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## 24. Big Bang Nucleosynthesis

Revised August 2023 by B.D. Fields (Astronomy, Illinois U.; Physics, Illinois U.), P. Molaro (INAF-OATS Trieste; IFPU) and S. Sarkar (Rudolf Peierls, Oxford U.).

### 24.1 Abstract

Big-Bang nucleosynthesis (BBN) offers the deepest reliable probe of the early Universe, being based on well-understood Standard Model physics [1]. Predictions of the abundances of the light elements, D, $^3$He, $^4$He, and $^7$Li, synthesized at the end of the first three minutes, are in good overall agreement with the primordial abundances inferred from observational data, thus validating the standard hot Big-Bang cosmology (see [2–5] for reviews). This is particularly impressive given that these abundances span nine orders of magnitude – from $^4$He/H~0.08 down to $^7$Li/H~10$^{-10}$ (ratios by number). Thus BBN provides powerful constraints on possible deviations from the standard cosmology, and on new physics beyond the Standard Model [6–9].

### 24.2 Theory

The synthesis of the light elements is sensitive to physical conditions in the early radiation-dominated era at a temperature $T \sim 1$ MeV, corresponding to an age $t \sim 1$ s. At higher temperatures, weak interactions were in thermal equilibrium, thus fixing the ratio of the neutron and proton number densities to be $n/p = e^{-Q/T}$, where $Q = 1.293$ MeV is the neutron-proton mass difference. As the temperature dropped, the neutron-proton inter-conversion rate per nucleon, $\Gamma_{n\leftrightarrow p} \sim G_F^2 T^5$, fell faster than the Hubble expansion rate, $H \sim \sqrt{g_* G_N} T^2$, where $g_*$ counts the number of relativistic particle species determining the energy density in radiation (see ‘Big Bang Cosmology’ — Sec. 22 of this Review). This resulted in departure from chemical equilibrium (freeze-out) at $T_{fr} \sim (g_* G_N / G_F^2)^{1/6} \simeq 1$ MeV. The neutron fraction at this time, $n/p = e^{-Q/T_{fr}} \simeq 1/6$, is thus sensitive to every known physical interaction, since $Q$ is determined by both strong and electromagnetic interactions while $T_{fr}$ depends on the weak as well as gravitational interactions. Moreover, the sensitivity to the Hubble expansion rate affords a probe of, e.g., the number of relativistic neutrino species [10]. After freeze-out, the neutrons were free to $\beta$-decay, so the neutron fraction dropped to $n/p \simeq 1/7$ by the time nuclear reactions began. A simplified analytic model of freeze-out yields the $n/p$ ratio to an accuracy of $\sim 1\%$ [11, 12].

The rates of these reactions depend on the density of baryons (strictly speaking, nucleons), which is usually expressed normalized to the relic blackbody photon density as $\eta \equiv n_b/n_\gamma$. As we shall see, all the light-element abundances can be explained with $\eta_{10} \equiv \eta \times 10^{10}$ in the range $6.143 \pm 0.190$. With $n_\gamma$ fixed by the present CMB temperature 2.7255 K (see ‘Cosmic Microwave Background’ — Sec. 29 of this Review), this can be stated as the allowed range for the baryon mass density today, $\rho_b = (4.2 \pm 0.1) \times 10^{-31} \text{ g cm}^{-3}$, or as the baryonic fraction of the critical density, $\Omega_b = \rho_b / \rho_{crit} \simeq \eta_{10} h^{-2} / 274 = (0.02244 \pm 0.00069) h^{-2}$, where $h \equiv H_0 / 100 \text{ km s}^{-1} \text{ Mpc}^{-1}$ is the present Hubble parameter (see ‘The Cosmological Parameters’ — Sec. 25 of this Review).

The nucleosynthesis chain begins with the formation of deuterium in the process $p(n, \gamma)D$. However, photo-dissociation by the high number density of photons delays production of deuterium (and other complex nuclei) until well after $T$ drops below the binding energy of deuterium, $\Delta_D = 2.23$ MeV. The quantity $\eta^{-1}e^{-\Delta_D/T}$, i.e., the number of photons per baryon above the deuterium photo-dissociation threshold, falls below unity at $T \simeq 0.1$ MeV; nuclei can then begin to form without being immediately photo-dissociated again. Only 2-body reactions, such as $D(p, \gamma)^3\text{He}$ and $^3\text{He}(D, p)^4\text{He}$ are important because the density by this time has become rather low – comparable to that of air!

Nearly all neutrons end up bound in the most stable light element $^4$He. Heavier nuclei do not form in any significant quantity both because of the absence of stable nuclei with mass number 5 or 8 (which impedes nucleosynthesis via $n^4\text{He}$, $p^4\text{He}$ or $^4\text{He}^4\text{He}$ reactions), and the large Coulomb barriers for reactions such as $^3\text{He}(^4\text{He}, \gamma)^7\text{Li}$ and $^3\text{He}(^4\text{He}, \gamma)^7\text{Be}$. Hence the primordial mass fraction of $^4\text{He}$, $Y_p \equiv \rho(^4\text{He})/\rho_b$, can be estimated by the simple

counting argument

$$Y_p = \frac{2(n/p)}{1 + n/p} \approx 0.25 \qquad (24.1)$$

where strictly speaking this gives the baryon fraction in $^4$He, which is what we will quote throughout. This differs slightly from the mass fraction due to small binding energy corrections.

There is little sensitivity here to the actual nuclear reaction rates for the production of $^4$He. Nuclear rates are, however, critically important in determining the other ‘left-over’ abundances: D and $^3$He at the level of a few times $10^{-5}$ by number relative to H, and $^7$Li/H at the level of about $10^{-10}$ (when $\eta_{10}$ is in the range 1–10). These values can be understood in terms of approximate analytic arguments [12, 13].

The elemental abundances shown in Fig. 24.1 as a function of $\eta_{10}$ were calculated using an updated version [14] of the Wagoner code [1]; other versions [15–17] too are publicly available. The $^4$He curve includes small corrections due to radiative processes at zero and finite temperatures [18], non-equilibrium neutrino heating during $e^\pm$ annihilation [19], and finite nucleon mass effects [20]; the range primarily reflects the $2\sigma$ uncertainty in the neutron lifetime. The spread in the curves for D, $^3$He, and $^7$Li corresponds to the $2\sigma$ uncertainties in nuclear cross sections, as estimated by Monte Carlo methods [21–24]. The input nuclear data have been carefully reassessed [2, 14, 21–31], leading to improved precision for the abundance predictions. In particular, the uncertainty in $^7$Li/H at interesting values of $\eta$ has been reduced recently by a factor $\sim 2$, a consequence of a similar reduction in the error budget [32] for the dominant mass-7 production channel $^3\text{He}(^4\text{He}, \gamma)^7\text{Be}$. Polynomial fits to the predicted abundances and the error correlation matrix have been given in refs. [23, 33]. The boxes in Fig 24.1 show the observationally inferred primordial abundances with their associated uncertainties, as discussed below.

The nuclear reaction cross sections important for BBN have all been measured at the relevant energies. Recently however there have been substantial advances in the precision of light element observations (e.g., D/H) and in the determination of cosmological parameters (e.g., from Planck). This motivates corresponding improvement in BBN predictions and thus in the key reaction cross sections. Recent measurements of $D(p, \gamma)^3\text{He}$ by the LUNA collaboration have significantly improved the precision of D/H predictions [34]. Even so, the nuclear uncertainties still leave D/H prediction errors larger than those of the observations [14, 30, 31]. The $D(D, n)^3\text{He}$ and $D(D, p)^3\text{H}$ reactions now not only dominate the uncertainty budget, but they can give significantly different D/H predictions depending on whether one uses just the empirical determination of the cross sections, or also uses theory to guide the functional form. Clearly, more experimental data is needed.

An additional experimental parameter important in determining the light element abundances is the neutron lifetime, $\tau_n$, which normalizes (the inverse of) $\Gamma_{n\leftrightarrow p}$. Its value has been revised downwards to $\tau_n = 879.4 \pm 0.6$ s (see N Baryons Listing).

### 24.3 Observations: the Light Element Abundances

BBN theory predicts the universal abundances of D, $^3$He, $^4$He, and $^7$Li which are essentially fixed by $t \sim 180$ s. However, abundances are derived at much later epochs, after stellar nucleosynthesis commenced. Stars produce heavy elements such as C, N, O, and Fe (“metals”), while the ejected remains of stellar processing alters the light element abundances from their primordial values. Thus, one seeks astrophysical sites with low metal abundances to measure light element abundances that are closer to primordial.

BBN is the only significant source of deuterium which is entirely destroyed when it is cycled into stars [35]. Thus, any detection provides a lower limit to primordial $D/H|_p$, and an upper limit on $\eta_{10}$. The best proxy to the primordial value of D is its measure in distant and chemically unprocessed matter, where stellar processing (astration) is minimal [35, 36]. This has become possible with the advent of large telescopes, but after nearly three decades
of observational efforts we have only 11 determinations listed in Table 24.1 [37–43].

<figure><img src="image_138.png" /><figcaption>Figure 24.1: The primordial abundances of $^4$He, D, $^3$He, and $^7$Li as predicted by the standard model of Big-Bang nucleosynthesis — the bands show the 95% CL range [44]. Boxes indicate the observed light element abundances. The narrow vertical band indicates the CMB measure of the cosmic baryon density, while the wider band indicates the BBN D+$^4$He concordance range (both at 95% CL).</figcaption></figure>

High-resolution spectra reveal the presence of D in high-redshift, low-metallicity quasar absorption systems via its isotope-shifted Lyman-α absorption features, though, unfortunately, these are often obscured or contaminated by the hydrogen features of the Lyman-α forest. A few DLA systems show D lines resolved up to the higher members of the Lyman series. Recent determinations [38, 40] and re-analyses [41–43] provide strikingly improved precision over earlier work. D/H shows no correlation with metallicity, redshift, or the neutral hydrogen column density $N(HI)$ ($= \int_{\text{los}} n_{\text{HI}} ds$) integrated over the line-of-sight through the absorber. Metallicities of the absorbers are $(0.001-0.03) \times \text{Solar}$, and the level astration is estimated at the 0.1-1% level [36]. In the Galaxy DI/HI measurements are anti-correlated with metal abundances, which suggests that interstellar D partly resides in dust particles [45]. However, in the absorbers where deuterium is measured, the dust content is small, as implied by Solar proportions of the abundances of refractory and non refractory elements. Thus, the neutral atomic ratio DI/HI ≡ D/H should match the underlying isotope ratio, and be truly representative of the primordial value $D/H|_p$. The weighted mean of the 11 most precise measurements in Table 24.1 is $D/H|_p \times 10^6 = (25.47 \pm 0.25)$. However, these measurements provide $\chi^2/(N-1) = 1.295$, showing that some error is underestimated. We thus increase the uncertainty by a factor $S = \sqrt{\chi^2/(N-1)} = 1.137$ and the recommended value is:

$$D/H|_p \times 10^6 = (25.47 \pm 0.29). \quad (24.2)$$

Considering also the previous determinations of [46–50], the weighted mean of 16 measurements is $D/H|_p \times 10^6 = (25.36 \pm 0.26)$, while different selections provide $D/H|_p \times 10^6 = (25.27 \pm 0.30)$ [43] or $D/H|_p \times 10^6 = (25.45 \pm 0.25)$ [42]. These are all consistent with each other, even without the S correction.

The primordial $^4$He abundance is best determined through recombination emission lines of He and H in the most metal-poor extragalactic HII (ionized) regions, viz. blue compact galaxies, generally found at low redshift. There is now a large body of data on $^4$He and CNO in these galaxies, with over 1000 such systems in the Sloan Digital Sky Survey alone [51]. These data confirm that the stellar contribution to the helium abundance is positively correlated with metal production, so extrapolation to zero metallicity gives the primordial $^4$He abundance $Y_p$. However, HII regions are complex systems and several physical parameters enter in the He/H determination, notably the electron density and temperature, as well as reddening. Thus, systematic effects dominate the uncertainties in the abundance determination [52, 53]. A major step forward has been the inclusion of the He $\lambda$10830 infrared emission line which shows a strong dependence on the electron density and is thus useful to break the degeneracy with the temperature, allowing for a more robust helium abundance determination. In recent works the underlying $^4$He stellar absorption, and/or the newly derived values of the HeI-recombination and H-excitation-collisional coefficients are addressed and the $^4$He abundances have increased significantly. Some recent results are reported in Table 24.2.

There is a reassuring convergence towards the value of $Y_p = 0.245$ between detailed analyses of specific extragalactic HII regions with other (statistically more significant) analyses of many systems. Thus our recommended $^4$He abundance is:

$$Y_p = 0.245 \pm 0.003. \quad (24.3)$$

The central value is close to the mean/weighted average of the values in Table 24.2, however we caution that combining these partially overlapping data sets is not straightforward. The uncertainty reflects the combined statistical and systematic errors, with the latter, estimated to be $\pm 0.002$ [59], being dominant.

The best suited objects for the determination of primordial $^7$Li are metal-poor stars in the Galactic halo, which have metallicities going down to $10^{-6}$ of the Solar value [61]. Observations have long shown [62–65] that $^7$Li does not vary significantly in halo dwarfs with metallicities $\lesssim 1/30$ of Solar — the Spite plateau [62, 66]. Recent observations show a puzzling drop in the Li/H abundance in metal-poor stars with [Fe/H] $< -3.0$ [67–69]. This becomes particularly acute at the very low metallicity end where only one star out of the seven dwarfs with metallicities [Fe/H] $\lesssim -4.5$ shows a $^7$Li abundance close to the Spite Plateau, while in the others where it ought to be present it is either lower or totally absent [61, 70]. The reason for the increase in scatter at low metallicity is unknown and prevents derivation of the primordial $^7$Li value by extrapolation to zero metallicity [68, 69].

To estimate the primordial $^7$Li value we consider only stars with metallicity in the range $-2.8 < [\text{Fe/H}] < -1.5$ [69], where no scatter in excess of the observational errors is observed. This yields:

$$\text{Li/H}|_p = (1.6 \pm 0.3) \times 10^{-10}. \quad (24.4)$$

Strictly speaking the suggested primordial $^7$Li abundance should be considered a lower bound rather than a measure. In fact, $^7$Li in Pop II stars may have been partially destroyed due to mixing of the outer layers with the hotter interior [71]. Such processes can be constrained by the absence of significant scatter in $^7$Li versus $T_{eff}$ [64], but $^7$Li depletion by a factor as large as $\sim 1.8$ may have occurred [64, 72]. Stellar determination of Li abundances typically sum over both $^6$Li and $^7$Li isotopes. However, high-precision measurements indicate $^6\text{Li}/^7\text{Li} \le 0.05$, thus confirming that $^7$Li is dominant [73].

The primordial abundance of $^3$He has the poorest observational determination of all of the light nuclides. The only data available come from the Solar system and from solar-metallicity HII regions in the Galaxy [74]. Therefore, inferring the primordial $^3$He abundance is problematic, compounded by the fact that stellar nucleosynthesis models for $^3$He are in conflict with observations. Consequently, we consider it inappropriate to use $^3$He (and also D+$^3$He) as a cosmological probe.
<table><caption>**Table 24.1:** D/H measurements. For systems with multiple measurements we used the most recent one which is generally more precise.</caption><thead><tr><th>QSO</th><th>z<sub>em</sub></th><th>z<sub>abs</sub></th><th>log N((HI))</th><th>[X/H]</th><th>(D/H) × 10<sup>6</sup></th><th>Ref</th></tr></thead><tbody><tr><td>SDSS J1419+0829</td><td>3.03</td><td>3.049</td><td>20.392±0.003</td><td>-1.92[O/H]</td><td>25.06 ± 0.52</td><td>[37]</td></tr><tr><td>HS 0105+1619</td><td>2.65</td><td>2.536</td><td>19.426±0.006</td><td>-1.77[O/H]</td><td>25.76 ± 1.54</td><td>[37]</td></tr><tr><td>QSO B0913+0715</td><td>2.78</td><td>2.618</td><td>20.312±0.008</td><td>-2.40[O/H]</td><td>25.29 ± 1.05</td><td>[37]</td></tr><tr><td>SDSS J1358+0349</td><td>2.89</td><td>2.853</td><td>20.524±0.006</td><td>-2.80[O/H]</td><td>26.18 ± 0.72</td><td>[38]</td></tr><tr><td>SDSS J1358+6522</td><td>3.17</td><td>3.067</td><td>20.495±0.008</td><td>-2.33[O/H]</td><td>25.82 ± 0.71</td><td>[37]</td></tr><tr><td>SDSS J1558-0031</td><td>2.82</td><td>2.702</td><td>20.75±0.03</td><td>-1.55[O/H]</td><td>24.04 ± 1.44</td><td>[37]</td></tr><tr><td>PKS 1937-1009</td><td>3.78</td><td>3.256</td><td>18.09±0.03</td><td>-1.87[O/H]</td><td>24.49 ± 2.80</td><td>[39]</td></tr><tr><td>QSO J1444+2919</td><td>2.66</td><td>2.437</td><td>19.983±0.010</td><td>-2.04[O/H]</td><td>19.68<sup>+3.3</sup><sub>-2.8</sub></td><td>[40]</td></tr><tr><td>PKS 1937-1009</td><td>3.78</td><td>3.572</td><td>17.925±0.006</td><td>-2.26[O/H]</td><td>26.24 ± 0.48</td><td>[41]</td></tr><tr><td>QSO 1009+2956</td><td>2.63</td><td>2.504</td><td>17.362±0.005</td><td>-2.50[Si/H]</td><td>24.77<sup>+4.1</sup><sub>-3.5</sub></td><td>[42]</td></tr><tr><td>QSO 1243+307</td><td>2.55</td><td>2.525</td><td>19.761±0.026</td><td>-2.77[O/H]</td><td>23.88 ± 0.82</td><td>[43]</td></tr></tbody><tfoot><tr><td colspan="7">Weighted mean, with S = 1.137</td><td>25.47 ± 0.29</td></tr></tfoot></table>

<table><caption>**Table 24.2:** Recent primordial <sup>4</sup>He measurements in extragalactic HII regions.</caption><thead><tr><th>Y<sub>p</sub>(<sup>4</sup>He)</th><th>±1σ<sub>stat</sub></th><th>±1σ<sub>sys</sub></th><th>±1σ<sub>tot</sub></th><th># systems</th><th>Ref</th></tr></thead><tbody><tr><td>0.2453</td><td>0.0034</td><td></td><td></td><td>16</td><td>[54]</td></tr><tr><td>0.2451</td><td>0.0019</td><td>0.0018</td><td>0.0026</td><td>1</td><td>[55]</td></tr><tr><td>0.243</td><td>0.005</td><td></td><td></td><td>16</td><td>[56]</td></tr><tr><td>0.2462</td><td>0.0022</td><td></td><td></td><td>120</td><td>[57]</td></tr><tr><td>0.2436</td><td>0.0040</td><td></td><td></td><td>54</td><td>[58]</td></tr><tr><td>0.2448</td><td>0.0027</td><td>0.0018</td><td>0.0033</td><td>7</td><td>[59]</td></tr><tr><td>0.2448</td><td></td><td></td><td>0.0033</td><td>17</td><td>[60]</td></tr></tbody></table>

## 24.4 Concordance, Dark Matter, and the CMB

We now use the observed light element abundances to test the theory. We first consider standard BBN, which is based on Standard Model physics alone, so $N_ν = 3$ and the only free parameter is the baryon-to-photon ratio $\eta$. (The implications of BBN for physics beyond the Standard Model will be considered below). Thus, any abundance measurement determines $\eta$, and additional measurements overconstrain the theory and thereby provide a consistency check.

While the $\eta$ ranges spanned by the boxes in Fig 24.1 do not all overlap, they are all within a factor $\sim 2$ of each other. In particular, the lithium abundance corresponds to $\eta$ values that are inconsistent with that of the (now very precise) D/H abundance as well as the less-constraining <sup>4</sup>He abundance. This discrepancy marks the *lithium problem*. The problem could simply reflect difficulty in determining the primordial lithium abundance, or could hint at a more fundamental omission in the theory. The possibility that lithium reveals new physics is addressed in detail in the next section. If however we exclude the lithium constraint because its inferred abundance suffers from systematic uncertainties, then D/H and <sup>4</sup>He are in agreement. The concordant $\eta$ range is essentially determined by D/H, and yields [75]

$$ \eta_{10} = 6.040 \pm 0.118 \qquad (24.5) $$

where the errors are $1\sigma$. Despite the lithium problem, the overall concordance remains remarkable: using only well-established microphysics we can extrapolate back to $t \sim 1$ s to predict light element abundances spanning nine orders of magnitude, in approximate agreement with observation. This is a major success for the standard cosmology, and inspires confidence in extrapolation back to such early times.

This concordance provides a measure of the baryon content:

$$ \Omega_b h^2 = 0.02205 \pm 0.00043 \qquad (24.6) $$

where again errors are $1\sigma$, a result that plays a key role in our understanding of the matter budget of the Universe. First of all $\Omega_b \ll 1$, i.e., baryons cannot close the Universe [76]. Furthermore, the cosmic density of (optically) luminous matter is $\Omega_{lum} \simeq 0.0024h^{-1}$ [77], so that $\Omega_b \gg \Omega_{lum}$: most baryons

are optically dark, probably in the form of a diffuse intergalactic medium [78]. Finally, given that $\Omega_m \sim 0.3$ (see the ‘Dark Matter’ and ‘Cosmological Parameters’ reviews), we infer that most matter in the Universe is not only dark, but also takes some non-baryonic (more precisely, non-nucleonic) form.

The BBN prediction for the cosmic baryon density can be tested through precision measurements of CMB temperature fluctuations (see the ‘Cosmic Microwave Background’ review). One can determine $\eta$ from the amplitudes of the acoustic peaks in the CMB angular power spectrum [79], making it possible to compare two measures of $\eta$ using very different physics, at two widely separated epochs. In the standard cosmology, there is no change in $\eta$ between BBN and CMB decoupling, thus, a comparison of $\eta_{BBN}$ and $\eta_{CMB}$ is a key test. Agreement would endorse the standard picture, while disagreement could point to new physics during/- between the BBN and CMB epochs.

The analysis described in the Cosmic Microwave Background review (Sec.29), based on Planck TT, TE, EE + lowE data and lensing, yields $\Omega_b h^2 = 0.02237 \pm 0.00015$ [80], which corresponds to $\eta_{10} = 6.12 \pm 0.04$ [81]. This result depends weakly on the primordial helium abundance, and the fiducial Planck analysis uses BBN theory to fix $Y_p(\eta)$. Without BBN theory, the Planck TT, TE, EE + lowE data plus lensing give $\Omega_b h^2 = 0.02230 \pm 0.00021$, corresponding to $\eta_{10} = 6.104 \pm 0.058$. As shown in Fig. 24.1, this CMB estimate of the baryon density (narrow vertical band) is consistent with the BBN range, i.e., in good agreement with the value inferred from high-redshift D/H measurements and local <sup>4</sup>He determinations; together these observations span diverse environments from redshifts $z \sim 1000$ to the present. Combining the CMB and BBN sharpens the baryon measures to $\eta_{10} = 6.115 \pm 0.038$ and $\Omega_b h^2 = 0.02233 \pm 0.00014$ [75].

The <sup>4</sup>He abundance is proportional to the $n/p$ ratio when the weak-interaction rate falls behind the Hubble expansion rate at $T_{fr} \sim 1$ MeV. The presence of additional neutrino flavors (or of any other relativistic species) at this time increases $g_*$, hence the expansion rate, leading to a larger value of $T_{fr}$, $n/p$, and therefore $Y_p$ [10, 82]. In the Standard Model at $T = 1$ MeV, $g_* = 5.5 + \frac{7}{4}N_ν$, where $N_ν$ is the effective number of (nearly) massless neutrino flavors. The helium curves in 24.1 were computed taking $N_ν = 3$; small corrections for non-equilibrium neu-
trino heating [19] are included in the thermal evolution and lead to an effective $N_ν = 3.044$ compared to assuming instantaneous neutrino freezeout (see 'Big Bang Cosmology' — Sec. 22 of this Review). The computed $^4$He abundance scales as $\Delta Y_p \simeq 0.013\Delta N_ν$ [11]. Clearly the central value for $N_ν$ from BBN will depend on $\eta$, which is independently determined (with weaker sensitivity to $N_ν$) by the adopted D or $^7$Li abundance. For example, if the best value for the observed primordial $^4$He abundance is 0.249, then, for $\eta_{10} \sim 6$, the central value for $N_ν$ is very close to 3. A maximum likelihood analysis on $\eta$ and $N_ν$ based on $^4$He and D abundances nearly identical to those above finds the (correlated) 68% CL ranges to be $\eta_{10} = 6.088\pm0.054$ and $N_ν = 2.898\pm0.141$ [44,75,83]. Identical results are obtained using a simpler method to extract such bounds based on $\chi^2$ statistics, given a set of input abundances [84].

The CMB damping tail is sensitive to the primordial $^4$He abundance independently of both BBN and local $^4$He measurements [85]. The Planck analysis using TT, TE, EE+lowE and lensing but not the BBN $Y_p(\eta)$ relation gives a $^4$He mass fraction $0.239_{-0.025}^{+0.024}$, and nucleon fraction $Y_p = 0.240_{-0.25}^{+0.24}$, both at 95% CL [80]. This is consistent with the HII region helium abundance determination. Moreover, this value is consistent with the Standard ($N_ν = 3$) BBN prediction for $Y_p$ with the Planck-determined baryon density. This concordance represents a successful CMB-only test of BBN.

The precision determination of the baryon density using the CMB motivates using this as an input to BBN calculations. Within the context of the Standard Model, BBN then becomes a zero-parameter theory, and the light element abundances are completely determined to within the uncertainties in $\eta_{CMB}$ and the BBN theoretical errors. Comparison with the observed abundances then can be used to test the astrophysics of post-BBN light element evolution [86]. Alternatively, one can consider possible physics beyond the Standard Model (e.g., which might change the expansion rate during BBN) and then use all of the abundances to test such models; this is discussed in 24.6 below.

## 24.5 The Lithium Problem and its Possible Resolution

As Fig. 24.1 shows, stellar Li/H measurements are inconsistent with the D/H (and CMB) measurements, given the error budgets we have quoted. For instance, the value of $Li/H|_p = (4.72\pm0.7) \times 10^{-10}$ [44] is a factor 3.1 higher than in eq.24.4 which is a 4.4 $\sigma$ discrepancy.

Stars that have been accreted by the Milky Way Gyrs ago from external galaxies show the same abundances as those in the Milky Way, showing that the Li problem is universal [87–89]. The question then becomes pressing as to whether this mismatch comes from systematic errors in the observed abundances, and/or uncertainties in stellar astrophysics or nuclear inputs, or whether there might be new physics at work [9]. Nuclear inputs (cross sections) for BBN reactions are constrained by extensive laboratory measurements; to increase $^7$Be destruction requires enhancement of otherwise subdominant processes that can be attained by missed resonances in a few reactions such as $^7$Be(d, p)2$\alpha$ if the compound nuclear state properties are particularly favorable [29, 90–92]. However, experimental searches have now closed off these possibilities [93–95], making a nuclear fix increasingly unlikely.

Another means to solve the lithium problem is by in situ destruction over the long lifetimes of the host halo stars. Stellar depletion mechanisms include diffusion, rotationally induced mixing, or pre-main-sequence depletion. These effects certainly occur, but to reduce lithium to the required levels generally requires some ad hoc mechanism and fine tuning of the initial stellar parameters [72, 96–98]. General features of diffusive models are a dispersion in the Li abundances and a pronounced downturn in the Li abundances at the hot end of the Li plateau. Some extra turbulence needs to be invoked to limit diffusion in the hotter stars and to restore uniform Li abundance along the Spite plateau [98]. Li abundances for over 100 000 field stars have been obtained in the GALAH (Galactic Archeology with HERMES) survey. Warm stars with [Fe/H] in between -1.0 and -0.5 form an elevated plateau

consistent with the BBN prediction and it has been suggested that this could have been also true for the more metal poor stars which have evolved further [99]. Li destruction in the pre-Main sequence phase has been also proposed [97].

Recent $^6$Li measurements have cast new light on the lithium problem. BBN production of $^6$Li is negligible [100]; instead it is created much later by energetic cosmic-ray p and $^4$He interactions with interstellar gas, in fusion reactions such as $^4$He+$^4$He $\rightarrow 6$, $^7$Li+... and spallation reactions on interstellar carbon, nitrogen, and oxygen such as $p_{cr} + ^{16}O \rightarrow 6$, $^7$Li$^9$Be$^{10,11}$B+... Thus $^6$Li, beryllium, and boron are co-produced, and the detection of Be and B in metal-poor stars thus requires cosmic-ray $^6$Li was present when these stars were formed. $^6$Li detections were reported for some of these stars [101–103], and because the weakly-bound $^6$Li nucleus is more easily destroyed than $^7$Li, the survival of $^6$Li implied that $^7$Li destruction should be small. But new observations place strong upper limits on $^6$Li/H in the plateau stars where this isotope had long been reported to exist [104]. This removes the $^6$Li argument against stellar depletion. Indeed, detailed models of cosmic-ray lithium production can quantify the $^6$Li depletion, and may suggest the $^7$Li destruction could be sufficient to explain the mismatch between the observed Li abundance and the expected BBN $^7$Li [105]. This line of argument supports the predictions of stellar depletion as the possible resolution of the lithium problem [72, 96–98]. Additional $^6$Li limits or detections, in concert with model calculations, can firm up this conclusion.

Observations of interstellar lithium in low-metallicity systems probe lithium abundances not subject to stellar depletion. Measurements of interstellar Li/H in the Small Magellanic Cloud lie near the primordial level, but also are consistent with Milky Way stellar abundances at that metallicity ($\sim 1/4$ solar) [106]. Additional such measurements in more metal-poor systems would be of great interest.

It remains possible that the lithium problem points to new physics. Nucleosynthesis models in which the baryon-to-photon ratio is inhomogeneous can alter abundances for a given $\eta_{BBN}$, but will overproduce $^7$Li [107]. Entropy generation by some non-standard process could have decreased $\eta$ between the BBN era and CMB decoupling, however the lack of spectral distortions in the CMB rules out any significant energy injection up to a redshift $z \sim 10^7$ [108]. The most intriguing resolution of the lithium problem thus involves new physics during BBN [7–9]

We summarize the general features of such solutions here, and later consider examples in the context of specific particle physics models. Many proposed solutions introduce perturbations to light-element formation during BBN; while all element abundances may suffer perturbations, the interplay of $^7$Li and D is often the most important i.e. observations of D often provide the strongest constraints on the allowed perturbations to $^7$Li. In this connection it is important to note that the new, very precise determination of D/H will significantly constrain the ability of such models to ameliorate or solve the lithium problem.

A well studied class of models invokes the injection of suprathemal hadronic or electromagnetic particles due to decays of dark matter particles. The effects are complex and depend on the nature of the decaying particles and their branchings and spectra. However, the models that most successfully solve the lithium problem generally feature non-thermal nucleons, which dissociate all light elements. Dissociation of even a small fraction of $^4$He introduces a large abundance of free neutrons, which quickly thermalize. The thermal neutrons drive the $^7$Be(n, p)$^7$Li conversion of $^7$Be. The resulting $^7$Li has a lower Coulomb barrier relative to $^7$Be and is readily destroyed via $^7$Li(p, $\alpha$)$^4$He [109, 110]. But $^4$He dissociation also produces D directly as well as via nonthermal neutron n(p, $\gamma$)d reactions. This introduces a tension between Li/H reduction and D/H enhancement that becomes increasingly restrictive with the increasing precision of deuterium observations. Indeed, this now forces particle injection scenarios to make very small $^7$Li perturbations — far short of the level needed. An exception is a recent model wherein MeV-scale decays by construction avoid $^4$He dissociation and associated D/H overproduction, instead borrowing neutrons by dissociating only deuterons [111].

Another important class of models retains the standard cosmic
particle content, but changes their interactions via time varia-
tions in the fundamental constants [112–118]. Here too, the de-
tails are model-dependent, but scenarios that solve or alleviate
the lithium problem often feature perturbations to the deuteron
binding energy. A weaker D binding leads to the D bottleneck
being overcome later, so that element formation commences at a
lower temperature and lower density. This leads in turn to slower
nuclear rates that freeze out earlier. The net result is a *higher*
final D/H, due to less efficient processing into $^4$He, but also *lower*
Li, due to suppressed production via $^3$He($\alpha$, $\gamma$)$^7$Be.

The cosmological lithium problem remains an unresolved issue
in BBN. Nevertheless, the remarkable concordance between the
CMB and the D (as well as $^4$He) abundance, is a non-trivial suc-
cess, and provides important constraints on the early Universe.

24.6 Beyond the Standard Model

Given the simple physics underlying BBN, it is remarkable that
it still provides the most effective test for the cosmological viabil-
ity of ideas concerning physics beyond the Standard Model. Al-
though baryogenesis and inflation must have occurred at higher
temperatures in the early Universe, we do not as yet have ‘stan-
dard models’ for these, so BBN still marks the boundary between
the established and the speculative in Big Bang cosmology. It
might appear possible to push the boundary back to the quark-
hadron transition at $T \sim \Lambda_{QCD}$, or electroweak symmetry break-
ing at $T \sim 1/\sqrt{G_F}$; however, so far no observable relics of these
epochs have been identified, either theoretically or observationally.
Thus, although the Standard Model provides a precise description
of physics up to the Fermi scale, cosmology cannot be traced in
detail before the BBN era.

The CMB power spectrum in the damping tail is independently sensitive to $N_\nu$ (e.g. [119]). The CMB value $N_\nu^{\text{CMB}}$ probes the cosmic radiation content at (re)combination, so a discrepancy would imply new physics or astrophysics. Indeed, observations by the South Pole Telescope implied $N_\nu^{\text{CMB}} = 3.85 \pm 0.62$ [120], prompting discussion of dark radiation such as sterile neutrinos [121]. However, Planck 2018 results give $N_\nu^{\text{CMB}} = 2.92^{+0.36}_{-0.37}$, 95% CL, when using Planck TT, TE, EE+lowE, a result quite consistent with 3 Standard Model neutrinos [80] (and adjusting for the CMB’s measurement of $N_{\text{eff}} = 3.044$ due to neutrino heading effects [122–124]). Indeed, the BBN and CMB constraints on $N_\nu$ are independent and of similar precision, so that it is now possible to limit any *change* in this parameter (and/or the baryon-to-photon ratio) between nucleosynthesis and recombination [75]. This can, for example, constrain models with late particle decays, or with early dark energy.

Just as one can use the measured helium abundance to place
limits on $g_*$ [82, 113, 125–127], any changes in the strong, weak,
electromagnetic, or gravitational coupling constants, arising *e.g.*,
from the dynamics of new dimensions, can be similarly con-
strained [128], as can any speed-up of the expansion rate in, *e.g.*,
scalar-tensor theories of gravity [129].

The limits on $N_\nu$ can be translated into limits on other types of
particles or particle masses that would affect the expansion rate of
the Universe during nucleosynthesis. For example, consider *sterile*
neutrinos with only right-handed interactions of strength $G_R <$
$G_F$. Such particles would decouple at higher temperature than
(left-handed) neutrinos, so their number density ($\propto T^3$) relative
to neutrinos would be reduced by any subsequent entropy release,
*e.g.*, due to annihilations of massive particles that become non-
relativistic between the two decoupling temperatures. Thus, (*rel-
ativistic*) particles with less than full strength weak interactions
contribute less to the energy density than particles that remain in
equilibrium up to the time of nucleosynthesis [130]. If we impose
$N_\nu < 4$ as an illustrative constraint, then the three right-handed
neutrinos must have a temperature $3(T_{\nu_R}/T_{\nu_L})^4 < 1$. Since the
temperature of the decoupled $\nu_R$ is determined by entropy con-
servation (see ‘Big Bang Cosmology’ — Sec. 22 of this Review),
$T_{\nu_R}/T_{\nu_L} = [(43/4)/g_*(T_d)]^{1/3} < 0.76$, where $T_d$ is the decoupling
temperature of the $\nu_R$. This requires $g_*(T_d) > 24$, so decoupling
must have occurred at $T_d > 140$ MeV. The decoupling tempera-
ture is related to $G_R$ through $(G_R/G_F)^2 \sim (T_d/3\text{MeV})^{-3}$, where
3 MeV is the decoupling temperature for $\nu_{\text{LS}}$. This yields a limit
$G_R \lesssim 10^{-2} G_F$. The above argument sets lower limits on the

masses of new $Z'$ gauge bosons to which right-handed neutrinos
would be coupled in models of superstrings [131], or extended
technicolour [132]. Similarly a Dirac magnetic moment for neu-
trinos, which would allow the right-handed states to be produced
through scattering and thus increase $g_*$, can be significantly con-
strained [133], as can any new interactions for neutrinos that have
a similar effect [134–136]. Right-handed states can be populated
directly by helicity-flip scattering if the neutrino mass is large
enough, and this property has been used to infer a bound of
$m_{\nu_\tau} \lesssim 1$ MeV (taking $N_\nu < 4$) [137]. If there is mixing be-
tween active and sterile neutrinos then the effect on BBN is more
complicated [138, 139].

BBN limits on the cosmic expansion rate constrain supersym-
metric scenarios in which the neutralino or gravitino are very light,
so that they contribute to $g_*$ [140]. A gravitino in the mass range
$\sim 10^{-4} - 10$ eV will affect the expansion rate of the Universe sim-
ilarly to a light neutralino (which is however now probably ruled
out by collider data, especially the decays of the Higgs-like bo-
son). The net contribution to $N_\nu$ then ranges between 0.74 and
1.69, depending on the gravitino and slepton masses [141].

The limit on the expansion rate during BBN can also be trans-
lated into bounds on the mass/lifetime of non-relativistic particles
that decay during BBN. This results in an even faster speed-up
rate, and typically also changes the entropy [142–144]. If the de-
cays include Standard Model particles, the resulting electromag-
netic [145] [86, 131, 146] and/or hadronic [147, 148] cascades can
strongly perturb the light elements, which leads to even stronger
constraints. Such arguments have been applied to rule out an
MeV mass for $\nu_\tau$, which decays during nucleosynthesis [149].

Decaying-particle arguments have proved very effective in prob-
ing supersymmetry. Light-element abundances generally are com-
plementary to accelerator data in constraining SUSY parameter
space, with BBN reaching to values kinematically inaccessible to
the LHC. Much recent interest has focused on the case in which
the next-to-lightest supersymmetric particle is metastable and de-
cays during or after BBN. The constraints on unstable particles
discussed above imply stringent bounds on the allowed abundance
of such particles [109]; if the metastable particle is charged (e.g.,
the stau), then it is possible for it to form atom-like electromag-
netic bound states with nuclei, and the resulting impact on light
elements can be quite complex [8, 90, 150]. Moreover, SUSY de-
cays can destroy $^7$Li and/or produce $^6$Li, leading to a possible su-
persymmetric solution to the lithium problems noted above [151]
(see [7] for a review).

These arguments impose powerful constraints on supersymmet-
ric inflationary cosmology [86, 131, 146–148], particularly thermal
leptogenesis [152]. These limits can be evaded only if the gravitino
is massive enough to decay before BBN, i.e., $m_{3/2} \gtrsim 50$ TeV [153]
(which would be unnatural), or if it is in fact the lightest super-
symmetric particle and thus stable [131, 146, 154, 155]. Similar
constraints apply to moduli – very weakly coupled fields in string
theory that obtain an electroweak-scale mass from supersymmetry
breaking [156].

Finally, we mention that BBN places powerful constraints on
the possibility that there are new large dimensions in nature, per-
haps enabling the scale of quantum gravity to be as low as the
electroweak scale [157]. Thus, Standard Model fields may be lo-
calized on a *brane*, while gravity alone propagates in the *bulk*.
It has been further noted that the new dimensions may be non-
compact, even infinite [158], and the cosmology of such models
has attracted considerable attention. The expansion rate in the
early Universe can be significantly modified, so BBN is able to set
interesting constraints on such possibilities [159, 160].

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## 25. Cosmological Parameters

Revised August 2023 by O. Lahav (UCL) and A.R. Liddle (ULisboa).

### 25.1 Parametrizing the Universe

Rapid advances in observational cosmology have led to the establishment of a precision cosmological model, with many of the key cosmological parameters determined to one or two significant figure accuracy. Particularly prominent are measurements of cosmic microwave background (CMB) anisotropies, with the highest precision observations being those of the Planck Satellite [1, 2], which supersedes the landmark WMAP results [3, 4]. However the most accurate model of the Universe requires consideration of a range of observations, with complementary probes providing consistency checks, lifting parameter degeneracies, and enabling the strongest constraints to be placed.

The term ‘cosmological parameters’ now has a wide scope, and may include the parameterization of some functions as well as simple numbers describing properties of the Universe. The original usage referred to the parameters describing the global dynamics of the Universe, such as its expansion rate and curvature. Now we wish to know how the matter budget of the Universe is built up from its constituents: baryons, photons, neutrinos, dark matter, and dark energy. We also need to describe the nature of perturbations in the Universe, through global statistical descriptors such as the matter and radiation power spectra. There may be additional parameters describing the physical state of the Universe, such as the ionization fraction as a function of time during the era since recombination. Typical comparisons of cosmological models with observational data now feature between five and ten parameters.

#### 25.1.1 The global description of the Universe

Ordinarily, the Universe is taken to be a perturbed Robertson-Walker space-time, with dynamics governed by Einstein's equations. This is described in detail in the Big-Bang Cosmology chapter in this volume. Using the density parameters $\Omega_i$ for the various matter species and $\Omega_\Lambda$ for the cosmological constant, the Friedmann equation can be written

$$ \sum_i (\Omega_i + \Omega_\Lambda - 1) = \frac{k}{R^2 H^2}, \quad (25.1) $$

where the sum is over all the different species of material in the Universe. This equation applies at any epoch, but later in this article we will use the symbols $\Omega_i$ and $\Omega_\Lambda$ to refer specifically to the present-epoch values.

The complete present-epoch state of the homogeneous Universe can be described by giving the current-epoch values of all the density parameters and the Hubble constant $h$ (the present-day Hubble parameter being written $H_0 = 100h \text{ km s}^{-1} \text{ Mpc}^{-1}$). A typical collection would be baryons $\Omega_b$, photons $\Omega_\gamma$, neutrinos $\Omega_\nu$, and cold dark matter $\Omega_c$ (given charge neutrality, the electron density is guaranteed to be too small to be worth considering separately and is effectively included with the baryons). The spatial curvature can then be determined from the other parameters using Eq. (25.1). The total present matter density $\Omega_m = \Omega_c + \Omega_b$ may be used in place of the cold dark matter density $\Omega_c$.

These parameters also allow us to track the history of the Universe, at least back until an epoch where interactions allow interchanges between the densities of the different species; this is believed to have last happened at neutrino decoupling, shortly before Big-Bang Nucleosynthesis (BBN). To probe further back into the Universe's history requires assumptions about particle interactions, and perhaps about the nature of physical laws themselves.

The standard neutrino sector has three flavors. For neutrinos of mass in the range $5 \times 10^{-4} \text{ eV}$ to $1 \text{ MeV}$, the density parameter in neutrinos is predicted to be

$$ \Omega_\nu h^2 = \frac{\sum m_\nu}{93.12 \text{ eV}}, \quad (25.2) $$

where the sum is over all families with mass in that range (higher masses need a more sophisticated calculation). We use units with $c = 1$ throughout. Results on atmospheric and Solar neutrino oscillations [5] imply non-zero mass-squared differences between the

three neutrino flavors. These oscillation experiments cannot tell us the absolute neutrino masses, but within the normal assumption of a mass hierarchy suggest a lower limit of approximately $0.06 \text{ eV}$ for the sum of the neutrino masses (see the Neutrino chapter).

Even a mass this small has a potentially observable effect on the formation of structure, as neutrino free-streaming damps the growth of perturbations. Analyses commonly now either assume a neutrino mass sum fixed at this lower limit, or allow the neutrino mass sum to be a variable parameter. To date there is no decisive evidence of any effects from either neutrino masses or an otherwise non-standard neutrino sector, and observations impose quite stringent limits; see the Neutrinos in Cosmology chapter. However, we note that the inclusion of the neutrino mass sum as a free parameter can affect the derived values of other cosmological parameters.

#### 25.1.2 Inflation and perturbations

A complete model of the Universe must include a description of deviations from homogeneity, at least statistically. Indeed, the most powerful probes of the parameters described above come from the evolution of perturbations, so their study is naturally intertwined with the determination of cosmological parameters.

There are many different notations used to describe the perturbations, both in terms of the quantity used and the definition of the statistical measure. We use the dimensionless power spectrum $\Delta^2$ as defined in the Big Bang Cosmology section (also denoted $P$ in some of the literature). If the perturbations obey Gaussian statistics, the power spectrum provides a complete description of their properties.

From a theoretical perspective, a useful quantity to describe the perturbations is the curvature perturbation $R$, which measures the spatial curvature of a comoving slicing of the space-time. A simple case is the Harrison-Zeldovich spectrum, which corresponds to a constant $\Delta_R^2$. More generally, one can approximate the spectrum by a power law, writing

$$ \Delta_R^2(k) = \Delta_R^2(k_*) \left[ \frac{k}{k_*} \right]^{n_s - 1}, \quad (25.3) $$

where $n_s$ is known as the spectral index, always defined so that $n_s = 1$ for the Harrison-Zeldovich spectrum, and $k_*$ is an arbitrarily chosen scale. The initial spectrum, defined at some early epoch of the Universe's history, is usually taken to have a simple form such as this power law, and we will see that observations require $n_s$ close to one. Subsequent evolution will modify the spectrum from its initial form.

The simplest mechanism for generating the observed perturbations is the inflationary cosmology, which posits a period of accelerated expansion in the Universe's early stages [6, 7]. It is a useful working hypothesis that this is the sole mechanism for generating perturbations, and it may further be assumed to be the simplest class of inflationary model, where the dynamics are equivalent to that of a single scalar field $\phi$ with canonical kinetic energy slowly rolling on a potential $V(\phi)$. One may seek to verify that this simple picture can match observations and to determine the properties of $V(\phi)$ from the observational data. Alternatively, more complicated models, perhaps motivated by contemporary fundamental physics ideas, may be tested on a model-by-model basis (see more in the Inflation chapter in this volume).

Inflation generates perturbations through the amplification of quantum fluctuations, which are stretched to astrophysical scales by the rapid expansion. The simplest models generate two types, density perturbations that come from fluctuations in the scalar field and its corresponding scalar metric perturbation, and gravitational waves that are tensor metric fluctuations. The former experience gravitational instability and lead to structure formation, while both generate CMB anisotropies. Defining slow-roll parameters (with primes indicating derivatives with respect to the scalar field, and $m_{\text{Pl}} \equiv \sqrt{\hbar c/G}$ the Planck mass) as

$$ \epsilon = \frac{m_{\text{Pl}}^2}{16\pi} \left(\frac{V'}{V}\right)^2, \quad \eta = \frac{m_{\text{Pl}}^2}{8\pi} \frac{V''}{V}, \quad (25.4) $$
which should satisfy $\epsilon, |\eta| \ll 1$, the spectra can be computed using the slow-roll approximation as

$$ \Delta_R^2(k) \approx \left. \frac{8}{3m_{\text{Pl}}^4} \frac{V}{\epsilon} \right|_{k=aH} , \quad \Delta_t^2(k) \approx \left. \frac{128}{3m_{\text{Pl}}^4} V \right|_{k=aH} . \quad (25.5) $$

In each case, the expressions on the right-hand side are to be evaluated when the scale k is equal to the Hubble radius during inflation. The symbol ‘≈’ here indicates use of the slow-roll approximation, which is expected to be accurate to a few percent or better.

From these expressions, we can compute the spectral indices [8]:

$$ n_s \approx 1 - 6\epsilon + 2\eta \quad ; \quad n_t \approx -2\epsilon. \quad (25.6) $$

Another useful quantity is the ratio of the two spectra, defined by

$$ r \equiv \frac{\Delta_t^2(k_*)}{\Delta_R^2(k_*)} . \quad (25.7) $$

We have

$$ r \approx 16\epsilon \approx -8n_t, \quad (25.8) $$

which is known as the consistency equation.

One could consider corrections to the power-law approximation, which we discuss later. However, for now we make the working assumption that the spectra can be approximated by such power laws. The consistency equation shows that $r$ and $n_t$ are not independent parameters, and so the simplest inflation models give initial conditions described by three parameters, usually taken as $\Delta_R^2$, $n_s$, and $r$, all to be evaluated at some scale $k_*$, usually the ‘statistical center’ of the range explored by the data. Alternatively, one could use the parametrization $V$, $\epsilon$, and $\eta$, all evaluated at a point on the putative inflationary potential.

After the perturbations are created in the early Universe, they undergo a complex evolution up until the time they are observed in the present Universe. When the perturbations are small, this can be accurately followed using a linear theory numerical code such as CAMB or CLASS [9]. This works right up to the present for the CMB, but for density perturbations on small scales non-linear evolution is important and can be addressed by a variety of semi-analytical and numerical techniques. However the analysis is made, the outcome of the evolution is in principle determined by the cosmological model and by the parameters describing the initial perturbations, and hence can be used to determine them.

Of particular interest are CMB anisotropies. Both the total intensity and two independent polarization modes are predicted to have anisotropies. These can be described by the radiation angular power spectra $C_\ell$ as defined in the CMB article in this volume, and again provide a complete description if the density perturbations are Gaussian.

### 25.1.3 The standard cosmological model

We now have most of the ingredients in place to describe the cosmological model. Beyond those of the previous subsections, we need a measure of the ionization state of the Universe. The Universe is known to be highly ionized at redshifts below 5 or so (otherwise radiation from distant quasars would be heavily absorbed in the ultra-violet), and the ionized electrons can scatter microwave photons, altering the pattern of observed anisotropies. The most convenient parameter to describe this is the optical depth to scattering $\tau$ (i.e., the probability that a given photon scatters once); in the approximation of instantaneous and complete reionization, this could equivalently be described by the redshift of reionization $z_i$.

As described in Sec. 25.4, models based on these parameters are able to give a good fit to the complete set of high-quality data available at present, and indeed some simplification is possible. Observations are consistent with spatial flatness, and the inflation models so far described automatically generate negligible spatial curvature, so we can set $k = 0$; the density parameters then must sum to unity, and so one of them can be eliminated. The neutrino energy density is often not taken as an independent parameter; provided that the neutrino sector has the standard interactions, the neutrino energy density, while relativistic, can be

related to the photon density using thermal physics arguments, and a minimal assumption takes the neutrino mass sum to be that of the lowest mass solution to the neutrino oscillation constraints, namely 0.06 eV. In addition, there is no observational evidence for the existence of tensor perturbations (with the upper limits now starting to become constraining on models), and so r could be set to zero. This leaves seven parameters, which is the smallest set that can usefully be compared to the present cosmological data. This model is referred to by various names, including ΛCDM, the concordance cosmology, and the standard cosmological model.

Of these parameters, only $\Omega_\gamma$ is accurately measured directly. The radiation density is dominated by the energy in the CMB, and the COBE satellite FIRAS experiment determined its temperature to be $T = (2.7255 \pm 0.0006) \text{ K}$ [10],<fnref n="1" /> corresponding to $\Omega_\gamma = 2.47 \times 10^{-5} h^{-2}$. It typically can be taken as fixed when fitting other data. Hence the minimum number of cosmological parameters varied in fits to data is six, though as described below there may additionally be many ‘nuisance’ parameters necessary to describe astrophysical processes influencing the data.

In addition to this minimal set, there is a range of other parameters that might prove important in future as the datasets further improve, but for which there is so far no direct evidence, allowing them to be set to specific values for now. We discuss various speculative options in the next section. For completeness at this point, we mention one other interesting quantity, the helium fraction, which is a non-zero parameter that can affect the CMB anisotropies at a subtle level. It is usually fixed in microwave anisotropy studies, but the data are approaching a level where allowing its variation may become mandatory.

In conventional parameter estimation, a set of parameters is chosen by hand and the aim is to constrain their values. The higher-level inference problem of model selection instead compares different choices of parameter sets, as is necessary to assess whether observations are pointing towards inclusion of new physical effects. Bayesian inference offers an attractive framework for cosmological model selection, setting a tension between model predictiveness and ability to fit the data [11], and its use is becoming widespread.

### 25.1.4 Derived parameters

The parameter list of the previous subsection is sufficient to give a complete description of cosmological models that agree with observational data. However, it is not a unique parameterization, and one could instead use parameters derived from that basic set. Parameters that can be obtained from the set given above include the age of the Universe, the present horizon distance, the present neutrino background temperature, the epoch of matter-radiation equality, the epochs of recombination and decoupling, the epoch of transition to an accelerating Universe, the baryon-to-photon ratio, and the baryon-to-dark-matter density ratio. In addition, the physical densities of the matter components, $\Omega_i h^2$, are often more useful than the density parameters. The density perturbation amplitude can be specified in many different ways other than the large-scale primordial amplitude, for instance, in terms of its effect on the CMB, or by specifying a short-scale quantity, a common choice being the present linear-theory mass dispersion in a radius of $8 h^{-1} \text{Mpc}$, known as $\sigma_8$.

Different types of observation are sensitive to different subsets of the full cosmological parameter set, and some are more naturally interpreted in terms of some of the derived parameters of this subsection than on the original base parameter set. In particular, most types of observation feature degeneracies whereby they are unable to separate the effects of simultaneously varying specific combinations of several of the base parameters.

## 25.2 Extensions to the standard model

At present, there is no positive evidence in favor of extensions of the standard model. These are becoming increasingly constrained by the data, though there always remains the possibility of trace effects at a level below present observational capability.

<footnote n="1">¹All quoted uncertainties in this article are 1σ/68% confidence and all upper limits are 95% confidence. Cosmological parameters sometimes have significantly non-Gaussian uncertainties. Values from original sources have been rounded according to the conventions of this volume.</footnote>
### 25.2.1 More general perturbations

The standard cosmology assumes adiabatic, Gaussian perturbations. Adiabaticity means that all types of material in the Universe share a common perturbation, so that if the space-time is foliated by constant-density hypersurfaces, then all fluids and fields are homogeneous on those slices, with the perturbations completely described by the variation of the spatial curvature of the slices. Gaussianity means that the initial perturbations obey Gaussian statistics, with the amplitudes of waves of different wavenumbers being randomly drawn from a Gaussian distribution of width given by the power spectrum. Note that gravitational instability generates non-Gaussianity; in this context, Gaussianity refers to a property of the initial perturbations, before they evolve.

The simplest inflation models, based on one dynamical field, predict adiabatic perturbations and a level of non-Gaussianity that is too small to be detected by any experiment so far conceived. For present data, the primordial spectra are usually assumed to be power laws.

#### 25.2.1.1 Non-power-law spectra

For typical inflation models, it is an approximation to take the spectra as power laws, albeit usually a good one. As data quality improves, one might expect this approximation to come under pressure, requiring a more accurate description of the initial spectra, particularly for the density perturbations. In general, one can expand $\ln \Delta_R^2$ as

$$ \ln \Delta_R^2(k) = \ln \Delta_R^2(k_*) + (n_{s,*} - 1) \ln \frac{k}{k_*} + \frac{1}{2} \left. \frac{dn_s}{d \ln k} \right|_* \ln^2 \frac{k}{k_*} + \dots, \quad (25.9) $$

where the coefficients are all evaluated at some scale $k_*$. The term $dn_s/d \ln k$ is often called the running of the spectral index [12]. Once non-power-law spectra are allowed, it is necessary to specify the scale $k_*$ at which the spectral index is defined.

#### 25.2.1.2 Isocurvature perturbations

An isocurvature perturbation is one that leaves the total density unperturbed, while perturbing the relative amounts of different materials. If the Universe contains $N$ fluids, there is one growing adiabatic mode and $N-1$ growing isocurvature modes (for reviews see Ref. [7] and Ref. [13]). These can be excited, for example, in inflationary models where there are two or more fields that acquire dynamically-important perturbations. If one field decays to form normal matter, while the second survives to become the dark matter, this will generate a cold dark matter isocurvature perturbation.

In general, there are also correlations between the different modes, and so the full set of perturbations is described by a matrix giving the spectra and their correlations. Constraining such a general construct is challenging, though constraints on individual modes are beginning to become meaningful, with no evidence that any other than the adiabatic mode must be non-zero.

#### 25.2.1.3 Seeded perturbations

An alternative to laying down perturbations at very early epochs is that they are seeded throughout cosmic history, for instance by topological defects such as cosmic strings. It has long been excluded that these are the sole original of structure, but they could contribute part of the perturbation signal, current limits being just a few percent [14]. In particular, cosmic defects formed in a phase transition ending inflation is a plausible scenario for such a contribution.

#### 25.2.1.4 Non-Gaussianity

Multi-field inflation models can also generate primordial non-Gaussianity (reviewed, e.g., in Ref. [7]). The extra fields can either be in the same sector of the underlying theory as the inflaton, or completely separate, an interesting example of the latter being the curvaton model [15]. Current upper limits on non-Gaussianity are becoming stringent, but there remains strong motivation to push down those limits and perhaps reveal trace non-Gaussianity in the data. If non-Gaussianity is observed, its nature may favor an inflationary origin, or a different one such as topological defects.

### 25.2.2 Dark matter properties

Dark matter properties are discussed in the Dark Matter chapter in this volume. The simplest assumption concerning the dark matter is that it has no significant interactions with other matter, and that its particles have a negligible velocity as far as structure formation is concerned. Such dark matter is described as 'cold,' and candidates include the lightest supersymmetric particle, the axion, and primordial black holes. As far as astrophysicists are concerned, a complete specification of the relevant cold dark matter properties is given by the density parameter $\Omega_c$, though those seeking to detect it directly need also to know its interaction properties.

Cold dark matter is the standard assumption and gives an excellent fit to observations, except possibly on the shortest scales where there remains some controversy concerning the structure of dwarf galaxies and possible substructure in galaxy halos. It has long been excluded for all the dark matter to have a large velocity dispersion, so-called 'hot' dark matter, as it does not permit galaxies to form; for thermal relics the mass must be above about 1 keV to satisfy this constraint, though relics produced non-thermally, such as the axion, need not obey this limit. However, in future further parameters might need to be introduced to describe dark matter properties relevant to astrophysical observations. Suggestions that have been made include a modest velocity dispersion (warm dark matter) and dark matter self-interactions. There remains the possibility that the dark matter is comprised of two separate components, e.g., a cold one and a hot one, an example being if massive neutrinos have a non-negligible effect.

#### 25.2.3 Relativistic species

The number of relativistic species in the young Universe (omitting photons) is denoted $N_{\text{eff}}$. In the standard cosmological model only the three neutrino species contribute, and its baseline value is assumed fixed at 3.044 (the small shift from 3 is because of a slight predicted deviation from a thermal distribution [16]). However other species could contribute, for example an extra neutrino, possibly of sterile type, or massless Goldstone bosons or other scalars. It is hence interesting to study the effect of allowing this parameter to vary, and indeed although 3.044 is consistent with the data, most analyses currently suggest a somewhat higher value (e.g., Ref. [17]).

#### 25.2.4 Dark energy and modified gravity

While the standard cosmological model given above features a cosmological constant, in order to explain observations indicating that the Universe is presently accelerating, further possibilities exist under the general headings of 'dark energy' and 'modified gravity'. These topics are described in detail in the Dark Energy chapter in this volume. This article focuses on the case of the cosmological constant, since this simple model is a good match to existing data. We note that more general treatments of dark energy/modified gravity will lead to weaker constraints on other parameters.

#### 25.2.5 Complex ionization history

The full ionization history of the Universe is given by the ionization fraction as a function of redshift $z$. The simplest scenario takes the ionization to have the small residual value left after recombination up to some redshift $z_i$, at which point the Universe instantaneously reionizes completely. Then there is a one-to-one correspondence between $\tau$ and $z_i$ (that relation, however, also depending on other cosmological parameters). An accurate treatment of this process will track separate histories for hydrogen and helium. While currently rapid ionization appears to be a good approximation, as data improve a more complex ionization history may need to be considered.

#### 25.2.6 Varying 'constants'

Variation of the fundamental constants of Nature over cosmological times is another possible enhancement of the standard cosmology. There is a long history of study of variation of the gravitational constant $G_N$, and more recently attention has been drawn to the possibility of small fractional variations in the fine-structure constant. There is presently no observational evidence for the former, which is tightly constrained by a variety of mea-
surements. Evidence for the latter has been claimed from studies of spectral line shifts in quasar spectra at redshift $z \simeq 2$ [18], but this is presently controversial and in need of further observational study.

### 25.2.7 Cosmic topology

The usual hypothesis is that the Universe has the simplest topology consistent with its geometry, for example that a flat Universe extends forever. Observations cannot tell us whether that is true, but they can test the possibility of a non-trivial topology on scales up to roughly the present Hubble scale. Extra parameters would be needed to specify both the type and scale of the topology; for example, a cuboidal topology would need specification of the three principal axis lengths and orientation. At present, there is no evidence for non-trivial cosmic topology [19].

## 25.3 Cosmological Probes

The goal of the observational cosmologist is to utilize astronomical information to derive cosmological parameters. The transformation from the observables to the parameters usually involves many assumptions about the nature of the data, as well as of the dark sector. Below we outline the physical processes involved in each of the major probes, and the main recent results. The first two subsections concern probes of the homogeneous Universe, while the remainder consider constraints from perturbations.

In addition to statistical uncertainties we note three sources of systematic uncertainties that will apply to the cosmological parameters of interest: (i) due to the assumptions on the cosmological model and its priors (i.e., the number of assumed cosmological parameters and their allowed range); (ii) due to the uncertainty in the astrophysics of the objects (e.g., light-curve fitting for supernovae or the mass-temperature relation of galaxy clusters); and (iii) due to instrumental and observational limitations (e.g., the effect of ‘seeing’ on weak gravitational lensing measurements, or beam shape on CMB anisotropy measurements).

These systematics, the last two of which appear as ‘nuisance parameters’, pose a challenging problem to the statistical analysis. We attempt a statistical fit to the whole Universe with 6 to 12 parameters, but we might need to include hundreds of nuisance parameters, some of them highly correlated with the cosmological parameters of interest (for example time-dependent galaxy biasing could mimic the growth of mass fluctuations). Fortunately, there is some astrophysical prior knowledge on these effects, and a small number of physically-motivated free parameters would ideally be preferred in the cosmological parameter analysis.

### 25.3.1 Measures of the Hubble constant

In 1929, Edwin Hubble discovered the law of expansion of the Universe by measuring distances to nearby galaxies. The slope of the relation between the distance and recession velocity is defined to be the present-epoch Hubble constant, $H_0$. Astronomers argued for decades about the systematic uncertainties in various methods and derived values over the wide range $40 \text{ km s}^{-1} \text{ Mpc}^{-1} \lesssim H_0 \lesssim 100 \text{ km s}^{-1} \text{ Mpc}^{-1}$.

One of the most reliable results on the Hubble constant came from the Hubble Space Telescope (HST) Key Project [20]. This study used the empirical period-luminosity relation for Cepheid variable stars, and calibrated a number of secondary distance indicators: Type Ia Supernovae (SNe Ia); the Tully-Fisher relation; surface-brightness fluctuations; and Type II Supernovae. Various systematics are under investigation, for example JWST is helping understanding of the effect of field crowding. This approach has been further extended, e.g. using HST observations of Cepheids in the hosts of 42 SNe Ia and exploiting Gaia EDR3 parallaxes, the SH0ES team derived $H_0 = (73.0 \pm 1.0) \text{ km s}^{-1} \text{ Mpc}^{-1}$ [21].

Three other methods have been used recently. One is a calibration of the tip of the red-giant branch applied to Type Ia supernovae, the Carnegie-Chicago Hubble Programme (CCHP), finding $H_0 = (69.8 \pm 0.6) \text{ (stat.)} \pm 1.6 \text{ (sys.)} \text{ km s}^{-1} \text{ Mpc}^{-1}$ [22]. The second uses the method of time delay in gravitationally-lensed quasars; Birrer et al. [23] find $H_0 = 67.4_{-3.2}^{+4.1} \text{ km s}^{-1} \text{ Mpc}^{-1}$ from a sample of 40 lenses. A third method that came to fruition recently is based on gravitational waves; the ‘bright standard siren’ method applied to the binary neutron star GW170817

yields $H_0 = 70_{-8}^{+12} \text{ km s}^{-1} \text{ Mpc}^{-1}$ [24]. Adding two ‘dark standard siren’ systems shifts this to $H_0 = 73_{-8}^{+11} \text{ km s}^{-1} \text{ Mpc}^{-1}$ [25], still dominated by the single bright siren system. The 47 sources in the third Gravitational-Wave Transient Catalog (GWTC-3), again combined with the bright siren, yield the similar result $H_0 = 68_{-8}^{+12} \text{ km s}^{-1} \text{ Mpc}^{-1}$ [26]. When many more gravitational-wave events have been acquired, the future uncertainties on $H_0$ from standard sirens will get smaller.

The determination of $H_0$ by the Planck Collaboration [2] gives a lower value than most of the above methods, $H_0 = (67.4 \pm 0.5) \text{ km s}^{-1} \text{ Mpc}^{-1}$. As they discuss, there is a strong degeneracy of $H_0$ with other parameters, particularly $\Omega_m$ and the neutrino mass. It is worth noting that using the ‘inverse distance ladder’ method gives a result $H_0 = (67.8 \pm 1.3) \text{ km s}^{-1} \text{ Mpc}^{-1}$ [27], close to the Planck result. The inverse distance ladder relies on absolute-distance measurements from baryon acoustic oscillations (BAOs) to calibrate the intrinsic magnitude of the SNe Ia (rather than nearby Cepheids and parallax). This measurement was derived from 207 spectroscopically-confirmed Type Ia supernovae from the Dark Energy Survey (DES), an additional 122 low-redshift SNe Ia, and measurements of BAOs. A combination of DES Year 3 (Y3) clustering and weak lensing with BAO and BBN (assuming $\Lambda$CDM) gives $H_0 = (67.6 \pm 0.9) \text{ km s}^{-1} \text{ Mpc}^{-1}$ [28]. The completed Extended Baryon Oscillation Spectroscopic Survey (eBOSS) [29] inverse distance ladder result, within an assumed extended cosmological model, is $H_0 = (68.2 \pm 0.8) \text{ km s}^{-1} \text{ Mpc}^{-1}$, also close to the Planck value.

The tension between the $H_0$ values from Planck and the traditional cosmic distance-ladder methods, with the SH0ES result deviating from Planck by about $5\sigma$, is under intense investigation for potential systematic effects. There is possibly a trend for higher $H_0$ derived from the nearby Universe and a lower $H_0$ from the early Universe, which has led some researchers to propose a time-variation of the dark energy component or other exotic scenarios. Ongoing studies are addressing the question of whether the Hubble tension is due to systematics in at least one of the probes, or a signature of new physics. Figure 25.1 shows a selection of recent $H_0$ values, summarizing the current status of the Hubble constant tension. See Ref. [30] for a review.

<figure><img src="image_139.png" /><figcaption>**Figure 25.1:** A selection of recent $H_0$ measurements from the various projects as described in the text. The standard-siren determinations are omitted as they are too wide for the plot. Figure courtesy of Pablo Lemos.</figcaption></figure>

### 25.3.2 Supernovae as cosmological probes

Empirically, the peak luminosity of SNe Ia can be used as an efficient distance indicator (e.g., Ref. [31]), thus allowing cosmology to be constrained via the distance-redshift relation. The favorite theoretical explanation for SNe Ia is the thermonuclear disruption of carbon-oxygen white dwarfs. Although not perfect ‘standard candles,’ it has been demonstrated that by correcting for a relation between the light-curve shape, color, and luminosity at maximum brightness, the dispersion of the measured luminosities can
be greatly reduced. There are several possible systematic effects that may affect the accuracy of the use of SNe Ia as distance indicators, e.g., evolution with redshift and interstellar extinction in the host galaxy and in the Milky Way.

In the late 1990s two major studies, the Supernova Cosmology Project and the High-z Supernova Search Team, found evidence for an accelerating Universe [32], interpreted as due to a cosmological constant or a dark energy component. When combined with the CMB data (which indicate near flatness, i.e., $\Omega_m + \Omega_\Lambda \simeq 1$), the best-fit values were $\Omega_m \simeq 0.3$ and $\Omega_\Lambda \simeq 0.7$. Most results in the literature are consistent with the $w = -1$ cosmological constant case. The leading sample currently is the Pantheon+ compilation [33]. This set of 1550 spectroscopically-confirmed SNe Ia gives $\Omega_m = 0.334 \pm 0.018$ (stat+sym) for an assumed flat $\Lambda$CDM model, while in combination with the CMB, for a flat wCDM model these data give $w = -0.98_{-0.3}^{+0.2}$ [34]. Future experiments (e.g. DES year 5 SNIa) will refine constraints on the cosmic equation of state $w(z)$.

### 25.3.3 Cosmic microwave background

The physics of the CMB is described in detail in the CMB chapter in this volume. Before recombination, the baryons and photons are tightly coupled, and the perturbations oscillate in the potential wells generated primarily by the dark matter perturbations. After decoupling, the baryons are free to collapse into those potential wells. The CMB carries a record of conditions at the time of last scattering, often called primary anisotropies. In addition, it is affected by various processes as it propagates towards us, including the effect of a time-varying gravitational potential (the integrated Sachs-Wolfe effect), gravitational lensing, and scattering from ionized gas at low redshift.

The primary anisotropies, the integrated Sachs-Wolfe effect, and the scattering from a homogeneous distribution of ionized gas, can all be calculated using linear perturbation theory. Available codes include CAMB and CLASS [9], the former widely used embedded within the analysis package CosmoMC [35] and in higher-level analysis packages such as CosmoSIS [36], CosmoLike [37], and Cobaya [38]. Gravitational lensing is also calculated in these codes. Secondary effects, such as inhomogeneities in the reionization process, and scattering from gravitationally-collapsed gas (the Sunyaev-Zeldovich or SZ effect), require more complicated, and more uncertain, calculations.

The upshot is that the detailed pattern of anisotropies depends on all of the cosmological parameters. In a typical cosmology, the anisotropy power spectrum [usually plotted as $\ell(\ell+1)C_\ell/2\pi$] features a flat plateau at large angular scales (small $\ell$), followed by a series of oscillatory features at higher angular scales, the first and most prominent being at around one degree ($\ell \simeq 200$). These features, known as acoustic peaks, represent the oscillations of the photon-baryon fluid around the time of decoupling. Some features can be closely related to specific parameters—for instance, the location in multipole space of the set of peaks probes the spatial geometry, while the relative heights of the peaks probe the baryon density—but many other parameters combine to determine the overall shape.

The 2018 data release from the Planck satellite [1] gives the most powerful results to date on the spectrum of CMB temperature anisotropies, with a precision determination of the temperature power spectrum to beyond $\ell = 2000$. The Atacama Cosmology Telescope (ACT) and South Pole Telescope (SPT) experiments extend these results to higher angular resolution, though without full-sky coverage. Planck and the polarization-sensitive versions of ACT and SPT give the current state of the art in measuring the spectrum of E-polarization anisotropies and the correlation spectrum between temperature and polarization. These are consistent with models based on the parameters we have described, and provide accurate determinations of many of those parameters [2]. Primordial B-mode polarization has not been detected (although the gravitational lensing effect on B modes has been measured).

The data provide an exquisite measurement of the location of the set of acoustic peaks, determining the angular-diameter distance of the last-scattering surface. In combination with other data this strongly constrains the spatial geometry, in a manner

consistent with spatial flatness and excluding significantly-curved Universes. CMB data give a precision measurement of the age of the Universe. The CMB also gives a baryon density consistent with, and at higher precision than, that coming from BBN. It affirms the need for both dark matter and dark energy. It shows no evidence for dynamics of the dark energy, being consistent with a pure cosmological constant ($w = -1$). The density perturbations are consistent with a power-law primordial spectrum, and there is no indication yet of tensor perturbations. The current best-fit for the reionization optical depth from CMB data, $\tau = 0.054$, is in line with models of how early structure formation induces reionization.

Planck also made the first all-sky map of the CMB lensing field, which probes the entire matter distribution in the Universe and adds some additional constraining power to the CMB-only datasets. The recent ACT results [39] demonstrate very well the reconstruction of the mass power spectrum at intermediate redshifts from CMB gravitational lensing. These measurements are consistent with Planck, supporting $\Lambda$CDM (see more in the CMB chapter).

### 25.3.4 Galaxy clustering

The power spectrum of density perturbations is affected by the nature of the dark matter. Within the $\Lambda$CDM model, the power spectrum shape depends primarily on the primordial power spectrum and on the combination $\Omega_m h$, which determines the horizon scale at matter-radiation equality, with a subdominant dependence on the baryon density. The matter distribution is most easily probed by observing the galaxy distribution, but this must be done with care since the galaxies do not perfectly trace the dark matter distribution. Rather, they are a ‘biased’ tracer of the dark matter [40]. The need to allow for such bias is emphasized by the observation that different types of galaxies show bias with respect to each other. In particular, scale-dependent and stochastic biasing may introduce a systematic effect on the determination of cosmological parameters from redshift surveys [41]. Prior knowledge from simulations of galaxy formation or from gravitational lensing data could help to quantify biasing. Furthermore, the observed 3D galaxy distribution is in redshift space, i.e., the observed redshift is the sum of the Hubble expansion and the line-of-sight peculiar velocity, leading to linear and non-linear dynamical effects that also depend on the cosmological parameters. On the largest length scales, the galaxies are expected to trace the location of the dark matter, except for a constant multiplier $b$ to the power spectrum, known as the linear bias parameter. On scales smaller than 20 Mpc or so, the clustering pattern is ‘squashed’ in the radial direction due to coherent infall, which depends approximately on the parameter $\beta \equiv \Omega_m^{0.6}/b$ (on these shorter scales, more complicated forms of biasing are not excluded by the data). On scales of a few Mpc, there is an effect of elongation along the line of sight (colloquially known as the ‘finger of God’ effect) that depends on the galaxy velocity dispersion.

#### 25.3.4.1 Baryon acoustic oscillations

The power spectra of the 2-degree Field (2dF) Galaxy Redshift Survey and the Sloan Digital Sky Survey (SDSS) are well fit by a $\Lambda$CDM model and both surveys showed first evidence for baryon acoustic oscillations (BAOs) [42, 43]. When eBOSS is combined with Planck, Pantheon Type Ia Supernovae and other probes the result is $w_p = -1.020 \pm 0.032$ at the pivot redshift $z_p = 0.29$ [29]. Similar results for $w$ were obtained e.g. by the WiggleZ survey [44]. Preliminary results from the DESI galaxy redshift survey [45] confirm the detection of the BAOs, in agreement with previous surveys.

#### 25.3.4.2 Redshift distortion

There is continuing interest in Kaiser’s ‘redshift distortion’ effect [46]. This distortion depends on cosmological parameters via the perturbation growth rate in linear theory $f(z) = d\ln\delta/d\ln a \simeq \Omega_m^\gamma(z)$, where $\gamma \simeq 0.55$ for the $\Lambda$CDM model and may be different for modified gravity models. By measuring $f(z)$ it is feasible to constrain $\gamma$ and rule out certain modified gravity models [47, 48]. We note the degeneracy of the redshift-distortion pattern and the geometric distortion (the so-called Alcock-Paczynski effect [49]), e.g., as illustrated by the
WiggleZ survey [50] and eBOSS [29].

### 25.3.4.3 Limits on neutrino mass from galaxy surveys and other probes

Large-scale structure data place constraints on $\Omega_{\nu}$ due to the neutrino free-streaming effect [51]. Presently there is no clear detection, and upper limits on neutrino mass are commonly estimated by comparing the observed galaxy power spectrum with a four-component model of baryons, cold dark matter, a cosmological constant, and massive neutrinos. Such analyses also assume that the primordial power spectrum is adiabatic, scale-invariant, and Gaussian. Potential systematic effects include biasing of the galaxy distribution and non-linearities of the power spectrum. An upper limit can also be derived from CMB anisotropies alone, while combination with additional cosmological datasets can improve the results.

The most recent results on neutrino mass upper limits and other neutrino properties are summarized in the Neutrinos in Cosmology chapter in this volume. The latest cosmological data constrain the sum of neutrino masses to be below 0.13 eV [28], consistent with Refs. [2, 29, 39]. Since the lower limit on this sum from oscillation experiments (assuming normal hierarchy) is 0.06 eV it is expected that future cosmological surveys will soon detect effects from neutrino mass. Also, current cosmological datasets are in good agreement with the standard value for the effective number of neutrino species $N_{\text{eff}} = 3.044$.

### 25.3.5 Clustering in the inter-galactic medium

It is commonly assumed, based on hydrodynamic simulations, that the neutral hydrogen in the inter-galactic medium (IGM) can be related to the underlying mass distribution. It is then possible to estimate the matter power spectrum from the absorption observed in quasar spectra, the so-called Lyman-$\alpha$ forest. The usual procedure is to measure the power spectrum of the transmitted flux, and then to infer the mass power spectrum. Photonization heating by the ultraviolet background radiation and adiabatic cooling by the expansion of the Universe combine to give a simple power-law relation between the gas temperature and the baryon density. It also follows that there is a power-law relation between the optical depth $\tau$ and $\rho_b$. Therefore, the observed flux $F = \exp(-\tau)$ is strongly correlated with $\rho_b$, which itself traces the mass density. The matter and flux power spectra can be related by a biasing function that is calibrated from simulations. There are two variants of Lyman-alpha analyses: 1-dimensional power spectra from individual lines-of-sight that probe small ($\sim$Mpc) scales; and 3-dimensional Lyman-alpha BAO analyses that measure large-scale correlations (over $\sim$100 Mpc scales) using neighboring quasar lines-of-sight.

The latest BOSS and eBOSS datasets have provided measurements of the BAO scale both in the Lyman-$\alpha$ absorption and in its cross-correlation with quasars at an effective redshift $z = 2.3$ [29, 52]. The results agree within $1.5-\sigma$ with the Planck CMB flat-$\Lambda$ estimation for BAO scale. Such measurements will improve with the Dark Energy Spectroscopic Instrument (DESI) and other future spectroscopic surveys. The Lyman-$\alpha$ flux power spectrum has also been used to constrain the nature of dark matter, for example limiting the amount of warm dark matter [53].

### 25.3.6 Weak gravitational lensing

Images of background galaxies are distorted by the gravitational effect of mass variations along the line of sight. Deep gravitational potential wells, such as galaxy clusters, generate 'strong lensing' leading to arcs, arclets, and multiple images, while more moderate perturbations give rise to 'weak lensing.' Weak lensing is now widely used to measure the mass power spectrum in selected regions of the sky (see Ref. [54] for reviews). Since the signal is weak, the image of deformed galaxy shapes (the 'shear map') must be analyzed statistically to measure the power spectrum, higher moments, and cosmological parameters. There are various systematic effects in the interpretation of weak lensing, e.g., due to atmospheric distortions during observations, the redshift distribution of the background galaxies (usually depending on the accuracy of photometric redshifts), the intrinsic correlation

<figure><img src="image_140.png" /><figcaption>**Figure 25.2:** Marginalized posterior contours (inner 68% confidence level, outer 95% confidence level) in the $\Omega_m-S_8$ plane. The plot shows KiDS weak lensing alone, DES Y3 weak lensing alone, DES and KiDS combined, and Planck CMB. Figure courtesy of Alexandra Amon and the DES and KiDS collaborations.</figcaption></figure>

of galaxy shapes, and non-linear modeling uncertainties.

Weak-lensing measurements from the Kilo-Degree Survey (KiDS) [55], the Subaru Hyper-Suprime-Cam (HSC) [56], and DES [57] have constrained the clumpiness parameter $S_8 \equiv \sigma_8(\Omega_m/0.3)^{0.5}$. Each of these surveys has yielded $S_8$ lower than that derived from Planck. Figure 25.2 shows the $\Omega_m-S_8$ constraints derived from a joint KiDS/DES analysis [58] versus the CMB constraint from Planck, where the tension is significantly reduced, to $1.7\sigma$, as compared to previous analyses. Results from weak lensing from DES, combined with other probes, are described in the next section.

### 25.3.7 Other probes

Other probes that have been used to constrain cosmological parameters, but that are not presently competitive in terms of accuracy, are the integrated Sachs-Wolfe effect [59,60], the number density or composition of galaxy clusters [61], and galaxy peculiar velocities that probe the mass fluctuations in the local Universe [62].

## 25.4 Bringing probes together

Although it contains two ingredients—dark matter and dark energy—which have not yet been verified by laboratory experiments, the $\Lambda$CDM model is almost universally accepted by cosmologists as the best description of the present data. The approximate values of some of the key parameters are $\Omega_b \simeq 0.05$, $\Omega_c \simeq 0.25$, $\Omega_\Lambda \simeq 0.70$, and a Hubble constant $h \simeq 0.70$. The spatial geometry is very close to flat (and usually assumed to be precisely flat), and the initial perturbations Gaussian, adiabatic, and nearly scale-invariant.

The most powerful data source is the CMB, which on its own supports all these main tenets. Values for some parameters, as given in Ref. [2], are reproduced in Table 25.1. These particular results presume a flat Universe. The constraints are somewhat strengthened by adding additional datasets, BAO being shown in the Table as an example, though most of the constraining power resides in the CMB data. Similar constraints at lower precision were previously obtained by the WMAP collaboration, and more recently by SPT.

If the assumption of spatial flatness is lifted, it turns out that the primary CMB on its own constrains the spatial curvature fairly weakly, due to a parameter degeneracy in the angular-diameter distance. However, inclusion of other data readily re-
<table><caption>**Table 25.1:** Parameter constraints reproduced from Ref. [2] (Table 2, column 5), with some additional rounding. Both columns assume the ΛCDM cosmology with a power-law initial spectrum, no tensors, spatial flatness, a cosmological constant as dark energy, and the sum of neutrino masses fixed to 0.06 eV. Above the line are the six parameter combinations actually fit to the data ($\theta_{MC}$ is a measure of the sound horizon at last scattering); those below the line are derived from these. The first column uses Planck primary CMB data plus the Planck measurement of CMB lensing. This column gives our present recommended values. The second column adds in data from a compilation of BAO measurements described in Ref. [2]. The perturbation amplitude $\Delta_R^2$ (denoted $A_s$ in the original paper) is specified at the scale $0.05 \text{ Mpc}^{-1}$. Uncertainties are shown at 68% confidence.</caption><thead><tr><th></th><th>Planck TT,TE,EE+lowE+lensing</th><th>+BAO</th></tr></thead><tbody><tr><td>Ω<sub>b</sub>h<sup>2</sup></td><td>0.02237 ± 0.00015</td><td>0.02242 ± 0.00014</td></tr><tr><td>Ω<sub>c</sub>h<sup>2</sup></td><td>0.1200 ± 0.0012</td><td>0.1193 ± 0.0009</td></tr><tr><td>100 θ<sub>MC</sub></td><td>1.0409 ± 0.0003</td><td>1.0410 ± 0.0003</td></tr><tr><td>n<sub>s</sub></td><td>0.965 ± 0.004</td><td>0.966 ± 0.004</td></tr><tr><td>τ</td><td>0.054 ± 0.007</td><td>0.056 ± 0.007</td></tr><tr><td>ln(10<sup>10</sup>Δ<sub>R</sub><sup>2</sup>)</td><td>3.044 ± 0.014</td><td>3.047 ± 0.014</td></tr><tr><td>h</td><td>0.674 ± 0.005</td><td>0.677 ± 0.004</td></tr><tr><td>σ<sub>8</sub></td><td>0.811 ± 0.006</td><td>0.810 ± 0.006</td></tr><tr><td>Ω<sub>m</sub></td><td>0.315 ± 0.007</td><td>0.311 ± 0.006</td></tr><tr><td>Ω<sub>Λ</sub></td><td>0.685 ± 0.007</td><td>0.689 ± 0.006</td></tr></tbody></table>

moves this degeneracy. Simply adding the Planck lensing measurement, and with the assumption that the dark energy is a cosmological constant, yields a 68% confidence constraint on $\Omega_{tot} \equiv \sum \Omega_i + \Omega_\Lambda = 1.011 \pm 0.006$ and further adding BAO makes it $0.9993 \pm 0.0019$ [2]. Results of this type are normally taken as justifying the restriction to flat cosmologies.

One derived parameter that is very robust is the age of the Universe, since there is a useful coincidence that for a flat Universe the position of the first peak is strongly correlated with the age. The CMB data give 13.797 ± 0.023 Gyr (assuming flatness). This is in good agreement with the ages of the oldest globular clusters and with radioactive dating.

The baryon density $\Omega_b$ is now measured with high accuracy from CMB data alone, and is consistent with and more precise than the determination from BBN. The value quoted in the Big-Bang Nucleosynthesis chapter in this volume is $\Omega_b h^2 = 0.0220 \pm 0.0004$.

While $\Omega_\Lambda$ is measured to be non-zero with very high confidence, there is no evidence of evolution of the dark energy density. As described in the Dark Energy chapter in this volume a combination of CMB, SN, RSD, and BAO measurements, assuming a flat Universe, yield $w = -1.03 \pm 0.03$ [28], consistent with the cosmological constant case $w = -1$. Allowing more complicated forms of dark energy weakens the limits.

The data provide strong support for the main predictions of the simplest inflation models: spatial flatness and adiabatic, Gaussian, nearly scale-invariant density perturbations. But it is disappointing that there is no sign of primordial gravitational waves; combining Planck and WMAP with BICEP2/Keck Array BK18 data (plus BAO data to help constrain $n_s$) gives a 95% confidence upper limit of $r < 0.036$ at the scale $0.05 \text{ Mpc}^{-1}$ [63]. The density perturbation spectral index is clearly required to be less than one by current data, though the strength of that conclusion can weaken if additional parameters are included in the model fits.

Tests have been made for various types of non-Gaussianity, a particular example being a parameter $f_{NL}$ that measures a quadratic contribution to the perturbations. Various non-Gaussian shapes are possible (see Ref. [64] for details), and current

constraints on the popular ‘local’, ‘equilateral’, and ‘orthogonal’ types (combining CMB temperature and polarization data) are $f_{NL}^{\text{local}} = -1 \pm 5$, $f_{NL}^{\text{equil}} = -26 \pm 47$, and $f_{NL}^{\text{ortho}} = -38 \pm 24$, respectively (these may appear weak, but prominent non-Gaussianity requires the product $f_{NL} \Delta_R$ to be large, and $\Delta_R$ is of order $10^{-5}$). Clearly none of these give any indication of primordial non-Gaussianity.

While the above results come from the CMB alone, other probes are becoming competitive (especially when considering more complex cosmological models), and so the combination of data from different sources is of growing importance. We note that it has become fashionable to combine probes at the level of powerspectrum data vectors, taking into account nuisance parameters in each type of measurement. Discussions on ‘tension’ in resulting cosmological parameters depend on the statistical approaches used. Commonly the cosmology community works within the Bayesian framework, and assesses agreement amongst data sets with respect to a model via Bayesian evidence, essentially the denominator in Bayes’ theorem.

As an example of results, combining DES Y3 (position–position clustering, galaxy–galaxy lensing, and weak lensing shear) with Planck, BAO, and RSD measurements from eBOSS, and Type Ia supernovae from the Pantheon dataset compilation, has shown the datasets to be mutually compatible and yields very tight constraints on cosmological parameters: $S_8 \equiv \sigma_8(\Omega_m/0.3)^{0.5} = 0.812 \pm 0.008$ and $\Omega_m = 0.306^{+0.004}_{-0.005}$ in ΛCDM, and $w = -1.03 \pm 0.03$ in wCDM [28] matching the constraint in Ref. [29]. The combined measurement of the Hubble constant within ΛCDM gives $H_0 = 68.0^{+0.4}_{-0.3} \text{ km s}^{-1} \text{ Mpc}^{-1}$, still leaving substantial tension with the SH0ES measurement described earlier. Future analyses and the next generation of surveys will test for deviations from ΛCDM, for example epoch-dependent $w(z)$ and modifications to General Relativity.

## 25.5 Outlook for the future

The concordance model is now well-established, and there seems little room left for any dramatic revision of this paradigm. A measure of the strength of that statement is how difficult it has proven to formulate convincing alternatives.

Should there indeed be no major revision of the current paradigm, we can expect future developments to take one of two directions. Either the existing parameter set will continue to prove sufficient to explain the data, with the parameters subject to ever-tightening constraints, or it will become necessary to deploy new parameters. The latter outcome would be very much the more interesting, offering a route towards understanding new physical processes relevant to the cosmological evolution. There are many possibilities on offer for striking discoveries, for example:

* the cosmological effects of a neutrino mass may be unambiguously detected, shedding light on fundamental neutrino properties;

* detection of primordial non-Gaussianities would indicate that non-linear processes influence the perturbation generation mechanism;

* detection of variation in the dark energy density (i.e., $w \neq -1$) would provide much-needed experimental input into its nature.

These supply more than enough motivation for continued efforts to test the cosmological model and improve its accuracy. Over the coming years, there are a wide range of new observations that will bring further precision to cosmological studies. Indeed, there are far too many to mention them all here, and so we highlight just a few areas.

The CMB observations will improve in several directions. A current frontier is the study of polarization, for which power spectrum measurements have now been made by several experiments. Detection of primordial B-mode anisotropies is the next major goal and a variety of projects are targeting this, though theory gives little guidance as to the likely signal level. Future CMB projects that are approved include the Simons Observatory and LiteBIRD.
An impressive array of cosmology surveys are already opera-
tional, under construction, or proposed, including the ground-
based Hyper Suprime Camera (HSC) and Rubin-LSST imaging
surveys, spectroscopic surveys such as DESI, and space missions
Euclid (succesfully launched in July 2023) and the Roman Space
Telescope.

An exciting area for the future is radio surveys of the redshifted
21-cm line of hydrogen. Because of the intrinsic narrowness of this
line, by tuning the bandpass the emission from narrow redshift
slices of the Universe will be measured to extremely high redshift,
probing the details of the reionization process at redshifts up to
perhaps 20, as well as measuring large-scale features such as the
BAOS. LOFAR and CHIME are the first instruments able to do
this and have begun operations. In the longer term, the Square
Kilometre Array (SKA) will take these studies to sophisticated
levels.

The development of the first precision cosmological model is a
major achievement. However, it is important not to lose sight
of the motivation for developing such a model, which is to un-
derstand the underlying physical processes at work governing the
Universe's evolution. From that perspective, progress has been
much less dramatic. For instance, there are many proposals for
the nature of the dark matter, but no consensus as to which is cor-
rect. The nature of the dark energy remains a mystery. Even the
baryon density, now measured to an accuracy of a percent, lacks
an underlying theory able to predict it within orders of magni-
tude. Precision cosmology may have arrived, but at present many
key questions remain, to motivate and challenge the cosmology
community.

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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 observa-
tions that can then be used to constrain neutrino properties. This
is a great example of the remarkable interconnection and inter-
play between nuclear physics, particle physics, astrophysics and
cosmology (for general reviews see e.g., [1–4]). Present cosmolog-
ical data are already providing constraints on neutrino properties
not only complementary but also competitive with terrestrial ex-
periments; for instance, upper bounds on the total neutrino mass
have shrunk by a factor of about 20 in the past two decades.
Forthcoming cosmological data may soon provide key informa-
tion, not obtainable in other ways like e.g., a measurement of the
absolute neutrino mass scale.

A relic neutrino background pervading the Universe (the Cos-
mic 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 di-
rectly, it has been indirectly confirmed by the accurate agreement
of predictions and observations of: a) the primordial abundance
of light elements (see Big Bang Nucleosynthesis – Chap. 24) of
this Review; b) the power spectrum of Cosmic Microwave Back-
ground (CMB) anisotropies (see Cosmic Microwave Background –
Chap. 29 of this Review); and c) the large-scale clustering of cos-
mological structures. Within the hot Big Bang model such good
agreement would fail dramatically without a CνB with proper-
ties matching closely those predicted by the standard neutrino
decoupling process (i.e., involving only weak interactions).

We will illustrate below that cosmology is sensitive to the fol-
lowing neutrino properties: their density, related to the number
of active (i.e., left-handed, see Neutrino Mass, Mixing, and Os-
cillations - Chap. 14 of this Review) neutrino species, and their
masses. At first order, cosmology is sensitive 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 comple-
mentary 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 case of anomalies in cosmological data, it might not be obvious to discriminate between interpretation in 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 decou-
pling.

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
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 neutrino-
to-photon density ratio between the time of electron-positron an-

nihilation and the non-relativistic transition of neutrinos would
be given by

$$
\frac{\rho_{\nu}}{\rho_{\gamma}} = \frac{7}{8} N_{\text{eff}} \left( \frac{4}{11} \right)^{4/3}, \qquad (26.1)
$$

with N<sub>eff</sub> = 3, and the last factor comes from the fourth power
of the temperature ratio T<sub>ν</sub>/T<sub>γ</sub> = (4/11)<sup>1/3</sup> (see Big Bang Cos-
mology – Chap. 22 in this Review). In the above formula, N<sub>eff</sub> 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 simula-
tions of neutrino decoupling and electron-positron annihilation,
taking into account flavor oscillations, provide precise predictions
for the actual phase-space distribution of relic neutrinos [12–18].
These distributions differ from the instantaneous decoupling ap-
proximation through a combination of a small shift in the photon
temperature and small flavor-dependent non-thermal distortions,
all at the percent level. The final result for the density ratio ρ<sub>ν</sub>/ρ<sub>γ</sub>
in the relativistic regime can always be expressed as in Eq. (26.1),
but with a different value of N<sub>eff</sub>. 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 and the most recent results on plasma thermody-
namics QED corrections, give<sup>1</sup> N<sub>eff</sub> = 3.044 [16,17]. The precise
number density ratio n<sub>ν</sub>/n<sub>γ</sub> can also be derived from such studies,
and is important for computing the ratio Ω<sub>ν</sub>h<sup>2</sup>/∑<sub>i</sub> m<sub>i</sub> (ratio of
the physical density of neutrinos in units of the critical density 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 distri-
butions remain constant when expressed in terms of the comoving
momentum q = pa, even when they become non-relativistic.

The neutrino temperature today, T₀ = 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 |Δm²₃₁|<fnref n="1" />/² > |Δm²₃₁|<fnref n="1" />/² > T₀
(see Neutrino mass, Mixing, and oscillations – Chap. 14 of this
Review). Thus at least two neutrino mass 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 fraction of 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 radi-
ation at early times (during the CMB acoustic oscillations) and
dark matter at late times (during structure formation), which
has consequences on the background evolution. Neutrinos be-
come non-relativistic when their mass is equal to their average
momentum, given for any Fermi-Dirac-distributed particle by
⟨p⟩ = 3.15 T. Thus the redshift of the non-relativistic transi-
tion is given by z_i^{nr} = m_i/(3.15 T_0^2) - 1 = m_i/[0.53 meV] - 1
for each eigenstate of mass m_i, giving for instance z_i^{nr} = 110 for
m_i = 60 meV, corresponding to a time deep inside the matter-
dominated regime. Second, until the non-relativistic transition,
neutrinos travel at the speed of light, and later on they move at
a typical velocity ⟨v_i/c⟩ = 3.15 T_0(z)/m_i = 0.53(1+z) meV/m_i,
which is several orders of magnitude larger than that of the dom-
inant cold (or even of possibly warm) dark matter component(s).
This brings their characteristic diffusion scale, called the “free-
streaming length”, to cosmological relevant values, with conse-
quences on gravitational clustering and the growth of structure.

Once neutrinos are non-relativistic, their energy density is given
by ρν ≃ ∑ mᵢnᵢ. Since the number densities nᵢ are equal to each
other (up to negligible corrections coming from flavour effects in
the decoupling phase), the total mass ∑ mν = m₁ + m₂ + m₃ can
be factorized out. It is possible 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

<footnote n="1">¹The last effects that potentially require some further checks in order to fully settle the fourth significant digit in N<sub>eff</sub> are temperature QED corrections; according to [18], the final result might be closer to 3.043 than to 3.044.</footnote>
<figure><img src="image_141.png" /><figcaption>**Figure 26.1:** Ratio of the CMB $C_{\ell}^{TT}$ (left, including lensing effects) and matter power spectrum $P(k)$ (right, computed for each model in units of $(h^{-1}\text{Mpc})^3$) for different values of $\Delta N_{\text{eff}} \equiv N_{\text{eff}} - 3.044$ over those of a reference model with $\Delta N_{\text{eff}} = 0$. In order to minimize and better characterise the effect of $N_{\text{eff}}$ on the CMB, the parameters that are kept fixed are $\{z_{\text{eq}}, z_{\Lambda}, \omega_b, \tau\}$ and the primordial spectrum parameters. Fixing $\{z_{\text{eq}}, z_{\Lambda}\}$ is equivalent to fixing the fractional density of total radiation, of total matter and of cosmological constant $\{\Omega_{\text{r}}, \Omega_{\text{m}}, \Omega_{\Lambda}\}$ while increasing the Hubble parameter as a function of $N_{\text{eff}}$. The statistical errors on the $C_{\ell}$ are $\sim 1\%$ for a band power of $\Delta\ell = 30$ at $\ell \sim 1000$. The error on $P(k)$ is estimated to be of the order of 5%.</figcaption></figure>

<figure><img src="image_142.png" /><figcaption>**Figure 26.2:** Ratio of the CMB $C_{\ell}^{TT}$ and matter power spectrum $P(k)$ (computed for each model in units of $(h^{-1}\text{Mpc})^3$) for different values of $\sum m_{\nu}$ over those of a reference model with massless neutrinos. In order to minimize and better characterise the effect of $\sum m_{\nu}$ on the CMB, the parameters that are kept fixed are $\omega_b, \omega_c, \tau$, the angular scale of the sound horizon $\theta_s$ and the primordial spectrum parameters (solid lines). This implies that we are increasing the Hubble parameter $h$ as a function of $\sum m_{\nu}$. For the matter power spectrum, in order to single out the effect of neutrino free-streaming on $P(k)$, the dashed lines show the spectrum ratio when $\{\omega_{\text{m}}, \omega_b, \Omega_{\Lambda}\}$ are kept fixed. For comparison, the error on $P(k)$ is of the order of 5% with current observations, and the fractional $C_{\ell}$ errors are of the order of $1/\sqrt{\ell}$ at low $\ell$.</figcaption></figure>

negligible. Using the expression for $n_i/n_\gamma$ obtained from precise neutrino decoupling studies [16, 17], and knowing $n_\gamma$ from the measurement of the CMB temperature, one can compute $\rho_\nu^0$, the total neutrino density today,<fnref n="2" /> in units of the critical density $\rho_{\text{crit}}^0$:

$$ \Omega_{\nu} = \frac{\rho_{\nu}^{0}}{\rho_{\text{crit}}^{0}} = \frac{\sum m_{\nu}}{93.12h^{2}\text{eV}}, \qquad (26.2) $$

and the total neutrino average number density today: $n_\nu^0 = 339.5 \text{ cm}^{-3}$ [19]. Here $h$ is the Hubble constant in units of $100 \text{ km s}^{-1} \text{ Mpc}^{-1}$.

## 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 for neutrinos with a mass

$m_i \ll 3.15 T_\nu^{\text{eq}} \sim 1.5 \text{ eV}$ (see Big Bang Cosmology, Chap. 22 in this Review), neutrinos enhance the density of radiation: this effect is parameterised by $N_{\text{eff}}$ and can be discussed separately from the effect of the mass that will be described later in this section. Increasing $N_{\text{eff}}$ impacts the observable spectra of CMB anisotropies and matter fluctuations through background and perturbation effects.

### 26.2.1 Effect of $N_{\text{eff}}$ on the CMB

The background effects depend on what is kept fixed when increasing $N_{\text{eff}}$. If the densities of other species are kept fixed, a higher $N_{\text{eff}}$ implies a smaller redshift of radiation-to-matter equality, with very strong effects on the CMB spectrum: when the amount of expansion between radiation-to-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 other parameters. Hence, to characterise the effect of $N_{\text{eff}}$, it is more useful and illuminating to enhance the density of total radiation, of total matter and of $\Lambda$ by exactly the same amount, in order to keep the redshift of radiation-to-matter equality $z_{\text{eq}}$ and matter-to-$\Lambda$ equality $z_\Lambda$ fixed [20–22]. The primordial spectrum parameters, the baryon

<footnote n="2">²The value 93.12 eV applies to the limit of large masses, in which $\rho_\nu^0 = \sum m_i n_i^0$ and neutrinos are degenerate in mass [19]. This number would be larger by $\sim 0.1\%$ in the limit of minimal normal hierarchy (or smaller by $\sim 0.06\%$ for minimal inverted hierarchy), due the highest mass state(s) being still relativistic today and to flavor-dependent non-thermal distortions in the neutrino phase-space distributions.</footnote>
density $\omega_b \equiv \Omega_b h^2$ and the optical depth to reionization $\tau$ can be kept fixed at the same time, since we can simply vary $N_{\text{eff}}$ together with the Hubble parameter $h$ with fixed $\{\omega_b, \Omega_c, \Omega_\Lambda\}$. The impact of such a transformation is shown in Fig. 26.1 for the CMB temperature spectrum $C_\ell^{TT}$ (defined in Chap. 29 in this Review) and for the matter power spectrum $P(k)$ (defined in Chap. 22 in this Review) for several representative values of $N_{\text{eff}}$. 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 the $C_\ell$'s is of the order of one per cent for a band power of $\Delta\ell = 30$ at $\ell \sim 1000$).

With this transformation, the main background effect of $N_{\text{eff}}$ is an increase in the diffusion scale (or Silk damping scale, see Cosmic Microwave Background – Chap. 29 in this Review) at the time of decoupling, responsible for the decrease in $C_\ell^{TT}$ at high $\ell$, plus smaller effects coming from a slight increase in the redshift of photon decoupling [20–22]. At the level of perturbations, a higher $N_{\text{eff}}$ implies that photons feel gravitational forces from a denser neutrino component; this tends to decrease 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 $N_{\text{eff}}$ on the polarization spectrum features are the same as on the temperature spectrum: an increased Silk damping, and a shift in 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 $N_{\text{eff}}$ and cannot be mimicked by other parameters; thus $N_{\text{eff}}$ can be accurately measured from the CMB alone. However, there are correlations between $N_{\text{eff}}$ and other parameters. In particular, we have seen (Fig. 26.1) that in order to minimise the effect of $N_{\text{eff}}$ on the CMB spectrum, one should vary $h$ at the same time, hence there is a correlation between $N_{\text{eff}}$ and $h$, which implies that independent measurements reducing the error bar on $h$ also reduce that on $N_{\text{eff}}$. Note that this correlation is not equivalent to a perfect degeneracy, so both parameters can be constrained with CMB data alone.

### 26.2.2 Effect of $N_{\text{eff}}$ on the matter spectrum

We have discussed the effect of increasing $N_{\text{eff}}$ while keeping $z_{\text{eq}}$ and $\omega_b$ fixed, because the latter two quantities are very accurately constrained by CMB data. This implies that $\omega_c$ increases with $N_{\text{eff}}$, and that the ratio $\omega_b/\omega_c = \Omega_b/\Omega_c$ decreases. However, the ratio of baryonic-to-dark matter has a strong impact on the shape of the matter power spectrum, because until the time of decoupling 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 decrease of $\Omega_b/\Omega_c$ following from the increase of $N_{\text{eff}}$ gives more weight to the most clustered of the two components, 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), clearly visible 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 from large-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 $m_i \ll 0.6$ 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 and dark matter ($\omega_b$ and $\omega_c$), the early cosmological evolution remains fixed and independent of the neutrino mass, until the time of the non-relativistic tran-

sition. 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, $\omega_m = \omega_b + \omega_c + \omega_\nu$, where $\omega_\nu \equiv \Omega_\nu h^2$ is given as a function of the total mass $\sum m_\nu$ by Eq. (26.2). The late background evolution impacts the CMB spectrum through the relation between scales 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, $d_\Lambda(z_{\text{rec}})$, and on the redshift of matter-to-Λ equality. Increasing $\sum m_\nu$ tends to modify these two quantities. By playing with $h$ and $\Omega_\Lambda$, it is possible to keep one of them fixed, but not both at the same time. Since the CMB measures the angular scale of acoustic 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 maintain a fixed scale $d_\Lambda(z_{\text{rec}})$. With such a choice, an increase in neutrino mass comes together with a decrease in the late ISW effect explaining the depletion of the CMB spectrum for $l \le 20$. The fact that both $\sum m_\nu$ and $h$ enter the expression of $d_\Lambda(z_{\text{rec}})$ implies that measurements of the neutrino mass from CMB data are strongly correlated with $h$. Second, the non-relativistic transition of neutrinos affects the total pressure-to-density ratio of the Universe, and causes a small variation of the metric fluctuations. If this transition takes place not too long after photon decoupling, this variation is observable through the early ISW effect [22, 28, 29]. It is responsible for the dip seen in Fig. 26.2 for $20 \le \ell \le 200$. Third, when the neutrino mass is higher, the CMB spectrum is less affected by the weak lensing effect induced by the large-scale structure at small redshift. This is due 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 $\ell \ge 200$. 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 with neutrinos. This leads to a small enhancement of $C_\ell^{TT}$ for $\ell \ge 500$, hardly visible on Fig. 26.2 because it is balanced by the lensing effect.

### 26.2.4 Effect of neutrino masses on the matter power 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 time scale $t_H = (a/\dot{a})$, 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 matter fluctuations, related to the squared fluctuation $\delta_m^2$ with $\delta_m \equiv \delta_b + \delta_c + \delta_\nu$, gets a negligible contribution from the neutrino component on small scales, and is reduced by a factor $(1 - 2f_\nu)$, where $f_\nu = \omega_\nu/\omega_m$. 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 the background density, but not to the density fluctuations. This changes the balance between the gravitational 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 $(1 - 6f_\nu)$. These two effects sum up to a factor $(1 - 8f_\nu)$ [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 $\{\omega_m, \omega_b, \Omega_\Lambda\}$ and varying $f_\nu$, 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 the redshift of equality $z_{\text{eq}}$ invariant, and has very large effects on the CMB spectra. If one follows the logic of minimizing CMB variations and fixing $z_{\text{eq}}$ like in the previous paragraphs, the increase in $\sum m_\nu$ must take place together with an increase of $h$, which tends to suppress the large-
<table><caption>Table 26.1: Summary of $N_{\text{eff}}$ constraints.</caption><thead><tr><th></th><th>Model</th><th>$N_{\text{eff}}$</th><th>Ref.</th></tr></thead><tbody><tr><td><strong>CMB alone</strong></td><td></td><td></td><td></td></tr><tr><td>Pl18[TT,TE,EE+lowE]</td><td>ΛCDM+$N_{\text{eff}}$</td><td>2.92<sup>+0.36</sup><sub>-0.37</sub> (95%CL)</td><td>[24]</td></tr><tr><td><strong>CMB + background evolution + LSS</strong></td><td></td><td></td><td></td></tr><tr><td>Pl18[TT,TE,EE+lowE+lensing] + BAO</td><td>ΛCDM+$N_{\text{eff}}$</td><td>2.99<sup>+0.34</sup><sub>-0.33</sub> (95%CL)</td><td>[24]</td></tr><tr><td>"</td><td>" +5-params.</td><td>2.85<sup>+0.23</sup><sub>-0.23</sub> (68%CL)</td><td>[25]</td></tr></tbody></table>

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 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 $A_s$, 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 $A_s e^{-2\tau}$. Hence a measurement of $\tau$ is necessary in order to fix $A_s$ from CMB data, and avoid a parameter degeneracy between $\sum m_\nu$ and $A_s$ [37–39].

A few of the neutrino mass effects described above –free-streaming scale, early ISW– depend on individual masses $m_\nu$, but most of them depend only on the total mass through $f_\nu$ – suppression of the matter power spectrum, CMB lensing, shift in angular diameter distance–. Because the latter effects are easier to measure, cosmology is primarily sensitive to the total mass $\sum m_\nu$ [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 the so-called Hubble tension which has now reached the ~5σ level (see Chap. 25 of this Review), it is important to bear in mind the following considerations. Inconsistent measurements should not be combined: bounds obtained from combination of discrepant data sets should be considered with extreme caution. Additionally, the constraints reported below assume that the solution of the $H_0$ tension –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 $N_{\text{eff}}$ obtained with several combination of data sets. ‘Pl18’ denotes the Planck 2018 data, composed of a high-$\ell$ temperature+polarization likelihood (TT,TE,EE), low-$\ell$ polarization (low E) and CMB lensing spectrum likelihood (lensing) based on lensing 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+$N_{\text{eff}}$), the constraint on $N_{\text{eff}}$ from the Planck 2018 data release [TT,TE,EE+lowE] is $N_{\text{eff}} = 2.92^{+0.36}_{-0.37}$ (95%CL). This number is perfectly compatible with the prediction of the standard neutrino decoupling model, $N_{\text{eff}} = 3.044$, 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 the growth rate and clustering amplitude of matter as a function of scale, although LSS data are not very constraining for the $N_{\text{eff}}$ parameter. The second result quoted in Table 26.1 shows that the $N_{\text{eff}}$ 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 and redshift-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 $H_0$ with distance ladders (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 $N_{\text{eff}}$ and $h$ means that inclusion of the $H_0$ measurement would tend to push $N_{\text{eff}}$ to higher values. However, the high values of $N_{\text{eff}}$ that would potentially reconcile the measured CMB or BAO peak scale with the Hubble parameter suggested by DLs, $H_0 \sim 73 \text{ km/s/Mpc}$, are excluded by the measurement of the damping tail of the CMB temperature spectrum, either by Planck itself (given the previously quoted $N_{\text{eff}}$ bounds) or by more recent and independent small-scale CMB data sets (as shown e.g. in [57]). Thus, the $N_{\text{eff}}$ extension to the ΛCDM model does not reduce the tension significantly, and the combination of Planck and DL data does not return any meaningful constraint on $N_{\text{eff}}$ in this context. It is currently unclear whether a resolution of the Hubble tension would require a departure from the ΛCDM (or ΛCDM+$N_{\text{eff}}$) model [11, 58–62].

As long as DL data are not included, the error bars on $N_{\text{eff}}$ degrade mildly when the data are analysed in the context of more extended cosmological scenarios. Adding only the total neutrino mass as an 8th free parameter has a negligible impact on the bounds. The authors of Ref. [25] take a more extreme point of view and fit a 12-parameter model to Pl18[TT,TE,EE+lowE+lensing] data; they obtain $N_{\text{eff}} = 2.95 \pm 0.24$ (68% CL), showing that it is very difficult with current cosmological data to accommodate shifts of more than 0.5 from the standard $N_{\text{eff}}$ 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 a large mixing angle, which would need to be thermalised unless non-standard interactions come into play [5]. In other words, cosmology disfavours the explanation of the oscillations anomalies in terms of 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 dark energy properties or modified gravity), it is in principle conceivable that $N_{\text{eff}}$ reaches larger values. However, to date, none of these ingredients has been shown to fully solve the Hubble tension and reconcile CMB, BAO, DL, and supernovae data.

### 26.3.2 Are they really neutrinos, as expected?

While a value of $N_{\text{eff}}$ significantly different from zero (at more than 15σ) and consistent with the expected number 3.044 yields a powerful indirect confirmation of the CνB, departures from standard $N_{\text{eff}}$ 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
<table><caption>Table 26.2: Summary of $\sum m_\nu$ constraints.</caption><thead><tr><th></th><th>Model</th><th>95% CL (eV)</th><th>Ref.</th></tr></thead><tbody><tr><td colspan="4"><strong>CMB alone</strong></td></tr><tr><td>Pl18[TT,lowE]</td><td>ΛCDM+<sum>m<sub>ν</sub></td><td>&lt; 0.54</td><td>[24]</td></tr><tr><td>Pl18[TT,TE,EE,lowE]</td><td>ΛCDM+<sum>m<sub>ν</sub></td><td>&lt; 0.26</td><td>[24]</td></tr><tr><td colspan="4"><strong>CMB + probes of background evolution</strong></td></tr><tr><td>Pl18[TT,TE,EE,lowE] + BAO</td><td>ΛCDM+<sum>m<sub>ν</sub></td><td>&lt; 0.13</td><td>[49]</td></tr><tr><td>Pl18[TT,TE,EE,lowE] + BAO</td><td>ΛCDM+<sum>m<sub>ν</sub>+5 params.</td><td>&lt; 0.515</td><td>[25]</td></tr><tr><td colspan="4"><strong>CMB + LSS</strong></td></tr><tr><td>Pl18[TT,lowE+lensing]</td><td>ΛCDM+<sum>m<sub>ν</sub></td><td>&lt; 0.44</td><td>[24]</td></tr><tr><td>Pl18[TT,TE,EE,lowE+lensing]</td><td>ΛCDM+<sum>m<sub>ν</sub></td><td>&lt; 0.24</td><td>[24]</td></tr><tr><td>Pl18[TT,TE,EE,lowE]+ ACT[lensing]</td><td>ΛCDM+<sum>m<sub>ν</sub></td><td>&lt; 0.12</td><td>[50]</td></tr><tr><td colspan="4"><strong>CMB + probes of background evolution + LSS</strong></td></tr><tr><td>Pl18[TT,TE,EE,lowE] + BAO + RSD</td><td>ΛCDM+<sum>m<sub>ν</sub></td><td>&lt; 0.10</td><td>[49]</td></tr><tr><td>Pl18[TT,TE,EE,lowE+lensing] + BAO + RSD + Shape</td><td>ΛCDM+<sum>m<sub>ν</sub></td><td>&lt; 0.082</td><td>[51]</td></tr><tr><td>Pl18[TT,lowE+lensing] + BAO + Lyman-α</td><td>ΛCDM+<sum>m<sub>ν</sub></td><td>&lt; 0.087</td><td>[52]</td></tr><tr><td>Pl18[TT,TE,EE,lowE] + BAO + RSD + SN + DES-Y1</td><td>ΛCDM+<sum>m<sub>ν</sub></td><td>&lt; 0.12</td><td>[49]</td></tr><tr><td>Pl18[TT,TE,EE,lowE] + BAO + RSD + SN + DES-Y3</td><td>ΛCDM+<sum>m<sub>ν</sub></td><td>&lt; 0.13</td><td>[53]</td></tr></tbody></table>

field with quartic potential, departures from Einstein gravity, or large extra dimensions are some of the possibilities for such ingredients. In principle one could even assume that the cosmic neutrino background never existed or has decayed (like in the “neutrinoless Universe” model of [63]) while another dark radiation component is responsible for $N_{\text{eff}}$. At least, cosmological data allow to narrow the range of possible interpretations of $N_{\text{eff}} \simeq 3$ 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 the metric 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 $G_{\text{eff}}$ for a Fermi-like four-fermions interaction at 95% confidence is $\log_{10}(G_{\text{eff}}\text{MeV}^2) < -1.47$ for Pl18+BAO [11]. Note however that neutrino self-interactions as strong as $\log_{10}(G_{\text{eff}}\text{MeV}^2) \simeq -1.4$ could reconcile CMB temperature and BAO data with the direct $H_0$ 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, $c_{\text{eff}}$ and $c_{\text{vis}}$ (see e.g., [65–67]): $c_{\text{eff}}^2$ generalizes the linear relation between isotropic pressure perturbations and density perturbations, while $c_{\text{vis}}^2$ modifies the neutrino anisotropic stress equation. While relativistic free-streaming species have $(c_{\text{eff}}^2, c_{\text{vis}}^2) = (1/3, 1/3)$, a perfect relativistic fluid would have $(c_{\text{eff}}^2, c_{\text{vis}}^2) = (1/3, 0)$. 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 $(c_{\text{eff}}^2, c_{\text{vis}}^2) = (1/3, 1/3)$ [68–70].

Finally, Ref. [23] (resp. [26]) shows that current data are precise enough to detect the “neutrino drag” 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 density of some relativistic relics, but also their anisotropies.

### 26.3.3 Neutrino masses

Table 26.2 shows a list of constraints on $\sum m_\nu$ 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-space distortions 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’ to the 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 $N_{\text{eff}}$ are compatible with the standard prediction, $N_{\text{eff}} = 3.044$, it is reasonable to adopt this value as a theoretical prior and to investigate neutrino mass constraints in the context of a minimal 7-parameter model, $\Lambda\text{CDM}+\sum m_\nu$. Under this assumption, the most robust constraints come from Planck 2018 temperature and polarization data alone: $\sum m_\nu < 0.26 \text{ eV (95\%CL)}$ [24]. Among the four effects of neutrino masses on the CMB spectra described before, current bounds are dominated by the first and the third effects (modified late background evolution, and distortions of the temperature and polarization spectra through weak lensing).

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 between $\sum m_\nu$ and $h$. The combination of Pl18[TT,TE,EE,lowE] with the most recent BAO measurements, including the eBOSS Data Release 16 (DR16), gives $\sum m_\nu < 0.13 \text{ eV (95\%CL)}$ [49]. Supernovae data are less constraining than BAO data for the neutrino mass determination.

Because the parameter correlation between $\sum m_\nu$ and $H_0$ is negative, the inclusion of DL data would tend to provide stronger bounds on neutrino masses. However, like for $N_{\text{eff}}$, 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 the small-scale suppression of the matter power spectrum due to neutrino free-streaming, although most 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 $0.12 \text{ eV}$. To get an even stronger bound, one can add data from the eBOSS survey, which infers the growth rate of structures from RSDs. The addition of BAO and RSD measurements from eBOSS tightens the bound down to $\sum m_\nu < 0.10 \text{ eV (95\%CL)}$ [49]. Further including information about the large-scale shape (‘Shape’) of the power spectrum, BOSS+eBOSS data alone give $\sum m_\nu < 0.40 \text{ eV (95\%CL)}$ with a BBN prior on $\omega_b$, while Planck+BOSS+eBOSS give $\sum m_\nu < 0.082 \text{ eV (95\% CL)}$ [51]. A similar bound arises from the most recent Lyα forest data from BOSS combined with Planck and BAO data, giving $\sum m_\nu < 0.087 \text{ eV (95\% CL)}$. Such bounds already challenge the
inverted hierarchy mass scheme, which predicts $\sum m_\nu \ge 0.11$ eV (e.g., [47] and refs. therein).

It should however be noticed that DES prefers a lower $\sigma_8$ value than the Planck best fit, relaxing the bound to $\sum m_\nu < 0.12$ eV (95%CL) [49] when using Year 1 data in combination with eBOSS and other datasets (Pl18+BAO+RSD+SN+DES). Using Year 3 data and a similar combination, the DES collaboration obtains $\sum m_\nu < 0.13$ eV (95%CL) [53]. Notice however that the DES analysis adopts a theory prior $\sum m_\nu \ge 0.06$eV while other constraints only assume $\sum m_\nu \ge 0$. For a direct comparison, the combined constraint of reference [49] can be recomputed with the same theory prior and becomes $\sum m_\nu < 0.13$ 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 $N_{\text{eff}}$ 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 cosmological parameters, and that neutrino mass bounds are becoming increasingly robust.

One should stress again that if the explanation of DL measurements requires a cosmological scenario with new – and yet unknown – 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.

## 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 – an effect 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, these bounds rely on the $\Lambda$CDM model or its most simple extensions, and thus, on the assumption that a resolution of the Hubble tension with DL measurements does not affect the modeling and interpretation of the data sets adopted here. If this is not the case, a shift of paradigm might be required, 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] and SKA [89], in combinations with new CMB experiments like Simons Observatory [90], CMB-S4 [91] or LiteBird [92]. If the $\Lambda$CDM model is confirmed, and if neutrinos have standard properties, the total neutrino mass should be detected at the level of at least $3-4\sigma$ 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 allowed by oscillations $\sum m_\nu \sim 0.06$ eV, neutrinos constitute $\sim 0.5\%$ of the Universe matter density, and their 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 and the possibility to perform consistency checks which in turns helps immensely with the control of systematic 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 a measurement of the sum of neutrino masses and possibly indication of their mass hierarchy.

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## 27. Dark Matter

Revised August 2023 by L. Baudis (Zurich U.) and S. Profumo (UC Santa Cruz).

### 27.1 The case for dark matter

Modern cosmological models invariably include an electromagnetically close-to-neutral, non-baryonic matter species with negligible velocity from the standpoint of structure formation, generically referred to as “cold dark matter” (CDM; see The Big-Bang Cosmology—Sec. 22 of this Review). For the benchmark $\Lambda$CDM cosmology adopted in the Cosmological Parameters—Sec. 25 of this Review, the DM accounts for 26.4% of the critical density in the universe, or 84.4% of the total matter density. The nature of only a small fraction, between at least 0.5% (given neutrino oscillations) and at most 1.6% (from combined cosmological constraints), of the non-baryonic matter content of the universe is known: the three Standard Model neutrinos (see the Neutrino Masses, Mixing, and Oscillations—Sec. 14 of this Review). The fundamental makeup of the large majority of the DM is, as of yet, unknown.

Assuming the validity of General Relativity, DM is observed to be ubiquitous in gravitationally collapsed structures of size ranging from the smallest known galaxies [1] to galaxies of size comparable to the Milky Way [2], to groups and clusters of galaxies [3]. The mass-to-light ratio is observed to saturate at the largest collapsed scales to a value indicative, and close to, what inferred from other cosmological observations for the universe as a whole [4]. In such collapsed structures, the existence of DM is inferred directly using tracers of mass enclosed within a certain radius such as stellar velocity dispersion, rotation curves in axisymmetric systems, the virial theorem, gravitational lensing, and measures of the amount of non-dark, i.e. baryonic, mass such as stellar number counts and tracers of gas density such as X-ray emission [5]. The global DM abundance as determined from cosmological probes is discussed in Sec. 22.

The picture of structure formation in modern cosmology heavily relies on, and can be considered an independent and exceptionally strong motivation for, DM. Baryonic density fluctuations at CMB decoupling are observed to be at most on the order of $\delta\rho_b/\rho_b|_{\text{rec}} \approx 10^{-5}$; since density perturbations grow linearly with the scale factor in the linear regime, absent any other matter fluid, one would predict that

$$ \delta\rho_b/\rho_b|_{\text{today}} \approx \frac{\delta\rho_b/\rho_b|_{\text{rec}}}{a_{\text{rec}}} \approx 10^{-2}, \quad (27.1) $$

at odds with the observed highly non-linear structures in the universe, $\delta\rho_b/\rho_b|_{\text{obs}} \gg 1$. The presence of a dominant non-relativistic (“cold”) pressure-less matter component decoupled from the thermal bath well before recombination allows instead for the prediction of a matter power spectrum in remarkable agreement with observations [6].

Assuming deviations of gravitational interactions on large scales from general relativity or from its Newtonian limit, certain effects, attributed in the standard scenario to DM, can be explained by modified gravity [7]. Usually such theories mimic the effects otherwise attributed to DM on a limited range of scales, but fail globally, and especially at the largest scales. Key issues that at present appear highly problematic in the framework of theories of modified gravity without DM include (i) predicting the correct spectrum of density perturbations, (ii) predicting the observed anisotropy power spectrum of the CMB, and (iii) explaining weak lensing and X-ray observations of merging clusters such as 1E 0657-558 (the “Bullet” cluster) [8]. The inferred relative speed of gravitational and electromagnetic radiation in GW170817 additionally excludes a significant swath of modified theories of gravity where the two speeds (of gravitational and electromagnetic waves) differ [9].

### 27.2 Properties of dark matter candidates

**Electric charge:** The “darkness” of DM can be quantified based on constraints from the CMB and large-scale structure: if the DM is charged, or “milli-charged” (for instance via a kinetic mixing with a dark photon field, producing an effective

suppressed coupling to the visible photon field), it might impact the baryon-photon plasma during recombination; in turn, DM density fluctuations can be suppressed by radiation pressure and photon diffusion, additionally altering the baryon acoustic peak structure. [10] finds that the most stringent constraints stem from the requirement that the DM be completely decoupled from the baryon-photon plasma at recombination, yielding a maximal “milli-electric” charge, in units of the electron charge, of $3.5 \times 10^{-7}(m_{\text{DM}}/1 \text{ GeV})^{0.58}$ for $m_{\text{DM}} > 1 \text{ GeV}$, and of $4.0 \times 10^{-7}(m_{\text{DM}}/1 \text{ GeV})^{0.35}$ for $m_{\text{DM}} < 1 \text{ GeV}$. Limits also exist from structure formation on how optically dark and dissipationless the DM should be.

**Self-interactions:** Observations of merging clusters [8] and of the ellipticity of certain galaxies as inferred from X-rays [11] constrain the level of DM-DM self interactions. The figure of merit is the ratio of the DM-DM cross section and the DM mass [12] (see Ref. [13] for a review), $\sigma_{\text{DM-DM}}/m_{\text{DM}} < 0.47 \text{ cm}^2/\text{g} \approx 0.84 \text{ barn/GeV}$ at 95% C.L.. Assuming a velocity dependence in $\sigma_{\text{DM-DM}}$, “self-interacting DM” has been advocated as a possible solution to certain possible small-scale structure issues in the standard non-collisional ($\sigma_{\text{DM-DM}} \simeq 0$) setup [13, 14] (see Sec. 27.4).

**Mass: Lower Limits:** Model-independent lower limits for very small DM masses are due to quantum effects: for fermionic DM particles, the phase-space density $f(\vec{x}, \vec{p})$ is bounded from above due to Pauli’s exclusion principle, $f < gh^{-3}$, with g the number of internal degrees of freedom and h Planck’s constant; observations of the velocity dispersion (or, equivalently, measures of the enclosed mass) and physical density in dwarf galaxies, lead to a lower limit on fermionic DM masses, sometimes known as the Tremaine-Gunn limit [15]. Using the Fornax dwarf, Ref. [16] finds $m_F > 70 \text{ eV}$. More stringent limits can be drawn from Lyman-$\alpha$ observations, although such limits depend on the thermal history of the DM. In the case of bosonic DM, the Compton wavelength of an ultra-light species might erase small-scale structure, in conflict with CMB and large-scale structure [17], Lyman-$\alpha$ observations [18, 19], and measurements of high-redshift galaxy luminosity functions and the Milky Way satellite luminosity function [20–22]: these observations indicate that $m_B \gtrsim 10^{-22} \text{ eV}$.

**Mass: Upper Limits:** General upper limits exist on the mass of the DM constituent from the stability against tidal disruption of structures immersed in DM halos, such as galactic disks and globular clusters, and of individual small galaxies. The most stringent limits can be derived using wide halo binaries [23] and the stability of the star cluster within Eridanus II [24]. Such limits constrain an individual, point-like DM constituent, assuming it makes up 100% of the DM, to be lighter than around 5 $M_{\odot}$. (Notice that the mass limits discussed here do not assume any specific production mechanism, and do not depend on the observed cosmological DM density).

**Stability:** The DM lifetime must be long compared to cosmological timescales [25].

### 27.3 Genesis of dark matter

The generation of DM in the early universe can proceed via thermal or non-thermal production, or both, or it may result from a particle-antiparticle asymmetry.

**Freeze-out:** The process of chemical decoupling from the high-temperature, high-density thermal bath (freeze-out) as a paradigm for particle production in the early universe is both a predictive and a successful one. The possibility that just like light elements, neutrinos, and CMB photons, particle DM also originated from a thermal decoupling process has thus garnered significant attention.

A particle species chemically decouples when the rate $\Gamma$ for the species’ number-changing processes drops below the Hubble rate $H$. Rough estimates for the abundance of relics can be obtained by (i) calculating the freeze-out (i.e. “decoupling”) temperature $T_{\text{f.o.}}$, corresponding to $H(T_{\text{f.o.}}) \sim \Gamma(T_{\text{f.o.}})$, (ii) equating the comoving number density at freeze-out and today, eventually (iii) obtaining the physical density of relic particles today. This procedure assumes that entropy is conserved between $T_{\text{f.o.}}$ and today, an assumption that could well be violated, especially for