1

26. Neutrinos in Cosmology

26. Neutrinos in Cosmology
Revised August 2023 by J. Lesgourgues (TTK, RWTH) and L. Verde (ICC, U. of Barcelona;
ICREA, Barcelona).

26.1 Standard neutrino cosmology
Neutrinos leave detectable imprints on cosmological observations that can then be used toconstrain neutrino properties. This is a great example of the remarkable interconnection andinterplay between nuclear physics, particle physics, astrophysics and cosmology (for general reviews
see e.g., [1–4]). Present cosmological data are already providing constraints on neutrino properties
not only complementary but also competitive with terrestrial experiments; for instance, upperbounds on the total neutrino mass have shrinked by a factor of about 20 in the past two decades.
Forthcoming cosmological data may soon provide key information, not obtainable in other ways
like e.g., a measurement of the absolute neutrino mass scale.
A relic neutrino background pervading the Universe (the Cosmic Neutrino background, CνB)
is a generic prediction of the standard hot Big Bang model (see Big Bang Nucleosynthesis – Chap.
24 of this Review). While it has not yet been detected directly, it has been indirectly confirmedby the accurate agreement of predictions and observations of: a) the primordial abundance of lightelements (see Big Bang Nucleosynthesis – Chap. 24) of this Review; b) the power spectrum of
Cosmic Microwave Background (CMB) anisotropies (see Cosmic Microwave Background – Chap.
29 of this Review); and c) the large-scale clustering of cosmological structures. Within the hot Big
Bang model such good agreement would fail dramatically without a CνB with properties matchingclosely those predicted by the standard neutrino decoupling process (i.e., involving only weakinteractions).
We will illustrate below that cosmology is sensitive to the following neutrino properties: theirdensity, related to the number of active (i.e., left-handed, see Neutrino Mass, Mixing, and Oscillations - Chap. 14 of this Review) neutrino species, and their masses. At first order, cosmology issensitive to the total neutrino mass, but is blind to the mixing angles and CP violation phase as
discussed in Neutrino Mass, Mixing, and Oscillations (Chap. 14 of this Review). This makes cosmological constraints nicely complementary to measurements from terrestrial neutrino experiments.
The minimal cosmological model, ΛCDM, currently providing a good fit to most cosmological
data sets (with the exception of some data in tension, discussed in The Cosmological Parameters
Chap. 25 of this Review), assumes that the only massless or light (sub-keV) relic particles since
the Big Bang Nucleosynthesis (BBN) epoch are photons and active neutrinos. Extended models
with light sterile neutrinos, light thermal axions or other light relics –sometimes referred to as“dark radiation”– would produce effects similar to, and potentially degenerate with, those of active
neutrinos. Thus neutrino bounds are often discussed together with limits on such scenarios. In caseof anomalies in cosmological data, it might not be obvious to discriminate between interpretationin terms of active neutrinos with non-standard decoupling, additional production mechanisms,
non-standard interactions, etc., or in terms of some additional light particles. Such extensions
have been explored as a possible way to resolve the H0 tension between late and early Universe
determinations [5–8], but are not widely favoured [9–11].
Hence neutrino density and mass bounds can be derived under the assumption of no additional
massless or light relic particles, and the neutrino density measured in that way provides a test of
standard (i.e., involving only weak interactions) neutrino decoupling.
In that model, the three active neutrino types thermalize in the early Universe, with a negligible
leptonic asymmetry. Then they can be viewed as three propagating mass eigenstates sharing the
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:17am

26. Neutrinos in Cosmology

5

5

2

Figure 26.1: Ratio of the CMB C`T T (left, including lensing effects) and matter power spectrum
P (k) (right, computed for each model in units of (h−1 Mpc)3 ) for different values of ∆Neff ≡Neff − 3.044 over those of a reference model with ∆Neff = 0. In order to minimize and bettercharacterise the effect of Neff on the CMB, the parameters that are kept fixed are {zeq , zΛ , ωb ,τ } and the primordial spectrum parameters. Fixing {zeq , zΛ } is equivalent to fixing the fractional
density of total radiation, of total matter and of cosmological constant {Ωr , Ωm , ΩΛ } while increasing
the Hubble parameter as a function of Neff . The statistical errors on the C` are ∼ 1% for a band
power of ∆` = 30 at ` ∼ 1000. The error on P (k) is estimated to be of the order of 5%.

CℓTT/CℓTT(Σmν = 0)

0.8

0.95

Σmν = 0.25eV
Σmν = 0.5eV
Σmν = 0.75eVΣmν = 1eV

0.90

0.85

P(k)/P(k)(Σmν = 0)

1.0

1.00

10

1

2

Multipole ℓ
10

10

3

0.6

Σmν = 0.25eV
Σmν = 0.5eV
Σmν = 0.75eVΣmν = 1eV

0.4

0.2
10

−4

10

−3

10

−2

10

k [h −1Mpc]

−1

10

0

Figure 26.2: Ratio of the CMB C`T T and matter power spectrum P (k) (computed for eachPmodel in units of (h−1 Mpc)3 ) for different values of mν over those of a reference model withP
massless neutrinos. In order to minimize and better characterise the effect of mν on the CMB,
the parameters that are kept fixed are ωb , ωc , τ , the angular scale of the sound horizon θs andthe primordial spectrum parameters (solid lines). This implies that we are increasing the Hubble
Pparameter h as a function of mν . For the matter power spectrum, in order to single out the effectof neutrino free-streaming on P (k), the dashed lines show the spectrum ratio when {ωm , ωb , ΩΛ }are kept fixed. For comparison, the error on P (k)√is of the order of 5% with current observations,and the fractional C` errors are of the order of 1/ ` at low `.

same temperature and identical Fermi-Dirac distributions, thus with no visible effects of flavour
oscillations. Neutrinos decouple gradually from the thermal plasma at temperatures T ∼ 2 MeV. In
the instantaneous neutrino decoupling limit, i.e., assuming that neutrinos were fully decoupled at
the time when electron-positrons annihilate and release entropy in the thermal bath, the neutrino31st May, 2024

26. Neutrinos in Cosmology

3to-photon density ratio between the time of electron-positron annihilation and the non-relativistictransition of neutrinos would be given byρν7= Neff
ργ
8



4
11

4/3

,

(26.1)

with Neff = 3, and the last factor comes from the fourth power of the temperature ratio Tν /Tγ =(4/11)1/3 (see Big Bang Cosmology – Chap. 22 in this Review). In the above formula, Neff is called
the effective number of neutrino species because it can be viewed as a convenient parametrisation of
the relativistic energy density of the Universe beyond that of photons, in units of one neutrino in the
instantaneous decoupling limit. Precise simulations of neutrino decoupling and electron-positronannihilation, taking into account flavor oscillations, provide precise predictions for the actual phasespace distribution of relic neutrinos [12–18]. These distributions differ from the instantaneous
decoupling approximation through a combination of a small shift in the photon temperature andsmall flavor-dependent non-thermal distortions, all at the percent level. The final result for thedensity ratio ρν /ργ in the relativistic regime can always be expressed as in Eq. (26.1), but with adifferent value of Neff . The most recent analyses, that includes the effect of neutrino oscillations
with the present values of the mixing parameters, an improved calculation of the collision terms andthe most recent results on plasma thermodynamics QED corrections, give1 Neff = 3.044 [16, 17].
The precise number density ratio nν /nγ can also be derived from such studies, and is important forP
computing the ratio Ων h2 / i mi (ratio of the physical density of neutrinos in units of the criticaldensity to the sum of neutrino masses) in the non-relativistic regime. Once neutrinos are decoupled,
their momentum redshifts like p ∝ 1/a but, due to the absence of interactions, their phase-space
distributions remain constant when expressed in terms of the comoving momentum q = pa, evenwhen they become non-relativistic.
The neutrino temperature today, Tν0 ' 1.7 × 10−4 eV ' 1.9 K, is smaller than at least two of
the neutrino masses, since the two squared-mass differences are |∆m231 |1/2 > |∆m221 |1/2 > Tν0 (seeNeutrino mass, Mixing, and oscillations – Chap. 14 of this Review). Thus at least two neutrinomass eigenstates are non-relativistic today and behave as a small “hot” fraction of the total dark
matter (they cannot be all the dark matter, as explained in Chap. 27 in this Review). This fractionof hot dark matter can be probed by cosmological experiments, for two related reasons, as we now
describe.
First, neutrinos are the only known particles behaving as radiation at early times (during the
CMB acoustic oscillations) and dark matter at late times (during structure formation), which hasconsequences on the background evolution. Neutrinos become non-relativistic when their mass
is equal to their average momentum, given for any Fermi-Dirac-distributed particle by hpi =3.15 T . Thus the redshift of the non-relativistic transition is given by zinr = mi /(3.15 Tν0 ) − 1 =mi /[0.53 meV]−1 for each eigenstate of mass mi , giving for instance zinr = 110 for mi = 60 meV, corresponding to a time deep inside the matter-dominated regime. Second, until the non-relativistictransition, neutrinos travel at the speed of light, and later on they move at a typical velocity
hvi /ci = 3.15 Tν (z)/mi = 0.53(1 + z) meV/mi , which is several orders of magnitude larger thanthat of the dominant cold (or even of possibly warm) dark matter component(s). This brings theircharacteristic diffusion scale, called the “free-streaming length”, to cosmological relevant values,
with consequences on gravitational clustering and the growth of structure.
P
Once neutrinos are non-relativistic, their energy density is given by ρν ' mi ni . Since thenumber densities ni are equal to each other (up to negligible corrections coming from flavour effects
Pin the decoupling phase), the total mass mν = m1 + m2 + m3 can be factorized out. It is
1

The last effects that potentially require some further checks in order to fully settle the fourth significant digit in
Neff are temperature QED corrections; according to [18], the final result might be closer to 3.043 than to 3.044.

31st May, 2024

26. Neutrinos in Cosmology

4possible that the lightest neutrino is still relativistic today, in which case this relation is slightly
incorrect, but given that the total density is always strongly dominated by that of non-relativistic
neutrinos, the error made is completely negligible. Using the expression for ni /nγ obtained fromprecise neutrino decoupling studies [16, 17], and knowing nγ from the measurement of the CMB
temperature, one can compute ρ0ν , the total neutrino density today,2 in units of the critical density
ρ0crit :Pρ0νmνΩν = 0 =,(26.2)
93.12h2 eVρcritand the total neutrino average number density today: n0ν = 339.5 cm−3 [19]. Here h is the Hubble
constant in units of 100 km s−1 Mpc−1 .

26.2 Effects of neutrino properties on cosmological observables
As long as they are relativistic, i.e., until some time deep inside the matter-dominated regime forneutrinos with a mass mi  3.15 Tνeq ∼ 1.5 eV (see Big Bang Cosmology, Chap. 22 in this Review),
neutrinos enhance the density of radiation: this effect is parameterised by Neff and can be discussed
separately from the effect of the mass that will be described later in this section. Increasing Neff
impacts the observable spectra of CMB anisotropies and matter fluctuations through backgroundand perturbation effects.
26.2.1 Effect of Neff on the CMB
The background effects depend on what is kept fixed when increasing Neff . If the densities ofother species are kept fixed, a higher Neff implies a smaller redshift of radiation-to-matter equality,with very strong effects on the CMB spectrum: when the amount of expansion between radiationto-matter equality and photon decoupling is larger, the CMB peaks are suppressed. This effect is
not truly characteristic of the neutrino density, since it can be produced by varying several otherparameters. Hence, to characterise the effect of Neff , it is more useful and illuminating to enhancethe density of total radiation, of total matter and of Λ by exactly the same amount, in order tokeep the redshift of radiation-to-matter equality zeq and matter-to-Λ equality zΛ fixed [20–22].
The primordial spectrum parameters, the baryon density ωb ≡ Ωb h2 and the optical depth toreionization τ can be kept fixed at the same time, since we can simply vary Neff together with the
Hubble parameter h with fixed {ωb , Ωc , ΩΛ }. The impact of such a transformation is shown in
Fig. 26.1 for the CMB temperature spectrum C`T T (defined in Chap. 29 in this Review) and for thematter power spectrum P (k) (defined in Chap. 22 in this Review) for several representative valuesof Neff . These effects are within the reach of cosmological observations given current error bars, as
discussed in Section 26.3.1 (for instance, with the Planck satellite data, the statistical error on theC` ’s is of the order of one per cent for a band power of ∆` = 30 at ` ∼ 1000).
With this transformation, the main background effect of Neff is an increase in the diffusion scale(or Silk damping scale, see Cosmic Microwave Background – Chap. 29 in this Review) at the timeof decoupling, responsible for the decrease in C`T T at high `, plus smaller effects coming from aslight increase in the redshift of photon decoupling [20–22]. At the level of perturbations, a higher
Neff implies that photons feel gravitational forces from a denser neutrino component; this tends todecrease the acoustic peaks (because neutrinos are distributed in a smoother way than photons)
and to shift them to larger scales / smaller multipoles (because photon perturbations traveling at
the speed of sound in the photon-baryon fluid feel some dragging effect from neutrino perturbations
travelling at the speed of light) [20,22,23]. The effect of increasing Neff on the polarization spectrum
features are the same as on the temperature spectrum: an increased Silk damping, and a shift in
2

The value 93.12 eV applies to the limit of large masses, in which ρ0ν = mi n0i and neutrinos are degenerate inmass [19]. This number would be larger by ∼0.1% in the limit of minimal normal hierarchy (or smaller by ∼0.06%
for minimal inverted hierarchy), due the lighest mass state(s) being still relativistic today and to flavor-dependent
non-thermal distortions in the neutrino phase-space distributions.

P

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526. Neutrinos in Cosmology

the acoustic peak amplitude and location - the latter effect is even more clear in the polarization
spectrum, in which the location of acoustic peaks does not get further influenced by a Doppler
effect like for temperature. The combination of these effects is truly characteristic of the radiation
density parameter Neff and cannot be mimicked by other parameters; thus Neff can be accurately
measured from the CMB alone. However, there are correlations between Neff and other parameters.
In particular, we have seen (Fig. 26.1) that in order to minimise the effect of Neff on the CMB
spectrum, one should vary h at the same time, hence there is a correlation between Neff and h,which implies that independent measurements reducing the error bar on h also reduce that on Neff .
Note that this correlation is not equivalent to a perfect degeneracy, so both parameters can beconstrained with CMB data alone.
Table 26.1: Summary of Neff constraints.

CMB alone
Pl18[TT,TE,EE+lowE]
CMB + background evolution + LSS
Pl18[TT,TE,EE+lowE+lensing] + BAO
”

Model

Neff

Ref.

ΛCDM+Neff

2.92+0.36−0.37 (95%CL)[24]

ΛCDM+Neff” +5-params.

2.99+0.34−0.33 (95%CL)+0.23
2.85−0.23 (68%CL)

[24]
[25]

26.2.2 Effect of Neff on the matter spectrum
We have discussed the effect of increasing Neff while keeping zeq and ωb fixed, because thelatter two quantities are very accurately constrained by CMB data. This implies that ωc increases
with Neff , and that the ratio ωb /ωc = Ωb /Ωc decreases. However, the ratio of baryonic-to-darkmatter has a strong impact on the shape of the matter power spectrum, because until the time ofdecoupling of the baryons from the photons, CDM experiences gravitational collapse, while baryons
are kept smoothly distributed by photon pressure and affected by acoustic oscillations. The decreaseof Ωb /Ωc following from the increase of Neff gives more weight to the most clustered of the twocomponents, namely the dark matter one, and produces an enhancement of the small-scale matter
power spectrum and a damping of the amplitude of baryon acoustic oscillations (BAOs), clearlyvisible in Fig. 26.1 (right plot). The scale of BAOs is also slightly shifted by the same neutrino
dragging effect as for CMB peaks [20, 26].
The increase in the small-scale matter power spectrum is also responsible for a last effect on
the CMB spectra: the CMB last scattering surface is slightly more affected by weak lensing fromlarge-scale structures. This tends to smooth the maxima, the minima, and the damping scale of
the CMB spectra [27].
26.2.3 Effect of neutrino masses on the CMB
Neutrino eigenstates with a mass mi  0.6 eV become non-relativistic after photon decoupling.
They contribute to the non-relativistic matter budget today, but not at the time of equality or
recombination. If we increase the neutrino mass while keeping fixed the density of baryons anddark matter (ωb and ωc ), the early cosmological evolution remains fixed and independent of theneutrino mass, until the time of the non-relativistic transition. Thus one might expect that the
CMB temperature and polarization power spectra are left invariant. This is not true for four
reasons.
First, the neutrino density enhances the total non-relativistic density at late times, ωm =P
ωb + ωc + ων , where ων ≡ Ων h2 is given as a function of the total mass mν by Eq. (26.2).
The late background evolution impacts the CMB spectrum through the relation between scales
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626. Neutrinos in Cosmology

on the last scattering surface and angles on the sky, and through the late ISW effect (see Cosmic
Microwave Background – Chap. 29 of this Review). These two effects depend respectively on the
angular diameter distance to recombination, dA (zrec ), and on the redshift of matter-to-Λ equality.
P
Increasing mν tends to modify these two quantities. By playing with h and ΩΛ , it is possible tokeep one of them fixed, but not both at the same time. Since the CMB measures the angular scale ofacoustic oscillations with exquisite precision, and is only loosely sensitive to the late ISW effect due
to cosmic variance, we choose in Fig. 26.2 to play with the Hubble parameter in order to maintaina fixed scale dA (zrec ). With such a choice, an increase in neutrino mass comes together with adecrease in the late ISW effect explaining the depletion of the CMB spectrum for l ≤ 20. The fact
Pthat both mν and h enter the expression of dA (zrec ) implies that measurements of the neutrinomass from CMB data are strongly correlated with h. Second, the non-relativistic transition ofneutrinos affects the total pressure-to-density ratio of the Universe, and causes a small variationof the metric fluctuations. If this transition takes place not too long after photon decoupling, thisvariation is observable through the early ISW effect [22, 28, 29]. It is responsible for the dip seenin Fig. 26.2 for 20 ≤ ` ≤ 200. Third, when the neutrino mass is higher, the CMB spectrum is lessaffected by the weak lensing effect induced by the large-scale structure at small redshift. This isdue to a decrease in the matter power spectrum described in the next paragraphs. This reduced
lensing effect is responsible for most of the oscillatory patterns visible in Fig. 26.2 (left plot) for` ≥ 200. Fourth, the neutrinos with the smallest momenta start to be non-relativistic earlier than
the average ones. The photon perturbations feel this through their gravitational coupling withneutrinos. This leads to a small enhancement of ClT T for ` ≥ 500, hardly visible on Fig. 26.2
because it is balanced by the lensing effect.
26.2.4 Effect of neutrino masses on the matter spectrum
The physical effect of neutrinos on the matter power spectrum is related to their velocity
dispersion. Neutrinos free-stream over large distances without falling into small potential wells.
The free-streaming scale is roughly defined as the distance travel by neutrinos over a Hubble timescale tH = (a/ȧ), and approximates the scale below which neutrinos remain very smooth. On
larger scales, they cluster in the same way as cold dark matter. The power spectrum of total
2 with δ ≡ δ + δ + δ , gets a negligible
matter fluctuations, related to the squared fluctuation δmmcνbcontribution from the neutrino component on small scales, and is reduced by a factor (1 − 2fν ),where fν = ων /ωm . Additionally, on scales below the free-streaming scale, the growth of ordinary
cold dark matter and baryon fluctuations is modified by the fact that neutrinos contribute to thebackground density, but not to the density fluctuations. This changes the balance between thegravitational forces responsible for clustering, and the Hubble friction term slowing it down. Thus
the growth rate of CDM and baryon fluctuations is reduced [30]. This results today in an additional
suppression of the small-scale linear matter power spectrum by approximately (1 − 6fν ). These
two effects sum up to a factor (1 − 8fν ) [31] (more precise approximations can be found in [2, 22]).
The non-linear spectrum is even more suppressed on mildly non-linear scales [3, 32–36].
This effect is often illustrated by plots of the matter power spectrum ratio with fixed parameters{ωm , ωb , ΩΛ } and varying fν , i.e., with the CDM density adjusted to get a fixed total dark matter
density [2, 22, 31] (see Fig. 26.2, right plot, dashed lines). This transformation does not leave theredshift of equality zeq invariant, and has very large effects on the CMB spectra. If one follows
the logic of minimizing CMB variations and fixing zeq like in the previous paragraphs, the increasePin mν must take place together with an increase of h, which tends to suppress the large-scale
power spectrum, by approximately the same amount as the neutrino free-streaming effect [37].
In that case, the impact of neutrino masses on the matter power spectrum appears as an overall
amplitude suppression, which can be seen in Fig. 26.2 (right plot, solid lines). The oscillations
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726. Neutrinos in Cosmology

on intermediate wavenumbers come from a small shift in the BAO scale [37]. This global effect is
not degenerate with a variation of the primordial spectrum amplitude As , because it only affects
the matter power spectrum, and not the CMB spectra. However, the amplitude of the CMB
temperature and polarization spectrum is given by the combination As e−2τ . Hence a measurementof τ is necessary in order to fix As from CMB data, and avoid a parameter degeneracy betweenPmν and As [37–39].
A few of the neutrino mass effects described above –free-streaming scale, early ISW– depend on
individual masses mi , but most of them depend only on the total mass through fν –suppression of
the matter power spectrum, CMB lensing, shift in angular diameter distance–. Because the latterPeffects are easier to measure, cosmology is primarily sensitive to the total mass mν [40, 41]. The
possibility that future data sets might be able to measure individual masses or the mass hierarchy,
despite systematic errors and parameter degeneracies, an active area of investigation [42–48].

26.3 Cosmological Constraints on neutrino properties
In this review we focus on cosmological constraints on the abundance and mass of ordinary
active neutrinos. Several stringent but model-dependent constraints on non-standard neutrinos(e.g., sterile neutrinos, active neutrinos with interactions beyond the weak force, unstable neutrinos
with invisible decay, etc.) can also be found in the literature.
We highlight that cosmological constraints on neutrino properties are always obtained within
the framework of a ΛCDM model or simple and popular extensions of this model as spelled out
in the following subsections. In light of the emergence of cosmological tensions and especially theso-called Hubble tension which has now reached the ∼ 5σ level (see Chap. 25 of this Review), it isimportant to bear in mind the following considerations. Inconsistent measurements should not be
combined: bounds obtained from combination of discrepant data sets should be considered withextreme caution. Additionally, the constraints reported below assume that the solution of the H0tension –whatever that may be– leaves the interpretation of the adopted probes unaffected.
26.3.1 Neutrino abundance
Table 26.1 shows a list of constraints on Neff obtained with several combination of data sets.
‘Pl18’ denotes the Planck 2018 data, composed of a high-` temperature+polarization likelihood(TT,TE,EE), low-` polarization (low E) and CMB lensing spectrum likelihood (lensing) based onlensing extraction from quadratic estimators [24]. ‘BAO’ refers to measurements of the BAO scale(and hence of the angular diameter distance) described in detail in the references given in the table.
Within the framework of a 7-parameter cosmological model (ΛCDM+Neff ), the constraint on
Neff from the Planck 2018 data release [TT,TE,EE+lowE] is Neff = 2.92+0.36−0.37 (95%CL). This number
is perfectly compatible with the prediction of the standard neutrino decoupling model, Neff = 3.044,
and can be viewed as a proof of self-consistency of the cosmological model.
The bounds can be tightened by adding information on the low-redshift background expansion
from BAOs. Finally one may also add information from large-scale structure (LSS), i.e., on thegrowth rate and clustering amplitude of matter as a function of scale, although LSS data are notvery constraining for the Neff parameter. The second result quoted in Table 26.1 shows that the
Neff bound tightens slightly when adding BAO information from the BOSS galaxy survey plus
LSS information from the Planck CMB lensing spectrum. Adding more recent data on BAOs andredshift-space distortions (RSDs) from the eBOSS galaxy survey [49] only provides a very marginal
improvement of the bounds [54]. All combinations of Planck 2018 data with BAO and/or LSS data
return measurements consistent with the standard expectation.
The situation is different with the inclusion of low-redshift measurements of H0 with distanceladders (DLs) [55, 56], now reaching a 5.7σ tension with Planck in the ΛCDM framework. As
explained in Section 26.2, the positive correlation between Neff and h means that inclusion of the
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826. Neutrinos in Cosmology
Table 26.2: Summary ofPmν constraints.

Model
CMB aloneP
Pl18[TT+lowE]
ΛCDM+ mνP
Pl18[TT,TE,EE+lowE]
ΛCDM+ mν
CMB + probes of background evolution
P
Pl18[TT,TE,EE+lowE] + BAO
ΛCDM+ mνP
Pl18[TT,TE,EE+lowE] + BAO
ΛCDM+ mν +5 params.
CMB + LSSP
Pl18[TT+lowE+lensing]ΛCDM+ mνP
Pl18[TT,TE,EE+lowE+lensing]ΛCDM+ mνP
Pl18[TT,TE,EE+lowE]+ ACT[lensing]ΛCDM+ mν
CMB + probes of background evolution + LSS
P
Pl18[TT,TE,EE+lowE] + BAO + RSD
ΛCDM+ mνP
Pl18[TT,TE,EE+lowE+lensing] + BAO + RSD + Shape ΛCDM+ mν
P
Pl18[TT+lowE+lensing] + BAO + Lyman-αΛCDM+ mνP
Pl18[TT,TE,EE+lowE] + BAO + RSD + SN + DES-Y1 ΛCDM+ mν
P
Pl18[TT,TE,EE+lowE] + BAO + RSD + SN + DES-Y3 ΛCDM+ mν

95% CL (eV)Ref.

< 0.54< 0.26[24][24]< 0.13< 0.515[49][25]< 0.44< 0.24< 0.12

[24]
[24]
[50]

< 0.10< 0.082< 0.087< 0.12< 0.13

[49]
[51]
[52]
[49]
[53]

H0 measurement would tend to push Neff to higher values. However, the high values of Neff thatwould potentially reconcile the measured CMB or BAO peak scale with the Hubble parametersuggested by DLs, H0 ∼ 73 km/s/Mpc, are excluded by the measurement of the damping tail of
the CMB temperature spectrum, either by Planck itself (given the previously quoted Neff bounds)
or by more recent and independent small-scale CMB data sets (as shown e.g. in [57]). Thus, theNeff extension to the ΛCDM model does not reduce the tension significantly, and the combinationof Planck and DL data does not return any meaningful constraint on Neff in this context. It iscurrently unclear whether a resolution of the Hubble tension would require a departure from theΛCDM (or ΛCDM+Neff ) model [11, 58–62].
As long as DL data are not included, the error bars on Neff degrade mildly when the data are
analysed in the context of more extended cosmological scenarios. Adding only the total neutrinomass as an 8th free parameter has a negligible impact on the bounds. The authors of Ref. [25] take amore extreme point of view and fit a 12-parameter model to Pl18[TT,TE,EE+lowE+lensing] data;
they obtain Neff = 2.95 ± 0.24 (68% CL), showing that it is very difficult with current cosmological
data to accommodate shifts of more than 0.5 from the standard Neff value, and to obtain good fits
with, for instance, a fourth (sterile) thermalized neutrino. This is interesting since the anomalies
in some oscillation data could be interpreted as evidence for at least one sterile neutrino with alarge mixing angle, which would need to be thermalised unless non-standard interactions come intoplay [5]. In other words, cosmology disfavours the explanation of the oscillations anomalies in termsof extra neutrinos if they are thermalized.
However, if a resolution to the Hubble tension requires a departure from the ΛCDM model or
its most simple extensions, the situation is more open. In the presence of new physical ingredients (such as, e.g., non-standard interactions, exotic dark matter candidates, non-minimal darkenergy properties or modified gravity), it is in principle conceivable that Neff reaches larger values.
However, to date, none of these ingredients has been shown to fully solve the Hubble tension andreconcile CMB, BAO, DL, and supernovae data.
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26.3.2 Are they really neutrinos, as expected?
While a value of Neff significantly different from zero (at more than 15σ) and consistent withthe expected number 3.044 yields a powerful indirect confirmation of the CνB, departures fromstandard Neff could be caused by any ingredient affecting the early-time expansion rate of the
Universe. Extra relativistic particles (either decoupled, self-interacting, or interacting with a dark
sector), a background of gravitational waves, an oscillating scalar field with quartic potential,
departures from Einstein gravity, or large extra dimensions are some of the possibilities for suchingredients. In principle one could even assume that the cosmic neutrino background never existedor has decayed (like in the “neutrinoless Universe” model of [63]) while another dark radiation
component is responsible for Neff . At least, cosmological data allow to narrow the range of possible
interpretations of Neff ' 3 to the presence of decoupled relativistic relics like standard neutrinos.
Indeed, free-streaming particles leave specific signatures in the CMB and LSS spectra, because
their density and pressure perturbations, bulk velocities and anisotropic stress also source themetric perturbations. These signatures can be tested in several ways.
A first approach consists of introducing a self-interaction term in the neutrino equations [6, 7].
Ref. [8] finds that current CMB and BAO data are compatible with no self-interactions. The
upper limit to the effective coupling constant Geff for a Fermi-like four-fermions interaction at
95% confidence is log10 (Geff MeV2 ) < −1.47 for Pl18+BAO [11]. Note however that neutrino selfinteractions as strong as log10 (Geff MeV2 ) ' −1.4 could reconcile CMB temperature and BAO datawith the direct H0 measurement of Ref [64], but such interactions seem to be hardly compatible
with BBN, laboratory constraints [10] and CMB polarization [9, 11].
A second approach consists of introducing two phenomenological parameters, ceff and cvis (seee.g., [65–67]): c2eff generalizes the linear relation between isotropic pressure perturbations and density perturbations, while c2vis modifies the neutrino anisotropic stress equation. While relativistic
free-streaming species have (c2eff , c2vis ) = (1/3, 1/3), a perfect relativistic fluid would have (c2eff , c2vis )= (1/3, 0). Other values do not necessarily refer to a concrete model, but make it possible to interpolate between these limits. Planck data, alone or in combination with galaxy clustering, strongly
suggests (c2eff , c2vis ) = (1/3, 1/3) [68–70].
Finally, Ref. [23] (resp. [26]) shows that current data are precise enough to detect the “neutrinodrag” effect mentioned in Sec. 26.2 through the measurement of the CMB peak (resp. BAO) scale.
These findings show that current cosmological data are able to detect not just the average densityof some relativistic relics, but also their anisotropies.
26.3.3 Neutrino masses
P
Table 26.2 shows a list of constraints on mν obtained with several combinations of data
sets. The acronyms ‘Pl18’ and ‘BAO’ have been described in the previous subsection, while ‘ACT’
refers to CMB observations from the Atacama Cosmology Telescope [50], ‘RSD’ to redshift-spacedistortions from the eBOSS galaxy survey [49], ‘Shape’ to the shape parameter of the matter
power spectrum measured by the BOSS (eBOSS) galaxy redshift survey [51, 71–73], ‘Lyman-α’ tothe one-dimensional flux power spectrum of eBOSS quasars [52], ‘SN’ to the Type Ia supernovae
compilation of [74], ‘DES-Y1’ to the weak lensing and galaxy auto- and cross-correlation measured
by the Dark Energy Survey (DES) using Year 1 data [75], and ‘DES-Y3’ for the same quantity
estimated from DES Year 3 data [53].
Given that most determinations of Neff are compatible with the standard prediction, Neff =3.044, it is reasonable to adopt this value as a theoretical prior and to investigate neutrino massP
constraints in the context of a minimal 7-parameter model, ΛCDM+ mν . Under this assumption,
the most robust constraints come from Planck 2018 temperature and polarization data alone:Pmν < 0.26 eV (95%CL) [24]. Among the four effects of neutrino masses on the CMB spectra
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1026. Neutrinos in Cosmology

described before, current bounds are dominated by the first and the third effects (modified latebackground evolution, and distortions of the temperature and polarization spectra through weaklensing).
Adding measurements of the BAO scale is crucial, since the determination of the angular diameter distance at small redshift allows us to break parameter degeneracies, for instance betweenPmν and h. The combination of Pl18[TT,TE,EE+lowE] with the most recent BAO measurements,
P
including the eBOSS Data Release 16 (DR16), gives mν < 0.13 eV (95%CL) [49]. Supernovae
data are less constraining than BAO data for the neutrino mass determination.
P
Because the parameter correlation between mν and H0 is negative, the inclusion of DL data
would tend to provide stronger bounds on neutrino masses. However, like for Neff , such bounds
would not be meaningful, since they would arise from a combination of discrepant data sets. Thus,
to constrain neutrino masses, it is once more crucial to understand whether the Hubble tension
arises from unmodeled aspects that have no effect on the current treatment and interpretation of
the datasets adopted here or new physical ingredients.
Taking the former point of view, it is interesting to add LSS data sets, sensitive also to thesmall-scale suppression of the matter power spectrum due to neutrino free-streaming, althoughmost of the constraining power for current data sets comes from the constraints on the expansion
history and growth-history signals [76]. The inclusion of CMB lensing data from Planck 18 improves
the CMB-only bound only by a small amount, but the inclusion of higher resolution CMB lensing
maps from ACT [50] brings the limit down to 0.12 eV. To get an even stronger bound, one can
add data from the eBOSS survey, which infers the growth rate of structures from RSDs. The
P
addition of BAO and RSD measurements from eBOSS tightens the bound down to mν < 0.10 eV(95%CL) [49]. Further including information about the large-scale shape (‘Shape’) of the powerP
spectrum, BOSS+eBOSS data alone give mν < 0.40 eV (95%CL) with a BBN prior on ωb , while
P
Planck+BOSS+eBOSS give mν < 0.082 eV (95% CL) [51]. A similar bound arises from the mostP
recent Lyα forest data from BOSS combined with Planck and BAO data, giving mν < 0.087 eV(95% CL). Such bounds already challenge the inverted hierarchy mass scheme, which predictsPmν ≥ 0.11 eV (e.g., [47] and refs. therein).
It should however be noticed that DES prefers a lower σ8 value than the Planck best fit, relaxingP
the bound to mν < 0.12 eV (95%CL) [49] when using Year 1 data in combination with eBOSS andother datasets (Pl18+BAO+RSD+SN+DES). Using Year 3 data and a similar combination, theP
DES collaboration obtains mν < 0.13 eV (95%CL) [53]. Notice however that the DES analysis
PP
adopts a theory prior mν ≥ 0.06eV while other contraints only assume mν ≥ 0. For a directcomparison, the combined constraint of reference [49] can be recomputed with the same theoryP
prior and becomes mν < 0.13 eV (95%CL).
Leaving aside DL measurements, one finds that upper bounds on neutrino masses become weaker
when other data are analysed in the context of extended cosmological models, but only by a small
amount. Floating Neff instead of fixing it to 3.044 has no significant impact on the neutrino mass
bounds reported in the previous paragraphs. Even in the extreme case considered by Ref. [25],with 12 free cosmological parameters, one can see in Table 26.2 that the bound from Planck 2018(without lensing) + BAO increases from 0.13 eV to 0.52 eV (95% CL) only. This shows that current
cosmological data are precise enough to disentangle the effect of several extended cosmologicalparameters, and that neutrino mass bounds are becoming increasingly robust.
One should stress again that if the explanation of DL measurements requires a cosmologicalscenario with new – and yet unkown – physical ingredients, the current neutrino mass bounds may
in principle be shifted or relaxed. However, in the absence of working scenarios that convincingly
explain the Hubble tension without raising other tensions, we do not at the moment have any
framework for quoting alternative mass bounds and assessing their model dependence.
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1126. Neutrinos in Cosmology

26.4 Future prospects and outlook
The cosmic neutrino background has been detected indirectly at very high statistical significance. Direct detection experiments are now being planned, e.g., at the Princeton Tritium Observatory for Light, Early Universe, Massive-neutrino Yield (PTOLEMY) [77]. The detection prospects
crucially depend on the exact value of neutrino masses and on the enhancement of their density at
the location of the Earth through gravitational clustering in the Milky Way and its sub-halos – aneffect however expected to be small [78–81].
Over the past few years the upper limit on the sum of neutrino masses has become increasingly
stringent, first indicating that the mass ordering is hierarchical and recently putting the inverted
hierarchy under pressure and favouring the normal hierarchy (although quantitative estimates of
how disfavoured the inverted hierarchy is vary depending on assumptions, see e.g. [44, 46, 47, 82–84]) which has consequences for planning future double beta-decay experiments. However, thesebounds rely on the ΛCDM model or its most simple extensions, and thus, on the assumptionthat a resolution of the Hubble tension with DL measurements does not affect the modeling andinterpretation of the data sets adopted here. If this is not the case, a shift of paradigm might berequired, potentially leading to weaker neutrino density and mass bounds.
Neutrino mass and density bounds are expected to keep improving significantly over the next
years, thanks to new LSS experiments like DESI [85], Euclid [86], LSST [87], SPHEREx [88] andSKA [89], in combinations with new CMB experiments like Simons Observatory [90], CMB-S4 [91]or LiteBird [92]. If the ΛCDM model is confirmed, and if neutrinos have standard properties, thetotal neutrino mass should be detected at the level of at least 3–4σ even at the minimum level allowed by oscillations. This is the conclusion reached by several independent studies, using different
dataset combinations (see e.g., [39, 93–98]). One should note that at the minimum level allowedPby oscillationsmν ∼ 0.06 eV, neutrinos constitute ∼ 0.5% of the Universe matter density, andtheir effects on the matter power spectrum is only at the 5% level, implying that exquisite control
of systematic errors will be crucial to achieve the required accuracy. At this level, the information
coming from the power spectrum shape is more powerful than that coming from geometrical measurements (e.g., BAO). But exploiting the shape information, especially on small, mildly non-linear
or non-linear scales, requires improved understanding of the non-linear regime, and of galaxy bias
for galaxy surveys. The fact that different surveys, different probes, and different data set combinations have enough statistical power to reach this level, offers a much needed redundancy andthe possibility to perform consistency checks which in turns helps immensely with the control ofsystematic errors and in making the measurement robust. Using the entire Universe as a particle detector, the on-going and future observational efforts hold the exciting prospect to provide ameasurement of the sum of neutrino masses and possibly indication of their mass hierarchy.
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