fiducial value of Σm_ν assumed when performing forecasts (Σmνfid=[0-0.66]eV). Larger fiducial values for the mass yield better sensitivity. These numbers include a marginalization over the uncertainties coming from an extended cosmological scenario, where a number of relativistic species different than 3.046, a non-zero curvature and a dynamical dark energy component are allowed. They also take into account the combination of the three-dimensional cosmic shear field as measured by a LSST-like survey with Planck-like CMB data and can be improved by a factor of 2 if either BAO or SNIa measurements are also considered, whereas a factor of 2 degradation could come from systematic effects. Interestingly enough, the observational strategy of LSST (large and deep survey) could provide the necessary sensitivity to explore the faint effects that the distinct neutrino mass eigenstates have on cosmological probes. This is a highly debated topic and we refer the reader to section 8 for related discussion.

Synergy between these large ground-based observatories and future space missions is expected. We consider here the ESA Euclid satellite¹⁴ and the NASA Wide Field Infrared Survey Telescope¹⁵ (WFIRST) as representative space-borne missions. Euclid will be a wide-field satellite that operates with imaging and spectroscopic instruments for 6 years and covers roughly 15,000 deg² in the optical and near-infrared bands, observing a billion galaxies and measuring ~100 million galaxy redshifts [37]. The redshift depth will be up to z ~ 2 for galaxy clustering and up to z ~ 3 for cosmic shear. The combination of the galaxy power spectrum measured with Euclid and primary CMB from Planck is expected to give σ(Σm_ν) = 0.04 eV; if instead the weak lensing dataset produced by Euclid is considered in combination with primary CMB, we expect σ(Σm_ν) = 0.05 eV [133]. Both combinations provide a ~ 1σ evidence in the minimal mass scenario. Some authors have also pointed out that weak lensing data as measured by Euclid could discriminate between the two neutrino hierarchies if the true value of Σm_ν is small enough (i.e., far enough from the degenerate region of the neutrino mass spectrum), see [133] and references therein¹⁶.

WFIRST is an infrared telescope with a primary mirror as wide as the Hubble Space Telescope's primary (2.4 m) and will operate for 6 years [38]. The primary instrument on board, the Wide Field Instrument, will be able to operate both in imaging and spectroscopic mode, observing a billion galaxies. The instrumental characteristics of WFIRST will more than double the surface galaxy density measured by Euclid. With this setup, WFIRST will test the late expansion of the Universe with great accuracy employing supernovae, weak lensing, BAO, redshift space distortions (RSD), and clusters as probes. From the BAO and broadband measurements of the matter power spectrum, WFIRST in combination with a Stage-III CMB experiment could provide σ(Σm_ν) < 0.03 eV [38].

We want to conclude this section by pointing out that the aforementioned missions will be extremely powerful if combined together. Indeed, they are quite complementary [137]. A significant example concerning the improvement of constraints on massive neutrinos is the combination of all the previously discussed surveys with the lensing reconstruction from CMB. The cross correlation of weak lensing (optical), CMB lensing power spectrum and galaxy clustering (spectroscopic) can highly reduce the systematics affecting each single probe, in particular the multiplicative bias in cosmic shear [138]. For example, a combination of WFIRST, Euclid, LSST, and CMB Stage-III can achieve σ(Σm_ν) < 0.01 eV [38]. Another example is the calibration of the cluster mass for SZ cluster count analyses. This calibration can be performed through optical surveys such as LSST or through CMB lensing calibration, with comparable results. In Madhavacheril et al. [139], the authors show that lensing-calibrated SZ cluster counts can provide a detection of the minimal neutrino mass Σm_ν at > 3σ level, also in extended cosmological scenarios.

In this section, we will briefly comment about the possibility to use 21-cm survey data to constrain Σm_ν. We refer the reader to the relevant papers for further readings. Measurements of the 21-cm signal such as those expected from the Square Kilometer Array¹⁷ (SKA) and the Canadian Hydrogen Intensity Mapping Experiment¹⁸ (CHIME) can shed light on the Epoch of Reionization, including a better determination of the reionization optical depth τ. In addition, they map the distribution of neutral hydrogen in the Universe, a tracer of the underlying matter distribution. Therefore, constraints on Σm_ν can benefit from 21-cm measurements in two ways: by breaking the degeneracy between Σm_ν and τ (see e.g., [140], where the authors report σ(Σm_ν) = 0.012 eV for a combination of CORE+Euclid lensing and FS+ a prior on τ compatible with expectations from future 21-cm surveys); by detecting the effect of Σm_ν on the evolution of matter perturbations (see e.g., [141–143]).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The reviewer SP declared a past co-authorship with one of the authors ML to the handling Editor.