<figure><img src="image_40.png" /><figcaption>Fig. 35. Impact of IA model choice to $S_8$ in recent WL analyses assuming the ΛCDM model. The DES Y3 + KiDS results are from Ref. [867], HSC Y3 from Ref. [864], DES Y3 cosmic shear (COSEBIS and $\tilde{\xi}_s$) + SR (shear-ratio) from Ref. [861,862] and DES Y3 3 × 2pt from Ref. [869]. The vertical dashed line is the Planck value with its corresponding uncertainty shaded in orange. All error bars correspond to 1-σ uncertainties.</figcaption></figure>

Exciting new approaches are also already emerging from recent observations. Measurements have consistently found IA amplitudes lower than theoretical expectations, with blue galaxies showing negligible alignment signals [1016]. Building on these findings, [878] proposed an innovative strategy of conducting cosmic shear analysis exclusively with blue galaxies – potentially eliminating the need for complex IA modeling entirely. While this approach sacrifices approximately half the galaxy sample, reducing statistical power, it could significantly mitigate systematic uncertainties. The viability of this or other compromises will become clearer as theoretical understanding advances and next-generation survey data becomes available.

### 2.2.1.6. Confirmation bias and blinding.
Modern cosmological analyses in surveys like DES, KiDS, and HSC are often highly complex and require the seamless interaction of large teams responsible for different parts of the analysis. Sources of epistemic error are difficult to control, which can lead to a dependence of cosmological parameter constraints on analysis choices. At the same time, complementary cosmological constraints like from the CMB might induce a subconscious bias in a select parameter subspace, like $\sigma_8$ and $\Omega_{m,0}$. In cases where the value of the cosmological parameters are revealed during the analysis, this might lead to ‘confirmation bias’, where analysis assumptions are subconsciously tuned toward a certain scientific narrative. The complexity of cosmological analysis combined with the significant dependency on analysis choices, therefore necessitates strict strategies to avoid this confirmation bias. The blinding strategies differ across surveys but generally consist of catalog-level blinding where a set of synthetic and encrypted multiplicative shear values are artificially added to obfuscate the true shear field and analysis-level blinding where the inferred parameter posteriors are altered to avoid a ready comparison with prior work [1017].

### 2.2.2. Galaxy cluster counts

**Coordinator:** Dominique Eckert

**Contributors:** Antonio da Silva, Filippo Bouchè, Francesco Pace, Iryna Vavilova, Jenny G. Sorce, Massimiliano Romanello, Radosław Wojtak, Sebastian Grandis, Shahab Joudaki, and Vittorio Ghirardini

#### 2.2.2.1. Methodology.
Galaxy clusters are the most massive gravitationally bound structures in the present Universe and originate from the largest fluctuations of the primordial matter distribution. Their mass and number density strongly depends on the underlying cosmological parameters [1018], such that galaxy cluster surveys are powerful probes of cosmological parameters. In this section, we describe the main steps that are needed to extract cosmological information from galaxy cluster surveys. We identify the major bottlenecks that need to be addressed to advance our cosmological knowledge with the use of this technique, and summarize the main recent results in the field.

**The halo mass function and its cosmological dependency.** In the hierarchical structure formation scenario, small structures formed at high redshifts progressively merge under the influence of gravity to form the massive systems we see today. As such, the number density of halos and its time evolution trace the growth of structures in the Universe, which depends on the underlying cosmological model. The halo mass function describes the number of collapsed halos at a given time per unit volume and halo mass. Its shape, normalization, and redshift evolution are highly sensitive to cosmological parameters.

In mathematical terms, the halo mass function can be expressed in the following way:

$$ \frac{dn}{d \ln M} = \frac{\rho_{m,0}}{M} f(\sigma) \left| \frac{d \ln \sigma}{d \ln M} \right|, \qquad (2.12) $$

with $\rho_{m,0}$ the mean matter density of the Universe at present day, $\sigma$ the mass variance on a mass scale $M$ and $f(\sigma)$ a function characterizing different fitting (or theoretical) expressions. Often, $f(\sigma)$ is written in as $\nu f(\nu)$, where the peak height $\nu = \delta_c/\sigma$ is a function of mass and time. Here, $\delta_c$ is the linearly extrapolated overdensity and can be evaluated within the framework of the spherical collapse model. Many different expressions exist for $f(\sigma)$ [1019–1024].

In Fig. 37 we present the dependence of the Tinker halo mass function [1022] on the matter power spectrum normalization $\sigma_8$ (left panel) and on the matter density $\Omega_{m,0}$. We see that the effect on the halo mass function is that of increasing the number of structures. A higher value of $\Omega_{m,0}$ implies also a higher $\sigma_8$ value. The two parameters are highly degenerate at the high-mass end, as shown by Fig. 36 where a 50 deg$^2$ mock cluster counts survey is compared with DES+KiDS WL and the latest Planck collaboration data. The degeneracy between $\Omega$, and $\sigma_8$ for galaxy cluster counts is clearly visible.

It is important to note that halo masses are usually defined within a spherical overdensity $\Delta$ with respect to either the critical density $\rho_c(z)$ or the mean density of the Universe $\rho_m(z)$. For a given overdensity $\Delta_c$ or $\Delta_m$, the corresponding mass $M_\Delta$ and radius $R_\Delta$ are defined as the mass and radius within which the condition

$$ \frac{M_\Delta}{4/3\pi R_\Delta^3} = \Delta_c/m \rho_c/m(z), \qquad (2.13) $$

is satisfied. The choice of the halo mass function and of the observational mass definition must therefore be adapted to the chosen spherical overdensity. Common choices for the definition of the overdensity are 500c, 200c, and 200m.

**Mass-observable scaling relations and the scatter around these relations.** While the halo mass function is highly sensitive to cosmological parameters, constraining it from galaxy cluster surveys is a challenging task. The main underlying issue is that halo mass is not a directly observable quantity. Samples of galaxy clusters are extracted from surveys of the baryonic component that fills these structures, whether it is in the form of hot gas (X-ray, Sunyaev-Zel'dovich effect) or stars (optical/IR). Galaxy cluster detection techniques and their primary
<figure><img src="image_41.png" /><figcaption>Fig. 36. $\Omega_{m,0} - \sigma_8$ degeneracy in typical cosmic shear and cluster counts experiments. The blue curve and shaded area show the degeneracy of the expected $\Omega_{m,0} - \sigma_8$ contours in a mock 50 deg$^2$ X-ray cluster count survey, whereas the green contours show the same degeneracy in the combined DES Y3 + KiDS-1000 cosmic shear estimate. We can see that the degeneracy is shallower in cluster counts experiments ($\sigma_8 \propto \Omega_{m,0}^{-0.3}$) compared to cosmic shear ($\sigma_8 \propto \Omega_{m,0}^{-0.5}$), such that any definition of $S_8$ that is optimal for either experiment will not perfectly follow the degeneracy of the other. For comparison, we add the Planck CMB $\Omega_{m,0} - \sigma_8$ contours in orange. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

observables are described in more detail in Section 2.2.2.2. In all cases, the primary observable of the survey is not directly the halo mass but rather a quantity that depends on it. The relation between survey observable and the halo mass, usually referred as a scaling relation, needs to be properly understood to turn the detected cluster samples into constraints on the halo mass function.

If galaxy clusters grow and evolve in a self-similar way [1025], the relations between observable quantities $\mathcal{O}$ and spherical overdensity halo mass $M_\Delta$ are expected to be power laws. While astrophysical effects such as mergers, cooling, and AGN feedback can cause deviations from the simple self-similar scenario, evidence suggests [1026–1029] that these relations can be well described by power laws with a log-normal intrinsic scatter $\sigma_{\log \mathcal{O}}$,

$$ \log \mathcal{O} = A_{\mathcal{O}M} + B_{\mathcal{O}M} \left( \frac{M_{\Delta}}{M_{\text{pivot}}} \right) + \gamma_{\mathcal{O}M} E(z) \pm \sigma_{\log \mathcal{O}}, \quad (2.14) $$

with $E(z) = H(z)/H_0$. Here the parameters $A_{\mathcal{O}M}$ and $B_{\mathcal{O}M}$ are the normalization and the slope of the mass-observable scaling relation and $\gamma_{\mathcal{O}M}$ governs its redshift evolution. The scaling relation allows us to construct the probability of detecting a system with a measured observable $\mathcal{O}$ given the source's mass and redshift, $P(\mathcal{O}|M, z)$. A precise knowledge of these parameters is required to make accurate cosmological inferences from galaxy surveys. We discuss the associated issues in more detail in.

**Selection function, forward modeling, likelihood function** The cosmological analysis of galaxy cluster catalogs follows a Bayesian inference approach. A generative model for the data given unknown model parameters has to be postulated, and the PDF of the actual data as a function of the model parameters, the likelihood, has to be computed. The generative model assumed for a cluster catalog with observable $\hat{\mathcal{O}}$ and redshift $z$ at a sky position $\theta$ is a Poisson realization of the density field

$$ \frac{d\bar{n}}{d\hat{\mathcal{O}}} = \frac{dn}{d\hat{\mathcal{O}}} (1 + b_{\text{eff}}\delta(\theta, z)), \quad (2.15) $$

with $b_{\text{eff}}$ the effective bias, $\delta(\theta, z)$ the matter density contrast, and

$$ \frac{dn}{d\hat{\mathcal{O}}} = \int d\mathcal{O} \int dM \frac{dn}{dM} P(\mathcal{O}|M, z) P(\hat{\mathcal{O}}|\mathcal{O}, z, \theta) P(\text{sel}|\hat{\mathcal{O}}, \mathcal{O}, z, \theta), \quad (2.16) $$

where $\frac{dn}{d\ln M}$ is the halo mass function (Eq. (2.12)), $P(\mathcal{O}|M, z)$ the observable mass relation and its scatter (Eq. (2.14)), $P(\hat{\mathcal{O}}|\mathcal{O}, z, \theta)$ the effect of instrumental noise of the intrinsic observable $\mathcal{O}$, and $P(\text{sel}|\hat{\mathcal{O}}, \mathcal{O}, z, \theta)$ the selection function. The latter takes values between 0 and 1 based on the probability of including the cluster with properties ($\hat{\mathcal{O}}, \mathcal{O}, z, \theta$).

The second term in Eq. (2.15) can be ignored for sufficiently high-mass and wide survey area samples, yielding the Poisson likelihood used by Refs. [736,1030–1036]. For smaller area surveys and lower mass surveys, the second term contributes significantly via the variance of the matter field. In such cases, a composite Gaussian-Poisson likelihood has to be used [1037–1044]. The resulting likelihood depends on the cosmological parameters via the halo mass function and the cosmological volume. It also depends on the nuisance parameters describing the observable mass relation and, potentially, the selection function. Marginalizing over sufficiently flexible observable mass relation parameterizations is crucial to accurately represent the cluster population, especially with regard to the halo mass.

### 2.2.2.2. Galaxy cluster surveys.

**X-ray** The vast majority of the baryonic content of galaxy clusters is in the form of a hot (> $10^7$ [K]), highly ionized plasma that fills the system's gravitational potential well. Given the high temperatures involved, the intracluster medium (ICM) shines predominantly in the X-ray range through thermal bremsstrahlung and line emission. As such, galaxy clusters are luminous ($L_X = 10^{44} - 10^{45}$ [erg/s]) X-ray sources spanning a diameter of several arcmin. The overwhelming majority of extended X-ray sources in the extragalactic sky are galaxy clusters, such that X-ray surveys are powerful cluster detection machines. In the early 1990s, Röntgensatellit (ROSAT) [1045–1047] performed an all-sky X-ray survey and several deeper but small area surveys, e.g., ROSAT Deep Cluster Survey (RDCS) [1048] which allowed for the detection of several thousand clusters. This spurred the build-up of several sub-samples, either volume-limited (REXCESS) [1049], or flux-limited (HIFLUGCS) [1050,1051] with a well-understood selection function.

Deep follow-up observations of ROSAT-selected clusters with XMM-Newton and Chandra allowed to constrain cosmological parameters with cluster number counts by measuring the mass of these systems with better precision [1052]. XMM-Newton, thanks to its larger field-of-view, enabled to scan deeply (30–40 ks) several tens of square degrees, enabling the serendipitous detection of the fainter galaxy group population, extending the mass range used in the cosmological analysis. The XXL survey [1053] detected ~450 clusters over an area of 50 deg$^2$ [1037,1041]. More recently, eROSITA undertook several all-sky surveys, with the ultimate goal of detecting about 100,000 galaxy clusters. The results from the first of these all-sky surveys were recently released [1036] and provided the detection of ~12,000 clusters over half the sky.

**Sunyaev-zel'dovich effect** The thermal SZ effect [tSZ;1054] describes the inverse-Compton scattering of CMB photons off the hot electrons of the ICM. Photons crossing the path of a hot electron cloud occasionally encounter hot electrons and gain energy, which induces a modification of the spectral shape of the CMB, see Ref. [1055] for a review. At the peak frequency of the CMB, the tSZ effect takes the form of a decrement in the temperature of the CMB, as a fraction of the photons are upscattered to higher frequencies. At first order, the spectral signal of the tSZ depends only on the Compton parameter, $y$, which describes the average energy gain over the line of sight. The total integrated signal over the area of a cluster, $Y_{SZ}$, is proportional to the integrated thermal energy of the ICM [1056].

The advantage of the tSZ effect as a cluster detection technique is twofold. First, the signal is essentially independent of redshift, which
<figure><img src="image_42.png" /><figcaption>**Fig. 37.** Dependence of the Tinker mass function on matter density $\Omega_m$ (left panel) and on the matter spectrum normalization $\sigma_8$ (right panel) at $z = 0$.</figcaption></figure>

<figure><img src="image_43.png" /></figure>

makes it very efficient to detect high-redshift systems. Second, the primary tSZ observable, $Y_{SZ}$, is believed to be an excellent proxy of cluster mass [1057–1060]. In the past decade, the Planck survey yielded the detection of > 2000 galaxy clusters with this technique extending out to $z \sim 1$ [737,1061]. While Planck was very sensitive to the tSZ thanks to its wide frequency range and all-sky coverage, the detection efficiency of high-redshift sources was hampered by the poor Planck beam, such that the bulk of the detected systems are at redshift $z < 0.5$. Conversely, ground-based experiments have larger collecting areas and better angular resolution. SPT [1062–1064] and ACT [1065,1066] were now able to gather catalogs of several thousand galaxy clusters out to $z \sim 2$.

**Optical/near-IR photometry** While in the past couple of decades, a great deal of attention was devoted to the selection of galaxy clusters through the ICM, there is now renewed interest in the detection of galaxy clusters as overdensities of galaxies thanks to the new generation of optical and near-IR surveys like *Euclid* and *Rubin*. As their name suggests, galaxy clusters were originally defined as such based on the detection of overdensities of galaxies in the same region of the sky. Based on visual inspection of photometric plates, Abell (1989) [1067] constructed a catalog of more than 4000 clusters that has been used for more than two decades as a reference of the local large-scale structure. Since that time, sophisticated galaxy cluster detection algorithms have been developed which allowed to extract large samples of clusters from galaxy surveys. Modern detection algorithms are mainly split into two classes: red sequence and photometric redshifts. Red sequence cluster finders [1068–1070] take advantage of the predominance of passive red galaxies in clusters with respect to the field to search for overdensities of red galaxies. This method reduces spurious detections thanks to the use of the red sequence, which is known to trace rich environments [1071,1072]. Alternatively, methods based on photometric redshifts [720,1073] make use of galaxy SEDs to search for galaxy overdensities in three-dimensional space. These techniques rely on the availability of high-quality photometric redshifts but make no assumption on the existence of a red sequence. For a detailed comparison of various cluster finding algorithm, we refer to Ref. [1074].

The main advantage of optical cluster catalogs is their high sensitivity, with the deepest current catalogs containing more than 100,000 clusters (e.g., see Refs. [1075–1079]). Modern algorithms also provide an estimate of the cluster redshift as a direct byproduct. The primary observable of optical cluster finders is the number of member galaxies, or *richness N*. A standardized richness estimate is computed iteratively to include cluster members above a given absolute magnitude. Richness is known to correlate with halo mass [1080,1081], although with a large intrinsic scatter induced by projection effects and miscentering [1082,1083].

**Shear selection** The techniques discussed above all rely on detection through the baryonic content of clusters, whether it is in the form of gas or galaxies, whereas the vast majority of the mass is made of DM. In the past decade, direct detection of galaxy clusters as peaks in projected WL mass maps has become possible [906,1084–1087]. This technique allows to detect clusters directly through their total mass content rather than through a proxy that is related to the halo mass in a complex way. Thanks to the depth of the Subaru/HSC-SSP survey (~30 galaxies per square arcmin), Ref. [1085] presented a sample of 65 shear-selected clusters within a ~160 deg$^2$ area with a S/N greater than 4.7. All the most significant peaks were associated with galaxy overdensities or X-ray signals [1088]. More recently, the first cosmological constraints from shear-selected cluster samples were extracted from an area of ~500 deg$^2$ [1086,1089]. While at the present day shear-selected cluster catalogs are much smaller than catalogs obtained from other techniques, upcoming lensing experiments like *Euclid* will yield much larger samples directly extracted from the projected mass distribution. The primary issue associated with shear selection is projection associated with correlated large-scale structure and halo orientation, such that the masses of the detected systems are expected to be over-estimated by ~50% [1090]. Future works will need to model projection effects in detail to make accurate cosmological inferences from large samples of shear-selected systems.

### 2.2.2.3. Mass calibration and systematics.

**Mass calibration techniques** The number of galaxy clusters as an observable displays a direct degeneracy between the mean mass scale of the sample and the cosmological parameters, which dictate the number density of halos via the mass function and the cosmological volume. Number counts of galaxy clusters can thus only constrain cosmological parameters if the relation between halo mass and cluster observables is determined, a problem called mass calibration. Our knowledge of subgrid astrophysical processes limits the direct calibration of this relation on hydrodynamical simulation. It is thus very challenging to derive the degree of uncertainty on the mass calibration from pure simulation-based studies. The latter is, however, the main contributor to the final cosmological constraining power of cluster number counts experiments and should, therefore, be assessed in a reliable, empirical fashion.

In the context of a CDM cosmology, most of the halo mass is composed of DM. Empirical mass calibration can thus only proceed via measurements of the cluster's gravitational potential. Three techniques have been established to observe the cluster gravitational potential: (i) hydrostatic equilibrium; (ii) cluster galaxy dynamics; (iii) WL. Owing to its independence on the dynamical state, WL has become in recent years the method of choice to calibrate galaxy cluster mass, see Ref. [1091] for a review.
While WL is expected to be close to unbiased on average, the signal-to-noise of individual cluster mass measurement is usually very low. Mass calibration is thus integrated into the likelihood framework discussed above by considering the marginal PDF of the observable directly linked to halos mass $\hat{\mathcal{O}}_M$ conditional on the observable one seeks to calibrate $\hat{\mathcal{O}}$ [1031,1032,1034,1035], reading

$$p(\mathcal{O}_M|\hat{\mathcal{O}}, z, \theta) = \left(\frac{dn}{d\hat{\mathcal{O}}}\right)^{-1} \frac{dn}{d\hat{\mathcal{O}}d\hat{\mathcal{O}}_M}, \qquad (2.17)$$

with

$$\frac{dn}{d\hat{\mathcal{O}}d\hat{\mathcal{O}}_M} = \int d\mathcal{O} \int d\mathcal{O}_M \int dM \frac{dn}{dM} P(\mathcal{O}, \mathcal{O}_M|M, z) P(\hat{\mathcal{O}}|\mathcal{O}, z, \theta) P(\hat{\mathcal{O}}_M|\mathcal{O}_M, z) P(\text{sel}|\hat{\mathcal{O}}, \mathcal{O}, z, \theta), \qquad (2.18)$$

where $P(\mathcal{O}, \mathcal{O}_M|M, z)$ is the joint scaling relation between mass calibration observable and primary observable, and $P(\hat{\mathcal{O}}_M|\mathcal{O}_M, z)$ parameterized the measurement uncertainty on the mass calibration observable. Note that the correlation between the observable mass relation is crucial to account for physical selection effects [1092,1093]. The marginal relation $P(\mathcal{O}_M|M, z)$ needs to be anchored, and its parameters tightly constrain within prior, such that the overall likelihood of the mass calibration depends only on the parameters of the primary observable mass relation, thus constraining them.

**Mass calibration uncertainties** For the sake of brevity, we shall focus on the systematic uncertainty affecting WL mass calibration. Given the advent of wide and deep photometric surveys, WL mass calibration has emerged as the principal way to calibrate cluster number count. Three sources of statistical uncertainty impact the WL signal of galaxy clusters: the shape noise due to the intrinsic ellipticity of source galaxies, the intrinsic heterogeneity of cluster mass profiles at the same mass (due to scatter around the concentration mass relation, varying degrees of substructure, orientation, triaxiality, etc.), and the statistical fluctuations in the line-of-sight integrated matter field [1094,1095]. Deeper and wider WL surveys can only reduce the first of these three noise sources.

WL mass calibration shares the systematics of the shear measurement and photometric redshift estimation inherent in the source catalogs from wide and deep photometric surveys. Furthermore, the displacement between the observed and true cluster position washes out the WL signal (mis-centering). As clusters are significant overdensities in the galaxy field, depending on the source background selection, a fraction of unlenased cluster members might contaminate the source sample and dilute the signal (cluster member contamination). Finally, baryonic effects do not only alter the total halo mass but also the halo mass profile that sources the lensing. The combined impact of these effects is summarized by establishing the relation between the WLL mass (that results from fitting the WL signal) and the halo mass on realistic WL simulations [1096–1098]. WL masses are biased low by 5%–10% depending on the fitting formula used and scatter around the true halo mass [1099–1101]. The systematic uncertainties of the aforementioned effects are distilled into an uncertainty on the bias of the WL mass [1034,1093,1102,1103]. Current understanding of baryon feedback puts a 2% lower limit on this uncertainty [1098]. Depending on the WL survey, photometric source redshift uncertainties increase this to 10% at high cluster redshifts. The other effects have a quantifiably smaller impact, as, for instance, mis-centering and cluster member contamination are partially controlled empirically [1034,1081,1093,1104].

**Halo mass function** The halo mass function described above only considers structures made by CDM. However, halos contain a certain amount of baryons. Neglecting this contribution leads to a systematic bias in determining the halo mass function. This can be easily understood by considering that baryons will change the total mass of the halo.

Hydrodynamical simulations found that the effects of baryons are different whether, in the simulations, radiative effects are taken into account or gas heating is only due to gravitational effects. For instance, models with AGN feedback decrease the abundance of clusters compared to DM-only simulations [1031]. The impact of baryons appears to depend on the halo mass. Considering an overdensity of 1500 times the critical density, baryons contribute up to 6%–7% to the total mass, while for 200 times the critical density, their contribution reduces to only about 1%.

In terms of number counts, [1105,1106] found that the number density of halos decreases up to 15% at low masses and redshifts, while being in broad agreement with that of DM-only simulations in the other cases. In addition, to better match X-ray and SZ observational studies, [1106] proposed fits to estimate the baryon contribution when moving from a halo defined as 200 times the critical density to one found using 500 times this density. The retrieved halo masses in the presence of baryons can also be corrected to match the expected halo masses in gravity-only simulations [1098,1107].

### 2.2.2.4. Results.

**Cosmological parameters measurements and tensions** Cosmological results on the mean matter density at present time, $\Omega_m$, from cluster number counts agree with CMB results that it should be around 0.3. Recent results from eROSITA (0.29±0.01 [1036]), SPT (0.29±0.03 [1035]), WtG (0.26 ± 0.03 [1030]), XXL (0.31 ± 0.03 [1041]), Planck SZ (0.33 ± 0.03 [736]) and DES (0.32±0.08$_{-0.07}^{+0.08}$ [1108]), and UNIONS (0.29±0.05 [1109]) are all consistent with the results from Planck CMB [192] of 0.315 ± 0.007.

Conversely, there is currently some disagreement between the values of $\sigma_8$ from cluster number counts obtained in various studies. While some studies find $\sigma_8$ in agreement with the CMB value of 0.811 ± 0.006 (eROSITA 0.88 ± 0.02 [1036]; SPT 0.82 ± 0.03 [1035]; WtG 0.83 ± 0.04 [1030]; XXL 0.84 ± 0.04 [1041]), Planck SZ found a value of 0.76±0.03 [736] which is in tension with the CMB at the level of ~ 2.5$\sigma$.

Similar to the case of cosmic shear, and as highlighted in Fig. 37, consistency with CMB should be assessed by considering the entire multi-dimensional posterior distribution. For this reason, it is customary to consider the quantity, $S_8 = \sigma_8 (\frac{\Omega_{m,0}}{0.3})^\alpha$, with $\alpha = 0.5$. We note, however, that the degeneracy between $\Omega_{m,0}$ and $\sigma_8$ in cluster count experiments follows a slightly different slope from cosmic shear, with an index in the range 0.2 (e.g., SPT [1035]) to 0.4 (e.g., eROSITA, [1036]).

Considering only experiments including an internally calibrated WL mass-observable scaling relation, i.e., eROSITA, SPT, WtG, and XXL, the tension between $\Omega_m$ and CMB measurements is at most 1.2-$\sigma$, thus showing a high level of consistency between experiments. The same exercise applied to $\sigma_8$ indicates a slightly higher level of tension (1.7 $\sigma$), such that potential systematics still remain at this level of precision.

**Constraints on dark energy** Cluster number counts are useful in providing constraints on the DE equation of state $w$. DE affects the growth of structures in the Universe at redshifts below ~ 1, thereby reducing the growth of massive halos. Most current studies on the DE equation of state from cluster counts assume a constant equation of state $w$ and a flat universe. All the results published thus far are consistent with $w = -1$ (Chandra $w = -1.14 \pm 0.21$ [1052]; WtG $w = -0.98 \pm 0.15$ [1030]; SPT $w = -1.45 \pm 0.31$ [1035]; eROSITA $w = -1.12 \pm 0.12$ [1036]). Similar results were obtained constraining the gas mass fraction to be constant with time ($w = -1.13^{+0.17}_{-0.20}$ [1110]).

**Alternative models** The halo mass function, as discussed above, is a great tool to investigate the underlying cosmological model. For this reason, several authors studied its behavior in MG models, such as the normal branch of Dvali–Gabadadze–Porrati theory (nDGP) [1111] and $f(R)$ [1112] simulations. To model the HMF in MG scenarios, N-body simulations are required to calibrate the difference between the HMF in the MG simulations with respect to $\Lambda$CDM. Ref. [1111] studied the evolution of perturbations for the nDGP model, which is characterized by a time-varying gravitational constant. The numerical nDGP mass
function predicts up to three times more objects at very high masses, for strong deviations from ΛCDM, explained by the fact that the halo mass function is very sensitive to $\sigma_8$.

A similar approach, but which combines the benefits of the previous ones is that used by Refs. [1112,1113] to study the halo mass function in $f(R)$ models. The authors parameterize the modifications induced by $f(R)$ models both in the spherical collapse model and in the HMF, calibrating these modifications on N-body simulations. These modifications are a function of mass, and redshift. Finally, Ref. [1114] showed that the halo mass function is universal when expressed in terms of $\ln(\sigma^{-1})$ rather than the halo mass M.

### 2.2.2.5. Other cosmological probes from galaxy clusters: cluster clustering.
The spatial distribution of galaxy clusters represents a valuable cosmological probe, depending on the composition of the Universe and on the nature of gravitational interaction. In particular, cluster clustering traces the growth rate of density perturbations, therefore it has been widely employed for inferring fundamental cosmological parameters, such as the matter density, $\Omega_{m,0}$, and the amplitude of mass fluctuations on scales of $8h^{-1}$ Mpc, $\sigma_8$ [1115–1118], as well as the neutrino properties [1119] and deviations from the ΛCDM model, possibly described by the DDE parameters, i.e., $w_0$ and $w_a$ [1120]. Moreover, it can be used to significantly improve the precision on the constraints from cluster weak gravitational lensing [1115] and number counts [1120], when analyzed in combination with them.

Despite the limit in the exploitation of the clustering properties due to the difficulty in collecting large homogeneous cluster samples, which leads to poorer statistics with respect to galaxy samples, cosmology with galaxy clusters presents a series of relevant advantages. First, galaxy clusters, being placed at the nodes of the filamentary structure of the cosmic web, are hosted by the latest and most massive virialized haloes formed by the hierarchical growth of cosmic structures. Therefore they are highly biased tracers, more clustered than galaxies [1018, 1121, 1122]. While, in general, the galaxy bias is difficult to model properly, especially on small physical scales, and it is usually seen as a nuisance parameter in cosmological analyses based on galaxy statistics, the effective bias of galaxy cluster samples can be theoretically predicted [1117, 1118, 1123, 1124], depending on the mass and the redshift of the hosting haloes [1125]. In turn, the cluster mass, mainly consisting of DM, can be linked to directly observable mass proxies, through the so-called mass-observable scaling relation [1018, 1104, 1126, 1127]. However, the uncertainties on scaling relation parameters and their redshift evolution affect our ability to recover the true cluster mass and represent one of the limits of cosmological inference from galaxy clusters. This highlights the importance of having a robust WL mass calibration, capable of assessing the systematics caused by halo orientation, selection, and projection effects [1040,1081]. A second advantage is that galaxy clusters present lower peculiar velocities with respect to their host galaxies, reducing the small-scale distortions in the clustering signal, namely the Fingers of God effect, caused by nonlinear dynamics and incoherent motion within virialized structures [1128,1129].

Cluster clustering is traditionally investigated through the analysis of the two-point correlation function, $\xi(r)$, which measures the excess probability of finding a pair of objects in the volume elements $\delta V_1$ and $\delta V_2$, at the comoving separation $r$, relative to that expected from a random distribution. Increasing attention is being paid to higher-order statistics, and in particular, to the three-point correlation function, $\zeta(r_{12}, r_{13}, r_{23})$, which provides the probability of finding triplets of objects at comoving separations $r_{12}$, $r_{13}$, and $r_{23}$, and has been successfully applied for the detection of the BAO peak of galaxy clusters [1130]. These probes require the assumption of a fiducial cosmology to convert the observed angular and redshift coordinates into distances, which might be different from the true one, resulting in geometric distortions [1131], which have to be taken into account in the modeling

of the clustering signal [1132,1133]. In recent years, the two-point redshift-space correlation function of a homogeneous sample of X-ray selected galaxy clusters from the XXL survey, carried out by the XMM-Newton satellite, has been used to measure a value of $\Omega_{m,0}$ in full agreement with the ΛCDM model [1134]. On the other hand, optically selected clusters from the Sloan Digital Sky Survey (SDSS) have been employed to get an estimate of $f\sigma_8$, which was found consistent with General Relativity predictions [1116], to determine the distance-redshift relation, $\Omega_{m,0}$ and $H_0$, from the BAO peak [1128,1135] and, in combination to stacked gravitational lensing, to constrain $\sigma_8$ [1115]. Moreover, from the 3D clustering of the Planck SZ selected galaxy clusters, the Planck mass bias, $b_{SZ}$, and $\Omega_{m,0}$, have been derived, while $\sigma_8$ was not constrained [1136]. The tomographic cluster autocorrelation from the Constrain Dark Energy with X-ray (CODEX) sample, after redshift-dependent richness selection, has been employed to explore the $\Omega_{m,0} - \sigma_8$ degeneracy, putting constraints on the structure growth parameter, $S_8$ [1137]. Since cluster clustering is particularly sensitive to these parameters, independent constraints on $S_8$ have been put by measuring the two-point correlation function of the photometric clusters detected in the third data release of KiDS-DR3 [1117]. An equivalent approach consists in the study of the angular correlation function, $w(\theta)$, and its spherical harmonic counterpart, the angular power spectrum, $C_\ell$ [1138]. In particular, a first measurement of the auto-correlation cluster power spectrum and of the galaxy-galaxy cluster cross-spectrum has been performed on the SDSS-DR8 photometric sample [1124]. Moreover, from the tomographic study of $w(\theta)$ and $C_\ell$ of the same KiDS-DR3 catalog, competitive constraints on $S_8$ have been found [1118]. These results, being based on angular positions alone, are not affected by geometric distortions, highlighting the potential of angular cluster clustering as a cosmological probe in future photometric redshift surveys.

### 2.2.3. Galaxy clustering — Other probes

**Coordinator:** Chandra Shekhar Saraf

**Contributors:** Kishan Deka, and PawełBielewicz

#### 2.2.3.1. Cross-correlations.
The $S_8$ parameter can be measured from the redshift space clustering of galaxies by estimating the two- and three-point correlation functions or their Fourier counterparts, the galaxy power spectrum, and bispectrum. These statistics can be used to gain insights into the growth rate of cosmic structures, $f\sigma_8(z)$, from RSD at the effective redshift of the galaxy sample [1139–1141]. The galaxy clustering data can be combined with CMB surveys to yield constraints on $\sigma_8$ and $S_8$. Several studies [1142–1145] have reported values of $\sigma_8$ consistent with the CMB-only analyses. An illustrative schematic of estimating $\sigma_8$ parameter from cross-correlation measurements between Planck CMB lensing convergence map and DESI's Legacy Imaging Survey (DESI-LIS; [1146]) is shown in Fig. 38 (see Ref. [1147] for details on cross-correlation analysis). A cross-correlation between unWISE galaxies and ACT DR6 CMB lensing measurements found $S_8 = 0.813 \pm 0.021$, and $S_8 = 0.810 \pm 0.015$ with a combination of Planck and ACT cross-correlations [1144]. These measurements are fully consistent with the predictions from primary CMB measurements within our standard cosmology. Another study correlating Gaia-unWISE QSO catalog with Planck PR4 CMB lensing found $\sigma_8 = 0.7766 \pm 0.034$ and $\Omega_{m,0} = 0.343^{+0.017}_{-0.019}$, which translates to $S_8 = 0.819 \pm 0.058$ [1143]. However, many cross-correlation analyses between CMB datasets and galaxy samples divided into narrow redshift bins [1148–1155] find low values of $\sigma_8$ and $S_8$ parameter (when $\Omega_{m,0}$ is generally assumed to be consistent with the well constrained values from SN and CMB analyses), resulting in mild tension with Planck analysis. A tomographic cross-correlation of DES-Y3 data and CMB lensing from SPT and Planck estimated $S_8 = 0.734^{+0.035}_{-0.028}$ [1149]. Another cross-correlation of DESI Legacy Imaging survey LRGs and CMB lensing from Planck found $S_8 = 0.765 \pm 0.023$ and $S_8 = 0.790^{+0.024}_{-0.027}$ with ACT DR6 [1156].
<figure><img src="image_44.png" /><figcaption>**Fig. 38.** An illustration of estimating $\sigma_8$ parameter from tomographic cross-correlation measurements between DESI-LIS galaxy survey and Planck lensing convergence map, assuming $\Lambda$CDM model [1157]. (a) The Planck lensing convergence map and DESI-LIS galaxy density maps in four tomographic bins, smoothed with a Gaussian beam of 60' FWHM (only to better show the large scale distribution). (b) The galaxy redshift distribution for the four redshift bins (solid blue lines) and the CMB lensing kernel (solid red line). The vertical dashed orange lines mark the boundaries of redshift bins. (c) The $\sigma_8$ parameter estimated from cross-correlation measurements of four tomographic bins. The red line is the evolution of $\sigma_8$ parameter as estimated from cross-correlation data. The dashed black line is the evolution of $\sigma_8$ parameter measured from the Planck CMB data alone. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

### 2.2.3.2. Full shape analyses.
The $\sigma_8$ parameter can also be measured by fitting the full shape of the observed power spectrum from spectroscopic surveys as well as the galaxy bispectrum [721,1158-1162]. Furthermore, the $\sigma_8$ parameter can be translated into $S_8$ parameter with additional measurements of $\Omega_{m,0}$ (from power spectrum shape, SN or CMB). The measurements of $S_8$ parameters employing full shape analysis of the galaxy power spectrum [1158,1162] are in tension with Planck measurements [192], but with typically less than $3\sigma$ statistical significance. Recent constraints on the $S_8$ parameter from the BOSS galaxy bispectrum multipoles found $S_8 = 0.774^{+0.056}_{-0.053}$ [721] and $S_8 = 0.77 \pm 0.04$ [1161], which are statistically consistent with both CMB and WL measurements.

### 2.2.3.3. 3 × 2 pt analyses.
The combined analysis of two-point statistics corresponding to galaxy clustering, galaxy-galaxy lensing, and cosmic shear is commonly known as the “3 × 2 pt” analysis. The joint analyses provide us with an improved control of the systematic uncertainties such as galaxy bias and intrinsic alignments. The first 3 × 2 pt analysis were published with the datasets from KiDS-450 × {2dFLenS + BOSS} [1163], KiDS-450 × GAMA [1164] and DES-Y1 [1165]. The DES-Y1 analysis used photometric galaxies as both lens and sources, finding $S_8 = 0.773^{+0.026}_{-0.020}$. The KiDS collaboration, on the other hand, used overlapping imaging and spectroscopic surveys, resulting in $S_8 = 0.742 \pm 0.035$ when combined with 2dFLenS + BOSS, and $S_8 = 0.800^{+0.029}_{-0.027}$ with GAMA survey.

The updated constraints on $S_8$ with 3 × 2 pt analyses combining KiDS-1000 with {2dFLenS + BOSS} datasets gives $S_8 = 0.766^{+0.020}_{-0.014}$ [874], at $3.1\sigma$ tension with Planck constraints within the $\Lambda$CDM model. The recent measurements from the DES-Y3 datasets resulted in $S_8 = 0.776 \pm 0.017$ in the $\Lambda$CDM cosmology and $S_8 = 0.775^{+0.026}_{-0.024}$ assuming a wCDM model [869]. However, a joint analysis of DES-Y3 3 × 2 pt data, Planck CMB anisotropy data (without lensing), eBOSS BAO and RSD measurements, and DES SNIa data provides $S_8 = 0.812 \pm 0.008$ in the $\Lambda$CDM model. A joint cosmic shear analysis of KiDS-1000 and DES Y3 data has obtained stringent constrain of $S_8 = 0.790^{+0.018}_{-0.014}$ [867]. Another 3 × 2 pt analyses performed with the HSC-Y3 imaging data and SDSS BOSS DR11 spectroscopic galaxies yielded $S_8 = 0.775^{+0.043}_{-0.038}$ for

the $\Lambda$CDM model [872]. The HSC-Y3 analysis is highly consistent with both KiDS-1000 and DES-Y3 measurements of $S_8$ within one standard deviation.

### 2.2.3.4. Systematics.
The measurements of cosmological parameters from large-scale structure evolution at low redshifts require state-of-the-art analysis pipelines to correct for a large number of systematic effects. The observation sector can include effects such as atmospheric and extragalactic extinction, stellar contamination, airmass and blurring [1166], fiber collisions [1167], angular and radial modes systematics [1168].

For the cross-correlation between galaxy clustering and CMB, galaxy survey systematics such as photometric calibration errors, foreground contamination and catastrophic errors need to be modeled and mitigated for unbiased inferences [1152,1155,1169-1171]. In order to allow tomographic cross-correlations with photometric galaxy surveys, both precise individual galaxy photometric redshifts and accurate redshift distributions for the ensemble, require techniques for redshift determination. Typically, a subset of spectroscopically observed sources are used to train, validate and calibrate photometric redshifts and redshift distributions. The accuracy of the redshift distribution directly impacts the accuracy of estimated cosmological parameters. Biases in the mean redshifts of the tomographic bins will cause a bias in $S_8$. Recently, [1147,1157] have shown the importance of precise modeling of redshift error distributions and redshift bin mismatch of objects, when estimating $\sigma_8$ and $S_8$ parameters.

In the theory sector, the largest uncertainty comes from the accuracy of the models for non-linear clustering of galaxies. The cross-correlation measurements, full shape analyses, and the bispectrum multipoles, all depend on the accurate modeling of the non-linear regime in the redshift-space galaxy clustering. This uncertainty has been improved after the significant progress from the effective field theory of the large-scale structure which provides accurate description of galaxy clustering at large scales [719,722,1172-1180].
## 2.3. Other challenges

### 2.3.1. Systematics and the $A_{\text{lens}}$ parameter

**Coordinator:** William Giarè

**Contributors:** Alessandro Melchiorri, Anto Idicherian Lonappan, Deng Wang, Elsa M. Teixeira, Enrico Specogna, Giulia Gubitosi, Ido Ben-Dayan, Matteo Forconi, Özgür Akarsu, and Rita B. Neves

On scales smaller than ten arcminutes, the interaction between CMB photons and the Universe's large-scale structure becomes significant, giving rise to second-order anisotropies. One of the most important contributions is the gravitational deflection, or lensing, experienced by CMB photons along their paths [1181]. At arcminute scales, lensing deflections distort the observed image of the CMB fluctuations, imprinting a distinctive non-Gaussian four-point correlation function (or trispectrum) in both the temperature and polarization anisotropies [1181]. This signal can be extracted with a high signal-to-noise ratio by correlating power in different directions on the sky, leading to a direct reconstruction of the power spectrum of the CMB lensing field, now measured at ~40σ [694,695,1182,1183].

Although gravitational lensing does not alter the overall distribution of primary CMB anisotropies, it leaves distinctive signatures in the spectra of temperature and polarization anisotropies. The most relevant effects are the conversion of E-mode polarization to B-modes, the transfer of power to the damping tail, and the lensing-induced smoothing of the acoustic peaks and troughs in the TT, TE, and EE spectra. The degree of smoothing is directly tied to the amplitude of the power spectrum of the lensing potential, which is derived from the six ΛCDM parameters. As a result, interesting consistency tests have been proposed to verify whether the amplitude of lensing inferred through the high-$\ell$ smoothing of acoustic peaks matches the theoretical predictions of the standard cosmological model.

In Ref. [724], the phenomenological parameter $A_{\text{lens}}$ was introduced to encompass various physical mechanisms that might affect the lensing amplitude. This parameter scales the amplitude of the lensing trispectrum, effectively accounting for the associated smoothing effects in the CMB temperature and polarization spectra. $A_{\text{lens}}$ is defined such that $A_{\text{lens}} = 1$ matches the standard ΛCDM prediction, whereas $A_{\text{lens}} = 0$ corresponds to a scenario where CMB lensing is disregarded altogether. By allowing $A_{\text{lens}}$ to remain a free parameter in theoretical models, its value can be directly constrained by data, potentially confirming or deviating from the ΛCDM predictions.

Focusing on the Planck-2018 data release (PR3), the analysis of the Planck plik likelihood for the TT, TE, and EE spectra at $\ell > 30$, combined with the Commander likelihood for the TT spectrum at $2 \le \ell \le 30$ and the SimAll likelihood for the EE spectrum at $2 \le \ell \le 30$, yields $A_{\text{lens}} = 1.180 \pm 0.065$ [192]. This finding suggests an excess lensing signal at approximately $2.8\sigma$, resulting from a significant improvement in $\chi^2$ of approximately $\Delta\chi^2 \sim 9.7$. This improvement primarily originates from high-$\ell$ temperature and polarization data, particularly within the multipole range $600 < \ell < 1500$. As shown in Fig. 24 of Ref. [192], there is a visible preference for increased lensing smoothing in the oscillatory residuals in the TT spectrum at $1100 < \ell < 2000$.

Since 2018, the Planck data has undergone substantial reanalyses. The new Planck PR4 (NPIPE) CMB maps incorporate significant improvements, such as increased coverage of sky area at high frequencies, improved processing of time-ordered data, and approximately 8% more data accounted for in the lensing trispectrum reconstruction. Updated likelihoods for temperature and polarization spectra have been released following these developments. The latest versions of CamSpec [700] and HiLLiPoP [701] — two likelihoods already employed in various studies by the Planck collaboration — are now based on the Planck PR4 (NPIPE) maps, reducing small-scale noise compared to plik and enhancing the constraints on cosmological parameters by up to 10%. Both likelihoods indicate a shift toward $A_{\text{lens}} = 1$. For CamSpec, the

constraints on the lensing amplitude inferred from high-$\ell$ smoothing of acoustic peaks are summarized in Table 6 of Ref. [700]. Temperature and polarization data reduce the preference for $A_{\text{lens}} > 1$ to less than $1.7\sigma$ while focusing solely on the TT spectrum increases the lensing anomaly to $2.3\sigma$. Constraints on $A_{\text{lens}}$ resulting from the new HiLLiPoP likelihood are summarized in Table 6 of Ref. [701]. They consistently agree with ΛCDM at $1\sigma$.

These new reanalyses suggest that the preference for excess power smoothing has decreased in the NPIPE maps, lending weight to interpreting the lensing anomaly as systematics in the Planck PR3 data. The latter interpretation finds support in CMB experiments other than Planck. ACT and the SPT have released precise small-scale measurements of the spectra of temperature and polarization anisotropies [704, 725, 732], as well as precise reconstructions of the lensing trispectrum [695,1183,1184]. The Data Release 4 of the ACT constrains $A_{\text{lens}} = 1.01 \pm 0.11$ in remarkable agreement with the baseline value (see Fig. 16 of Ref. [702]). Similarly, the SPT TT, TE, and EE spectra analysis gives $A_{\text{lens}} = 0.87 \pm 0.11$, consistent with ΛCDM at $1.2\sigma$. In this case, we refer to Table V of Ref. [704] and Table VII of Ref. [725] for earlier analyses involving only the EE and TE spectra (yielding $A_{\text{lens}} = 0.98 \pm 0.12$).

Fig. 39 summarizes the 68% CL intervals for the parameter $A_{\text{lens}}$, inferred over the years from the analysis of lensing-induced smoothing of the acoustic peaks and troughs in the TT, TE, and EE spectra measured by various experiments and likelihoods discussed so far.

### 2.3.2. Evidence for a nonvanishing $\Omega_k$

**Coordinator:** Will Handley

The spatial curvature of the Universe, parameterized by $\Omega_k$, is a fundamental parameter in cosmology, intimately tied to the geometry and fate of the Universe. While the inflationary paradigm predicts a value extremely close to zero, it is important to observationally test this prediction. In the standard cosmological model, $\Omega_k$ is degenerate with other parameters when using CMB data alone, such that CMB data do not provide strong constraints on curvature. However, the addition of external data sets, such as BAO measurements, can break this degeneracy.

The Planck 2018 data release presented an intriguing hint of a possible departure from a flat universe. Analyzing the TT,TE,EE+lowE data, the Planck team found a preference for negative values of $\Omega_k$ in the range $-0.095 < \Omega_k < -0.007$ at 99% confidence level, with $\Delta\chi^2_{\text{eff}} = -11$ compared to the baseline ΛCDM model [192]. This preference was attributed to a combination of volume effects and better fits both the high-$\ell$ and low-$\ell$ data in closed models. Notably, closed models predict a higher lensing amplitude ($A_L$), which can explain the preference for $A_L > 1$ in Planck data, and also a better fit to the low-$\ell$ temperature power spectrum. Adding Planck lensing data to TT,TE,EE+lowE, the constraint on $\Omega_k$ shifts to $\Omega_k = -0.0106 \pm 0.0065$ at 68% confidence level, reducing the preference for closed models to less than $2\sigma$. Finally, including BAO data results in $\Omega_k = 0.0007 \pm 0.0019$, consistent with a flat universe at $1\sigma$ [192].

However, this conclusion has been challenged in the literature. Ref. [1185] and Ref. [1186] show that there is Bayesian evidence for a closed universe, albeit not quantified in the Planck paper, and that the Planck lensing reconstruction is in tension with TT,TE,EE+lowE data at 2.5$\sigma$. In particular, the tension is driven by the preference for a higher value of $A_L$ in closed models, which is not supported by the Planck lensing reconstruction. They argue that BAO data are also in tension with TT,TE,EE+lowE data, and that combining these datasets is not justified. Ref. [1187] conversely shows that the preference for $\Omega_k < 0$ is reduced when using the CamSpec likelihood, which uses more data than the Planck baseline likelihood, and argue that BAO data strongly prefer flat universes.

More recently, Ref. [1188] showed that the BAO scale measurements implicitly assume a fiducial ΛCDM cosmology, and that the
<figure><img src="image_45.png" /><figcaption>**Fig. 39.** The whisker plot summarizes the 68% CL intervals for the parameter $A_{\text{lens}}$, inferred over the years from the analysis of lensing-induced smoothing of the acoustic peaks and troughs in the TT, TE, and EE spectra measured by various experiments with different likelihoods. Constraints based on the Planck-PR3 measurements indicate an excess of lensing, which is reduced in the recent Planck-PR4 updated likelihoods. Small-scale temperature and polarization spectra from ACT-DR4 and SPT-3G are in overall agreement with $A_{\text{lens}} = 1$, consistently recovered within one standard deviation (or slightly more).</figcaption></figure>

preference for a flat universe can be weakened when relaxing these assumptions. They find that using the full shape of the galaxy power spectra, the tension between BAO and Planck data in the $\Omega_k$ parameter is reduced to ~1.5σ.

The tension between Planck CMB data and BAO measurements in the context of non-vanishing curvature has motivated numerous works that explore its possible origin and its implications for cosmology. For example, Ref. [1189] used CC as an alternative dataset to break the geometric degeneracy, finding $\Omega_k = -0.0054 \pm 0.0055$, consistent with a flat universe. However, the tension between different $H_0$ measurements persists even in the context of non-flat models [1190–1192].

The curvature tension therefore remains an open problem in cosmology. Future observations, such as those from DESI, Euclid, and the Vera Rubin Observatory will provide more precise BAO measurements, which will allow us to test the robustness of the flat Universe assumption and potentially shed light on the origin of this discrepancy.

### 2.3.3. Anisotropic anomalies in the cosmic microwave background radiation

**Coordinator:** Leandros Perivolaropoulos

**Contributors:** András Kovács, Anto Idicherian Lonappan, Emanuela Dimastrogiovanni, Eoin Ó Colgáin, Frode K. Hansen, Giulia Gubitosi, Laura Mersini-Houghton, Marina Cortês, Nils A. Nilsson, Shahin Sheikh-Jabbari, and Venus Keus

The CMB, a relic from the early Universe, provides crucial insights into cosmology. Observations by COBE, WMAP, and Planck have revealed several anisotropic anomalies that may challenge the prevailing $\Lambda$CDM cosmological model and the cosmological principle, prompting a reevaluation of our understanding of the early Universe's structure and the inflationary paradigm [1193,1194].

One such anomaly is the unusually low quadrupole moment ($\ell = 2$) in the CMB power spectrum, which deviates from the predictions of cosmic variance and the $\Lambda$CDM model [726,1195]. This anomaly has implications for inflationary cosmology and may require theoretical adjustments. The low quadrupole moment was first observed by COBE and later confirmed by WMAP and Planck, with increasing precision. The statistical significance of this anomaly has been thoroughly investigated, and its persistence across multiple observations challenges our understanding of the primordial power spectrum and the inflationary paradigm. Theoretical efforts to explain the low quadrupole moment have included modifications to the inflation model, such as a running

spectral index or a cutoff in the primordial power spectrum at large scales [1196,1197].

Another striking feature is the alignment of the quadrupole and octopole moments, which appears to be oriented with the ecliptic plane and the motion of the Solar System [1198,1199]. This alignment violates statistical isotropy and suggests potential cosmological or local explanations. The quadrupole–octopole alignment was first reported using WMAP data and later confirmed by Planck, with both missions providing strong evidence for its existence. The alignment has been studied extensively, with some proposing that it could be a signature of cosmic topology [1200], while others have investigated the possibility of local foreground contamination or systematic effects [1201,1202]. The statistical significance of this alignment and its potential origins remain active areas of research.

The upper and lower panels of Fig. 40 illustrate the derived quadrupole and octopole moments, respectively. The alignment of these moments with the ecliptic plane highlights one of the key anisotropic anomalies in the CMB. This alignment suggests a possible violation of statistical isotropy and indicates that there may be underlying cosmological or local causes for this observed pattern. Understanding the significance and origins of these alignments is crucial for interpreting the implications for the standard cosmological model and exploring potential new physics.

The CMB also exhibits a hemispherical power asymmetry, known as the dipolar power distribution, where one half of the celestial sphere has significantly more temperature fluctuations than the other [1204,1205]. The scale and statistical significance of this asymmetry raise questions about the density variations in the early Universe. The hemispherical power asymmetry was first detected in WMAP data and later confirmed by Planck. The hemispherical asymmetry can be considered as two different anomalies as described in detail in Ref. [1205]. For larger scales, $\ell < 100$, the asymmetry can be modeled as a dipolar modulation of an isotropic field. The p-value of this modulation asymmetry has been reported as 0.1–1%, but considerably less significant when correcting for the choice of $\ell_{\text{max}}$. The second hemispherical asymmetry, the angular clustering asymmetry, extends to much smaller scales. By estimating the CMB power spectrum locally in different parts of the sky, maps of the spatial distribution of the power spectrum can be estimated for different multipole ranges. When estimating a dipole for these local power spectrum maps at different scales, these dipoles are found to cluster. This clustering is found to persist at least to $\ell = 1000$ and only 1/1000 simulations show similar clustering. The clustering direction is close to the direction of the large scale dipolar asymmetry. The origin of
<figure><img src="image_46.png" /><figcaption>**Fig. 40.** Upper Panel: Derived quadrupole moment from the Planck SMICA map, showing the preferred axis of the quadrupole. Lower Panel: Derived octopole moment, indicating the alignment with the quadrupole. These alignments suggest a significant anomaly in the CMB, challenging the assumption of statistical isotropy. The plus and star symbols indicate the axes of the quadrupole and octopole, respectively, around which the angular momentum dispersion is maximized. The diamond symbols correspond to the quadrupole axes after correction for the kinematic quadrupole. From Ref. [1203].</figcaption></figure>

this asymmetry remains unclear, with some proposing that it could be a signature of non-Gaussianity in the primordial perturbations [1206–1208], while others have explored the possibility of a super-horizon scale mode modulating the primordial power spectrum [1209].

Additionally, there is a parity asymmetry in the CMB, with a preference for odd-parity modes over even-parity modes, challenging the scale invariance of primordial fluctuations [1210,1211]. The parity asymmetry was first reported using WMAP data and later confirmed by *Planck*, with both missions providing evidence for a statistically significant difference between the power in odd and even multipoles. This asymmetry has been studied in the context of the primordial tensor-to-scalar ratio, with some suggesting that it could be a signature of chiral gravity [1212,1213]. The physical origin of this asymmetry and its implications for the inflationary model remain open questions.

The Cold Spot is another intriguing anomaly — a large, unusually cold region in the CMB, surrounded by a hot ring [1214–1216]. Both the WMAP and *Planck* missions identified it as a significant deviation from the Gaussian random field expectation (~3σ), and its region has been studied extensively. To explain it, novel cosmological mechanisms [1217,1218], a cosmic texture [1219], a large void in the line-of-sight [1220], and a systematic effect [1214,1221] have all been investigated. Importantly, the *Eridanus supervoid* (*R* ≈ 200 h⁻¹Mpc, δ ≈ −0.2) was detected at *z* < 0.3 aligned with the Cold Spot, based on galaxy counts from multiple surveys [1222,1223]. This evidence is supported by reconstructions of the cosmic velocity field [1224] and WL analyses by the DES team [1225], suggesting a causal relation between these individually rare objects in the CMB and in the cosmic web. Yet, the statistical significance of the Cold Spot and its potential origins remain active areas of research.

Another very significant CMB anomaly, which potentially could give rise to many of the above mentioned anomalies, is the cooling of CMB

photons around nearby galaxies. First discovered in Ref. [1226], the stacking of CMB temperatures in areas around the halos in nearby *z* < 0.015 late type spiral galaxies showed lower CMB temperatures compared to elsewhere on the sky. This was followed up in Ref. [1227] where it was shown that an attempt at modeling the discovered cooling of CMB photons around galactic halos gives rise to several of the observed anomalies. In particular, the hemispherical asymmetries and the cold spot arise naturally in such a scenario with the correct directions on the sky. Correlations are found between the largest scales of the CMB and the nearby galaxy distribution obtained from the 2MRS galaxy catalog [1228]. In Ref. [1221] a detailed study of the galaxies in the cold spot areas showed that the shape and size of the cold spot to a large degree could be explained by cooling of the galaxies in this area. In particular, the nearby large Eridanus group of galaxies is located at this position. In Ref. [1229], the cooling was found to correlate to the nearby cosmic density field. In Ref. [1230], the cooling is detected at the 5.7σ significance level when looking at galactic halos in the most massive nearby cosmic filaments. Even when correcting for the lookelsewhere-effect by allowing for different choices of galaxy properties, the detection is stronger than in any of 10.000 simulated CMB skies. Finally, Ref. [1231] shows that by masking the affected area on the sky, the estimate of the cosmological parameters is not significantly altered. The origin of the cooling of CMB photons in galactic halos is unknown, but in Ref. [1230], it is speculated whether DM could be involved.

The statistical independence and potential interconnections between these anomalies are crucial areas of investigation. For example, the relationship between the low quadrupole moment and the lack of large-angle correlations in the CMB is of particular interest [1232,1233]. The lack of large-angle correlations was first reported using COBE data and later confirmed by WMAP and *Planck*, with all three missions providing evidence for a significant deficit of correlations on angular scales greater than ~60°. This anomaly has been studied in the context of cosmic topology [1234,1235], with some proposing that it could be a signature of a non-trivial topology of the Universe. The relationship between the lack of large-angle correlations and the low quadrupole moment has also been investigated, with some suggesting that they may share a common origin [1236].

Collectively, these anomalies pose a significant challenge to the standard model of cosmology. While some of these anomalies may be the result of foreground contamination or systematic effects, their persistence across multiple observations and their statistical significance suggests that they may have a cosmological origin. The standard ΛCDM model, which assumes a flat, homogeneous, and isotropic Universe, struggles to explain these anomalies, and their existence may require modifications to the model or the development of new theoretical frameworks.

Upcoming observational missions, such as the *Euclid* space telescope and next-generation ground-based CMB experiments, have the potential to shed new light on these anomalies [437,682]. These missions will provide unprecedented sensitivity and angular resolution, allowing for a more detailed study of the CMB and its anomalies. In particular, the *Euclid* mission will provide detailed measurements of the large-scale structure of the Universe, which can be used to test models of the early Universe and the origin of the CMB anomalies. Next-generation CMB experiments, such as CMB-S4, will provide a significant improvement in sensitivity and angular resolution, allowing for a more detailed study of the CMB power spectrum and its anomalies.

Several theoretical frameworks and models have been proposed to explain these anomalies, such as the role of topology in large-angle CMB correlations [1235], the impact of a cosmological GW background [1237], and the effects of superhorizon isocurvature DE [1209]. Other models, including anisotropic k-essence [1238], loop quantum cosmology [1239], non-canonical anisotropic inflation [1240], and unexpected topology of temperature fluctuations [1234], have also been explored. These models aim to provide a theoretical framework that
can account for the observed anomalies while maintaining consistency with other cosmological observations.

The development of new theoretical models and advanced simulations will also be crucial in understanding the early Universe's complexities. Theoretical efforts to explain the CMB anomalies will require a deep understanding of the physics of the early Universe, including the dynamics of inflation, the generation of primordial perturbations, and the evolution of the Universe in the presence of DE and DM. Advanced simulations, such as those performed with the Planck Sky Model [1241], will be essential in understanding the impact of foreground contamination and systematic effects on the observed CMB anomalies.

In the early 2000's in the program of investigation of the selection of the initial conditions of our Universe [1242–1245], authors proposed to allow the wave-function of the Universe to propagate on the landscape of string theory and used quantum cosmology to derive the probability of our origin. This was the first work where the answer was derived from an underlying fundamental theory. It showed that in contrast to previous beliefs, the most likely Universe to spontaneously come into existence is the one that starts at very high energy. The authors proposed to use the quantum entanglement between branches of the wave function as a way to test the theory and, for the first time to have a handle to test the existence of the quantum multiverse [1218,1246]. Traces of earlier entanglement are imprinted as anomalies in our sky. The authors calculated and predicted a series of seven anomalies, including the Cold Spot, power asymmetry, alignments, and suppression of power in lowest multipoles, etc., all of which are by now observed. Status of the predicted anomalies against observations showed perfect agreement [1247–1250].

As cosmology continues to evolve, the anisotropic anomalies in the CMB serve as a reminder of the importance of critically examining our models and assumptions, driving us to deepen our understanding of the cosmos. The existence of these anomalies suggests that our current understanding of the early Universe and the origin of cosmic structure may be incomplete, and that new theoretical frameworks and observational efforts may be necessary to fully explain the observed features of the CMB. As we continue to study these anomalies and their implications for cosmology, we may uncover new insights into the fundamental nature of the Universe and the physical processes that shaped its evolution.

### 2.3.4. Hints of dynamical dark energy in DESI baryon acoustic oscillations and beyond

**Coordinator:** William Giarè

**Contributors:** Anton Chudaykin

DESI has recently released data from its first (DR1) [699,745,783, 809,1251] and second (DR2) [717,1252–1254] year of BAO measurements, based on observations of tens of millions of extragalactic objects, including galaxies, QSOs, and Lyman-α forest tracers. These DESI BAO observations provide precise constraints on the transverse comoving distance, the Hubble rate, and their combination (all relative to the sound horizon at the drag epoch) across seven redshift bins in the range $0.1 < z < 4.2$.

One of the most notable results from DESI BAO observations concerns the nature of DE. As initially highlighted by the DESI collaboration's DR1 results [699,1251] and recently confirmed by DR2 results [717,1255], the combination of DESI BAO with CMB data from the Planck satellite and SNIa distance moduli measurements from three independent SNIa samples (i.e., the Pantheon-plus catalog [32, 33], the Union3 compilation [1256], and five-year observations from DESy5 [1257–1259]) provides moderate to strong evidence for a time-evolving DE component, commonly referred to as DDE. One important aspect of this preference is that the data indicates a phantom crossing scenario, in which the dark energy equation of state crosses the $w = -1$ [1260]. While this behavior presents theoretical challenges for most

simple scalar-field models, it can be realized in multi-field scenarios, alternative models of gravity [1261–1264]. The DESI preference for an evolving DE has been explored in the context of general scalar-tensor Horndeski theories [1265–1268] and in modified gravity models with non-minimal coupling [1266,1269].

Given the potential impact of these results on our understanding of the Universe, caution is essential, and thoroughly testing the robustness of these findings is of paramount importance. Unsurprisingly, a significant portion of the cosmology and high-energy physics community has actively engaged with this issue, clarifying and/or bringing up several aspects and concerns surrounding this preference toward DDE. In the following, we review some of the key results that have emerged from the consistency checks conducted over the past few months, highlighting both the strengths and weaknesses of the observed preference, while stressing aspects that warrant further clarification.

* **Parameterization of the Equation of State – Barring, for the** moment, any potential systematic issues in the different datasets involved in the analyses, a first key aspect that has undergone significant cross-checking [809,1269–1281] is the parameterization used to describe the DE EoS. Originally, the DESI collaboration parameterized the time evolution of the EoS using the linear CPL form, $w(a) = w_0 + w_a(1 - a)$, arguing that various combinations of data indicate a consistent preference for a present-day quintessence-like EoS ($w_0 > -1$) that crosses the phantom barrier ($w_a < 0$). Specifically, within this parameterization, DESI DR1 BAO measurements, when combined with Planck CMB and DESy5 SNIa, yield a preference for DDE at a significance level of ~3.9σ. This preference decreases to ~2.5σ (~3.5σ) when replacing DESy5 with Pantheon-plus (Union3) [699]. Notably, within the same CPL parameterization, these results remain stable when substituting DESI DR1 BAO with the latest DR2 DESI measurements. The latter not only confirms the preference for DDE but also consistently strengthens it by approximately ~0.3σ across all dataset combinations with BAO and SNIa. Although the CPL parameterization has been shown to match the background evolution of distances arising from the exact DE equations of motion with about 0.1% accuracy for viable cosmologies across a broad range of physics (including scalar fields, MG, and phase transitions see, e.g., Refs. [298,1282]) alternative parameterizations that deviate from CPL at both $z \ll 1$ and $z \gtrsim 1$ remain consistent with current observations. In Ref. [1283], it was argued that assuming the CPL parameterization is not the primary driver of this preference. When combining CMB, DESI BAO, and SNIa measurements within various EoS parameterizations, $w_0$ consistently remains in the quintessence regime, while the constraints on $w_a$ indicate a preference for a dynamical evolution crossing into the phantom regime (see also Ref. [1280]). This result holds for both DESI DR1 and the latest DESI DR2 BAO measurements, as confirmed by the DESI collaboration in Ref. [1255]. In this sense, the preference is to be considered robust against different models.

* **DESI BAO measurements (and beyond)** – Since the preference for DDE was first highlighted by the DESI collaboration, the impact of DESI BAO DR1 data on this preference has been carefully scrutinized, with systematic effects in DESI BAO measurements remaining a topic of debate, see, e.g., Refs. [1265,1284–1287]. Notably, the DESI BAO measurement at $z = 0.71$ exhibits a ~3σ tension with predictions from the Planck best-fit ΛCDM cosmology, making it a key driver of several hints for new physics, including (part of) the preference for DDE. However, it is crucial to emphasize that (i) the latest DESI DR2 BAO measurements not only confirm the preference for DDE but also consistently strengthen it by approximately ~0.3σ across all dataset combinations with BAO and SNIa. While this increase in preference from DESI DR1 to DESI DR2 is not highly statistically significant, it definitively reinforces the signal, suggesting that the
<figure><img src="image_47.png" /><figcaption>**Fig. 41.** Observational constraints in the $w_0-w_a$ plane (at 68% and 95% CL) assuming a CPL parameterization for DE across different datasets. The left panel shows constraints from a combination of the Planck Plik likelihood for high-$\ell$ TT, TE, and EE PR3 spectra, the Commander and SimAll likelihoods for low-$\ell$ temperature and polarization (TT and EE), and Planck PR4 NPIPE combined with ACT-DR6 lensing likelihoods, along with DESI DR1 BAO and Pantheon-plus / DESy5 SNIa data. The middle panel examines the impact of replacing DESI BAO with SDSS BAO, while the right panel explores how the constraints shift when Planck is replaced with ACT-DR4 for temperature and polarization spectra and ACT-DR6 for lensing.</figcaption></figure>

initial preference for DDE observed in DESI DR1 may not be merely a statistical fluctuation resulting from a preliminary one-year observation. (ii) In combined analyses of CMB, BAO, and SNIa data, the preference for DDE is not exclusively driven by DESI BAO, whether from DR1 or DR2. In fact, as reported in Ref. [1288], a similar shift toward DDE is observed when replacing DESI BAO measurements with SDSS BAO. Planck CMB combined with SDSS BAO and DESy5 SNIa indicates a preference for DDE at a statistical significance exceeding $2.5\sigma$, see also Fig. 41. Likewise, Planck CMB combined with SDSS BAO and Union 3 points to a DDE component at $\gtrsim 2\sigma$ significance. The only combination where this preference is notably weakened below $2\sigma$ is when SDSS BAO and Pantheon-plus SNIa are analyzed together with Planck CMB. Importantly, across all independent datasets — each showing hints of deviation from a cosmological constant at varying levels of statistical significance — a consistent shift in parameter space is observed in the same direction, namely toward a quintessence-like equation of state in the present epoch that transitions into a phantom-like regime in the past. Overall, while DESI BAO data undoubtedly strengthen the preference for DDE, the other datasets involved in the analysis — most prominently SNIa distance modulus measurements — play an equally significant role in shaping this result. For further discussions on BAO data, we refer to Section 2.1.19.

* **SNIa-** It quickly became clear that SNIa data play a major role in driving the preference for DDE. Excluding DESI BAO and considering data combinations involving Planck CMB and either DESy5 or Union3 (i.e., without including any BAO surveys) already leads to a preference for DDE at ~2–2.5σ (see, e.g., Refs. [1288, 1289]). Given the crucial role of SNIa measurements in this trend, the potential impact of systematic effects has been widely investigated [523,1290–1294], particularly for the DESy5 sample, which exhibits the strongest shift toward DDE. For instance, Ref. [1295] highlighted the influence of SNIa measurements, reporting a cross-correlation between the PantheonPlus and DESy5 supernova samples that suggested a calibration difference of ~0.04 mag between low and high redshifts. Correcting for this offset brings the DESy5 sample into closer agreement with Planck's ΛCDM cosmology. Since the parameter space favored by the uncorrected DESy5 sample diverges from many other cosmological datasets, it has been suggested that the apparent evidence for DDE might primarily arise from systematic effects in DESy5 SNIa. In response to these concerns, the DES collaboration in Ref. [1296] showed that the debated ~0.04 mag offset between Pantheon+ and DESy5 at low and high redshift is partly due

to improvements in the modeling of supernova intrinsic scatter and host galaxy properties in DESy5 (which account for up to ~43% of the offset) and partly (~38%) due to a misleading comparison. The latter arises because different selection functions characterize the DES subsets included in Pantheon+ and DES-SN5YR, leading to differences in individual supernova distance measurements due to distinct bias corrections. Therefore, while SNIa data do play a pivotal role in shaping the preference for DDE, these findings might warrant additional tests to carefully assess potential systematics before drawing definitive conclusions. For further discussions on SNIa data, we refer to Section 2.1.7.

* **Planck CMB data (and beyond):** The final dataset considered in these analyses is the CMB angular power spectra, measured by the Planck Collaboration. While a DDE component primarily affects the late-time evolution of the Universe, it also has a non-negligible impact on the CMB spectra. The most significant effect of DE dynamics on the CMB angular power spectrum is observed in the amplitude of the ISW plateau at very large angular scales. This amplitude is mainly determined by primordial inflationary parameters ($A_s$ and $n_s$) and contributions from the late-time ISW effect, which is sensitive to DE dynamics (see also Section 2.3.9). Specifically, a phantom (quintessence) DE component suppresses (enhances) the decay of gravitational potentials that source the late ISW effect. Therefore, assessing the impact of Planck measurements on the preference for DDE is certainly worthwhile. As discussed in Ref. [1297], Planck data — particularly the large-scale temperature and E-mode polarization measurements — play a crucial role in reinforcing the preference for DDE reported by the DESI collaboration. Several studies in the literature [192, 1298–1303] have highlighted that temperature and polarization data at large angular scales ($\ell \le 30$) drive a number of mild anomalies, including a latent preference for a phantom-like DE component [192,1298,1299]. As discussed in Ref. [1297], temperature and E-mode polarization anisotropy measurements at $\ell \lesssim 30$ are precisely the subsets of Planck data strengthening the shift toward DDE, as well. Notably, when Planck's large-scale data are excluded from the analysis or when considering alternative CMB experiments that are independent of Planck, the preference for DDE diminishes significantly. In these cases, no strong preference for DDE is observed when combining DESI with Pantheon-plus SNIa data, and the ΛCDM model remains consistent within (or close to) the 95% confidence level results. For instance, no convincing preference for DDE is found in any combination involving SPT data, whether using DESI BAO together with Pantheon-plus or DESy5, or even when including
Planck/WMAP large-scale temperature and polarization measurements. Similarly, in analyses involving ACT data, a preference for DDE is observed only when combining ACT with DESI BAO and DESY5 SNIa, while it gets diluted when replacing DESY5 with Pantheon-plus, see also Fig. 41. However, when Planck large-scale temperature and polarization measurements are combined with ACT small-scale temperature and polarization data, a preference for DDE (re-)emerges. Long story short, in the CMB front of the analysis, the shift toward DDE is primarily strengthened by Planck’s large-scale data, while other CMB experiments generally weaken the case for DDE. Taking a conservative stance, this adds to broader concerns about potential systematic issues raised in other datasets discussed earlier.

### 2.3.5. Neutrino tensions

**Coordinator:** Gabriela Barenboim

**Contributors:** Adèle Poudou, Janusz Gluza, Mariana Melo, Matteo Forconi, Olga Mena, Rasmi Hajjar, Rishav Roshan, and Stefano Gariazzo

Neutrino physics is in its Golden Age era. Neutrinos possess unique characteristics that set them apart. Firstly, they are significantly lighter than other fermions by several orders of magnitude. Remarkably, no direct mass measurement has yet provided evidence for a non-zero neutrino mass; for all detected neutrinos, $E = pc$ within errors. Secondly, they have their own cosmological background. Neutrinos are often described as “elusive” due to their nature, yet they are incredibly abundant throughout the Universe. Their near absence of interaction is why we seldom detect their presence. This relic neutrino background has never been detected directly. Nevertheless, neutrinos are hot thermal relics, and their masses, even if tiny, affect the Universe’s evolution, having its largest impact on the growth of structure at small scales due to the neutrino free steaming nature. Therefore, albeit indirectly, we can set an upper bound on them by cosmological observations. Although official Planck CMB results report $\sum m_i < 0.24$ eV [192], the addition of LSS data in the form of BAO allows reaching $\sum m_i < 0.09$ eV [1304] or even $\sum m_i < 0.072$ eV from the very recent DESI BAO survey observations [699,1305], all at 95% CL. Similar bounds are obtained when different likelihood approaches are adopted for analyzing Planck data [701], while the limits are relaxed when high-multipole CMB data are considered instead of Planck [1306].

Nevertheless, the tightest limits to date are those reported in Ref. [1307] after the addition to Planck and DESI observations of other background probes, such as CC, galaxy clusters angular diameter distances, and GRBs distance moduli. For instance, the combination of CMB with GRBs and DESI BAO provides $\sum m_v < 0.049$ eV. The most constraining bound is obtained when CMB, SNIa luminosity distances, DESI BAO, and all background probes are combined: this limit is 0.043 eV at 95% CL. Background probes are therefore able to provide strong bounds on the neutrino mass due to both the preferred higher mean value of the Hubble constant and the smaller errors on both $H_0$ and the matter mass-energy density $\Omega_m$. Consequently, cosmological limits currently possess the highest constraining power on neutrino masses if the standard scenario is assumed for neutrino decoupling and if neutrinos are massive stable particles (see Fig. 42).

In the past, neutrinos were supposed to provide a possible explanation to the $H_0$ and $\sigma_8$ tensions, see e.g., [1308]. More recently, however, it has been shown that an increase in $H_0$ due to neutrino physics is normally related to an increase also in $\sigma_8$, see e.g., Fig. 34 in Ref. [192]. A solution to both tensions thanks to neutrinos seems therefore unlikely.

On the other hand, neutrino oscillations measured at terrestrial experiments while being insensitive to the overall neutrino mass scale, determine the value of two squared mass differences, the atmospheric $|\Delta m_{31}^2| \approx 2.55 \cdot 10^{-3}$ eV$^2$ and the solar $\Delta m_{21}^2 \approx 7.5 \cdot 10^{-5}$ eV$^2$ splittings [1309,1310]. Since the sign of $|\Delta m_{31}^2|$ is unknown, two mass

<figure><img src="image_48.png" /><figcaption>**Fig. 42.** The sum of the neutrino masses as a function of the lightest neutrino mass for normal (red) and inverted (blue) orderings. Cosmological constraints from CMB, CMB plus DESI, and including measurements from CC, distance moduli from GRBs, and angular diameter distances from galaxy clusters are also depicted by the shaded green and yellow regions. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

orderings are possible, the normal (NO) and the inverted (IO) orderings:

$$ \sum m_i = \begin{cases} \text{NO:} & m_v^2 + \sqrt{m_v^2 + \Delta m_{21}^2} + \sqrt{m_v^2 + \Delta m_{31}^2} \ge 0.058 \text{ eV}, \\ \text{IO:} & m_v^2 + \sqrt{|\Delta m_{31}^2| + m_v^2} + \sqrt{|\Delta m_{31}^2| + \Delta m_{21}^2 + m_v^2} \ge 0.101 \text{ eV}. \end{cases} \qquad (2.19) $$

Notice that not only some of the above-mentioned current cosmological limits are below the minimum sum of the neutrino masses allowed in the inverted hierarchical scenario, but also some of them are below the minimum mass required from neutrino oscillation probes. Very interestingly, several of the possible data combinations imply a neutrino mass limit at $2\sigma$ smaller than the minimum expected from oscillation experiments, i.e., $\sum m_v \le 0.06$ eV, pointing toward a clear tension between cosmological and oscillation neutrino mass limits, see Ref. [1307]. Therefore, despite the fact that such tension has so far not been present [1311], not only one but several sets of cosmological measurements indicate a clear problem between these cosmological and terrestrial searches of neutrino masses. Cosmological constraints, therefore, at present, reflect some tension regarding neutrino masses:

* Through cosmological observations, we are about to rule out the minimum value of $\sum m_i$ allowed by neutrino oscillations. It is important to remember that cosmology does not directly measure neutrino masses but rather constrains the neutrino energy density, which is proportional to neutrino masses in the standard scenario. This relation could be altered by the presence of new physics: current cosmological bounds may point to the existence of very exotic cosmological scenarios (possibly related to dark sector physics) and/or non-standard neutrino physics since these bounds are extremely robust within simple extensions of the $\Lambda$CDM model; see, e.g., [1312]. Possible scenarios to relax the cosmological neutrino mass bounds include time-varying neutrino masses [1313], decaying neutrinos [1314–1317], strongly-interacting neutrinos [1318,1319], long-range neutrino forces [1320], among others [1321–1325].

* Using Planck 2018, BOSS Lyman-alpha, and DESI data, if negative neutrino masses are allowed in the fitting pipeline, a slightly better fit is obtained with $\sum m_i < 0$ than with $\sum m_i \ge 0$ [1324]. It should be noted, however, that the absolute sign of the mass of a fermion is a phase (i.e., is unobservable) and therefore this
preference for negative masses should be taken as an indication of an incomplete model rather than an actual physical measurement.

These tensions will need to be addressed in the coming years, and upcoming cosmological measurements, such as those from e.g., future observations by DESI or CMB-S4, will be able to test all the current tensions (and possibly additional ones; see below) while sharpening the cosmological neutrino mass limits.

Last but not least, the Karlsruhe Tritium Neutrino (KATRIN) experiment, a tritium beta decay experiment, may directly discover the electron neutrino mass, finding a positive signal.<fnref n="9" /> Such a putative neutrino mass detection by KATRIN will show an additional clear tension with current cosmological mass limits. As previously stated, possible inconsistencies among laboratory and cosmological searches would definitely point to a much richer neutrino sector, to a departure from the standard model of cosmology, or to a combination of both. Such a situation would offer exciting possibilities to discover new physics. Neutrino physics has brought the first departure from Standard Model Physics and may also be the first one to falsify the current standard cosmological scenarios where they are hot thermal stable relics interacting exclusively via weak interactions.

### 2.3.6. Cosmic dipoles

**Coordinator:** M.M. Sheikh-Jabbari

**Contributors:** Dinko Milaković, Eoin Ó Colgáin, Francesco Sorrenti, Iryna Vavilova, Jenny Wagner, José Pedro Mimoso, Laura Mersini-Houghton, Leandros Perivolaropoulos, Lu Yin, Maciej Bilicki, and Manolis Plonis

Cosmological principle, the assumption that the Universe around us at cosmological scales is (statistically or on the average) homogeneous and isotropic, is the cornerstone of the modern cosmology, see Ref. [1328] for a recent review. CMB temperature fluctuations [192] have been regarded as a very precise verification of cosmic isotropy at the surface of last scattering at around $z \sim 1100$. However, the existence of a very small deviation from isotropy in the CMB is not ruled out theoretically [1329] and references therein or observationally [1328,1330–1335] and references therein. Such small anisotropies, if they exist and are not due to systematics, in principle can grow in time through the large structure formation, especially when we enter a nonlinear growth regime. It is therefore, prudent to thoroughly explore evidence for traces of anisotropy especially in late cosmology.

Any physical observable may be expanded in terms of spherical harmonics on the celestial sphere. The lowest mode is $ℓ = 0$ which preserves isotropy. The next mode is $ℓ = 1$, corresponding to dipoles of the physical observable in question. Given that deviations from spherical symmetry are expected to be small, one expects higher multipoles to have smaller contributions and hence harder to detect. On the other hand, the dipole component, which is the largest mode beyond isotropy, is observer dependent and can be a kinematical effect. For example, it is well established that the dipole in the CMB, which corresponds to $\sim 10^{-4} - 10^{-3}$ fluctuation in the CMB temperature (compared to usual $10^{-5}$ fluctuations), can be attributed to peculiar motion of the observer compared to the “CMB frame” that moves with the Hubble flow. So, exploring the dipoles in cosmic observables one should always note this frame dependence. The very local Universe is obviously not isotropic around us. One can ask how far one should look to be confident in the Hubble/CMB frame. It is established that this distance is larger than $\sim 100$ Mpc, as indicated also by the convergence scale of the acceleration dipole due to matter fluctuations traced by galaxies and clusters of galaxies distributions, first at Ref. [1336–1338] and later at Ref. [1339–1342]. The “bulk flow” in the Local Universe

is a collective phenomenon due to the peculiar motions of matter structures [1343] that instead of moving in random directions, appear to follow an approximate dipole velocity flow, see Ref. [1344] and references therein. The existence of bulk flows that are too large for $\Lambda$CDM expectations may be taken as a sign of departure from isotropy that shows itself as a dipole in various cosmological observables.

#### 2.3.6.1. Non-comoving cosmology, tilted cosmology.
The cosmological dipole is a frame dependent notion. To distinguish kinematical and a non-kinematical (not removable by the choice of frame) one needs to study usual FLRW cosmological models in a non-comoving frame, see Ref. [1345–1349]. A related, but conceptually different, notion is the “tilt” introduced by King and Ellis [1350]. Suppose we choose a comoving frame, a frame in which metric does not have time-space off-diagonal elements, $g_{ii} = 0$. In this frame cosmic fluids may exhibit a momentum flow, i.e., the time-space component of energy momentum tensor $T^i_i$ may be nonzero. Tilt may be different for different components of cosmic fluid, e.g., (pressureless) matter or radiation may have different tilts and hence there does not exist a Hubble or CMB frame. Tilt is a dynamical variable in cosmology, like the shear or Hubble parameter, e.g., see Ref. [1329,1351–1355] and references therein. This is in contrast with the non-comoving cosmology in which the dipole components are not dynamic ones. The existence of tilts yields dipole anisotropy in various cosmological observables.

#### 2.3.6.2. Observational hints for cosmic dipoles.
We start by recalling that typical cosmological observables are redshift (as a measure of distance), look-back time, Hubble expansion rate, deceleration/acceleration parameter, luminosity distance, angular diameter distance, and number counts. In an anisotropic cosmology, these quantities besides the redshift $z$ also depend on line-of-sight. Redshift, the ratio of the frequency of the photon emitted by a source and the one observed by the observer minus one, receives a Doppler shift if there is a bulk flow of velocity $v$:

$$1+z = (1+z_0)\Delta, \quad \Delta := (1-v/c \cos \theta_{\text{LOS}})/\sqrt{1-v^2/c^2}, \qquad (2.20)$$

where $1+z_0 = 1/a$ is redshift when the source and observer have zero relative velocity and $\theta_{\text{LOS}}$ denotes the line of sight angle. For more formal and detailed derivations and analysis see Refs. [1354,1356–1361].

#### 2.3.6.3. Ellis-Baldwin test.
A standard technique for searching for a dipole component in the distribution of sources in the sky is the seminal Ellis-Baldwin test [1362]. The number $N$ per solid angle of a local source in the sky (say radio galaxies or QSO) in two frames moving with a relative velocity $v$ and angle $\cos \theta_{\text{LOS}}$ is given by

$$\left. \frac{dN}{d\Omega} \right|_{\text{obs}} = \left. \frac{dN}{d\Omega} \right|_{\text{rest}} \Delta^{2+k}, \qquad (2.21)$$

where $k$ is a constant parameterizing astrophysical characteristics of the source population.

#### 2.3.6.4. Cosmic microwave background dipole.
A dipole component in the CMB is customarily treated as purely kinematical, due to the motion of the observer. This may be viewed as the definition of “CMB frame”, a frame in which the CMB has no dipole component. While one can always find such a frame, only in an isotropic universe all other cosmological observables are necessarily dipole free in the CMB frame. The heliocentric frame, in which our cosmological observations are made, is moving with respect to the CMB frame with velocity $v = (369.82 \pm 0.11) \text{ kms}^{-1}$ along the direction (in degrees) $(l, b) = (264.02 \pm 0.0085, 48.253 \pm 0.004)$ [192]. While the CMB dipole may be treated as kinematical, there have been persistent CMB quadrupole and octopole components which may be hints of deviations from the cosmological principle, see Refs. [1328,1330,1363] for reviews.

The Planck CMB data in the CMB frame may still show traces of a dipole once one makes a hemispherical split analysis (HSA) of the data. This method involves splitting the celestial sphere into two

<footnote n="9">⁹ The current limit is $m_\beta < 0.45$ eV (90% CL) [1326], and the expected sensitivity is 0.2 eV (90% CL) [1327]</footnote>
hemispheres and fitting ΛCDM to CMB data in each hemisphere. One then reads the values of $H_0$ and $\Omega_{m,0}$ in each hemisphere, rotates the hemisphere separation plane in the sky, and records the ratio of the difference between the two values of $H_0$ in each hemisphere, $\Delta H_0$, over the mean $H_0$ value over the whole sky, $\bar{H}_0$. This analysis has yielded a maximal value of $\Delta H_0/\bar{H}_0 \simeq 10\%$, roughly along 45° to the direction of the CMB kinematic dipole [1334], a direction close to the axis of the hemispherical power asymmetry, a well-recognized CMB anomaly [1330]. Intriguingly, this is the same level as the discrepancy in Hubble tension; however see Ref. [1364].

### 2.3.6.5. Maximum temperature asymmetry (MTA).

is another dipolar anomaly in the CMB which is identified by examining the temperature differences between opposite pixels in the CMB sky map after subtracting the dipole component. MTA selects a preferred axis by maximizing the temperature difference between these opposite pixels and indicates a preferred direction in the CMB temperature fluctuations, signature of a dipole. See Ref. [1365] for MTA analysis in the WMAP 7-year data. The proximity of the MTA axis to the $\alpha$ dipole, DE dipole, and dark flow directions (see below) further strengthens the case for a potential underlying physical cause for these anisotropies.

### 2.3.6.6. Fine structure constant dipole.

Initially reported by Ref. [1366], was identified through the analysis of QSO absorption spectra using the Very Large Telescope (VLT) and the Keck Observatory. The dipole suggests a spatial variation in the fine structure constant $\alpha$ across the sky, with a dipole axis pointing toward $(l, b) = (331^\circ, -14^\circ)$, and an amplitude $A = (0.97\pm0.21)\times10^{-5}$. However, recent studies have reduced the significance of the $\alpha$ dipole to less than $4\sigma$ [1363,1367].

### 2.3.6.7. Type Ia supernovae dipole.

SNIa are widely accepted as standard candles with a fairly good distribution over the sky [JLA, Pantheon(+), Union3, DES-5Y, ZTF]. The SNIa observations are of course made in the heliocentric frame, correcting for the presumed (peculiar) motion w.r.t. the CMB frame, assuming that a Hubble flow frame exists and that the far enough (farther than ~ 100 Mpc SNIa) are already in the Hubble flow frame w.r.t us. This may obscure possible dipole components in the data in the heliocentric frame [1368]. SNIa data may be explored for a possible cosmological dipole component in 3 different kind of analyses:

(I) Ellis-Baldwin test for SNIa. We still do not have enough observed SNIa to carry out this test. It may be feasible in some years.

(II) HSA using a cosmographic expansion, dipole fitting method. For relatively low redshift data, $z \lesssim 1$, one can expand Hubble parameter and luminosity distance in powers of $z$,

$$H = H_0 [1 + (1 + q_0)z + \mathcal{O}(z^2)], \quad D_L(z) = \frac{cz}{H_0} [1 + \frac{1}{2}(1 - q_0)z + \mathcal{O}(z^2)]. \qquad (2.22)$$

One may now consider a dipole in $H_0$, i.e., $H_0(\theta) = H_0(1 + D_H \cos \theta)$ or similarly in acceleration, $q_0(\theta) = q_0(1 + D_q \cos \theta)$ and perform HSA. This test has been carried out for JLA [1369,1370] and Pantheon(+) [1348, 1368,1371–1373] datasets claiming for a small dipole in $H_0$ and a notable dipole for $q_0$ roughly along the CMB dipole in the heliocentric frame. See also, [1344,1374]. There have been analyses that claim not seeing a dipole see e.g., [33,1375–1382]. We need to wait for more SNIa data, e.g., the DES data and ZTF to confirm or refute these claims.

(III) HSA and dipole fitting of ΛCDM into SNIa data. This analysis started in 2010 in Ref. [1383] with the Union2 SNIa dataset and identified a significant anisotropy in the accelerating expansion rate in direction $(l, b) = (309^\circ+23^\circ, 18^\circ+11^\circ)$ and $\Omega_{0m,max} = 0.43 \pm 0.06$ was consistent with statistical isotropy in about 30% of simulations. The alignment with other observations like bulk velocity flows and CMB low multipole moments, fine structure constant, and DE dipole obtained through MTA method [1384] suggested a potential underlying physical cause.

Similar recent approaches use the luminosity distance and distance ladder relation $\mu = \frac{5 \log D_L(z) + 25}{\frac{c}{H_0} \int^z dz' / \sqrt{1 - \Omega_{m,0} + (1 + z')^3 \Omega_{m,0}}$ and perform HSA. This has been carried out yielding a small variation $\Delta H_0 \lesssim \text{ few km s}^{-1}\text{Mpc}^{-1}$ over the sky [1385], see also Refs. [1374,1386–1390] and Refs. [1372,1374, 1391] for further analysis. It would be instructive to repeat a similar analysis with other datasets like DES 5Y [1374,1389] or ZTF data release. Alternatively, one may repeat a similar analysis with absolute magnitudes of SNIa (instead of $H_0$ and $\Omega_{m,0}$) [1392]. Analyzing the latest ZTF data release, particularly at low redshifts, may yield further valuable information. Ultimately, the combination of these different approaches and datasets will enhance our understanding of cosmic anisotropies and contribute to a more comprehensive picture of the Universe’s large-scale structure.

### 2.3.6.8. Quasar (QSO) dipoles.

QSO catalogs contain about 1000 times more data points compared to SNIa datasets, nonetheless, unlike the SNIa their standardizability [182,1393,1394] is still a matter of debate, see Refs. [524,1395–1397] and references therein. The 3 anisotropy/dipole searches mentioned above using SNIa data, can be repeated for the QSO samples. Ellis-Baldwin test for QSO has been carried out yielding a significant (up to $4.9\sigma$) departure from isotropy [1398]: QSO of redshifts $z \gtrsim 1$ are not in Hubble/CMB frame, while in the heliocentric frame they move in the same direction as the CMB dipole, their peculiar velocity is few orders of magnitude bigger than the CMB velocity $370 \text{ kms}^{-1}$, see also Refs. [1399,1400]. This result is still under debate e.g., see Refs. [1401,1402] and references therein.

The HSA for QSO data with the dipole fitting method has also been carried out, in a cosmographic expansion and/or using Risailit-Lusso [1394] X-ray/UV flux relation. This has also led to a $2-3\sigma$ level result for a dipole component in $H_0$ [1403].

### 2.3.6.9. Radio galaxy dipole.

Radio galaxies are typically located at cosmological distances and hence within the standard cosmological principle paradigm, they are expected to be moving with the Hubble flow; they should not exhibit a flow in the CMB frame and CMB frame should be their rest frame. This was first put to the test in 2002 [1404], confirming the expectation. However, the same observation has been repeated with better accuracy, refuting the earlier result in about $3\sigma$ level [1405], finding radio galaxies moving with speed 4 times bigger than the CMB velocity in the heliocentric frame, and roughly in the same direction. Similar results have been reported in Refs. [1406–1413], see however Refs. [1414–1419].

### 2.3.6.10. Gamma-ray burst dipole.

GRBs may also be used to perform similar dipole tests as in QSO or SNIa. However, a relatively low number of data points in GRB samples and their still debated standardizability [542,1420–1422] (see Ref. [516] for a recent review) puts limits on using GRBs as reliable cosmological observables. See Refs. [1403,1423] for the search of a dipole in $H_0$ through an HSA within ΛCDM and GRB data, reporting $2-3\sigma$ deviation from isotropy.

### 2.3.6.11. Cosmicflow-4 (CF4) bulk flow.

A measure for a possible (non)kinematic dipole may be inferred from peculiar velocities of large catalogs of distant galaxies with known distances (which typically means cosmologically close $\lesssim 500$ Mpc). Such large distance, non-random and coherent in direction velocity fields, if they exist, are called bulk flows. Bulk flows may be compared to the expected Hubble flow. CF4 (superseded CF2 and CF3) [1424] complies $\sim 56000$ galaxies to this end, while reporting $H_0 = 74.6 \pm 0.8 \pm 3(Sys)$ km s$^{-1}$Mpc$^{-1}$ also reports their peculiar velocities.<fnref n="10" /> The amplitude and alignment of the inferred velocity field from the CF4 data is at $\sim 2-3\sigma$ discrepancy with respect to the ΛCDM model [1425]. While closer galaxies at around 50–100

<footnote n="10">¹⁰ It is notable that this value for $H_0$ has independent systematics from those based on cosmic distance ladder.</footnote>
Mpc seem to show no excess or deficit in their velocity fields compared to expected Hubble flow, farther ones show sizable excesses (by more than 3 times) [1426,1427] and in a direction compatible with the CMB dipole, confirming similar anomalies seen in CF3 [1428]. Similar bulk flows have also been reported from the low redshift part of Pantheon+ compilation (0.015 $\le$ z $\le$ 0.06); these flows are toward the Shapley supercluster [1344] (see also Ref. [1374]).

A specific kind of bulk flow (as defined above) has been given a different name, *dark flows*. Dark flows are associated with peculiar velocities of galaxy clusters that can be measured through fluctuations in CMB generated by the CMB photons off X-ray emitting gas inside clusters [1429]. Dark flows are found in scales larger than 300 Mpc and the dark flow dipole is a dipole found at the position of galaxy clusters in filtered maps of CMB temperature anisotropies [1430] see also Refs. [1431–1433], see however, Ref. [1434]. It is hard to accommodate dark flow dipoles in the $\Lambda$CDM cosmology.

### 2.3.6.12. Cosmological models accommodating the presence of cosmic dipoles.
As discussed, dipoles can appear in various cosmological observables, such as the distribution of QSO and SNIa, distances, acceleration parameter, $H_0$, the fine-structure constant $\alpha$. One can construct models that accommodate dipoles in some of these observables and/or introduce mechanisms to yield large-scale inhomogeneities or anisotropies in the Universe. Below, we discuss some of the prominent theoretical models that could explain the observed cosmic dipoles.

*Off-Center observers in spherical cosmological scale inhomogeneities.* Off-center observers situated in a spherically symmetric inhomogeneous universe can see a dipolar anisotropy due to the observer's position relative to the center of the inhomogeneity. This idea is explored in models where large-scale structures or voids introduce anisotropies due to the observer's peculiar location within the cosmic structure [1435,1436].

*Primordial dipolar horizon scale perturbations.* Primordial perturbations generated during the inflationary epoch can also have dipole anisotropy that can be imprinted on the largest scales and re-enter the horizon at later times, leading to observable dipolar anisotropies in the CMB and other cosmological observables. Such mechanisms can be related to specific inflationary models that predict large-scale anisotropies [1206,1437].

*Nontrivial topology of the cosmos.* The spatial topology of the Universe may play a role in creating cosmic dipoles. Nontrivial topologies, such as those involving multiply connected spaces, can introduce preferred directions and anisotropies on large scales which may be detectable in the distribution of galaxies, the CMB, and other cosmological datasets [1234,1438]. (Non-trivial topology may induce anisotropy through parity-violation without involving a dipole [1332,1439].)

*Hubble scale topological defects.* Hubble scale topological defect, such as a global monopole created by the DE scalar field at recent cosmological times, known as topological quintessence, can be another source of dipolar anisotropy in various observable (from the off-center observer viewpoint) [1363,1384,1440,1441]. This scenario may also explain the alignment of various observed dipoles, such as the QSO dipole, the SNIa dipole, and the $\alpha$ dipole.

*Other physical mechanisms.* Other physical mechanisms that can induce cosmic dipoles include:

- *Anisotropic Dark Energy:* Models with an anisotropic equation of state for DE can lead to directional dependencies in the expansion rate of the Universe, thereby creating dipolar anisotropies [1442–1444].

- *Large-Scale magnetic fields:* can introduce anisotropies in the CMB and other observables, potentially explaining some of the observed dipoles [1445,1446].

- *Vector field models:* Inflationary models involving vector fields can naturally generate anisotropic perturbations [1447], leading to observable dipoles in various cosmological datasets [1448,1449].

- *Tilted models:* Tilted cosmological models, where observers have peculiar velocities relative to the cosmic rest frame, can lead to locally

observed acceleration and dipole-like anisotropies in the deceleration parameter [1347,1352–1356,1369,1401,1450–1453].

These theoretical models provide a rich framework for understanding the observed cosmic dipoles. Future observations and more refined data will be crucial in testing these models and determining the true nature of the large-scale anisotropies in our Universe.

### 2.3.6.13. Conclusion and future directions.
We briefly discussed dipoles in various cosmological observables, from the dipole in the distribution of astrophysical sources like SNIa, QSO, GRB and radio galaxies over the sky, to dipole in the coherent flows of matter (clusters and superclusters), to dipoles in the cosmological model observables like $H_0$, $\Omega_{m,0}$. Besides the dipoles we discussed here, there have been reports on an intrinsic (non-kinematical) dipole in the CMB [1454–1458], the dipole in the fine structure constant [1366,1384,1459] and in the distribution of baryon matter in the Universe using FRB [1460,1461]. Moreover, if the CMB dipole is interpreted as our departure from the Hubble flow it will induce ‘time dilation dipole’, a directionally-dependent time dilation over the sky. Detection of time dilation dipole can provide a new assessment of the cosmological principle [1462,1463].

These dipoles, if of a cosmological origin and if not kinematical, hint at a breakdown of the cosmological principle, the very pillar of modern cosmology. Most of the dipoles discussed here may be individually of not a large statistical significance, nonetheless, the synergy between them suggests that they may not be dismissed. Future data with better sky coverage will be instrumental in establishing/refuting true cosmological dipoles. If established, one should take more serious steps in accommodating a cosmological dipole in the cosmological models, such first steps have been taken in [1352,1354].

## 2.3.7. Big bang nucleosynthesis

**Coordinator:** Nils Schöneberg

**Contributors:** Dinko Milaković, Ismailov Nariman Zeynalabdi, John Webb, Luca Izzo, and Venus Keus

BBN describes the process of generating the atomic nuclei of light elements shortly after the big bang as a result of the Universe cooling down beyond the point at which photo-disintegration of these nuclei becomes inefficient. Since the heavier elements need to be built up from lighter components,<fnref n="11" /> the process of the nucleosynthesis only begins when the Deuterium photo-disintegration becomes inefficient, which occurs when the photon temperature is around $T \sim 100$ keV (corresponding to redshift $z \sim 5 \cdot 10^8$). For reviews see Refs. [1464–1467]. Not all stable elements are readily produced from BBN due to the instability of isobars with 5 or 8 baryons. In particular, the stable elements produced in significant proportion from BBN are hydrogen ($^1$H and $^2$H), Helium ($^3$He and $^4$He), and Lithium ($^6$Li and $^7$Li), with the remaining elements produced only in tiny proportions compared to later stellar and cosmic-ray induced generation. As such, the primordial abundances of these heavier elements are almost impossible to measure, given that already for the small abundance of Lithium the post-BBN generation presents a significant systematic, see Section 2.3.7.1. Despite generally showing an excellent agreement between predicted and observed abundances, we discuss hints at possible tensions for these different elements in Section 2.3.7.1–Section 2.3.7.3. Furthermore, BBN can play a critical role in other tensions, see Section 2.3.7.4.

### 2.3.7.1. Lithium tension.
The two stable isotopes of Lithium ($^6$Li and $^7$Li) are produced only in very subdominant amounts, with $^7$Li at

<footnote n="11">¹¹ The reason for this is that the direct n-particle fusion is suppressed by a power of $\eta_b^n$ where $\eta_b \sim 6 \cdot 10^{-10}$ is the tiny baryon-to-photon ratio. As such, subsequent progressive 2-nuclei fusion processes are more relevant for the fusion of heavier elements, which makes the isobar stability gaps crucially important.</footnote>
a level 10⁻⁹ times below that of hydrogen (¹H). At face value, the prediction of the ⁷Li abundance from standard BBN is about 2–4 times higher than that measured in the atmospheres of metal-poor stars in the galactic halo, for example, see Ref. [1468] computes ⁷Li/H=(4.94 ± 0.72)·10⁻¹⁰ (for a CMB-motivated baryon density, see also Refs. [1469–1473] for other such high computations) while Ref. [1474] measured ⁷Li/H=(1.58±0.35)·10⁻¹⁰ (see Ref. [1475] for a higher measurement in the SMC, possibly related to nova enrichment [1476]). See Ref. [1477] for a review. Naïvely this appears to be a large issue for standard BBN. However, there are a few caveats to take into account.

The observations of the ⁷Li abundances are extrapolated to zero metallicity (iron abundance), but the plateau originally observed (the so-called Spite plateau) Ref. [1478,1479] appears to not persist at lower metallicities (increased scatter falling on average below the plateau), see for example Refs. [1477,1480,1481]. The issue is that Lithium can be burnt up in stellar environments, possibly even in metal-poor stars. In particular, convective mixing in metal-poor stars could bring the Lithium down to layers with a temperature above 3·10⁶ K sufficient for further burning, see for example Ref. [1482]. This stellar depletion argument is still under investigation, for a recent discussion see Ref. [1483] — In this case ⁶Li would be burnt up and thus its abundance is used as a useful argument on whether such burning is expected to have taken place. However, the observation of ⁶Li is even more difficult due to its even smaller abundance (and its ready generation through cosmic rays). Initial observations indicating a plateau at ⁶Li/⁷Li~0.05 (BBN predictions are at ~10⁻⁴ Refs. [1465,1469]) would indicate that stellar burning could not have taken place [1484–1490], but newer observations currently only put upper bounds [1491–1494], overall making stellar depletion a likely explanation. See Ref. [1495] for another argument toward stellar processes and Ref. [1496] for possible observations of a truly primordial plateau.

Another argument could be made on the aspect of the nuclear reaction rates, though the rates appear to be consistent, see Ref. [1483] for a discussion. As such, while nuclear rates are unlikely to be the cause of the Lithium tension, stellar depletion could very well be an entirely sufficient explanation. Given this unclear footing of the Lithium tension, any claims of a “required” modification of standard BBN should be seen with great caution. Despite this, a range of non-standard BBN models have been proposed as possible solutions to the Lithium tension, such as for example Refs. [1481,1497–1509], and Ref. [1510] for a review.

### 2.3.7.2. Helium anomaly.
Primordial ³He is remarkably difficult to measure [1467] (see also Ref. [1511]) since ³He is generally readily produced and depleted in stars. It is thus rarely used to constrain BBN. Conversely, ⁴He is a crucial part of the BBN predictions that can be readily measured in ionized metal poor extragalactic HII regions. Since ⁴He is produced in stellar cores, the primordial Helium mass fraction $Y_p$ is usually found by extrapolating the ratio of ⁴He/¹H from the measured systems as a function of metallicity to zero metallicity, which is presumed to represent the primordial value. These determinations from 214 systems [1512–1517] (which roughly average to $Y_p = 0.245 \pm 0.003$ [1467]) generally agree extremely well with the value predicted from baryon abundance and neutrino abundance determinations from the Planck satellite under the standard BBN assumptions (giving $Y_p = 0.241 \pm 0.025$ [192]). The recent determination from the EMPRESS survey on the Subaru telescope by Ref. [1518] returns a much lower value ($Y_p = 0.2370 \pm 0.0033$) compared to all previous results, including that of Ref. [1516] (which gets $Y_p = 0.2436 \pm 0.0040$). This is interesting because Ref. [1518] uses the entire sample 1 of Ref. [1516] and the only difference is the addition of five additional extremely metal-poor galaxies (J1631+4426, J0133+1342, J0825+3532, J0125+0759, and J0935-0115). Four out of five of these galaxies lie significantly below the expected Helium mass fraction (from the fit of Ref. [1516]) given their metallicity, see Fig. 43. Taking the measurements at face value, one would conclude a non-zero lepton asymmetry [1519,1520] and a

slight preference for additional relativistic degrees of freedom [1518, 1519]. However, such an analysis is yet to be independently confirmed. We also note that Ref. [1521] finds a higher-than-mean value.

### 2.3.7.3. Deuterium — A new tension?
The final element that is produced in decent quantity from BBN is Deuterium (²H, alternatively D). While the Deuterium measurements are largely consistent, see Fig. 43, its abundance can also be seen to be in tension with the baryon abundance determined from CMB data, depending on which modeling of the nuclear rates is performed [1471,1522,1523], thus requiring new measurements of nuclear rates to clear up the posited tension. In particular, the crucial rates for Deuterium are

$$ ^2\text{H} + ^2\text{H} \rightarrow ^3\text{H} + p^+ \text{ (ddp)}, \qquad ^2\text{H} + ^2\text{H} \rightarrow ^3\text{He} + n \text{ (ddn)}. \qquad (2.23) $$

Conversely, to probe the consistency with the CMB one can also check what nuclear rates would be preferred given the baryon abundance and neutrino number from the CMB (see Section 2.1.18) [1522,1524].

### 2.3.7.4. Impact of BBN on other tensions.
While BBN is extremely self-consistent, it can also be used to support other tensions. For example, the combination of BAO+BBN data provides one of the tightest non-Planck determinations of a low $H_0$ value, see for example Refs. [699, 705,789,1525–1529] (from DESI DR1 [699], 68.51±0.58 km s⁻¹ Mpc⁻¹ from DESI DR2, or 67.6±0.9 km s⁻¹ Mpc⁻¹ from eBOSS (and 68.3±0.7 km s⁻¹ Mpc⁻¹ when additionally including eBOSS ShapeFit [1526]), see also Section 2.1.19), recently reaching a precision of $H_0 = 68.52 \pm 0.62$ km s⁻¹ Mpc⁻¹ [699], which puts a significant strain on solutions of the Hubble tension that focus solely on the CMB. See also in particular Section 2.1.19. BBN data also aids in constraining dark radiation or varying constant solutions, see Sections 4.8 and 4.1.3 (although the latter is not tightly enough constrained to exclude it as a solution to the Hubble tension, see Ref. [1530]).

## 2.3.8. Anomalies with Lyman-α measurements

**Coordinator:** Vid Iršič

**Contributors:** Simeon Bird

Similarly to how galaxies trace the matter distribution at lower redshifts, the IGM has been used to trace the matter distribution at the time when first galaxies are forming ($z > 2$) (e.g., Ref. [1532]). The most well-studied observable of the IGM is the Lyman-α (Lyα) forest — a series of absorption lines in the spectra of QSOs, that arise due to the scattering of QSO light on the intervening neutral hydrogen atoms in the ground state. In the last two decades the Lyα forest has been measured in large spectroscopic surveys to study the precise position of the BAO [745,1533], full-shape RSD [1534,1535], as well as small-scale 1D clustering statistics [1536,1537]. The latter relies on the estimation of the 1D power spectrum along each QSO spectrum and then averaged over all the QSO spectra in the data sample [1538,1539]. This methodology has also been applied to smaller samples, with 10-100's of high-resolution and high signal-to-noise QSO spectra [1540–1543]. The highest precision constraints from the Lyα forest on the amplitude and shape of the matter power spectrum come from the measurements of the 1D flux power spectra.

The cosmological parameter inference from the 1D flux power spectra relies on state-of-the-art cosmological simulations [1544–1547] as well as a data compression technique [1548]. This method assumes that at $2 < z < 4$ and small scales ($< 10$ Mpc/h) the Lyα forest is constraining only two effective parameters — the amplitude and slope of the linear matter power spectrum at a pivot wavenumber and redshift, with the exact pivot point depending on the survey. This data compression has been shown to be valid also for some of the extensions of the ΛCDM model, such as cosmology with massive neutrinos or running of the spectral index [1549].

However, this methodology is typically valid when limited to scales $> 1$ Mpc/h or cosmological models that do not drastically change
<figure><img src="image_49.png" /><figcaption>**Fig. 43.** Left: A comparison of the Helium abundance measurements from Ref. [1516] (Hysu+2020) to some recent fits of Ref. [1518] (Mastumoto+2022). For the fits, S1 is sample 1, S2 is sample 2, and E denotes the EMPRESS data. Right: A comparison of Deuterium abundance measurements from Ref. [1467](PDG) compared to the corresponding summary value and the Planck prediction according to Ref. [1531] (Fields+2019).</figcaption></figure>

the behavior of the small-scale linear matter power spectrum (such as light DM models). In those cases, at least one more parameter can be extracted from the data corresponding, for example, to the scale of suppression of the matter power spectrum due to free-streaming nature of DM [1550]. This has been successfully used to produce bounds on various types of alternative DM models [1551–1555].

While the ongoing DESI survey will provide updated constraints, the current most precise measurements of the 1D flux power spectrum at 0.1–1.0 h Mpc⁻¹ come from the SDSS-IV/eBOSS spectroscopic survey [1556]. A subsequent cosmological inference analysis [1553] reported a tension with the CMB inferred cosmological parameters, suggesting a hint of new physics in the form of a running of the spectral index. No non-statistical effects in the data analysis were identified that could explain the signal and significant effort was invested toward mitigation strategies of systematic effects. A later re-analysis of the eBOSS DR14 measurements used improved cosmological simulations and sampling techniques [1557] and pointed to possible internal tension in the data between lower redshift ($z < 2.5$) and higher redshift ($z > 2.5$) Ly$\alpha$ data. The analysis also proposed that the previous tension driving the constraints on the running of the spectral index can be completely alleviated at the expense of the increased tension in the amplitude of the matter clustering $\sigma_8$. The S8 tension between the amplitude of structure expected by the CMB and the amplitude of structure detected by WL at lower redshifts has also been detected at $z = 2$ by the Ly$\alpha$ analysis of Ref. [1557], with agreement between the measurements of the Ly$\alpha$ forest and WL.

Similarly, several analyses of high-resolution data pointed to a small tension between the CMB and Ly$\alpha$ forest from the high-resolution observations of QSO spectra [1558,1559]. A 2$\sigma$ tension was reported between Planck CMB analysis and high redshift ($3 < z < 5.5$) Lyman-$\alpha$ forest data from XQ-100/MIKE/HIRES in Ref. [1560]. The study argued that several systematic effects in the data, such as QSO continuum mis-estimation, residual high column density systems not masked in the data selection, and astrophysical uncertainties in the mean transmission of the high redshift Lyman-$\alpha$ scattering could potentially alleviate the tension. A further study combining both high redshift XQ-100/MIKE/HIRES and low redshift eBOSS DR14 data [1561] showcased that the internal tension is strongest in the compressed parameter space of amplitude and slope of the linear matter power spectrum at pivot scale ($k_p \sim 1$ h Mpc$^{-1}$) and redshift ($z \sim 2.5$) as probed by the Lyman-$\alpha$ forest data. Both studies [1560,1561] further proposed that systematics in the data would pull in the direction of higher slope and lower amplitude (or vice-versa). Study of Ref. [1561] then argued that

for that reason EDE models are severely penalized in this part of the parameter space and can thus be robustly excluded. A recent study by Ref. [1562] summarized these findings and proposed a new physics model of mixed ultra-light axions to solve the tension between eBOSS and Planck CMB data. While successful in explaining the results of the eBOSS survey, such a model would induce heavy suppression of matter power spectrum on smaller scales $< 1$ h$^{-1}$ Mpc, that would be detected high redshift data [1541,1542]. These data sets access smaller scales and put stringent constraints on such on DM suppression, including mixed DM models [1563,1564].

The dependence of the Ly$\alpha$ 1D analysis on the cosmological simulations makes independent tests costly. Moreover, the cosmological parameter inference is not immune to astrophysical effects. While certain physical properties of the IGM are becoming well understood, and can be measured independently, such as the thermal history evolution [1565]; the impact of other effects, such as inhomogeneous helium reionization, on the 1D flux power spectrum is not yet fully understood [1566,1567]. Aside from the uncertainty of the astrophysics of the IGM, several observational systematic effects can potentially explain the differences between the current measurements.

Nevertheless, these anomalies in the Ly$\alpha$ forest data could offer a fascinating hint of new physics that can be tested and explored in a unique range of redshifts and clustering scales probed.

### 2.3.9. Cosmic superstructures and the ISW anomalies

**Coordinator:** Istvan Szapudi

**Contributors:** András Kovács, Christine Lee, Deng Wang, Maret Einasto, Mina Ghodsi, and Pekka Heinämäki

#### 2.3.9.1. Galaxy superclusters.
The largest structures in the cosmic web are superclusters of galaxies — overdensity regions that embed the richest galaxy clusters connected by filaments of poor clusters and galaxy groups [1568–1570]. Galaxies and galaxy systems form due to initial density perturbations of different scales. Perturbations of a scale of about 100h⁻¹ Mpc give rise to the largest superclusters, the largest coherent systems in the Universe [1571]. The sizes of the richest and largest superclusters are up to almost 200h⁻¹ Mpc [1570,1572,1573]. With their galaxies, groups, clusters, filaments, and gas, superclusters can be considered as miniature versions of universes. Therefore, they are ideal laboratories to study the properties and evolution of various elements of the cosmic web [1574,1575].

Superclusters as the connected overdensity regions in the cosmic web have been defined based on individual objects (typically optical or
<figure><img src="image_50.png" /><figcaption>Fig. 44. The first stacking measurement of WMAP toward SDSS LRGs [1583] from 2008. The significance exceeded 4.4σ with A ≈ 4. The anomalous signal has been reproduced with Planck CMB maps and in most wide-angle LSS maps over different areas of the sky.</figcaption></figure>

X-ray groups and clusters of galaxies) or luminosity-density or velocity fields [1570,1572,1576–1578].

The extreme cases of observed objects usually provide the most stringent tests for theories; this motivates the need for a detailed understanding of various properties of the richest superclusters [1579]. Various properties of superclusters can be used to test cosmological models, such as their abundance, masses, sizes, shapes, morphology, the properties of their high-density cores, and so on [1574,1580]. Recent observational results (e.g., DESI Collaboration [699]) have shown a tension in the Hubble value based on different observational probes, challenging the concordance model of the DE equation of state. Superclusters, representing the grand finale of hierarchical merging events, evolve at the forefront under the influence of two competing forces driven by DE and DM. They may provide a useful probe through their superior mass and extent. For example, the abundance of rich superclusters at a given epoch and the stacked signal through the ISW effect generated by large superclusters can be useful to constrain the DE equation of state [1581–1585]. In what follows, we focus on supercluster imprint on the CMB, and on the properties of supercluster planes, shells, and on the regularity in the supercluster distribution.

2.3.9.2. The ISW puzzle. When photons cross the decaying potential well of a supercluster, they become hotter; in voids, they become colder. This phenomenon is the ISW effect [1586]. The ISW puzzle consists of anomalous signal levels detected when stacking the CMB aligned with supervoids and superclusters. The effect has persisted since 2008 [1583].

The linear contribution to the late-time ISW CMB T ISW is integrated over the line-of-sight from the present time (z = 0) to the surface of last scattering (z LS ) [1586],

$$
\frac{\Delta T_{\text{ISW}}}{\bar{T}}(\hat{n}) = 2 \int_{0}^{z_{\text{LS}}} \frac{a}{H(z)} \Phi(\hat{n}, \chi(z)) \, dz = -2 \int_{0}^{z_{\text{LS}}} a(1 - f(z)) \Phi(\hat{n}, z) \, dz, \qquad (2.24)
$$

where a is the scale factor, χ is the comoving distance, Φ is the gravitational potential, and H ( z ) is the Hubble parameter. The second equality reveals the connection with the logarithmic growth function f ( τ ) ≡ d ln D/d ln a [14]. While the analogous non-linear Rees-Sciama effect [1587] happens in all models, the ISW effect gauges the diver- gence of growth history from the Einstein–de Sitter (EdS) model where it is zero.

The CMB stacked toward superclusters and supervoids creates hot and cold spots, respectively (c.f., Fig. 44). The ISW amplitude relative to concordance expectations, A$_{ISW}$ ≡ Δ T$_{obs}$/ Δ T$_{CDM}$ quantifies any anomalies when A$_{ISW}$ ̸ = 1. The first measurement to stack superstruc- tures [1583] found A$_{ISW}$ ≈ 4 with the correct signs for supervoids and superclusters.

2.3.9.3. Summary of observations. Later stacking measurements, mainly based on voids, are consistent with A$_{ISW}$ ≈ 4–5 [1582,1588–

1591]. Cross-correlating the CMB with large-scale structure is typically consistent with the concordance model [1153,1592], although with less statistical power from a given data set. Stacking superstructures focuses on the highest signal-to-noise data in contrast with the unweighted average of the two-point statistics.

The BOSS redshift survey updated the void stacking measurements of the SDSS. Void finding techniques solving the “void-in-void” prob- lem by combining voids into larger structures confirmed the earlier excess [1593,1594]. Algorithms with no merging identified no ex- cess [1595]. Stacking [1596,1597] DES Year-1 and Year-3 photometric redshift data sets further corroborated earlier results by Ref. [1583]. Since DES covers a different part of the sky than the original SDSS, it lessens the likelihood of a statistical fluke. The combined DES and BOSS data sets [1594] yield the ultimate result A$_{ISW}$ ≈ 5.2 ± 1.6 in the redshift range of 0.2 < z < 0.9. Fig. 45 summarizes the principal observations of the ISW tension.

2.3.9.4. The cold spot. The stacking measurements draw attention to the largest anomaly of the CMB, the Cold Spot (CS), an exceptionally cold of approximately 70 µ K area centered on Galactic coordinates (l, b) ≈ (209°, − 57°) [1193,1214,1598]. Could it be due to the ISW effect? The Eridanus void, found by Refs. [1222,1599] targeting an extended underdensity aligned with the CS, is the likely cause. While the full CS signal with substructures taken into account predicted by the concordance model A$_{ISW}$ = 1 [1223] is not enough to explain observations, the excess A$_{ISW}$ ≈ 4–5 would suffice. Bayesian hypothesis testing overwhelmingly favors the hypothesis of the Eridanus void causing the CS over chance alignment [1222].

2.3.9.5. Avera and the ISW and hubble puzzles. The Average Expan- sion Rate Approximation (AvERA) model [13,13,14] simultaneously explains the excess ISW signal and Hubble constant anomaly, see Ref. [1600], where they fit H$_{0}$ = 71 . 99 +1 . 05 − 1 . 03 km s − $^{1}$Mpc − 1 to SNIa. AvERA approximates emerging curvature models [1601–1603] with a modified N -body simulation. AvERA reverses the order of averaging and solving the Friedman equation; a minor but surprisingly consequential modifi- cation of the standard N -body algorithm. Its motivation is the Separate Universe Hypothesis (SUH), stating that an under- or overdense region evolves like a universe with modified cosmological parameters. Recent general relativistic simulations [1604] confirmed that emerging curva- ture in voids dominates the late cosmic expansion history as predicted by AvERA.

The final Hubble constant in AvERA depends slightly on the coarse- graining parameter. Between the extreme scales of the box size (no effect) and resolution (SUH breaks down), the final results are only mildly sensitive to coarse-graining: an order of magnitude change will alter the results by a few percent. The best fit, expressed as a mass scale of ≈ 1 − 2 × 10 $^{11}$M ⊙ , reproduces the locally observed higher Hubble constant [13].
<figure><img src="image_51.png" /><figcaption>**Fig. 45.** These figures from Ref. [1605] sum up the ISW tension as of today. Left: Observed ISW amplitudes in different redshift bins. The excess ISW signal transitions around $z \le 1.5$ and changes sign. The bottom panel displays the significance of the deviations compared to $\Lambda$CDM predictions ($1\sigma$ and $2\sigma$ correspond to shaded bands). Right: Solid and dashed lines show the cosmic growth history in Planck 2018 $\Lambda$CDM cosmology and in AvERA as used by Ref. [1153]. Points with an asterisk display $f\sigma_8(z)$ constraints divided by the Planck $\sigma_8(z)$ value. The DES, BOSS, and eBOSS ISW anomalies have a consistent trend in terms of the re-scaled $\Lambda$CDM growth rate values ($A_{\text{ISW}}(z) \times [1 - f^{\Lambda\text{CDM}}(z)]$). The CMASS [1595] and eBOSS LRGs ISW amplitude is not significantly anomalous. Nevertheless, the eBOSS ELG, QSO, and high-$z$ constraints from the FastSound and SDSS Ly$\alpha$ lean toward the late complexity trend predicted by the AvERA.</figcaption></figure>

<figure><img src="image_52.png" /></figure>

The AvERA expansion history is similar to concordance expectations with minor differences in detail. By definition, it starts on the same trajectory as a $\Lambda$CDM or EdS model at high $z$. Around $z \simeq 4$, the initial collapse of high-density regions accelerates growth compared to the concordance model. At lower redshifts, $z \simeq 1.5$, the voids start dominating. Their expansion stunts growth more effectively than $\Lambda$, thus the “void effect” explains the local higher Hubble constant. The higher derivative also produces an excess ISW effect [14] comparable to observations. Thus, the AvERA explains the Hubble and ISW puzzles simultaneously. In AvERA, the coarse-graining scale drives the local Hubble constant. While boosting the coarse-graining scale by a factor of two induces only a few percent change in $H_0$, a reasonable choice of a few times $10^{11} M_{\odot}$ (Lagrangian) achieves perfect consistency with the local measurements. The late complexity of AvERA expansion history affects $S_8$ as well, although this has not been investigated in detail.

AvERA has a robust and surprising prediction: a sign change of the ISW effect above $z \simeq 1.5$ [14]. In contrast, the concordance DE models predict an unmeasurably small ISW signal at the same redshift.

### 2.3.9.6. Opposite sign ISW effect.
The eBOSS DR16 QSOs [1606] enabled the first look for ISW sign reversal [1605]. The data cover a crucial redshift range $0.8 < z < 2.2$ where $\Lambda$CDM and AvERA diverge: the first tends to zero while the latter predicts a sign change around $z \simeq 1.5$.

The concordance model signal was estimated with a QSO mock catalog from the Millennium-XXL for comparison with stacking 800 supervoids of the eBOSS DR16 QSO catalog. The excess signal $A_{\text{ISW}} \approx 3.6 \pm 2.1$ in the redshift range of $0.8 < z < 1.2$ is comparable with earlier observations. At $1.5 < z < 2.2$, the AvERA-predicted opposite-sign ISW signal emerged in $2.7\sigma$ tension with the $\Lambda$CDM. Despite the moderate significance of these measurements, taken together with the excess at low redshift and the CMB CS, it suggests a more complex growth history than any variations of the concordance model. The observed late complexity (c.f., Fig. 45) consistent with AvERA implies that alternative models, such as emerging curvature, will explain the late expansion history of the Universe.

### 2.3.9.7. Supercluster complexes and planes.
In several cases, the richest superclusters form complexes in which several very rich superclusters are almost connected. These are, for example, the Sloan Great Wall

at redshift $z = 0.08$ and the BOSS Great Wall at redshift $z = 0.47$ with a length of over $250h^{-1}$ Mpc [1573,1607–1609]. It has been questioned whether the presence of several such rich complexes in the local Universe is in agreement with the standard cosmological model [1583,1610–1612].

Also, in the local Universe galaxy superclusters are arranged on huge planes which span many hundreds of Mpc in space. Among these are the Local Supercluster Plane, discovered by Ref. [1613] in the distribution of superclusters. Rich optical and X-ray clusters, as well as luminous galaxies and radio galaxies in the local Universe are located on this plane [1572,1614–1616]. Perpendicular to the Local supercluster plane, [1617] discovered in the distribution of rich superclusters another plane — the Dominant supercluster plane. It is not yet clear whether the presence of such planes are in agreement with our standard cosmological model. For example, using constrained simulations of the local Universe in the $500h^{-1}$ Mpc box [1618] found that a plane with the extent of approximately $100h^{-1}$ Mpc can be recovered in simulations. However, 0.28% of random realizations of simulations only matched both underdense and overdense regions of the local Universe, and their simulation box was not large enough to test the presence of a supercluster plane hundreds of Mpc long. Also, they did not analyze the probability of finding two perpendicular supercluster planes. Refs. [1616,1619] speculate that such planes may be a signature of long, nearly straight strings.

### 2.3.9.8. Regularity in the distribution of rich superclusters and supercluster shells.
Ref. [1572] analyzed for the first time the spatial distribution of superclusters based on the all-sky catalog of superclusters using various methods. In their study [1572] discovered that local rich superclusters are arranged in an almost regular pattern with the characteristic distance between superclusters $120–140h^{-1}$Mpc, see also Ref. [1620] on the regularity periodogram of the local structures. The power spectrum of rich clusters has a bump at this scale, and the correlation function has a series of wiggles [1621–1623]. Rich superclusters are separated by giant voids or supervoids [1624,1625]. Superclusters are not arranged randomly in the cosmic web, and this makes projections smaller. As a result, supervoids are in some cases connected and form very large voids, as the Eridanus supervoid. Such huge structures can leave their signature to the CMB. These signatures are discussed in Section 2.3.9.2. In addition, Ref. [1626] detected a hint of the regularity with the scale
of about 400h⁻¹Mpc in the distribution of QSO systems at redshift range 1 < z < 1.8.

The Sloan Digital Sky Survey covers a part of the regular patterns of superclusters. Ref. [1627] showed that rich superclusters in the sky region, covered by the SDSS, are arranged in six intertwined shell-like structures with the same characteristic radius as found for the whole pattern of superclusters, approximately 120–140 h⁻¹ Mpc [1627]. The centers and walls of these shells are marked by rich galaxy clusters in superclusters such as the Bootes supercluster, the Sloan Great Wall, the Corona Borealis supercluster, the Ursa Major supercluster, and other superclusters, see also Ref. [1628]. The most prominent of these shells was recently determined also using the CosmicFlows 4 data, and named “Ho'oleilana” [1629].

The huge sizes of these shells tell us that the origin of these shells, and regular pattern in the supercluster distribution should come from the very early Universe. Ref. [1627] concluded that the process behind these shells is still unknown. Ref. [1629] interpreted the shell which they named as HoÂ'oleilana as BAO shell. However, in the BAO shells the central mass is always much higher than the mass in shell walls [1630]. Analysis of the luminosities and masses of superclusters have shown that rich superclusters typically have mass-to-light ratios M/L ≈ 300 [1608,1631,1632]. Therefore, we can estimate masses of superclusters in the HoÂ'oleilana using their luminosities provided by Ref. [1627] and this M/L value. The mass of the central cluster in the Bootes supercluster, Abell cluster A1795 is M ≈ 6.6–11.2 × 10¹⁴ M☉ [1633,1634]. The mass of the Bootes supercluster itself is the lowest among superclusters in this shell, MBoo ≈ 0.1 × 10¹⁶ M☉. Masses of superclusters in the walls of this structure are more than a hundred times higher than the mass of the central cluster, A1795, of the order of Msc1 ≈ 0.2–3.5 × 10¹⁶ M☉, and their total mass is at least Mtot ≈ 25 × 10¹⁶ M☉.

Therefore, masses of superclusters contradict the interpretation of Ho'oleilana as BAO shell, as proposed in Ref. [1629]. Ref. [1627] already provided arguments against BAO interpretation. Namely, the radius of the shell is larger than ≈ 109 h⁻¹ Mpc, the BAO scale, and shell is wide, being in the interval of 120–140 h⁻¹ Mpc. Ref. [1629] proposed that for the interpretation that this structure represents a BAO shell, the value of the Hubble constant should be approximately 77 km s⁻¹Mpc⁻¹, see Ref. [1629] for details. However, so high value of H0 is very unlikely. Therefore it is still, even thirty years after discovery, an open question whether the presence of such structures can be explained within the ΛCDM cosmological model, and which processes in the very early Universe are behind this regularity.

### 2.3.10. JWST anomalies

**Coordinator:** Matteo Forconi

**Contributors:** William Giarè

One of the major challenges in testing the ΛCDM model is the lack of direct observations at high redshift (z ≳ 10). Current probes, such as BAO and SNIa, do not reach this regime, making it difficult to study the accuracy of the ΛCDM model around these epochs; this is particularly important considering that the large-scale structures we observe today largely emerged in the early stages of the Universe's evolution. The JWST, however, has been able to achieve this goal, offering a glimpse into these distant epochs of galaxy evolution history. Because of the expansion of our Universe, photons emitted by distant sources experience a non-negligible redshifting, such that UV light falls into the infrared spectrum today. Moreover, the farther away the source (i.e., the earlier in cosmic time), the fainter the observed signal. To address these observational challenges, JWST Near Infrared Camera (NIRCam) [1635] covers a wavelength range of [0.6–5]µm, improving the coverage of its predecessor, HST, that was optimized for visible wavelengths. Coupled with a primary mirror which is nearly three times larger than that of HST, JWST significantly increases our ability to detect and resolve faint and very distant targets.

Recent JWST observations have identified a large population of photometric galaxy candidates [1636–1645]. Many of these galaxies are very bright [1637,1639,1646], with UV LFs that differ from previous theoretical and observational predictions [1641,1647–1651]. They are found at remarkably high redshifts (z ≥ 12) [1641,1652–1655] and some exhibit extremely large stellar masses (M ≥ 10¹⁰.⁵ M☉) [1656–1660]. The unveiling of these extreme objects is difficult to reconcile within the standard galaxy formation models and, by extension, the assumed underlying cosmology (ΛCDM) [1661–1663]. Moreover, JWST's ability to discover new SN candidates [1664–1669] introduces additional prospects for tackling tension in cosmology [1670,1671].

It is important to emphasize that these preliminary findings are from photometric data which, alone, are not enough to draw definitive conclusions. A major source of uncertainty is the difficulty in differentiating early star-forming galaxies from quiescent galaxies at lower redshifts [1672], as well as the spectral energy distribution templates used to interpret photometric bands may be unsuitable for very massive galaxies in the early Universe — though updated templates have been suggested, e.g., see Refs. [1673,1674]. Nonetheless, comparisons of photometric and spectroscopic redshifts in overlapping samples of galaxies with both measurements further validate these observations [1675–1682].

A way to alleviate this emerging tension is to reconsider the physics and assumptions behind the structure evolution and star formation process at high redshift [1683–1694]. Because measured stellar masses strongly depend on the Initial Mass Function (IMF), adopting a top-heavy IMF [1691,1695,1696] instead of the typical standard assumption of a Salpeter IMF [1697] could shift the inferred stellar masses. Additionally, increasing the gas temperatures can lead to a greater contribution to the brightness of early galaxies and consequently lowering the stellar mass estimates, although feedback effects might compensate this mechanism [1698]. Another critical factor is the star formation history and star formation efficiency ϵ. The results are highly sensitive to the model adopted [1699–1702]; a higher ϵ can explain large early galaxies by accelerating the reionization process [1703–1707] though this must be reconciled with constraints from lower-redshift data. Indeed, extremely high ϵ with minimal feedback is required to explain bright z ~ 16 JWST candidates [1708,1709].

These possible explanations deal with the complex physics relating star formation to the evolution of DM halos at high redshifts. However, emerging evidence suggests the issue might lie within our cosmological model. While it is premature to draw any definitive conclusions from these preliminary observations, if neither of the aforementioned possibilities can reconcile theory with the JWST data, it may be necessary to rethink the assumptions of the underlying cosmological model itself.

A common way to quantify discrepancies with JWST data is through the Cumulative Stellar Mass Density (e.g., see Refs. [1636,1656,1657, 1710]), defined as

$$ \rho_{*}(\bar{M}) = \epsilon f_{b} \int_{z_{1}}^{z_{2}} \int_{M}^{\infty} \frac{dn_{h}}{dM} M dM \frac{dV}{V(z_{1}, z_{2})} \quad (2.25) $$

where f_b = Ω_b/Ω_m is the cosmic baryon fraction, ϵ is the efficiency of converting baryons into stars, V is the comoving volume of the Universe between redshift z_2 and z_1 and dn_h/dM is the mass function. For computing the mass function, the halo multiplicity [1019] is needed and one of the most used is the Sheth-Tormen alternative [1020,1122] due to its theoretical motivations for collapsed halos [1711,1712] and multiple successful N-body tests [1024,1713] (even though other phenomenological fitting mass functions exist [1714,1715]). A further assumption often made is to use a Top-Hat window function for smoothing the density field. In focusing on the cosmological framework, one typically sets ϵ to a fixed value (for a conservative approach usually ϵ = 0.2 is enough). In principle, star formation efficiency can be a function of the halo mass [1716] and further adjustments to star formation physics might be needed for more precise computations [1660]. Nevertheless, for massive halos (e.g., CEERS observations [1656]), one
can approximate $\epsilon$ as a smoothly varying power law. If the mass range is short, this approximation does not affect greatly the assumptions for the model. Lastly, the cosmic baryon fraction is usually considered, instead of computing the baryon evolution in different halos [1717–1723]. If $f_b$ is not a function of the halo mass, it plays a role of a multiplicative factor and a greater $f_b$ can only push the theoretical predictions toward the observed data points. All these methodological choices and simplifications are widely used in the literature and allow to present conservative results that can be directly compared with similar works following the same approach.

One strategy for reconciling JWST data is to boost the halo mass function at the tail, thereby producing enough massive galaxies at early-times. This can be achieved with massive primordial black holes [1724,1725] with a fractional contribution of $f_{\text{PBH}} \sim 10^{-3}$, consistent with current constraints [1726,1727]. Super massive primordial black holes not only can accelerate the structure formation process but they could also explain the observed super massive black holes that, otherwise, would have insufficient time to form from Pop III stellar remnants [1728–1736]. Other approaches involve modifying the primordial fluctuations through, for instance, rare density fluctuations from the inflaton field [1737], a blue tilted [1738,1739] or modified [1740] primordial power spectrum, or also introducing non-Gaussianities in the initial condition, affecting the abundance of DM halos [1741]. Barring self-interactions, the properties of DM halos around galaxies imply that $m_a > 10^{-21}$ eV i.e. that this particle is indistinguishable from a cold one on galactic scale [1742].

Axion-related solutions offer a further possibility for understanding JWST's detection of unusually massive galaxies [1743]. For example, axion miniclusters can boost early galaxy formation [1744] though their required masses exceeded super-radiance constraints. Alternatively, axions with masses between $10^{-22}$eV $\le m_a \le 10^{-19}$ eV with delayed oscillations, can produce efficient axion field fragmentation driving the formation of more massive galaxies [1745]. Introducing a fuzzy DM component, usually modeled with axions, in the structure formation process can lead to suppression of small halos and galaxies, thus potentially alleviating tension if paired with altered star formation efficiency [1746]. Yet another axion-like solution is found in axion quark nuggets [1747].

These JWST observations can also be interpreted by invoking PMF [1748] or cosmic string loops with a tension below the limit imposed by CMB observations but above constraints from astrophysical probes (that can be avoided by adjusting the effective low radius cut-off) [1749–1751]. Observed massive galaxies also allow us to revisit the proprieties of Warm Dark Matter (WDM) particles [1752–1754]. Explored in the context of EDE, the fit to JWST observations is improved with respect to the one in the minimal $\Lambda$CDM framework. In fact, EDE leads to an increased $n_s$ and simultaneously raises the fraction of EDE, that allows for a larger number of halos and massive galaxies, and, at the same time, achieving a non-negligible improvement toward easing the Hubble tension [1755–1758]. Another possibility to increase the predicted cumulative stellar mass density in agreement with JWST observations, is to employ alternatives in the DE sector such as phantom crossing in the DE equation of state [1759], a negative cosmological constant [1760–1762] a sign-switching model [1763] or a simple CPL parameterization [1764]. Conversely, interactive dark energy appears to exacerbate the tension [1755].

### 2.3.11. Cosmic voids

**Coordinator:** Dante Paz

**Contributors:** Alice Pisani, Carlos Correa, Cora Uhlemann, Maret Einasto, Mina Ghodsi, Nico Hamaus, Nico Schuster, Sofia Contarini, and Umut Demirbozan

#### 2.3.11.1. Properties and identification of cosmic voids.
Cosmic voids are vast underdense regions of the Universe. Since their discovery [1765–

1768], they have been recognized as powerful cosmological laboratories [1769–1772]. Hence, they encode key information about the expansion history and geometry of the Universe [1772,1773]. The statistical properties of voids depend on two factors: (i) the matter tracers used to map the large-scale structure, namely, galaxies, and (ii) the method used to identify them from the spatial distribution of these tracers. Regarding the first factor, the non-trivial biased relation between galaxies and matter distribution in low-density environments has been studied intensively in recent years [1624,1774–1777]. Regarding (ii), there are different classes of void finders; for a comparison see Ref. [1778]. Some methods exploit the watershed transform technique, see for instance Refs. [1779,1780], others are based on dynamical properties [1781], and some methods identify low integrated density in spherical or free-shape regions, see for instance Ref. [1782]. Despite the intrinsic differences between all the methods, there is a consensus on the basic statistical properties of voids. Roughly speaking, voids are underdense regions with densities as low as 10–20% of the average density of the Universe, their properties described here are almost exclusively governed by gravity [1604,1783–1785], and the motion of tracers around them is consistent with linear dynamics [1786,1787]. They can have diameters starting from few Mpc, spanning several tens of Mpc and the sizes of the largest voids may even exceed hundreds of Mpc [1572,1612,1788]. Such huge voids are delineated by the richest structures - galaxy superclusters, connected by a hierarchical pattern of filaments within these large voids [1612,1624].

#### 2.3.11.2. Void size function as cosmological test.
The evolution of underdense regions in the Universe, particularly isolated spherically symmetric density perturbations, involves several key stages. Initially, these regions expand until internal matter overtakes the outer shells, transitioning from linear to non-linear growth. As voids expand, their density decreases, forming spherical shapes with reverse top-hat density profiles, leading to super-Hubble bubbles with suppressed structure growth and boundary ridges. There are two modes of this void evolution: the void-in-void mode, where regions expand at all scales, and the void-in-cloud mode, where voids expand but surrounding matter density leads to contraction and eventual collapse [1786,1789]. The spherical expansion model, combined with excursion set theory, can model the void size function (VSF) — the number density of voids as a function of their comoving size [1789–1791]. The VSF depends on the power spectrum of fluctuations, the growth rate of structures, and the critical density threshold when expanding shells cross. Using the cosmological model, we can predict the abundance of voids of a given comoving size. By comparing this modeled function with observations of abundances at different redshift bins, it is possible to conduct cosmological tests.

#### 2.3.11.3. The void-galaxy cross-correlation function.
The cross-correlation function for voids and galaxies can also be used to perform an Alcock-Paczynski test, similar to the galaxy correlation function at BAO scales. The one void term can be interpreted as the stacked mean void profile and serves as a standard sphere at different redshifts to test the Universe's geometry and the relations between the angular diameter and the comoving distances. This profile, in combination with a model for RSD, can be used to test cosmological parameters and gravity models [1792–1795].

#### 2.3.11.4. Redshift-space distortions in voids.
The assumption of spherical symmetry for the galaxy distribution around voids breaks down in observations due to RSD acting along the line of sight. The general characteristics of RSD include an elongation of inner correlation contours toward larger scales (due to void expansion) and a flattening of the correlation levels at large scales due to the infall and lower expansion rates at the void outskirts. These anisotropic patterns encode valuable information about void dynamics and the galaxy velocity field. The first model for RSD in voids is the linear model, equivalent to the Kaiser factor in the galaxy autocorrelation [1794]. The pairwise velocity distribution can also be described with the Gaussian streaming
<figure><img src="image_53.png" /><figcaption>**Fig. 46.** Confidence contours from void counts (light blue, [1798]) and void shape distortions (green, [1799]) of the BOSS DR12 voids, and their combination (black) as independent constraints on Ω_m, H₀ and σ_8. Adapted from [1798], licensed under Creative Commons Attribution 4.0 International (CC BY 4.0). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

model [1786]. For the impact of RSD in VSF and void-galaxy cross correlation see for instance [1796,1797].

**2.3.11.5. An independent probe in the landscape of cosmic tensions.** Recent studies have highlighted the potential to test the standard cosmological model using voids in upcoming redshift surveys, such as Euclid [1800–1802]. In a recent analysis presented in Ref. [1798], the void size function in the BOSS DR12 galaxies were examined. Through a combined analysis of void abundances and void-galaxy cross-correlations, these findings proved to be both competitive and compatible with cosmological constraints derived from other probes (see Fig. 46). Furthermore, recent studies of the CMB lensing imprints of cosmic voids have become a valuable test for the cosmological model. In particular, the BOSS survey reported up to 5.3σ [1803], DES reached up to 5.9σ [1804], the WISE–PanSTARRS combination obtained 13.3σ [1805], and the DESI Legacy Survey DR9 achieved up to 17σ [1806], all in excellent agreement with ΛCDM predictions. However, the achieved precision on the parameters does not yet allow for a decisive position in the context of current cosmological tensions. Nevertheless, these results emphasize the significance of cosmic voids as they provide an independent cosmological probe that comes at no extra costs in modern cosmological surveys.

### **2.3.12. Fast radio burst probes of cosmic tensions**

**Coordinator:** Amanda Weltman

**Contributors:** Anthony Walters, Bing Zhang, Christo Venter, Shruti Bhatporia, and Surajit Kalita

**2.3.12.1. Introduction to fast radio burst cosmology.** FRB observations are relatively recently discovered bright transient events (typically observed for a few ms duration) detected in the radio spectrum [1807]. Of particular use to us in cosmology, is the observation that they originate at cosmological distance, apparently from host galaxies at these distances [1808], and thus they provide us with potentially very powerful probes of cosmology [1809,1810] as well as the IGM [1811, 1812].

To date, there have been approximately 800 observed FRBs with information available in the public domain. The majority of these have

been detected by the Canadian Hydrogen Intensity Mapping Experiment (CHIME).<fnref n="12" /> The primary open problem of FRB science is understanding their origins and the progenitor mechanism at work [1813]. Compelling evidence has accumulated for a magnetar engine, as exemplified by the only Galactic FRB 20200428D detected from a magnetar in the MW. On the other hand, some other engines for cosmological FRBs are still possible (see Refs. [1813–1815] for detailed reviews on FRB progenitor mechanisms). Perhaps the greatest open puzzle is the curious observation that while some FRBs appear to repeat, not all of them do, and there does not appear to be any pattern or periodicity to so-called FRB repeaters. While there may be some selection effects at work, it is likely that there are simply different classes of bursts and thus even more to discover about FRBs and with FRBs. Hence, multi-wavelength observations or detections of GWs [1816] and neutrinos from FRB sites will be important to better understand their progenitor mechanisms.

Recent searches for persistent radio emission from one-off and repeating FRBs identified a nearly unresolved source for FRB 20190714 A, with a peak brightness of 53 µJy/beam, marking the first detection of persistent radio emission potentially linked to a non-repeating FRB. Multi-wavelength follow-ups in ultraviolet, optical, X-ray, and gamma-ray bands set upper limits [1817], aiding the distinction between repeating and one-off FRBs. Persistent emission modeling can constrain central source energetics and test magnetar-driven emission scenarios. Follow-ups also refine host galaxy properties, localization, and redshift-dependent FRB characteristics, enhancing their use in cosmology.

**2.3.12.2. Fundamental parameter inference from fast radio bursts.** One of the key parameters in understanding FRBs is the dispersion measure (DM), which gives information about the ionized plasma along the path of the light ray. Broadly it gets contributions from each of the 4 environments along the line of sight, our Galaxy, its halo, the IGM, and the host galaxy as illustrated in Fig. 47. The (1 + z_S) factor in the denominator of the last term accounts for the observed value of redshifted DM<sub>Host</sub>. In general, DM<sub>IGM</sub> contains the information of the underlying cosmology. The average DM<sub>IGM</sub> is given by the following

<footnote n="12">¹² https://www.chime-frb.ca/catalog</footnote>
<figure><img src="image_54.png" /><figcaption>Fig. 47. Illustrative diagram indicating key components of contributing to measured DM of FRBs.</figcaption></figure>

equation (now known as the Macquart relation [1812])

$$ \langle \mathrm{DM}_{\mathrm{IGM}}(z_{\mathrm{S}}) \rangle = \frac{3c\Omega_{\mathrm{b}}H_{0}^{2}}{8\pi Gm_{\mathrm{p}}} \int_{0}^{z_{\mathrm{S}}} \frac{f_{\mathrm{IGM}}(z)\chi(z)(1+z)}{H(z)} \mathrm{d}z, \qquad (2.26) $$

where $\Omega_{\mathrm{b}}$ is the baryonic matter density, $m_{\mathrm{p}}$ is the proton mass, $f_{\mathrm{IGM}}$ is the baryon mass fraction in the IGM, and $\chi(z)$ is the ionization fraction along the line of sight. As $\langle \mathrm{DM}_{\mathrm{IGM}} \rangle$ contains cosmological parameters, this equation can be used to constrain their values. For instance, utilizing a set of mock FRBs, DE equation of state for $\omega$CDM cosmology was constrained [1818,1819] and predicted that with $\mathcal{O}(10^4)$ localized FRBs, this constraint can be tighter than the same obtained from BAO measurements.

FRBs have been observed over a relatively narrow redshift range to date, 100 MHz–8 GHz [1820,1821], and partly this is most likely a result of observation bias, where we can more easily observe the bursts that are closest to us. This has limited their potential in constraining cosmological parameters directly [1809], though using cosmology as a prior allows for IGM constraints, specifically a potential solution to the missing baryon problem [1811,1812]. Moreover, using a specific case of FRB 150418, earlier studies established a limit on the photon mass of $m_{\gamma} < 1.8 \times 10^{-14} \mathrm{eV} \mathrm{c}^{-2}$ [1822] and this bound has been continuously strengthened with the inclusion of additional FRBs [1823,1824]. Furthermore, localized FRBs have been used to constrain the parameterized post-Newtonian (PPN) parameter [1825] and other fundamental constants like the fine-structure constant and the proton-to-electron mass ratio [1826,1827].

### 2.3.12.3. Hubble constant estimations using fast radio bursts.
Several recent studies have leveraged FRBs with measured redshifts to constrain $H_0$ within a Bayesian framework. This approach compares the measured DM$_{IGM}$ of localized FRBs with their theoretical predictions, yielding constraints on $H_0$ and other cosmological parameters. Ref. [1809] was among the first to investigate the utility of FRBs as cosmological probes through simulations. Their mock FRB catalog highlighted the challenge posed by DM variations due to IGM inhomogeneities. Notably, the most significant improvement was observed in the $\Omega_{b,0}h^2$ parameter when combining FRBs with CMB, BAO, SN, and $H_0$ data. Additionally, they found that FRBs offered limited constraints on the DE equation of state, while the curvature parameter ($\Omega_{\kappa}$) showed some improvement when combined with other data. Moving forward, Ref. [1812] leveraged 8 localized FRBs to constrain $\Omega_{b,0} = 0.051^{+0.021}_{-0.025}h^{-1}$ where $h_{70} = H_0/(70 \mathrm{km s}^{-1} \mathrm{Mpc}^{-1})$, consistent with values derived from CMB and BBN data. Another interesting work using a strong lensing effect with 10 FRBs found $H_0 \approx 70 \mathrm{km s}^{-1} \mathrm{Mpc}^{-1}$ [1828].

More recently, as more FRB data are available, these constraints have been revised to a great extent. Ref. [1829] analyzed 9 localized FRBs and found $H_0 = 62.3 \pm 9.1 \mathrm{km s}^{-1} \mathrm{Mpc}^{-1}$, assuming a homogeneous host galaxy contribution, which may limit the complete validation of

the result. Subsequently, Ref. [1830] classified 18 localized FRBs based on host galaxy morphology and employed the IllustrisTNG simulation to estimate individual host contributions. This approach yielded $H_0 = 68.81^{+4.99}_{-4.33} \mathrm{km s}^{-1} \mathrm{Mpc}^{-1}$. However, their model-based host DM values were lower than the actual reported values for some FRBs. Moreover, Ref. [1831] included 16 localized and 60 unlocalized FRBs detected by ASKAP, estimating $H_0 = 73^{+12}_{-8} \mathrm{km s}^{-1} \mathrm{Mpc}^{-1}$. Their larger error bars, despite a substantial number of FRBs, likely reflect the inclusion of systematic uncertainties. Moving beyond, Ref. [1832] incorporated a scatter in the Macquart relation with 78 FRBs (21 localized) and obtained $H_0 = 85.3^{+9.4}_{-8.1} \mathrm{km s}^{-1} \mathrm{Mpc}^{-1}$. Ref. [1833] employed a model-independent method with 18 localized FRBs and found $H_0 = 71 \pm 3 \mathrm{km s}^{-1} \mathrm{Mpc}^{-1}$. Conversely, Ref. [1834] estimated $H_0 = 95.8^{+7.8}_{-9.2} \mathrm{km s}^{-1} \mathrm{Mpc}^{-1}$ using 24 localized FRBs with a wide flat prior on $DM_{\mathrm{Halo}}$. Furthermore, Ref. [1835] combined 12 unlocalized FRBs with BBN data, resulting in $H_0 = 80.4^{+24.1}_{-19.4} \mathrm{km s}^{-1} \mathrm{Mpc}^{-1}$. Additionally, Ref. [1836] used a combination of 18 localized FRBs and the Pantheon dataset to estimate $H_0 = 65.5^{+6.4}_{-5.3} \mathrm{km s}^{-1} \mathrm{Mpc}^{-1}$. Ref. [1837] propose a novel method using ANN architectures for 23 localized FRBs to estimate $H_0 = 67.3 \pm 6.6 \mathrm{km s}^{-1} \mathrm{Mpc}^{-1}$. Furthermore, utilizing Bayesian analysis with different likelihood functions and distinct host distributions for 64 localized FRBs, Ref. [1838] obtained $H_0$ values well above 70 $\mathrm{km s}^{-1} \mathrm{Mpc}^{-1}$ aligning with the late-Universe $H_0$ values with $1\sigma$ error bars no longer overlap with those obtained from early-Universe measurements. More recently, accounting for 98 localized FRBs, Ref. [1839] showed multiple estimates of $H_0$ utilizing different methodologies.

As we see, FRBs do offer preliminary constraints on cosmological parameters, although they currently lack the power to definitively resolve existing tensions in cosmology. While not as stringent as constraints derived from established probes like CMB and SNIa, FRB-derived constraints are significantly tighter than those obtained from GW observations. A primary contributor to the discrepancies in $H_0$ values reported by different researchers lies in the choice of models for IGM and host DM contributions, which remain largely unconstrained from observations. Future surveys aiming to detect a larger number of FRBs, with a substantial fraction localized to their host galaxies, hold promise for significantly improved constraints. These advancements could potentially contribute to resolving the current Hubble tension.

### 2.3.12.4. Understanding nature of dark matter using fast radio bursts.
Beyond constraining the Hubble constant and other cosmological parameters related to the evolution of the Universe, FRBs hold promise for constraining the fraction of primordial mass black holes made up of DM ($f_{\mathrm{PBH}}$). In this context, strong gravitational lensing of FRBs plays a crucial role. When light rays from FRBs pass near a massive object with mass $M_L$, they can be deflected, potentially producing multiple, time-delayed copies of the original burst. The time delays and magnification ($\mu$) of these lensed images provide an estimate of the source's optical depth ($\tau$), which is linked to $f_{\mathrm{PBH}}$ as follows

$$ \tau(M_{\mathrm{L}}, z_{\mathrm{S}}) = \frac{3}{2} f_{\mathrm{PBH}} \Omega_{\mathrm{c}} \int_{0}^{z_{\mathrm{S}}} dz_{\mathrm{L}} \frac{H_{0}^{2}}{c H(z_{\mathrm{L}})} \frac{D_{\mathrm{L}} D_{\mathrm{LS}}}{D_{\mathrm{S}}} (1+z_{\mathrm{L}})^{2} [y_{\mathrm{max}}^{2}(\mu) - y_{\mathrm{min}}^{2}(M_{\mathrm{L}}, z_{\mathrm{L}})] \qquad (2.27) $$

where $D_L$ and $D_S$ are respectively angular diameter distances to the lens object and the source from the observer, $D_{LS}$ is the same between lens and source, $y_{min}$ and $y_{max}$ are minimum and maximum impact parameters, respectively. Ref. [1840] pioneered this method, setting an initial constraint of $f_{\mathrm{PBH}} < 0.08$ assuming FRB lensing by black holes exceeding mass $M > 20 M_\odot$. Subsequent studies incorporated FRB microstructure [1841] and extended mass functions [1842] to refine these bounds. Notably, Ref. [1843] visualized using a null detection of lensed FRBs within a sample of 110 real FRB observations. Further investigations explored the impact of intervening plasma, which mimics the lensing effects. Ref. [1844] demonstrated this using 172 CHIME
FRBs, highlighting the need to account for such decoherence or scattering screens. More recently, Ref. [1845] analyzed 636 FRBs from CHIME and the absence of confirmed lensing events suggesting that MG might introduce a screening effect akin to the plasma scenario. In summary, FRBs offer constraints on $f_{PBH}$ in the mass range of approximately $10^{-4} - 10^4 M_{\odot}$ and these constraints are comparable or even surpassing existing experiments like OGLE, EROS, Icarus, and MACHO.

The landscape of FRB-based constraints is constantly evolving as new and more powerful radio telescopes come online. Arrays such as Hydrogen Intensity and Real-time Analysis eXperiment (HIRAX) [1846], Deep Synoptic Array (DSA)-2000 [1847], Bustling Universe Radio Survey Telescope in Taiwan (BURSTT) [1848] and Square Kilometer Array Observatory (SKAO) [1849] promise to detect a significantly larger number of FRBs, including potentially localized ones. This will undoubtedly lead to further refinements in our understanding of the Universe.

### 2.3.13. Radio background excess

**Coordinator:** Jack Singal

**Contributors:** Alan Kogut, Marco Regis, and Nicolao Fornengo

Throughout most of the electromagnetic spectrum the observed level of surface brightness and anisotropy of the sky background radiation is roughly consistent with that expected from known emission mechanisms from astrophysical and cosmological sources. This is the case for the backgrounds at infrared, e.g., see Ref. [1850], microwave, e.g., see Ref. [1434], optical/UV, e.g., see Ref. [1851], X-ray, e.g., see Ref. [1852], and gamma-ray, e.g., see Ref. [1853] wavelengths. However, in the radio region of the electromagnetic spectrum, both the observed surface brightness level of the photon background and the level of its anisotropy angular power are seemingly much larger than that which can be produced by known source classes, resulting in a tension that is, as of now, an anomaly.

As recently summarized in Ref. [1854], with the known structure resulting from Galactic diffuse emission subtracted, the surface brightness of the level of diffuse background radiation on the sky, from at least ~20 MHz to 3 GHz, as a function of frequency $v$ is, in radiometric temperature units

$$T_{BGND}(v) = 30.4 \pm 2.6K \left( \frac{v}{310 \text{ MHz}} \right)^{-2.66 \pm 0.04} + T_{CMB}, \quad (2.28)$$

where $T_{CMB}$ is the frequency-independent contribution of 2.725 K due to the CMB. This background level has been measured by the Absolute Radiometer for Cosmology, Astrophysics, and Diffuse Emission 2 (ARCADE 2) [1855,1856] at the high frequency end, and several radio maps at lower frequencies from which an absolute zero level calibration can be determined, including recently the Long Wavelength Array (LWA) [1857]. Because the spectral index of -2.6 is consistent with synchrotron radiation, this has been referred to as the radio synchrotron background (RSB). The RSB surface brightness level is shown in radiometric temperature units and spectral energy density units in Fig. 48.

This level of surface brightness is several times higher than can be produced by known classes of radio sources in the Universe. Studies of radio source counts show that the total emission from the known discrete extragalactic sources, particularly AGN and star-forming galaxies, is around a factor of five lower than the surface brightness level of the RSB [1863-1865]. Producing the surface brightness of the RSB with discrete extragalactic sources would require an enormous number of a new class of low-flux sources [1863].

For lines of sight far away from the Galactic plane, the RSB is considerably brighter than the expected contribution from models of Galactic diffuse emission [1866]. The possibility that the RSB originates from a larger spherical halo of radio emission surrounding our Galaxy is highly disfavored by several considerations, e.g., see Ref. [1854,1866]

including that it would make our Galaxy highly atypical [1867]. Observational constraints on the polarization structure of the diffuse radio emission rule out the Local Bubble as an origin scenario [1868].

In a parallel situation to the surface brightness, the anisotropy level of the RSB, as measured at 140 MHz, is also higher than that which could result from models of known source classes [1869,1870]. The measured angular power spectrum of the diffuse radio emission is a power law (in $K^2$ or “$(\Delta T)^2$” units) with increasing $\ell$ with index $\beta = 2.17 \pm 0.08$ from $700 \lesssim \ell \lesssim 4000$ (angular scales from ~2' to ~20'), indicative of unclustered point sources, diffuse sources, or a large number of faint, clustered point sources [1870]. However, its level of anisotropy power rules out the former, given that the density of high flux point sources is constrained by source counts observations such as e.g., [1864]. This leaves very numerous but highly clustered point sources or diffuse sources as the source classes that can seemingly reproduce the level of the observed angular power spectrum of the RSB.

There are other observational constraints on possible origin scenarios for the RSB. Any possible source class and emission mechanism must not:

* overproduce the observed level of the background radiation at far-infrared wavelengths, given the known correlation between radio and far-infrared emission in galaxies, e.g., see Ref. [1871]. Any origin scenario involving galaxies or their components would need this correlation to evolve with redshift [1872,1873].

* overproduce the observed level of the X-ray background through inverse-Compton upscattering of the CMB and/or optical/UV background [1872]. If the RSB indeed originates from a synchrotron process, this puts a lower limit on the strength of the magnetic field in the sources from which the RSB originates of around $1\mu G$ at redshift $z=0$. At higher redshifts this lower limit increases by a factor of up to $(1+z)^4$ due to the increase in the surface brightness level of the CMB at higher redshifts.

* overproduce the observed limits on the 21-cm absorption trough, due to the possible presence of the RSB at redshifts $z \sim 10$ increasing the temperature of the background relative to the temperature of the 21-cm transition, e.g., see Ref. [1874].

* overproduce the observed cross-correlation angular power spectrum between the RSB and optical source catalogs, indicating that the vast majority of its sources, if discrete, must be at $z \gtrsim 0.5$ [1873].

Some RSB origin scenarios that have been suggested include SN of massive population III stars [1875], emission from Alfvén reacceleration in merging galaxy clusters [1876], annihilating DM in halos or filaments [1877-1880] or ultracompact halos [1881], “dark” stars in the early Universe [1882,1883], dense nuggets of quarks [1884], accretion onto primordial black holes [1885-1887], decays to dark photons [1888], dark photon decays [1889,1890], and other injections of photons in the early Universe from processes [1891] including superconducting cosmic strings [1892] and decays of relic neutrinos [1893]. Some of these models likely violate one or more of the observational constraints discussed here, while others contain parameter spaces where they do not.

Upcoming observatories, and observations possible with current facilities, have the potential to further constrain and test RSB origin scenarios. A measurement, such as with the space-based LuSSE-Night [1894] to determine if the spectrum hardens below 10 MHz would be useful in this regard. The radio SZ effect [1895,1896] when combined with the well-known CMB SZ effect, would result in observed cluster radio emission having a null frequency at $700 \lesssim v \lesssim 800$ MHz, below which there would be an increment in the observed surface brightness in the direction of a cluster and above which there would be a decrement, each on the order of $\lesssim 1$ mk (both on top of the contribution from the background RSB). A detection or lack thereof at the required sensitivity in clusters of various redshifts would constrain the redshift(s) of the origin of the RSB. The radio background being far in excess, in both surface brightness and anisotropy angular power,
<figure><img src="image_55.png" /><figcaption>**Fig. 48.** *Left:* The radio sky zero level in radiometric temperature units, reproduced from Ref. [1857], as measured by several different instruments or surveys reporting an absolute zero-level calibration. Results are shown for ARCADE 2 at 3–10 GHz [1855,1856], Reich & Reich at 1.4 GHz, [1858], Ref. [1859] at 408 MHz, Ref. [1860] at 45 MHz, and Ref. [1861] at 22 MHz, as well as several points reported by Ref. [1857]. **Right:** The photon backgrounds in the Universe in units of spectral energy surface brightness density. Reproduced from Ref. [1862].</figcaption></figure>

of that which is expected to be produced by known source classes and processes in the Universe is an intriguing issue.

### 2.3.14. Tension between the large scale bulk flow and the standard cosmological model

**Coordinator:** Richard Watkins

**Contributors:** Hume A. Feldman

The Cosmological Principle requires that the motions of galaxies averaged over a sphere of radius $R$ (bulk flow) should go to zero as $R$ becomes cosmologically large. Thus the determination of the Large-Scale Bulk Flow is an important probe of this principle and of the Standard Cosmological Model more generally. Estimating the bulk flow requires large catalogs of galaxy peculiar velocity measurements; in the last decade, the quantity and quality of this type of data has improved to where bulk flows can provide an important test of our understanding of the Universe.

#### 2.3.14.1. Analyzing the bulk flow using the cosmic flows 4 catalog.
The Cosmic Flows 4 (CF4) peculiar velocity catalog [1424] contains velocities and their uncertainties for 38,057 groups and individual galaxies; this represents most of the peculiar velocity data in existence. Analyzing the bulk flow using this data presents several significant challenges. First, we can only measure the radial component of the peculiar velocity. Second, peculiar velocities typically have large uncertainties; objects in the CF4 catalog typically have uncertainties that are around 15% of the redshift. This means that since distant galaxies have much larger absolute uncertainties, most of the information in the catalog is concentrated nearby. Thus care must be taken to ensure that the bulk flow estimate is not dominated by objects near the center of the volume. Finally, the CF4 survey has a very irregular distribution both radially and on the sky. Thus analyzing the bulk flow using the CF4 requires a method that can “even out” the distribution of the information in the volume of interest both so that the volume is uniformly sampled and so that radial flows do not make spurious contributions. The latter is particularly important given the current tension in the value of the Hubble constant. The use of an incorrect value of the Hubble constant in calculating peculiar velocities can result in phantom radial flows that can contribute to the bulk flow if the volume of interest is not sampled in an isotropic manner.

Ref. [1426] deals with these challenges by analyzing the CF4 catalog using the Minimum Variance (MV) method [1428,1897,1898]. The MV method generates estimates of the bulk flow components, in a volume of radius $R$, that is as close as possible to those that would

be measured from an ideal (isotropic and well sampled) survey; thus it balances the information in the catalog so that the volume is evenly sampled. In addition, a constraint is put on the MV bulk flow estimates so that they are completely independent of the value of the Hubble constant. It should be noted that even though the analysis uses only radial velocity measurements, if it is assumed that the velocity field is irrotational (as it should be if it were generated via gravitational instability), then the MV method estimates reflect averages of the full three-dimensional velocities.

#### 2.3.14.2. The bulk flow tension.
Ref. [1426] determined that the largest radius for which the bulk flow could be estimated with the CF4 catalog is around $200h^{-1}$Mpc. Their bulk flow estimates as a function of radius are shown in Fig. 49 and displayed in Table 3. The figure shows both the estimates and uncertainties for the components of the bulk flow (in galactic coordinates) as well as the magnitude. Included in the figure (in red) is the expected magnitude of the bulk flow as a function of radius calculated using the Cosmological Standard Model. Contrary to expectations, they find that the bulk flow magnitude increases with increasing radius. In addition, they find that the bulk flow has a much larger magnitude than expected in the Cosmological Standard Model [192]. Indeed, they found the probability of obtaining a bulk flow as large or larger in the Standard Model for $R = 200 h^{-1}$Mpc to be $1.5 \times 10^{-6}$, equivalent to about $4.8\sigma$. Ref. [1427] find a similar magnitude for the bulk flow (also using the MV method) but estimate larger uncertainties using cosmological simulation data. Ref. [1425] also analyzes the bulk flow from the CF4 catalog using a very different method and obtains a result that is consistent with the Standard model. However, it is difficult to discern the sensitivity of their bulk flow to different scale motions. Thus the large-scale bulk flow potentially poses a significant challenge to the standard model, but determining the strength of this challenge will require resolving differences in analysis methods.

### 2.3.15. Ultra long period cepheids

**Coordinator:** Ilaria Musella

**Contributors:** Giuliana Fiorentino, Marcella Marconi, Roberto Molinaro, and Vincenzo Ripepi

The Ultra Long Period Cepheids are pulsating stars characterized by periods longer than ~ 80 days with a mean absolute magnitude in the I band $-9 < M_I < -7$ mag. Thanks to their significant brightness they are observable at very long distances, up to cosmologically interesting scales. Thus they could represent competitive standard candles, not
<figure><img src="image_56.png" /><figcaption>**Fig. 49.** The green points with error bars show the bulk flow components and magnitude estimated from the CF4 catalog as a function of radius *R*. The error bars indicate the uncertainty in the estimates due to measurement noise. The dotted blue lines show the theoretical standard deviation of the expected differences between the bulk flow estimates and the bulk flow from an ideal survey calculated using the cosmological standard model (not including measurement noise). The red dashed line indicates the theoretical expectation for the magnitude of the bulk flow calculated using the Cosmological Standard Model.

*Source:* The figure is reproduced with permission from Ref. [1426].</figcaption></figure>

<table><caption>**Table 3**

Summary of Bulk Flows for $R = 150h^{-1}$ Mpc and $R = 200h^{-1}$ Mpc. The uncertainties include both the theoretical difference between the estimate and the bulk flow from an ideal survey and the measurement noise.</caption><thead><tr><th></th><th>R = 150h<sup>-1</sup> Mpc</th><th>R = 200h<sup>-1</sup> Mpc</th></tr></thead><tbody><tr><td>Expectation (km/s)</td><td>139</td><td>120</td></tr><tr><td>Bulk Flow (km/s)</td><td>395 ± 29</td><td>427 ± 37</td></tr><tr><td>Direction</td><td><i>l</i> = 297° ± 4°<br><i>b</i> = -4° ± 3°</td><td><i>l</i> = 298° ± 5°<br><i>b</i> = -7° ± 4°</td></tr><tr><td>χ<sup>2</sup> with 3 d.o.f.</td><td>20.19</td><td>29.84</td></tr><tr><td>Probability</td><td>1.54 × 10<sup>-4</sup></td><td>1.49 × 10<sup>-6</sup></td></tr></tbody></table>

requiring the combination with secondary distance indicators [1899–1905] to reach the Hubble flow. On this basis, using ULPs might reduce the possible effect of systematic errors on the calibration of the extragalactic distance scale and, in turn, on the local determination of $H_0$. Despite their expected important role in distance scale studies, their application as standard candles is still challenging due to the relatively small sample of ULPs with well-sampled light curves covering more than one cycle and some inconsistencies between theoretical evolutionary and pulsational predictions and the observed properties. Indeed, the number of known ULPs is 73 observed in different galaxies. In particular, only one has recently been discovered in our Galaxy in the Gaia DR3 catalog. Their position in the period–luminosity planes and the color-magnitude diagrams suggest that these objects represent the extension at higher luminosity and mass of the Classical Cepheids, but their mass–luminosity relation could be different according to their evolutionary stage. As recently suggested in Ref. [1905], the consistency between ULP and Classical Cepheids relations increases when the photometry accuracy improves, thus supporting the hypothesis that they are the same type of pulsating variables but with different mass and period ranges.

However, at such high luminosity and mass levels, stellar evolution does not predict the occurrence of the blue loop typical of the central helium-burning phase of intermediate-mass stars, so the population of the Instability Strip might correspond to the first crossing toward the RGB. Moreover, the observed pulsation period provides a strong constraint on the stellar mass at fixed temperature and luminosity. Current evolutionary tracks matching the observed position of the ULPs

do not always provide a combination of mass, luminosity, and effective temperature consistent with the observed periodicity.

The agreement between the ULP and Classical Cepheid Wesenheit relations tends to confirm the Hubble constant value obtained based on Classical Cepheids, even if the remaining uncertainties on these objects’ evolutionary and pulsational properties need to be solved and the dispersion of their Wesenheit relation has to be reduced to contribute to the understanding of the Hubble tension. To this aim, future theoretical and observational investigations will include different steps: to better understand the evolutionary status of the ULPs through a detailed comparison with updated sets of evolutionary tracks and isochrones, to produce an extended grid of nonlinear convective pulsation models covering the high mass and luminosity regime expected for these variable start to increase the number of ULPs and improve their photometric accuracy. In this respect, exploring the Gaia database to find new ULPs will be fundamental. Recently, the first MW ULP was found among the variables classified as Long Period Variables in Gaia DR3 and there are likely other misclassified ULPs. In addition, the forthcoming Rubin-LSST survey will also represent a fundamental opportunity to improve the photometry of the known Local Group ULPs and/or increase the sample.

## 3. Data analysis in cosmology

**Coordinator:** Agnieszka Pollo

The volume and complexity of data available for observational cosmology has increased greatly during the last years, and in the coming years, the amount of data generated by different ground-based and space observatories and experiments is expected to grow by orders of magnitude.

For example, the Early Data Release of DESI comprising 2 million spectra of extragalactic sources amounts to 80 TB, and it is only 2% of the expected full final DESI catalog [1906–1908]. Imaging data are even more voluminous: a single exposure of the Dark Energy Camera (DECam) produces an image of the size of a gigabyte [1909]; and the latest DES Data Release 2 [1910] was built upon 76,217 such single-epoch images. Similarly, one night of observations with the Subaru HSC camera results in several hundred gigabytes of data [1911], which translated into ~ 70 TB database already with the first data release of
the Hyper Suprime-Cam Subaru Strategic Program [1912], and several hundreds of terabytes after the third data release [1913].

All these numbers are expected to fade soon with the expected arrival of Euclid [164], Vera Rubin Observatory [897], or SKAO data.<fnref n="13" /> In particular, Vera Rubin Observatory is expected to deliver more than 500 petabytes of imaging data per year, while for SKAOthe data volume is expected to exceed 700 petabytes per year.<fnref n="14" />

In addition to the expected multiwavelength and multimessenger observational or experimental data, there exists a whole additional realm of data useful for cosmology, i.e., simulations. Simulated catalogs are being used to test different cosmological scenarios, to probe mechanisms of baryonic interactions in the dark-matter-dominated Universe, especially at small, non-linear scales; and finally, to create mock realizations of real catalogs which can be used both for physical predictions and for tests of specific measurement techniques against a variety of biases introduced by observational strategy and technique. The creation of simulated catalogs is usually a multi-level process, requiring a lot of computational power and creating amounts of data much larger than the corresponding observational data themselves.

Using all types of data expected in the near future, both observed and simulated, to provide reliable scientific results, requires developing new approaches to data handling and analysis at different levels. Firstly, all these data need to be properly stored and transferred. Ideally, they should also be made publicly accessible in formats that are easy to handle for users from different scientific communities. Datasets observed at different wavelengths or coming from different experiments need to be cross-correlated to facilitate multi-messenger studies. Finally, scientific analysis of these data results in the creation of derived data sets, in amounts often exceeding the original data. Storage, transfer, and subsequent analysis of large amounts of data is not only costly but also requires a well thought out technological and strategic approach. ASTRONET Roadmap for years 2022-2035¹⁵ indicates near-future big data as one of the biggest challenges that astronomy in Europe will face in the coming decade and recommends the development of a “Tiered” approach for data infrastructure, to optimize the use of available resources.

With new big data, unavoidably, come new methods, ideally fully automated and requiring little human intervention (e.g., see Ref. [1914]). It is therefore not surprising that a majority of sections in this chapter deal with different aspects of ML-based approaches to the data analysis, in order to speed up and optimize the science output, and to probe all possible parameter spaces. However, with the growing popularity of these methods, new challenges arise.

An old saying popular in ML community says “rubbish in, rubbish out”; it concisely summarizes the fact that the errors or biases that exist in the original data, also those unknown to the users, are unavoidably transferred to the results. Informed application of any machine-learning methods requires an in-depth understanding of the properties and limitations of the data. A related problem is a common “black box” approach to ML applications. Insufficient understanding of algorithms used may result in users not being aware of their limitations and applicability to a given scientific problem and given data. The next big challenges are related to the interpretability and reproducibility of the results obtained with the aid of machine-learning-based methods. In particular, reproducibility strengthens the importance of keeping codes and data open to the scientific community so that all the scientific results obtained can be independently verified. However, it cannot be forgotten that storage — especially long term storage — and computing power come with a cost, both monetary, and environmental. The

latter aspect draws more and more attention especially now, when commercial ML and artificial intelligence projects fuel an increasing demand for resources.

Observational cosmology has already become an extremely data-intensive endeavor, and the future large, complex and interdependent data sets that will be generated by observatories, space missions, real and mock experiments, theoretical model simulations, and different types of numerical simulations will require new tools and approaches, and close co-operation between different fields of science. It is safe to say that cosmology and extragalactic astrophysics are now only at the beginning of this road.

### 3.1. Bayesian inference methods in cosmology

**Coordinator:** Jesús Torrado

**Contributors:** Alessandro Vadalà, Benjamin l'Huillier, Denitsa Staicova, Guadalupe Cañas-Herrera, Jenny G. Sorce, Laura Herold, Matías Leizerovich, Matteo Martinelli, Ruth Lazkoz, and Susana J. Landau

Modern cosmology has entered an era of precision measurements, where extracting robust constraints on theoretical models from increasingly complex datasets demands sophisticated statistical approaches. Monte Carlo (MC) methods have emerged as the cornerstone of parameter inference in cosmology, allowing researchers to efficiently explore high-dimensional parameter spaces and quantify uncertainties in a fully Bayesian framework. This section presents an overview of the most commonly used MC techniques in cosmological analyses, their strengths and limitations, and practical considerations for their implementation.

The goal of Bayesian parameter inference is to determine the posterior $p(\theta|d, M)$, which is the probability of the parameters of model $M$ taking some value $\theta$, given the data $d$. The posterior can be related to the likelihood of the data $d$ being a realization of the model with given parameter values, i.e., $\mathcal{L}(d|\theta, M)$, via the Bayes theorem

$$p(\theta|d, M) = \frac{\mathcal{L}(d|\theta, M) \pi(\theta, M)}{\mathcal{Z}(d, M)}, \qquad (3.1)$$

where $\pi(\theta, M)$ is the prior and $\mathcal{Z}(d, M)$ the model evidence. Since one typically cares about the constraints imposed on the model parameters, which are independent of the overall normalization of the posterior, the model evidence is often neglected as it contributes only a constant factor. However, as we will discuss below, this evidence becomes important for model selection. In standard parameter estimation contexts, we usually drop the explicit conditioning on the model $M$. When required, lower-dimensional constraints, e.g., Bayesian credible intervals, are easily obtained by marginalization, i.e., integration of the posterior over the remaining parameters. By construction, the model evidence is the marginal likelihood over the full parameter space, $\mathcal{Z}(d) = \int \mathcal{L}(d|\theta)\pi(\theta)d\theta$, determining the probability of the data having been realized under the given model, whatever the values of its parameters (hence their usefulness for model comparison).

There are various Monte Carlo (MC) techniques aimed at determining the posterior $p(\theta|d)$ by means of obtaining a fair sample from it. Below we offer a short review of the most popular MC algorithms used by the astrophysical and cosmological community, which is illustrated in Fig. 50. These approaches have diverse strengths, but also limitations, which we refer to as “sampling problems” below.

Moreover, the choice of prior $\pi(\theta)$ can lead to unwanted (and sometimes unknown) effects like prior dependence or projection/prior volume effects, which we refer to as “modeling problems” and will be discussed below in the context of cosmological tensions.<fnref n="16" />

<footnote n="13">¹³ https://www.skao.int/</footnote>

<footnote n="14">¹⁴ https://www.skao.int/sites/default/files/documents/SKA-TEL-SKO-0001818-01_DataProdSummary-signed_0.pdf</footnote>

<footnote n="15">¹⁵ https://www.astronet-eu.org/wp-content/uploads/2023/05/Astronet_RoadMap2022-2035_Interactive.pdf</footnote>

<footnote n="16">¹⁶ Of course, the modeling of the likelihood $\mathcal{L}(d|\theta)$ is also often based on approximations and can lead to uncertainties. Since the likelihood is the basis of all parameter inference, not only MC approaches, we will not discuss those here.</footnote>