**Particle DM** An intriguing possibility is that DM results from a completely new sector of physics not comprised within the SM. This includes candidates such as sterile neutrinos, axions, supersymmetric particles (gravitinos, neutralinos, axinos), weakly/feebly/strongly interacting massive particles, Q-balls, or ultralight scalars. Generally speaking, DM particles should be (i) non-relativistic at recombination, (ii) neutral or feebly charged, (iii) stable over cosmological timescales, (iv) feebly interacting with the SM. Interactions between the dark and visible sectors can help alleviate cosmological tensions. For example, a model in which dark matter interacts with the baryon sector yields $S_8 = 0.794 (0.804)^{+0.009}^{-0.010}$ for the marginalized posterior (maximum of the full posterior), when fit to Planck 2018, SDSS BAO, and DES-Y3 data at $1\sigma$ CL [2327]. In a scenario where dark matter feebly interacts with photons, a fit to Planck 2018 + SDSS BAO leads to $S_8 = 0.803 \pm 0.021$ and $H_0 = 67.70 \pm 0.43 \text{ km s}^{-1}\text{Mpc}^{-1}$ at $1\sigma$ CL [3396]. While the upper limit for the mass *m* of a particle candidate is the Planck mass, the lower limit for a fermionic species is provided by the Gunn–Tremaine condition [3397], which reads $m \gtrsim 100$ eV from consideration on dwarf spheroidal galaxies [3398]. For a bosonic species, the lower mass limit is often considered as the Hubble rate at matter–radiation equality, $m \gtrsim 10^{-27}$ eV, from the consideration that the ultralight field should oscillate by the time at which the matter component becomes relevant over radiation. Some particle models that have been explored include the following. Explaining the missing mass across cosmic times and scales by a single extension of the known cosmological model has motivated the search for further corroboration of its existence and led particle physicists to investigate potential particle candidates.

**Flavor discrete symmetry models** The SM fails to explain neutrino masses, mixings, and DM with the WIMP paradigm comprehensively. Standing at this juncture, it is certainly a tempting challenge to find a common origin of these two seemingly uncorrelated sectors and much effort goes beyond the SM of particle physics to explore scenarios that can accommodate a candidate of DM and explain non-zero neutrino masses and mixing [3399]. Typically, models based on non-Abelian discrete flavor or modular symmetries are considered to explain observed neutrino mixing, and breaking of the same symmetry may lead to the residual symmetry, which at the same time stabilizes the DM [3399,3400]. For example, in models based on the $A_4$ discrete group, light neutrino masses can be associated with tree-level type-I seesaw mechanism and the one-loop contributions accommodating viable DM candidates responsible for observed relic abundance of DM [3401]. In such models, constraints coming from the lepton flavor violating processes are also important, restricting DM masses and leaving ranges of DM parameters that can be probed in collider studies.

**Supersymmetry** Since *B* and *L*-violating operators in the SM have dimension *d* ≥ 5, the renormalizable Lagrangian possesses an accidental global *B*-*L* symmetry [3402–3405], which is imposed in supersymmetric extensions of the SM to avoid experimental consequences such as a large proton decay rate. Such a symmetry leads to an R-parity which is conserved in scattering and decay processes, leading to supersymmetric particles to be produced in pairs. The same R-parity leads to the stability of the lightest supersymmetric particle, which could be a DM candidate if it is neutral. Supersymmetry, first postulated to address the hierarchy problem, has thus long provided us with a long list of DM candidates [3406]. These include neutralinos [3407–3413], sneutrinos [3414,3415], gravitinos [3416–3426], and axinos [3427, 3428], massive spin-2 particles [3429–3433] resulting from modifications of gravity, and Kaluza–Klein excitations [3434–3436] found in theories incorporating extra dimensions. However, some of the simplest versions of the theory are under stress due to the non-observation of any supersymmetric particles in collider experiments. The lightest neutralino, among others, is an example of a much broader category of particle DM candidates dubbed weakly interacting massive particles (WIMPs), produced thermally in the early Universe.

**Weakly interacting massive particles (WIMPs)** WIMPs, some of the most studied DM candidates, are hypothesized to have been produced in the early Universe as they were initially in thermal equilibrium with the primordial bath. As the Universe expanded and cooled, the production of WIMPs ceased and they “froze out” of thermal equilibrium. The freeze-out temperature is defined as the point when the Hubble expansion rate exceeds the WIMP annihilation rate. The leftover density of WIMPs after freeze-out, i.e., their relic density, depends on their annihilation cross-section. The annihilation cross-section needed for WIMPs to achieve the observed DM density today is naturally of the same order of magnitude as the weak nuclear force cross-section. This coincidence is known as the “WIMP miracle” because it suggests that WIMPs could naturally account for DM without needing highly tuned parameters. Since the WIMP miracle provides a compelling argument for them as the leading DM candidate, many direct and indirect detection experiments are built to probe the typical WIMP mass range of ~$\mathcal{O}(\text{GeV})$ to ~$\mathcal{O}(100 \text{GeV})$ [3375,3376]. Even though direct and indirect experiments continue to constrain the parameter space for the vanilla WIMPs scenario - as for most DM models - exciting new directions in the WIMP paradigm, such as CP-violating DM [3437], multi-component DM [3438], pseudo-Goldstone DM [3439], have been proposed which not only provide viable DM candidates but also address other shortcomings of the SM [3440]. Even if the WIMP miracle is not realized, loopholes exist such as coannihilation [3407,3441,3442], annihilation to slightly heavier states [3441], p-wave annihilation, resonances [3441,3443], and Sommerfeld enhancement [3444,3445].

**Sub-gev thermal DM** The WIMP paradigm can be generalized to lower DM masses and coupling constants, and various sub-GeV DM candidates have been proposed that can also be produced thermally in the early Universe [3446–3448]. In this case, their mass is typically bounded from below to $m > \mathcal{O}(\text{MeV})$ by astrophysical constraints and BBN, while even stronger indirect detection bounds are present for light DM candidates annihilating to SM species via s-wave channels, see the recent review in Ref. [3449]. In other models, however, dedicated direct detection techniques are needed to probe them [3450], and multiple accelerator-based searches are ongoing and planned [3451].

**Sterile neutrinos** A minimal extension of the SM with the addition of a Weyl fermion *N* as a sterile neutrino could provide viable DM candidates. This particle couples to the SM through a Yukawa term such as ~$yN\overline{L}H$, where *L* is a left-handed lepton doublet and *H* is the SM Higgs doublet, which could give rise to the small neutrino masses through the seesaw mechanism [3452–3455] and possible lead to baryogenesis via the leptogenesis mechanism [3456]. Sterile neutrinos of mass $m < m_e$ might decay into three active neutrinos, thus constraining the sterile mixing angle $\theta$ for a given *m* [3457]. The abundance of sterile neutrino DM is fixed through a freeze-in mechanism in which active neutrinos oscillate into *N* before decoupling [3458]. Possible caveats include a finite lepton asymmetry [3459] or the presence of self-interactions [3460–3462].

**The QCD axion** The QCD axion [3463,3464] is a light pseudoscalar coupled with the SM gluons, introduced to solve the strong-CP problem through the Peccei–Quinn mechanism [3465]. The interaction with the gluon leads to a mixing of the axion with the neutral pions and to a small mass $m_a \sim 6 \mu\text{eV}(10^{12} \text{GeV}/f_a)$, where $f_a$ is the energy scale at which the infrared description breaks. Because of its interaction with the gluon that induces a coupling with the charged pions, this particle inevitably couples with the SM photon. Models that generalize the Peccei–Quinn mechanism leading to a very light axion and a prediction for the axion–photon coupling $g_{a\gamma\gamma}$ have been long studied and constitute the benchmark for laboratory axion searches, such as the KSVZ [3466,3467] and the DFSZ axion [3468,3469]. A nonrelativistic population of cosmic QCD axions is produced through the vacuum misalignment mechanism [3470–3472], leading to a number density in axions with a large occupation number at the time $T_{\text{osc}} \sim \text{GeV}$ at which coherent oscillations in the field begin. This population of
axions can possibly address the DM abundance for given values of the mass $m_a$ and of the initial value of the axion field $\theta_i f_a$ [3473–3475]. The uncertainties in the assessment of the axion parameters can be divided into cosmological [3476,3477], particle content [3478,3479], and theoretical [3480], see Refs. [2591,3481] for reviews. A robust experimental search is currently ongoing worldwide to try and detect this particle [2596,3482].

**Ultralight bosons** Ultralight bosons or “fuzzy” DM (FDM) with masses in the range $m \sim 10^{-23}–10^{-18}$ eV would possess a de Broglie wavelength of astronomical size [2371,3483,3484]. An ultralight scalar, naturally generated in string theory, is generally considered to be real or pseudo-scalar field minimally coupled to the metric, and acquires a small mass from various UV mechanisms [3331,3485]. When the Universe cools below a certain critical temperature, the field starts to roll down and oscillates about the minimum of its potential [2590,3486–3488]. N-body cosmological simulations have shown the formation of a DM core within each virialized halo, which is supported against gravitational collapse by the internal quantum pressure due to the Heisenberg uncertainty principle on the de Broglie scale, surrounded by an interference pattern representing the oscillation in the density and velocity fields of the DM particles [2383,3489–3493]. A lower bound on the boson mass is derived by considering the quantum pressure introduced at cosmological scales, which suppresses the power spectrum of linear inhomogeneities at large wavelengths and leads to a lower bound $m \gtrsim 10^{-21}$ eV [1551,1552,1563,3494,3495]. Stronger constraints on the mass and the abundance of ultralight scalars are placed from a variety of observations which involve detection strategies at astronomical or cosmological scales [2336,3378,3496–3504], which could nevertheless depend on the details of cosmology and the astrophysics involved. An issue with FDM candidates lies in that such particles cannot produce the very large regions of about constant density observed at the center of large galaxies [1742,3505]. Ultralight bosons coupled to a dark energy fluid can help relieve both the $H_0$ and $S_8$ tensions. For instance, a model fit to Planck 2018, lensing, Pantheon, SH0ES, and WiggleZ LSS data results in $H_0 = 68.81_{-0.67}^{+1.60}$ km s$^{-1}$Mpc$^{-1}$ and $S_8 = 0.7993_{-0.0140}^{+0.0410}$ at $1\sigma$ CL [3506].

#### 4.4.1.3. Experimental efforts.

**Laboratory searches** Given the vast landscape of particle DM candidates, several techniques have been implemented for their searches. Here, we highlight the detection of WIMPs and axions in underground detectors.

If DM is in the form of WIMPs with a small coupling to SM nucleons, it is possible to detect a small event rate $R$ based on DM-nucleon elastic scattering, totaling $R \sim N_T n_{DM} \sigma_A v$. Here, $N_T$ is the number of nucleon targets in the experiment, $n_{DM}$ is the number density of DM in the Solar system, $v \sim 220$ km/s is the relative velocity, and $\sigma_A$ is the scattering cross section off a nucleon of atomic number $A$. For spin-independent (SI) interactions, the DM scattering cross section is often quoted in terms of $\sigma_{SI} \approx \sigma_A/A^4$ [3375,3507]. For spin-dependent (SD) scatterings, the enhancement with the target mass is more modest and reads $\sigma_A \approx \sigma_{SD} A^2(J+1)/J$, where $J$ is the spin of the nucleus. The most sensitive experiments include liquid xenon-based detectors: XENONnT [3508], LUX-ZEPLIN (LZ) [3509], and PandaX-4T [3510], which all set a bound on the SI DM-nucleon cross section $\sigma_{SI} \lesssim 10^{-47}$ cm$^2$ for the DM mass $m \sim 40$ GeV. The argon-based detector DarkSide-50 [3511] is the most sensitive in targeting the SI cross section for WIMP masses below about 3 GeV, with proposed experiments including DarkSide-20k [3512] and ArDM [3513]. Note, that the results depend on several assumptions on: (i) the distribution of DM in the solar system and around the Earth, (ii) the universal coupling between the WIMP and the nucleons, (iii) the mass range of the mediator, (iv) the knowledge of the nuclear form factors, (v) the effective theory used to model the interaction, (vi) the absence of self-interactions, (vii) the scattering process being elastic. All of these caveats have been thoroughly investigated in the literature in relation to realistic DM models.

The laboratory search for the QCD axion and other light bosons proceeds by means of resonant cavities in a “haloscope” [3514,3515], light shining through wall experiments [3516–3519], dielectric haloscopes [3520], and DM radio searches [3521]. In a resonant cavity, a galactic axion converts into a microwave photon through the interaction with the virtual photon of an external magnetic field [3522–3528]. DM radio searches are employed in DMRadio/ABRACADABRA [3529] and, for light axions not related to the QCD theory, with SuperMAG [3530]. Scalar fields produced in the Sun are searched through helioscopes: CAST [3531] and IAXO [3532].

**Indirect detection** Indirect detection of DM involves observing the potential byproducts of DM interactions rather than revealing a new particle. This method relies on DM particles annihilating or decaying into SM particles, which are then detected in astronomical and terrestrial instruments. The byproducts of these interactions typically include (1) gamma rays [3533,3534], high-energy photons that can travel vast distances and are easier to trace back to their source; (2) Neutrinos [3535–3537], which are nearly massless and interact very weakly with matter, making them hard to detect but potentially useful for indirect detection; and (3) Charged cosmic rays [3538–3540] such as electrons, positrons, protons, and antiprotons. An excess of these particles in cosmic rays could indicate DM interactions [3541].

Different detectors and observatories are used to identify the potential signals from DM interactions. Instruments like the Fermi Gamma-ray Space Telescope observe the sky for gamma-ray emissions, focusing on high DM density regions like the Galactic Center or dwarf spheroidal and dwarf spirals [3542–3546]. Facilities such as IceCube in Antarctica detect neutrinos by observing the Cherenkov radiation produced when neutrinos interact with ice [3547,3548]. Instruments like the Alpha Magnetic Spectrometer (AMS-02) on the International Space Station measure the flux of cosmic rays, looking for anomalies that could indicate DM annihilation or decay [3549].

One key challenge for indirect detection is to distinguish potential DM signals from astrophysical backgrounds. For instance, gamma rays can be produced by various astrophysical sources like pulsars, supernova remnants, or AGN, while neutrinos are produced in the Sun and in the atmosphere. Cosmic rays have numerous sources, including the Sun and SN, making it difficult to attribute any observed excess to DM. For this, fingerprints are searched through cross-correlation studies using data from different wavelengths and types of particles to identify coincident excesses that could point at DM, and spectrum analysis where the energy spectrum of detected particles is analyzed to identify features characteristic of DM annihilation or decay.

Currently, there is no definitive detection of DM through indirect methods, but some intriguing signals and excesses warrant further investigation. For instance, Fermi has observed a gamma-ray excess in the Galactic Center which could be due to DM annihilation [3550]. Also, the AMS-02 has detected an excess of positrons in cosmic rays, which might indicate DM [3549], although astrophysical sources like pulsars are a competing explanation. Future instruments and missions, such as the Cherenkov Telescope Array (CTA) [3551–3555] and the next-generation neutrino observatories will provide more sensitive measurements and potentially offer clearer insights into the nature of DM.

Inferring the distribution and amount of DM by its gravitational interactions is a complementary route to the detection of byproducts. Gravitational lensing is a very suitable probe because it only relies on light deflection, with no further need for the modeling of luminous matter effects. Since the first detection of a strongly lensed QSO [3556, 3557], strong gravitational lensing has successfully constrained masses of galaxies and galaxy clusters to about 10% [3558,3559]. While the initial lens modeling of the light-deflecting mass density profiles overestimated the DM inferred from strong lensing, increasing complexity of the lens models has reduced the amount of DM necessary to produce the observed, highly magnified and distorted images of background
galaxies [3560]. Yet, open questions still remain whether the amount of substructures observed in galaxy clusters agree with $\Lambda$CDM [3561–3565].

### 4.4.2. Warm dark matter

**Coordinator:** Supriya Pan

**Contributors:** Biswajit Karmakar, Branko Dragovic, Cláudio Gomes, Cora Uhlemann, Davide Pedrotti, Emmanuel N. Saridakis, Ioannis D. Galamas, Janusz Gluza, Massimiliano Romanello, Mehmet Demirci, Pran Nath, Reggie C. Pantig, Riccardo Della Monica, Torsten Bringmann, Venus Keus, Weiqiang Yang, and Wojciech Hellwing

#### 4.4.2.1. Constraints on WDM from observation of high-redshift galaxies and reionization.
The introduction of WDM particles, with a mass of the order of a few keV, can alleviate some of the challenges of the $\Lambda$CDM model at the kpc scales, namely the Missing Satellites problem or the Too-Big-to-Fail problem [2354]. Indeed, during the structure formation process, the higher thermal velocity of non-collisional WDM particles determines a flow from overdense to underdense regions, canceling the cosmic perturbations with a physical size smaller than the so-called free streaming length, $\lambda_{\text{FS}}$. This leads to a suppression in the matter power spectrum, $P(k)$, which becomes more important with decreasing WDM particle mass, $m_\chi$, and has macroscopic consequences on the number density of low-mass haloes.

In addition to baryonic effects related to the star formation efficiency in DM haloes, small-scale modifications in the halo mass function can also alter the high-$z$ number density of faint galaxies, resulting in a turn-over in the rest frame galaxy ultraviolet LF (UV LF) [3566,3567] and in a delay in the galaxy formation [3568,3569]. From the comparison between the abundance of faint galaxies and the number density of DM haloes predicted in the context of WDM cosmologies, it is possible to derive a lower limit on $m_\chi$, independent of the baryonic physics involved in the galaxy formation process [3570].

In turn, the progressive steepening of the UV LF faint-end slope at high redshift confirms the fundamental role of faint galaxies during the EoR, when the intergalactic hydrogen passed from a neutral to an ionized state for the effect of energetic photons emitted by the primeval sources of light, at $z \gtrsim 6$. Indeed, the UV LF is directly linked to the star formation rate (SFR) [3571] and so to the existence of a hot stellar population that acts as a source of ionizing photons responsible for the reheating of the IGM [3572–3574]. In particular, their number density is given by $\dot{N}_{\text{ion}} = f_{\text{esc}} \dot{\epsilon}_{\text{ion}} \rho_{\text{UV}}$, where the UV luminosity density, $\rho_{\text{UV}}$, derived from the integral of the UV LF, is multiplied by two quantities, namely the ionizing photon production efficiency, $\dot{\epsilon}_{\text{ion}}$, and the escape fraction, $f_{\text{esc}}$. The former describes how efficiently it is possible to get ionizing photons from a UV continuum radiation field and depends on different astrophysical quantities, such as the initial mass function (IMF), the metallicity and the age of the stellar population. The latter is defined as the fraction of ionizing photons that escape from the emitting galaxies and contribute to the phase transition of the IGM. Due to the lack of information about the geometry of the ISM in high-$z$ galaxies, $f_{\text{esc}}$ is not well constrained and it is difficult to model properly. Moreover, as a multiplicative factor of the total ionizing photons budget, it degenerates with the WDM particles mass, thus summarizing most of our uncertainties about the reconstruction of the reionization history in both $\Lambda$CDM and $\Lambda$WDM scenarios [3575,3576]. Despite these astrophysical uncertainties, constraints on $m_\chi$ can be obtained through observations of the LF [3577–3579] and the SFR density [1754] up to high redshift, and from the comparison of the expected electron scattering optical depth, $\tau_{\text{es}}$, positive with an ionized IGM, with empirical data from Planck [3580].

#### 4.4.2.2. Candidates.
Traditionally, WDM is defined as a fermion with 2 degrees of freedom that decouples relativistically and has a temperature to match the observed DM relic density with zero chemical potential.

This allows to express observational limits on the free-streaming length as constraints on the mass of the DM particle. Lyman-$\alpha$ observations, for example, firmly exclude $m_{\text{WDM}} > 1.9$ keV [3581] (though sometimes more aggressive bounds have been derived, reaching up to $m_{\text{WDM}} > 5.3$ keV [1553]).

From the theoretical point of view, an excellent WDM candidate consists of a fermion that is a singlet under the standard model gauge group, commonly referred to as sterile (or right-handed) neutrino [3582,3583]. The resulting mixing with active neutrinos, through the neutrino portal, is constrained to mixing angles too small for sterile neutrinos to fully thermalize in the early Universe. Instead, DM production proceeds via oscillations, combined with neutral and charged current interactions of the active neutrinos with the SM heat bath [3458]. Parameter regions for which this mechanism produces the correct DM abundance in its simplest form are excluded [3584]. These bounds can be evaded by enhancing the production through oscillations by introducing a large primordial lepton asymmetry [3459], new self-interactions of the SM [3461,3585] or sterile neutrinos [3462,3586]. Further alternative scenarios that remain viable include the decay of a new scalar state combined with subsequent entropy injection [3587–3589] and thermal production via extended gauge sectors [3590–3593]. Common to these scenarios is that the WDM mass bounds in general become model-dependent, and therefore have to be re-expressed by using model-independent bounds on the free-streaming length (for the case of self-interacting sterile neutrinos, notably, the bound on the sound-horizon due to late kinetic decoupling can be more stringent [3462]).

Fuzzy dark matter (FDM): Fuzzy DM (FDM), also known as wave DM, quantum wave dark matter, or ultra-light axion DM, is an interesting area of research that bridges concepts from quantum mechanics, astrophysics, and cosmology, which offers potential solutions to some longstanding puzzles in our understanding of the Universe. It is a theoretical model proposing that DM consists of ultra-light particles with boson masses on the order of $m_b = 10^{-22}$ eV [2383,3486].

Unlike traditional particle DM models, fuzzy DM behaves more like a wave due to its extremely low mass. This wave-like nature leads to unique quantum mechanical effects on large scales [3594,3595]. This wave-like behavior of FDM results in a quantum pressure that opposes gravitational collapse on small scales. This can suppress the formation of small-scale structures, potentially addressing the “missing satellites problem” observed in the MW and other galaxies. In terms of halo structure, the density of DM halos is predicted to have a central core with a soliton-like structure. This differs from the sharp cusps predicted by CDM models and can help resolve some discrepancies between observations and simulations of galactic cores. While ultra-light, FDM behaves like a CDM on cosmological scales, providing a good fit for large-scale structure formation and CMB observations. Some works show hints that ultralight axions can improve consistency between CMB and galaxy clustering data, potentially reducing the $S_8$ tension [3596]. Tensions between the former and Lyman-alpha constraints could be alleviated with a component of ultralight or WDM [1562]. Regarding astrophysical tests, precise measurements of the distribution and movement of stars in galaxies, the structure of DM halos, and gravitational lensing effects can be done. The low mass end for ultralight DM has been ruled out by the dynamics of ultra-faint dwarf galaxies based on stellar kinematics in virialized halos [3502] and the Lyman-alpha forest [1551] using hydrodynamical simulations, where uncertainties in the modeling remain. Recently, constraints on the soliton mass using the Event Horizon Telescope (EHT) results for the black hole shadows of M87* and Sgr. A* was found [3597], confirming the theoretical model of FDM. It was also tested and constrained using the motion of the S2 star around Sgr. A* [3504]. How the FDM soliton evolves while being accreted on a black hole helps determine the stages of accretion flow [3598]. Not only for black holes, the effect of solitonic quantum wave DM was also explored through wormhole
solutions [3599]. Indeed, these studies explored how the effects of FDM can manifest through astrophysical objects, acting as an alternative method for Earth-based laboratory detection.

Based on the existing literature, it is clear that WDM models are potentially reached since some indications for alleviating the $S_8$ tension have been noted. However, the models in this category are not fully explored widely unlike other cosmological models. We anticipate that WDM remains a potentially rich area that needs to be investigated widely aiming to understand how the tensions on $H_0$ and $S_8$ can be relieved.

### 4.4.3. Interacting and decaying dark matter

**Coordinator:** Elsa M. Teixeira

**Contributors:** Amare Abebe, Amin Aboubrahim, Andrzej Borowiec, Anil Kumar Yadav, Branko Dragovic, Brooks Thomas, Davide Pedrotti, Emmanuel N. Saridakis, Emre Özülker, Gaspard Poulot, Ioannis D. Gialamas, Ivonne Zavala Carrasco, Jose A. R. Cembranos, Keith R. Dienes, Luis Anchordoqui, Marcin Postolak, Nikolaos E. Mavromatos, Paolo Salucci, Pran Nath, Rafael C. Nunes, Reggie C. Pantig, Riccardo Della Monica, Rishav Roshan, Simony Santos da Costa, Tomi Koivisto, Torsten Bringmann, Víctor H. Cárdenas, Vasiliki A. Mitsou, Venus Keus, Vivian Poulin, William Giarè, and Wojciech Hellwing

#### 4.4.3.1. Interacting dark matter.
Models of IDM replace the CDM component in ΛCDM by posing either a self-interaction between DM particles or interactions between DM and the other species in the Universe. Consequently, the mass of the DM particles bears some dependence on such interaction, defined in an FLRW background as

$$ \dot{\rho}_{\text{IDM}} + 3H\rho_{\text{IDM}} = Q_i, \qquad (4.20) $$

where $Q_i$ embodies the interaction between DM and the ith species with energy density $\rho_i$ (whose conservation equation will include a symmetric interacting term to ensure energy conservation). One key feature in favor of several models of IDM is that they can exhibit suppression of the matter power spectrum in relation to its ΛCDM counterpart. This effect is relevant in terms of attempts to address the cosmological tensions, particularly the $S_8$ tension, even upon including constraints from the CMB data. In the following, we present various types of IDM models and discuss their relevance for the resolution of cosmic tensions.

**Dark matter interactions with baryonic matter** Theoretical models often consider DM-baryonic matter (DM-BM) scattering to modify the evolution of density perturbations. For instance, velocity-dependent cross-sections in these interactions can impact the growth rate of cosmic structures, affecting the matter power spectrum and, hence, the derived value for the $S_8$ parameter [2327,3600,3601]. These interactions may also modify the reionization history through changes in the ionization fraction, optical depth, and CMB data-derived cosmological parameters [3602]. Recent simulations suggest even weak DM-BM interactions can leave observable effects on large-scale structures and the Lyman-α forest [3603]. Screened DM models propose local environment-dependent interaction strengths [3263] while models like matter bounce scenarios propose alternatives to the standard inflationary paradigm [3604,3605]. These DM-BM interaction models must balance improving cosmological fits without violating constraints from other astrophysical observations, such as galaxy clustering and WL surveys, thereby representing a promising but challenging research avenue [3606]. Further observational data from upcoming missions and theoretical refinement are needed to assess their viability as a potential solution to the cosmological discrepancies [3607].

Recent astrophysical evidence shows discrepancies between observed mass distributions in galaxies and predictions from the collisionless ΛCDM model [3375]. Notably, a constant density region is often found in galaxies' innermost areas, contrary to the expected cuspy density profile [358,2355,2357,2364,3608,3609]. Additionally,

DM halo parameters correlate with the structural parameters of the luminous component, such as the half-light radius $R_{50}$ and stellar mass $M_*$ [3610–3614]. This entanglement between the dark and luminous components suggests some interaction between DM and BM [3615], occurring only where the product of DM and BM densities exceeds a threshold [3615,3616]. Outside these regions, DM halos retain their original NFW profiles. The DM-BM interaction, observed over several Gyrs, does not affect the formation of galaxy dark halos but gradually modifies the halo density over time, erasing the initial cusp. This process may be observable in the recent mass distribution of high-redshift galaxies [3615]. While the macroscopic effects of this interaction are clearer, the microscopic processes remain uncertain. Numerous candidates for IDM have been proposed (e.g., see Refs. [3617–3620]), and investigations continue. Observations suggest these interactions could occur in secluded regions of galaxies, such as stars, molecular clouds, neutron stars, black holes, and central supermassive black holes, aligning with various studies (e.g., see Refs. [3621,3622]).

**Dark matter interactions with photons** In Ref. [2674], an extension of the ΛCDM model is proposed where DM couples with photons ($\gamma$), resulting in a nonconservation of particle numbers. This model suggests that DM particles gradually decay into photons over time without significant scattering processes, slightly deviating from the standard cosmic evolution. It offers a potential solution to the cosmic tensions (see Fig. 4). Based on the theoretical study of couplings between DM and relativistic relics in Ref. [3623], a lower limit on the DM mass has been derived [3624] for DM-$\gamma$ interactions: $m_\gamma > 8.7$ keV at 95% CL (assuming it saturates 100% of the observed relic density). Alternatively, one could derive a bound on the abundance of such DM candidates depending on the mass [3593]. Observational constraints on DM-$\gamma$ scattering cross-section were derived in Ref. [3625], with $\sigma_{\text{DM-$\gamma$}} \le 2.25 \times 10^{-6} \sigma_{\text{Th}}$ ($m_{\text{DM}/\text{GeV}}$) at 95% CL. A multi-IDM framework, including interactions with photons, was explored in Ref. [3396], simultaneously addressing the $H_0$ and $S_8$ tensions. The prospects for testing DM-$\gamma$ interactions through CMB spectral distortions have been discussed in Ref. [3626]. The black hole shadow, a dark region observed against a bright background, is closely related to the photonsphere, the region where photons orbit the black hole. By performing backward tracing techniques, one can study how the shadow deviates from the classical prediction due to DM effects. Studies like Refs. [3627,3628] have shown that DM-$\gamma$ interactions in the photonsphere can alter the shadow's shape, illustrating how astrophysical objects can be used as independent DM probes. Numerous studies have since explored different DM models and their effects on black hole properties [3504, 3597,3629–3658], concluding that significant DM concentrations are necessary for observable effects on a black hole's shadow [3659].

**Dark matter interaction with dark radiation** The possibility that the multiple cosmic discrepancies are tied to the existence of a dark sector filled with a thermal bath of dark relativistic species, also known as dark radiation (DR), interacting with DM has been widely explored (e.g., see Ref. [2561,2565,2673,2674,3660–3667]). In the standard model, the effective number of neutrinos accounting for the three neutrino species, $N_{\text{eff}} = 3$. Still, the presence of DR can modify this value, reducing the sound horizon and adjusting the inverse distance ladder calibration of the BAO and SNIa. This is discussed extensively in Section 4.1.3.

The DR-DM interaction has two primary effects: it introduces a drag term in the DM Euler equation, reducing the growth of DM perturbations and potentially explaining the low $S_8$ values measured by WL surveys; and it influences DR perturbation dynamics, reducing their free-streaming, which can result in a larger contribution to $N_{\text{eff}}$ and align with $H_0$ measurements from the $SH_0\text{ES}$ experiment. Various models embody these ideas, such as the “non-Abelian DM model” [3660,3661,3663], “partially acoustic DM” [3662], “cannibal DM” [3664], stepped-dark radiation [2561,2565,2673,3665–3667], and neutrino-DM interaction [2329]. Interestingly, these models may
also address other CDM issues, such as the cusp/core problem, the so-called “too big to fail” (TBTF) issue, and the missing satellites problem (see Ref. [2354] for a review). Indeed, Ref. [3668] is the first particle-physics proposal to address all three problems simultaneously. The model in Ref. [3669] integrates this into DR as sterile neutrinos, introducing late-time effects that influence $H_0/\sigma_8$. Ref. [3670] offers a comprehensive Lagrangian-level classification of DM-DR interaction models, showing how CDM can mimic a WDM cut-off in the matter power spectrum. Alternatively, model-independent analyses can be conducted through the “effective field theory of structure formation” (ETHOS) [2308,2309], which provides a mapping from Lagrangian parameters of DM-DR models to phenomenological parameters of the linear power spectrum, with numerical simulations extending this to the full non-linear power spectrum.

**Dark matter interaction with dark energy** Interactions within the dark sector may help resolve various tensions simultaneously by leveraging on the unknown nature of DE and DM (see Section 4.2.3 for a more thorough account of IDE models). For instance, in Refs. [3671–3673], the authors show that by applying fluid approximation methods for rapidly oscillating scalar field DM coupled to scalar field DE through a Lagrangian approach, DM can effectively be treated as a fluid on cosmological scales under the appropriate Hubble-averaged behavior. These models are expected to help alleviate cosmic tensions, namely on $S_8$, due to their contributions in suppressing the matter power spectrum on small scales, and on $H_0$ through the modified background dynamics.

A recent paper proposed a cosmological model of DM and DE based on a particle physics Lagrangian [3506] to address cosmic tensions. The model involves two interacting ultralight scalar fields, one for DM and one for DE, with a coupling between them and was tested against various cosmological data sets, including Planck [192, 729,1434], BAO [777,797,1145,2515,3674,3675], and Pantheon and SH0ES data [33,34]. It demonstrated an ability to alleviate the $H_0$ tension, reducing the discrepancy to about ~2σ, and also resolved the $S_8$ tension.

### 4.4.3.2. Decaying dark matter.
Another approach posits instead that a fraction of the DM sector is undergoing a decay process into another species, falling under the broad heading of decaying DM (DDM) models. This scenario is motivated by theoretical considerations, can explain specific experimental results, and addresses small-scale issues in the CDM paradigm. Initially developed in the 1980s, the DDM model has gained renewed interest in light of cosmic tensions. Accordingly, such models conventionally contain two extra parameters: the lifetime of DDM $\tau$ plus the initial fraction of DDM to total DM $f_{DDM}$. The decay is expressed in FLRW as

$$ \dot{\rho}_{\text{DDM}} + 3H\rho_{\text{DDM}} = -\Gamma_i\rho_{\text{DDM}}, \quad (4.21) $$

where $\Gamma_i$ is the decay width of DDM (related to its lifetime as $\Gamma = \tau^{-1}$) into the species $\rho_i$. There are several variations of the DDM hypothesis. However, these scenarios alone are unlikely to resolve the $H_0$ tension, which would require, for example, assuming an earlier decay of DM.

**Dark matter decays into dark decay products** Various scenarios involving the decay of an unstable component of multicomponent DM into dark radiation have been proposed to address the $H_0$ and $S_8$ tensions [2571,3676–3683]. For short-lived particles ($\Gamma \gtrsim 10^6$ Gyr$^{-1}$), DM decays into dark radiation at early-times ($t \ll t_{LS} \sim 1.17 \times 10^{13}$ s), where $t_{LS}$ denotes the time of last scattering), increasing the expansion rate and reducing the sound horizon size $r_s$, thus increasing $H_0$ [2571,3676,3677]. Since the angular size of the sound horizon at the last scattering surface $\theta_{LS} \equiv r_{LS}/D_M(t_{LS})$ is a CMB observable that must be kept fixed, a reduction of $r_{LS}$ simultaneously decreases the comoving angular diameter distance from a present-day observer to the last scattering surface $D_M(t_{LS})$, and increases $H_0$. For long-lived particles, DM is depleted into radiation after $t_{LS}$, shifting matter-DE equality to earlier times and allowing a late-time increase in $H_0$ [3678, 3679]. Moreover, two-body decays that transfer energy from DM to DR,

reducing the late Universe matter content, help accommodate local $S_8$ measurements [3680–3682]. For $\Gamma \gtrsim H_0 \sim 0.7$ Gyr$^{-1}$, most unstable DM particles have decayed by redshift $z = 3$, impacting observations like those from IceCube if sterile neutrinos play the role of DR [3684, 3685]. If $\Gamma \lesssim H_0$, only a fraction of DM particles have decayed. Recent CMB data severely constrain the fraction of unstable DM in these scenarios [3593,3686–3691]. Short-lived particles are constrained by CMB polarization [3690,3691], and low redshift DM depletion reduces CMB lensing power, conflicting with Planck data [3686–3688]. Including BAO measurements further tighten constraints on the fraction of long-lived particles [3689,3690]. Current bounds make a decaying DM solution to the $H_0$ tension unlikely, though a combination of scenarios with multiple decaying DM particles (for example decays into a combination CDM and WDM [3692]) at different epochs may effectively ease this tension.

More promising is the role that DDM can play regarding the $S_8$ tension, especially in models where the (dark) decay products are massive particles. These products get a velocity kick $\epsilon \equiv v/c$ from the decay, suppressing structure growth on scales below their free-streaming length while behaving like regular CDM at the background level, thus avoiding the strong constraints from BAO data. These scenarios have been tested against various cosmological data [3681,3682], including the full shape of BOSS power spectrum [3693] and KiDS WL measurements [3694], and shown to reduce the $S_8$ tension to ~1.5σ for lifetimes $\Gamma^{-1} \approx 120$ Gyr and velocity kick $\epsilon \approx 1.2\%$ [3693]. In addition, depending on the lifetime and kinetic energy of the decay, DDM can address or mitigate various tensions related to small-scale structures and galaxy formation [3695–3699] such as the missing satellites problem [2360,2361], by estimating the power spectrum cut-off scale.

It is also worthwhile to move beyond the specific idea of decaying particle DM and consider a more generic parameterization to describe the potential conversion of a DM component to DR, with a transition rate that may not necessarily follow exponential decay [2326,2410]. This approach includes traditional decaying DM but also encompasses scenarios such as merging primordial black holes (PBHs) or late-time DM self-annihilation in the presence of long-range forces. In these generalized scenarios, the $S_8$ and $H_0$ tensions can be mitigated in somewhat different ways, though a complete resolution of these tensions remains unlikely.

**Dynamical dark matter** DDM models the decay of a DM ensemble across epochs, balancing the collective lifetimes against cosmological abundances among various DM components [3700]. One realization of such a DDM ensemble that has been extensively studied consists of a tower of dark Kaluza-Klein states associated with a large extra dimension; such a tower then allows for the possibility of “intra-ensemble” decays from heavier to lighter dark states [3701,3702]. Such decays can modify the DM velocity distribution over time [3702] and potentially address the $S_8$ tension [3703,3704]. This idea has recently found new relevance within the Swampland program [3705], which suggests an extra mesoscopic dimension (“dark dimension”) in the micron range [2923], leading to a tower of weakly interacting light DM particles which are KK excitations of the graviton [3706]. While DDM scenarios of this sort may potentially address the $S_8$ tension, they are also subject to constraints on the expansion rate at late-times from, e.g., SNIa [3707].

**Dark sectors with varying equations of state** In the case of soft DM and soft DE [3708,3709] one considers that effectively the dark sectors have a different equation of state at large scales, namely at scales entering the Friedmann equations, and a different one at intermediate scales, namely at scales entering the perturbation equations, features that are typical in soft materials [3710]. In this case, the perturbative-level properties are slightly changed, and one can easily alleviate the $S_8$ tension. It has also been shown that such scenarios can be compatible with JWST observations, offering a potential explanation
for the unexpectedly rapid emergence of massive galaxies at high redshifts [3711].

Some applications of decaying dark matter DM decaying during or after recombination can alter the CMB power spectrum by injecting energy that reionises the IGM, constraining the DM lifetime to $\tau_{DM} \gtrsim 10^{25}$ s for photon or $e^+e^-$ decay products [192,3593,3712, 3713]. Additionally, null detections of diffuse X/γ-rays impose strict constraints on the DM lifetime. In DM-dominated galaxies or clusters, $e^+e^-$ decay products can produce radio waves through electromagnetic interactions, observable by radio telescopes like the SKAO [3714], providing a better probe for decaying DM compared to current gamma-ray observations. A recent study [3715] found that DM decay width $\Gamma_{DM} \gtrsim 10^{-30}$ s$^{-1}$ is detectable with SKAO. These constraints were used in Refs. [3716,3717] to understand the scale of Quantum Gravity, assuming that it breaks all approximate global symmetries, including DM stability. Moreover, null detection of X-ray signals has ruled out minimal DM models like vMSM [3718] for DM produced from active-sterile oscillation and has severely constrained the DM mass above the MeV scale for feebly interacting massive particles produced from gauge boson decays [3719].

## 4.5. Other solutions

### 4.5.1. Estimates based on a possible local void

**Coordinator:** Indranil Banik

**Contributors:** Alireza Talebian, Bahman Khanpour, Ebrahim Yusofi, Harry Desmond, Paolo Salucci, Richard Stiskalek, Sahar Mohammadi, Sandeep Haridasu, Sergij Mazurenko, Sveva Castello, and Vasileios Kalaitzidis

Cosmic voids may help to alleviate the Hubble tension. A void effectively adds negative mass to a localized region of an otherwise homogeneous universe. The repulsive gravitational effect of this negative mass causes matter to flow away from the void, further deepening it and enhancing its repulsive effect. Recent studies in a void-dominated late universe [3720–3723] indicate that cosmic voids achieve maximum entropy and thermal equilibrium [3724,3725]. This hierarchical evolution may lead to the formation of large spherical cosmic voids through the merging of smaller sub-voids via the void-in-void process [1789,1798,3726]. Studies suggest that clusters and voids can exert positive pressure individually [3722,3725]. Their coexistence in the Universe’s web-like structure results in opposing pressures: overdensities generate positive pressure that pulls them closer together, while voids create negative expansion pressure that separates galaxies. The outflow from a local void could inflate observed redshifts and thus estimates of $H_0$, though this effect is not sufficient to solve the Hubble tension in ΛCDM [3727,3728].

Near-infrared observations covering 90% of the sky suggest that we live inside the Keenan–Barger–Cowie (KBC) void [3729]. The results are deep enough to cover most of the galaxy LF, making them representative of the underlying matter distribution. The KBC void or Local Hole is evident throughout the electromagnetic spectrum, including in X-rays [3730,3731], optical [3732,3733], infrared [3729,3734–3741], and radio wavelengths [1407,3742]. Comparison with the Millennium XXL simulation [3743] suggests that the KBC void is 6σ discrepant with ΛCDM [3744]. Since the underlying observations are galaxy number counts in redshift space, the results are unaffected by the assumed $H_0$.

Various proposals have been made arguing that gravitationally driven outflows from the KBC void could solve the Hubble tension [3745–3749]. In a homogeneously expanding universe, we would expect that $cz' = \dot{a} = H_0$, where $z' \equiv dz/dr$ is the rate at which $z$ rises with $r$. However, if we are located near the center of a large void, the outflow velocity would rise from zero at the void center to some maximum further out, before gradually decaying away. The initial rise with $r$ would inflate $cz'$. To estimate how much, we note that the KBC void has an underdensity in redshift space of $\delta_{\text{obs}} = 46 \pm 6\%$ (see

figure 11 of Ref. [3729]). The actual underdensity is about half as much because of RSD [3750], the reduction in distance to any fixed redshift due to outflow from a local void inflating $cz'$. The reduced volume reduces the number counts, making the underdensity appear deeper in redshift space than it actually is. Accounting for this and assuming the required reduction in comoving density from the nearly homogeneous early Universe must be due to an increase in comoving volume, [3744] argued in their Eq (5) that

$$ \frac{cz'}{H_0} = (1 - \delta_{\text{obs}})^{-1/6}. \qquad (4.22) $$

Thus, we would generally expect the locally estimated $H_0$ (actually $cz'$) to exceed $H_0$ estimated from the CMB or other high-redshift probes by $11 \pm 2\%$. This is only a rough estimate — the actual result depends on how the void has evolved over cosmic history. This must differ from the growth of voids in ΛCDM given that density fluctuations on the KBC void’s comoving scale of 300 Mpc are very well observed at the time of recombination and match ΛCDM expectations [701], implying that density perturbations on these scales grow faster than in ΛCDM. It is suggested that this is caused by a modification to gravity on length scales $\gtrsim 100$ Mpc, which would not affect the early Universe as its smaller age reduced the cosmic horizon, the distance that light and gravity could have traveled [3744].

A further argument for this scenario is the observed bulk flow curve out to $z = 0.083$ or about 350 Mpc [1426] based on the CF4 galaxy catalog [1424]. The bulk flow measures the average vector velocity of tracers within a spherical region centered on the Sun, albeit using only line of sight velocities. Ref. [1426] argue that the observed bulk flow on the largest probed scales is roughly quadruple the ΛCDM expectation and thus in $> 5\sigma$ tension with it (see also Section 2.3.14). The bulk flow curve in a full-sky survey is independent of the assumed $H_0$, which sets the monopole of the velocity field, while the bulk flow is a function of the orthogonal dipole. There are additional subtleties when the sky coverage is not complete, but since CF4 covers most of the sky, a slight adjustment to the estimator is sufficient to deal with the small unobserved regions of the sky, which mainly lie at low Galactic latitudes (see figure 2 of Ref. [1424]). However, the sky coverage becomes less complete at $z \gtrsim 0.05$ as CF4 mostly relies on SDSS. There can be spurious effects even at lower $z$ because CF4 is reliant on combining surveys, each with their own zero-point and limited sky coverage.

The local void scenario was explored in great detail by Ref. [3744], who constructed semi-analytic models of a small initial underdensity at $z = 9$ and evolved it to today. Those authors used Milgromian MOND [364–366] to enhance the gravity from any given (under)density distribution, allowing the exploration of self-consistent models with enhanced structure growth on the relevant scales. A good joint fit was obtained to the density profile of the KBC void and the observed magnitude of the Hubble tension. An important conclusion was that our location in the void needs to be fine-tuned at the 2% level, which is not a strong argument against the model [3744]. The velocity field in their model was recently calculated in more detail [3751]. The predicted bulk flow curve was found to agree well with the observations of Ref. [1426], which are themselves in good agreement with the measurements reported by Ref. [1427]. The rising bulk flow curve implies we are located fairly close to the void center, minimizing any impact on the CMB quadrupole due to gravitational lensing [3752,3753]. Ongoing work aims to perform this peculiar velocity inference at the field level rather than relying on the bulk flow summary statistic.

The $H_0$ tension should disappear at high $z$ once the density returns to the cosmic mean and the void’s impact gradually decays. Indeed, the ages of the oldest Galactic stars and CCs (Section 2.1.13) suggest a low $H_0$ consistent with the CMB value and well below the local $cz'$ [438,441,3754,3755]. In fact, the Hubble tension is largely a mismatch between the local $cz'$ and $H_0$ estimated in other ways, typically from higher $z$ data [3756]. This can be done assuming ΛCDM,
leading to the concept of $H_0(z)$, the value of $H_0$ inferred from data in a narrow redshift range centered on $z$. Although $\Lambda$CDM predicts a flat curve, there is evidence for a declining trend [516,1290,3103, 3104,3110,3757,3758]. The most recent analyses carefully minimize covariance between $H_0$ estimates from different redshift bins, finding $6\sigma$ evidence for a declining trend [3759,3760]. Similar techniques can be applied to construct simulated $H_0(z)$ curves that map the decay of the void's impact on $z$ [3761]. Their figure 3 shows good agreement with the observed $H_0(z)$ [3759,3760].

BAO observables also deviate from $\Lambda$CDM in the manner predicted by the local void scenario [3762]. These discrepancies (especially in DESI data; [699]) have alternatively been interpreted in terms of the DE density varying with time [1283,1284,3763]. Regardless of the interpretation, the Hubble tension seemingly becomes weaker at high redshift, and not merely due to larger uncertainties [2689]. A recent study examined a variable DE fluid with a quadratic EoS in a late-time context [3723,3725,3764]. Such an EoS may emerge from cluster/void mergers and could represent a redshift-dependent EoS parameter for cosmic acceleration [3721–3723,3725].

The main issue facing the void model at the moment is that the Hubble tension does not clearly decay at high $z$ in the analysis of SN [3765], though there are some hints of such a decay [516,3110, 3759]. While the BAO ruler has a fixed comoving size at $z \lesssim 1000$, SN are not standard candles but merely standardizable [3766]. The intrinsic SN luminosity depends on color and stretch, or how rapidly the light curve decays [33,218]. There are not enough SN in host galaxies with Cepheid distances to reliably constrain these dependencies, which must be obtained jointly with the other cosmological parameters in the Hubble flow sample. This is usually done assuming some parameterized form for the relation between luminosity distance and redshift derived from an isotropic and homogeneous Friedmann cosmological model [3767,3768], though with an additional free parameter in the DE equation of state $w$ [3769–3771]. SN calibrated through the distance ladder, in combination with BAO data and the Planck distance priors [192], suggest compatibility with a homogeneous cosmology and indicate that a potential local void would not significantly impact the measurement of $H_0$ from SN [3772–3774] — but this would need to be confirmed by less model-dependent approaches as a cross-check [3775,3776]. A particular concern is that the typical stretch and color of SN exhibit strong trends with $z$ [3777–3779]. Due to the need to calibrate how both affect the intrinsic SN luminosity, these trends may mask the decaying away of the excess redshift induced by a local void [3761]. One might think that neglecting an actually present local void would cause the optimal homogeneous Friedmann model to provide a poor fit to the data, but the SN magnitude uncertainties are artificially inflated such that the reduced $\chi^2 = 1$, completely hiding any such issue [3780]. The results also change slightly if we assume that it is the color rather than magnitude that has an extra unknown source of noise, for instance, due to dust [3781]. There is ongoing work on revising the SN fitting procedure to allow a local void to inflate redshifts in the nearby Universe, jointly considering cosmological and standardization parameters.

While the KBC void could potentially solve the Hubble tension, it is not evident in the reconstructed local density field [2317] and its existence has not been validated kinematically [3765,3782]. A recent analysis of galaxies using the Radial Tully–Fisher (RTF) relation [355] provides a very promising way to investigate the possibility [356]. They demonstrated comprehensively that a local void capable of alleviating the Hubble tension is highly unlikely to reside at $z \lesssim 0.01$ (as suggested by Ref. [3739]) because it would have been evident in the analysis of the local cosmic expansion through the RTF distance indicator. This agrees with previous studies over the same redshift range using standard SN distance indicators and datasets [3765,3782] — though a plausible local void would inflate redshifts much further out [3761]. Remarkably, the RTF indicator could easily become the best distance

indicator for $z \lesssim 0.15$, where it could be applied to a galaxy sample several times larger than SN datasets. In each galaxy, the RTF method would provide multiple distance measurements, each as accurate as those obtained from SN.

The KBC void is unique among solutions to the Hubble tension in that it was proposed long before the Hubble tension became known. Its large size and depth and the anomalous bulk flow curve all suggest that structure formation on scales of several hundred Mpc is more efficient than expected in $\Lambda$CDM, as also suggested by the high redshift, mass, and collision velocity of the El Gordo interacting galaxy clusters [3783, 3784]. If so, the Hubble tension might be resolved without adjusting the background cosmology, preserving its excellent fit to high-redshift datasets and the ages of the oldest stars [441,456,3754]. In the future, important tests will be provided by techniques that extend beyond the void radius. For instance, peculiar velocity reconstructions using galaxy scaling relations already constrain $cz'$ out to $z = 0.1$ [389,390,394,397]. Extending them to $z \gtrsim 0.3$ would test the predicted decaying of the void's effect at larger $z$ [3761]. BAO measurements suffer the opposite problem: they are most accurate at $z \gtrsim 0.5$, well beyond the void's influence [699]. It is crucial to extend BAO measurements to $z \lesssim 0.2$, where a local void should have an appreciable impact [3762]. More detailed analyses of existing datasets would also help, for instance obtaining bulk flow measurements using SN to check results obtained so far using galaxies and extend them further out [1388,1390,1391, 2844,3785,3786]. The inferred bulk flows can serve as a consistency check on measurements using the far more numerous galaxies.

### 4.5.2. Primordial magnetic fields

**Coordinator:** Karsten Jedamzik

**Contributors:** Alireza Talebian, Anto Idicherian Lonappan, Cláudio Gomes, Gaetano Lambiase, Iryna Vavilova, Levon Pogosian, and Tom Abel

The possible existence of PMFs in the early Universe has been a research topic for decades (for further reviews see Refs. [3787–3789]). The underlying question is if magnetic fields observed in galaxies, clusters of galaxies, filaments, and in particular voids of the extragalactic medium [3790–3796] could be the result of a magnetogenesis process in the very early Universe which magnetized the Universe already well before recombination. The alternative would be that all/or most of the observed magnetic fields are due to astrophysical processes such as dynamos and outflows. The impact of a putative PMFs on the CMB has therefore been studied in great detail [687,3797–3838,3838–3843]. Most proposed effects result in upper limits around comoving $\sim 1$ nG, which is however a field strength about two orders of magnitude stronger than that required for cluster magnetic fields $\sim 1$ µG to be explained entirely by a PMFs [3844]. It has been realized that the anisotropies in the CMB are particularly sensitive to one effect, the creation of small-scale baryon inhomogeneities by PMFs [3845,3846].

A PMF of comoving field strength $\sim 0.05$ nG on comoving length scales $\sim$ kpc produces slightly non-linear baryon inhomogeneities on such scales during recombination. This is possible since kpc scales are well below the photon mean free path such that magnetic pressure is counter-acted only by the small baryon pressure and not the photon pressure. This process is often referred to as “baryon clumping”.

Note that any PMF is characterized by a continuous spectrum, containing initially magnetic power on a large range of scales, either with a scale-invariant spectrum due to inflationary magnetogenesis or a very blue Batchelor spectrum due to magnetogenesis during cosmic phase transitions. In the latter scenario the bulk of the PMF is dissipated as the Universe expands, and a comoving $\sim 0.05$ nG field left shortly before recombination results in a $\sim 0.01$ nG field at the present epoch, approximately the value to explain cluster magnetic fields. The process of PMF-induced baryon clumping has been studied numerically and allowed to set stringent upper limits on the final total present PMF strength of $\sim 0.01$ nG for a Batchelor spectrum [3847]. This upper limit
<figure><img src="image_85.png" /><figcaption>**Fig. 78.** Left: Results of an MHD simulation [3852] starting at redshift $z = 4500$ with an initial comoving magnetic field of 0.526 nG with Batchelor spectrum and initially vanishing density perturbations and peculiar flows. The figure shows the (projected) density fluctuations generated at $z = 1000$ by the stochastic magnetic field. Right: Preliminary results of a MCMC comparing the theoretical prediction of a ΛCDM model with a PMF with Batchelor spectrum to the combination of Planck CMB data [192], DESI-1Y BAO data [699], and the SHOES [34] Hubble constant determination. It is seen that a final magnetic field of $B \approx 0.01$ nG is preferred leading to Hubble constants of $H_0 \approx 70 \text{ km s}^{-1} \text{Mpc}^{-1}$, significantly larger than those predicted in ΛCDM without a PMF. This field strength is close to that needed to explain observed cluster magnetic fields entirely by a PMF.</figcaption></figure>

may be subject to factor two changes when the newer Planck CMB data and/or a more realistic treatment of the PMF effects on the CMB are employed.

Recombination in a clumpy baryon medium proceeds faster than in a homogeneous medium due to the non-linearity in the recombination rate (i.e., $\alpha_e\langle n_e n_p \rangle > \alpha_e\langle n_e \rangle\langle n_p \rangle$ where $\alpha_e$ is the recombination coefficient and $n_e, n_p$ are electron- and proton- density, respectively, with $n_e \approx n_p$). Recombination therefore occurs earlier when PMFs are present, such that the sound horizon $r_*$ is reduced [3848]. A reduction of the sound horizon compared to ΛCDM is the ingredient utilized by essentially all early-time solutions for the Hubble tension (see Section 4.1.1 for an explanation). The effect of baryon clumping on the CMB has been analyzed by several groups [3848–3851]. Here the clumping was treated in toy three-zone models, averaging the ionization fraction from three different and independent regions with the average density the same as in an homogeneous universe. Due to the absence of detailed knowledge of the PMFs generated baryon PDF function, i.e., the probability to find a baryon at a certain density, and its evolution, non-evolutionary models with an ad hoc PDF were chosen. In Ref. [3848] it was found that when Planck data in combination with three late-time $H_0$ determinations was confronted to such clumping models, clumping was preferred at the $\sim 3\sigma$ CL with inferred $H_0 \approx 69.8–71 \text{ km s}^{-1} \text{Mpc}^{-1}$ depending on the chosen baryon PDF. Though the fit to the CMB data was slightly worse than in ΛCDM it was still reasonably good (a PTE value of 0.17 compared to 0.20 for ΛCDM without PMFs). On the other hand, when other data is included, such as BAO and SN, slightly lower $H_0 \approx 69.7–70.6 \text{ km s}^{-1} \text{Mpc}^{-1}$ were inferred. It is now known that the combination of Planck CMB data and BAO data essentially disallows $H_0$ as high as the SHOES value $H_0 \approx 73 \text{ km s}^{-1} \text{Mpc}^{-1}$ in all early-type solutions to the Hubble tension if the physical matter density parameter $\Omega_m h^2$ has a value comparable to the one in the best-fit ΛCDM model [804]. When the model was confronted only to CMB data (Planck in combination with SPT-3G and/or ACT), but no late-time $H_0$ measurements were used, clumping was no further detected but only limited from above [3849–3851]. It was argued that clumping is heavily constrained by the reduced Silk damping in conflict with the ACT data [3849], see below however.

These promising trends tempted a recent much more sophisticated study of recombination in the presence of PMFs [3852]. Instead of

using three-zone models, MHD simulations of PMFs before, during, and after recombination were performed automatically leading to the correct baryon PDF and its evolution. The simulations were performed down to redshift $z = 10$, providing a direct connection between the magnitude of clumping during recombination and the PMF total strength at the onset of structure formation. A chemistry solver coupled to the MHD simulations was used to self-consistently compute the ionization fraction in each volume element. The average perturbation of the ionization fraction compared to a homogeneous no-PMF model was computed and can be used to compute the anisotropies in the CMB in a magnetized universe. The study revealed the possible importance of two further effects, namely the mixing of Lyman-alpha photons between different regions, and enhanced Lyman-alpha photon loss due to peculiar motions of the baryon fluid. Whereas the second effect was found to be subdominant for weak magnetic fields (final field $\sim 0.01$ nG for Batchelor spectrum) the almost complete mixing of Lyman-alpha photons from regions at different densities leads to a further reduction of the ionization fraction during recombination. It also results in an enhancement of Silk damping such that magnetized models often have virtually identical Silk damping to non-magnetized ones.

The left panel of Fig. 78 shows the projected baryon overdensity, i.e., $(\rho - \langle\rho\rangle)/(\langle\rho\rangle)$ of a magnetized universe at $z = 1000$, with density clumps entirely created by the existing stochastic PMF with Batchelor spectrum. There is currently no other than PMF explanation for the putative existence of baryon isocurvature density fluctuations before recombination as other scenarios (pre-existing isocurvature fluctuations) are ruled out by BBN or are not volume-filling enough (isocurvature fluctuations due to cosmic strings). The right panel of Fig. 78 shows the result if a magnetized ΛCDM model is compared to Planck, DESI, and SHOES data. It is found that a universe without PMFs is essentially ruled out by this data combination. Moreover, the predicted Hubble constant is around $70 \text{ km s}^{-1} \text{Mpc}^{-1}$. Though this value does not match the SHOES value it is in accordance with other observational determinations of $H_0$ [174,252]. It is intriguing to realize that the final present day magnetic field $B_0 \approx 0.01$ nG is essentially that required to explain the origin of cluster magnetic fields. It furthermore is also an explanation of the putative observations of an extragalactic field by γ-ray observations. Further observational advances in small-scale CMB anisotropy experiments (ACT, SPT-3G, S4) and γ-ray observations of a
compact-size halo of powerful blazars (CTA with sensitivity less 0.1 TeV and sufficient angular resolution [3853]) have the potential to establish that the early Universe had already been magnetized.

### 4.5.3. Feasibility of inflationary models to ameliorate the cosmic tensions

**Coordinator:** Jaume de Haro, Luis A. Anchordoqui

**Contributors:** Ali Övgün, Alireza Talebian, Andrew R. Liddle, Andrzej Borowiec, Anil Kumar Yadav, Anto Idicherian Lonappan, Antonio Racioppi, Benjamin L'Huillier, Carsten Van De Bruck, Cláudio Gomes, Cristian Moreno-Pulido, Dario Bettoni, David Benitsy, Deng Wang, Denitsa Staicova, Diego Rubiera-Garcia, Elias C. Vagenas, Emanuela Dimastrogiovanni, Emmanuel N. Saridakis, Giulia Gubitosi, Giuseppe Fanizza, Goran S. Djordjević, Ido Ben-Dayan, Ignatios Antoniadis, Ioannis D. Galamas, Javier Rubio, Joan Solà Peracaula, Juan Garcia-Bellido, Laura Mersini-Houghton, Leila L. Graef, M.M. Sheikh-Jabbari, Margus Saal, Marina Cortês, Micol Benetti, Milan Milošević, Nikolaos E. Mavromatos, Nils A. Nilsson, Octavian Postavaru, Oem Trivedi, Özgür Akarsu, Petar Suman, Rishav Roshan, Salvatore Samuele Sirletti, Saurya Das, Simony Santos Da Costa, and Venus Keus

#### 4.5.3.1. Modeling the inflationary expansion and its generalities in connection to the cosmic tensions.
Inflation was first suggested to solve the flatness and horizon problems [3854]. The simplest inflationary model, known as “vanilla” or single-field inflation, describes the expansion of the Universe using a single scalar field $\phi$ with a suitable potential $V(\phi)$. The dynamics of this field are governed by the conservation and Friedmann equations. The most popular approximation for the scalar field is the so-called slow-roll approximation, with the potential energy dominating the energy density of the Universe, i.e., $\dot{\phi}^2 \ll V(\phi)$ and $\ddot{\phi} \ll 3H\dot{\phi}$, where $H$ is the Hubble rate. This leads to an approximately exponential expansion, with the scale factor evolving as $a(t) \sim e^{Ht}$. It is possible to define the slow roll parameters: $\epsilon = M_{\text{Pl}}^2 [V'(\phi)/V(\phi)]^2/2$ and $\eta = M_{\text{Pl}}^2 [V''(\phi)/V(\phi)]$. Using them, one can relate inflationary observables, such as the scalar spectral index $n_s \approx 1-6\epsilon+2\eta$ and the tensor-to-scalar ratio $r \approx 16\epsilon$, to the shape of the inflaton potential $V(\phi)$ [3855]. A point worth noting at this juncture is that an approach to addressing the $H_0$ tension involves studying the slow-rolling dynamics of a self-interacting scalar field, which can lead to an emerging $H$ as a function of redshift; however, it should also be noted that while the scalar field could be responsible for a quintessence slow-rolling, it has not been associated thus far to inflationary models [3107].

Vanilla inflation predicts a nearly scale-invariant spectrum of scalar perturbations with a slight red tilt, quantified by the scalar spectral index $n_s \approx 0.96-0.98$. However, the slow-roll approximation is only one way to go, there are other examples such as fast-roll approximation [3856], constant-roll [3857,3858], rapid-turn inflation [3859, 3860], scalar field inflation in the presence of a non-minimal coupling between matter and curvature [3861] symmergent inflation [3862, 3863], gauge-field inflation [1447,3864,3865,3865-3889], s-dual inflation [3890,3891], nonlinear electrodynamics inflation [3892-3900], and symmergent gravity theory [3862,3863].

Many inflationary models predict primordial non-Gaussianities, which play a crucial role in understanding the early Universe's evolution and testing the inflationary paradigm. Non-Gaussianities are quantified by the non-linearity parameter $f_{\text{NL}}$, which measures the amplitude of non-Gaussian fluctuations in the primordial curvature perturbations. These fluctuations require analysis of higher-order correlation functions, such as the bispectrum, beyond the two-point correlation function used for Gaussian fluctuations. Previous CMB experiments, including WMAP and early releases of the Planck data, aimed to detect these non-Gaussianities. The most recent constraints from the Planck 2018 results provide $f_{\text{NL}}$ values for the standard templates: $f_{\text{NL}}^{\text{local}} = -0.9 \pm 5.1$, $f_{\text{NL}}^{\text{equilateral}} = -26 \pm 47$, and $f_{\text{NL}}^{\text{orthogonal}} = -38 \pm 24$, what indicates no statistically significant detection of primordial

non-Gaussianities yet. However, it is important to keep in mind that multi-field or exotic single field inflationary models can lead to strongly scale-dependent non-Gaussianity [3901], which could evade large-scale constraints while still significantly affecting small scales. Such scale-dependent primordial non-Gaussianities could actually alleviate the $S_8$ tension in the non-linear regime of structure formation, without affecting the linear regime [3902].

Future CMB surveys, such as those planned with the Simons Observatory, CMB-S4, and the Euclid mission, promise significantly improved constraints on $f_{\text{NL}}$. The Simons Observatory aims to achieve sensitivity levels that could reduce the uncertainties on $f_{\text{NL}}$ to around $\Delta f_{\text{NL}} \sim 1$. Similarly, CMB-S4, with its planned deep and wide CMB observations, is expected to further tighten these constraints and provide unprecedented precision in measuring primordial non-Gaussianities. The Euclid mission, primarily designed for large-scale structure surveys, will complement CMB observations by probing the distribution of galaxies and the large-scale structure of the Universe, thereby offering additional avenues to constrain $f_{\text{NL}}$. These upcoming observations are expected to enhance our ability to test a broader range of inflationary models and deepen our understanding of models that could alleviate the $S_8$ tension.

#### 4.5.3.2. Classes of inflationary models with ramifications for the cosmic tensions.
There are numerous extensions or alternatives to vanilla inflation, justified in various ways. In this subsection, we summarize the generalities of inflationary models that have a connection to the cosmic tensions and call attention to the role these specific models have in ameliorating the tensions.

**Higgs inflation** In the simplest Higgs inflation model [3903], the Standard Model Higgs field is coupled non-minimally to gravity through a large dimensionless parameter, leading effectively to an almost flat Einstein-frame potential able to support inflation for a sufficient number of e-folds (for a review, see Ref. [3904]). One of the appealing aspects of this scenario is its minimalist approach, as it does not require the introduction of new fields or couplings beyond the Standard Model content. In particular, the reheating stage can be computed in detail, leading, in terms of the number of last e-folds $N_*$, to precise predictions for the spectral tilt $n_s = 1-2/N_* \approx 0.966$ and the tensor-to-scalar ratio $r = 12/N_*^2 \approx 0.0033$ [3905-3907]. Additionally, the scenario offers a compelling connection with accelerator experiments, providing a detailed description of the power spectrum across a vast range of scales [3908,3909]. The MCMC analysis of current cosmological data confirms the model's observational viability and demonstrates that for the interval $N_* \in [54.5, 60]$, it can break down the $H_0 - \sigma_8$ correlation [3910]. Specifically, considering an instantaneous transition to radiation-domination, the $H_0$ tension is reduced to approximately $3\sigma$ [76], constraining $H_0 = 67.94 \pm 0.45 \text{ km s}^{-1}\text{Mpc}^{-1}$, while the value of $\sigma_8 = 0.793 \pm 0.003$ is in complete agreement with the KiDS-1000 results [3911]. The constrained $S_8 = 0.8019 \pm 0.0097$ is in $2\sigma$ with Planck, DES and KiDS-1000 data.

Several extensions of the minimal Higgs inflation scenario have been considered in the literature [3912-3919]. The so-called Higgs-Dilaton model addresses the current accelerated expansion of the Universe by introducing a unimodular constraint that manifests itself as the strength of a runaway Einstein-frame potential for a dilaton field [3912]. A significant feature of this scenario is the existence of specific consistency relations between the spectral tilt of primordial density perturbations, the tensor-to-scalar ratio, and the DE equation of state [3913]. Predictions of these relations are in tension with direct measurements of $H_0$ using low-redshift SN, but they align well with current CMB observations and large-scale structure data, offering distinctive signatures that could be tested by future galaxy surveys [3920-3922].

**Starobinsky inflation** Starobinsky cosmology is renowned as an excellent large-field slow-roll inflationary model [2074], aligning very well with CMB observations [3923,3924]. The key feature of the
Starobinsky potential, essential for achieving slow-roll inflation, is its asymptotic behavior, where $V_s(\phi) \to \text{const.}$ as $\phi \to \infty$. Despite its celebrated success, the model does not seem to align well with proposals that could resolve the cosmic tensions. For example, it was noted in Ref. [3925] that Starobinsky inflation can hardly coexist with an EDE fraction ~ 0.06, which is only able to reduce the $H_0$ tension down to about $3\sigma$.

**Brane inflation** A theory that stems from the string theory, involves the interaction between branes in higher-dimensional space. Inflation occurs due to the motion of branes within a compactified extra-dimensional space, such as in the Klebanov-Strassler throat. The separation between branes acts as the inflaton, and their eventual collision ends inflation. This scenario can lead to observable levels of non-Gaussianity in the CMB power spectrum and the distribution of large-scale structures in the Universe, as well as potentially detectable GWs. The dynamics of the inflaton in brane inflation are governed by the Dirac-Born-Infeld (DBI) action, which is a type of non-canonical action [3926-3930]. Matrix inflation (a.k.a. M-flation) [3931] may also be viewed as a multi-brane version of the brane-inflation scenarios.

Tachyon cosmological models of inflation, based on the dynamics of a D3-brane in the bulk of the second Randall-Sundrum model, have been extensively considered [3932]. Particle creation and reheating in a braneworld inflationary scenario with a tachyon field were extended to include matter in the bulk. It has been demonstrated how the interaction of the tachyon with the U(1) gauge field drives the cosmological creation of massless particles and where estimates have been found for the resulting reheating at the end of inflation [3933]. A holographic braneworld scenario with a D3-brane located at the holographic boundary of an asymptotic AdS$_5$ bulk [3934] has shown significantly better agreement with Planck data [3923].

The cosmic tensions have been addressed in several models based on brane inflation. For example, in Ref. [2882] it was shown that phantom braneworld prefers a higher value of $H_0$ providing a much better fit to the local measurements. In addition, in Ref. [3935] it was pointed out that the simplest D-term inflation can be consistent with the cosmic string bound provided by observations of gravitational. Consistency with observations requires a spectral index $n_s = 1$ in accordance with the pre-recombination proposals to alleviate the $H_0$ tension (e.g., EDE).

**Quintessential inflation** After the discovery of the current cosmic acceleration [48,2980,3936,3937], several theoretical mechanisms were developed to explain it. Quintessence models broadly encompass scenarios with one or more scalar fields with appropriate potential/dynamics that can drive the late-time cosmic acceleration, allowing for much richer possibilities beyond that simple case, in which one gets a dynamical equation of state parameter. An important feature of quintessential models is the need to examine ghosts and other non-physical features of their perturbations [1264]. For recent applications in cosmology, see Refs. [3938-3940]. One of the simplest ways to do it is the so-called *quintessential inflation* [3941-3947], where the inflaton field is the only responsible for both accelerated phases. Several authors developed and improved the original Peebles-Vilenkin model [3948-3951], also embedding it in more fundamental frameworks [3952-3955]. In particular, $\alpha$-attractors quintessential-inflation parameters can be constrained by stage IV surveys [3956] and they have been tested using Planck and low-redshift data [3957] and scale invariant models [3922]. Of particular interest here, the quintessential inflation, coming from the Lorentzian distribution introduced in Refs. [3958, 3959], agrees with the recent observations and is in agreement with the SH0ES $H_0$ measurement [3960]. Indeed by confronting the model to cosmological observations, one obtains $H_0 = 72.25 \pm 0.74 \text{ km s}^{-1} \text{Mpc}^{-1}$ at 68% CL [3960,3961].

**K-essence, multifield inflation, and their generalizations** K-essence extends the standard model by including a non-canonical kinetic term in the Lagrangian, $P(\phi, X)$, where $X = \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi$ [2070,2772,3962-3964]. K-essence models can help address cosmic tensions through

their unique equation of state properties and field dynamics affecting both early and late Universe evolution. In Ref. [3965] researchers demonstrate how k-essence can modify the expansion history in ways that specifically target both the Hubble and structure growth tensions. Similarly, Ref. [3966] examines how an EDE model triggered by the radiation-matter transition can be realized in k-essence. K-essence's theoretical flexibility makes it particularly valuable for tension resolution, as shown in Refs. [3967,3968], where k-essence is shown to maintain consistency with both observational constraints and theoretical requirements. The multifield dynamics have been further studied in Ref. [3969]. The generalized k-essence can further contribute to models affecting the sound speed of perturbations and the equation of state, which can modify cosmological distance ladder and thus influence tension-related parameters [2780,3970,3971].

**Uniform five-dimensional inflation** Compact extra dimensions can obtain large size by higher dimensional inflation, relating the weakness of the actual gravitational force to the size of the observable Universe [2924,3703]. A solution to the horizon problem implies that the fundamental scale of gravity must be smaller than $10^{13}$ GeV which can be realized in a brane-world framework for any number of extra dimensions. However, the requirement of (approximate) flat power spectrum of primordial density fluctuations consistent with present observations make this simple proposal possible only for one extra dimension at around the micron scale [2924,3972]. Consistency with 2018 Planck CMB data is supported by numerical simulations [3973-3975]. After the end of five-dimensional inflation, the radion modulus can be stabilized at a vacuum with positive energy of the order of the present DE scale. An attractive mechanism for radion stabilization is based on the contribution to the Casimir energy of fields propagating in the bulk: the graviton, a real scalar field, and three right-handed neutrinos [2899]. The scalar field has a potential holding two local minima with very small differences in vacuum energy and bigger curvature (mass) of the lower one, and thus when the false vacuum tunnels to its true vacuum state, the field becomes more massive and its contribution to the Casimir energy becomes exponentially suppressed [2900]. The tunneling process then changes the difference between the total number of fermionic and bosonic degrees of freedom contributing to the quantum corrections of the vacuum energy, yielding a sign-switching cosmological constant, $\Lambda_s$, see Section 4.2.2. Despite the fact that when inflation comes to an end radion stabilization necessitates a dark sector with $\Delta N_{\text{eff}} \sim 0.25$ (i.e., saturating Planck's upper limit [192]), in the style of Ref. [812], uniform five-dimensional inflation could help simultaneously resolve the $H_0$, $S_8$, and $M_B$ tensions [2701,2926].

**Running vacuum models** The RVM approach (see Refs. [2477, 2793,2794] and references therein), is a framework for an effective description of cosmology, and it has implications for the time evolution of the fundamental constants [2797,2802]. The RVM framework [2473] is based upon the idea that the cosmological vacuum energy densities and pressure are functions of even powers of the Hubble parameter $H(t)$ (and, under circumstances, also of logarithmic non-polynomial corrections of $H(t)$). In the cosmic vacuum, which characterizes inflation, a de Sitter equation of state $p(H(t)) = -\rho(H(t))$, with

$$ \rho(H(t)) = \frac{\Lambda(H(t))}{\kappa^2} = \frac{3}{\kappa^2} \left( c_0 + \nu H^2 + \alpha \frac{H^4}{H_I^4} + \dots \right), \quad (4.23) $$

is satisfied, although, during each post-inflationary era, the total energy density and pressure is supplemented by the corresponding contributions of matter and/or radiation, as recent quantum-field-theory calculations in the RVM context have shown [2790-2792,3976]. We remark that the coefficients $\nu$ and $\alpha$ are considered as constants during each epoch of the Universe's evolution; $H_I$ is the scale of inflation, which, according to data in Ref. [3923], is currently set to $H_I \lesssim 10^{-5} M_{\text{Pl}}$, in order of magnitude. The important feature of the RVM framework is that inflation does not require any external inflaton fields, but it is mainly due to the non-linear gravitational dynamics due to the $H^4$ and higher-order terms in the expression of $\rho(H(t))$ [2475,2476],
which dominate the early Universe eras. Note that the RVM can provide
a smooth evolution of the Universe, and can also account for its
thermodynamic history and properties [2478,3977].

This generic framework finds a concrete realization in the context of Quantum Field Theory (QFT) in curved spacetime when the vacuum energy density is renormalized using the adiabatic renormalization method, in which a subtraction is performed at an off-shell scale *M* subsequently identified with the Hubble rate *H*(*t*) at each cosmic epoch. This eliminates the quartic mass contributions ~ *m*⁴ (hence the need for fine tuning) and induces a dynamics of the vacuum energy of the above form [2790–2792,3976]. An alternative realization, termed “Stringy RVM” (StRVM), is found when RVM is embedded in string theory cosmologies, under the assumption [2471,2473,2479] that the early Universe consists of fields appearing in the bosonic massless ground-state multiplet of superstrings. The model assumes that the dilaton is stabilized to a constant value, through the minimization of an appropriate potential, possibly induced by string loops [2472]. The pertinent string-inspired, low-energy gravitational theory is a Chern–Simons (CS) gravity [2469,3978]. The CS nature of gravity arises from the linear coupling of the string-model independent axion [2470, 2480] to the CS gravitational anomaly, which is due to the Green–Schwarz counterterms [3979] that modify the antisymmetric-tensor field strength as a consequence of the requirement of cancellation of gauge and gravitational anomalies in the extra dimensional string space. In the StRVM, the gravitational anomalies in the primordial Universe are not assumed to be canceled, although such a cancellation occurs at the exit from RVM inflation, after the generation of chiral fermionic matter [2471,2473,2479], due to the decay of the RVM vacuum. The RVM inflation in the StRVM is induced by condensates of the gravitational CS term due to condensation of primordial GW, which are formed during a pre-inflationary epoch [2473,2479]. An estimate for the CS condensate can be provided within a second quantization of weak GW formalism [2481,3275,3276]. It is found to be proportional to *H*⁴(*t*), which during inflation varies very slowly with the cosmic time. As shown in Ref. [2481], the formation of the CS condensate provides an approximately linear KR-axion monodromy potential, leading to inflation in the sense that the latter corresponds to a saddle point of the evolution of an appropriate dynamical system of variables. This result is general and refers to all scenarios with linear axion potentials, such as those obtained, for instance, from appropriate compactifications of string theories [2483].

Finally, let us notice the intriguing potential implication of this
framework on the alleviation of cosmological tensions [2484,2485].
Indeed, in the context of StRVM, quantum graviton corrections induce
in general corrections to the effective vacuum energy density of the
non-polynomial form $H^{2n} \ln(H)$, $n \in \mathbb{Z}^+$. Such corrections have been
computed explicitly in the context of dynamically broken supergravity
models [2819], which can also be cast in an RVM format [3980] and
can describe a pre-RVM inflationary epoch of StRVM. In cases such
corrections survive in modern epochs, they can provide a resolution
of the Hubble and, under some circumstances, also of the growth-of-
structure tension: $H_0 = 71.27_{-0.73}^{+0.76}$ and $\sigma_8 = 0.816_{-0.015}^{+0.017}$ [2484]. See
also the “phantom matter” proposal of Ref. [2485], inspired by the
StRVM approach and providing a potential resolution of both tensions.
Subdominant logarithmic corrections on $H$ can also arise within the
conventional RVM, by integrating out matter fields in QFT of fermions
and bosons in expanding spacetimes [2790–2792,3976].

In a related context to the RVM, we can mention the works from
Ref. [3981,3982] on the back-reaction effect of super-Hubble cosmolog-
ical fluctuations. Ref. [3981] examines a scenario in which a large bare
cosmological constant drives early accelerated expansion and shows
that the backreaction effect of infrared fluctuations can reduce it. Such
kind of models are expected to have an impact on the $H_0$ tension
problem, as discussed in Ref. [2863].

4.6. *The cosmological principle*

Coordinator: Eoin Ó Colgáin

**Contributors:** Alexander Bonilla Rivera, András Kovács, Anil Kumar Yadav, Antonio da Silva, Asta Heinesen, Daniele Oriti, Dario Bettoni, David Benisty, David L. Wiltshire, Diego Rubiera-Garcia, Dinko Miklovic, Duško Borka, Ebrahim Yusofi, Elias C. Vagenas, Giuseppe Fanizza, Goran S. Djordjević, Hassan Abdalla, Ido Ben-Dayan, Iryna Vavilova, Jenny G. Sorce, Jenny Wagner, Jessica Santiago, John K. Webb, José Pedro Mimoso, Juan Garcia-Bellido, Laura Mersini-Houghton, Leandros Perivolaropoulos, M.M. Sheikh-Jabbari, Maciej Bilicki, Marie-Noëlle Cé lérier, Marina Cortês, Milan Milošević, Nihan Katırcı, Nils A. Nilsson, Özgür Akarsu, Predrag Jovanović, Saurya Das, Tajron Jurić, Víctor H. Cárdenas, Valerio Marra, Vesna Borka Jovanović, and Wojciech Hellwing

In recent years the status of the cosmological principle (CP), a fun-
damental assumption in modern cosmology, has become less clear cut.
However, what stymies debate is the absence of a consensus definition
of the CP. The CP is commonly presented as the idea that the Universe
is isotropic (the same in every direction for a comoving observer who is
at rest with respect to the cosmic fluid of the expanding Universe) and
homogeneous (the same at every point on a hypersurface corresponding
to a given cosmic time) at suitably large scales. Homogeneity implies
that there is no special location, such as a preferred observing position
or center, while isotropy states that there is no special direction, such as
an axis. These two statements do not automatically imply one another,
and the possible departures from each of them are currently among the
most prominent research topics in cosmology.

Unfortunately, the terminology is loose and it is always possible
to find a definition that is perfectly untestable. First, what is meant
by large scales is rarely discussed. Secondly, given that we have only
one vantage point in the Universe, testing homogeneity is difficult. In
particular, depending on how one defines the latter, one could end up
with a claimed “large scale” varying from ~ 70 h⁻¹ Mpc [3983–3985]
up to ~ 260 h⁻¹ Mpc [3986]. Thirdly, even when scientists argue that
there is an anomaly with the CP, great care is required to separate
ΛCDM predictions from generic predictions of isotropic/homogeneous
Universes. For example, ~ 260 h⁻¹ Mpc [3986] is a claimed upper
bound for the homogeneity scale in a ΛCDM universe. Moreover,
homogeneity is also challenged by large structures [1607,3987–3990],
but one inevitably must compare to the predictions of a specific FLRW
model, i.e., ΛCDM, and often, one either finds comparable structures in
simulations or one can question the statistics [1611,1612,3991–3993].
Finally, a point very rarely discussed, and which may pass by as a
minimal assumption, is the very existence of a “cosmic fluid”. Besides
how harmless those words might sound, what this actually assumes is
the existence of a *unique reference frame* which is at rest with respect
to the average distribution of the CMB, baryonic matter, DM and DE
simultaneously in different time periods of the history of the Universe.
Looking closely at this definition, one then opens Pandora’s box, in
which to start, we have a lack of proper and robust definition of
averages in GR, followed by questions regarding averaging scales and
even if the averaging scales should be the same or not for different
fluids.<fnref n="61" /> Admittedly, these are excellent theoretical questions, but if
one approaches cosmology at a granular scale, constructing an observa-
tionally testable model is impossible. In cosmology one should not lose

<footnote n="61">⁶¹ Applying the CP for a void-dominated cosmology [3720–3723], involving superclusters and cosmic voids, necessitates that cosmic voids tend toward achieving maximum symmetry. The result of such an evolution would be a vast spherical cosmic void, which is formed by the merging of smaller sub-voids (void-in-void process) [1789,1798,3726]. The density of the largest cosmic voids will be closer and closer to that of the Universe, making the CP more applicable to the present void-dominated Universe.</footnote>
<figure><img src="image_86.png" /><figcaption>Fig. 79. Left: Angular variations of $H_0$ on the sky in galactic coordinates in the aftermath of the HST Key Project. Reproduced from Figure 1 (a) of Ref. [3994]. Right: Angular variations of $H_0$ on the sky in galactic coordinates from galaxy cluster scaling relations. Reproduced from Figure 7 of Ref. [3995].</figcaption></figure>

track of the grossly simplified assumptions - justifiable on the relatively poor quality of astronomical data - being made.

The Universe is a gravitating system. Gravity is GR, at least in a minimal setting. This necessitates a choice for a spacetime metric $g_{\mu\nu}$ to insert on the left hand side of Einstein's equation. Here the modus operandi is to start from the most symmetric possibility. This leads one to a maximally symmetric spacetime in 4D, namely Minkowski, anti-de Sitter, and de Sitter metrics. From these three, the latter is the only physically relevant possibility and appears in the asymptotic future of any universe with late-time accelerated expansion. The pursuit of symmetry also leads one to the perfect cosmological principle (PCP), according to which the Universe is homogeneous and isotropic in space and time. This means that the Universe in the large is also unchanging with time. The PCP was the basis of the steady-state cosmology, an alternative to the Big Bang theory [3996,3997], but this required a physically contrived scenario whereby the density of matter remained unchanged due to a continuous creation of matter.

### 4.6.1. FLRW spacetime

In pursuit of a more physical model, one can always pursue classes of less restrictive spacetimes. The concession one makes is reducing maximal symmetry in the 4D spacetime to maximal symmetry in a 3D space. This leads one to the FLRW metric, a cornerstone of modern cosmology. The FLRW spacetime metric $g_{\mu\nu}$ is given by Ref. [3767]

$$ds^2 = g_{\mu\nu} dx^\mu dx^\nu = -c^2 dt^2 + a(t)^2 \left( \frac{dr^2}{1 - kr^2} + r^2(d\theta^2 + \sin^2\theta d\phi^2) \right), \quad (4.24)$$

where $a(t)$ is the scale factor, $k$ represents the spatial curvature, and $(t, r, \theta, \phi)$ are the comoving coordinates. Combining FLRW with GR, the Einstein equation reduces to the Friedmann equation,<fnref n="62" /> which has an important consequence that the Hubble constant $H_0 = H(z = 0)$ arises as an integration constant [3758]. Note, Eq. (4.24) simply specifies the spacetime and not the model. However, once one fully specifies the model, e.g., ΛCDM, the onus is on the astronomy and cosmology communities to demonstrate that $H_0$ is a constant in cosmic time or redshift. If $H_0$ is not constant in a given model, one runs into a “Hubble tension” problem, e.g., see Refs. [34,192].

This then brings us to the simplest way to test the metric in Eq. (4.24). $H_0$ can be determined model independently in the local Universe $z \lesssim 0.2$ using Cepheids, TRGB, or other possible primary distance indicators in order to calibrate SNIa [34,144]. Note that working locally, one can safely decouple the cosmological model [220]. In short,

one should make sure that there are no statistically significant angular variations of $H_0$ between different directions on the sky, because as discussed, $H_0$ must be a constant. Already, galaxy cluster scaling relations point to ~9% changes in $H_0$ in the local Universe [3995,4000,4001]. Evidently, if the precision on $H_0$ is ~5–10%, this is no problem, but it may become problematic if one pursues ~1% precision. Since nobody builds the distance ladder on galaxy clusters, it is imperative to recover this result in SNIa. Preliminary results exist, but given the relatively small size of SNIa samples, which become worse when restricted to lower redshifts, angular $H_0$ variations at low redshifts are consistent with statistical fluctuations (< 2σ) [1344,1374] (see also Refs. [1380,1385,1390,3994,4002]).<fnref n="63" /> Given the better statistics in cosmicflows (CF) data, a data set comprising SNIa alongside other distance indicators, the observation of an anisotropic $H_0$ has recently been pushed to 3.9σ [4006]. It should be stressed that there is no guarantee that one can get to a > 3σ variation in $H_0$ from SNIa alone, but the test can and should be done. Note that a statistically significant angular variation in $H_0$ would make both the Hubble constant and the Hubble tension ill-defined. Since, $S_8$ is a parameter within the ΛCDM model, and the model builds upon FLRW, a breakdown of FLRW would also make $S_8$ tension ill-defined<fnref n="64" />.

In addition, locally, there are also claims from the CF program that there are anisotropic bulk flows [1343], which may no longer converge to ΛCDM expectations [1426] (see also Refs. [1425,1427]), thereby echoing a kinematic SZ anomaly, one disputed by the Planck collaboration [4008], from a decade previously [1429,1431,1432,4009]. An upgrade of CF data to include WALLABY [391,4010,4011], FAST [393] and DESI data [394] recently found that the homogeneity scale is not yet recovered at 300 h⁻¹ Mpc [4012], thereby strictly violating the 260 h⁻¹ bound [3986]. In tandem, the inferred bulk flow was larger than ΛCDM predictions, but remained consistent with the model. Note, in Ref. [1426] an isotropic expansion or constant $H_0$ is assumed, but the bulk flow becomes anomalously large. CF data makes use of a host of distance indicators, including SNIa, so one can in principle recover the same result from SNIa alone.⁶⁵ Here it is worth noting that SNIa redshifts are typically corrected for peculiar velocities of the host galaxy, so it is imperative to make sure that anomalously

<footnote n="62">⁶² The Friedmann equation can be derived by moving to the accelerated frame using the scale factor [3998,3999].</footnote>

<footnote n="63">⁶³ One can also fit a dipole to SNIa [1391,4003–4005]. One may find [4005] that SNIa and CMB are not in the same frame, but that the difference can be interpreted as a bulk peculiar velocity $v_{\text{bulk}}$ consistent with ΛCDM expectations. Note, these analyses are typically biased toward the lowest redshift SNIa, so one needs to study the redshift dependence of $v_{\text{bulk}}$ in shells, where overlap is expected with the cosmicflows program [1426].</footnote>

<footnote n="64">⁶⁴ As an aside, since an anisotropy makes a contribution to the energy budget of the Universe scaling as $a^{-6}$, which is relevant in the early Universe, it is plausible a new anisotropic cosmological model can be found that alleviates the tension with larger local $H_0$ determinations, e.g., see Refs. [1364,4007].</footnote>
large bulk flows are not simply swept under the rug to reinforce an isotropic expansion (see criticism of SNIa peculiar velocity corrections in Refs. [4013,4014]). Another cautionary note on making peculiar velocities corrections regards the non-linearities that arise on local scales. This is an extremely delicate subject, especially given that the boundary scales where linear and non-linear regimes are applicable, together with the regime where perturbation theory is no longer valid, are still under investigation. In order to avoid this point, one must go to high redshift sources.

The other way to test the CP model independently is through aberration and Doppler effects with a large sample of sources. Although the Ellis-Baldwin test [1362] was outlined last century, the test only became feasible this century [1398,1404-1407,1410-1414,1419, 4015,4016]. While there may be outliers [1414,1417,1418,1463,4017, 4018], there is a growing consensus that radio galaxy and QSO data sets do not inhabit the same rest frame as the CMB [1398,1405-1407, 1411-1413,1419,4015,4016]. Nevertheless, these are challenging studies, both technically and sociologically, thus it is always conceivable that observational [1402,1463,4018,4019] and or theoretical systematics [1360,4020] (however see Ref. [4021]) are at play. The earliest studies have typically employed frequentist statistics, but later papers have provided a Bayesian interpretation [1400,1419,1463]. Taken at face value, the disagreement between CMB and radio galaxies/QSOs on the cosmic dipole points to a breakdown of the CP — and, therefore, of FLRW. The problem is interpreting this result. At what scales or redshifts does this happen? How badly is FLRW broken? None of this would be obvious without corroborating FLRW anomalies in the local Universe.

So where are we now on FLRW? Both in the local Universe $z \lesssim 0.2$ and large samples of radio galaxies/QSOs with assumed median redshifts of $z \sim 1$ we see hints of a violation of FLRW. Given the rich structures in the local Universe [1343,4022,4023], variations in $H_0$, or alternatively bulk flows, are expected. See Fig. 79. The important question is whether these effects deviate from the expectations of the $\Lambda$CDM model. On the other hand, a mismatch in the cosmic dipole, could easily be explained by a relatively local effect and not an FLRW violation at large scales. Thus, it is of interest to study the Hubble diagram out to $z \sim 1$ in order to check if one recovers an FLRW universe. One could adopt the $\Lambda$CDM model, e.g., see Refs. [1385,1403,4002,4024], and check that variations in $H_0$ become less significant as effective redshifts of SNIa samples, etc, increase, but one can also tackle this problem model independently through cosmography [2086] up to $z = 1$, the radius of convergence of the expansion [2126]. See Refs. [1382,4025] for preliminary studies in this direction. Note, cosmography introduces additional parameters, especially if one wants to describe models beyond FLRW [1358], so one will require large data sets to conduct these studies. Nevertheless, the motivation should be clear. Current hints of FLRW violation at low and high redshift need to be connected in a common framework. The big question that needs to be addressed is how good is FLRW as an approximation to the physical Universe? While few scientists outside the field of cosmology, especially relativists, would challenge the assertion that FLRW is an approximate symmetry, since there is no argument that it is fundamental, this may be cold comfort for precision cosmology. $\sim 9\%$ variations in $H_0$ [3995,4000,4001], suggest that FLRW is a good approximation to $\lesssim 10\%$. This is easy to square with the perceived isotropy of the CMB, when one factors in residual asymmetries in CMB data or CMB anomalies [1330]. A subset of the latter has persisted through three independent satellite experiments, namely COBE [677], WMAP [2754] and Planck [192], allowing one to make the case that the Universe is unlikely to be statistically isotropic based

on CMB data alone [1332]. Unsurprisingly, $H_0$ can vary by up to $\sim 10\%$ along the axis of the hemispherical power asymmetry [1333,1334], a recognized CMB anomaly, and attempts to recover the CMB dipole from aberration and Doppler boosts are contaminated at lower multipoles (larger scales) by this anomaly (see Fig. 3 of Ref. [4026]).$^{66}$ In addition, it is evident that WMAP/Planck CMB data prefers a phenomenological Bianchi model over $\Lambda$CDM [4029-4032],$^{67}$ which is only possible if there is a residual anisotropy in the data. Nevertheless, it is not clear if beyond FLRW models, especially the models in Section 4.6.2, can rival FLRW cosmology. This is not a statement about fits to data, this is the statement that at a purely technical level one would need a complete understanding of cosmological perturbation theory in these models. Given the anomalies in this white paper, it is hard not to contemplate that precision cosmology, defined through models where fitting parameters are constrained at the $\sim 1\%$ error, may be at a crossroads. See Ref. [1328] for a recent overview of observations and the key claim that we have exceeded the precision where the approximate nature of the FLRW assumption is evident.

### 4.6.2. Moving beyond FLRW

In light of these FLRW anomalies, there is a growing interest in exploring more general metrics that relax the assumptions of homogeneity and isotropy. In this respect, if the CP is relaxed so that the homogeneity holds but not isotropy, then the corresponding exact solutions of Einstein's field equations are given by the class of cosmological models called Bianchi universes, named after the Italian mathematician Luigi Bianchi who first classified these spaces [1364,4033-4035]. Homogeneity in space implies that Einstein's field equations reduce from partial to ordinary differential equations in time, making Bianchi models exact solutions. A complete list of all such solutions for all Bianchi cosmological models from type I to type IX and for perfect fluid is given by Ref. [4036]. Bianchi universes also contain, as a subclass, the standard isotropic FLRW universes. Although Bianchi models were previously considered to be inconsistent with observations, recent studies highlighted above challenge the isotropy assumption and revive interest in these models, which therefore remain widely studied [4037, 4038]. Ref. [1364,4035] presents an observational analysis of Bianchi type-I, -V, and -XI spacetime extensions of the $\Lambda$CDM model. This section discusses some of the alternative metrics that have been proposed to address these issues.

#### 4.6.2.1. Lemaître-tolman-bondi (LTB) metric.
The LTB metric [4039-4041] describes a spherically symmetric but radially inhomogeneous universe. It is given by

$$ds^2 = -c^2dt^2 + \frac{R^*(r, t)^2}{1 + 2E(r)} dr^2 + R(r, t)^2(d\theta^2 + \sin^2\theta d\phi^2), \quad (4.25)$$

where $R(r,t)$ is the radius of the spherical shell at time $t$ and radial coordinate $r$, and $E(r)$ is an arbitrary function related to the energy of the shells. The LTB metric allows for variations in the density and expansion rate along different radial directions, providing a more flexible framework to model inhomogeneities in the Universe [4039-4041]. The series of N-body simulations discussed in Ref. [4042] explores the development of large-scale structures against an LTB background that incorporates a cosmological constant, specifically within the $\Lambda$LTB model framework.

Spherical inhomogeneities in LTB models can emerge either due to matter voids or due to Hubble scale spherical inhomogeneities of DE. In the latter case, such inhomogeneities can be supported by topological considerations, particularly in the context of recently formed global monopoles. These topological defects arise when the vacuum

<footnote>$^{65}$ Indeed, restricting the CF data to SNIa, one confirms the large bulk flow, but large errors make this consistent with $\Lambda$CDM expectations. We thank Rick Watkins for private communication.</footnote>

<footnote>$^{66}$ For this reason, it is customary to remove large scales, e.g. see Refs. [4027, 4028].</footnote>

<footnote>$^{67}$ These phenomenological Bianchi models cannot be seen as deformations of the $\Lambda$CDM model since the cosmological parameters adopt different values.</footnote>
manifold of the quintessence scalar field has non-trivial $\pi_2$ topological properties [4043,4044]. This scenario corresponds to the “Topological Quintessence” class of models, which may support deviations from the FLRW metric on cosmological scales.

In the context of topological quintessence, a global monopole formed during a recent phase transition with a core size comparable to the present Hubble scale could induce the observed accelerated expansion of the Universe. The monopole's scalar field is trapped near a local maximum of its potential in a cosmologically large region of space. This setup leads to an inhomogeneous but isotropic DE distribution, where the core of the monopole exhibits an effective cosmological constant-like behavior, while the outer regions revert to a matter-dominated Einstein-de Sitter universe.

An off-center observer within such a topological quintessence framework would naturally observe cosmic dipoles due to the asymmetric positioning relative to the monopole core. This asymmetry can manifest as dipoles in various cosmological observations, including the CMB and large-scale velocity flows [1361,4045,4046].

Recent numerical simulations have explored the dynamics of such global monopoles minimally coupled to gravity in an expanding universe with initially homogeneous matter. These studies show that when the energy density of the monopole core starts dominating the background density, the spacetime in the core begins to accelerate its expansion in accordance with a $\Lambda$CDM model with an effective inhomogeneous spherical DE density parameter $\Omega_\Lambda(r)$ [1440]. Away from the core, the Universe appears as an Einstein-de Sitter Universe, while near the core, $\Omega_\Lambda(r)$ reaches a maximum.

These findings suggest that topological quintessence models could provide viable explanations for certain cosmological observations and anomalies. The key is that the presence of a large-scale topological defect could naturally introduce the observed anisotropies without requiring exotic modifications of gravity or the introduction of multiple new fields.

### 4.6.2.2. Bianchi metrics.
Bianchi models [4033,4047–4052] generalize the FLRW metric by allowing for anisotropies while maintaining homogeneity. They are classified into nine different types based on their symmetry properties. For example, the Bianchi I metric is given by

$$ds^2 = -c^2 dt^2 + a(t)^2 dx^2 + b(t)^2 dy^2 + c(t)^2 dz^2, \qquad (4.26)$$

where $a(t)$, $b(t)$, and $c(t)$ are scale factors along the $x$, $y$, and $z$ axes, respectively. These models can describe universes that have directional dependencies in their expansion rates, which might help explain certain observed anisotropies [1328,4053–4055]. See Ref. [1364,4035] for observational analysis of Bianchi type-I extension of the $\Lambda$CDM model.

### 4.6.3. Mc-Vittie spacetime

The McVittie metric is the exact solution of Einstein's field equations describing a black hole or a massive object immersed in an expanding cosmological spacetime [4056,4057]. The metric can be written by

$$ds^2 = -(1 - \Phi_N) dt^2 + a(t)^2 \left[ \frac{dr^2}{1 - \Phi_N} + r^2 d\Omega^2 \right], \qquad (4.27)$$

where $\Phi_N = 2GM/(rc^2)$. In the low-energy limit ($\Phi_N \ll 1$ and $rH \ll 1$) the equation of motion reads [4058–4060]: $\ddot{r}/r = -GM/r^3 + \ddot{a}/a$. For the de Sitter-Schwarzschild metric the $\ddot{a}/a$ changes to $\Lambda c^2/3$. For $z \approx 0.67$ the $\ddot{a}/a$ acts as an attractive force for early cosmic times while for the de Sitter-Schwarzschild metric only the constant term $\Lambda c^2/3$ remains. Galaxy groups in the local Universe are bounds on the interplay between the expansion force and the Newtonian attraction [4061]. This gives another setup to determine the Hubble constant [4062–4068].

### 4.6.3.1. Inhomogeneous general relativity solutions.
Beyond the specific cases of LTB and Bianchi metrics, there are more general inhomogeneous solutions in GR that do not assume any specific symmetry. These solutions can be constructed using various techniques, such as analytic or perturbative methods or numerical relativity. One notable

example of an exact solution of the GR field equations, obtained with a dust gravitational source, is the Szekeres model, which generalizes the matter (+ cosmological constant) dominated FLRW solution by dropping any symmetry in the metric

$$ds^2 = -c^2 dt^2 + e^{2\alpha} dx^2 + e^{2\beta} (dy^2 + dz^2), \qquad (4.28)$$

where $\alpha$ and $\beta$ are functions of both spatial and temporal coordinates. These models allow for full inhomogeneity and anisotropy, providing a more general framework to describe the Universe [4069]. The Szekeres models can be divided into three classes: quasi-spherical, quasi-plane and quasi-hyperbolic. The quasi-spherical Szekeres class of solutions, which includes the LTB model as a subclass, can therefore be viewed as a generalization of this LTB model where the spheres of constant $\{t, x\}$ coordinates are non-concentric and exhibit each a given mass dipole whose direction is rotated from one sphere to the other. However, the surfaces $x = \text{const}$ within a space $t = \text{const}$ might be quasi-spherical in regions of the space and quasi-hyperbolic elsewhere, with a zero curvature at the boundaries. Every class possesses the FLRW solutions as a limit for particular forms of the functions defining its metric which can be obtained at some large scale for any of the classes, quasi-spherical, quasi-plane and quasi-hyperbolic. Hence, the possibility to use the Szekeres model for representing the inhomogeneous late Universe with a transition to homogeneity (FLRW) at some larger scale. These models have been used to study various cosmological phenomena, including the formation of large-scale structures and the impact of inhomogeneities on the CMB. Now, in the era of precision cosmology, their application to the analysis of the largest cosmological data sets will have to be performed [4070].

#### 4.6.3.2. Fully model independent cosmographic approach.
Another possibility to test the CP in a fully covariant, model independent way without imposing any a priori metric, is using the general cosmographic approach developed by Kristian and Sachs [4071] (see also Refs. [1358, 1361,4072]). Its only assumption is that the galaxies can be described in terms of a pressureless dust fluid following flow lines that can be described in terms of a unique congruence — an assumption made by all of the other theoretical approaches as well. The idea then lies on the fact that the matter flow, and its kinematical parameters — the expansion $\Theta$, the shear $\sigma_{ab}$ and the vorticity $\omega_{ab}$ — are a direct probe to the expansion of the Universe. One then can Taylor expand the generalized luminosity distance in redshift and obtain the generalized Hubble and deceleration parameters, which are now naturally directional dependent quantities. For an observer boosted with respect to the matter frame (one must always assume this is the case and directly measure the boost velocity), the generalized Hubble parameter has been proven to contain a monopole, a dipole induced only by the boost velocity, and a quadrupole component [1361]. This approach also provides a fully covariant non-perturbative generalization of the perturbative result obtained by Ref. [4073] in which they develop a way to disentangle the intrinsic dipole from a kinematically originated one [1361], a very important need in order to test the CP.

#### 4.6.3.3. Timescape cosmology.
A new study of the Pantheon+ catalog has now claimed that an inhomogeneous alternative [4074–4077] to the models discussed above fits better than $\Lambda$CDM with strong to very strong Bayesian evidence (ln $B \sim 3–5$) [3776]. Positive evidence (ln $B \sim 1–2$) remains even when restricted to SNIa with $z > z_{\text{min}} = 0.06$. In contrast to models based on a single metric, timescape combines small scale FLRW geometries — regionally valid on scales of $\sim 3–30$ Mpc — via a Buchert average [4078–4080]. A closure condition is needed to supplement the Buchert equations, without which its physical interpretation is open to debate [1602,4081]. Timescape extends the Strong Equivalence Principle to a Cosmological Equivalence Principle to apply on small scales over which average isotropic motion in empty space is operationally indistinguishable from average isotropic expansion in nonempty space [4082], leading to the quasilocal Hubble
expansion condition as a closure relation for Buchert averages. The predicted variance in local Hubble expansion can be calibrated [4076] and tested observationally, leading to a potential resolution of the Hubble tension [3775], and a natural framework for resolving dipole anomalies in terms of small scale non-kinematic differential expansion [4083]. This appears to be consistent with new analysis of void statistics in numerical relativity simulations using the full Einstein equations [1604].

The timescape and $\Lambda$CDM expansion histories differ by ~1–3% over small redshift ranges, but can be distinguished with a long enough lever arm. Independent projections made for the Euclid mission in 2014 [4084] show that with Euclid BAOs plus 1000 independent SNIa distances, the FLRW expansion history and the timescape alternative can be definitively tested via the Clarkson–Bassett–Lu (CBL) test [4085]. The new Pantheon+ results for SNIa [3776] have been independently confirmed independently with the DES survey [4086], where it was found that for events with $z > z_{min} = 0.033$ timespace is preferred over $\Lambda$CDM with $\ln B = 1.7$. Based on simple geometric scaling arguments, the DES analysis [4086] finds that BAOs strongly favor $\Lambda$CDM over timescape. However, in timescape the BAO must be extracted directly from raw galaxy clustering data [4087], and independently recalibrated from the CMB anisotropy spectrum and the sound speed in the primordial plasma. The nonbaryonic to baryonic matter densities ratio for timescape still has large uncertainties [4076]. Thus reanalysis of the matter model in the early Universe in conjunction with constraints from a variety of forthcoming datasets is an urgent goal for implementing the CBL test to definitively decide between FLRW and timescape by 2030.

#### 4.6.3.4. Constraints from observations.
Detailed cosmological observations have imposed constraints on some of the alternative models described above. For instance, the Generalized LTB model with inhomogeneous isotropic DE has been studied to understand its consistency with observations such as the Union2 SNIa data and the CMB multipoles [1435]. More recently, Ref. [3774] confronted $\Lambda$LTB models to a host of data sets finding that the models could not resolve Hubble tension. It has been shown that for such models to be consistent with observations, the size of the inhomogeneity must be large, typically on the order of a few Gpc. Additionally, the observer must be located relatively close to the center of the inhomogeneity to avoid large dipole anisotropies.

The exploration of these alternative metrics is crucial for addressing the current tensions in cosmology. By considering more general spacetimes, we can test the robustness of the CP and potentially uncover new physics that could reconcile discrepancies between observations and the standard model. As high-precision data continue to pour in, the development and testing of these models will be an essential part of the future of cosmology.

### 4.6.4. What needs to happen going forward

Even though a more precise definition of the CP took a little while to be developed [4088,4088] following the pioneering works of Vesto Slipher [4089], Henrietta Swan Leavitt [4090], Georges Lemaître [4091] and Edwin Hubble [4092], it was formulated in a time when, more than a principle, it was a *need* for cosmology to go forward. Its historical importance and scientific contribution to the advancement of cosmology is undeniable. But, given the amount of present and future data, continuing to apply it without a proper investigation of its validity and limitations is no longer a matter of science, but faith.

Here we develop a brief list of the necessary requirements in order to properly test the CP:

* Robust data with large sky coverage, in order to test for possible directional dependent effects;

* Model independent techniques for cleaning up possible foreground contamination which does not assume (or imposes) the CP;

* Model independent analysis and parameter estimation;

* The practice of clearly and forthrightly stating all of the (explicit and implicit) assumptions made throughout the analysis;

* A robust model independent theoretical method connected to observations, in a way that allows for the interpretation of the data.

Note that angular variations of $H_0$ on the sky and the cosmic dipole can both be studied independently. The former will benefit from large SNIa samples with excellent sky coverage from ZTF, Vera Rubin observatory, and Romans space telescope. The SKAO will provide large radio galaxy samples allowing a definitive conclusion on the cosmic dipole anomaly [1408,4093].

## 4.7. Quantum gravity phenomenology

**Coordinator:** Giulia Gubitosi

**Contributors:** Christian Pfeifer, Elias C. Vagenas, Gaetano Lambiase, Manuel Hohmann, Nikolaos E. Mavromatos, Saurya Das, and Vasiliki A. Mitsou

One might reasonably conjecture that at least some of the MG models discussed in the previous sections of this review might emerge in an appropriate limit from quantum gravity. In fact, for some of the models the connection is more apparent, as is the case, for example, of Hořava-Lifshitz gravity [4094], that introduces anisotropic scaling between space and time at high energies. This leads to a power-counting renormalizable theory of gravity that deviates from GR at short distances. This model has been proposed by Hořava in Ref. [4094], where an effective Quantum Gravity approach not requiring the Lorentz invariance at fundamental ultraviolet scales has been formulated. This invariance, however, emerges at large distances. It mainly aims to solve the high-energy issues suffered by GR through a spacetime foliation capable of reproducing the causal structure out of the quantum regime. Basic foundations and applications of this approach can be found e.g., in Refs. [3604,4095-4105]. This model is potentially capable of addressing the $H_0$ tension, as demonstrated in Ref. [4106]. Specifically, the authors obtain a positive result on the cosmic tensions between the Hubble constant $H_0$ and the cosmic shear $S_8$ due to a shift of $H_0$ toward a higher value. Moreover, in Ref. [4107] the authors show that up to 36% of the Hubble tension can be explained by Lorentz-violating effects in a Hořava-Lifshitz scenario. This, in fact, is a common feature of theories involving gravitational Lorentz violation, where local $G$ no longer equals cosmological $G$ [4108].

However, a full-fledged fundamental theory of quantum gravity is still elusive. This motivates the adoption of a bottom-up approach, that is complementary to the top-down approach attempting to formulate fundamental theories and then working out their predictions in specific limits. In this bottom-up approach, possible features of the quantum interaction and dynamics of gravity as well as particles and fields propagating on a quantum spacetime are described at an effective level via phenomenological models. This field of research, called quantum gravity phenomenology, successfully leads to observable predictions whose confirmation or constraints serve as guidelines for building a fundamental theory of quantum gravity, as it is explained in detail in the review in Ref. [3067] and white papers in Refs. [3068,4109].

In the following, we will outline how phenomenological models of quantum gravity can affect the understanding of cosmological tensions and how cosmological observations may lead to the discovery of quantum aspects of gravity.
### 4.7.1. QG modified gravitational dynamics

#### 4.7.1.1. Hubble tension and generalized uncertainty principle.
The GUP was originally motivated by considerations coming from string theory [4110] and from the analysis of the relation between gravitational and quantum theories in black hole physics [4111]. Subsequent work found relations to noncommutative geometry [4112,4113]. Recently, in Ref. [4114] it was shown that the deformation parameter entering the GUP can be related to the coefficients of the standard model extension (SME) in the gravity sector [4115] and in this case, stringent bounds on the GUP parameter can be inferred from SME parameters.

The idea of GUP was suggested as a possible explanation of the Hubble tension in Refs. [4116,4117]. GUP introduces Planck-scale corrections to the standard relations between canonical variables, due to the interplay between the quantum theory and general relativity. Therefore, it is expected that such GUP corrections will be relevant during the very early/Planck epochs of cosmology, and leave their fingerprints on the quantum fluctuations. As these GUP-modified quantum fluctuations propagate in a cosmological spacetime, they affect primordial fluctuations during cosmic inflation. Since these fluctuations are encoded in the CMB anisotropies [4118,4119], one expects to read the GUP corrections in the CMB power spectrum. In particular, one can select the cosmological spacetime to be the isotropic and homogeneous FLRW Universe with Hamiltonian, in natural units, of the form [4120,4121]

$$H_{\text{FLRW}}(p_a, a) = N \frac{p_a^2}{24a} + 6Nka - N\rho a^3 + \kappa\Pi, \qquad (4.29)$$

where $a$ and $p_a$ are, respectively, the scale factor which plays the role of the generalized coordinate operator and the generalized canonical momentum conjugate to the scale factor. Note that $N$ is the lapse function and $\Pi$ is its conjugate momentum, while $\kappa$ is its corresponding coefficient. Then, one introduces the GUP-modified canonical momentum [4122–4125]

$$P_a = p_a(1 + \lambda_1 p_a + \lambda_2 p_a^2 + \mathcal{O}(p_a^3)), \qquad (4.30)$$

and substitutes it in the Hamiltonian $H_{\text{FLRW}}(p_a, a)$, in order to obtain the GUP-modified Hamiltonian (keeping terms up to 2nd order in momentum)

$$H_{\text{FLRW}}^{\text{GUP}}(p_a, a) = \frac{1}{24} \frac{p_a^2(1 + 2\lambda_1 p_a + 2\lambda_2 p_a^2 + \lambda_1^2 p_a^2)}{a} + 6ka - \rho a^3 + \kappa \Pi. \qquad (4.31)$$

By combining the GUP-modified Hamiltonian equations, one finds the GUP-modified Hubble function to be [4126]

$$H_{\text{GUP}} = H \left[ 1 + 48\lambda_1 a^2 H + (864\lambda_2 + 576\lambda_1^2) a^4 H^2 \right]^{1/2}, \qquad (4.32)$$

where $H$ is the standard Hubble function.

As already mentioned, CMB will include signatures of the GUP corrections, and thus, the fingerprints of quantum gravity. Therefore, on the one hand, the GUP-modified Hubble parameter, i.e., $H_{\text{GUP}}$, can represent the one obtained by the Planck collaboration which utilizes the CMB data [192], $H_{\text{CMB}}$. On the other hand, the unmodified Hubble parameter $H$, can be assumed to be the one obtained from the HST which utilizes the SNIa data [48]. In this case, we dub it $H_{\text{SN}}$. Based on this, the above expression for the GUP-modified Hubble parameter becomes

$$H_{\text{CMB}} = H_{\text{SN}} \left[ 1 + 48\lambda_1 a^2 H_{\text{SN}} + (864\lambda_2 + 576\lambda_1^2) a^4 H_{\text{SN}}^2 \right]^{1/2}. \qquad (4.33)$$

It is evident that the GUP can, in principle, provide at least a partial explanation of the Hubble tension problem. Detailed analyses are currently ongoing.

### 4.7.2. Quantum gravity effects on the physics of particles and fields

#### 4.7.2.1. Propagation of particles and fields on quantum spacetime.
One intensively studied aspect of the interaction between quantum spacetime and particles and fields is a modification of their propagation properties. Usually, this is studied in terms of a modified dispersion

relation (MDR) of the particles, which encodes a modified light cone and mass shell structure. One distinguishes two cases: the Lorentz invariance violating (LIV) case [4127], where just an MDR is considered, and the deformed relativity (DSR) case [4128,4129], where the MDR is supplemented by a modified energy–momentum conservation law and deformed (most often non-linear) Lorentz transformations between observers which leave the MDR invariant and transform the deformed energy–momentum conservation in a covariant way.

In the context of cosmology, the most interesting consequence of an MDR is a different time of arrival of photons of different energy, when they are emitted simultaneously at the same spacetime event at redshift $z$ [4130], such as GRBs [4131,4132] or AGNs [4133]. Such a time delay can be thought of analogously as to what happens when electromagnetic radiation propagates through an optical non-trivial medium and photons of different energy are affected differently by the medium.

In general, such an MDR for photons can be parameterized in the form [4134]

$$E^2 = p^2(1 + f(E, \mathbf{p}, E_{\text{QG}}, z)), \qquad (4.34)$$

where $f$ is a function that parametrizes the Planck scale modification in terms of the energy $E$ of the photon at emission, its comoving momentum $\mathbf{p} = P/a(t)$, the quantum gravity energy scale $E_{\text{QG}}$ (which might or might not be the Planck scale) and possibly the redshift. For two different photons, emitted with an energy $E_1$ and $E_2$, to leading non-vanishing order in $E_{\text{QG}}$, this leads to a time delay of the form

$$\Delta t_{\text{QG}} = t_2 - t_1 = \frac{1}{H_0} \frac{E_2^n - E_1^n}{E_{\text{QG}}^n} \kappa(z), \qquad (4.35)$$

where $\kappa(z)$ is the redshift distance function to the source at redshift $z$. It depends on the choice of the MDR model, i.e., in the choice of the function $f$ and on the choice of the cosmological model and its parameters. Some examples are

* non-critical string-inspired models with redshift dependent $E_{\text{QG}}$ [4135–4137] or the Jacob-Piran model [4138] with constant $E_{\text{QG}}$, with $\kappa(z) = \int_0^z \frac{(1+z')^n}{E_{\text{QG}}^n(z')H(z')} dz'$.

* For $n = 1$ in Eq. (4.35), DSR models like $\kappa$-Poincare in the bicrossproduct basis [4139], where the redshift distance function reads $\kappa(z) = \int_0^z \frac{1}{(z'+1)H(z')} dz'$ or in more general DSR models in FLRW spacetime, where the redshift distance function is described by the three-parameter $(\eta_1, \eta_2, \eta_3)$ model [4140]

$$\kappa(z) = \int_{0}^{z} dz' \left( \frac{(z'+1)}{H(z')} \left[ \eta_1 + \eta_2 \left( 1 - \left( 1 - \frac{H(z')I(z')}{z'+1} \right)^2 \right) + \eta_3 \left( 1 - \left( 1 - \frac{H(z')I(z')}{z'+1} \right)^4 \right) \right] \right), \quad I(z') = \int_{0}^{z'} \frac{dz''}{H(z'')}$$

Taking into account these time delay effects in the analysis of the Hubble tension might lead to an alleviation of the present tensions [4141–4143]. More detailed analyses remain a future prospect.

#### 4.7.2.2. Interactions of particles and fields on quantum spacetime.
Another relevant feature related to quantum gravity effects on particles concerns modifications of relativistic interactions, that might induce changes to the predictions of the particle content of the Universe.

In LIV models, the combination of MDR with the expected standard conservation of energy and momentum gives rise to strong modifications to interaction thresholds [4127], with effects for example on the opacity of the Universe to high-energy particles [4144,4145].

Going beyond LIV, an important ingredient for self-consistent DSR models is a modified energy momentum conservation. In a process in which two particles with 4-momenta $p$ and $q$ collide, the center of mass energy is not given by the simple sum of their momenta, since this would not be invariant under modified Lorentz transformations, but
given by a modified addition law, to first order of the form [4129]

$$ (p \oplus q)_\mu = p_\mu + q_\mu + \frac{1}{E_{QG}} f(p, q)_\mu, \qquad (4.36) $$

where the precise expression of $f(p, q)_\mu$ depends on the model under consideration. In general, it is a non-linear function of $p$ and $q$.

This modified energy–momentum conservation law leads as well to modifications in relativistic threshold reactions, however they would be much weaker than in the LIV case, typically only relevant for particles of Planck-scale energy. While these effects have no direct link to cosmological tensions, their concomitant presence with the propagation effects discussed above might help constrain the specific form of MDR modification that is compatible with observations.

## 4.8. Varying fundamental constants and their role in the hubble tension

**Coordinator:** Jens Chluba

**Contributors:** Catarina Marques, Dan Grin, Gabriel Lynch, Leo Vacher, Nils Schöneberg, Ruchika, and Vitor da Fonseca

Fundamental physical constants need not be constant, neither spatially nor temporally. This seemingly simple statement has profound implications for a wide range of physical processes and interactions, and can be probed through a number of observations, see Ref. [4146–4148] for a broad review. Studies of fundamental constants (FCs) and their possible temporal and spatial variations are thus of utmost importance, and could provide a glimpse at physics beyond the standard model, possibly shedding light on the presence of additional scalar fields and their couplings to the standard sector, e.g., see Ref. [4149, 4150].

In the cosmological context, the fine-structure constant, $\alpha_{EM}$, and electron rest mass, $m_e$, are the most interesting, although variations of Newton’s constant have also been considered [4151–4153] subject to some recent theoretical and observational constraints [4154,4155]. The former can be directly probed with measurements of the CMB temperature and polarization anisotropies, e.g., see Ref. [4156–4163] through their effect on the cosmological recombination history and photon scattering rate. A detailed description of individual physical effects on the CMB power spectra is given in Ref. [4164], with calculations of the cosmological recombination history carried out using CosmoRec [4165]. In short, increasing $\alpha_{EM}$ and/or $m_e$ leads to earlier recombination. This is primarily driven by the changes to the atomic energy levels, which scale as $E \propto \alpha_{EM}^2 m_e$, thereby enforcing a higher temperature for recombination to occur. Beyond this leading dependence, several subtle effects are encountered leading to differences in how $\alpha_{EM}$ and $m_e$ variations affect the CMB signals. Crucially, the effect of $\alpha_{EM}$ and $m_e$ on the Thomson scattering rate, $\sigma_T \propto \alpha_{EM}^2/m_e^2$, has to be carefully taken into account to yield consistent constraints [4164, 4166]. For additional discussion of the effects on recombination see Ref. [4167].

Analyzing Planck 2013 data, the values of $\alpha_{EM}$ and $m_e$ around recombination were proven to coincide with those obtained in the lab to within $\approx 0.4\%$ for $\alpha_{EM}$ and $\approx 1\%–6\%$ for $m_e$ [4168]. These limits are $\approx 2–3$ orders of magnitude weaker than constraints obtained from other “local” measurements [4169–4172] and those from BBN [4173]; however, the CMB is sensitive to very different phases in the history of the Universe, centered around the time of last scattering some 380,000 years after the Big Bang, thereby complementing these measurements. In addition, CMB measurements can be used to probe spatial variations of the FCs at cosmological distances [4174,4175], opening yet another avenue for exploration.

With the Planck 2013 results in mind, no significant surprises were expected from the analysis of improved CMB data of the Planck 2015 and 2018 releases. It, however, turned out that when considering models with varying $m_e$, the geometric degeneracy becomes significant and can accommodate shifts in the value of the Hubble parameter when

multiple probes are combined [4166]. The same geometric freedom is not encountered when varying $\alpha_{EM}$ due to the modified dependence of the visibility function on this parameter. Indeed, when allowing $m_e$ to depart from the standard (local) value, a non-standard value of $\Delta m_e/m_e = 1.0191 \pm 0.0059$ ($\approx 3.2\sigma$ significance) can be traded for a reduction of the Hubble tension, suggesting that new physics may be at work (see Fig. 80 for illustration). The addition of extra degrees of freedom that influence the post-recombination Universe such as $\Omega_k$ or $w_0/w_a$ to variations of the electron mass allows to further ease the Hubble tension with a $\sim 3.6\sigma$ preference, robust to SN data and BBN constraints [710,1530,2456].

This finding has spurred an increased interest in studying VFCs in this context, with scenarios that allow for varying $m_e$, e.g., see Ref. [4166,4176–4179] for additional discussion, ranking high in model comparisons [710]. Since VFCs can be caused by the presence of scalar fields, e.g., see Ref. [4149,4180,4181], a natural question is whether the same scalar field could also be causing effects relating to EDE, possibly indicating a ‘two sides of the same coin’ interplay. Constraints using the synergy of cosmological and local data have been put on a variety of well-motivated scalar fields models allowing for physical changes in $\alpha_{EM}$ [4182–4186]. However, local data, such as atomic clocks, are putting very strong constraints on such models such that they are mostly unable to produce significant variations of $\alpha_{EM}$ during the recombination era [4187], while such constraints on varying $m_e$ models are less restrictive [1530]. In addition, VFCs could play a role in solving the Hubble tension even in light of the recent DESI measurements [1530,2902,4188–4190], although a general mechanism causing early recombination could simply be the main cause of the tensions [4190].

Given the state of affairs, it will be extremely important to ask how different cosmological probes can be combined to shed light on the physical origin of the Hubble tension. One important avenue forward is to directly constrain the electron recombination history, given its crucial role in the formation of the CMB anisotropies [739,4191–4196]. This has recently been achieved using a non-perturbative and model-independent approach [4188], providing a clear target for theoretical exploration. Indeed, the cosmological data prefers early recombination over the standard recombination history obtained using CosmoRec, very much like what is caused by varying $m_e$ or models with early structure formation [4188]. This data-driven result may therefore indicate new physics in the redshift one thousand Universe, but at this point cannot distinguish the physical cause of this finding.

One way of directly probing the recombination history is through measurements of the cosmological recombination radiation (CRR) [2544,4175,4197–4199]. This tiny spectral distortion signal is created by photons emitted in the hydrogen and helium recombination eras, and thus directly depends on the time and duration of the recombination process. It was shown in Refs. [2544,4175] that various extensions to $\Lambda$CDM can in principle be distinguished with future CMB spectrometer measurements as envisioned for the ESA Voyage 2050 program [4200]. Should the Hubble tension persist, then this will provide the ultimate test for various theoretical models. In addition, one can expect the observational uncertainties in the cosmological recombination history to hamper our ability to answer questions about extensions to $\Lambda$CDM. A measurement of the CRR is therefore highly motivated even beyond questions about the Hubble tension, and provides a direct probe of one of the main pillars in our interpretation of CMB data.

## 4.9. Local new physics solutions to the Hubble and growth tensions

**Coordinator:** Leandros Perivolaropoulos

**Contributors:** Bhuvnesh Jain, Harry Desmond, Indranil Banik, Jeremy Sakstein, Nick Samaras, Radosław Wojtak, and Ruchika
<figure><img src="image_87.png" /><figcaption>**Fig. 80.** Constraints on the fundamental constants using various combinations of Planck data together with their $H_0$ values and errors. Left: results from the fine structure constant $\alpha_{EM}$. Right: similar results but for the effective electron mass $m_e$. Here, we have redacted the constraint for $H_0$ from CMB data only because the error bars are so large. For the $m_e$ MCMC analysis, we have widened the prior on the Hubble constant such that $H_0 > 20 \text{ km s}^{-1} \text{Mpc}^{-1}$. The figure is from Ref. [4166], which illustrated that varying $m_e$ can alleviate the Hubble tension.</figcaption></figure>

A recent comprehensive analysis provides new insights into the Hubble tension, suggesting that the core of the problem may lie in the distance ladder measurements rather than in conflicts between early and late Universe observations [3756]. This study compiled and analyzed two distinct groups of $H_0$ measurements: those based on the distance ladder approach, and those derived from one-step methods independent of both the distance ladder and the sound horizon scale. The analysis revealed a significant discrepancy between these two groups. Distance ladder-based measurements yielded a best-fit $H_0 = 72.8 \pm 0.5 \text{ km s}^{-1} \text{Mpc}^{-1}$, while one-step measurements resulted in $H_0 = 69.0 \pm 0.48 \text{ km s}^{-1} \text{Mpc}^{-1}$. Notably, when two outlier measurements were removed from the one-step sample, the best-fit value reduced to $H_0 = 68.3 \pm 0.5 \text{ km s}^{-1} \text{Mpc}^{-1}$, which is fully consistent with sound horizon-based measurements like those from Planck CMB observations. A Kolmogorov–Smirnov test yielded a *p*-value of 0.0001, indicating that the two samples are not drawn from the same underlying distribution.

Since the distance ladder is the only method for measuring $H_0$ that is based on local physics (via calibrators of SNIa), these findings lend support to the hypothesis of local physics solutions to the Hubble tension. They suggest that the discrepancy may not be between early and late-time measurements, but rather between distance ladder measurements and all other $H_0$ determinations. This points to either a systematic effect influencing all distance ladder measurements or new physics differentially impacting rungs of the ladder.

Such an intriguing class of potential solutions to the Hubble tension involves new physics acting in the local part of the distance ladder (first and second rungs). This idea was first developed by Ref. [4201], who identified that screened fifth forces (for reviews see Refs. [4202–4206]) may have a differential impact on different rungs of the ladder such that their neglect could bias the inference of $H_0$. Ref. [4201] developed a range of phenomenological screening models, in which the degree of screening (and hence the effective value of Newton’s constant, $G_{eff}$) is set by various gravitational properties of the galaxies used to calibrate SNIa and their environments. This included a novel screening mechanism governed by the local DM density arising from the viable DE candidate of baryon–DM interactions [3605,4207]. Ref. [4201] determined ranges for the screening proxies such that the anchor galaxies of the Cepheid PLR (the MW, LMC, and N4258) are screened while some of the galaxies used to infer the SNIa absolute magnitude are not. Since Cepheid pulsation periods are reduced and luminosities increased by an

unscreened fifth force, the standard analysis neglecting the fifth force would then underestimate the distance to the SNIa-calibrator galaxies, which at fixed redshift would overestimate $H_0$.

[235,4208] present another perspective by analyzing the extinction models used for SNIa in the calibration galaxies. Their study suggests that the standard extinction model developed by [232] for the Pantheon+ SNIa compilation underestimates the brightness of reddened supernovae in high stellar-mass calibration galaxies. They propose a modified extinction model that assumes a Milky Way-like distribution of the total-to-selective extinction coefficient $R_B$ in all calibration galaxies, which is also consistent with the extinction corrections employed for Cepheids, and a modified shape of the reddening distribution. This approach yields a lower value of the derived Hubble constant, reducing the tension with the Planck measurement assuming a flat $\Lambda$CDM cosmology from 5.2$\sigma$ to 2.8$\sigma$. This result highlights the importance of accurately modeling local astrophysical effects, such as dust extinction, to resolve the Hubble tension. Different extinction properties in the calibration galaxies do not necessitate violations of the Copernican principle, but they may likely result from the fact that the calibration galaxies are not a representative sample of galaxies in the Hubble flow (the former are solely late-type galaxies, whereas the latter include all morphological types). This selection bias can propagate to the SNIa sector via environment-dependent properties, whose physical origin is currently the subject of intense research.

A detailed modification to the SHOES analysis pipeline allowing for this effect revealed that the Hubble tension could be eliminated with a fifth-force strength ($G_{eff} - G)/G \approx 0.1$ in some of the screening models. However, an important constraint derives from the consistency of Cepheid and TRGB distances to galaxies where both can be measured. The fifth force affects TRGB distances oppositely to Cepheid distances [4201], so the fact that these distances agree under GR implies that they will disagree under a strong fifth force. This requires ($G_{eff} - G)/G \lesssim 0.05$ in the most effective models, preventing this theory from reducing the $H_0$ tension below the $2\sigma$ level. Other constraints were studied, but found not to yield such a strong bound. Nevertheless, in conjunction with some other effects (of which a myriad are presented in this white paper), the tension could be resolved.

Ref. [4201] only addressed the Cepheid-calibrated distance ladder. The principal alternative is TRGB, which yields a slightly smaller $H_0$ value of $70–72 \text{ km s}^{-1} \text{Mpc}^{-1}$ under the assumption of GR. The model was
extended to the TRGB calibration in Ref. [4209], where it was shown to be fully effective at solving the Hubble tension in that case. This requires the LMC (the anchor galaxy of the TRGB absolute magnitude) to be less screened than the SNIa calibrators. As the LMC is the least massive of the SH0ES anchors — and TRGB stars tend to be found in higher-mass galaxies than Cepheids, which are young stars found in star-forming disks — significant regions of the MG parameter space can solve both the TRGB tension and reduce the Cepheid tension to ~2.5σ. Note that the SNIa themselves are assumed screened in all these analyses, as would typically be expected in viable MG models due to their dense environments — discussion of possible effects on SNIa may be found in Appendix B of Ref. [4201]. Variants of the model are studied in Ref. [3267,4210].

An alternative to screening is the more recent idea that new physical phenomena or transitions in the form of physical laws occur locally, specifically at distances ≲ 40 Mpc or redshifts $z \lesssim 0.01$, affecting the connection between the distance ladder's second rung (SNIa calibration by Cepheids or TRGB) and third rung (Hubble flow SNIa). These Local Physics Transitions (LPTs) are assumed to involve abrupt changes of fundamental constants like the gravitational constant $G$ or the fine structure constant. They can affect the calibrators and SNIa used in the first and second rungs of the distance ladder, leading to a different behavior compared to that in the Hubble flow (at larger scales).

The simplest class of this paradigm involves an abrupt transition in the value of $G$ while the other fundamental constants remain fixed. Refs. [2682,2695] propose that a rapid transition in $G$ at a redshift $z_t \approx 0.01$ could resolve the Hubble tension. This $G$-step model (GSM) could imply that SNIa have lower luminosity $L_<$ at local scales (second rung of distance ladder) than the luminosity $L_>$ of distant SNIa in the Hubble flow (third rung). The ratio that would be required to solve the Hubble tension is $L_>/L_<= 1.15$, leading to a higher inferred value of $H_0$ due to the degeneracy between the SNIa $L_>$ and $H_0$ in the Hubble flow.

The LPT paradigm was studied in Ref. [2695] through a reanalysis of the SH0ES data. By allowing for a transition in the absolute magnitude of SNIa at a distance of about 50 Mpc, they find that the best-fit value of $H_0$ drops significantly from $H_0 = 73.04 \pm 1.04 \text{ km s}^{-1}\text{Mpc}^{-1}$ to $H_0 = 67.32 \pm 4.64 \text{ km s}^{-1}\text{Mpc}^{-1}$, in full consistency with the Planck value. This model also shows a substantial improvement in the fit to the data when an additional constraint from the inverse distance ladder is included. Another study further tested and confirmed that the Cepheid and SNIa datasets do not disfavor such a transition [4211].

Assuming $L - A \propto G^\gamma$ (where $A$, $\gamma$ are constants and $L$ is the SNIa absolute luminosity), an exponent $\gamma < 0$ ($\gamma > 0$) would mean that a lower (higher) value of $G$ is required at $z \gtrsim 0.01$ to solve the Hubble tension, leading also to a potential resolution (exacerbation) of the growth rate tension [2682].

Recent constraints on $G$ variation between the present time and recombination suggest that $|\Delta G/G| < 0.05$ at $2\sigma$, where $\Delta G \equiv G_> - G_<$ [4153]. These studies assume $\Lambda$CDM expansion and that all other fundamental constants remain unchanged; relaxing these could weaken the constraints. Using the strong constraints of Ref. [4153], requiring $L_>/L_<= 1.15$ with $A=0$ requires $\gamma \notin [-4.5, 2.8]$.

Early studies assumed that $L \propto M_{\text{Ch}}$ (where $M_{\text{Ch}}$ is the Chandrasekhar mass), which implies $\gamma = -3/2$ [4212–4214]. However, this does not account for the standardization procedure required for turning SNIa into standard candles; building a semi-analytic model to account for this, more recent studies use a simplified SNIa standardization procedure (no full use of light curve stretch and no use of color) and approximate semi-analytical arguments (no hydrodynamical simulation) to suggest that $\gamma \simeq 1.46$ with $A=0$ [4201,4215,4216]. Both approaches make significant simplifications to the underlying physics; a fully reliable analysis should use full 3D hydrodynamical simulations of SNIa with a range of properties and obtain their light curves, properly treat MG effects on white dwarf structure, and adopt

a standardization protocol identical to that employed empirically, also producing uncertainties on $\gamma$ and $A$.

It is thus possible that $\gamma \notin [-4.5, 2.8]$ could be viable in the GSM. $\gamma < 0$ is however also strongly constrained: a sharp rise in $G$ up to 150 Myr ago ($z \lesssim 0.01$) would have dramatic unobserved effects on neutron stars by causing them to contract, thereby releasing vast amounts of energy [4217]. This is however also based on significant simplifying assumptions that may not hold in practice. In particular, roughly constant neutron star and binary formation rate over time is assumed, all galaxies are assumed similar to the MW, higher black hole formation at early-times due to more massive stars and low metallicity is ignored, etc.

Further strong challenges to the GSM are presented in Ref. [4155]. Their main criticisms relate to the expansion of the Earth's orbit around the Sun coupled with lower $G$ leading to a 10% increase in the number of days per year that is not seen in the geochronometric and cyclostratigraphic records, a higher helioseismic age of the Sun due to higher $G$ over the vast majority of its history creating a mismatch with the ages of the oldest meteorite samples from the early Solar System, and a sharp drop in Solar insolation on the Earth in the geologically recent past due to a drop in $G$, leading to a large temperature drop.

Even if $\gamma \notin [-4.5, 2.8]$ were to be conclusively ruled out or SNIa modeling found to require $\gamma < 0$ which is then ruled out by neutron stars, or the tests in Ref. [4155] were verified, the possibility of a simultaneous transition of $G$ with other fundamental constants [4218–4220] like the fine structure constant would remain, keeping the general LPT paradigm alive. Transitions of the type envisaged by the paradigm could be produced by theories such as Dilaton and Kaluza-Klein theories [4180], TeVeS-like theories [4221], or varying $\alpha$ theories [4222]. Note that the above constraints do not apply to the screening models discussed at the start of this section because screening naturally hides the fifth force locally and there is no universal change to fundamental constants like $G$.

In conclusion, the proposed local physics modifications discussed here, including both the screened fifth force model and transitions in the effective gravitational constant, appear to constitute a promising approach to the resolution of the Hubble tension. These results underscore the importance of further investigation of new physics in the local Universe that could affect the calibration of the cosmic distance ladder, and of constraints on such physics from other sources.

## 5. Discussion and future opportunities

The problem of cosmic tensions poses several challenges and a number of opportunities in more deeply assessing possible systematics in observational surveys, developing new data analysis tool kits to refine our treatment of these data products, and developing more robust physical models that can build on the successes of the concordance model and pose new questions and give new solutions to the next decade of cosmology. To meet these challenges, a community-wide effort will be needed involving advances in each separate field, but also strongly and more interconnected relationships between these disciplines.

The coming decade will bring a number of pivotal surveys and observatories that aim to confront the central questions posed by the last decades of concordance physics. This may bring additional unforeseen discoveries in the cosmic history of the Universe, as well as new tools from the increasingly dominant spectrum of ML and statistical physics toolkits, and possibly a new paradigm in our perspective of fundamental physics as an explanation of the driver of cosmic evolution. In Section 5.1 the most significant observatories and survey prospects are discussed in the context of the coming decade of observational cosmology. In Section 5.2 several key areas of development are identified which will be crucial to meeting the growing observational, statistical, and fundamental physics challenges of the upcoming decade of cosmology. Finally, we close in Section 5.3 with a summary of the
central themes of this work and an outlook perspective on the impact of cosmic tensions over the next few years.

## 5.1. Future survey prospects

Cosmology has undergone astounding advancements in the last few decades, due in part to the rapid progress of robust surveys, including unprecedented instrument precision and impressive new approaches to statistical analyses. These advancements have enabled a wider spectrum of tests of theoretical models, as well as unforeseen discoveries such as the accelerating expansion of the Universe in the late 1990s and the unexpectedly high number of galaxies imaged in the Hubble Deep Field. The next decade is set to extend this discovery potential with a swath of exciting missions nearing completion.

The dynamics of future surveys have pivoted to extreme precision surveys with vastly expanded mission objectives. These include measurements reaching much closer to the early Universe, such as the potential detection of the first luminous objects in the Universe through the 21-cm hydrogen line, the possible detection of signatures of inflation in the CMB, and significant reductions in the uncertainty of current CMB measurements. There are also a host of surveys that will track the spatial distribution of galaxies in the Universe to unprecedented levels in terms of the number of galaxies and the precision of these measurements. These surveys will facilitate even deeper investigations into the large-scale structure of the cosmos, providing more information on the nature and evolution of DE and DM.

There is a vast array of planned observational surveys and measurements for the coming decade, which will add to the already considerable number of active survey collaborations. Below, we describe some of the major planned and ongoing missions and survey analyses.

### 5.1.1. Cosmic microwave background

A key focus of the next generation of cosmological surveys is the study of the CMB. Building on decades of groundbreaking discoveries, future CMB experiments are poised to address some of the most pressing challenges in modern cosmology, including the persistent tensions in the Hubble constant and the amplitude of matter fluctuations ($S_8$). Upcoming CMB missions aim to achieve unprecedented precision in measurements of the CMB's temperature and polarization anisotropies, offering unique insights into the physics of the early Universe and its evolution.

The $H_0$ tension stems from a significant discrepancy between the value of the Hubble constant inferred from early Universe observations, such as the CMB, and local measurements using the cosmic distance ladder. Future CMB experiments will refine measurements of the sound horizon at the baryon drag epoch — a critical calibration scale for early-time $H_0$ estimates. These high-precision data will test proposed solutions to the $H_0$ tension, including EDE models, extra relativistic species, or modifications to pre-recombination physics, by either validating or ruling out scenarios that alter the expansion history before recombination.

The $S_8$ tension, which reflects a discrepancy in the amplitude of matter clustering, is another key focus of upcoming CMB experiments. By producing high-resolution maps of the CMB lensing power spectrum, they will directly probe the distribution of DM and the growth of large-scale structures. These measurements will be critical for cross-checking WL and galaxy clustering observations, helping to clarify whether the $S_8$ tension arises from unknown systematics in late-time surveys or signals a breakdown in the standard cosmological model.

Additionally, CMB experiments will probe primordial B-mode polarization, providing a unique window into the inflationary era and the energy scale of the early Universe. These observations could also inform models that connect the physics of inflation to current cosmological tensions, such as scenarios involving new scalar fields or MG.

Future CMB missions will also contribute to our understanding of DE and neutrino physics. Improved measurements of secondary

anisotropies, such as the Sunyaev–Zel'dovich effect, will enable detailed studies of galaxy clusters and baryonic feedback processes, while precise constraints on the sum of neutrino masses and extra relativistic species will help refine models of the early Universe. By combining these capabilities with cross-correlations to other cosmological probes, such as BAO and WL surveys, CMB experiments will play a pivotal role in addressing cosmological tensions and advancing our understanding of the fundamental properties of the Universe.

Below, we highlight some of the major planned and ongoing CMB-focused missions and their scientific goals:

* **South Pole Telescope — 3rd Generation (SPT-3G):** The SPT-3G [4223] is the latest upgrade to the SPT, focusing on high-resolution studies of the CMB from the exceptional observational site at the South Pole. This third-generation camera, featuring over 16,000 detectors, has significantly improved sensitivity, enabling deeper and more precise measurements of CMB temperature and polarization anisotropies. What sets SPT-3G apart is its ability to perform high-resolution surveys over small sky patches, optimized for studies of galaxy clusters via the Sunyaev–Zel'dovich effect, as well as gravitational lensing of the CMB. These measurements are critical for understanding the distribution of DM and large-scale structure formation. Additionally, SPT-3G contributes to constraints on the sum of neutrino masses and potential new physics beyond the standard cosmological model. The compact field-of-view and deep integration capabilities make SPT-3G uniquely suited for detecting subtle signals, such as CMB lensing, and studying small-scale anisotropies.

* **Atacama Cosmology Telescope (ACT):** Located in the Atacama Desert of Chile, ACT [4224] is a ground-based observatory dedicated to high-resolution studies of the CMB. The telescope has been instrumental in mapping CMB temperature and polarization anisotropies and probing large-scale structure. The ACT Data Release 4 (DR4) provided high-sensitivity maps of the CMB, including measurements of temperature, E-mode polarization, and cross-correlations, covering over 17,000 square degrees of the sky. DR4 data significantly refined estimates of cosmological parameters, including constraints on $H_0$, $\sigma_8$, and the sum of neutrino masses. It also included improved measurements of the lensing power spectrum, advancing our understanding of the distribution of matter in the Universe. The forthcoming ACT Data Release 6 (DR6) is expected to feature even more precise maps with reduced noise levels and expanded sky coverage. DR6 aims to provide improved constraints on primordial B-mode polarization, enhancing our ability to test inflationary physics. Additionally, it will enable more detailed studies of secondary anisotropies, such as the Sunyaev–Zel'dovich effect, and cross-correlations with galaxy surveys.

* **PolarBear and Simons Array:** PolarBear, along with its successor, the Simons Array [4225], is focused on high-resolution measurements of the CMB's polarization anisotropies. Located in the Atacama Desert, Chile, these experiments aim to detect B-mode polarization caused by gravitational lensing and primordial GWs. The Simons Array, an upgraded version of PolarBear, consists of multiple telescopes with advanced detectors for improved sensitivity. These measurements will refine constraints on the tensor-to-scalar ratio $r$, test inflationary models, and map the lensing power spectrum, shedding light on the distribution of DM and the evolution of large-scale structure.

* **SPIDER:** SPIDER [2394] is a balloon-borne experiment designed to detect the large-scale polarization of the CMB, focusing on B-modes linked to primordial GWs. By operating above the atmosphere, SPIDER achieves reduced contamination from ground-based noise and atmospheric effects. Its payload consists of multiple telescopes equipped with cryogenic polarimeters, optimized
for observing large angular scales. SPIDER has already completed successful flights, with future missions planned to improve sensitivity and expand sky coverage.

* **Ali Cosmic Polarization Telescope (AliCPT):** AliCPT [4226] is a ground-based CMB experiment located at the high-altitude Ali Observatory in Tibet. AliCPT focuses on measuring the polarization of the CMB at large angular scales, particularly the B-modes associated with primordial GWs. Its high-altitude location minimizes atmospheric contamination, allowing for precise measurements.

* **Q&U Bolometric Interferometer for Cosmology (QUBIC):** QUBIC [4227] is an international collaboration designed to measure the B-mode polarization of the CMB using bolometric interferometry. It uniquely combines the sensitivity of bolometric detectors with the spatial filtering capabilities of interferometry, allowing for precise measurements of polarization anisotropies. Located at high altitude in Argentina, QUBIC targets primordial B-modes associated with inflation, as well as lensing-induced B-modes, providing critical constraints on the tensor-to-scalar ratio $r$. The first module began operations in 2022.

* **The Simons Observatory (SO):** The Simons Observatory [681] is a next-generation ground-based experiment located in the Atacama Desert, Chile. It will consist of three small-aperture telescopes (SATs) and one large-aperture telescope (LAT), all equipped with advanced cryogenic detector arrays. The primary goal of the observatory is to measure the CMB temperature and polarization anisotropies with high precision across a wide range of angular scales. SO will target the primordial B-mode polarization of the CMB, providing critical tests of inflationary physics and insights into the early Universe's energy scale. It will also probe secondary anisotropies, such as the Sunyaev–Zel'dovich effect, to map the growth of large-scale structure. The observatory will place constraints on the sum of neutrino masses and search for new physics beyond the standard model. Additionally, SO will refine measurements of cosmological parameters like $H_0$ and $S_8$, addressing key tensions in the ΛCDM model. The Simons Observatory is expected to begin operations in the mid-2020s.

* **Cosmology Large Angular Scale Surveyor (CLASS):** The CLASS experiment [4228] is designed to study the polarization of the CMB on large angular scales. Located in the Atacama Desert, Chile, CLASS employs a series of telescopes equipped with cryogenic detectors to measure the faint polarization signals from the early Universe. The experiment focuses on detecting the reionization and recombination bumps in the CMB polarization power spectrum, aiming to constrain primordial GWs and the optical depth of reionization. CLASS's innovative strategy includes observing large sky patches with rapid rotation to minimize systematic errors caused by atmospheric and instrumental noise.

* **GroundBIRD:** GroundBIRD [4229] is a ground-based experiment designed to measure the polarization of the CMB on large angular scales. Located at the Teide Observatory in Spain, it uses fast rotation and superconducting detectors to reduce atmospheric noise and systematic errors. The experiment focuses on detecting B-mode polarization signals, particularly those associated with primordial GWs. GroundBIRD's innovative observing strategy, which involves continuous scanning with a rotating cryostat, enhances its sensitivity to large-scale polarization while minimizing contamination.

* **BICEP/Keck Array and BICEP Array:** The BICEP/Keck Array and its successor, the BICEP Array [4230], are focused on detecting B-mode polarization of the CMB with high sensitivity. Operating from the South Pole, these experiments target large-scale polarization signals to constrain the tensor-to-scalar ratio $r$ and probe the inflationary epoch. The BICEP Array incorporates advanced

detector technology and expanded frequency coverage to improve sensitivity and reduce systematic uncertainties.

* **The Lite satellite for the study of B-mode polarization and Inflation from cosmic background Radiation Detection (Lite-BIRD):** The LiteBIRD probe [2393,4231–4234] aims to measure the unique imprints of B-mode polarization in CMB photons, which are related to primordial GWs and inflation. It will succeed the Planck mission in conducting full-sky surveys. Expected to launch in late 2029, LiteBIRD is a Japanese initiative by JAXA involving collaborations with agencies in North America and Europe. Positioned at the Lagrange point L2 in the Sun-Earth system, it will conduct a three-year survey. The relic CMB radiation features E-mode polarization, linked to scalar perturbations, and B-mode polarization, associated with tensor perturbations. While the Cosmic Background Imager provided the first detailed E-mode polarization map, the B-mode signal remains undetected. LiteBIRD will take cosmic-variance-limited measurements of E-mode polarization to study large-scale correlations, shedding light on initial conditions for cosmological perturbations and providing insights into physics beyond the standard model. Regarding B-mode polarization, LiteBIRD will aim for a tensor-to-scalar ratio limit of $r < 0.001$. A positive detection would have a profound impact on fundamental physics, offering insights into inflationary physics, parity violation, and primordial cosmological magnetism.

* **CMB — Stage 4 (CMB-S4):** The CMB-S4 observatory [682] will operate telescopes at both the South Pole and the Atacama Desert in Chile, enabling deep microwave observations over small and large sky fields. It will employ advanced superconducting detector array technologies to reduce galactic foreground contamination. Expected to start operations in the late 2020s, though currently on hold, CMB-S4 aims to measure primordial GWs associated with the early rapid expansion of density fluctuations, characterized by the B-mode polarization of the CMB. This would provide critical information about cosmic inflation and fundamental physics, targeting a tensor-to-scalar ratio of $r < 0.002$, an order of magnitude improvement over current limits. CMB-S4 will constrain the sum of neutrino masses, critical for understanding the sterile neutrino theory and the inverted neutrino mass hierarchy. Beyond neutrino physics, it will impose stringent limits on possible light particles beyond the standard model, addressing extra effective degrees of freedom. On DM, CMB-S4 will probe CMB anisotropies for potential signals of WIMP annihilation, placing constraints on their masses, while exploring non-thermal DM effects on lensing power spectra. Another key goal is probing DE, with precision measurements of ΛCDM cosmological parameters such as $H_0$ and $S_8$, enabling comparisons with other early-time probes. Additionally, CMB-S4 will generate higher-resolution maps of matter distribution in the Universe by measuring distortions in CMB photons caused by gravitational lensing from the surface of last scattering. This will significantly enhance future galaxy surveys.

* **Cosmic Microwave Background - High Definition (CMB-HD):** CMB-HD [4235,4236] is a proposed next-generation ground-based observatory that aims to revolutionize cosmology with its unparalleled resolution and sensitivity. Unlike other CMB experiments, CMB-HD focuses on small-scale anisotropies, achieving an angular resolution of 0.5 arcminutes and surveying over 50% of the sky. This capability allows it to probe previously inaccessible signals and extend our understanding of the Universe to finer detail. CMB-HD's primary science goals include precise measurements of the Sunyaev–Zel'dovich effects to map galaxy clusters, their gas content, and baryonic physics. It will also produce detailed maps of the small-scale CMB lensing power spectrum, offering new insights into the distribution of DM and the growth of large-scale structure. By detecting primordial B-mode polarization, CMB-HD will constrain the tensor-to-scalar ratio $r$.
to test inflationary physics at energy scales far beyond current limits. Additionally, CMB-HD will place stringent constraints on the sum of neutrino masses, light relics, and possible deviations from the $Λ$CDM model.

* **Probe of Inflation and Cosmic Origins (PICO):** PICO [4237] is a proposed next-generation satellite mission designed to provide a comprehensive, all-sky measurement of the CMB. Operating from space, PICO will avoid the challenges of atmospheric contamination and cross-calibration issues that affect ground-based experiments. Its observations will span a broad frequency range from 20 GHz to 800 GHz, enabling highly accurate foreground removal and calibration consistency across the entire sky. A key advantage of PICO's space-based platform is its ability to precisely measure the optical depth to reionization, $\tau$. This measurement, critical for understanding the early history of star formation and the growth of cosmic structures, is less accessible to ground-based telescopes due to the need for absolute calibration. By combining this capability with its unparalleled sensitivity to B-mode polarization, PICO will constrain the tensor-to-scalar ratio $r$ to levels below 0.001, providing definitive tests of inflationary models. In addition to probing inflation, PICO will deliver high-resolution maps of the E-mode polarization and CMB lensing power spectrum, offering insights into the distribution of DM and large-scale structure. It will also place stringent constraints on the sum of neutrino masses, detect potential signals from light relics, and test extensions to the $Λ$CDM model. If approved, PICO will complement ground-based experiments by addressing systematic challenges and providing the calibration accuracy and sensitivity only possible from space.

* **Polarized Radiation Imaging and Spectroscopy Mission (PRISM):** PRISM [4238] is a proposed satellite mission designed to provide extremely sensitive, all-sky measurements of the CMB's temperature and polarization anisotropies. With its advanced instrumentation and broad frequency coverage, PRISM aims to enhance our understanding of both early and late-time cosmology. A key goal of the mission is to detect primordial B-mode polarization, which would provide direct evidence of inflation and constrain the energy scale of the early Universe. PRISM's ability to refine measurements of the CMB power spectrum and polarization will enable tighter constraints on the sound horizon at the baryon drag epoch, directly addressing the $H_0$ tension by testing EDE models and other modifications to pre-recombination physics. Additionally, PRISM's high sensitivity to secondary anisotropies, such as CMB lensing, will improve measurements of the lensing power spectrum, aiding investigations into the $S_8$ tension and the growth of large-scale structure. If approved, PRISM would complement ground-based and balloon-borne experiments by avoiding atmospheric contamination, offering a clean and highly detailed dataset for cosmological studies.

### 5.1.2. Baryon acoustic oscillations

The study of BAO has become a cornerstone of modern cosmology, providing a powerful method for measuring the expansion history of the Universe. BAO are the relic imprints of sound waves that propagated in the early Universe, leaving a characteristic scale in the large-scale distribution of galaxies and matter. This standard ruler has proven invaluable for calibrating cosmic distances and constraining key cosmological parameters, including the Hubble constant and the DE equation of state.

As the precision of cosmological measurements continues to improve, future BAO experiments are poised to deliver new insights into the late-time Universe. By mapping the three-dimensional distribution of galaxies, QSOs, and other tracers across an extensive redshift range, these experiments will probe the dynamics of cosmic expansion and the growth of large-scale structure. This enhanced precision will enable a deeper exploration of the $H_0$ and $S_8$ tensions, testing whether they

arise from unaccounted systematics or new physics beyond the $Λ$CDM model.

In addition to refining our understanding of DE and its influence on cosmic acceleration, upcoming BAO surveys will provide complementary constraints on modifications to gravity and potential extensions to standard cosmology. Synergies with other probes, such as WL and CMB lensing, will further bolster the ability to test fundamental physics and the consistency of the cosmological model.

Below, we describe some of the major planned and ongoing BAO-focused missions and their scientific objectives:

* **Dark Energy Spectroscopic Instrument (DESI):** DESI [814] is a state-of-the-art spectroscopic survey currently operating on the Mayall 4-meter telescope at Kitt Peak National Observatory. Its primary goal is to create the most detailed three-dimensional map of the Universe by observing tens of millions of galaxies and QSOs across a wide redshift range. DESI's high-precision BAO measurements will provide critical constraints on the cosmic distance scale and the expansion history of the Universe, directly addressing the $H_0$ tension. Additionally, RSD analyses from DESI will help probe the growth rate of large-scale structures, contributing to our understanding of the $S_8$ tension and testing potential modifications to gravity.

* **Canadian Hydrogen Intensity Mapping Experiment (CHIME):** CHIME [4239] is a revolutionary radio interferometer located at the Dominion Radio Astrophysical Observatory in Canada. Designed to operate in the 400–800 MHz band, it maps the large-scale structure of the Universe using the 21 cm hydrogen line, covering a redshift range of $0.8 < z < 2.5$. By using intensity mapping techniques, CHIME provides precise measurements of the BAO scale, complementing optical surveys like DESI and HIRAX. The experiment's wide field-of-view and innovative cylindrical reflector design enable it to conduct highly efficient surveys, making CHIME a key player in refining the cosmic distance scale.

* **BAO from Integrated Neutral Gas Observations (BINGO):** BINGO [4240] is a 21 cm intensity mapping experiment specifically designed to measure BAO in the redshift range $0.13 < z < 0.48$. Located in Brazil, it utilizes a compact array of radio antennas to map the large-scale distribution of neutral hydrogen. BINGO is optimized to minimize instrumental noise and systematics, providing precise constraints on the expansion history of the Universe and DE. The project is currently under construction and expected to begin operations in the mid-2020s.

* **Euclid:** The Euclid mission [4241], led by ESA, is designed to create a detailed three-dimensional map of the Universe, enabling precise measurements of the BAO scale across a wide redshift range ($0.7 < z < 2.0$). Its near-infrared spectroscopic survey will provide accurate redshifts for tens of millions of galaxies, establishing a robust standard ruler for cosmological distances. By focusing on high-redshift galaxy clustering, Euclid will refine the expansion history and DE equation of state. Its spectroscopic data will also complement optical BAO studies, bridging gaps in redshift coverage and enhancing multi-probe analyses with other cosmological surveys.

* **Prime Focus Spectrograph (PFS):** The PFS survey [4242, 4243] is a spectroscopic survey operating on the 8-meter Subaru Telescope. It is designed to observe 2,400 objects simultaneously within a $1.2 \text{ deg}^2$ field of view, covering wavelengths from the near-ultraviolet to the near-infrared. With its wide field and spectral coverage, PFS focuses on three primary science programs: cosmology, galaxy evolution, and galactic archeology. For cosmology, PFS will observe approximately 4 million emission-line galaxies (ELGs) over $1,200 \text{ deg}^2$, covering redshifts from 0.8 to 2.4. Unlike other surveys, it will uniquely map ELGs at $2.0 < z < 2.4$, complementing DESI. PFS will provide high-precision measurements of BAO and the Alcock–Paczyński (AP) effect, enabling
constraints on the Hubble expansion history and testing potential evolution in dark energy out to $z = 2.4$. By combining these results with lower-redshift BAO constraints, PFS will determine the dark energy density to approximately 7% accuracy per redshift bin. Additionally, RSD measurements will reconstruct the growth rate of cosmic structure, $f\sigma_8(z)$, with 6% accuracy up to $z = 2.4$, allowing for a precise determination of the sum of neutrino masses, with an uncertainty of $\sigma(\sum m_\nu) = 0.02$.

* **Roman Space Telescope (formerly WFIRST):** The Roman Space Telescope [896], led by NASA, is a highly ambitious mission designed to address key questions about DE, exoplanets, and the structure of the Universe. Through its High Latitude Survey, Roman will map the large-scale distribution of galaxies and perform WL and BAO analyses with unprecedented precision. This mission will provide robust constraints on the expansion history, DE dynamics, and the growth of cosmic structures.

* **Rubin Observatory's Legacy Survey of Space and Time (LSST):** The Rubin Observatory [4244], in its final construction phase in northern Chile, will conduct the LSST, a 10-year survey of $18,000\,\text{deg}^2$ of the sky across six wavelength bands. Public data release is expected approximately two years after first light in 2025. LSST will address a broad range of fundamental questions, including: (1) Is DE dynamical, as characterized by the $w_0-w_a$ parameterization? (2) Can the expansion history and large-scale structure help distinguish between exotic energy densities and MG? (3) What are the properties of DM, as revealed by microlensing searches? (4) Do DM halos exist without hosting galaxies? (5) Are matter fluctuations in the late Universe consistent with CMB-derived constraints? LSST is expected to profoundly impact our understanding of DE, DM, and fundamental physics.

* **SKAO (Square Kilometer Array Observatory):** SKAO [4245] is a next-generation radio telescope array under development, with sites in South Africa and Australia. Its unprecedented sensitivity and angular resolution will enable precise 21-cm intensity mapping, allowing for detailed BAO measurements across a wide redshift range. These data will be instrumental in tracing the evolution of cosmic expansion and structure growth, providing crucial information for resolving the $H_0$ and $S_8$ tensions and testing the consistency of the ΛCDM model. (see Section 5.1.5)

* **The Hydrogen Intensity and Real-time Analysis eExperiment (HIRAX):** HIRAX [1846] is a radio interferometer array under construction in South Africa, optimized for intensity mapping of the 21-cm hydrogen line. HIRAX will measure BAO across a redshift range of $0.8 < z < 2.5$, directly probing the expansion history during the period of cosmic acceleration driven by DE. These measurements will complement optical BAO surveys, refining constraints on DE models and addressing potential new physics.

* **Dark Energy Spectroscopic Instrument - Phase II (DESI-II):** DESI-II [4246] is a proposed extension of the highly successful DESI survey, aiming to build on its existing infrastructure to further refine cosmological measurements. By expanding its redshift coverage and increasing the volume of observed galaxies and QSOs, DESI-II will provide even more precise measurements of the BAO and RSD. The extended survey will target fainter galaxies and higher redshift objects, allowing for a deeper exploration of the late-time Universe's expansion history and the growth of cosmic structures.

* **The 4-meter Multi-Object Spectroscopic Telescope (4MOST):** 4MOST [4247], based at the ESO's VISTA telescope in Chile, is designed to conduct massive spectroscopic surveys of galaxies and QSOs over large sky areas. Its multi-object spectrograph will enable precise BAO measurements and RSD analyses, improving constraints on the cosmic expansion rate and structure growth. 4MOST will work synergistically with imaging surveys like LSST to provide redshift information critical for cosmological studies, including cross-correlation analyses to probe the $S_8$ tension.

* **Spectro-Photometer for the History of the Universe, Epoch of Reionization, and Ices Explorer (SPHEREX):** SPHEREX [4248] is a NASA mission designed to perform an all-sky spectral survey. By mapping galaxies across a wide range of redshifts, SPHEREX will measure BAO and RSD signals, enabling precise constraints on the expansion history and the growth of cosmic structure. Its unique spectral coverage will complement optical surveys and provide new insights into the physics.

* **CO Mapping Array Project (COMAP):** COMAP [4249] is a pioneering experiment focused on intensity mapping of carbon monoxide (CO) lines at high redshift. COMAP will trace the large-scale structure of the Universe during the epoch of galaxy formation, providing a complementary approach to BAO measurements. These data will help refine constraints on the expansion history and test models of DE and MG.

* **Packed Ultra-Wideband Mapping Array (PUMA):** PUMA [4250] is a proposed next-generation radio interferometer optimized for detecting BAO and measuring RSD over a wide redshift range of $2 < z < 6$. Using 21 cm intensity mapping, PUMA will map the large-scale distribution of neutral hydrogen, providing unprecedented insights into the Universe's expansion history and the growth of cosmic structure. Its innovative design, featuring a densely packed array of antennas, will enable high sensitivity and wide bandwidth, making it a critical experiment for studying DE, testing modifications to gravity, and resolving tensions in cosmological parameters. Expected to begin operations in the 2030s, PUMA will complement optical BAO surveys and push the boundaries of cosmological research.

### 5.1.3. Weak lensing experiments

WL is a cornerstone tool for investigating cosmological tensions, particularly the $S_8$ discrepancy, which reflects a persistent difference between the amplitude of matter fluctuations inferred from early- and late-Universe observations. WL measures the subtle distortions in galaxy shapes caused by the gravitational lensing effect of intervening mass distributions, providing a direct probe of the growth of cosmic structures. By combining the clustering amplitude $\sigma_8$ with the matter density parameter $\Omega_{m,0}$, the derived parameter $S_8 = \sigma_8\sqrt{\Omega_{m,0}/0.3}$ becomes a critical test of the ΛCDM model.

In addition to its crucial role in addressing the $S_8$ tension, WL data also contribute to resolving the $H_0$ tension. Synergistic analyses that combine WL with other cosmological probes, such as CMB lensing and BAO measurements, enable a multi-probe approach to jointly constrain both early- and late-Universe parameters. This integrated strategy improves the robustness of results and provides deeper insights into the underlying physics driving these tensions.

Below, we highlight planned and ongoing WL experiments and their contributions to addressing the $S_8$ tension:

* **Kilo-Degree Survey (KiDS):** KiDS [860] provided high-precision WL and photometric redshift measurements across $1350\,\text{deg}^2$, with a strong focus on controlling systematics such as shear calibration, photometric redshifts, and intrinsic galaxy alignments. Using high-quality imaging from the Very Large Telescope (VLT) and tomographic redshift binning, KiDS delivered some of the most precise $S_8$ constraints, highlighting a persistent tension with ΛCDM predictions. Its integration with external datasets, such as galaxy clustering and CMB lensing maps, enabled multi-probe analyses and established a strong foundation for addressing the $S_8$ tension in future surveys.

* **Dark Energy Survey (DES):** DES [4251] observed $5000\,\text{deg}^2$ of the southern sky, combining WL and galaxy clustering to provide robust $S_8$ constraints. With precise photometric redshift calibration and careful control of systematics, DES advanced studies to provide redshift information critical for cosmological studies, including cross-correlation analyses to probe the $S_8$ tension.
of large-scale structure growth. Its cross-correlations with CMB lensing maps and tomographic analyses tested $\Lambda$CDM predictions and explored extensions like evolving DE. DES's extensive data and methodological innovations set a high standard for future WL surveys.

* **Hyper Suprime-Cam (HSC):** HSC [854], conducted on the Subaru Telescope, delivered WL data over 1400 deg² with exceptional resolution and depth, enabling precise studies of cosmic shear. Its high-resolution imaging allowed detailed analyses of smaller-scale structures and provided key insights into $S_8$, revealing persistent tensions with $\Lambda$CDM predictions. The survey's innovative techniques for photometric redshift estimation and systematic error control ensured high accuracy in its results. HSC also contributed significantly to cross-correlation studies with galaxy clustering and CMB lensing, further refining constraints on the growth of structure and testing extensions to the standard cosmological model.

* **Super-pressure Balloon-borne Imaging Telescope (SuperBIT):** SuperBIT [4252] is a stratospheric, balloon-borne telescope designed for high-resolution, wide-field imaging, enabling precise WL measurements of galaxy clusters. By operating above most of Earth's atmosphere, it minimizes atmospheric distortions, delivering exceptional data quality. SuperBIT's observations are critical for mapping the distribution of DM and studying large-scale structure formation, providing complementary insights to ground- and space-based surveys. Its unique capabilities contribute to addressing the $S_8$ tension by offering an independent probe of structure growth.

* **Euclid:** Euclid's high-resolution optical and near-infrared imaging capabilities [4241] are designed to map the weak gravitational lensing of billions of galaxies over 15,000 deg² of the sky. Its ability to probe redshifts up to $z \sim 2.5$ makes it uniquely suited for tomographic studies of structure growth, providing tight constraints on the $S_8$ parameter. By achieving unparalleled depth and resolution in WL, Euclid will play a crucial role in testing modifications to $\Lambda$CDM, such as evolving DE models and MG theories. Its combination of imaging and spectroscopic data will enable synergy with Rubin LSST and CMB lensing maps, further enhancing our understanding of cosmic structure formation.

* **Rubin Observatory's Legacy Survey of Space and Time (LSST):** The Rubin Observatory [4244] will survey 18,000 deg² of the sky over a 10-year period, providing deep, multi-band imaging across six optical filters. Rubin's LSST high precision in WL and galaxy clustering measurements will allow detailed tomographic studies of structure growth and yield tighter constraints on $S_8$. Its ability to detect millions of faint galaxies at high redshifts will improve our understanding of the evolution of cosmic structures and test potential extensions to $\Lambda$CDM. Rubin's LSST rich dataset will also facilitate cross-correlations with CMB and other WL surveys, strengthening the multi-probe approach to resolving cosmological tensions.

* **Roman Space Telescope (formerly WFIRST):** The Roman Space Telescope's High Latitude Survey [896] will deliver high-resolution WL data over 2000 deg², focusing on the distribution of DM and the growth of cosmic structures. Its combination of near-infrared imaging and spectroscopy enables precise photometric redshift estimation, essential for tomographic studies of WL. Rubin's unparalleled sensitivity at high redshifts ($z \sim 2$) will refine measurements of the $S_8$ parameter, providing stringent tests of $\Lambda$CDM and its alternatives, such as evolving DE and MG models. Its synergy with Euclid and Rubin's LSST will enhance cross-calibration efforts and improve constraints on cosmic structure formation across a wide range of scales and epochs.

* **Square Kilometer Array Observatory (SKAO):** SKAO [4245], utilizing 21-cm intensity mapping and WL, will provide unique

and independent constraints on $S_8$ by probing the distribution of DM and the growth of structures. Its radio-based approach will extend WL studies to higher redshifts and larger scales, offering a complementary perspective to optical surveys. By addressing potential systematics in traditional probes, SKAO will play a crucial role in testing deviations from $\Lambda$CDM and enhancing multi-probe strategies to resolve cosmological tensions.

* **Einstein Telescope (ET) and Cosmic Explorer (CE):** These proposed third-generation GW observatories [594,4253-4255], expected to begin operations in the mid to late 2030s, will achieve unprecedented sensitivity in detecting GWs. In addition to their primary focus on GW astrophysics, they will enable measurements of WL effects on GW signals. This innovative approach provides a novel and independent method to probe the $S_8$ parameter, offering insights into the growth of cosmic structures and testing extensions to the standard cosmological model. Their unique capabilities will complement traditional WL surveys, further enriching multi-probe cosmological studies. (see Section 5.1.4)

### 5.1.4. Gravitational waves as probes of cosmological tensions

GWs offer a revolutionary perspective in addressing cosmological tensions, functioning as “standard sirens” that enable the measurement of cosmic distances independently of electromagnetic calibrations. By directly determining the luminosity distance to GW events, particularly those accompanied by electromagnetic counterparts (e.g., binary neutron star mergers), this method bypasses systematics associated with traditional distance ladder techniques. For a recent review on the capabilities of GW observatories see [654].

In addition to these “bright sirens”, GWs from events lacking identifiable electromagnetic counterparts, termed “dark sirens”, can also contribute to cosmology. By correlating the GW signal with galaxy catalogs to infer host redshifts, dark sirens expand the scope of GW cosmology, providing complementary constraints on the Hubble constant and other cosmological parameters. Furthermore, advancements in spectroscopic observations of host galaxies have introduced “spectroscopic sirens”, which refine redshift measurements associated with GW events. These high-precision techniques enhance the reliability of constraints on the expansion history and reduce uncertainties linked to host galaxy identification.

Beyond $H_0$, GWs contribute to understanding large-scale structure formation through WL of GWs by intervening matter. These lensing effects provide unique insights into the amplitude of matter fluctuations, addressing the $S_8$ tension. Additionally, third-generation observatories such as the ET and the Cosmic Explorer will extend GW observations to higher redshifts ($z \sim 10$), probing the early Universe's expansion history and potential deviations from the $\Lambda$CDM model. The integration of GW observations with traditional cosmological probes, such as BAO, CMB, and WL, opens new pathways for multi-messenger cosmology. Together, these complementary approaches promise to resolve key tensions and deepen our understanding of the fundamental physics of the Universe.

Below, we highlight ongoing and planned GW experiments and their contributions to addressing cosmological tensions.

* **LIGO-Virgo-KAGRA Network:** The collaboration between the LIGO, Virgo, and KAGRA interferometers [590] has already achieved a remarkable number of significant detections, including the merger of a neutron star with an unknown compact object. This success is attributed to substantial sensitivity improvements, stemming from upgrades to individual detectors and the synergistic effect of combined observations. The fourth observation run (O4) is scheduled to conclude in the summer of 2025, after which further upgrades will be implemented. These upgrades will include advancements such as reduced thermal noise, new test mass mirrors, and the installation of a larger beamsplitter in the LIGO detectors. These enhancements will significantly increase