between EDE and DM. The coupling modifies the scalar potential, including a matter contribution and a modulation of the DM mass. In a similar spirit, the coupled scalar DM to EDE has been studied in Refs. [2428,2500,2501]. The waterfall EDE model was explored in Ref. [2502], in which the EDE phase ends through spontaneous symmetry breaking. This mechanism has the potential to generate a dark-photon-DM component. In Ref. [2503], the EDE phase is realized and ended similarly to the warm inflation model. The possibility of EDE coupling to neutrinos was studied in Refs. [2504,2505] with the coupling to neutrinos being a natural trigger for the EDE field at around the matter–radiation equality when the neutrinos become non-relativistic. The possibility of these models not being able to solve the Hubble tension was raised in Ref. [2506] and later disputed in Ref. [2507]. Emerging DE models consider a non-negligible contribution of the DE density during the radiation domination era. This effect is introduced through an evolving equation of state that, at early-times, mimics radiation and at late-times asymptotically reaches $\omega \to -1$ [2508–2510]. This phenomenological emergent DE model [2511] shows promising results when confronted with data [808].

Instead of using a specific EDE model, one can use a **model-independent reconstruction** of the EDE expansion history. Ref. [2495] uses a tomographic approach, in which $\rho_{EDE}$ is grouped into redshift bins $z$ and constrained by confronting this model with data. They find that the tightest upper bound on $\rho_{EDE}$ occurs around the CMB decoupling, whereas much weaker constraints are obtained before and after recombination, leaving room for the EDE model to alleviate the Hubble tension. Ref. [2108] determine $\rho_{EDE}$ in scale-factor bins, finding that the maximum $\rho_{EDE}$ is reached at about $z \sim 10^5$, which is different from the $z_c$ derived with the standard approach. Alternatively, a parametric equation of state can be fit [2512]. Moreover, different microphysics of EDE could have important effects [2513]. In Ref. [2514], a phenomenological model is proposed that can mimic the EDE phase through multiple transitions in the vacuum DE density.

#### 4.1.1.4. Constraints on EDE from CMB, BAO and SN.
In the following sections, we will discuss observational constraints on the axion-like EDE model from different data sets (see Fig. 70, right, for a whisker plot). When analyzing EDE with Planck primary and lensing CMB [726], BAO [777,2515,2516], SNIa [2033], and the $H_0$ measurement of the SHOES collaboration [48,2517], these data sets generally prefer high values of $f_{EDE}$ and high values of $H_0$ consistent with direct measurements.<fnref n="51" /> The goodness of fit, i.e., the $\chi^2$, reduces compared to $\Lambda$CDM, albeit with three extra parameters compared to $\Lambda$CDM. This is mainly driven by the ability of EDE to accommodate higher $H_0$ values by reducing $r_s$, which in turn leads to higher values of $n_s$ and $\omega_{cdm}$. Ref. [2454] uses the effective fluid description of the axion-like EDE model, Planck 2015 [726] and SHOES 2018 [2517] and obtains $f_{EDE} \sim 5\%$ at $z_c \sim 5000$ and $H_0 = 70.6 \pm 1.3 \text{ km s}^{-1}\text{Mpc}^{-1}(\Delta\chi^2 = -14.2$, first errorbar in Fig. 70), whereas in Ref. [2455] using the scalar field description and updating to the SHOES 2019 value [48] results in $f_{EDE} \sim 10\%$ at $z_c \sim 3700$ and $H_0 = 71.49 \pm 1.20 \text{ km s}^{-1}\text{Mpc}^{-1}(\Delta\chi^2 = -20.3)$.<fnref n="52" /> Updating to Planck 2018 data, [1965,2459] also use the scalar field description yielding $f_{EDE} \sim 14\%$ at $z_c \sim 3800$ and $H_0 = 71.2 \pm 1.1 \text{ km s}^{-1}\text{Mpc}^{-1}(\Delta\chi^2 = -16.2)$ while updating to DDESI BAO yields $H_0 = 71.7 \pm 0.76 \text{ km s}^{-1}\text{Mpc}^{-1}$. Finally, those constraints were recently updated with Planck 2020 NPIPE data [1970] yielding $H_0 = 71.24 \pm 0.77 \text{ km s}^{-1}\text{Mpc}^{-1}$ for $f_{EDE} = 0.107 \pm 0.023$ at $\log_{10}(z_c) = 3.585_{-0.15}^{+0.049}$ ($\Delta\chi^2 = -28.0)$.

#### 4.1.1.5. EDE and LSS.
Confronting the EDE model with LSS data can impose stringent constraints on $f_{EDE}$. This can be understood as LSS

data typically prefer lower values of the amplitude of clustering $\sigma_8$ and the related parameter $S_8$ compared to CMB data (the so-called “$S_8$ discrepancy”). Ref. [2454] find $S_8 = 0.838 \pm 0.015$ for Planck, BOSS BAO, Pantheon and SHOES data, and more recently Ref. [2442] find $S_8 = 0.831 \pm 0.012$ for Planck, DESI BAO, Pantheon+ and SHOES data. However, $f_{EDE}$ typically correlates positively with the amount of late-time clustering due to increased $\omega_{cdm}$ in models with higher $f_{EDE}$ which compensates for the EDE-induced boost to the early Integrated Sachs-Wolfe (eISW) effect to maintain a good fit to the CMB [2455,2518–2520]. Including LSS data thus weakens the evidence for EDE compared to using CMB data alone by providing an independent constraint on the clustering amplitude [2457].

Several studies have shown that the combination of BOSS full-shape and BAO data [777,1527,2515,2521–2523] (based on the effective field theory of LSS, e.g., see Refs. [1177,1178]) alongside the CMB give tight upper bounds on $f_{EDE}$ [1173,2524,2525] when not including any direct measurements of $H_0$. WL data have also been explored in the context of the EDE model. For example, see Ref. [2457] where DES-YI $3 \times 2$pt data [1165], as well as $S_8$ priors from KiDS and HSC [855,858,859], were used to derive tight upper limits of $f_{EDE} < 0.060$ and $H_0 = 68.92_{-0.59}^{+0.57} \text{ km s}^{-1}\text{Mpc}^{-1}$ while giving high values of $S_8 = 0.8060 \pm 0.0082$ when combined with Planck [192] leading to the conclusion that LSS data can rule out the EDE model (fifth errorbar in Fig. 70). In contrast, [1965] examine the EDE model in the context of KiDS [859] and DES [698] data and find that a one-parameter EDE model, which fixes $\{z_c, \theta_l\}$, does not lead to a significantly worsened $S_8$ discrepancy compared to $\Lambda$CDM and advocate for caution when combining datasets that are statistically discrepant. Furthermore, [1966] argue that the stringent constraints found when combining CMB data with BOSS full-shape and BAO likelihoods can be traced to a potential tension in the power spectrum amplitude $A_s$ between Planck and BOSS which is present even in $\Lambda$CDM. Thus, while it is clear that LSS data can play an important role in constraining the EDE model, the existence of subtle inconsistencies between CMB and LSS data, present even within the $\Lambda$CDM framework, complicates the interpretation.

It is also worth mentioning the important role that the use of $\sigma_8$ as a cosmological parameter might play in this discussion. The $\sigma_8$ parameter is computed as a convolution of the matter power spectrum and a window function at a characteristic scale $R_8 \equiv 8 \text{ h}^{-1} \text{ Mpc}$. Thus, $\sigma_8$ provides information about the amplitude of the power spectrum at a scale that changes in models with different values of $H_0$. To address this issue, [2439] suggested using the parameter $\sigma_{12}$, which is computed at a fixed scale $R_{12} = 12 \text{ Mpc}$ that does not depend on $h$. In Ref. [1976] it is shown that despite the positive correlation between $\sigma_8$ and $f_{EDE}$ in fitting analyses with CMB+BAO+SNIa, the correlation between $\sigma_{12}$ and $f_{EDE}$ is negligible.

Attempts have been made to address both the $H_0$ and $S_8$ tensions simultaneously by expanding the EDE model parameter space to include components that reduce the amplitude of clustering at late-times. For example, Refs. [1969,2428] allow for a free neutrino mass in the EDE model finding that current data limits the effectiveness of massive neutrinos to suppress structure formation at small scales due to the stringent constraint placed on the sum of neutrino masses from the Planck 2018 data which is not much weakened in the context of EDE. Other works have also considered a coupling between EDE and DM in order to try to decrease the value of $\omega_{cdm}$ and make the growth of large-scale structures less efficient at late-times [2428,2459,2498].

#### 4.1.1.6. Prior effects and frequentist constraints.
Due to the complicated parameter structure of the EDE model, the tight upper limits on EDE from CMB and LSS are not only driven by physical effects but also affected by prior volume (or projection) effects in the MCMC posterior. These prior effects arise due to the nested parameter structure of the model: while $f_{EDE}$ controls the fraction of EDE, $z_c$ and $\theta_l$ are auxiliary parameters describing the model in more detail. When $f_{EDE}$ approaches zero, the $\Lambda$CDM limit is recovered, and $z_c, \theta_l$ becomes redundant and

<footnote n="51">⁵¹ All data combinations and model implementations in this section use the same SNIa sample and BAO data, but differ in $H_0$ and Planck data.</footnote>

<footnote n="52">⁵² All central intervals in this section are quoted at 68% CL while all upper limits are quoted at 95% CL.</footnote>
unconstrained. This leads to a larger prior volume at $f_{EDE} \approx 0$ than at $f_{EDE} > 0$ and a non-Gaussian posterior, which in turn can lead to a preference for the $\Lambda$CDM limit in the marginalized posterior. In Refs. [1965,1966] (and Ref. [1971] for NEDE) the influence of prior effects was illustrated by fixing $z_c$, $\theta_l$ in the analysis and varying only $f_{EDE}$ along with the $\Lambda$CDM parameters, which leads to larger allowed values of $f_{EDE} = 0.0523^{+0.026}_{-0.036}$ when using the same BOSS full-shape data [720,1145,1172] with the effective field theory of LSS [1177, 1178] combined with BAO and CMB [1434].

Since these effects are related to the prior, which is an inherently Bayesian quantity, Refs. [1967,1968] constructed frequentist constraints based on the profile likelihood, which are independent of priors and only depend on the likelihood (top three dashed error bars in Fig. 70). Using Planck CMB [1434] and BOSS full-shape power spectrum data [720,1145,1172,2526] based on the effective field theory of LSS, the profile likelihood yields higher values of $f_{EDE} = 0.087 \pm 0.037$ and $H_0 = 70.57 \pm 1.36 \text{ km s}^{-1}\text{Mpc}^{-1}(\Delta\chi^2 = -5.6)$ than the constraints from Bayesian posteriors. Additionally, including SHOES data pushes the inferred $H_0 = 72.1 \pm 0.8 \text{ km s}^{-1}\text{Mpc}^{-1}$ even further up. Even when additionally including 3 × 2pt data from DES [869], the profile likelihood yields moderately high values of $f_{EDE} \approx 6\%$ and $H_0 = 70.28 \pm 1.33 \text{ km s}^{-1}\text{Mpc}^{-1}$. This shows that the Bayesian constraints from CMB and LSS are partially driven by prior effects and indicates that more data is necessary to fully rule out or detect EDE. These results agree with Ref. [1976], which uses an approximate profile distribution obtained from MCMC samples, which is computationally cheaper than computing a profile likelihood. In a Bayesian context, [1173] explores the impact of priors on constraints from the effective field theory of LSS on EDE. Using a Jeffreys prior on nuisance parameters mitigates projection effects from the effective field theory of LSS, resulting in smaller overall volume projection effects on EDE when combined with other datasets. Ref. [2527] utilizes a neural network to obtain a data-driven parameterization of EDE.

### 4.1.1.7. Ground-based CMB experiments and alternative CMB pipelines.
The Planck experiment offers high-precision measurements of the large-to intermediate-scale CMB temperature and polarization power spectra. Ground-based CMB experiments such as ACT and SPT can improve over Planck data, especially at small scales ($\ell \gtrsim 2000$) and in polarization measurements [732]. Constraining EDE with data from the ACT Data Release 4 (TT, TE, and EE power spectra) [732] combined with large-scale Planck TT [192,729,1434], Planck CMB lensing [694], and BAO data [777,1145,2515] yields $f_{EDE} = 0.091^{+0.020}_{-0.036}$ and $H_0 = 70.9^{+1.0}_{-2.0} \text{ km s}^{-1}\text{Mpc}^{-1}$ [710,2458,2528]. The combination of ACT, WMAP [1193], BAO, Pantheon [2033] in a Bayesian analysis leads to $f_{EDE} = 0.158^{+0.015}_{-0.094}$ and $H_0 = 73.43^{+2.6}_{-3.4} \text{ km s}^{-1}\text{Mpc}^{-1}$ [2458] (seventh errorbar in Fig. 70). The combination of data sets ACT DR4, SPT-3G TEE 2018 [725,2529], and Planck TT650TEE 192] gives a similar preference for EDE [2529–2531]. Hence, replacing Planck data with ACT data leads to a preference of high values of $f_{EDE}$ and $H_0$ without the inclusion of any $H_0$ prior [2532–2534]. However, the inclusion of SPT temperature and Planck temperature TT at $\ell > 650$ strongly decreases this preference, with $f_{EDE} < 0.071$ in a Bayesian analysis [1964].

This preference for EDE diminishes with recent ACT DR6 data, finding $f_{EDE} < 0.12$ for ACT combined with Planck, and even lower for ACT alone [703,706]. For ACT DR6 alone, the EDE model does not show any improvement in fit compared to $\Lambda$CDM, suggesting that the preference for EDE in the previous ACT DR4 could be due to a statistical fluctuation in the EE and TE power spectra.

The most commonly used Planck likelihood for multipoles $\ell > 30$ is the Plik likelihood [729]. The CAMSPEC [2535] and Hillipop [701] likelihoods are alternative likelihoods, which differ in data cuts and analysis choices. Ref. [1970] use the CAMSPEC likelihood with the Planck PR4 NPIPE maps [2536] along with BAO data from BOSS [1145], SDSS [2515] and 6dFGS [777] and supernova

data from Pantheon+ [33] to constrain the EDE model. They find low values of $H_0 = 68.11^{+0.47}_{-0.82} \text{ km s}^{-1}\text{Mpc}^{-1}$ and $f_{EDE} < 0.061$ in a Bayesian analysis, and similarly $H_0 = 68.37 \pm 0.75 \text{ km s}^{-1}\text{Mpc}^{-1}$ and $f_{EDE} < 0.094$ in a frequentist analysis (bottom two errorbars in Fig. 70). Using the Hillipop likelihood, Ref. [2446] find similarly tight constraints on EDE, as well as Ref. [2537]. Hence, both alternative Planck likelihoods pose a challenge to the axion-like EDE model’s ability to solve the $H_0$ tension.

### 4.1.1.8. Complementary EDE observables.
If EDE existed in the early Universe, it is natural to expect evidence other than the $H_0$ tension of this deviation from $\Lambda$CDM. In this section, we discuss possible such signatures in data sets different from the commonly used data discussed above. Galaxies and small-scale LSS probes could provide important insight into EDE models. For example, EDE cosmologies predict more massive clusters and an increased abundance of galaxy-mass haloes at higher redshifts [1756]. Refs. [1755,1757,2538] propose that EDE cosmologies could accommodate the massive high-redshift JWST galaxies. However, while the fit to JWST data improves, EDE possibly worsens the fit to Hubble Space Telescope data. Lyman-$\alpha$ forest data from SDSS eBOSS [1556] and MIKE/HIRAS [2333] give tight upper limits on EDE [1561] due to the preference of lower values of $n_s$ than Planck. However, the Lyman-$\alpha$ forest data shows a discrepancy with Planck data [1562], which warrants further investigation. Furthermore, an expansion rate differing from $\Lambda$CDM would affect the relation between redshift and age, which can be constrained by the age of ancient objects in the Universe [2539]. Hence, improved age constraints will be able to constrain EDE cosmologies.

Measurements of the CMB anisotropies along with other CMB observables could hold clues about deviations from $\Lambda$CDM. For example, if EDE would couple to photons in a parity-violating manner, it could explain the tentative evidence of cosmic birefringence observed in Planck data [689,693,2540]. The EDE field with the potential in Eq. (4.4) predicts a unique shape of the EB power spectra, which is not favored by Planck data and hence constrains a possible parity-violating coupling of EDE to photons [691,2541–2543]. Moreover, spectral distortions are sensitive to physics in the (pre-) recombination era. Since EDE is dynamical during this epoch, spectral distortion measurements could be sensitive to distinct EDE signatures [2544].

Theoretical considerations and the requirement of a consistent UV completion can place constraints on EDE models. For example, Ref. [2545] employed the axion weak gravity conjecture<fnref n="53" /> to derive an upper bound on the axion decay constant. Specifically, for $n = 1$ in the potential Eq. (4.4), this leads to $f < 0.008 M_{\text{Pl}}$, which is not consistent with the typically assumed parameter values. Potential ways to circumvent or relax this constraint exist [2545]. Moreover, EDE cosmologies typically prefer higher values of $n_s$ close to unity, which would point to a scale invariant Harrison–Zeldovich spectrum [2519,2531,2548–2550] and would have important implications for inflation [2551,2552].

Hence, while EDE models provide a promising class of models to resolve the Hubble tension, theoretical considerations and new data provide challenges to these models.

### 4.1.2. New early dark energy and variants

Coordinator: Florian Niedermann

Contributors: Martin S. Sloth, and Mathias Garny

NEDE is a theoretical framework to address the Hubble tension where a scalar field $\psi$ undergoes a phase transition between BBN

<footnote n="53">⁵³ The axion weak gravity conjecture is part of a broader program known as the “swampland”. This program aims to define the boundary between effective field theories that are consistent with quantum gravity and those that exist in the “swampland” of incompatible theories. The precise statement of the axion weak gravity conjecture remains a topic of debate [2546,2547].</footnote>
<figure><img src="image_78.png" /><figcaption>**Fig. 71.** Schematic representation of the NEDE framework, explaining its microphysics and phenomenology. Left: The NEDE field decays in a triggered first-order phase transition, giving rise to an energy injection before recombination. EDE is identified with the field's vacuum energy before the transition, and DE with the true vacuum after the transition. Right: On cosmological length scales, a fluid description can be applied. Cold and Hot NEDE energy densities are represented as the solid and dashed lines, respectively.</figcaption></figure>

and recombination on a time scale that is short compared to the Hubble time, such as, e.g., a first-order phase transition [2553,2554]. In contrast to other phenomenological approaches to the Hubble tension, NEDE is rooted in concrete microphysics, which is operative at the eV scale. Current CMB temperature and polarization measurements are sensitive to the precise mechanism triggering the phase transition, and, therefore, the phenomenology differs depending on the concrete NEDE realization. The two main examples are Cold and Hot NEDE, where the trigger of the phase transition is a second scalar field or the temperature of a dark sector, respectively. In the case of Cold NEDE, the phenomenology shares some features with other EDE-type models, but differs both at background and perturbation level, particularly after the phase transition. On the other hand, Hot NEDE is more similar to strongly interacting dark radiation (SIDR) models, and can be viewed as a UV completion of these types of models, explaining naturally the creation of extra radiation after BBN in a supercooled phase transition. In a nutshell, the idea is summarized in Fig. 71, where the left panel explains the field-theoretic idea and the right panel depicts the cosmological model. Initially, the NEDE field $\psi$ is stuck in a false minimum of its potential. It is separated from the true minimum by a large potential barrier. The (false) vacuum energy associated with $\psi$ constitutes an early DE component (orange line). However, as the trigger mechanism kicks in, the barrier shrinks, initiating a strong first-order phase transition. This corresponds to the nucleation of bubbles that separate the true from the false vacuum. Eventually, all of space is converted to the true vacuum. In the case of a fast phase transition, this can be described in a fluid model in terms of an abruptly decaying early DE component (red line). It leaves behind a residual vacuum energy, which can be identified with DE (green line). As a closely related idea, we mention ChainEDE, which relies on a series of a large number of first-order phase transitions to address the $H_0$ tension [2555].

#### 4.1.2.1. Cold new early dark energy.
Cold NEDE introduces an additional, ultralight scalar field $\phi$ to trigger the phase transition [1971, 2554]. Its effective potential valid at low energies is

$$V(\psi, \phi) = \frac{\lambda}{4}\psi^4 + \frac{\beta}{2}M^2\psi^2 - \frac{1}{3}\alpha M\psi^3 + \frac{1}{2}m^2\phi^2 + \frac{1}{2}\bar{\lambda}\phi^2\psi^2 \dots, \quad (4.5)$$

where $\alpha, \beta, \lambda$, and $\bar{\lambda}$ are dimensionless EFT parameters. The model introduces a hierarchy of scales, where the trigger field $\phi$ is an ultralight field with mass $m \sim 10^{-27}$ eV and the tunneling field $\psi$ is much heavier with mass $M \sim$ eV. A possible high-energy completion has been argued to be possible within a multi-axion framework, where the small masses are protected through an approximate shift symmetry [2556]. In terms of its cosmological fluid description, Cold NEDE introduces

four additional parameters: the fluid's maximal energy fraction, $f_{\text{NEDE}}$, the redshift $z_*$ of the phase transition, the equation of state parameter of the post-phase transition fluid, $w_{\text{NEDE}}$, and the relic abundance of the trigger field today $\Omega_{\phi}$.

The phenomenological model has been implemented in the Boltzmann code TriggerCLASS<fnref n="54" /> and tested extensively against cosmological data in the literature. Most work has been done in the limit where $\Omega_{\phi} \ll 1$, corresponding to a subdominant trigger field. For example, using Planck 2018 [192], Pantheon [2033], and BAO [777, 1145,2515] data, the $H_0$ tension was found to be reduced to $2\sigma$ when employing the $Q_{\text{DMAP}}$ tension measure [710] (with a value for $H_0$ taken from Ref. [78]). This result was supported later in Ref. [1972] in a profile likelihood approach, which gets around prior volume issues, that are otherwise known to drive the posteriors for $f_{\text{NEDE}}$ toward small values [1971] (see also the discussion in Section 4.1.1 for more details). Without including a prior on $H_0$, it was found that $H_0 = 69.56^{+1.16}_{-1.29}$ km s$^{-1}$Mpc$^{-1}$ and $f_{\text{NEDE}} = 0.076^{+0.040}_{-0.035}$ at 68% CL, amounting to a $2\sigma$ indication for a non-vanishing fraction of NEDE. Including the SHOES prior, $H_0 = 71.62^{+0.78}_{-0.76}$ km s$^{-1}$Mpc$^{-1}$ and $f_{\text{NEDE}} = 0.136^{+0.024}_{-0.026}$ was obtained at 68% CL, which amounts to a $5\sigma$ evidence for a non-vanishing fraction of NEDE. The model has also been studied in the presence of additional LSS [1972,2557] and ground-based CMB data [2458,2558,2559], broadly matching the baseline results cited above, although it should be noted that axiEDE and Cold NEDE respond differently to ACT and SPT data [2458]. Recently the assumption $\Omega_{\phi} \ll 1$ was dropped in Ref. [2556]. In that case, the trigger field can make a sizable contribution to the energy budget today in the form of a fuzzy DM component (for a phenomenologically similar model see also Ref. [2560]). With their non-vanishing pressure fluctuations, the trigger field's perturbations act against gravitational collapse on small scales as required for solving also the $S_8$ tension. As a result, the model was found to simultaneously reduce both tensions<fnref n="55" /> below $2\sigma$ (with the above choice of datasets $S_8 = 0.818^{+0.023}_{-0.017}$ was obtained).

#### 4.1.2.2. Hot new early dark energy.
In Hot NEDE [2562,2563], the phase transition is triggered by the decreasing dark sector temperature $T_d$, and the sound horizon is lowered due to the dark radiation

<footnote n="54">⁵⁴ https://github.com/NEDE-Cosmo/TriggerCLASS.git</footnote>

<footnote n="55">⁵⁵ This is another important difference at the phenomenological level between axiEDE (and other EDE type models) and NEDE. While axiEDE is known to worsen the $S_8$ tension, NEDE has the potential to resolve both the $H_0$ tension and the $S_8$ tension.</footnote>
<figure><img src="image_79.png" /><figcaption>Fig. 72. Evolution of $\Delta N_{\text{eff}}$ in Hot NEDE with a supercooled Coleman–Weinberg phase transition (PT) between the BBN and recombination epochs, leading to a strong increase ($\Delta N_{\text{BBN}} \to \Delta N_{\text{NEDE}}$) due to the latent heat being converted into relativistic $SU(2)$ dark gauge and Higgs bosons during the PT, and a second slight increase due to the dark Higgs becoming non-relativistic ($\Delta N_{\text{NEDE}} \to \Delta N_{\text{IR}}$). This evolution is intimately linked to the underlying Coleman–Weinberg model. Also shown are bounds from BBN, and the values required to solve the $H_0$ tension around recombination, as well as two models featuring extra dark radiation but no phase transition (SIDR and WZDR [2561]).</figcaption></figure>

generated from the latent heat, i.e., the false vacuum energy released into the dark sector during the transition [2564] (for a comparison to Cold NEDE see Table 5). If the phase transition occurs between BBN and recombination, this setup naturally reconciles SIDR-type solutions to the Hubble tension with BBN bounds on extra radiation [2564] (see Fig. 72). It furthermore differs from SIDR at the perturbation level, and makes complementary predictions. In particular, the model predicts oscillatory features on small scales in the matter power spectrum alongside a Stochastic Gravitational Wave Background (SGWB) signal that can be looked for in Lyman-$\alpha$ forest data and pulsar timing arrays, respectively.

Hot NEDE can be realized by the well-known Coleman–Weinberg mechanism of spontaneous symmetry breaking within a dark sector featuring a dark Higgs field $\psi$ that transforms under a dark gauge symmetry $SU(N)$. This setup naturally leads to the required supercooled first-order phase transition, breaking the initial $SU(N)$ to $SU(N-1)$, as well as to self-interactions described by the non-Abelian gauge symmetry. The dynamics is captured by the effective potential [2564]

$$V(\psi; T_d) = B\psi^4 \left( \ln \frac{\psi^2}{v^2} - \frac{1}{2} \right) - \frac{\mu_{\text{eff}}^2}{2} \psi^2 \left( 1 - \frac{\psi^2}{2v^2} \right) + \Delta V_{\text{thermal}}(\psi; T_d), \quad (4.6)$$

where the dimensionless parameter $B \sim g^4$ depends on the gauge coupling $g$. During the NEDE phase transition, $\psi$ picks up a super-eV expectation value $v$. Due to an approximate conformal symmetry (controlled by the soft breaking parameter $\mu_{\text{eff}} \sim \text{eV}$), the phase transition exhibits a significant amount of supercooling, where the latent heat dominates over the dark radiation plasma, giving rise to an EDE component. In a more fundamental picture, $\psi$ is the modulus of a dark sector Higgs multiplet, breaking an initial $SU(N)$ gauge symmetry to $SU(N-1)$. The corresponding massless gauge bosons alongside the light Higgs degrees of freedom are then populated during the phase transition, reheating the dark sector and introducing a sizable increase in the effective number of relativistic degrees of freedom, $\Delta N_{\text{eff}}$. Although both Cold and Hot NEDE feature an EDE component, their phenomenology differs significantly. In particular, the Hot NEDE phase transition occurs before the CMB epoch at redshifts $z_* \gtrsim 10^5$ (see the right panel in Fig. 71). As a result, Hot NEDE lowers the sound

horizon $r_s$ (and thus increases $H_0$ as explained in Section 4.1.1) through the energy injection provided by the strongly interacting dark sector plasma after the phase transition and, on a phenomenological level, is thus more akin to SIDR models (with mass threshold effects) such as Ref. [2561]. However, its main phenomenological advantage is that it gets around BBN constraints on $\Delta N_{\text{eff}}$, which otherwise would rule out SIDR models as solutions to the Hubble tension [2564,2565], see Fig. 72 (for other ideas to avoid BBN bounds see Refs. [2566–2574]). A first test against cosmological data [2564] showed that the model reduces the $Q_{\text{DMAP}}$ tension to $2.8\sigma$. Moreover, it was found that $H_0 = 96.13^{+0.62}_{-1.00} \text{ km s}^{-1}\text{Mpc}^{-1}$ and $H_0 = 71.17 \pm 0.83 \text{ km s}^{-1}\text{Mpc}^{-1}$ without and with including a Gaussian prior on $H_0$, respectively (datasets as for cold NEDE except for the prior on $H_0$ taken from Ref. [34]).

### 4.1.3. Extra relativistic degrees of freedom

Coordinator: Sunny Vagnozzi

Contributors: Adèle Poudou, Alexander Bonilla Rivera, Biswajit Kar-makar, Branko Dragovic, Davide Pedrotti, Diego Rubiera-García, László Árpád Gergely, Leila L. Graef, Luca Visinelli, Luis Anchordoqui, Marcin Postolak, Margus Saal, Mariana Melo, Matteo Forconi, Özgür Akarsu, Salvatore Capozziello, Sebastian Bahamonde, Shouvik Roy Choudhury, Simony Santos da Costa, Stefano Gariazzo, Thejs Brinckmann, Utkarsh Kumar, Vivian Poulin, William Giarè, and Wojciech Hellwing

Introducing extra relativistic species (ERS, also referred to as “dark radiation”) is one of the most well-motivated and straightforward extensions of the $\Lambda$CDM model, but also one of the simplest ways of raising the inferred value of $H_0$ [2575]. The ESR energy density is most easily parameterized via the effective number of neutrino species $N_{\text{eff}}$ [2576], controlling the relation between the total radiation energy density $\rho_r$ and the photon energy density $\rho_\gamma$ [2577,2578]

$$\rho_r = \rho_\gamma \left[ 1 + \frac{7}{8} \left( \frac{4}{11} \right)^{\frac{4}{3}} N_{\text{eff}} \right]. \quad (4.7)$$

In the presence of only three standard neutrinos undergoing instantaneous decoupling, $N_{\text{eff}} = 3$. However, in the standard cosmological model, $N_{\text{eff}}^{\text{SM}}$ is slightly larger than 3, owing to the fact that the neutrino decoupling process is not instantaneous. The latest determinations yield $N_{\text{eff}}^{\text{SM}} \simeq 3.044$ [2579–2582]. After electron-positron annihilation, the only relativistic particles in the standard cosmological model are neutrinos and photons. However, several well-motivated extensions of the Standard Model of particle physics predict the existence of additional particles which could be relativistic at epochs of cosmological interest, including but not limited to light sterile neutrinos [2583–2587], Goldstone bosons [2588,2589], axions, axion-like particles, and ultra-light DM [2453,2453,2590–2600], light or massless dark photons [2601–2609], and so on [2610,2611]: such particles lead to a value of $\Delta N_{\text{eff}} = N_{\text{eff}} - N_{\text{eff}}^{\text{SM}} \neq 0$. It is worth noting that a non-zero value of $\Delta N_{\text{eff}}$ does not necessarily imply the existence of new particles. For instance, it could be associated with a non-standard distribution of standard neutrinos, or the presence of neutrino asymmetries (i.e., a non-zero chemical potential) [2612,2613]. In addition, a SGWB, possibly at the origin of the signal detected in pulsar timing array data [2614–2617], could also provide a significant contribution to $N_{\text{eff}}$ [2618–2628]. Moreover, $\Delta N_{\text{eff}}$ can in principle be negative, for instance, in low-reheating scenarios following inflation, where the reheating temperature $T_{\text{rh}}$ can be as low as a few MeV, leading to incomplete thermalization of standard neutrinos [2629–2632].

A non-zero value of $\Delta N_{\text{eff}}$ leads to a host of signatures at both the times of BBN and recombination. ERS increases the physical radiation density and thereby raises the pre-recombination expansion rate $H(z)$ before recombination (as well as after, although by then the radiation component is sub-dominant). At the BBN epoch, this pushes the freeze-out of nuclear interactions toward higher temperatures, raising the yield of light elements. At recombination, the presence of ERS leads to
<table><caption>**Table 5**
Comparison of Cold and Hot NEDEs, both relying on a fast-triggered phase transition before recombination to address the Hubble tension.</caption><thead><tr><th>NEDE model</th><th>Cold NEDE</th><th>Hot NEDE</th></tr></thead><tbody><tr><td>model family</td><td>EDE</td><td>(stepped) strongly interacting dark radiation</td></tr><tr><td>physics lowering r<sub>s</sub></td><td>false vacuum energy</td><td>latent heat released as SIDR</td></tr><tr><td>trigger</td><td>ultralight field</td><td>dark sector temperature</td></tr><tr><td>redshift of phase transition</td><td>10<sup>3</sup> &le; z<sub>s</sub> &le; 10<sup>4</sup></td><td>10<sup>5</sup> &le; z<sub>s</sub> &le; 10<sup>9</sup></td></tr><tr><td>equation of state parameter</td><td>w<sub>NEDE</sub> &gt; 1/3 (~2/3)</td><td>w<sub>NEDE</sub> = 1/3</td></tr><tr><td>microphysics</td><td>axion-like particle</td><td>dark sector Higgs</td></tr></tbody></table>

<figure><img src="image_80.png" /><figcaption>**Fig. 73.** Posterior distributions for $H_0$ inferred from a combination of Planck 2018, SDSS BAO, and PantheonPlus SNIa data for different fixed values of $N_{\text{eff}}$. The green shaded region corresponds to the R21 measurement $H_0 = 73.04 \pm 1.04 \text{ km s}^{-1} \text{Mpc}^{-1}$ as inferred by the SH0ES team in Ref. [34].</figcaption></figure>

a variety of direct and indirect signatures on the CMB: however, once the redshift of matter–radiation equality $z_{eq}$ and the acoustic angular scales $\theta_s$ are fixed, the only remaining effects of $\Delta N_{\text{eff}}$ on the CMB power spectra consist of an increased (Silk) damping of the higher acoustic peaks and a phase-shift of the acoustic peaks toward larger scales (e.g., see Refs. [2633–2635]). All these effects indeed lead to some of the tightest cosmological constraints on $\Delta N_{\text{eff}}$: current BBN and CMB inferences both broadly indicate $\Delta N_{\text{eff}} \lesssim 0.2–0.3$, while next-generation cosmological surveys will be able to improve the sensitivity of $\Delta N_{\text{eff}}$ by almost an order of magnitude [681,682].

Increasing $N_{\text{eff}}$ while keeping the physical baryon and DM densities $\omega_b$ and $\omega_c$ fixed has the effect of increasing the pre-recombination expansion rate, thus decreasing the comoving sound horizon at recombination $r_s$, which then requires a higher value of $H_0$ in order to keep $\theta_s$ fixed. This is the reason why ERS are one of the most economical ways of raising $H_0$ and are generally considered a benchmark model in this sense. These effects lead to a positive correlation between $N_{\text{eff}}$ and $H_0$ [712,2636], with an increase in the former leading to an increase in the latter, as captured by the following linear relation [2637], as can be seen in Fig. 73

$$ \Delta H_0 = H_0|_{\Delta N_{\text{eff}} \neq 0} - H_0|_{\Delta \text{CDM}} \approx 5.9 \Delta N_{\text{eff}}, \quad (4.8) $$

as calibrated to Planck 2018 CMB, BAO, and Pantheon SNIa measurements in 2020. This relation, first obtained in Ref. [2637], was illustrated in Ref. [2520] in the $H_0-\sigma_8$ plane, together with the analogous relation for several tension-resolving candidate models.

From Eq. (4.8) we see that increasing $H_0$ to a level that solves the Hubble tension requires $N_{\text{eff}} \gtrsim 4$, corresponding to the effect of an extra fully thermalized neutrino which, importantly, is completely excluded by BBN and CMB considerations. The main reason why the CMB excludes high values of $N_{\text{eff}}$ is that they disproportionately alter the damping scale $r_d$, in turn altering the ratio $r_s/r_d$, which is tightly

constrained especially by high-multipole polarization measurements to be close to its $\Delta \text{CDM}$ value (see e.g., the discussion in Ref. [713]). Moreover, the neutrino drag/phase shift effect [2638,2639] is also altered to an extent that is incompatible with the data. This is why the simplest, 7-parameter $\Delta \text{CDM}+N_{\text{eff}}$ model fails to solve the Hubble tension, despite constituting a useful benchmark [709,2640,2641]. Within the simplest 7-parameter $\Delta \text{CDM}+N_{\text{eff}}$ model one infers $N_{\text{eff}} = 2.89 \pm 0.19$, $H_0 = 66.3 \pm 1.4 \text{ km s}^{-1} \text{Mpc}^{-1}$, and $S_8 = 0.831 \pm 0.013$ from Planck 2018 temperature, polarization, and lensing data, whereas further adding SDSS BAO and Pantheon SNIa data change these numbers to $N_{\text{eff}} = 3.00 \pm 0.17$, $H_0 = 67.4 \pm 1.1 \text{ km s}^{-1} \text{Mpc}^{-1}$, and $S_8 = 0.823 \pm 0.011$ respectively (in what follows, the switch from Pantheon to PantheonPlus SNIa data is not expected to bring about significant changes).

The cosmological evolution of ERS characterized by Eq. (4.7) is de facto equivalent to that of massless neutrinos, and for this reason we can refer to the $\Delta \text{CDM}+N_{\text{eff}}$ model as describing free-streaming ERS. It is worth noting that the above considerations hold for some of the simplest dark radiation models, including light sterile neutrinos [2642], models with lepton asymmetries/non-zero neutrino chemical potential [2643,2644], thermal axions [2593], and so on. Relaxing the assumption that ERS are free-streaming can potentially help address the above shortcomings. The simplest possibility for such a self-interacting dark radiation is one where the ERS component is strongly self-coupled and therefore constitutes a perfect fluid [2645], whose Boltzmann hierarchy can be truncated to the first moment (with vanishing anisotropic stress). Nevertheless, when confronted with cosmological data, such a scenario fails to solve the Hubble tension [710], as it is unable to reduce the neutrino drag and Silk damping effects to an extent that is allowed by the data while raising $H_0$ sufficiently. When confronted with Planck 2018 and SDSS BAO data, one finds $H_0 = 68.67 \pm 0.84 \text{ km s}^{-1} \text{Mpc}^{-1}$. Other natural extensions envisage the possibility of both free-streaming and self-interacting ERS, for instance within the “dark sector equilibration” model of Ref. [2646], and the “recoupling” or “instantaneous decoupling” models (such as in atomic DM [2647–2649] or twin Higgs [2650]) of Ref. [2325], which, however, can only partially alleviate the tension, leading to figures very similar to the previous ones. Yet another possibility to undo the unwanted effects of free-streaming ERS and improve the tension-solving ability of self-interacting dark radiation features scattering between the latter and the DM component: however, this model can only partially alleviate the Hubble tension and is disfavored from a model-comparison perspective due to its higher statistical complexity [710]. In this case, one finds $H_0 = 68.55 \pm 0.92 \text{ km s}^{-1} \text{Mpc}^{-1}$ from a fit to Planck 2018 and SDSS BAO data [710].

A class of models that has received considerable interest in this context features non-standard interactions that do not involve the dark radiation components, but the otherwise standard neutrino species. Non-standard neutrino interactions, for instance via a four-point contact interaction mediated by a sufficiently massive ($m \gtrsim 1$ keV) mediator, as studied in [1319,2325,2651–2663] can suppress or delay neutrino free-streaming, and, with it, the free-streaming-induced phase shift. For fixed values of the sound horizon $r_s$ (which is mostly unaffected by these interactions), the position of the first acoustic peak would therefore be at higher multipoles. Reversing it to the observed position requires a higher value of $H_0$. It has been shown that a
combination of self-interacting neutrinos, additional dark radiation components ($N_{\text{eff}} \sim 4$), and large neutrino masses ($M_\nu \sim 1 \text{ eV}$) can offset the unwanted effects of dark radiation on the damping tail of the CMB, and allow for large values of $H_0$ while not spoiling the fit to CMB temperature data [1318]. A fit to the latter and BAO data reveals a bi-modal distribution for the interaction strength, with stronger values of the interaction strength ($\sim 10^9$ times that of the weak interaction strength) being the tension-solving ones. Nevertheless, this solution is strongly challenged from both the data and theoretical perspectives. From the data side, small-scale polarization data strongly disfavor these models with strong self-interaction strength [2657,2658], which are also partially challenged by BAO data (due to the fact that $r_s$ is unchanged) [1319]. In particular, when confronted against Planck 2018, SDSS BAO, and PantheonPlus SNIa data, one infers $H_0 = 67.3_{-2.1}^{+2.2} \text{ km s}^{-1}\text{Mpc}^{-1}$ and $H_0 = 66.7_{-2.1}^{+2.2} \text{ km s}^{-1}\text{Mpc}^{-1}$ for the moderately interacting and strongly interacting modes respectively [2657,2658]. Finally, from the model-building side, a model in which all three neutrinos self-interact equally is excluded by laboratory constraints [2664], while constructing a phenomenologically viable model has proven challenging [2663]. EFT analyses of BOSS data reveal support for a strongly interacting neutrino mode [2661], although an analysis combined with CMB data still disfavors the strongly self-interacting neutrino model where all 3 neutrinos are interacting [1319]. Interestingly, parameter degeneracies allow for a lower value of the scalar spectral index in the interacting neutrino model, which can be leveraged to allow for inflationary models that are otherwise ruled out within $\Lambda$CDM, such as natural inflation and Coleman–Weinberg inflation [2655,2659,2663]. It is worth noticing that large values of $N_{\text{eff}}$ are in tension with BBN, unless this extra dark radiation is generated after the BBN epoch. This is the scenario studied in Ref. [2665], where a fraction of the DM abundance results from the decay of an unstable and initially thermally decoupled heavy particle, which decays into a DM particle (initially relativistic and therefore contributing to $N_{\text{eff}}$, before becoming cold once the Universe has expanded sufficiently) and a Standard Model particle such as neutrinos or photons. In this case, one finds in the best case $H_0 = 69.08 \pm 0.71 \text{ km s}^{-1}\text{Mpc}^{-1}$ and $S_8 = 0.850 \pm 0.008$ from a fit to Planck 2018, SDSS BAO, and Pantheon SNIa data.

Although the simplest self-interacting neutrino models appear to be ruled out, the framework remains a very interesting benchmark one and gives a good idea of possible ingredients one might want to consider in order to make non-minimal ERS models viable. Motivated by these considerations, various works have explored models of eV-scale Majorons, pseudo-Goldstone bosons associated with the spontaneous breaking of a global $U(1)$ lepton number symmetry. The Majoron enjoys weak couplings to neutrinos, although in the theoretically best motivated coupling limit, it cannot induce the four-point contact interactions mentioned earlier. The resulting phenomenology is instead dominated by inverse neutrino decays and Majoron decays. Overall, these effects still damp neutrino free-streaming, although with a different time dependence, which leads to a better performance of the model when confronted against cosmological data. It should be noted that Majoron decays also lead to a value of $\Delta N_{\text{eff}} \sim 0.11$ [2666]. Nevertheless, although relatively successful, Majoron models fall short of completely solving the Hubble tension [2667], with $H_0 = 70.06_{-2.31}^{+2.21} \text{ km s}^{-1}\text{Mpc}^{-1}$ from a fit to Planck 2018, SDSS BAO, and Pantheon SNIa data. A fit to Planck 2018 and SDSS BAO data instead leads to $H_0 = 70.18 \pm 0.61 \text{ km s}^{-1}\text{Mpc}^{-1}$, whereas the $S_8$ values inferred within the model are not available [2668].

The above are only examples of some of the ERS models considered in the literature, and it is beyond our scope to give a full list of examples. Other interesting possibilities which have been considered in the literature involve self-interacting sterile neutrinos [2669–2672], or the so-called stepped fluids [2561,2673–2675]. The latter are a particular class of self-interacting ERS models, consisting of a mix of massless and massive particles: when the temperature of the dark sector

drops below the mass of a massive species, the latter deposits entropy in the lighter ones, leading to a “step” in the ERS energy density. Concrete Lagrangian realizations of these classes of models have been studied, for instance, within the context of “Wess-Zumino dark radiation”, and have been found to be relatively promising in alleviating the Hubble tension, although current data sets do not show a strong preference for the model [2565]. From a fit to Planck 2018, SDSS BAO, and Pantheon SNIa data, one finds $H_0 = 69.3_{-1.3}^{+0.9} \text{ km s}^{-1}\text{Mpc}^{-1}$ and $S_8 = 0.829 \pm 0.011$ for the Wess-Zumino dark radiation model, whereas for a more general stepped dark radiation model these change to $H_0 = 68.8_{-1.5}^{+1.3} \text{ km s}^{-1}\text{Mpc}^{-1}$ and $S_8 = 0.823 \pm 0.013$ respectively [2561].

Overall, despite the fact that there is no consensus ERS model that solves the Hubble tension, many of these models possess interesting features that could guide the community in the right direction. It would therefore not be surprising if ERS were to play at least some role in the Hubble tension. These and related aspects are being actively investigated.

## 4.2. Late-time proposals

### 4.2.1. Late dark energy

Coordinator: Rafael C. Nunes

**Contributors:** Abdolali Banihashemi, Adrià Gómez-Valent, Anil Kumar Yadav, Anjan Ananda Sen, Anne Christine Davis, Arianna Favale, Benjamin L'Huillier, Branko Dragovic, Brooks Thomas, Carlos G. Boiza, Carsten van de Bruck, Celia Escamilla-Rivera, Cláudio Gomes, Cristian Moreno, Daniele Oriti, David Benisty, David Tamayo, Davide Pedrotti, Elsa Teixeira, Emmanuel N. Saridakis, Gaetano Lambiase, Giulia De Somma, Hsu-Wen Chiang, Hussain Gohar, Ilim Cimdiker, Jaume Haro, Javier Rubio, Joan Solà Peracaula, Juan García-Bellido, Jurgen Mifsud, Kathleen Sammut, Keith R. Dienes, Laur Järv, Laura Mersin, Leandros Perivolaropoulos, Leila L. Graef, Lilia Anguelova, Luis A. Escamilla, Luz Ángela García, Mahdi Najafi, Margus Saal, Mariam Bouhmadi-López, Masoume Reyhani, Mina Ghodsi Yengejeh, Nikolaos E. Mavromatos, Nima Khojrani, Oem Trivedi, Özgür Akarsu, Paloma Morilla, Paolo Salucci, Purba Mukherjee, Rahul Shah, Ronaldo C. Batista, Ruth Lazkoz, Salvatore Capozziello, Sanjay Mandal, Sebastian Bahamonde, Simony Santos da Costa, Suresh Kumar, Utkarsh Kumar, Víctor H. Cárdenas, Vasiliki A. Mitsou, Vincenzo Salzano, Vivian Poulin, and Wojciech Hellwing

#### 4.2.1.1. Current status of simplified DE model parameterizations circa 2024
The key property of DE is its EoS, defined as $w \equiv \frac{P_s}{\rho_s}$. Extensions of the $\Lambda$CDM model, where $w$ can either be a constant or a dynamical function of cosmic time, represent the simplest parametric approaches for testing deviations from the $\Lambda$CDM framework. Fig. 74 presents the current state-of-the-art constraints on the EoS, $w$, at the 68% CL, inferred from various dataset combinations under the assumption of the wCDM model [2676]. The figure is adapted from Ref. [1298]. As widely discussed in the literature, models with $w < -1$, when constrained using CMB data alone, tend to predict higher values of $H_0$, due to a strong degeneracy between $w$ and $H_0$ [2677]. To achieve a more robust analysis, a joint dataset combination of CMB+BAO+SN+CC yields $H_0 = 68.6_{-1.5}^{+1.7} \text{ km s}^{-1}\text{Mpc}^{-1}$, which is in good agreement with the $\Lambda$CDM.

Robust and up-to-date analyses of simple extensions to the well-known dynamic $w(z)$CDM models are presented in Refs. [1283,2678]. The results include the most recent BAO measurements obtained by DESI. The preference for the $w(z)$CDM model remains robust, regardless of the parameterization used, but none of the parameterizations are able to resolve the $H_0$ tension. Similar or identical simple parametric models within the $w(z)$CDM class have been discussed in the literature. In general, while certain datasets may favor a dynamic parameterization of DE, these models are not capable of resolving the tension in $H_0$.
<figure><img src="image_81.png" /><figcaption>Fig. 74. The whisker plot summarizes the 68% CL intervals for the DE EoS parameter, $w$, inferred from various dataset combinations under the assumption of the $w$CDM model. Results that incorporate CMB data are represented by blue bars, except for the final consensus dataset combination (CMB+BAO+SN+CC), which is shown in green. In contrast, results obtained without the CMB data are depicted in red. The green band highlights the 68% CL interval derived from the final consensus dataset combination, yielding $w = -1.013^{+0.038}_{-0.043}$. The gray vertical dashed line marks the cosmological constant value, $w = -1$. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

Source: Figure taken from Ref. [1298].

### 4.2.1.2. Challenges for late dark energy models from BAO and SNIa data on deformations of $H(z)$ and the ultra-late physics transition approach.
LDE models aim to resolve the Hubble tension by introducing degrees of freedom that deform the Planck18 ΛCDM form of $E(z) \equiv H(z)/H_0$, reconciling the early sound horizon scale with late-time Cepheid calibrators. However, these models face significant challenges in simultaneously fitting the SH0ES $H_0$ value, the sound horizon scale at recombination, and distances from SNIa and anisotropic BAO at $z \in [0.01, 2.5]$. Many models also exacerbate the growth tension [3,769, 1298, 1363, 2466, 2532–2534, 2679–2693].

Recent analyses also propose a rapid transition in the effective gravitational constant, $G_{\text{eff}}$, at ultra-late times ($z_t \lesssim 0.01$) to address the tension [72,386,2682,2684,2694–2698]. This is based on the mismatch in absolute magnitude ($M_B$) between CMB and local Universe observations. The luminosity distance-redshift relation

$$ \mu(z) = m_B(z) - M_B = 5 \log_{10} \left( \frac{d_L(z)}{10 \text{pc}} \right), $$

suggests SNIa luminosities at $z > z_t$ are consistent with constant $G_{\text{eff}}$ [1363]. A sudden transition in $G_{\text{eff}}$ at $z_t$ could explain the $\Delta M_B \approx -0.2$ mag deviation in local SN needed to reconcile high- and low-redshift $H_0$ values. This model addresses the $H_0$ tension and suggests reduced growth of density perturbations without affecting ΛCDM background expansion [386,1363,2694,2695]. Future GW, astrophysical, and cosmological perturbation analyses may test this hypothesis in the context of physical models [2695,2696,2699]. The ultra-late $G_{\text{eff}}$ transition provides a compelling alternative framework warranting further investigation.

On the other hand, when the inverse distance ladder is built with angular (or 2D) BAO, instead of anisotropic (or 3D) BAO, the phenomenology required to solve the Hubble tension is completely different. Angular BAO data are claimed to be less subject to model dependencies, since in contrast to anisotropic BAO analyses, no fiducial model is used to convert redshifts and angles into distances of a tracer map. Despite being obtained from the same parent catalogs, 2D and 3D BAO data are known to be in tension [764–766,768,769,2190], pointing to the existence of unaccounted-for systematic errors in one

or both data sets or an underestimation of their uncertainties. When angular BAO is employed in the fitting analysis, it is possible to find a late-time solution to both the Hubble and growth tensions without violating the constancy of $M_B$ [769]. The latter requires the effective DE density to become negative at $z \gtrsim 1.5-2$ to compensate for the late-time increase of $H(z)$ and not spoil the description of the CMB. This possibility has been recently realized in the context of a model with a sign-switching cosmological constant [801,802,811,812,2700–2702] and even more effectively in the model presented in Ref. [2485].

### 4.2.1.3. Holographic dark energy and gravity–thermodynamics correspondence models.
The holographic principle, rooted in quantum gravity, proposes that a system's entropy is determined by its surface area rather than its volume. Holographic dark energy (HDE) utilizes this principle to describe late-time DE as an infrared cutoff for specific HDE models. Various alternative HDE formulations exist, including the Tsallis and Barrow models, each with distinct energy density equations and parameters [2703–2721]. Moreover, choices for the infrared cutoff can vary, ranging from simple options such as the Hubble horizon to more intricate ones like the Granda–Oliveros or Nojiri–Odintsov cutoffs. Beyond theoretical motivations, HDE models potentially resolve tensions in $H_0$ measurements. Fitting these models to comprehensive datasets yields values consistent with recent observations, possibly enhanced by incorporating sterile neutrinos [2722].

On the other hand, modified cosmology based on gravity-thermodynamics correspondence [2723–2726] is another way to explore the Universe's evolution. The relationship between black hole quantities and conventional thermodynamics [2727,2728] has been extended to cosmic horizons [2729], with a range of applications [2723, 2725,2726]. In particular, these concepts have been used for entropic force cosmological models [2730–2735], in which entropic force terms account for the Universe's accelerated expansion. Furthermore, modifications to holographic dark energy models have been made using different notions of entropy [2736–2738] coming from statistical mechanics and thermodynamics. Indeed, the fundamental limit established by the Bekenstein entropy bound [2739] has had important consequences for understanding fundamental aspects of geometry and possible ways of reducing the Hubble tension [2740] consistent with data from Refs. [2741,2742].

The $H_0$ and $\sigma_8$ tension can be concurrently relieved by the modified cosmology employing Tsallis–Cirto nonextensive entropy [2736], as demonstrated in Ref. [2743]. The authors demonstrate how a phantom effective DE, which is recognized as one of the adequate processes that might relieve $H_0$ tension, can be obtained for specific Tsallis–Cirto parameter choices. Furthermore, given the same parameter choice, they find an enhanced friction term and an effective Newton's constant smaller than the conventional one, therefore solving the $\sigma_8$ tension. In Ref. [2744], the $\sigma_8$ tension is examined using Jacobson's method [2723] in the nonextensive setting. Additionally, Hubble tension can be relieved in entropic force cosmological models [2731, 2745] because the interaction terms naturally arise from reversible and irreversible processes across the horizons in the evolution equations. Nevertheless, further work needs to be done to develop these approaches. Moreover, a wide range of holographic dark energy models have been utilized to examine these cosmological tensions, for example, Tsallis holographic dark energy [2746]. Addressing these cosmological challenges will require further research on gravity–thermodynamic relationship to cosmological applications.

### 4.2.1.4. Modeling late dark energy: Examining the phantom, tracker, quintom, multifield descriptions and their clustering properties.
Phantom-scalar-field DE cosmologies are usually denoted by an EoS that satisfies $w < -1$ [2064,2747]. However, the current observations have shown a good agreement with the standard ΛCDM model. Within these observational analyses, some suggestions invoke the presence of a phantom divide boundary. Furthermore, these phantom scalar field DE models
have introduced scalar fields $\phi$ to study the dynamics of evolving DE through this phantom divided boundary, however, within this scheme, we experiment with issues like fine-tuning [2748]. To find a viable solution, tracker DE scenarios have been analyzed [295,2749], where $\phi$ controls the energy density and it is possible to obtain attractor background solutions. Some parametrizations in this regard [2750–2752] are used to verify if this model can solve (or relax) the $H_0$ tension. To introduce a phantom scalar field term, we consider a gravitational action as

$$S = \frac{1}{2\kappa^2} \int d^4x \sqrt{-g} (R + \mathcal{L}_{\phi}(\phi, \partial_{\mu}\phi) + \mathcal{L}_{m}), \quad (4.9)$$

where $\mathcal{L}_{\phi}$ is the Lagrangian for a phantom scalar field. Considering a flat FLRW metric, we can vary Eq. (4.9) with respect to the metric and $\phi$ to obtain the gravitational field equations and the Klein–Gordon equation. In this scheme, it is standard to consider a new set of hyperbolic polar coordinates which can ease the numerical solutions [2751].

From the perspective of this ansatz, it was reported that this kind of phantom/tracker model does not address the Hubble tension when considering a compressed Planck likelihood, given a $H_0 = 69.1^{+0.5}_{-0.6} \text{ km s}^{-1}\text{Mpc}^{-1}$ [2751] value, which is in a $3\sigma$ CL tension with the latest local result [34]. However, Ref. [2753] explored the possibility of reconstructing the constraints using the full Planck [192] likelihood in addition to model-independent CMB baselines, e.g., ACTPol DR4 [702], the SPT-3G [725] and the WMAP9 [2754] datasets, showing that phantom tracker scalar field cosmologies can reduce the statistical tension on $H_0$ to below $3\sigma$.

However, quintom DE, featuring two scalar fields with distinct kinetic energies, blends quintessence and phantom characteristics, and has been shown to be a framework of interest. Model-independent techniques reveal a fluctuating EoS, crossing the phantom divide line ($w_{DE} = -1$) multiple times. Quintom models offer a promising alternative to standard quintessence and phantom models, potentially providing a more accurate fit with fewer parameters. Several studies explore quintom's potential in resolving the $H_0$ tension, with analyses coupling the phantom component to DM and examining various quintessence, phantom, and quintom models. These studies yield various estimates of the Hubble constant, suggesting the possibility of reducing the tension by up to $2.6\sigma$ [1263,2066,2067,2069,2091,2103, 2106,2755–2765].

Multifield models of cosmic acceleration arise from the coupling of several scalar fields to gravity. The standard action that describes this system is

$$S = \frac{1}{2\kappa^2} \int d^4x \sqrt{-g} [R - G_{IJ}((\phi^L))\partial_\mu\phi^I \partial^\mu\phi^J - V((\phi^L))], \quad (4.10)$$

where $g_{\mu\nu}$ is the four-dimensional space–time metric and $G_{IJ}$ is the metric on the manifold parameterized by the scalars $\phi^I$ with $I = 1, 2, \dots, n$. When the field-space metric $G_{IJ}$ has curvature, such multifield models can lead to novel effects, compared to single-scalar field models, in the context of both cosmological inflation and LDE. Specifically, for LDE models of this type, see Refs. [2766–2771].

The qualitatively new features of the multifield case arise from solutions of the background equations of motion, whose field-space trajectories are (strongly) non-geodesic. In the case of two fields, that is, when $n = 2$, the deviation from a geodesic is measured by the turning rate of the background trajectory ($\phi_0^I(t), \phi_0^J(t)$) in field space, $\Omega = -N_I D_i T^I$, where $T^I$ and $N_I$ are unit vectors, respectively, tangent and normal to the trajectory; also $D_i T^I \equiv \phi_0^I \nabla_J T^I$. Strongly non-geodesic trajectories are characterized by large turning rates. Multifield models of LDE, whose background solutions have such rapid-turning field-space trajectories, can have an equation of state parameter $w$ very close to $-1$ and, despite that, can be observationally distinct from a cosmological constant [2768–2771]. This is because the speed of sound $c_s$ of the dark-energy perturbations around the background solution can be (significantly) reduced compared to the speed of light.

DE perturbations around the exact background solution of Ref. [2770] were studied in Ref. [2771]. It was shown there that the speed of sound of these perturbations is significantly reduced. Combining that with an equation of state parameter $w \approx -1$, as well as incorporating matter in the exact background solution, it was further argued in Ref. [2771] that this model of multifield DE is very promising for alleviating simultaneously the $\sigma_8$ and Hubble tensions.

More generally, a low sound speed of DE perturbations is a natural possibility of any k-essence model [1263,2772,2773] and also phantom models [1264]. In this case, DE perturbations can grow on small scales and impact the structure formation, see Ref. [2774] for a review. Recent efforts in understanding the effects of DE perturbations in structure formation include Refs. [2775–2778], but we still need an accurate determination of their impact on nonlinear observables.

With more freedom in the perturbative sector, clustering DE models could circumvent the usual trend seen in late DE models of worsening the growth tension. However, some explorations of this case were conducted only for non-phantom EoS. Recent analyses of this scenario indicate that DE with low sound speed worsens the $S_8$ tension [2779] and the very low $c_s$ values are not allowed by current data [2780].

#### 4.2.1.5. Running vacuum models.
Quantum fluctuations in the expanding Universe induce the running of the vacuum energy density (VED). An example, motivated by dualities in string theory is Ref. [2781] where the spacetime vacuum is such that its trans-Planckian modes are coupled and track the Hubble expansion rate [2782–2786]. The running character of the vacuum was originally motivated using renormalization group arguments in curved spacetime [2787–2789], but more formal calculations have been recently performed in the context of QFT in curved spacetime using adiabatic regularization and renormalization techniques [2790–2792] as well as effective low-energy string theory [2471,2472,2479]. See Refs. [2793,2794] for dedicated reviews of the RVM within the QFT approach and [2473] for the stringy version. The running VED takes the following form

$$\rho_{\text{vac}}(H) = \frac{3}{8\pi G}(c_0 + \nu H^2 + \bar{\nu} \dot{H}) + \mathcal{O}(H^4). \quad (4.11)$$

In contrast to the rigid energy density $\rho_A = \Lambda/8\pi G$ associated with the cosmological constant in $\Lambda$CDM, $\rho_{\text{vac}}$ also receives dynamical contributions from the quantized matter fields. For $\nu, \bar{\nu} \neq 0$ the VED becomes dynamical. Eq. (4.11) lies at the core of the class of RVMs. The higher-order terms $\mathcal{O}(H^4)$ can explain inflation with a graceful exit without the need of ad hoc scalar fields [2475–2477,2479,2795,2796]. Here we just focus briefly on the low-energy terms appearing in Eq. (4.11), which are controlled by the parameters $\nu$ and $\bar{\nu}$. Their values depend on the masses and non-minimal couplings (in the case of the scalar fields) of the various particle species of the theory, and they are expected to be $\lesssim \mathcal{O}(10^{-4} - 10^{-2})$, although they are ultimately determined from observations. In spite of being small, the running of the vacuum can leave a non-negligible imprint on the post-inflationary stages of cosmic expansion both at the background and perturbation levels. The time evolution of $\rho_{\text{vac}}$ may imply either the non-conservation of matter and/or the time-evolution of the gravitational coupling $G$ and other fundamental “constants” of nature, and this can lead to a variety of scenarios explored in the literature [2797–2802]. The RVMs are able to accommodate the wealth of cosmological data at our disposal and have a positive bearing on the cosmological tensions. The RVMs were actually the first to point out significant hints of DE (vacuum) dynamics in the Universe using a large amount of cosmological data, almost one decade before the advent of DESI [2803,2804], see also Refs. [2803–2809] and Refs. [2810–2813]. The most updated constraints on these models can be found in the recent works Refs. [2814,2815], where the authors analyze the RVMs in light of a complete dataset under the simplified assumption $\bar{\nu} = \nu/2$ and show that coupling between DM and the running vacuum can significantly loosen the growth tension, especially when the vacuum dynamics is activated at $z \sim 1$, and also that a joint
running of $\rho_{vac}$ and G can strongly mitigate both the growth and $H_0$ tensions. This is also true in the Brans–Dicke model [2418–2420,2816].

The stringy version of the RVM [2471–2473,2479] provides both early- and late-dark-energy models within string theory [2817,2818]. This model, a Chern–Simons modification of GR [2469], results in a vacuum energy density with logarithmic corrections in the Hubble parameter, $H^2 \ln H$. These corrections, arising from quantum graviton effects in an expanding Universe [2819], help alleviate the $H_0$ and growth-of-structure tensions [2484], and can dominate logarithmic corrections from matter QFT effects [2790–2792].

### 4.2.1.6. Emergent dark energy.
Dynamic DE models are gaining attention over the simple cosmological constant, $\Lambda$, to reconcile discrepancies between CMB and $SH_0$ES data [2508,2820]. One such model is Critically Emergent Dark Energy (CEDE), where DE emerges as a critical phenomenon [2821–2823]. In CEDE, DE density grows from zero in a phase transition, with an order parameter sensitive to photon thermal baths. Below a critical redshift, $z_c$, DE self-interaction dominates, leading to universal order. This model uses Ginzburg–Landau theory to describe phase transitions and derives DE density dependence below $z_c$ as [2823,2824]

$$ \kappa^2 \rho_{\text{DE}}(z)/3H_0^2 = (1 - \Omega_{m,0} - \Omega_{r,0})\sqrt{1 - z/z_c} \quad (4.12) $$

Confronting this phantom-like model with the CMB data [729], yields $H_0 = 70.0^{+1.2}_{-2.7}$ km s$^{-1}$Mpc$^{-1}$with less $\chi^2$ than $\Lambda$CDM. Adding SN data [2033] and/or BAO data [776,777,2433,2515], pushes $z_c$ toward larger values where practically $\Lambda$CDM is restored; and the $H_0$ posterior shrinks to the tension zone of Planck18. CEDE offers richer predictions beyond phantom-like DE. Considering the specific scale dependence of DE density fluctuations in this model [2825], it alleviates both low-$\ell$/$\ell$ inconsistencies in the CMB angular power spectrum and the $A_L$ anomaly reported by Planck [192]. Assuming CEDE, $1\sigma$ posterior intervals of cosmological parameters are constrained with $\ell < 800$ and $\ell > 800$ overlap. Additionally, in this framework, $A_L$ is consistent with unity well within the $1\sigma$ region.

Another Phenomenologically Emergent Dark Energy (PEDE) model developed to address the Hubble tension is [2508]

$$ \Omega_{\text{DE}}(z) = \Omega_{\text{DE},0} \left[ 1 - \tanh \left( \log_{10}(1+z) \right) \right] \quad (4.13) $$

The lack of additional parameters makes the model especially intriguing. However, Ref. [2826] argues that while the PEDE model fits background data well, it fails to match cluster-scale observations compared to $\Lambda$CDM at the perturbation level. Various authors further analyzed this model statistically with diverse cosmological datasets and in extended frameworks [1270,2187,2192,2511,2690,2827–2834]. However, it offers a worse fit to the combined datasets in comparison to $\Lambda$CDM [710]. As extensions of the PEDE model, include the generalized emergent dark energy (GEDE) [2511,2820] and the modified emergent dark energy (MEDE) models [2835]. Both the MEDE & GEDE frameworks can help in reducing the $H_0$ tension to $\lesssim 3\sigma$ CL.

### 4.2.1.7. Vacuum metamorphosis.
The Vacuum Metamorphosis (VM) model, proposed to explain the late-time accelerated expansion of the Universe, is based on non-perturbative quantum gravitational effects [2836–2838]. It features a minimally coupled, ultra-light scalar field with mass $m \sim 10^{-33}$ eV. The model’s key aspect is a gravitational phase transition occurring when the Ricci scalar curvature $R$ reaches the order of the scalar field’s squared mass, $m^2$, which is related to the current matter density parameter $\Omega_{m,0}$. In this framework, the gravitational phase transition occurs at the critical redshift $z_t = -1 + \frac{3\Omega_{m,0}}{4(1-M)}$, where $M$ is a free parameter of the theory.

This transition leads to an effective radiation component in the Universe post-transition, resulting in a De-Sitter phase, distinct from the $\Lambda$CDM scenario. In the original VM scenario, there is no cosmological constant at high redshifts, and the matter density parameter $\Omega_{m,0}$ is given by $\left[\frac{4}{3}(3M(1-M-\Omega_k-\Omega_{r,0}))^3\right]^{1/4}$, where $\Omega_k = -k/H_0^2$ and $\Omega_{r,0}$

is the radiation density parameter. Joint analysis with Planck 2018 + BAO + Pantheon yields $H_0 = 74.21 \pm 0.66$ km s$^{-1}$Mpc$^{-1}$ [2839], which resolves the $H_0$ tension within $1\sigma$. An extension of the VM model, namely the VM-VEV model, includes a cosmological constant at high redshifts, represented by the vacuum expectation value of the massive scalar field. This extension requires additional conditions such as the transition occurring in the past ($z_t \ge 0$) and the DE density being non-negative post-transition i.e., $\Omega_{\text{DE}}(z > z_t) \ge 0$, respectively. Joint analysis with Planck 2018 + BAO + Pantheon leads to $H_0 = 73.26 \pm 0.32$ km s$^{-1}$Mpc$^{-1}$ [2839]. While the VM & VM-VEV models can align $H_0$ with the SH0ES value, they exhibit a poorer fit to low-redshift data, such as BAO and Pantheon, compared to $\Lambda$CDM.

### 4.2.1.8. Others DE frameworks and their implications for the $H_0$ tension.
Rather than initially adopting a particular physical model for DE, given our lack of clear insight into its origin, we can opt for an exploratory approach utilizing a probing function to represent the DE density. The DDE model, generalized to $w(z) = \sum_n w_n x(z)$, where the parameters $w_n$ are fixed by observations and $x(z)$ is a function of redshift that can correspond to both phantom and non-phantom fields, has been shown to alleviate the $\sigma_8$ tension [2840] as well as the $H_0$ tension. It is worth noting that the $H_0$ tension improves further when the neutrino mass $\sum_v m_v$ is taken into account [2841]. Another approach was initiated in Refs. [2055,2056,2842], where the authors used both linear and quadratic probe functions for $X(z) = \frac{\rho_{\text{DE}}(z)}{\rho_{\text{DE}}(0)}$. For $\Lambda$CDM, $X(z) = 1$, while $X(z) \ne 1$ for any redshift $z$ indicates DE evolution. A quadratic probe function used in the literature is proposed in Ref. [2057]. This perspective shows that such an approach may prefer $X(z)$ decreasing with $z$, even taking negative values for $z > 1$. Using the most recent BAO observations from the DESI [699], the same trend was found in Ref. [809]. This trend, including the possibility of a negative $X(z)$, poses a challenge for DE modeling and may have profound implications in light of recent cosmological tensions and the foundations of standard cosmology.

A pseudo-Rip DE scenario, leading to asymptotically de Sitter behavior, is proposed in Ref. [2843]. This model appears to be statistically favored over the consensus $\Lambda$CDM model according to some Bayesian discriminators. The equation of state parameter, typically evaluated at the present time, is given by $w_0 = -1 - \frac{\eta\eta_{\text{sech}}(2\eta)}{3\text{arctan}(\sinh(\eta))}$, where $\eta$ is a free parameter of the model. It is concluded that the pseudo-Rip $\Lambda$CDM model significantly improves the description of late-time ($z \le 2.5$) data more decisively than that at higher redshifts.

Late-time DE transitions at redshifts $z \ll 0.1$ [2532] can make the predicted value of $H_0$ compatible with SH0ES measurements. Conversely, in Ref. [2844], a model-independent constraint on late-time models shows strong evidence against homogeneous new physics over the $\Lambda$CDM model. Surprisingly, despite the absence of $H_0$ and $M_B$ tensions in the local Universe, Ref. [2845] argues that late-time data solutions to the $H_0$ tension require a smaller sound horizon at the recombination era. Additional discussions on $r_d$ scales and $H_0$ tension can be found in Refs. [769,2846]. The authors in Ref. [2847] argue that new physics at low redshifts cannot resolve the $H_0$ tension.

It is important to emphasize that simple scalar field quintessence models have been widely studied for their ability to address the coincidence problem through tracking behavior. A possibility is to use generalized axion-like models with inverse-cosine-like potential [2848–2851], which provides natural entrance and exit of the tracking regime within a single-field framework. While it is not ruled out, the Hubble tension can be slightly reduced [2852]. Also, a different approach to tackle the late acceleration of the Universe is by invoking a modified Chaplygin gas [2853–2856], being the simplest case when its equation of state deviates from that of a cosmological constant by a simple power law of its energy density. This was analyzed from a perturbation point of view in Ref. [2857].

We also highlight that the cosmological stasis framework [2858–2860] is a phenomenon arising within many BSM cosmologies in which
the abundances $\Omega_i$ of multiple cosmological energy components with different equation of state parameters $w_i$ remain constant across extended periods despite cosmic expansion. This occurs due to dynamical feedback within the equations of motion for the corresponding energy densities, with the stasis state serving as a global dynamical attractor. Such systems evolve toward stasis irrespective of initial conditions and remain in stasis until some underlying mechanism alters the dynamical feedback. Most importantly, the effective equation of state parameter $w_{\text{eff}} = \sum_i \Omega_i w_i$ for the Universe remains constant during stasis but nevertheless generically differs from the canonical values associated with, e.g., matter, radiation, or vacuum energy. Stasis at either early or late-times (relative to recombination) can therefore potentially broaden the scope of possibilities for reconciling cosmological tensions. In a similar context, we can also mention the effects of the backreaction of super-Hubble cosmological fluctuations on the late-time accelerated expansion [2861]. Some works suggested that this mechanism could drive a dynamical self-regulating relaxation of the cosmological constant, as first speculated in Ref. [2862]. This behavior could potentially result in an oscillatory effective DE [2863]. A discussion on the tension problem within these scenarios can be found in Ref. [2863], which also provides further references on these models.

### 4.2.2. Dark energy models exhibiting a rapid density transition from negative to positive values in the late universe

**Coordinator:** Özgür Akarsu, Rafael C. Nunes

**Contributors:** Adrià Gómez-Valent, Alexander Zhuk, Anil Kumar Yadav, Anjan Ananda Sen, Antonio De Felice, Branko Dragovic, Davide Pedrotti, Emmanuel N. Saridakis, Emre Özüller, Eoin Ó Colgáin, Hanyu Cheng, J. Alberto Vázquez, Joan Solà Peracaula, Jurgen Mifsud, Laur Järv, Leandros Perivolaropoulos, Leila L. Graef, Luca Visinelli, Luis Anchordoqui, M.M. Sheikh-Jabbari, Nihan Katirci, Ruth Lazkoz, and Suresh Kumar

The possible need for DE assuming negative density values at high redshifts was first highlighted by BOSS collaboration [789], detecting $\Lambda > 0$ for $z < 1$ but favoring $\rho_{\text{DE}} < 0$ for $z > 1.6$, particularly based on Ly-$\alpha$ BAO measurements at $z_{\text{eff}} \approx 2.34$, which exhibited a $2.5\sigma$ tension with Planck-$\Lambda$CDM [192] and suggested a non-monotonic $H(z)$ evolution that is difficult to reconcile with strictly $\rho_{\text{DE}} \ge 0$ in GR. Independently, Ref. [444] proposed that this discrepancy with Planck-$\Lambda$CDM, known as the Ly-$\alpha$ BAO anomaly, could be explained by an MG model in which $\Lambda > 0$ is dynamically screened, leading to an effective $\rho_{\text{DE}}$ passing below zero, accompanied by a singularity in its EoS parameter $w_{\text{DE}}$, at $z \sim 2.3$. This discrepancy was later reduced to $\sim 1.5\sigma$ with the completed eBOSS [1533]. However, model-independent/non-parametric reconstructions of $\rho_{\text{DE}}$, using the BAO data, consistently favor $\rho_{\text{DE}}$ crossing below zero or vanishing for $z \gtrsim 1.5-2$. Moreover, such reconstructions of $w_{\text{DE}}$ — though inherently unable to capture a sign change in DE density — persistently favor $w_{\text{DE}} \sim -1$ for $z \lesssim 1-1.5$ but trend toward large negative values below $-1$ at $z \sim 1.5-2$ — which is expected right after a DE density smoothly transitions from negative to positive as the Universe expands, therefore potentially indicating a sign change in $\rho_{\text{DE}}$. The recent DESI Ly-$\alpha$ BAO data shows no tension with Planck-$\Lambda$CDM [699]. However, while DESI BAO data — when analyzed using the CPL parameterization — provides more than $3\sigma$ evidence for dynamical DE [699], a less noted but significant finding is that non-parametric reconstructions of DE density also indicate the possibility of vanishing or negative values for $z \gtrsim 1.5-2$ [809,2864], consistent with trends observed in pre-DESI BAO data, e.g., SDSS BAO [2145,2162,2864].

It was not immediately recognized that addressing the Ly-$\alpha$ anomaly could naturally be linked to the $H_0$ and $S_8$ tensions, both of which emerged sometime after the Ly-$\alpha$ anomaly was first identified. A brief explanation of this connection is as follows [801,811]: If a DE model leaves the pre-recombination Universe remains unaltered, as in the

standard $\Lambda$CDM model, then the comoving sound horizon at last scattering, $r_{*} = \int_{z_{*}}^{\infty} c_{s} H(z)^{-1} dz$, is expected to remain effectively unchanged from its $\Lambda$CDM counterpart. The Planck CMB spectra provide precise, nearly model-independent measurements of the angular scale of the sound horizon, $\theta_{*} = r_{*}/D_{M}(z_{*})$, and the present-day physical matter density, $\Omega_{m,0}h^2$, derived from the peak structure and damping tail. Consequently, in DE models featuring negative DE densities at high redshifts, say, for $z > z_{\dagger}$, both the comoving angular diameter distance to last scattering, $D_{M}(z_{*}) = c \int_{0}^{z_{*}} H(z)^{-1} dz$, and $\Omega_{m,0}h^2$ must remain consistent with their Planck-$\Lambda$CDM inferred values. This requires that any suppression of $H(z)$ at $z > z_{\dagger}$, due to negative DE density, must be compensated by an enhancement of $H(z)$ at $z < z_{\dagger}$ to maintain consistency with the Planck-$\Lambda$CDM-inferred $D_{M}(z_{*})$. As a result, this mechanism naturally increases $H_0$ (implying a fainter $M_B$) and decreases $\Omega_{m,0}$ relative to Planck-$\Lambda$CDM. A later transition (i.e., smaller $z_{\dagger}$) leads to a prolonged phase of suppression, or similarly, more negative DE density values lead to more suppression; both effects amplify the enhancement in $H_0$ and reduction in $\Omega_{m,0}$, provided the transition occurs before the negative DE density becomes dominant — beyond which expansion would halt and contraction would ensue. This framework also predicts higher $\sigma_8$ values because the suppressed $H(z)$ at $z > z_{\dagger}$ reduces cosmic friction, thereby enhancing structure formation at high redshifts [2696, 2865]. However, the lower present-day matter density leads to reduced $S_8 = \sigma_8 \sqrt{\Omega_{m,0}/0.3}$ values, potentially resolving the $S_8$ tension. This reduced cosmic friction enhances structure formation for $z > z_{\dagger}$ [801], potentially explaining the JWST anomaly [1755,1759,2866,2867] — where deep-space observations at $z \gtrsim 5$ indicate stronger structure growth than predicted by Planck-$\Lambda$CDM.

In the rapidly growing literature, fitting within the framework outlined above, Refs. [2700,2868] are the earliest studies explicitly connecting the Ly-$\alpha$ anomaly with the $H_0$ tension. Ref. [2868] assumed that the Universe is consistent with Planck-$\Lambda$CDM for $z \gtrsim 4$ and reconstructed $H(z)$ using low-redshift data, including Ly-$\alpha$ BAO. They found that the DE density reaches a minimum within a certain redshift range and becomes negative for $z \gtrsim 2$, accompanied by higher $H_0$ values, and argued that this behavior can be most simply explained by an AdS-like cosmological constant ($\Lambda < 0$) combined with an evolving DE component. Ref. [2700] proposed that inertial mass density, $\rho \equiv \rho + p$, maybe more fundamental than energy density and introduced a DE parameterization with a minimal dynamical deviation from the usual vacuum energy/cosmological constant ($\rho = 0$) in the form $\rho \propto \rho^{\lambda}$, referred to as graduated dark energy (gDE) ($\rho = \text{const}$, simple-gDE [2869]). gDE exhibits a wide range of behaviors depending on $\lambda$; notably, for large negative $\lambda$, it serves as a phenomenological model for an AdS-to-dS-like transition in DE. It was shown via gDE that joint observational data, including but not limited to Planck-CMB and Ly-$\alpha$ BAO, suggest $\lambda \sim -18$, indicating a behavior analogous to a rapid AdS-to-dS transition at $z_{\dagger} \sim 2$, consequently alleviating both the $H_0$ tension and the Ly-$\alpha$ anomaly. With this constraint on $\lambda$, gDE resembles a smooth step-like function, yielding $\rho_{\text{gDE}}/\rho_{c0} \sim -0.7$ with $w_{\text{DE}} \gtrsim -1$ for $z_{\dagger} \gtrsim 2$ before rapidly switching sign at $z_{\dagger} \sim 2$ and settling at $\rho_{\text{gDE}}/\rho_{c0} \approx 0.7$ with $w_{\text{DE}} \lesssim -1$ for $0 \le z \le 2$.<fnref n="56" /> This led to the conjecture that around $z_{\dagger} \sim 2$, the Universe underwent a period of rapid

<footnote n="56">⁵⁶ For a minimally interacting DE with $\rho_{\text{DE}}$ transitioning from negative to positive at $z_{\dagger}$ as the Universe expands ($z$ decreases), physical consistency demands $\rho_{\text{DE}} < 0$ with $w_{\text{DE}} > -1$ just before the transition and $\rho_{\text{DE}} > 0$ with $w_{\text{DE}} < -1$ just after, ensuring a smooth crossing through $\rho_{\text{DE}} = 0$ while maintaining the continuity equation. Since $\rho_{\text{DE}}$ remains finite, $w_{\text{DE}}$ exhibits a singularity at $z = z_{\dagger}$, with $\lim_{z \to z_{\dagger}^{-}} w(z) = \pm\infty$. However, this is a safe singularity — $\rho_{\text{DE}}$ remains finite, and $w_{\text{DE}}$ diverges solely because $\rho_{\text{DE}}$ crosses zero while $\rho_{\text{DE}}$ remains finite and nonzero. Consequently, given that $H^2 \propto \rho_m + \rho_{\text{DE}}$, H and other kinematical parameters evolve smoothly; in scalar field realizations, the sound speed remains luminal, further reinforcing the regularity of the crossing [2870]. By contrast, the alternative scenario of sign-changing $\rho_{\text{DE}}$ — where $\rho_{\text{DE}} < 0$ with $w_{\text{DE}} < -1$ just before the transition and</footnote>
mirror AdS-to-dS transition in vacuum energy — rapid sign-switching cosmological constant, $Λ_s$, from negative to positive while preserving the magnitude — or a similar phenomenon.

The $Λ_s$CDM framework [801,802,811,812,1325,2872] extends ΛCDM by replacing the usual cosmological constant ($Λ$) with a dynamically evolving counterpart ($Λ_s$) that undergoes a mirror AdS-to-dS transition in the late Universe, while leaving other standard cosmological components — such as CDM, baryons, pre-recombination physics, BBN, and inflation paradigm — unchanged. This transition can typically be described using sigmoid-like functions, such as the smooth approximation of the signum function, $sgn x \approx \tanh(kx)$, where $k > 1$ and $x$ represents either redshift ($z$) or scale ($a$). An example is $Λ_s(z) = Λ_{s0} \tanh[v(z_↑ - z)] / \tanh[vz_↑]$, where $v > 1$ controls the sharpness of the transition, $Λ_{s0} > 0$ is the present-day value, and $z_↑$ denotes the transition redshift. For a rapid transition (e.g., $v \gtrsim 10$) occurring at $z_↑ \sim 2$, this function effectively behaves as $Λ_s \approx Λ_{s0}$ for $z \lesssim 2$ and $Λ_s \approx -Λ_{s0}$ for $z \gtrsim 2$. In the limiting case $v \to \infty$, the transition becomes instantaneous, $Λ_s(z) \to Λ_{s0} \operatorname{sgn}[z_↑ - z]$, defining the abrupt $Λ_s$CDM model, which extends ΛCDM by a single additional parameter, serving as an idealized representation of a rapid mirror AdS-to-dS transition. Abrupt $Λ_s$CDM [801,802,811,812,1325,2872] has emerged as one of the most promising and economical extensions of ΛCDM; introduces only a single additional parameter, $z_↑ \sim 2$ (estimated through robust observational analyses), beyond ΛCDM, resolving major cosmological tensions — including those in $H_0$, $M_B$, and $S_8$ — as well as the BAO Ly-$\alpha$ anomaly, while yielding an age of the Universe consistent with estimates from the oldest globular clusters. Additionally, when allowing variations in $m_v$ and $N_{\text{eff}}$, it predicts values consistent with the standard model of particle physics [1325], suggesting that it may avoid the recently proposed anomaly in which cosmological data (such as DESI BAO), within the ΛCDM framework, appears to favor $m_v < 0$ [1307,1324,2386,2873–2875] (see Section 2.3.5). From a physical perspective, $Λ_s$CDM is identical to ΛCDM for $z < z_↑$, featuring a dS-like cosmological constant after the transition, but introduces a minimal modification by adopting an AdS-like cosmological constant for $z > z_↑$, i.e., for all redshifts prior to the transition. However, from a phenomenological perspective — viz., in terms of the Universe’s expansion dynamics and observational signatures — the impact of this modification is effectively confined to redshifts $z \lesssim z_↑ \sim 2$. Specifically, $Λ_s$CDM replicates the $H(z)$ of ΛCDM for $z < z_↑$ — albeit with systematically larger values — introduces $H(z)$ deformation around $z \sim z_↑$, and becomes nearly indistinguishable from ΛCDM at higher redshifts ($z \gtrsim 3$). Consequently, from a phenomenological standpoint, $Λ_s$CDM is a post-recombination/late-time modification to ΛCDM. Abrupt $Λ_s$CDM, the simplest phenomenological realization of the $Λ_s$CDM framework, has been extensively studied [801,802,811,812,1325]. However, its abrupt transition at $z = z_↑$ introduces a discontinuity, leading to a type II (sudden) singularity [2876] at $z = z_↑$, though Ref. [2696] has shown this has negligible impact on cosmic structure formation and evolution. Nonetheless, this singularity suggests that the abrupt $Λ_s$CDM model should be interpreted as an idealized approximation, effectively serving as a proxy for a rapid yet smooth transition.<fnref n="57" />

We refer readers to Refs. [425,444,769,789,802,1288,1291,1292,1325,2046,2103,2104,2224,2248,2465,2485,2508,2689,2701,2821,2822,2868–2870,2872,2878–2910] for further theoretical and observational studies — including model agnostic reconstructions (see Section 3.3) — that explore DE with negative densities, often consistent with an AdS-like cosmological constant, at $z \gtrsim 1.5–2$, and aimed at addressing major cosmological tensions. Phantom DE models, which typically feature $ρ_{DE}$ that decreases with redshift but are conventionally assumed to yield $ρ_{DE} > 0$, are known to mitigate the $H_0$ tension. Among these, the phantom crossing model, proposed phenomenologically in Ref. [2911] (DMS20 [813]), stands out. A recent analysis, which considered this model as a particular example of the broader Omnipotent DE class [813], reaffirmed its success while also revealing that its ability to assume negative densities for $z \gtrsim 2$ — mimicking an AdS-like cosmological constant beyond sufficiently high $z$ — is central to its effectiveness. IDE models [768,800,2427,2912–2922] (see Section 4.2.3) offer an alternative approach to resolving the $H_0$ tension; however, model-independent reconstructions of the IDE kernel [2162] suggest that negative DE densities at $z \gtrsim 2$ persist as a possibility.

While late-Universe rapid AdS-to-dS (or analogous) transitions in DE, as proposed by $Λ_s$CDM, were initially viewed as challenging to reconcile with a robust physical mechanism, the remarkable phenomenological success of this approach — despite its simplicity — has prompted deeper theoretical inquiries. Even established frameworks, upon re-examination, have been found to accommodate such transitions within previously overlooked solution spaces, prompting researchers to adopt a fresh perspective on familiar theories. For instance, $Λ_s$CDM⁺ [2701,2899,2900] proposes a stringy realization of $Λ_s$CDM; it was shown that, despite the AdS swampland conjecture suggesting that a late-universe AdS-to-dS transition is unlikely due to the arbitrarily large separation between AdS and dS vacua in moduli space, such a transition can nonetheless be realized through the Casimir forces of fields inhabiting the bulk.⁵⁸ Building on the same theoretical framework, $Λ_s$CDM∓ [2926] extends $Λ_s$CDM⁺ by allowing the AdS phase to have arbitrary depths, considering different curvature radii in the AdS and dS phases. It was demonstrated in Ref. [2897] that, in various formulations of GR, a $Λ_s$ can arise naturally through an overall signature change of the metric. $Λ_s$VCDM [802,2872] advances the $Λ_s$CDM framework into a theoretically complete physical cosmology, offering a fully predictive description of the Universe, including the AdS-to-dS transition epoch itself. It was shown that

GR, is often linked to ghost instabilities and WEC violations. These impose constraints on the transition’s rapidity in GR but may be circumvented in type-II minimally MG theories such as VCDM [2877,2878]. For further discussion, see [802,2870,2872].

⁵⁸ By combining swampland conjectures with observational data, it was proposed that the cosmological hierarchy problem — i.e., the smallness of DE density in Planck units — could be understood as an asymptotic field-space limit corresponding to the decompactification of a micron-sized extra (dark) dimension [2923]. Within this framework, Casimir forces from fields inhabiting the dark dimension can drive an AdS-to-dS transition [2899], forming the basis of the $Λ_s$CDM∓ and $Λ_s$CDM⁺. Specifically, a 5D Einstein-de Sitter gravity action compactified on a circle induces a runaway potential inherited from the 5D cosmological term [2924]. If the 5D cosmological constant is small, the quantum contribution of the lightest 5D modes — identified with Casimir energy [2925] — becomes significant. A minimal setup requires a 5D mass spectrum comprising the graviton, three generations of light right-handed neutrinos, and a real scalar field $\varphi$ with a potential featuring two local minima [2899]. At $z_↑ \sim 2$, $\varphi$ undergoes quantum tunneling from the false to the true vacuum [2900], acquiring a larger mass and suppressing its Casimir energy contribution. This modifies the balance of fermionic and bosonic degrees of freedom, triggering the AdS-to-dS transition. The deep infrared fields of the dark sector contribute to the effective number of relativistic neutrino-like species.

<footnote>$ρ_{DE} > 0$ with $w_{DE} > -1$ just afterward — leads to singularities in both $w_{DE}$ and $ρ_{DE}$, as $\lim_{z \to z_↑} w_{DE}(z) = \mp\infty$ and $\lim_{z \to -z_↑} ρ_{DE}(z) = \mp\infty$, thereby leading to a breakdown of the spacetime metric. Thus, only the first scenario is physically viable, predicting a phantom-like phase ($w_{DE} < -1$) following the transition. Consequently, the tendency of model-agnostic reconstructions of $w_{DE}(z)$ and various EoS parameterizations (e.g., CPL) to favor large negative values beyond $-1$ for $z \sim 1.5–2$ may indicate negative or vanishing DE densities for $z \gtrsim 1.5–2$. For further discussion, see Refs. [813,2696,2870,2871]</footnote>

<footnote n="57">⁵⁷ A well-defined formulation necessitates a smooth transition, enabling a detailed study of perturbations but introducing theoretical challenges. A rapid AdS-to-dS transition can sharply increase $\dot{H}$, inducing transient super-acceleration ($\dot{H} > 0$), which may affect cosmological observables and, within</footnote>
the mirror AdS-to-dS transition can be effectively realized within a type-II minimally MG framework called VCDM [2877,2878], through a specific Lagrangian incorporating an auxiliary scalar field with a smoothly sewed two-segmented linear potential. Ref. [2903] demonstrated that the teleparallel $f(T)$ gravity, specifically its exponential infrared form [2927], which has shown significant promise in addressing the $H_0$ tension [2928,2929] in its solution space giving phantom-like effective DE, admits previously overlooked solution spaces, which accommodate an alternative scenario where the effective DE transitions smoothly from negative to positive at $z_+ \sim 1.5$, while remaining consistent with CMB power spectra. Building on these insights, $f(T)-\Lambda_s$ CDM successfully maps the background dynamics of $\Lambda_s$ CDM into the $f(T)$ gravity framework [2904], further establishing a theoretical foundation for AdS-to-dS-like transitions in the late Universe. Ph-$\Lambda_s$ CDM [2870], introduced a phantom DE model, which is a specific realization of a general scalar field with a hyperbolic tangent potential that induces smooth AdS-to-dS, 0-to-dS, and dS-to-dS transitions. Despite its negative kinetic term, the step-like potential prevents pathologies like unbounded energy growth, Big Rip, and WEC violations, ensuring smooth evolution of cosmological parameters. Notably, the AdS-to-dS transition in DE density does not precisely parallel that of the potential, persisting longer due to the kinetic term's contribution. Even if different realizations of $\Lambda_s$ CDM yield identical background dynamics, they still exhibit differences. GR-based $\Lambda_s$ CDM models [801,811,812,2870], $\Lambda_s$ VCDM [802,2872], and $f(T)-\Lambda_s$ CDM differ in their predictions for linear perturbations, while the string-inspired $\Lambda_s$ CDM$^+$ [2701,2899,2900,2926] predicts a modest excess in the total effective number of neutrino species, with $N_{\text{eff}} = 3.294$. Such distinguishing features are invaluable, as they provide a means to compare and ultimately discriminate between these alternative realizations using observational data. Other than the models explicitly realizing/resembling the background dynamics of $\Lambda_s$ CDM, there exist various other approaches that fit within the broader paradigm outlined earlier. These include brane-world models [444,2881,2882], multiple axion models [2930, 2931], energy-momentum log gravity [2887], bimetric gravity [2909], Horndeski gravity [2898], holographic DE [2906], Granda-Oliveros holographic DE [2907], composite DE ($w$XCDM) [2485,2908], inspired by the $\Lambda$XCDM framework [2932] — extends the abrupt $\Lambda_s$ CDM model by introducing two free parameters that allow $w_{\text{DE}}$ to take arbitrary constant values before and after the transition$^{59}$ —, the Omnipotent DE concept [813], with its specific realization DMS20 [2911], represents a class of DE models characterized by non-monotonic densities and transitions across $w_{\text{DE}} = -1$, DE pressure parametrizations [2933, 2934], which yield $w_{\text{DE}}$ similar DMS20, the Lotka-Volterra model of two interacting fluids [2896], running Barrow entropy [2895], multiple-transition vacuum DE models [2514], and scenarios invoking a modification of the gravitational constant between super- and sub-horizon regimes, motivated by the Hořava-Lifshitz proposal or the Einstein-aether framework [2935].

Thus, a rapidly growing body of literature has emerged within this class of models — particularly in recent years — with many having been tested against a variety of observational datasets, consistently revealing a strong correlation between the redshift of the

sign change in the DE density, $z_+$, and the model's ability to address major cosmological tensions, especially those involving $H_0$ and $S_8$. To provide insight, we highlight several representative observational constraints. The abrupt $\Lambda_s$ CDM model, using the combined Planck+BAOtr+KiDS1000+SNIa+SH0ES dataset, yields $z_+ = 1.72^{+0.09}_{-0.12}$, predicting $H_0 = 73.16 \pm 0.64 \text{ km s}^{-1} \text{ Mpc}^{-1}$ and $S_8 = 0.774 \pm 0.009$, thereby fully resolving both tensions [812]. When analyzed with the Planck+SNIa+SH0ES combination, the same model gives $z_+ = 1.83^{+0.11}_{-0.19}$, $H_0 = 72.07 \pm 0.88$, and $S_8 = 0.786 \pm 0.011$, still strongly alleviating both tensions, albeit trending slightly toward $\Lambda$CDM values — as expected for higher $z_+$, which reduces the impact of the AdS-like phase [802]. The $\Lambda_s$ CDM$^+$, the string-theory-inspired realization, analyzed with the same combined dataset as the first case, yields a higher transition redshift of $z_+ = 2.105 \pm 0.264$; yet, due to its enhanced $N_{\text{eff}} = 3.249$, it predicts an even larger Hubble constant, $H_0 = 74.04 \pm 0.71$, thus also fully resolving the Hubble tension [2701]. The $w$XCDM model, the two-parameter extension of the abrupt $\Lambda_s$ CDM, yields a slightly lower transition redshift $z_+ = 1.46 \pm 0.02$, predicting $H_0 = 70.94 \pm 0.56 \text{ km s}^{-1} \text{ Mpc}^{-1}$ and $S_8 = 0.784 \pm 0.009$ based on the Planck+BAOtr+DESY5 SN+CC+$f\sigma_{12}$+SH0ES dataset, thereby successfully addressing both tensions, while differing in that it features an EoS parameter crossing from $w_{\text{DE}} < -1$ to $w_{\text{DE}} > -1$ across the transition from negative to positive energy density [2908]. These examples show that, despite variations in dataset and model details, scenarios featuring AdS-to-dS or analogous transitions with $z_+ \sim 1.5-2$ consistently succeed in addressing both the $H_0$ and $S_8$ tensions. Further insights come from $\Lambda_s$ CDM$^-$, the extension of $\Lambda_s$ CDM$^+$ that allows arbitrary AdS depths, revealing a degeneracy whereby deeper AdS phases permit higher $z_+$ values while still resolving the $H_0$ tension [2926]. Even the few observational findings highlighted here provide a compelling case for continued theoretical and observational investigation of this class of models.

If future observations continue to support these models, the implications for theoretical physics would be profound, as an AdS-like cosmological constant is a theoretical sweet spot, favored by the AdS/CFT correspondence [2936] and string theory frameworks [2937]. Recent work [2938] has demonstrated that a supersymmetric vacuum in string theory can naturally produce an AdS-like cosmological constant at present-day energy scales, motivating DE scenarios where the field evolves on a potential with an AdS minimum rather than the standard dS minimum. Such a negative cosmological constant could shape both the current cosmic acceleration and the Universe's long-term evolution, offering a compelling link between fundamental theory and observations. In this direction, Ref. [2888] examined a DE model consisting of an AdS-like cosmological constant and a DE component with an EoS parameter $w_\phi$, finding no evidence for an AdS-like cosmological constant and a mild preference for an effective phantom DE component, though $\Lambda$CDM remains favored. Ref. [2939] (see also [2891]) extends $w$CDM and CPL-CDM by introducing an AdS-like cosmological constant, demonstrating improvements in both data fits and alleviation of the $H_0$ tension, while further studies [1760–1762] show that this model can also address the JWST anomaly. Indeed, the models discussed in this section are expected to generally enhance structure formation at high $z$, making them natural candidates for explaining the JWST anomaly [1755,1759,2866,2867]. However, their rigorous quantitative analysis of perturbation evolution and cosmic structuring is necessary. The evolution of cosmic structures and linear perturbations in the (abrupt) $\Lambda_s$ CDM model has been studied in Refs. [1763,2696,2865]. Specifically, Ref. [1763] found that for $\Omega_{A,0} = -0.7$ and $\Omega_{m,0} = 0.3$, $\Lambda_s$ CDM predicts up to an 80% increase in cluster density for turnaround redshifts $z_{\text{max}} \gtrsim 2$, suggesting a potential explanation to the JWST anomaly. Ref. [2696] demonstrated that even in abrupt $\Lambda_s$ CDM (the most extreme case), the transition itself has no significant impact on bound structures, preserving model viability. Additionally, Ref. [2865] showed that the growth index remains $\gamma \sim 0.55$ as in

<footnote n="59">59 The $w$XCDM model [2485,2908], inspired by $\Lambda$XCDM [2932], introduces a composite DE scenario with two components: $X$ (for $z > z_+$) and $Y$ (for $z < z_+$). $Y$ behaves as running vacuum energy with a quintessence-like EoS ($w_Y \gtrsim -1$), while $X$, termed 'phantom matter' (PM), has a phantom-like EoS ($w_X \le -1$) but negative energy density, mimicking effective string action terms at low energies [2473]. The Kalb-Ramond axion and gravitational Chern-Simons term generate a 'phantom vacuum' [2479], where PM enhances structure formation at high $z$, potentially explaining the JWST anomaly. The de Sitter vacuum is restored via gChS condensates, and data support $w_Y > -1$, $w_X < -1$, aligning with DESI results [699].</footnote>
ΛCDM. However, Planck CMB data predicts $\Omega_{m,0} = 0.28$ for $\Lambda$CDM and $\Omega_{m,0} = 0.32$ for ΛCDM, leading to growth rates of $f = 0.49$ and $f = 0.53$ at $z = 0$, respectively. Notably, $\Lambda$CDM predicts a value closer to $f = 0.48$, recently obtained from LSS data when $\gamma$ is treated as a free parameter in ΛCDM [2060], indicating its potential to resolve the structure growth anomaly. DESI data (combination with other datasets) have further validated DE models incorporating an AdS-like cosmological constant [2901,2905], while obviating the need for phantom DE in the DE sector [2905]. The post-reionization HI 21-cm signal is explored as a probe for an AdS-like cosmological constant in the DE sector [2940]. In summary, a negative cosmological constant in the DE sector is physically motivated, consistent with current data, and may yield distinctive cosmological and astrophysical signatures — including the possibility of a bouncing future universe [2941].

Ref. [2942] suggested that a promising approach to resolving cosmological tensions may involve combining early- and late-time new physics to better fit the data. Ref. [2902] explored a hybrid model integrating abrupt $\Lambda$CDM with a varying electron mass mechanism, identifying it as a promising candidate. However, they found that this combination does not improve the tension, as the two models push $\Omega_{m,0}$ in opposite directions. Ref. [1325] analyzed the $\Lambda$CDM+$N_{\text{eff}}+\sum m_{\nu}$ and $\Lambda$CDM+$N_{\text{eff}}+\sum m_{\nu}$ models, allowing variations in $N_{\text{eff}}$ and $\sum m_{\nu}$ within abrupt $\Lambda$CDM and $\Lambda$CDM. They found that $\Lambda$CDM+$N_{\text{eff}}+\sum m_{\nu}$ consistently fits the data better while preserving the success of $\Lambda$CDM and predicting standard neutrino properties. In contrast, when $\Lambda$CDM+$N_{\text{eff}}+\sum m_{\nu}$ yields high $H_0$ values, it does so at the cost of large $\Delta N_{\text{eff}}$, whereas $\Lambda$CDM+$N_{\text{eff}}+\sum m_{\nu}$ achieves similarly high $H_0$ values while remaining consistent with the standard $N_{\text{eff}}$. This suggests that the $\Lambda$CDM+$N_{\text{eff}}+\sum m_{\nu}$ alleviates the need for pre-recombination new physics, at least concerning neutrino properties.

### 4.2.3. Interacting dark energy

**Coordinator:** Carsten van de Bruck

**Contributors:** Amare Abebe, Dario Bettoni, David Benisty, David Tamayo, Denitsa Staicova, Diego Rubiera-Garcia, Emmanuel Saridakis, Emre Üzülker, Kay Lehnert, Leila L. Graef, Lu Yin, Luis Anchordoqui, Marcel A. van der Westhuizen, Marco de Cesare, Nikolaos E. Mavromatos, Oem Trivedi, Oleksii Sokoliuk, Purba Mukherjee, Rahul Shah, Sveva Castello, Vitor da Fonseca, and Yuejia Zhai

#### 4.2.3.1. Introduction.
In this subsection, we discuss models with an interaction between DE and DM. Such IDE models have been introduced to address problems of the $\Lambda$CDM model, including the $H_0$ and $S_8$ tensions (for comprehensive reviews, e.g., see Ref. [709,2943,2944]). Some IDE models can reduce the $H_0$ tension from $5\sigma$ to $3.6\sigma$ [709].

In a general model, the conservation equations for DM and DE take the form

$$ \nabla_{\mu} T^{\mu\nu}_{(\text{DM})} = Q^{\nu}, \quad \text{and} \quad \nabla_{\mu} T^{\mu\nu}_{(\text{DE})} = -Q^{\nu}, \qquad (4.14) $$

ensuring that the total energy–momentum tensor of DM and DE remains conserved, namely, $\nabla_{\mu}(T^{\mu\nu}_{(\text{DM})} + T^{\mu\nu}_{(\text{DE})}) = 0$. Often, the resulting background equations are written in the form

$$ \dot{\rho}_{\text{DM}} + 3H\rho_{\text{DM}} = Q, \quad \text{and} \quad \dot{\rho}_{\text{DE}} + 3H(\rho_{\text{DE}} + p_{\text{DE}}) = -Q. \qquad (4.15) $$

In order for IDE models to be viable candidates for addressing the $H_0$ and $S_8$ tensions, special attention must be given to the physicality of the parameter space [2945]. In these models, there is not always a mechanism to halt the energy transfer when either the DM or DE density becomes zero (i.e., $Q \neq 0$ in Eq. (4.15) when $\rho_{\text{DE}/\text{DM}} = 0$), which can lead to negative energies. The case of $Q > 0$ corresponds to an energy transfer from DE to DM. It has been reported that CMB observations seem to favor an energy transfer from DE to DM, WL measurements and thermodynamical considerations suggest an energy transfer from DM to DE [2050,2946–2948].

One well studied class of models is coupled quintessence, in which $Q^{\nu} = \beta\phi^{\nu}T^{\mu\nu}_{(\text{DM})\mu}$ and $Q = -\beta T^{\mu}_{(\text{DM})\mu}$, where $\phi$ is the DE scalar field and $T^{\mu}_{(\text{DM})\mu}$ denotes the trace of the DM energy–momentum tensor [2051, 2422,2949]. There also exists an extension of these models in which DM and DE are disformally coupled [2950–2953]. Such scalar field theories of IDE may alleviate the Hubble tension but do not fully solve it [2427,2429,2954].

Other, more phenomenological approaches propose a coupling of the form $Q^{\mu} = Q^{\mu}_{\text{DM}}/a$ or $Q^{\mu} = Q^{\mu}_{\text{DE}}/a$ [2955,2956] (in these expressions, $a$ is the scale factor and $u^{\mu}_{\text{DM}}$ or $u^{\mu}_{\text{DE}}$ are the four-velocity of DM or DE fluid, respectively). The interaction $Q$ is usually written as $Q = \xi H \rho_{\text{DE}}$ or $\xi H \rho_{\text{DM}}$, where $\xi$ is a dimensionless coupling constant, determining the size and direction of energy/momentum flow. The case $Q = \xi H \rho_{\text{DE}}$ was recently studied in Ref. [2957] (see also [2913] for a previous study), assuming an equation of state of DE $w_{\text{DE}} = -0.999$, using the CMB data provided by Planck, WMAP and ACT. Different dataset combinations resulted in consistent results, showing evidence for a non-zero coupling and a Hubble expansion rate $H_0$ consistent with local measurements. Using Planck alone, the analysis results in $H_0 = 71.6 \pm 2.1 \text{ km s}^{-1}\text{Mpc}^{-1}$, ACT alone results in $H_0 = 72.6_{-2.6}^{+3.4} \text{ km s}^{-1}\text{Mpc}^{-1}$ and the combination of ACT and Planck results in $H_0 = 71.4_{-2.8}^{+2.5} \text{ km s}^{-1}\text{Mpc}^{-1}$. In Ref. [2958], the assumption about the equation of state was relaxed, allowing for a dynamical evolution via the parameterization $w_{\text{DE}}(a) = w_0 + w_a(1-a)$. It was found that models featuring a dynamical phantom equation of state ($w_0 < -1$) perform worse than the non-dynamical case. On the other hand, models featuring a dynamical quintessence equation of state ($w(a) > -1$ at any redshift) perform better in attempting to increase the value of $H_0$ compared to the respective non-dynamical case. However, when considering the joint analysis of CMB, BAO SDSS, and SN data, no significant increase in $H_0$ to solve the Hubble tension was found, while it is a promising solution considering DESI BAO data [800].

In the following, we present a number of other IDE models.

#### 4.2.3.2. Models.

**Nonlinear interacting dark energy** This class of models is an extension of the models discussed above. A general form of the nonlinear IDE models is given by $Q = 3H\xi F(\rho_{\text{DE}}, \rho_{\text{DM}})$, where $F$ is a nonlinear function of the energy densities and $\xi$ is a free constant parameter. Generally, when $Q > 0$, the $H_0$ tension worsens and the $S_8$ alleviates; conversely, when $Q < 0$, the $H_0$ tension alleviates and the $S_8$ worsens [2945]. The authors of Ref. [2959] investigate nonlinear IDE as an interaction between DM and a scalar field, in Ref. [2960] a model is discussed that could be interpreted as a particular case of a running vacuum model $\Lambda(H)$.

Power laws IDE models with the general form $Q = 3H\xi \rho_{\text{DM}}^{p} \rho_{\text{DE}}^{q} (\rho_{\text{DM}} + \rho_{\text{DE}})^r$, where $p, s$ and $r$ integers, were studied in Ref. [2961], and observational constraints for specific cases provided in Ref. [2962]. Observational constraints of $Q = 3H\xi (\rho_{\text{DM}} + \rho_{\text{DE}} + \frac{\rho_{\text{DM}}\rho_{\text{DE}}}{\rho_{\text{DM}}+\rho_{\text{DE}}})$ are discussed in Ref. [2963]. The model $Q = \gamma \rho_{\text{DM}}^{\alpha} \rho_{\text{DE}}^{\beta}$, using CMB, BAO, RSD, and SNIa data, predicts lower values of $f(z)\sigma_{8}(z)$ at $z < 1$ comparing to $\Lambda$CDM, which alleviates the tension of $\Lambda$CDM with various RSD data [2964]. Other model types, such as exponential models $Q = 3H\xi \rho_{\text{DE}} \exp(\rho_{\text{DE}}/\rho_{\text{DM}} - 1)$ [2965], and logarithmic models $Q = 3H\xi \rho_{\text{DE}} \log(\rho_{\text{DE}}/\rho_{\text{DM}})$ and $Q = 3H\xi \rho_{\text{DE}} \log(\rho_{\text{DM}}/\rho_{\text{DE}})$ [2966], were studied but showed no significant promising results.

Models with the forms $Q = 3H\xi \rho_{\text{DE}} \sin(\rho_{\text{DE}}/\rho_{\text{DM}} - 1)$ and $Q = 3H\xi \rho_{\text{DE}}[1 + \sin(\rho_{\text{DE}}/\rho_{\text{DM}} - 1)]$ have gained special attention for their potential to alleviate the $H_0$ tension [2967]. Analysis with Planck 2018 gives $H_0 = 72.67_{-8.43}^{+5.43} \text{ km s}^{-1}\text{Mpc}^{-1}$ at 68% CL, and Planck 2018+BAO gives $H_0 = 69.17_{-1.71}^{+8.26} \text{ km s}^{-1}\text{Mpc}^{-1}$ at 68% CL. Hence, for Planck 2018 and Planck 2018+BAO, the $H_0$ tension with respect to SH0ES ($H_0 = 73.0 \pm 1.0 \text{ km s}^{-1}\text{Mpc}^{-1}$) is reduced down to $0.05\sigma, 2\sigma$ respectively; notably, for Planck 2018 alone case the tension is completely solved.
However, these results should be interpreted with caution due to potential biases and uncertainties. Additionally, a theoretical justification for the specific form of $Q$ is necessary to ensure that it is not merely an ad hoc model.

**Diffusion interactions** A diffusive interaction between DE and DM was introduced in Refs. [2968–2970]. The diffusion of energy density between DE into DM uses a non-conserved stress energy tensor $T^{\mu\nu}$ with a source current $j^\mu$, $\nabla_\mu T^{\mu\nu}_{(m)} = \gamma^2 j^\nu$, where $\gamma^2$ is the coupling diffusion coefficient of the fluid. The current $j^\mu$ is a time-like covariant conserved vector field $\nabla_\mu j^\mu = 0$ which describes the conservation of the number of particles in the system. In a homogeneous expansion, the modified Friedman equations read

$$ \dot{\rho}_{\text{DM}} + 3H\rho_{\text{DM}} = \frac{\gamma}{a^3}, \quad \dot{\rho}_{\Lambda} = -\frac{\gamma}{a^3}, \qquad (4.16) $$

The contribution of the current goes as $\sim a^{-3}$ since the current is covariantly conserved. In this way, there is a compensation between DE and DM. Ref. [2971] introduces cases with a diffusion constant that could be represented for a scalar field $\phi$ or a perfect fluid, leading to late-time forms of DE and a DE density parameter that could be interpreted as a perturbation of the $\Lambda$CDM model.

Another example of diffusion interactions is represented by models where the energy–momentum transfer vector comes from a potential, $Q_\mu = \nabla_\mu J$. This class of models finds a natural embedding within unimodular gravity. In fact, in unimodular gravity the total energy–momentum of matter fields need not be conserved in general [2972]. The energy density of DE is then identified with the potential $J$ up to an arbitrary integration constant, $\rho_{\text{DE}} = J + \Lambda_0$, and the equation of state is $w_{\text{DE}} = -1$ identically. Models of this kind have been proposed to describe effective diffusion processes due to an underlying discrete spacetime structure [2973,2974]. In Ref. [2889] it was argued that such diffusive interactions — with energy transfer from DM to DE — can alleviate the Hubble tension, see Refs. [2975–2977] for a more detailed analysis. This class of models is also formally equivalent to “interacting vacuum models”, which have been studied within the context of general relativity [2978,2979]. Embedding minimally IDE with $w_{\text{DE}} = -1$ within UG has the following advantages: (i) the vacuum energy does not gravitate in UG removing the age-old problem regarding the puzzling absence of the vacuum energy density in cosmological observations [2980,2981], (ii) the well-known perturbative large-scale instability studied in Ref. [2955] does not exist in UG [2982], (iii) since the DE is an effective source in UG rather than a physical one, $\rho_{\text{DE}}$ can attain negative values opening up a large phenomenological landscape to address the cosmological tensions. Moreover, it opens up a path for quantum gravity motivated interaction terms; e.g., see Refs. [2972–2974].

An important aspect often forgotten in the study of such diffusive models is to study the effect of large-scale structure formation in such a fluid environment. Growth-rate analyses can shed light on whether such a model alleviates the $\sigma_8$ tension or not.

**Metastable dark energy** Models of metastable DE have been analyzed in several works, see for instance Refs. [709,2983–2993]. In particular, in Refs. [2983,2985], phenomenological models in this context were investigated, considering the case of a constant DE decay rate, depending only on the intrinsic properties of DE and the type of decay channel. The following cases of metastable DE decaying in three distinct ways were considered: (I) exponentially; (II) into DM; (III) into dark radiation. Among these models, model II showed slightly better consistency [2985]. This model is sometimes called metastable IDE [709], since it is described by Eq. (4.15) with $Q = \Gamma \rho_{\text{DE}}$, where $\Gamma$ is the DE decay rate.

In Ref. [2985] it was shown that despite the fit for this model against Pantheon + BAO data providing higher values for $H_0$, when the CMB distance priors from Planck 2018 are included, the Hubble tension is restored. Later on, in Ref. [2986], a full analysis was performed using different combinations of data from Planck 2018, BAO, DES,

R19 [48,192]. A larger value of $H_0$ was then supported, solving the tension with R20 [78] within $1\sigma$. Future data with higher precision must provide us with a clearer understanding of the performance of these models concerning the tension. Some discussion on motivations for metastable IDE arising from quantum mechanics can be found in Refs. [2987,2988]. We can also find some examples from field theory descriptions, as in Refs. [2984,2989–2991], for instance.

Another model of metastable EDE is provided by the so-called Chern–Simons gravity inspired from string theory, studied in Refs. [2471,2473,2481]. It has been shown that, under some conditions, condensation of chiral GWs in early epochs of the Universe, can lead to a non-trivial condensate of the Chern–Simons gravitational anomaly term. The latter is characterized by the presence of imaginary parts, which are such that they lead to a metastable EDE, with a lifetime that can be in the phenomenologically right ballpark of larger than 50–60 e-foldings. The model, which also leads to RVM type inflation, due to the non-linear $H^4$ dependence of the (real part of the) pertinent vacuum energy density, can also help alleviate the Hubble and growth-of-structure tensions in modern eras [2484].

**Elastic dark energy interactions** In DE-DM elastic interactions models the two species interact via velocity or momentum transfer. There are various ways of implementing these exchanges [2994–2996] but here, for the sake of concreteness, we focus on the particular realization presented in Ref. [2996] and further developed in Refs. [2997–2999] (see Ref. [3000] for a short review), which is based on the following phenomenological implementation of the coupling

$$ \nabla_{\nu} T^{\mu\nu}_{\text{DM}} = \alpha (u_{\text{DM}}^{\mu} - u_{\text{DE}}^{\mu}), \quad \nabla_{\nu} T^{\mu\nu}_{\text{DE}} = -\alpha (u_{\text{DM}}^{\mu} - u_{\text{DE}}^{\mu}), \qquad (4.17) $$

where $\alpha$ measures the strength of the interaction. It is clear that the interaction is active only when there is relative motion between the two species, i.e., at sub-Hubble scales while, at the largest scales, where the two fluids share a common reference frame, the interaction is absent. This means, in turn, that no modifications to the background evolution are produced. As a consequence, these models are not designed to address the $H_0$ tension but have an impact on relaxing the $\sigma_8$ one. In fact, at the level of linear perturbation, Euler equation receives a new contribution proportional to the relative velocity between the two species with a time dependent interaction rate $\Gamma_\alpha = \alpha a^4 / \Omega_{\text{DM}}$ in the case of DM and further weighted by the relative abundance of DM to DE $R = a^3 w \Omega_{\text{DM}} / (1+w) / \Omega_{\text{DE}}$ in the case of DE. Since the interaction grows with the Universe expansion as $a^4$, eventually, $\Gamma_\alpha \sim H$ and the interaction with DE becomes relevant starting to exert a drag on DM which suppresses its clustering. This allows for the onset of a dark-coupling epoch at late-time as desired and as suggested by data that points toward a suppression in the clustering at $z \lesssim 2$. By considering Planck 2018 data releases, SNIa and Baryon Acoustic Oscillations data it has been shown that the interaction can indeed reduce the value of $\sigma_8$ and does so without affecting $\Omega_m$ ($\sigma_8 = 0.753^{+0.011+0.022}_{-0.011-0.021}$, $\Omega_m = 0.311^{+0.007+0.014}_{-0.007-0.013}$) and that the presence of the interaction is strongly favored [2997,2998]. The fact that no correlations between the interacting parameter and background ones is introduced is a particularly welcome feature since it implies no worsening of the $H_0$ tension. Remarkably, the inclusion of $S_8$ data allows for a constraint at percent level of the interaction by surveys like JPAS, Euclid, and DESI [2997] while SKAO-like experiments should have enough sensitivity to detect the interaction parameter [3001]. Finally, it is worth mentioning that the inclusion of massive neutrinos does not spoil the detection prospects for this model [3002] and that other implementations and analyses reach similar conclusions [2999,3003,3004].

**Fading dark matter** If the swampland conjecture for DE is correct, it would lead to the prediction that the $\Lambda$CDM model, where DE is constant, cannot be veracious [3005,3006]. Quite independently of swampland considerations, there have been proposals for a rolling scalar field as the source of DE. If this were the case, the swampland
distance conjecture suggests that the rolling field would lead at late-
times to an exponentially light tower of states. Identifying this tower as
residing in the dark sector yields a natural coupling of the scalar field
to the DM, leading to a continually reducing DM mass as the scalar
field rolls in the recent cosmological epoch. It has been shown that
the way in which the tower of light states evolves over time could
help to reduce (though not fully eliminate) the $H_0$ tension [3007].
More unambiguously, the coupling of the scalar field to the DM leads
to a reduction of mass, or fading of DM, which is compensated by
a bigger value of DE. The latter becomes more noticeable in the
present accelerating epoch, leading to an increase in $H_0$. Salam-Sezgin
6-dimensional supergravity [3008] and its string realization of Cvetič-
Gibbons-Pope [3009] can bring to fruition the fading DM model [3010,
3011].

**Late-time interacting constant/dynamical dark energy** Late-time dark energy interacting with DM has been extensively explored in the context of the $H_0$ and $\sigma_8$ tensions. Numerous studies have been carried out in the context of the Hubble tension using different forms of the interaction term and $\Lambda$/constant EoS late-time dark energy, and by imposing various theoretical bounds on the parameter space to prevent instabilities [768,2913-2916,2918-2920,2942,2944,2967,3012-3028]. Similarly in the context of the clustering tension significant studies include [2716,2916-2918,2920,3016,3026,3029-3033]. In Ref. [2958] an interacting setup with interaction proportional to the DE density ($Q \propto \rho_{de}$), was investigated in the light of late-time DDE assuming a CPL parameterization in the context of the Hubble tension, and found no significant scope of alleviation of the tension with various combinations of datasets in either the phantom or quintessence regimes. In Ref. [2828], using the same form of interaction, various other parametrizations of LDE (MEDE, CPL, JBP) were considered. It was reported here that a quintessence EoS in such an interacting scenario does not help with the Hubble tension, and noticeably makes the $\sigma_8$ tension worse. However, a phantom EoS, although did not affect the Hubble tension, showed significant scope in alleviating the $\sigma_8$ tension (especially for the CPL and JBP parametrizations). The inclusion of RSD data in this scenario helped obtain tighter constraints on certain parameters, but resulted in a relatively less pronounced relaxation of the clustering tension, which might indicate $\Lambda$CDM bias in RSD data. A phenomenological parameterization, where a scalar field depends linearly on the number of e-folds, was used to test its interaction with DM [2948]. However, the constrained interacting dark energy (CIDER) model, with a background identical to the standard $\Lambda$CDM model, appears to better address the $\sigma_8$ tension [2426,3034].

**Late-time singularities** Depending on the choice for the interacting term $Q$, late-time singularities, for which infinities in the Hubble factor (or on its derivatives) appear, may arise at finite future times [2123,3035–3041]. Since some of them represent possible ends for the Universe, this undermines the consistency of the theory. Furthermore, they have been shown to occur for both linear and non-linear forms of $Q$.

### 4.2.3.3. Modeling and observational aspects of IDE models.
In this section, we present some other aspects of IDE models, such as N-body structure formation simulations of certain IDE models, and comparing IDE models to data.

**Marginalization approach to IDE** A recent estimate of IDE contribution from Pantheon and Pantheon Plus, CC and transversal BAO [3012] has shown evidence for up to $2\sigma$ deviation from $\Lambda$CDM. This study has been done in the framework of the marginalization approach which aims to circumvent the Hubble tension by integrating $H_0$ and $r_d$ out of the model, leaving only $\Omega_m$, $w$ and $\xi$ as parameters. The results show a strong dependence on the used SNIa dataset and whether it is Cepheids calibrated or not, with the latter remaining compatible with $\Lambda$CDM.

**N-Body simulations of IDE** Several simulations of the structure formation were run up to date under the IDE ansatz, namely works of

Refs. [2299,3022], where they have considered several cases of IDE and derived such probes of the LSS as matter power spectrum, halo mass function, concentration-mass relation, and halo bias. The most interesting parameter is the concentration-mass relation, namely $c_{200}$ that can be related to the X-ray data and galaxy rotation curves, which via the maximum likelihood method imposes the redshift evolving constraints on the DE-DM coupling parameter $\xi$, with the joint posterior distribution giving a best fit of $\xi = 0.071 \pm 0.034$.

**Distinguishing IDE from modified gravity** A word of caution is required when constraining IDE models based on large-scale structure observables. Models involving a fifth force acting on DM lead to a breaking of the weak equivalence principle for this component and thus a deviation in the Euler equation, affecting the growth of cosmic structure $f$. However, the resulting observational signatures on current and forecasted measurements of $f\sigma_8$ present a strong degeneracy with gravity modifications in the Poisson equation [3042,3043]. WL, which is sensitive to the sum of the two gravitational potentials describing the geometry of the Universe, is also generically unable to break this degeneracy. This indicates that standard large-scale structure analyses may not be able to distinguish IDE from gravity modifications, leading to a potential ambiguity among models proposed to address cosmological tensions.

Luckily, the two scenarios can be disentangled by considering measurements of gravitational redshift [3044,3045], an observable accessible in two-point correlations by the coming generation of galaxy surveys [3046,3047]. This effect is sensitive to the gravitational potential encoding the distortion of time and appearing in the Euler equation, thus providing a direct test of the weak equivalence principle and a clear way to discriminate between the two scenarios.

**Model-independent reconstructions of IDE kernel** In Ref. [2162], the interaction kernel $Q(z)$, which governs the energy exchange between DM and DE, is reconstructed within a fully model-independent framework, without imposing any predefined functional form. Employing both binned step-function reconstructions and GP interpolation, the authors infer the redshift evolution of $Q(z)$ directly from observational data, including BAO, CC, and the Pantheon+ supernova sample. The reconstructed kernel displays nontrivial features — most notably, multiple sign changes and mild oscillatory behavior at the $1\sigma$ level — suggesting dynamic and potentially reversible energy transfer between the dark components. As a result of this evolving interaction, the effective DE density is found to cross zero and become negative at redshifts $z \gtrsim 2$, demonstrating that IDE scenarios do not necessarily exclude negative DE densities at high redshift (see Section 4.2.2), despite the conventional assumption that DE remains positive in IDE models.

**Application of the parameterized post-Friedmann framework in IDE** Understanding the interaction of DE necessitates examining both the expansion history of the Universe and the growth of cosmic structures [3048,3049]. It has been observed that in the IDE scenario, calculating perturbations often results in divergent curvature perturbations on superhorizon scales, posing a significant problem for IDE cosmology [3050–3052].

To address this issue, the parameterized post-Friedmann (PPF) framework is an effective theory that treats DE perturbations based solely on the fundamental properties of DE [3053–3057]. This approach extends the parameterized PPF framework, originally designed for uncoupled DE, and can mitigate the instability in IDE cosmology [3058]. Consequently, the entire parameter space of IDE models can be explored using observational data [2722,3023,3059–3063].

A generalized framework to study IDE models is also provided by the effective theory of IDE [3064]. This formalism relies on a model-independent approach at the level of the action and provides a description of linear cosmological perturbations in scalar-tensor theories of gravity with a homogeneous and isotropic background. This allows for a direct comparison with cosmological data to constrain a rich non-standard phenomenology [3043,3065].
## 4.3. Modified gravity

### 4.3.1. Modified gravity in light of cosmic tensions

**Coordinator:** Francesco Bajardi, Micol Benetti, Salvatore Capozziello

**Contributors:** Adrià Gómez-Valent, Alessandro Vadalà, Ali Övgün, Amare Abebe, Andronikos Paliathanasis, Anil Kumar Yadav, Araceli Soler Oficial, Christian Pfeifer, Daniel Blixt, David Benisty, Diego Rubiera-Garcia, Duško Borka, Emmanuel N. Saridakis, Erik Jensko, Francisco S. N. Lobo, Gabriel Farrugia, Gaetano Lambiase, Giovanni Montani, Giuseppe Sarracino, Hussain Gohar, Ilim Cimdiker, Inês S. Albuquerque, Ismael Ayuso, Joan Solà Peracaula, Konstantinos F. Dialektopoulos, László Á. Gergely, Marcin Postolak, Marco de Cesare, Maria Caruana, Mariaveronica De Angelis, Nihan Katirci, Nils A. Nilsson, Noemi Frusciante, Pierros Ntelis, Predrag Jovanović, Rebecca Briffa, Rocco D'Agostino, Saeed Rastgoo, Sanjay Mandal, Sergei D. Odintsov, Tiago B. Gonçalves, Tiziano Schiavone, Vesna Borka Jovanović, and Vincenzo Salzano

The shortcomings exhibited by GR on different energy scales question its validity as the best theory to describe the gravitational interaction [3066]. Indeed, at the quantum level it shows problems with its extension to a quantum theory of gravity [3067,3068], since it is non-renormalizable [3069], and the prediction of infinite gravitational tidal forces, by the existence of singularities [3070]. In the large-scale scenario, the observed Universe's accelerating expansion [27,34] cannot be predicted within the context of the standard model of cosmology without introducing the cosmological constant, whose theoretical value assessed by Quantum Field Theory differs from the value inferred by the Friedman equations [2980,3071]. The repulsive force driving the Universe expansion is dubbed “Dark Energy”, whose existence, to date, is only based on indirect observations [3072,3073]. In addition, on galactic scales, the dynamics of the farthest stars orbiting around the center of galaxies (and, generally, incompatibilities on the LSS) represents a further open issue [3074,3075] currently addressed to a never-detected form of matter called “Dark Matter” [3076,3077].

These incompatibilities therefore open up the possibility of new (or rather, extended) scenarios, and here we will examine possible modifications to gravity.

Among the most important and widely considered approaches to address the aforementioned problems related to the unknown nature of DE and DM, as well as cosmological tensions, are modifications to GR itself. These can broadly be split into two main categories, whose features will be considered in detail in the next sections. The first category, referred to as *extended theories of gravity*, includes extensions of the Einstein-Hilbert action by other curvature terms (see Ref. [3078] for a review on the topic). The second category, referred to as *alternative theories of gravity* consists of models modifying the basic principles of GR, such as the Equivalence principle or Lorentz invariance. The main purpose of modified theories of gravity is to address GR shortcomings on different scales, providing a different view to describe gravity, potentially able to fix issues related to cosmology [3079] and the quantum formalism of the gravitational interaction [3067]. Extended and alternative theories of gravity naturally introduce extra degrees of freedom with respect to GR, which might potentially accommodate cosmic tensions. Several promising MG models involve modifications to the Einstein-Hilbert gravitational action, such as incorporating higher-order curvature invariants [3080–3082], establishing connections between geometry and scalar fields [2072,3083–3085], or introducing additional geometric features to spacetime beyond curvature, such as torsion (Poincaré gauge gravity and teleparallel gravity) [3086,3087] or non-metricity (metric-affine gravity and symmetric teleparallel gravity) [3088,3089], or non-linear connections (Finsler gravity) [3090, 3091]. Different classes of MG theories are outlined in Fig. 75.

#### 4.3.1.1. *Extended theories of gravity.*
One of the famous examples of GR extensions is the $f(R)$ gravity models, where the Ricci scalar $R$

in the Einstein-Hilbert action is replaced by a general function $f(R)$. This modification leads to fourth-order field equations, which can explain cosmic acceleration without invoking DE [3038,3092–3091]. Various formulations of this theory have been explored, some of which can produce deviations in the Newtonian potential [3102], potentially addressing phenomena like the galaxy rotation curves without resorting to DM. Notably, the Starobinsky model [2074], which introduces a quadratic term in the scalar curvature to explain cosmic inflation, has attracted significant attention within this framework. Extra degrees of freedom in $f(R)$ gravity affect the Hubble function $H(z)$, providing deviations from the $\Lambda$CDM paradigm. In the equivalent scalar-tensor representation in the Jordan frame of $f(R)$ gravity, in Ref. [3103] it is shown that the scalar field dynamics may lead to a definition of an effective Hubble constant $H_0^{\text{eff}}(z)$ that depends on the redshift $z$. In this regard, the non-minimal coupling between the scalar field and the metric plays a crucial role. Notably, $H_0^{\text{eff}}(z)$ may successfully address the Hubble tension since it could reconcile measurements and estimates of $H_0$ obtained at different redshifts, from local probes at $0 \lesssim z \lesssim 2$ to CMB at $z \sim 1100$. Accordingly, the general scalar field potential is computed in Ref. [3103], as well as the respective $f(R)$ functional form in the low-redshift regime within $f(R)$ quadratic gravity. Furthermore, Ref. [3104,3105] shows that a rescaling of the Universe's expansion rate can emerge without presupposing a specific dependence on the redshift (as done in Ref. [3103]). Indeed, it is just the non-minimally coupled scalar field of the scalar-tensor representation responsible for the Hubble constant scaling. Then, the additional DDE source was implemented in the $f(R)$ gravity to provide a smooth transition between a quintessence regime and phantom fluids, resulting in an effective Hubble constant that exhibits a plateau behavior for $z \gtrsim 5$ matching the Planck prediction. For equivalent approaches, see also Refs. [3106–3109]. These scenarios with $H_0^{\text{eff}}(z)$ were motivated by binned analysis of the SNIa Pantheon sample and BAOS [516,1670,3110], pointing out a slowly decreasing trend of the Hubble constant. In Ref. [3111], by exploiting the $f(R)$ gravity correspondence with a scalar-tensor theory, the authors provide a condition in which the $H_0$ tension is alleviated. Specifically, this condition is based on the existence of a metastable de Sitter point that occurs for redshifts near the recombination.

The Einstein-Hilbert action could be extended also by introducing dynamical scalar fields non-minimally coupled with the geometry [3112]. Scalar-tensor theories are often taken into account to address the dynamics of the early Universe, in the framework of the inflationary paradigm [3113]. In fact, inflation is usually conceived as generated by a scalar field $\phi$, called the inflaton, which is supposed to be responsible for the accelerating expansion of the early Universe. According to this picture, inflation should be led by some scalar field driving the cosmic acceleration between $10^{-34}$ and $10^{-35}$ s after the Big Bang, generating an isotropic and homogeneous Universe. The theory of inflation describes also the production of particles after the early-time accelerating expansion (reheating) [3114–3116]. Moreover, to tackle the cosmological constant problem or the evolution of cosmological vacuum energy, new degrees of freedom for the gravitational field have to be considered.

In this scenario, Horndeski theory represents the most general scalar-tensor theory with second-order field equations in four dimensions [2071,3117]. It encompasses a wide range of models and offers a comprehensive framework for modifying the gravitational interaction by incorporating an extra degree of freedom through a scalar field [3118–3120]. This results in significant and intriguing consequences for both inflation and DE physics. The detection of the GW170817 event together with the electromagnetic counterpart [587] provides one of the most stringent constraints on scalar-tensor theories that propose an unusual speed of GWs. This imposes severe limits on Galileons and extends to other scalar-tensor theories, including the quartic and quintic sectors of Horndeski's theory [3121–3124]. Nevertheless, it has been noted that since the energy scales observed at
<figure><img src="image_82.png" /><figcaption>Fig. 75. A summary of the MG roadmap, outlining the potential extensions of GR.</figcaption></figure>

LIGO lie very close to the typical cutoff of these types of theories, the translation of the bound on the speed of GWs to a cosmological setting might be non-trivial [3125].

Scalar-Tensor models have been taken into account also to address the $H_0$ tension problem. For instance, in Ref [3126]. Planck 2018 data are used to constrain the simplest models of scalar-tensor theories, accommodating a higher value for $H_0$ and therefore alleviating the tension between Planck/BAO and distance-ladder measurement from SNIa data. The capability of scalar-tensor and bi-scalar-tensor modified theories to alleviate the tension is also considered in Refs. [3127,3128], and other works [3129–3131].

Even when limiting the analysis to Horndeski models that respect the GWs' speed bound, observational constraints on $H_0$ using Planck data alone have been shown to yield values that are consistent with local determinations at $2\sigma$ for the Galileon Ghost Condensate [3132] and $1\sigma$ for the Generalized Cubic Covariant Galileon [2909,3133]. More recent studies have also considered Horndeski gravity to specifically address cosmic tensions, showing that particular setups free of ghost instabilities could lead to interesting late-time features possibly alleviating the $H_0$ tension [2898,3134].

Cosmic tensions have also been addressed in the context of the time-honored Brans–Dicke (BD) theory [3135,3136], which is the prototype of the large family of scalar-tensor theories (see also Ref. [2072]). BD gravity embodies the first historical attempt at incorporating a dynamical gravitational coupling to GR. It also constitutes the simplest scalar-tensor theory which can be embedded within the much broader class of Horndeski models [2071] and, in fact, many of these models reduce to BD at cosmological scales [2417]. Apart from the metric tensor, BD gravity involves a new (scalar) degree of freedom, $\phi$, which controls the strength (and the possible time evolution) of the gravitational interaction $G_{\phi} = G/\phi$, with G being the (locally measured) Newton constant transforming the gravitational coupling constant into a new dynamic variable [3137]. The scalar field $\phi$ is non-minimally coupled to gravity via the Ricci scalar and its dynamics is controlled by a dimensionless parameter $\omega_{BD}$, the BD parameter (or equivalently $\epsilon_{BD} \equiv 1/\omega_{BD}$).

Let us denote by BD-ΛCDM the Brans–Dicke counterpart of the standard ΛCDM, which represents the BD model with a cosmological constant, $\Lambda$. The cosmological term $\Lambda$ is actually not present in the original BD theory, but in modern times $\Lambda$ is obviously needed to trigger the late-time observed acceleration of the Universe. The BD-ΛCDM extension of the original BD model has been recently tested in the light of a large set of cosmological data of various sorts [2420], such as SNIa+BAO+$H(z)$+CMB+LSS, where $H(z)$ may include or not the SH0ES prior on $H_0$. If the latter is included, one finds [2420] that the BD-ΛCDM model is remarkably successful in reducing the $H_0$ and $\sigma_8$

tensions to inconspicuous levels of ~ $1.5\sigma$ (when the high-l polarization and lensing data from Planck 2018 are not considered) [2420]. Notice that The RVM is mimicked by the BD-ΛCDM model, particularly the type-II RVM version with variable $G_{\phi}$ [2418,2814–2816]. The BD-ΛCDM turns out to fit the cosmological data significantly better than the standard ΛCDM model, and this is confirmed by different statistical criteria (see Refs. [2419,2420,2814,2816] for details). On the other hand, adding a constant potential for the scalar field does not correspond to the usual vacuum energy in BD theory. An explicit detailed theoretical and observational investigation of this BD extension of ΛCDM model, studied in Ref. [3138], reveals that the model does not alleviate the tensions with no significant deviations from ΛCDM.

Values of the effective cosmological gravitational coupling $G_{\phi}$ about ~ 7–9% larger than G are preferred at ~ $3\sigma$ CL. This is possible thanks to the fact that $\phi$ remains below 1 throughout the entire cosmic evolution, while preserving the matter and radiation energy densities very close to the typical ΛCDM values. As a result, one finds higher $H(z)$ values during all the stages of the cosmic history without changing dramatically the abundances of the matter species in the pre- and post-recombination epochs. The lowering of the sound horizon at the baryon-drag epoch, $r_d$, is accompanied by an increase in the Hubble parameter. This helps to alleviate the $H_0$ tension in good agreement with other relevant background datasets, such as e.g., BAO and SNIa. In order not to spoil the correct fit to the CMB temperature data, the model is prone to yield values of the spectral index $n_s$ that are considerably larger (viz. closer to 1 from below) than the standard ΛCDM model. This causes, however, no problem since the dynamics of $\phi$ can compensate for the changes in the matter power spectrum introduced by this fact (cf. Sec. 3 of Ref. [2420]). Remarkably, the model is able to cut back the $\sigma_8$ tension at the same time. The relief of this tension can be accomplished by means of a negative value of $\epsilon_{BD}$, or alternatively through a positive value, provided that sufficiently massive (light) neutrinos are allowed [2420]. In both cases one finds $|\epsilon_{BD}| \lesssim \mathcal{O}(10^{-3})$. The BD parameter has a direct impact on the LSS data through the linear perturbation equations. For instance, for $\epsilon_{BD} < 0$ the friction term in the equation of the matter density contrast grows, and the source term decreases. Both effects contribute in sync to the lowering of $\sigma_8$, thus providing a physical explanation for a possible solution to the growth tension. It goes without saying that the compatibility between the larger cosmological value obtained for the gravitational coupling and the value measured locally, G, should be possible provided one can find an appropriate screening mechanism. Different studies show that these mechanisms are possible, although other works find that their implementation may not be so straightforward [3139].

Another way to extend the GR action is to introduce higher-order curvature invariants [3140–3150]. This is the case of Gauss–Bonnet
gravity, where the action includes a specific combination of curvature invariants known as the Gauss-Bonnet term. In this context, particularly intriguing are theories incorporating quadratic curvature terms, as they stem from GR viewed as an Effective Field Theory aiming to construct effective models toward quantum gravity. Notably, certain combinations of higher-order contractions of the Riemann tensor yield a topological surface term in four dimensions, known as the Gauss-Bonnet topological scalar, $G$. In four dimensions, this scalar equates to the Euler density, which, according to the generalized Gauss-Bonnet theorem, provides the Euler characteristic when integrated over the manifold. The Gauss-Bonnet invariant is often utilized for its topological properties to simplify dynamics. However, in four dimensions, to render its contribution non-trivial, it is typically coupled to a dynamical scalar field [3151–3155] or incorporated as a function in the gravitational action [3156–3159]. This approach, where a function $f(G)$ is added to the scalar curvature, can mimic the behavior of a cosmological constant at late-times and, in the limit $f(G) = 0$, GR is restored. Another avenue involves considering higher dimensions where $G$ is non-trivial and coupling the Gauss-Bonnet invariant to a constant diverging when $D = 4$. Gauss-Bonnet gravity has been studied to address the $H_0$-tension [3160,3161], finding that can be greatly resolved within $2\sigma$ level.

Bimetric gravity [3162] is a consistent theory of non-linearly interacting spin-2 fields, one massless and one massive, which allows for a broad range of cosmological expansion histories and is compatible with cosmological as well as local tests of gravity [3163–3165]. In its most general form, the theory features four additional parameters in addition to the standard $\Lambda$CDM ones. There are no free functions, unlike several other MG theories. Studies based on a restricted class of two-parameter models have shown that the value of $H_0$ inferred from the inverse distance ladder is only slightly increased compared to $\Lambda$CDM [2884]; however, such models do not comply with the constraints required for a working screening mechanism [3164]. A more recent analysis [2909] based on more general three-parameter models has shown that, using (model-independent) transverse 2D BAO data in combination with SN and CMB, results in a value of $H_0$ which is closer to the SH0ES value. On the other hand, using 3D BAO data produces a lower $H_0$ value, suggesting a potential bias in the latter. A study of the $S_8$ tension has not been performed at present, since a framework for structure formation in bimetric gravity has not yet been developed.

Functors of actions theories (FAT), is part of the extensions of gravity theories, and they predict the existence of *actionions*, i.e., the actionic fluctuations and field-particles, which are analogs of energetic fluctuations and field particles. In light of deviations from DE equations of state, $w_0 \simeq -1.1$, FAT predicts the existence of actionic fluctuations which are of the order of one tenth of the observed volume [3166, 3167].

### 4.3.1.2. Alternative theories of gravity.
The second class of modifications revises the basic foundations of GR, such as the Equivalence Principle, metric compatibility, Lorentz invariance, etc. One such approach involves considering a more general geometric framework with an affine connection different from the Levi-Civita one, thereby introducing both torsion and non-metricity into spacetime. Torsion is associated with the antisymmetric part of the connection, while non-metricity arises from the non-vanishing covariant derivative of the metric tensor. Within this framework, the Riemann tensor and its contractions in GR can be expressed in terms of torsion, non-metricity, or a combination of both. The gravity formulations characterized by curvature, torsion, and non-metricity are dynamically equivalent, differing only by a boundary term in their respective actions, reason for which they are often termed the “geometric trinity of gravity” [3168]. This classical equivalence involving boundary terms that do not affect the field equations (and are thus omitted from the action formulation), is viewed from the perspective of GR as motivated by the possibility of formulating GR as a gauge theory [3086,3087,3169,3170].

In this framework, a consistent gravitational theory can be formulated by promoting torsion over curvature as the sole governing factor of spacetime, resulting in a theory that precisely mirrors the dynamics of GR. This theory, known as the “Teleparallel Equivalent of General Relativity” (TEGR) [3171], has garnered significant attention in recent years, undergoing extensive analysis [3086,3087,3170,3172–3174]. TEGR presents a theoretical framework that interprets gravity as a consequence of torsion within the fabric of spacetime. In this context, gravitational interactions are described through a set of tetrad fields, also referred to as “vierbeins”, which form the basis for depicting spacetime geometry. These tetrad fields define a torsion tensor, serving as the source of gravity in the theory and representing the antisymmetric component of the Christoffel connection.

An extensively studied model alongside TEGR is the “Symmetric Teleparallel Equivalent of General Relativity” (STEGR), which describes spacetime through non-metricity. Non-metricity allows for the consideration that spacetime may not adhere to the metric compatibility condition, a fundamental assumption in GR. While torsion arises directly from the antisymmetry of the affine connection, non-metricity emerges when the covariant derivative of the metric tensor is non-zero, denoted as $\nabla_{\sigma}g_{\mu\nu} \neq 0$. As consisting of field equations entirely equivalent to those of GR, both TEGR and STEGR fall short of addressing the limitations GR imposes on larger scales. Consequently, similar to $f(R)$ gravity within the metric formalism, modifications to the Lagrangian density of TEGR can be explored in various ways [3175], including introducing new torsion [3176,3177] and non-metricity invariants [3178], coupling these to additional scalar or axion fields [3179] or introducing an arbitrary function of the torsion scalar, leading to $f(T)$ gravity [3180,3181]. The latter has been proposed as a potential solution to late-time cosmological issues, such as the Universe's accelerated expansion [3182,3183]. The characteristics of gravity theories including torsion and non-metricity, such as $f(T)$, $f(Q)$ (with $T$ being the torsion scalar and $Q$ the non-metricity scalar) gravity, new general relativity and many more, are currently under investigation, particularly concerning their applications in cosmology and astrophysics [2127,3184–3207]. Specifically, in the context of cosmological tensions, Refs. [2175,2176,3208] explores methods to address the $H_0$ tension within $f(T)$ models; in Refs. [3209,3210] it is shown how to alleviate both the $H_0$ and $\sigma_8$ tensions simultaneously within torsional gravity from the perspective of effective field theory; in Ref. [3211] the evolution of scalar perturbations in $f(T)$ gravity and its effects on the CMB anisotropy is evaluated to show that $f(T)$ models do not provide tension on the Hubble constant that prevails in the $\Lambda$CDM cosmology; in Ref. [3212] the authors demonstrate that the $f(T)$ models prefer a higher value of $H_0$ with respect to the Planck prediction, in better agreement with local estimates. Similar effects can be achieved in some classes of the beyond-generalized Proca model that, at the background level, resembles $f(T)$ theory [3213]. Finally, in Refs. [3214–3216] solutions to the tensions were proposed in the framework of $f(Q)$ gravity.

In another class of these models, Finsler gravity [3090,3217–3221] offers an intriguing new way to derive the gravitational field of many particle systems, when they are modeled as a kinetic gas, instead of as a (perfect) fluid [3222,3223]. The dynamics of a kinetic gas is described by a 1-particle distribution function (1PDF) on the 1-particle phase space of spacetime, which takes the velocity distribution of the gas particles into account. Usually, when the gravitational field of a kinetic gas is derived, one obtains the energy–momentum tensor of the gas by averaging the 1PDF over the velocity distribution of the particles. By means of this averaging procedure, the information about the velocity distribution of the gas particles are lost and the gravitational field is derived from an effective fluid energy momentum-tensor as a source term in the Einstein equations. Finsler spacetime geometry describes gravity in terms of a curved 1-particle phase space of a curved spacetime (technically the tangent or co-tangent bundle). Here, the
1PDF directly sources the gravitational field of the kinetic gas without losing information through velocity averaging. The contribution of the velocity (kinetic energy) distribution of the gas to its gravitational field is fully taken into account in Finslerian gravity, rather than being averaged away [3222,3223]. The first promising solutions of Finsler gravity in the cosmological context have been obtained [3224], which show a linearly expanding Universe as a vacuum background solution. Further studies are ongoing to demonstrate that the contribution of the kinetic energy of a many-particle system to its gravitational field can be the solution to the Hubble tension and our understanding of DE.

It is worth pointing out that some alternative theories propose that gravity is mediated by a graviton with a small non-zero mass $m_g$. Experimental limits on $m_g$ are set using models such as the Yukawa potential and dispersion relations, tested on astrophysical and cosmological scales. In addition to detectors like LIGO/Virgo or LISA, graviton mass limits can be derived from electromagnetic observations of gravitational systems, such as constraints on S-stars orbits [3225,3226], or from Schwarzschild precession of S2 in Yukawa gravity [3227], consistent with LIGO data.

Some theories of gravity relax the assumption of nonminimal coupling between the geometry and matter sources. This happens, for instance, in $f(R, \mathcal{L}_m)$ [368,3228], $f(R, \mathcal{T})$ [3229], $f(R, T_{\mu\nu}T^{\mu\nu})$ [3230–3233], and $f(R, \mathcal{T}, R_{\mu\nu}T^{\mu\nu})$ [3234,3235] theories of gravity, where $\mathcal{L}_m$ is the matter Lagrangian density, and $\mathcal{T}$ the trace of the energy–momentum tensor $T_{\mu\nu}$ (in its standard definition). Additionally, one can construct similar nonminimal coupled theories in the torsional framework, such as in $f(T, \mathcal{L}_m)$ [3236,3237] and in $f(T, \mathcal{T})$ [3238] theories of gravity, where $\mathcal{T}$ is the torsion scalar.

A characteristic of these theories with nonminimal geometry–matter coupling is that the covariant divergence of the matter energy–momentum tensor can in general be nonzero (though the standard continuity equation can be recovered in particular models). A new avenue has been recently opened by modifying the introduction of the material source in the usual EH action, such as matter-type modified theories of gravity with $f(\mathcal{L}_m, \mathcal{T}, T_{\mu\nu}T^{\mu\nu})$, since these theories are equivalent to nonminimal interaction models in GR [3239]. Depending on the form of interaction determined by $f$ function, some nontrivial dynamics easing cosmological tensions [2162], difficult to achieve via simple interaction kernels can be achieved, these DE models have effects on early and late-times of the Universe and hence may address the current tensions within $\Lambda$CDM model. The extra two degrees of freedom have the potential to offer more room to accommodate cosmic tensions. However, this also provides a challenge, given that Einstein–Boltzmann codes are commonly prepared to deal with at most one extra degree of freedom, it is, at the moment, more challenging to compute the evolution of perturbations and constrain these theories.

So far, studies have focused on the simplest classes of models. An example of studying dynamics of scalar perturbations, using the quasistatic approximation, is in Ref. [3240] for $f(R, \mathcal{T}) = R + f_2(\mathcal{T})$ models with $f_2(\mathcal{T}) \propto \mathcal{T}^{1/2}$ to ensure the usual matter continuity equation. This work finds that there is a strong scale $k$ dependence of the matter perturbations, not supported by observations.

A more recent work Ref. [3241] reports on the possibility of $f(R, \mathcal{T}) = R + \lambda\mathcal{T}$ alleviating the $\sigma_8$ tension, while increasing the $H_0$ tension. In the Hu–Sawicki model of $f(R)$ gravity the $\sigma_8$-tension observations worsen [2840], while it might be alleviated in viscosity in the DM model (in modified cosmological models, massive neutrinos suppress the matter power spectrum on the small length scales, which implies that the bounds on neutrino mass get modified too) [3242–3245].

To conclude, it is worth noticing that most alternative/extended theories of gravity may be subject to screening mechanisms, such as the chameleon mechanism [3246,3247], which can be encountered in several scalar-tensor theories [3248,3249] as well as in MG models [3250–3262] that can be recast in terms of an additional scalar field

(e.g., see Ref. [3263]), like $f(R)$ gravity. In this case, the scalar field is coupled to the matter fields so that its mass is environment-dependent. As long as the matter density is high enough, the scalar field acquires a heavy mass around the potential minimum, strongly reducing the range of the fifth force it mediates, and making it essentially unobservable, see however [3264,3265].

On the contrary, on cosmological scales, far from regions with higher densities, the scalar field has a lighter mass, effectively mediating the additional gravitational interaction. Another screening mechanism is the symmetron screening [3266], where the potential of the scalar field $V(\phi)$ breaks the parity symmetry in low-density regions, which instead is restored at smaller scales. In this mechanism, the scalar field has a vacuum expectation value (VEV) that depends on local matter density, becoming larger in low-density regions and smaller in high-density regions. Moreover, the coupling of the scalar field to matter is proportional to the VEV, so that the scalar couples with gravitational strength in the low-mass-density regions, while it is decoupled, and then screened, in the high-density ones. In Ref. [3267] it has been concluded that symmetron screening cannot be used to alleviate the Hubble tension.

Finally, the Vainshtein mechanism [3268] is important for some MG models, such as massive gravity, Dvali–Gabadadze–Porrati (DGP), or Galileon theory. The Vainshtein mechanism manifests itself at non-linear scales, below a certain radius known as the Vainshtein radius. Nearby massive bodies GR are recovered through a strong kinetic self-coupling, weakening the interaction with matter and reducing the propagation of the extra degree of freedom. Recently, it was shown that bimetric gravity can alleviate the $H_0$ tension (refer to Section 4.2), and could potentially alleviate also the $S_8$ tension [2909].<fnref n="60" />

According to the latest observations, one of the most promising extended gravity models is the generalized cubic covariant Galileon model. Using only CMB data, provides $H_0 = 72_{-8}^{+8} \text{ km s}^{-1} \text{Mpc}^{-1}$, $\sigma_8 = 0.88_{-0.05}^{+0.07}$ at a 95% confidence level [3133] and $S_8 = 0.84 \pm 0.10$. At the same time, the tensions are restored by combining the CMB data with external datasets.

In the framework of alternative theories, particularly encouraging is the $\Delta_4$ model, which is a specific non-local gravity model providing non-local quantum corrections to standard GR. This model predicts the values $H_0 = 70.3 \pm 0.9 \text{ km s}^{-1} \text{Mpc}^{-1}$ and $\sigma_8 = 0.82 \pm 0.02$ [3271], using the CMB, BAO and SN datasets, which is higher than Planck predictions, in better agreement with the local estimates. The related $S_8$ value is $S_8 = 0.81 \pm 0.02$.

All in all, although there is not a universally accepted MG model that resolves the cosmological tensions, many of these models offer promising aspects that could help guide future research. Most of the models considered in this section provide a higher value for $H_0$, though implications on $\sigma_8$ should be further explored. As cosmological data continue to improve, these theories will be subject to increasingly stringent tests, potentially leading to a deeper understanding of the fundamental nature of gravity and the Universe. Therefore, it is quite possible that extended/alternative theories could contribute, at least in part, to addressing the $H_0$ and $\sigma_8$ tensions.

### 4.3.2. Early modifications to gravity

**Coordinator:** Emmanuel N. Saridakis, Konstantinos F. Dialektopoulos

**Contributors:** Andrzej Borowiec, Alfio Maurizio Bonanno, Ali Övgün,

<footnote n="60">⁶⁰ However, it must be noted that drawing accurate predictions for the formation of LSS in bimetric gravity is challenging, so at present the $S_8$ tension can only be addressed under some simplifying assumptions. Regarding the $H_0$ tension the alleviation comes with the drawback that more parameters are introduced and bimetric gravity is preferred by AIC [3269] but $\Lambda$CDM is preferred by BIC [3270] with the latter giving a stronger penalty for introducing more parameters to the model in consideration.</footnote>
Andreas Lymeris, Andreas Papatriantafyllou, Anil Kumar Yadav, Antonio Racioppi, Athanasios Bakopoulos, Athanasios Chatzistavrakidis, Branko Dragovic, Carlos G. Boiza, Charalampos Tzerefos, Chiara De Leo, Christian Pfeifer, Cláudio Gomes, Damianos Iosifidis, Daniele Oriti, David Benisty, Davide Pedrotti, Despoina Totolou, Diego Rubiera-Garcia, Elias Vagenas, Elisa Fazzari, Elvis Baraković, Flavio Bombacigno, Fotios K. Anagnostopoulos, Francisco S. N. Lobo, Gaetano Lambiase, Genly Leon, Gerasimos Kouniatis, Giannis Papagiannopoulos, Giovanni Otalora, Giulia De Somma, Giuseppe Gaetano Luciano, Gonzalo J. Olmo, José Pedro Mimoso, Jurgen Mifsud, Kathleen Sammut, László Árpád Gergely, Laur Järv, Manolis Plionis, Manuel Gonzalez-Espinoza, Manuel Hohmann, Margus Saal, Maria Petronikolou, Miguel A. S. Pinto, Nikolaos E. Mavromatos, Nikolaos Petropoulos, Oem Trivedi, Özgür Akarsu, Petros Asimakis, Saeed Rastgoo, Salvatore Capozziello, Saurya Das, Sebastian Bahamonde, Simony Santos da Costa, Spyros Basilakos, Stylianos A. Tsilioukas, Supriya Pan, Thanasis Karakasis, Tiago B. Gonçalves, Tomi Koivisto, Vasilios Zarikas, Vedad Pasic, Xin Ren, Yuejia Zhai, and Yu-Min Hu

MG is a quite general framework that can offer alleviation to the tensions, since apart from late-time solutions (which were analyzed in the previous subsection) one can apply it at early-times too [3272]. In order to solve the Hubble tension, it is possible to introduce modifications to the standard cosmological model. The general idea is that new assumptions at pre- and post-recombination are needed. For this reason, changing some of these assumptions, i.e., changing the cosmological model before or after the recombination epoch, may be the solution to obtain a different value of $H_0$. One possible solution is to consider a MG scenario before the recombination epoch. Since the BAO data constrain $Hr_d$, the final goal is to obtain a lower value of the sound horizon at the drag epoch $r_d$ and thus a higher value of $H_0$. Some constraints from models adhering to this approach are shown in Table 6.

The expression of the sound horizon is $r_d = \int_{z_d}^{\infty} dz c_s(z)/H(z)$, where at sufficiently high redshift the sound speed $c_s(z)$ can be considered almost constant and equal to $c/\sqrt{3}$. To decrease the value of $r_d$ we need that, in a certain redshift interval $z \in (z_1, z_2)$, the expansion of the Universe is higher with respect to the $\Lambda$CDM case in the same value. An example of this approach can be found in Ref. [3273], where it is shown that the Hubble tension can be reduced by assuming a MG model acting at early-times with a scalar field non-minimally coupled to the Ricci scalar. Furthermore, in Ref. [3274], they explore the possibility of obtaining a lower value of the sound horizon, and consequently a higher value of $H_0$, by using a MG model that allows for a phenomenological shift in the effective Planck mass on cosmological scales before recombination, through a non-minimal coupling of a scalar field with gravity.

In Refs. [2471,2473,2479] it was proposed that the dynamics of the early Universe is described by a Chern-Simons (CS) gravitational model, characterized by gravitational anomalies, which arises in the low-energy limit of string theory [2470,2480], after appropriate compactification to $(3+1)$ spacetime dimensions. The model involves the graviton and Kalb-Ramond (KR), or gravitational, axion field (assuming a constant dilaton). The latter couples to the anomalous terms. Chiral primordial GW can condense under some circumstances, leading to a non-trivial condensate of the CS terms [2481,3275,3276], and hence a linear axion potential, resembling the axion monodromy situation in string-theory models [2483]. Such linear-axion potentials lead to inflation [2481,2818], which however are of the so-called RVM type [2476]. This means that the corresponding vacuum energy density contains dominant terms of order $H^4$ (where $H$ is the Hubble parameter, which is almost constant during inflation).

It is such non-linear terms that drive the RVM inflation. In Ref. [2481], it was shown that the CS condensates are metastable, in the sense that they contain imaginary parts [2482], which can determine the duration of the RVM inflation. The model contains a pre-RVM-inflationary era characterized by stiff-KR-axion-matter dominance. It

is during such an epoch that chiral GWs are formed, by either non-spherically-symmetric mergers of primordial (rotating) black holes, or collapses of domain walls, the latter arising, for instance, from dynamical breaking of supergravity [3277,3278], which can characterize the early epochs of such string-inspired models after the Big Bang [2473].

The detailed transition from such an era to the RVM inflation is discussed in Ref. [2481], using a dynamical system approach. The model can also lead to post-inflationary late-eras DE modifications, involving vacuum energy terms of the form $H^2\ln(H)$, which are held responsible for the simultaneous alleviation of $H_0$ and growth-of-structure tensions [2484].

The simplest generalization of Einstein's theory are the so-called $f(R)$ theories, which can be further generalized by including a non-minimal coupling between the matter Lagrangian and another generic function of the curvature [368]

$$S = \int d^4x \sqrt{-g} (f_1(R) + f_2(R)L).$$

This model gives rise to an extra force term in the geodesics for a perfect fluid. In fact, stemming from the Liouville theorem in phase space, the flux density of particles is shown to be covariantly conserved, and the Boltzmann H-theorem is still preserved as the entropy vector flux is a non-decreasing function of the spacetime coordinates likewise in general relativity. Despite the distribution function being formally equivalent to Einstein theory for a homogeneous and isotropic universe, some quantities differ, namely as the effects of the non-minimal coupling appear on the radiation density evolution and on the matter Lagrangian choice, which is no longer degenerate, for instance, Ref. [3279]. In these theories, the baryon asymmetry is generated through an effective coupling between the Ricci scalar and the net baryon current, which dynamically breaks CPT invariance, thus matching the observed asymmetry for very small deviations from general relativity. Moreover, the resulting temperatures are compatible with the subsequent formation of the primordial abundances of light elements [3280]. Hence, among others, this scenario allows for a good agreement with SN distance data and the BAO data for different exponents in a power-law type non-minimal coupling function, thus alleviating the Hubble tension [372].

A consistent gauge theory of gravity and spacetime based on the Lorentz symmetry was found rather recently [3281–3283]. Addressing the problem of time, the theory predicts that the space that emerges via spontaneous breaking of the Lorentz symmetry could be massive [3281, 3284,3285]. This would be the minimal explanation of the missing mass in the Universe, and a slight extension of the scenario might alleviate the $H_0$ tension through a cosmic spin as has been discussed also in the context of Poincaré models of gravity [3285–3288]. An ultraviolet completion mentioned in Ref. [3289] leads to modifications of early cosmology, whereas the relevance of a different description of nonperturbative effects to the cosmological tensions was speculated in Ref. [3290]. Ref. [3291] rewrites the higher curvature theories using a gauge theory of gravity.

As a subcase of metric-affine and gauge theories of gravity, teleparallel gravity has also been well-studied in the early Universe. The Teleparallel Equivalent of General Relativity (TEGR) describes gravitational interactions as the torsion of the connection of spacetime. Several teleparallel inflationary scenarios have been proposed in the literature, in order to change the early-Universe evolution and eventually alleviate the $H_0$ tension, and here we briefly summarize some of them. Born-Infeld inspired inflation originates from the work of Born and Infeld, who proposed a finite self-energy for point-like particles to avoid divergences in physics. This approach has influenced various gravity theories, including Eddington-inspired Born-Infeld gravity, which addresses singularities like black holes and the big bang. In teleparallel cosmology, Born-Infeld inspired models use an $f(T)$ gravity Lagrangian, featuring a scaling parameter $\lambda_{BI}$ that induces inflation without an inflaton field [3182,3292,3293]. This model aligns with
the CMB and transitions to standard $Λ$CDM at later times, providing a comprehensive framework for early Universe dynamics [3294–3298].

In Loop Quantum Cosmology within the FLRW framework, the matter bounce scenario is described by a modified Friedmann equation, resulting in a non-singular bounce at $t = 0$. This model can be reformulated in terms of $f(T)$ gravity [3299–3307]. The Lagrangian can be adjusted to fit observations by varying the critical density $\rho_{cr}$. Scalar fields, affecting spectral indices and power spectra, are crucial for matching CMB observations and ensuring reheating. Perturbations reveal additional degrees of freedom, impacting the scalar and tensor power spectra, with the tensor-scalar ratio potentially exceeding observed bounds, dependent on scalar field dynamics.

Higgs inflation explores the coupling of the Higgs field with gravitational theories like GR, TEGR, and STEGR. The Higgs potential $V(\phi) = \frac{\Lambda^4}{4}(\phi^2 - v^2)^2$ can be coupled with the torsion scalar $T$ to investigate inflationary scenarios. In this context, a particular action is used, incorporating kinetic and potential functions [3308]. Some studies reveal issues, such as a high tensor-to-scalar ratio or failing to achieve inflationary models. Despite these challenges, alternative scalar field couplings and potential functions may yield more practical inflationary models, suggesting further exploration is necessary for natural inflation in teleparallel gravity.

When the canonical quantization techniques of Loop Quantum Gravity are applied to cosmological scenarios, one finds that the big bang singularity is generically resolved by a big bounce [3309–3314]. This happens even when different quantization prescriptions are used, which leads to various nonsingular models of loop quantum cosmologies [3315–3319] (known as LQC, mLQC-I, and mLQC-II [3317]) that differ in their concrete pre-bounce and post-bounce dynamics. In Ref. [3320], it has been found that the effective dynamics of those three quantum-corrected models can be accurately approximated by a 3-parameter family of metric-affine $f(R)$ theories of gravity [3321], where two of the parameters can be set by constraints at the bounce and at low curvatures (GR limit), while the third one can be determined (numerically) by fitting the background evolution. Interestingly, the non-perturbative dynamics of all three models are dominated by a logarithmic correction [3322]. Additionally, the best fit value of the free parameter can be well approximated by elementary combinations of the bounce density and the Barbero-Immirzi parameter. On the other hand, it is possible to recover regular big-bounce scenarios also in the presence of a dynamical Barbero-Immirzi field, when a purely torsional Nieh-Yan term on a FLRW background [3323,3324] and its non-metrical generalization in Bianchi-I cosmologies [3325] are taken into account. This idea naturally leads to projective-invariant metric-affine formulations of Chern-Simons MG, for which big bounce and de Sitter solutions arise [3326]. A peculiar aspect of these models is that they predict the coupling of metric tensor modes to torsion tensor components, leading to torsional birefringence and the possibility of distinguishing the usual metric version of Chern-Simons gravity from its metric-affine counterpart via the quasinormal mode emission of compact objects [3327] (see also Ref. [3328]).

Axions and Axion-Like Particles (ALPs) provide well-motivated candidates to address important problems in cosmology, maintaining at the same time a significant discovery potential in current and future experiments. Among their uses, they can serve as candidates for DE within the quintessence scenario [3329,3330]. ALPs arise naturally in string-inspired models [2480], often not single but as a plethora of light pseudoscalar fields [3331]. Their characteristic sinusoidal potential generates models of EDE that can potentially resolve the Hubble tension [2454,2455,3332]. Remarkably, although these models are loosely inspired by string theory, recent attempts highlight the potential of embedding them into stable type IIB string compactifications [2461,2462], which would provide a fundamental starting point. The appearance of multiple ALPs in stringy scenarios allows for alignment mechanisms that combine them and can lead to models where at different stages

of the cosmological evolution different ALPs act as inflaton, QCD axion, and quintessence [3333,3334]. Pseudoscalar fields were also considered in the context of teleparallel and symmetric teleparallel gravity, where they couple to the CP-violating quadratic terms that can be formed with the torsion tensor or/and the non-metricity tensor [3179,3335,3336]. Such additional fields could contribute to the cosmological dynamics in the early Universe [3179,3337] and can also lead to GW velocity birefringence [3338,3339], which can be tested using multi-messenger approaches [3340].

DM creation during or after inflation can be associated with irreversible thermodynamic processes [3341–3343]. Refs. [3344,3345] proposed an alternative cosmology based on the irreversible thermodynamics of open systems, in which the explanation for the production of macroscopic matter and entropy in the early Universe relies on a reinterpretation of the matter energy–momentum tensor that includes an irreversible creation term. However, this contrasts with the covariant conservation of the energy–momentum tensor in GR.

In the last decade, several MG theories that contain nonminimal couplings between geometry and matter have been proposed, such as $f(R, L_m)$ [3228], $f(R, T)$ [3229], $f(R, T_{\mu\nu} T^{\mu\nu})$ [3230–3233] and $f(R, T, R_{\mu\nu} T^{\mu\nu})$ [3234] theories of gravity, where $R$ and $R_{\mu\nu}$ are the Ricci scalar and tensor, $L_m$ the matter Lagrangian density, and $T$ the trace of the energy–momentum tensor $T_{\mu\nu}$. A property of all these theories is that the matter energy–momentum tensor is not conserved. This feature allowed Harko [3346] to physically interpret such non-conservation as an irreversible energy flow from the gravitational field to the matter sector that could result in particle creation, by using the thermodynamics of open systems approach. The effects and implications of the irreversible matter creation processes on the late cosmological evolution have been studied on some of these modified theories of gravity [3346–3349] (see Ref. [3350] for a review). It was shown in Ref. [3239] that if such theories include the Ricci scalar $R$ in the Einstein-Hilbert form, i.e., assuming minimal geometry–matter coupling — they are effectively equivalent to nonminimal interaction models within the framework of GR. These MG theories could be further explored in the context of the early Universe DM creation that can eventually lead to alleviation of the $H_0$ tension.

$Λ$CDM paradigm is very efficient in describing our Universe; however, it is broadly acknowledged that the model ought to break down and exhibit limitations in at least the two extreme phases of the Universe, namely its very early and very late phases. In the very early Universe, Planck scale or quantum gravity effects are expected to set in, resulting in potentially significant conclusions, some of which may be measurable. While proposals for quantum gravity abound, and one can in principle take any theory and examine its implications for the above stages of the Universe, here we will primarily be interested in a virtual model-independent prediction of theories of quantum gravity, namely the Generalized Uncertainty Principle (GUP). The GUP predicts significant changes to several thermodynamic quantities, while it leaves some others intact [3351–3354].

A generic and often-used version of GUP is

$$[x_i, p_j] = i\hbar \left[ 1 - \alpha \left( p\delta_{ij} + \frac{p_i p_j}{p} \right) + \alpha^2 \left( p^2 \delta_{ij} + 3p_i p_j \right) \right] \quad (4.18)$$

$$\Delta x \Delta p \ge \frac{\hbar}{2} [1 - 2\alpha\langle p \rangle + 4\alpha^2\langle p \rangle^2] , \quad (4.19)$$

where the dimensional parameter $\alpha$ can be traded off with a dimensional one $\alpha_0$, via $\alpha = \alpha_0 \ell_{PI}/\hbar$, where $\ell_{PI} \simeq 10^{-35}$ m is the standard Planck length and $1 \le \alpha_0 \le 10^{15}$, the upper bound dictated by the fact that no Planck scale effects have been observed at the LHC, which has a characteristic length scale of $10^{-20}$ m $\simeq 10^{15} \ell_{PI}$ (corresponding to $10$ TeV of energy), which the “new length scale” $\alpha_0 \ell_{PI}$ should not exceed.

It follows that quantum gravity effects can in principle take effect right from the electroweak to the Planck scale. Since our focus is in the very early Universe, we will be concerned near the Planck
scale. As explained in Ref. [3351] the standard Heisenberg Uncertainty Principle and the GUP can be translated into a corresponding time-energy uncertainty relation. This, coupled with the first Friedmann equation, can in turn be used to estimate the Planck energy, mass, density, time, temperature, and effective number density. Since the GUP terms can differ significantly from the standard Heisenberg one near the Planck scale, it is not surprising that many of the quantities alluded to above undergo significant changes in the early Universe. Perhaps the most significant among them is the Planck energy density, whose ratio to the observed DE density increases by at least a factor of 4 to about $10^{119}$.

Additionally, the same uncertainty principle can in fact affect the Friedmann equations and cosmological predictions in the radiation dominated era, as shown in Refs. [3352,3353]. There, an interaction term coupling baryon current and space-time was used to satisfy the first two Sakharov conditions, which along with the modified Friedmann equations break thermal equilibrium and thus satisfy the third and last Sakharov condition. The results can provide a rich phenomenology.

In the GUP approach, where a modification to the canonical algebra of the system leads to minimal uncertainties in the configuration and/or momenta, the evolution of the FLRW Universe with canonical variables $(c, p)$ in the presence of scalar matter field $\phi$ with its momentum $p_{\phi}$, can be described by the action of the GUP modified Wheeler-deWitt equation on the wave function of the Universe $\Psi$, $(\partial_{\phi}^2 + \hat{\Theta}_{\text{GUP}})\Psi = 0$, where $\hat{\Theta}_{\text{GUP}}$ is a self-adjoint operator. In one approach [3355], which is equivalent to a cut-off in the length, $[q, p] = i(1 + \beta p^2)$, using suitable wave-packets for the deep Planckian regime, it is found that the wave-packets do not fall in the classical singularity and the GUP Universe exhibit a stationary behavior in approaching the Planckian region. This implies that no bounce is present in this model. In another approach using GUP-modified Friedmann equations [3356], it is found that the DE density scales as $\tilde{\alpha}H^4$, where $\tilde{\alpha}$ is a GUP parameter associated to the minimum length, and $H$ is the Hubble parameter. This means that this approach to GUP cannot explain the acceleration of the present Universe, since the energy density decreases very quickly. It is worth mentioning that GUP comes in various forms and a recent proposal based on a different modification of algebra (minimal momenta) [3357] together with an improved prescription may be a better approach. The reason is that (1) the momenta of gravitational field are related to the metric or triads so it is more natural to have minimal values in those quantities, and (2) without an improved scheme, just like LQC, one might obtain incorrect classical or asymptotic limits and hence the theory might not also be correct in the Planckian regime.

The Asymptotic Safety approach to Quantum Gravity offers a potential solution to the problem of quantizing the gravitational field. This approach is grounded in the existence of an ultraviolet non-Gaussian fixed point for the couplings of the $\sqrt{g}R$ and $\sqrt{g}$ operators in the Euclidean theory [3358]. A direct consequence of this fixed point is the dynamic suppression of Newton constant $G$ at very high energies, which has significant implications for early cosmological models. In Ref. [3359], a comprehensive cosmological evolution from the inflationary phase to a late de Sitter phase is discussed. Additionally, [3360] proposes the idea that the Universe originates from a state of zero entropy and evolves toward an accelerated expansion phase.

Following Ref. [3361], one can also expect various modifications to the Effective Action at the inflationary scale. In particular, it has been shown in Ref. [3362] that an $f(R)$ Lagrangian of the form $L = R + \alpha R^{3/2} + \frac{R^2}{6m^2} - A$ represents a modification of the well-known Starobinsky quadratic Lagrangian. This modification arises due to the presence of higher curvature relevant operators generated by the renormalized flow near the non-Gaussian fixed point. The predicted scalar-to-tensor ratio $r$ is significantly higher than in the standard Starobinsky model, making it a potential target for testing in future CMB missions like LiteBIRD.

The cosmological evolution of the early Universe is tested at temperature $T \lesssim 1$ MeV through the prediction of BBN. For temperatures $T \gg 1$ MeV, it is not excluded that new physics occurred, offering the possibility that non-standard cosmologies might have played a relevant role in the interplay between particle physics and cosmology. The Ice Cube experiment, for example, has revealed a high-energy astrophysical neutrino flux, with energies ~ $O(1)$ PeV [3363,3364]. Among the various scenarios to explain this phenomenon, it has been proposed the minimal model of DM decay [3365]. This interaction is also able to generate the correct abundance of DM in the Universe [3365]. In the standard cosmological model, the rate of DM decay, needed to obtain the correct DM relic abundance, differs by several orders of magnitude as compared with that needed to explain the Ice Cube data, making the four-dimensional operator unsuitable. However, such a discrepancy can be reconciled in modified cosmologies, so that the Ice Cube neutrino rate and DM relic density can be consistently explained [3366–3368].

In Fig. 76 we summarize the various early modified gravity theories that have been proposed to alleviate the tensions, while some corresponding predictions for $H_0$ and $\sigma_8$ are summarized in Table 6.

## 4.4. Matter sector solutions

### 4.4.1. Cold dark matter

Coordinator: Luca Visinelli

**Contributors:** Andrzej Borowiec, Anil Kumar Yadav, Branko Dragovic, Davide Pedrotti, Goran S. Djordjevic, Ioannis D. Gialamas, Ippocratis Saltas, Ivan De Martino, Janusz Gluza, Jenny G. Sorce, Jenny Wagner, Krishna Naidoo, Marcin Postolak, Martti Raidal, Mehmet Demirci, Paolo Salucci, Pran Nath, Reggie C. Pantig, Riccardo Della Monica, Rishav Roshan, Sebastian Trojanowski, Torsten Bringmann, Venus Keus, and Wojciech Hellwing

#### 4.4.1.1. Motivation and evidences.
The circular velocity of stars in a spiral galaxy can be inferred from the measurements of the Doppler shift of atomic lines at different distances $R$ from the galactic center, leading to a velocity distribution $v = v(R)$ known as the *rotation curve*. The distribution of the luminous matter in virialized objects does not match that of the gravitational matter, demanding an additional non-luminous component.

Historically, this “missing mass problem” in the late Universe led to the proposition that there is a yet undetected form of matter [3369–3373]. Subsequently, CMB observations revealed a missing mass in the power spectrum already at redshift $z = 1100$, which could also be attributed to the same form of matter we call DM today. Evidence for the presence of DM in the Universe spans from galactic (rotation curves of spiral galaxies) to extra-galactic (clusters of galaxies, weak gravitational lensing observations) to cosmological scales (LSS, acoustic peaks in the CMB, BBN, non-linear growth of inhomogeneities. See Ref. [3374] for a historic reconstruction and Refs. [358,3375–3379] for reviews. N-body simulations with DM only reproduce well the LSS [2355,3074,3380] and are advancing to incorporate the essential *gastrophysics* effects [2280,3381].

The role of DM in addressing cosmic tensions is of profound interest from both theoretical and observational perspectives. If discovered, a DM particle would be a relic from a period before BBN, for which we currently have no direct data. The properties of DM — such as its clustering power spectrum, free-streaming velocity, and particle characteristics — would enable us to better model the pre-BBN epoch and differentiate between various “early Universe” proposals aimed at resolving these tensions. Furthermore, some theories that address the $H_0$ tension explicitly involve interactions between DM and other components, including DE. Pinning down the properties of DM is crucial for understanding its interactions with other particles, including those in exotic DE models.
<table><caption>**Table 6**
Early modifications to gravity: predictions.</caption><thead><tr><th>Model</th><th>H<sub>0</sub></th><th>σ<sub>8</sub></th></tr></thead><tbody><tr><td>Early MG and the sound horizon [2445]</td><td>&gt; 70</td><td>&lt; 0.8</td></tr><tr><td>String-inspired Chern-Simons modifications [2484]</td><td>71.27<sup>+0.76</sup><sub>-0.73</sub></td><td>0.816<sup>+0.017</sup><sub>-0.015</sub></td></tr><tr><td>Non-minimally coupled models [372]</td><td>70.59<sup>+0.52</sup><sub>-0.53</sub></td><td>—</td></tr><tr><td>Axions and Axion-like particles [3340]</td><td>70.6 ± 1.3</td><td>—</td></tr></tbody></table>

<figure><img src="image_83.png" /><figcaption>**Fig. 76.** Possible early modifications to gravity.</figcaption></figure>

#### 4.4.1.2. Theory.
The evidence supporting the existence of non-baryonic DM is compelling across all observed astrophysical scales. While alternative theories are still under consideration, the notion of CDM has emerged as the dominant framework. DM might exist as macroscopic objects or as a particle, either fundamental or composite, proposed in extensions beyond the SM framework. A non-exhaustive summary of the properties (scattering cross section and target mass) of some candidates is provided in Fig. 77, inspired by previous work in Refs. [3382–3384].

**Macroscopic DM** One of the earliest propositions on the nature of DM is that it could appear in the form of known astrophysical objects of mass *M* such as planets or stellar black holes [3385,3386]. This conjecture has been challenged by gravitational microlensing surveys, which collectively exclude these objects as the main DM component over the mass range $10^{-11} \lesssim M/M_{\odot} \lesssim 10$ [3387–3390].

Along with known astrophysical objects, massive DM candidates consist of primordial black holes which are produced in the early Universe by a plethora of mechanisms at different mass scales. Stable primordial black holes possess a mass $M \gtrsim 5 \times 10^{14}$ g so that their Page time [3391] exceeds the age of the Universe, making them suitable DM candidates. A more stringent bound $M \gtrsim 10^{17}$ g is obtained from the non-observation of Hawking emission in the extragalactic $\gamma$-ray flux and in X-ray detectors [3392–3395]. Lensing constraints also apply to primordial black holes, leaving an open window of opportunity where primordial black holes are the DM in the mass range $M \in [10^{17}-10^{22}]$ g under the assumption of a monochromatic mass function.

<figure><img src="image_84.png" /><figcaption>**Fig. 77.** An estimate of the scattering cross section for some DM candidates, as a function of the DM mass. The plot is inspired from Refs. [3382–3384]. As a comparison, the SM neutrino is also included.</figcaption></figure>