In the realm of the Hubble tension—a review of solutions^*

We consider here as 'early' predictions for H₀ those relying, in principle or in practice, on the accuracy of a number of assumptions of the ΛCDM model used to describe the Universe at z > 1000, including a number of ansatzes about the properties of neutrinos (e.g. there are 3 active species known with a minimal total mass of 0.06 eV assuming normal hierarchy [21]), particle interactions, the absence of primordial magnetic fields (PMFs), a null running of the scalar spectral index, no additional relativistic particles or degrees of freedom, etc. Certain types and scales of breakdowns in these assumptions may be apparent within the CMB power spectra (and are not seen) though others may not. Many of these same ansatzes are used to relate local measurements of 'primordial' abundances to the baryon density [22]. The ΛCDM model is further used to describe the evolution of the Universe at 0 < z < 1000 to predict the expansion rate, H(z) and its present value, H₀, from the parameters derived from the CMB data and the early model. The late Universe form of ΛCDM makes use of different ansatzes than at early times including descriptions of dark matter (no interactions, stable, cold) and DE (as a cosmological constant). Again, some of these are tested but not to the precision with which they are relied upon in the model. For this reason the Hubble constant tension can identify a failure of the standard ΛCDM scenario at early or late epochs.

First we review the status of H₀ predictions from a variety of CMB experiments beginning with Planck which is the de-facto 'gold standard' experiment. The most widely cited prediction from Planck in a flat ΛCDM model for the Hubble constant is H₀ = 67.27 ± 0.60 km s⁻¹ Mpc⁻¹ at 68% confidence level (CL) for Planck 2018 [11], while it is H₀ = 67.36 ± 0.54 km s⁻¹ Mpc⁻¹ at 68% CL for Planck 2018 + CMB lensing [11], i.e. with the inclusion of the four-point correlation function or trispectrum data. ¹⁴ The previous CMB satellite experiment Wilkinson microwave anisotropy probe (WMAP) [23], in its nine-year data release, assuming the same ΛCDM model, preferred a value for the Hubble constant H₀ = 70.0 ± 2.2 km s⁻¹ Mpc⁻¹ at 68% CL, a value that can be in agreement with both Planck and R20 because of its very large error bars. This conclusion used to apply to another CMB experiment from the ground, South Pole Telescope (SPTPol) [24], that reports a value of H₀ = 71.3 ± 2.1 km s⁻¹ Mpc⁻¹ at 68% CL, considering the full datasets in TE and EE. However, the result from SPT-3G [25] improves from those in reference [24] and leads to a value of H₀ = 68.8 ± 1.5 km s⁻¹ Mpc⁻¹ at 68% CL. The recent SPTPol result is competitive with those from other ground-based experiments such as the combination of the Atacama Cosmology Telescope (ACT), a ground based telescope, and WMAP. Indeed, the combination of ACT (from ℓ = 600 in TT and ℓ = 350 in TE/EE) and WMAP data, with a Gaussian prior on τ instead of the low-ℓ polarization likelihood, results in H₀ = 67.6 ± 1.1 km s⁻¹ Mpc⁻¹ at 68% CL [26], always assuming a ΛCDM model, or H₀ = 67.9 ± 1.5 km s⁻¹ Mpc⁻¹ at 68% CL for ACT alone. Finally, a combination of ground based CMB experiments SPT, SPTPol, and the Atacama Cosmology Telescope polarimeter (ACTPol) gives H₀ = 69.72 ± 1.63 km s⁻¹ Mpc⁻¹ at 68% CL [27], while SPTPol + ACTPol, when combined with the Planck dataset, gives H₀ = 67.49 ± 0.53 km s⁻¹ Mpc⁻¹ at 68% CL [28].

We may also consider less precise constraints that arise exclusively from measurements of the polarization of the CMB, i.e. from the EE CMB power spectra [29], always assuming a ΛCDM model: Planck EE gives H₀ = 70.0 ± 2.7 km s⁻¹ Mpc⁻¹ at 68% CL, ACTPol ${H}_{0}=72.{4}_{-4.8}^{+3.9}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, and SPTPol ${H}_{0}=73.{1}_{-3.9}^{+3.3}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, but their combination finds H₀ = 68.7 ± 1.3 km s⁻¹ Mpc⁻¹ at 68% CL for the different directions of correlations [29].

Measurements of baryon acoustic oscillations (BAO) (or other features in Galaxy power spectra) at any redshift are 'scale-free', primarily constraining the product of the sound horizon and the H₀ value, but neither without a prior on the other. When the prior comes from the CMB, or baryon abundance estimates, the determination of H₀ depends on the above ansatz at z > 1000 and we will consider the result as belonging to the early or indirect class. As such, there are H₀ estimates from a reanalysis of the Baryon Oscillation Spectroscopic Survey (BOSS) data release 12 (DR12) on anisotropic Galaxy clustering in Fourier space [30], that provide H₀ = 67.9 ± 1.1 km s⁻¹ Mpc⁻¹ at 68% CL using a prior on the physical baryon density ω_b, derived from measurements of primordial deuterium abundance [22] (D/H = (2.527 ± 0.030) × 10⁻⁵) assuming the standard big bang nucleosynthesis (BBN) picture, and a ΛCDM model with a total neutrino mass free to vary in a small CMB-motivated range and a fixed primordial power spectrum (PPS) tilt n_s to the Planck best-fit. The same lower H₀ is confirmed also from a reanalysis of the BOSS DR12 data using the effective field theory (EFT) of large-scale structure (EFTofLSS) formalism [31], predicting the clustering of cosmological large-scale structure in the mildly non-linear regime, that results in H₀ = 68.5 ± 2.2 km s⁻¹ Mpc⁻¹ at 68% CL, always assuming BBN, and fixing the values of the baryon/dark-matter ratio, Ω_b/Ω_c, and n_s to the Planck 2018 best-fit. A companion paper [32] gives instead H₀ = 68.7 ± 1.5 km s⁻¹ Mpc⁻¹ at 68% CL, assuming a BBN prior on Ω_b h² instead of Ω_b/Ω_c. In addition, the combination of BAO from main Galaxy sample (MGS) [33], BOSS Galaxy and extended BOSS (eBOSS), with the BBN prior independent from the CMB anisotropies, provides H₀ = 67.35 ± 0.97 km s⁻¹ Mpc⁻¹ at 68% CL in a ΛCDM scenario [34]. Moreover, a lower Hubble constant H₀ = 68.19 ± 0.36 km s⁻¹ Mpc⁻¹ at 68% CL [34] is also obtained within the ΛCDM scheme when combining together Planck 2018, the Pantheon sample [35] of 1048 type Ia supernovae (SNIa), Sloan Digital Sky Survey (SDSS) BAO + redshift space distortions (RSD), and the Dark Energy Survey (DES) 3 × 2 pt data [16, 36, 37]. We have to note here that SNIa data is similar to BAO in that it is scale-free and cannot directly measure H₀ nor is early or late until its luminosity is calibrated at one end or the other. These lower Hubble constant values are in agreement with previous estimates, when other BAO data [38–40] were included in the dataset combinations (see also references [41–46]). For a flat ΛCDM model, the combination of WMAP + BAO (6dF Galaxy Survey, MGS, the BOSS DR12 galaxies and the eBOSS DR14 quasars) also gives a lower value ${H}_{0}=68.3{6}_{-0.52}^{+0.53}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [47]. Lastly, a combination of Galaxy cluster sparsity, cluster gas mass fraction and BAO gives H₀ = 69.6 ± 1.7 km s⁻¹ Mpc⁻¹ at 68% CL [48].

By combining the unreconstructed BOSS DR12 Galaxy power spectra P_ℓ (k), modeled using the EFTofLSS, assuming a weak Gaussian prior on the amplitude of the scalar PPS A_s centered on the Planck best-fit, and a Ω_m prior from Pantheon, reference [49] finds ${H}_{0}=65.{1}_{-5.4}^{+3.0}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL. In addition, the same analysis is performed with a Ω_m prior from uncalibrated BAO (6dFGS, MGS, and eBOSS DR14 Lyman-α measurements) giving ${H}_{0}=65.{6}_{-5.5}^{+3.4}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [49]. Finally, considering the combination of P_ℓ (k) with the Planck 2018 CMB-marginalized lensing likelihood [50], and a prior on A_s twice tighter than before, reference [49] obtains ${H}_{0}=70.{6}_{-5.0}^{+3.7}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL. This result is shifted and slightly stronger (for the addition of Galaxy information) with respect to another sound horizon independent measurement as obtained in reference [51], that, analysing the same CMB lensing data from Planck, using conservative external priors on Ω_m from Pantheon and A_s from Planck 2018, and varying the total neutrino mass, finds H₀ = 73.5 ± 5.3 km s⁻¹ Mpc⁻¹ at 68% CL. Finally, for the combination P_ℓ (k) + BAO + BBN, reference [52] finds H₀ = 68.6 ± 1.1 km s⁻¹ Mpc⁻¹ at 68% CL within a ΛCDM model plus a total neutrino mass free to vary, using a prior on the physical baryon density ω_b but neglecting any knowledge on the power spectrum tilt n_s.

Using the latest BAO data, including the eBOSS DR16 measurements [34], and a prior on Ω_m h² based on the Planck 2018 best fit in a ΛCDM model, reference [53] finds H₀ = 69.6 ± 1.8 km s⁻¹ Mpc⁻¹ at 68% CL. Considering Pantheon SNIa apparent magnitude + DES-3yr binned SNIa apparent magnitude + H(z) + BAO in reference [54] the authors find H₀ = 68.8 ± 1.8 km s⁻¹ Mpc⁻¹ at 68% CL. In reference [55] the authors apply the inverse distance ladder to fit a parametric form of H(z) to BAO and SNIa data, using priors on the sound horizon at the drag epoch r_d from Planck, obtaining H₀ = 68.42 ± 0.88 km s⁻¹ Mpc⁻¹ at 68% CL, and from WMAP, obtaining H₀ = 67.9 ± 1.0 km s⁻¹ Mpc⁻¹ at 68% CL.

It may be worth noting that early inferences of H₀ tend to increase (rather than decrease) from the baseline value derived from the Planck 2018 temperature anisotropy data with the inclusion of polarization data, BAO data, or additional freedom in ΛCDM (see figure 1).

2.1.1. CMB—systematics in Planck?

The best-established and only strictly empirical method to measure H₀ locally comes from measuring the distance–redshift relation, usually undertaken by building a 'distance ladder'. The most often utilized approach is to use geometry (e.g. parallax) to calibrate the luminosities of specific star types (e.g. pulsating Cepheid variables and exploding type Ia supernovae or SNIa) which can be seen at great distances where their redshifts measure cosmic expansion. Cepheids are most often used to reach distances of 10–40 Mpc because they are the brightest objects in the optical with luminosities reaching in excess of 100 000 solar luminosities and offer the highest precision per object of about 3% in distance at a given pulsation period. ¹⁵ SNIa exceed a billion solar luminosities and are nearly as precise per object but they are rare in any volume, such as the local one, thus often serve as the last rung on the distance ladder. These methods treat stars as empirical, standardized candles, i.e. the premise that once empirically standardized, the same type has the same luminosity, without reference to stellar modeling or astrophysics theory. One may consider the failure of this premise to be anti-Copernican and harder to imagine than a failure of ΛCDM!

The Hubble Space Telescope (HST) provided the first capability to measure Cepheids beyond a few Mpc to reach the nearest SNIa hosts (and the hosts of other long-range distance indicators) and the final result of the HST Key Project was (72 ± 8) km s⁻¹ Mpc⁻¹ [61], a result later recalibrated to use improved geometric distance calibration to the large magellanic cloud (LMC) to yield (74.3 ± 2.2) km s⁻¹ Mpc⁻¹ [62], see also reference [63]. However, these efforts were severely limited by the reach of the first generation of Hubble instruments to observing Cepheids in the hosts of just a few well-observed, well-standardizable SNIa.

The SH0ES Project started in 2005 and advanced this approach by

• (a)  
  Increasing the sample of high quality calibrations of SNIa by Cepheids from a few to 19 (R16) [64],
• (b)  
  Increasing the number of independent geometric calibrations of Cepheids to five (R18) [65] including by extending the range of parallax measurements to Cepheids using spatial scanning of HST,
• (c)  
  Measuring the fluxes of Cepheids with geometric distance measurements and those in supernova hosts with the same instrument to negate calibration errors (R19) [66],
• (d)  
  Measuring Cepheids in the near-infrared to reduce systematics related to dust and reddening laws.

Improved geometric distance estimates to the LMC using detached eclipsing binaries [67], to NGC 4258 using water masers [68] and to Milky Way Cepheids from European Space Agency (ESA) Gaia parallaxes [69] have greatly advanced this work in recent years. The values of H₀ by this route have ranged between 73–74 km s⁻¹ Mpc⁻¹, with the present status based on the improved ESA Gaia mission early data release 3 (EDR3) of parallax measurements using 75 Milky Way Cepheids with HST photometry and EDR3 parallaxes [70], that gives H₀ = 73.2 ± 1.3 km s⁻¹ Mpc⁻¹ at 68% CL [2], in tension at 4.2σ with the Planck value in a ΛCDM scenario. We will refer to this new measurement as R20 and this will be a reference throughout the review. This value is also close to the conservative average (excludes R20) and optimistic average (includes R20) we present later in this section so this is a reasonable overall benchmark.

There have been numerous reanalyses of the SH0ES data using different formalisms, statistical methods of inference, or replacement of parts of the dataset, but none has produced a significant indication of a change in H₀. The larger value of H₀ is seen in the reanalysis of the R16 Cepheid data by using Bayesian hyper-parameters [71] H₀ = 73.75 ± 2.11 km s⁻¹ Mpc⁻¹ at 68% CL, and the local determination of the Hubble constant [72] achieved using the cosmographic expansion of the luminosity distance, that gives H₀ = 75.35 ± 1.68 km s⁻¹ Mpc⁻¹ at 68% CL. There is a measurement obtained replacing the sample of SNIa measured in the optical with that measured in the near-infrared (NIR) where SNIa are better standard candles [73], i.e. H₀ = 72.8 ± 1.6(stat) ± 2.7(sys) m s⁻¹ Mpc⁻¹ at 68% CL. Other measurements based on the Cepheids–SNIa include reference [74], that finds H₀ = 73.2 ± 2.3 km s⁻¹ Mpc⁻¹ at 68% CL, analysing the final data release of the Carnegie Supernova Project I and a different method for standardizing SNIa light curves. A number of reanalyses including a notable one that leaves the reddening laws in distant galaxies uninformed by the Milky Way is performed in reference [75], that finds H₀ = 73.3 ± 1.7 km s⁻¹ Mpc⁻¹ at 68% CL. These are in agreement with R16, showing that systematic bias or uncertainty in the Cepheid calibration step of the distance ladder measurement cannot explain the Hubble tension. Reference [76] produces an estimate of the Hubble constant based on a Bayesian hierarchical model of the local distance ladder, that gives H₀ = 73.15 ± 1.78 km s⁻¹ Mpc⁻¹ at 68% CL, allowing outliers to be modeled. These measurements generally made use of the Cepheid photometry presented by the SH0ES Team. However, the previously cited result for H₀ of 74.3 ± 2.2 km s⁻¹ Mpc⁻¹ from reference [62] used an independent set of Cepheid data from that of the SH0ES Team, obtained with different instruments on HST, and with photometry measured with different algorithms (and by different investigators) which removes the dependence of the tension on any one set of Cepheid measurements. Similarly, reference [77] has undertaken a complete reanalysis of SH0ES Cepheid measurements starting at the pixel level from the HST data and using different methods for measuring Cepheid photometry, correcting for bias, developing new Cepheid light curve templates, etc, and the result agreed with the prior SH0ES analysis in R16 to 0.5σ or 0.02 mag (1% in distance) indicating that the measurements are robust.

Using the Gaia Data Release 2 parallaxes [78] of Cepheid companions (in binaries or host clusters rather than of the Cepheids themselves) to obtain a Galactic calibration of the Leavitt law in the V, J, H, K_S, and Wesenheit W_H bands, it is possible to derive a Hubble constant measurement anchored to Milky Way Cepheids. When all Cepheid companions are considered, the authors in reference [79] obtain H₀ = 72.8 ± 1.9(stat + sys) ± 1.9(parallax zero-point) km s⁻¹ Mpc⁻¹ at 68% CL.

There have been alternative distance ladders which substitute another type of star for Cepheids. There are such measurements obtained using the tip of the red giant branch (TRGB) in lieu of Cepheids, performed by different teams, and these are in the range of ∼70–72 km s⁻¹ Mpc⁻¹. We have the 2017 measurement of the Hubble constant based on the calibration of the SNIa using the TRGB obtained by reference [80], that is H₀ = 71.17 ± 1.66(random) ± 1.87(sys) km s⁻¹ Mpc⁻¹ at 68% CL. There is the 2019 determination made by reference [81] which measures TRGB in a nine SNIa hosts, adds 5 from [80], and calibrates TRGB in the LMC which yields H₀ = 69.8 ± 0.8(stat) ± 1.7(sys) km s⁻¹ Mpc⁻¹ at 68% CL and reference [82] (F20), for which H₀ = 69.6 ± 0.8(stat) ± 1.7(sys) km s⁻¹ Mpc⁻¹ at 68% CL, or the same but with a different accounting of the LMC extinction of the TRGB using reddening maps derived from red clump stars by [83] gives H₀ = 72.4 ± 2.0 km s⁻¹ Mpc⁻¹ at 68% CL. A value of H₀ ∼ 72 km s⁻¹ Mpc⁻¹ also results from the revised OGLE Team LMC reddening maps [84, 85]. The addition of two new TRGB measurements in NGC 1404 and NGC 5643, host to 4 SNIa [86] appears to raise the F20 value of H₀ by ∼1% to ∼70 km s⁻¹ Mpc⁻¹ but the revised value is not tabulated. Even the lower mean value from F20 from the higher LMC extinction gives H₀ measurements in agreement with both Planck and R20 estimates within 95% CL, and therefore cannot discriminate between the two. Furthermore, if the luminosity of SNIa is calibrated with the TRGB luminosity, that is, calibrated with the Gaia EDR3 trigonometric parallax of Omega Centauri, in reference [85] is obtained the Hubble constant H₀ = 72.1 ± 2.0 km s⁻¹ Mpc⁻¹ at 68% CL. Another determination of H₀ using velocities and TRGB distances to 33 galaxies located between the local group and the Virgo cluster is given by reference [87] and it is equal to H₀ = 65.9 ± 3.5(stat) ± 2.4(sys) km s⁻¹ Mpc⁻¹ at 68% CL, i.e. in agreement with both Planck and R20 within 2σ.

An alternative to either Cepheids or TRGB is MIRAS (variable red giant stars) [88]. These stars come from older stellar populations than Cepheid variables and have been calibrated directly in the maser host, NGC 4258 and used to calibrate SNIa in the host NGC 1559, to yield H₀ = 73.3 ± 4.0 km s⁻¹ Mpc⁻¹ at 68% CL.

There has been some discussion of whether the SNIa used at either ends of the distance ladder are consistent because of the possibility of differences in the SNIa environments and related impact on their luminosity (references [89–91]). Such differences will depend on the specific samples used to measure H₀. In reference [92] the authors analysed the residual, host dependencies on the sample used by the SH0ES Team and found expectable deviations in H₀ at the level of 0.3% and thus which do not appear to encompass a large fraction of the difference.

There are also distance ladders which substitute SNIa for another long range indicator calibrated by Cepheids and TRGB such as the use of the surface brightness fluctuations (SBF) method, which gives H₀ = 70.50 ± 2.37(stat) ± 3.38(sys) km s⁻¹ Mpc⁻¹ at 68% CL [93] from legacy SBF data and H₀ = 73.3 ± 0.7 ± 2.4 km s⁻¹ Mpc⁻¹ at 68% CL [94] from a new sample of NIR data from HST. Moreover, in reference [94] a reanalysis of the result obtained by reference [93] is performed, improving the LMC distance, and finding H₀ = 71.1 ± 2.4(stat) ± 3.4(sys) km s⁻¹ Mpc⁻¹ at 68% CL. Likewise is the use of Tully–Fisher relation, i.e. on the correlation between the rotation rate of spiral galaxies and their absolute luminosity, used to measure the distances after calibration from TRGB and Cepheids. Considering the optical and the infrared bands, reference [95] finds H₀ = 76.0 ± 1.1(stat) ± 2.3(sys) km s⁻¹ Mpc⁻¹ at 68% CL, while using the baryonic Tully–Fisher relation, reference [96] finds H₀ = 75.1 ± 2.3(stat) ± 1.5(sys) km s⁻¹ Mpc⁻¹ at 68% CL. Lastly, the authors of reference [97] have presented another measurement of H₀ independent of SNIa using type II supernovae (SN II) as standardisable candles, providing the result ${H}_{0}=75.{8}_{-4.9}^{+5.2}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL. A further Hubble constant determination is given in reference [98], that uses as a standard candle the relation between the integrated Hβ line luminosity and the velocity dispersion of the ionized gas of HII galaxies and giant HII regions, finding H₀ = 71.0 ± 2.8(random) ± 2.1(sys) km s⁻¹ Mpc⁻¹ at 68% CL.

Finally, the Megamaser Cosmology Project (MCP) [99] measures the Hubble constant using geometric distance measurements to six megamaser-hosting galaxies. This approach avoids any distance ladder (i.e. multiple objects) by providing geometric distance directly into the Hubble flow and finds H₀ = 73.9 ± 3.0 km s⁻¹ Mpc⁻¹ at 68% CL for maser host redshifts in the CMB rest frame, and a value of a few higher or lower for different methods of mapping peculiar velocities. The use of the 2M++ peculiar velocity maps in particular gives a value that is lower than this by ∼2–3 km s⁻¹ Mpc⁻¹ [100].

The above methods have been fully or largely empirical and we may view these as being largely independent of astrophysical modeling other than the assumptions of a FLRW metric for computing distances. Although the systematic uncertainty of the distance ladder measurement has also been debated, recent surveys including various H₀ measurements robustly conclude that the discrepancy in the value of H₀ between early- and late-Universe observations ranges between 4σ and 6σ [101–103]. The distance ladder method also seems to be insensitive to the choice of the cosmology underlying Cepheids calibration [75]. Now we consider late Universe approaches to measuring H₀ with some dependence on astrophysical modeling problems, though the models are not the same as ΛCDM.

2.2.1. (Astrophysical) model-dependent

2.2.2. Standard sirens

2.2.3. Systematics

An underdense local Universe, corresponding to the simplest possibility for solving the Hubble constant tension for a sample-variance effect, has been definitely ruled out, because empirical and theoretical estimates of such fluctuations are a factor of ∼20 too small. Such a void would need to extend to z > 0.5 or higher to not be apparent in the Hubble diagram of SNIa or BAO measurements. Considering a large-volume cosmological N-body simulation ¹⁹ to model the local measurements and to quantify the variance due to local density fluctuations and inhomogeneous selection of SNIa, in reference [155] it has been found that the extreme underdensity required for such a void is very unlikely to exist in the LSS fluctuations of a ΛCDM Universe aside from the conflict with the observations. In reference [156] the evidence in the Hubble diagram of large scale outflows caused by local voids has been studied, finding that the SNIa luminosity distance–redshift relation is in disagreement at 4–5σ with large local underdensities that can explain the Hubble tension. These findings agree with reference [157], that concludes that a large local void alone is a very unlikely explanation, and with reference [158], where the void matter distribution is described by an inhomogeneous but isotropic Lemaître–Tolman–Bondi metric.

Previous work has questioned the isotropy of the expansion of the Universe by estimating the anisotropy in the Hubble constant from SNIa data [159–161] and from large samples of galaxies and clusters [162, 163], coming to diverse conclusions regarding the level of anisotropy. When the Pantheon dataset is analysed, a non-zero anisotropy is found which is mostly due to the non-uniform angular distribution of SNIa in the sample [164].

In references [165, 166], a consistent analysis that does not take into account an underlying FLRW metric has computed the luminosity distance cosmography for a general spacetime under a minimal set of assumptions. This is achieved by a series expansion of the luminosity distance for a general spacetime with no assumptions on the metric tensor and allows to relax the assumptions of an isotropic expansion rate. In this metric-free analysis, the effective deceleration parameter can be negative without the need for a cosmological constant. A direct testing of the geometric assumptions for the FLRW metric using this method has yet to be carried out. This framework has been recently tested against cosmological numerical simulations [167–169].

A different type of inhomogeneity relates to the non-linear time evolution in GR. Local inhomogeneities could drive a portion of volume away from an initial 'background' FLRW model, which would serve as an approximation to the actual spacetime metric. How well the FLRW metric approximates the actual lumpy spacetime metric, the 'fitting problem', was first discussed in references [170, 171]. Inhomogeneities back-react on the large scale metric to produce an effective stress–energy tensor that adds up to the large scale stress–energy tensor. Different studies that attempt to assess the magnitude of such a backreaction of local structure on large scale cosmological dynamics reach conflicting results [172, 173], with the discrepancy being partly due to the differences in the quantification of backreaction in the different schemes [174]. Various frameworks for investigating the fitting problem have been proposed, see e.g. references [175–180], including the Buchert's scheme [181–184] which is treated in relation to the Hubble tension in section 14.3.

In the following sections we will briefly review some of the most discussed models in the literature. Before going through all the possibilities, a word of caution is mandatory here: the solution to the Hubble constant tension can introduce a further disagreement with the BAO data, or the so-called 'sound horizon problem'.

The Hubble constant value is estimated from the CMB data, assuming a model, in three passages:

• (a)  
  From the measurements of the baryon density and the matter density, derivation of the sound horizon at the CMB last-scattering ${r}_{\mathrm{s}}^{{\ast}}$ at redshift z_*,
• (b)  
  From the position of the CMB acoustic peaks, derivation of the comoving angular diameter distance to last scattering ${D}_{\mathrm{A}}^{{\ast}}={r}_{\mathrm{s}}^{{\ast}}/{\theta }_{\mathrm{s}}^{{\ast}}$,
• (c)  
  From ${D}_{\mathrm{A}}^{{\ast}}={\int }_{0}^{{z}_{{\ast}}}\mathrm{d}z/H(z)$, a derivation of H(z) is available for all the redshifts z.

BAO data can also provide a measurement of the Hubble constant, since these measurements constrain the product Hr_d. ²⁰ This implies that in order to be in agreement with the CMB, which requires a low value of the Hubble constant value, the BAO constraints on the sound horizon at the baryon drag epoch lie on the high allowed region, i.e. around 147 Mpc. Contrarily, to be in agreement with R20, BAO data prefer a lower value for the sound horizon, i.e. around 137 Mpc. Therefore, to reach an agreement among all the datasets, both a larger H₀ value and a lower sound horizon are needed from the CMB assuming a specific model, see reference [186].

In reference [187] it has been argued that late time DE modifications of the expansion history are slightly disfavoured. Instead, in a pre-CMB decoupling scenario, an extra DE component can better solve the H₀ tension. The same thing happens if modified gravity modifications are accounted for, see e.g. reference [188].

Following this direction, guidance to model building can instead be found in reference [189]. If different solutions are divided into post-recombination and pre-recombination solutions of the Hubble tension, the post-recombination modifications of the expansion history, such as the wCDM model where the DE equation of state is free to vary (see section 5.1), do not change the sound horizon, therefore they are unlikely to be a possible direction for fitting all the datasets. More promising are instead the pre-recombination solutions, as extra radiation at recombination as parameterized by N_eff or an early DE component, since these non-standard cosmologies can increase H₀ while reducing r_s. Unfortunately, these solutions are unable to solve completely the H₀ tension with R20 [190].

Many modifications to the ΛCDM model have been proposed in order to solve the Hubble constant tension, focusing on the scenarios that can reduce the sound horizon r_s at recombination. Nevertheless, it has been pointed out in a recent article [191] that models which only reduce r_s can never fully resolve the Hubble constant tension, if they are expected to be in agreement at the same time with the other cosmological datasets, such as BAO or weak lensing observations. For this very same reason different proposed models in the literature are often classified as either early or late time modifications of the expansion history, in order to take into account the sound horizon problem appearing when BAO data are considered [189, 190]. ²¹

'Late time solutions' of the Hubble constant tension refer to the modifications of the expansion history after recombination, that increase the H₀ value leaving the sound horizon unaltered. These late solutions are well-known for solving successfully the Hubble constant tension, but being in disagreement with the BAO + Pantheon data [189, 190]. In the following sections we shall present some of the most studied models in the literature belonging to this class of solutions.

We offer a brief comment that some local determinations of H₀ and constraints on H(z) that use SNIa (e.g. from SH0ES and Pantheon SNIa) have covariance, sharing SNIa and light curve parameters which define the Hubble expansion at 0.02 < z < 0.15 and that it is not strictly valid to use both constraints simultaneously and independently without proper account of their interdependence [60, 194]. This is likely to have consequences particularly for late-time solutions that allow for a sudden or rapid change in H(z) at z < 0.1 which would impact both constraints. There are two approaches that may be used in principle to account for the covariance. One may use an inverse distance ladder starting in the early Universe to calibrate SNIa in the Hubble flow (in the context of any cosmological model to predict H(z)) and thus predict the absolute peak magnitude M_B of SNIa needed to match its empirical calibration from the local distance ladder. However, we caution that the value of M_B derived is specific to a SNIa light curve fitting formalism and therefore it is crucial to measure M_B consistently and to account for the covariance of SNIa data in both the local and Hubble flow samples. Alternatively one may use the SNIa distance ladder to directly calibrate Hubble flow SNe so that their constraining power and covariance are fully contained in the SNIa sample, i.e. a single set of distances, redshifts and their covariance which may then be used to constrain a cosmological or cosmographic model as done in [60]. This approach will be formally included in a future SH0ES + Pantheon data release. A good approximation to this latter approach (neglecting only the SNIa–SNIa data covariance) is to (i) subtract from Pantheon distance moduli the quantity 5 log₁₀(H₀/70.0) in magnitudes, where e.g. H₀ = 73.2 km s⁻¹ Mpc⁻¹ [2] as 70.0 was the Pantheon reference; (ii) include covariance between every SN, namely a coherent 1.7% (the uncertainty on H₀ from the calibration procedure only), which corresponds to a magnitude of 0.037. This later step adds a fixed quantity (0.037)² to the covariance matrix of errors which is already provided by the Pantheon collaboration. Here we note that the benchmark local H₀ determination from R21 uses a value of q₀ = −0.55 derived from Pantheon, so this approximation is not strictly combining independent information, but any non-pathological alternative expansion of H(z) consistent with either BAO, SNIa or CMB + ΛCDM would affect H₀ at the ⩽1% level.

'Early time solutions', instead, modify the expansion history before the recombination period, changing both H₀ and r_s in the appropriate direction to solve the Hubble tension and the sound horizon problem simultaneously. Namely, a lower value of the sound horizon r_s is needed to allow H₀ to be in agreement with R20 and BAO + Pantheon at the same time. This can be achieved by increasing the expansion rate H(z) before decoupling by, for instance, allowing an energy injection around the recombination epoch [195, 196]. This class of early time solutions is known to be able to alleviate, but not to solve, the H₀ tension below the 3σ significance [190, 197].

Finally, while in reference [198] the authors explored a set of 7 assumptions that a model needs to break in order to alleviate the Hubble tension, in reference [199] the authors propose the use of new cosmic triangle plots to simultaneously represent independent constraints on key quantities related to the Hubble parameter (t_U, r_s, and Ω_m) useful to find its solution.

An injection of energy at early times (approximately at z ≳ 3000), where the DE component behaves like a cosmological constant and then dilutes away as radiation, has been shown to be an effective possibility for reducing the H₀ tension. For example, the authors in reference [212] proposed a physical EDE model based on a scalar field ϕ with a potential having an oscillating feature of the form [213]:

$$\begin{equation}V(\phi )\propto {\left[1-\mathrm{cos}\left(\phi /f\right)\right]}^{n},\end{equation} \tag{ 1 }$$

where f is an unknown energy scale and n > 0. At early times, the scalar field is frozen and behaves like a cosmological constant until it starts to oscillate at a critical redshift z_c, after which it behaves as a fluid with an equation of state w_n = (n − 1)/(n + 1) [214]. The energy density parameter and the equation of state of the scalar field as a function of the scale factor ${a}_{\mathrm{c}}\equiv {(1+{z}_{\mathrm{c}})}^{-1}$ at which the transition occurs are, respectively [215]:

$$\begin{equation}{{\Omega}}_{\phi }(a)=\frac{2{{\Omega}}_{\phi }({a}_{\mathrm{c}})}{{\left(a/{a}_{\mathrm{c}}\right)}^{3(1+{w}_{n})}+1},\end{equation} \tag{ 2 }$$

$$\begin{equation}{w}_{\phi }(a)=-1+\frac{1+{w}_{n}}{1+{({a}_{\mathrm{c}}/a)}^{3(1+{w}_{n})}}.\end{equation} \tag{ 3 }$$

At early times a → 0, the scalar field behaves as a cosmological constant with the equation of state w_ϕ (a) → −1, while for a ≫ a_c we have w_ϕ (a) → w_n . Hence, the energy density is constant at early times, and decays as ${a}^{-3(1+{w}_{n})}$ when the scalar field becomes dynamical [216]. The EDE component dilutes like matter (w_n = 0) for n = 1, like radiation (w_n = 1/3) for n = 2, and faster than radiation for n ⩾ 3; for n → ∞, the scalar field behaves like a stiff fluid with the equation of state w_n → 1, and corresponds to a scalar 'kination' field [217] whose energy density is dominated by its kinetic term and dilutes as a⁻⁶.

The authors of reference [212] showed that n = 3 is the solution preferred by the data, and Planck 2015 + CMB lensing + BAO + Pantheon + R18 gives H₀ = 70.6 ± 1.3 km s⁻¹ Mpc⁻¹ at 68% CL, solving the Hubble tension within 2σ. We should stress here that this result includes the R18 prior on the Hubble constant.

Extremely light pseudoscalar particles known as 'axions' can arise from various mechanisms such as the breaking of 'accidental' symmetries [218, 219] or from manifold compactification within string theory [220–224]. We discuss the QCD axion in section 7.3, while for now we consider an axion-like field ϕ of mass m that does not necessarily relate to QCD. Axion-like particles can explain the DM observed [225, 226] and, at a different mass scale, they are a candidate for DE [227].

Reference [228] attempts to alleviate the Hubble tension by considering sub-dominant oscillating scalar field moving under a potential inspired by the one that generically arises in string theory for an axion-like field:

$$\begin{equation}V(\phi )={m}^{2}{f}^{2}{\left[1-\mathrm{cos}\left(\phi /f\right)\right]}^{n},\end{equation} \tag{ 4 }$$

where f is an energy scale. The axion-like potential is recovered for the case n = 1. A fit to the Planck 2015 + CMB lensing + BAO + Pantheon + R19 datasets gives H₀ = 71.49 ± 1.20 km s⁻¹ Mpc⁻¹ at 68% CL for n = 3, and ${H}_{0}=71.4{5}_{-1.40}^{+1.10}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL for free n [228], apparently reducing the Hubble tension at one standard deviations. Indeed, as in the previous case, we should stress that the R19 prior is included in the analysis, possibly biasing the final result towards higher H₀ values.

Although the expressions in equations (1)–(4) share a similar dependence on the field ϕ, the results presented in reference [228] differ from those in reference [212] because in the latter an approximate form of the scalar field evolution equations was used, while the authors in reference [228] investigate the scenario by directly solving the linearized scalar field equations without relying on approximations.

An update of these results that considers more recent data is performed in reference [229]. In this case, while the fit of a full combination Planck 2018 + CMB lensing + BAO + RSD + Pantheon + R19 gives H₀ = 70.98 ± 1.05 km s⁻¹ Mpc⁻¹ at 68% CL, in tension at 1.3σ with R20, also including a prior on the Hubble constant, Planck 2018 data alone provides a value of ${H}_{0}=68.2{9}_{-1.00}^{+1.02}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, in disagreement at 2.9σ with R20. The authors therefore conclude that this EDE model, apart from showing a disagreement with all current cosmological datasets, does not solve the H₀ tension. These findings are confirmed by references [230, 231], where additional dataset combinations and model extensions are considered, and in reference [232], where Planck 2018 + CMB lensing + BOSS DR12 gives ${H}_{0}=68.5{4}_{-0.95}^{+0.52}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, with a disagreement at 3.3σ with R20.

A different conclusion is instead reached in reference [233], where the authors revisit the impact of EDE on Galaxy clustering using BOSS Galaxy power spectra, properly analysed adopting the EFTofLSS, and Planck 2018. They found that the conclusions can change with the choice of priors on the EDE parameter space, and that EDE and ΛCDM provide a statistically indistinguishable fits, with almost the same χ², for EFTofLSS + Planck 2018 + SNIa. Unfortunately, a Bayesian model comparison accounting for the numbers of extra parameters in the EDE model is missing. However, in reference [234] the authors analyse the same model, finding for Planck 2018 + CMB lensing + BAO + Pantheon + full shape (FS) of BOSS DR12 ${H}_{0}=67.7{2}_{-1.00}^{+0.42}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, in disagreement with R20 at 3.9σ.

In reference [235], moreover, it has been pointed out that the one-parameter EDE cosmology can solve the tension between Planck and R20 and be favoured by the full dataset combination. In particular, Planck 2018 gives ${H}_{0}=70.1{0}_{-1.6}^{+1.4}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [235], alleviating the tension with R20 at 1.6σ, and Planck 2018 + CMB lensing + BAO + Pantheon + FS of BOSS DR12 + R19 gives ${H}_{0}=71.7{1}_{-0.95}^{+1.0}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [235], in full agreement with R20.

A complementary analysis is performed in [236], that for Planck 2018 TT (up to ℓ = 1000) + SPTPol (TE and EE) + SPT lensing gives H₀ = 70.79 ± 1.41 km s⁻¹ Mpc⁻¹ at 68% CL, solving the tension with R20 within 1.3σ.

Finally, in reference [237] it is argued that a mechanism in which an EDE dumps most of its energy content into radiation in the redshift range z = [3000, 5000] can solve the Hubble tension, and this might be an observational signal of the weak gravity conjecture.

4.2.1. Dissipative axion

4.2.2. Axion interacting with a dilaton

In reference [240], the authors consider an alternative EDE scenario with a potential of the form:

$$\begin{equation}{V}_{n}(\phi )={V}_{0}\frac{{\phi }^{2n}}{{2}^{n}},\end{equation} \tag{ 5 }$$

where V₀ is the amplitude of the potential and n is a power-law index. This potential approximates the anharmonic potential in equation (1) in the limit ϕ/f ≪ 1. A fit to the Planck 2018 TT (up to ℓ = 1000) + SPTPol (TE and EE) + SPTLensing + S₈ prior (from KiDS, VIKING-450 and DES of [241]) + R19 datasets with n = 3 gives H₀ = 73.06 ± 1.26 km s⁻¹ Mpc⁻¹ at 68% CL [240], solving the Hubble tension within 1σ. However, since the derived H₀ value is obtained assuming the R19 prior, it is difficult to properly assess the ability of the model to solve the tension.

Reference [242] considers a scenario in which a scalar field evolves under a potential of the form V ∝ ϕ²ⁿ . Depending on the value of the index n, the scalar field asymptotically evolves to either an oscillatory (rocking) behavior or to a rolling solution with a nearly constant equation of state. The presence of the scalar field injects energy close to recombination, effectively reducing the sound horizon and increasing the Hubble constant value. The potential of the model is parameterized as:

$$\begin{equation}V(\phi )={V}_{0}{\left(\frac{\phi }{{M}_{\text{Pl}}}\right)}^{2n}+{V}_{{\Lambda}},\end{equation} \tag{ 6 }$$

with a constant value of V₀ and V_Λ, and where ${M}_{\text{Pl}}=1/\sqrt{8\pi {G}_{\mathrm{N}}}$ is the reduced Planck mass. Within this model, and for n = 2, the Planck data and the R20 measurement are in better agreement than in the canonical ΛCDM framework, provided a modest tuning to justify the absence of lower orders in the potential [242].

Indeed, for this scenario, Planck 2015 + BAO + Pantheon + R18 data provides the constraint ${H}_{0}=70.{1}_{-1.2}^{+1.0}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [242], apparently reducing the Hubble tension at 1.9σ. However, again, we note the presence of the R18 prior in the analysis. An updated analysis is performed in reference [234], where Planck 2018 + CMB lensing + BAO + Pantheon gives ${H}_{0}=68.5{2}_{-0.89}^{+0.55}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, in disagreement with R20 at 3.3σ.

In reference [243] the authors propose a model in which a first-order phase transition occurs in a dark sector before recombination and avoid to imprint unobserved large-scale anisotropies in the CMB. Such a transition would produces a short phase of new EDE (NEDE) which could address the Hubble tension. Similarly to previously considered mechanisms for ending inflation [244–246], the potential considered involves two scalar fields and it is of the form:

$$\begin{equation}V(\psi ,\phi )=\frac{\lambda }{4}{\psi }^{4}+\frac{1}{2}\beta {M}^{2}{\psi }^{2}-\frac{1}{3}\alpha M{\psi }^{3}+\frac{1}{2}{m}^{2}{\phi }^{2}+\frac{1}{2}\tilde {\lambda }{\phi }^{2}{\psi }^{2},\end{equation} \tag{ 7 }$$

where ψ is the field responsible for the tunneling and ϕ is the trigger field required to modulate the tunneling. The parameters of the potential are subject to the restrictions α² > 4βλ and β > 0. The background field changes from the cosmological constant equation of state w_Λ = −1 to a constant ${w}_{\text{NEDE}}^{{\ast}}$ around the time t_tr. Such a sudden transition can be modeled through the equation of state:

$$\begin{equation}{w}_{\text{NEDE}}(t)=\begin{cases}-1\enspace ,\quad \quad \quad \text{for}\enspace t{\leqslant}{t}_{\text{tr}};\\ {w}_{\text{NEDE}}^{{\ast}}\enspace ,\quad \text{for}\enspace t{ >}{t}_{\text{tr}},\end{cases}\end{equation} \tag{ 8 }$$

where we expect that the NEDE energy density redshifts faster than radiation, as $1/3{\leqslant}{w}_{\text{NEDE}}^{{\ast}}{\leqslant}1$. In the approximation of a sudden transition, the background energy density evolves as:

$$\begin{equation}{\bar{\rho }}_{\text{NEDE}}(t)={\bar{\rho }}_{\text{NEDE}}^{{\ast}}{\left(\frac{a({t}_{{\ast}})}{a(t)}\right)}^{3[1+{w}_{\text{NEDE}}(t)]},\end{equation} \tag{ 9 }$$

with a constant parameter ${\rho }_{\text{NEDE}}^{{\ast}}$.

A fit to Planck 2018 + CMB lensing + BAO + Pantheon data gives ${H}_{0}=69.{6}_{-1.3}^{+1.0}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, reducing the Hubble tension within 2.3σ [247]. Including R19, the Hubble constant becomes H₀ = 71.4 ± 1.0 km s⁻¹ Mpc⁻¹ at 68% CL, with an evidence at 4σ for NEDE, and reducing the tension with R20 at 1.1σ.

Chain EDE proposes an alternative mechanism in which a scalar field tunnels rapidly via a series of first order phase transitions through many (N ≫ 1) successive metastable minima of ever lower energy. This kind of model was previously employed as a mechanism for inflation [248]. Building on this, an alternative model of EDE called chain EDE has been proposed in reference [249] as a solution to the Hubble constant tension. In the model, the Hubble tension could be resolved without inducing large anisotropies in the CMB by invoking N ≳ 10⁴ such phase transitions [249]. However, a full data analysis for this model is currently missing.

In reference [250] the authors propose a phenomenological EDE model with an Anti-de Sitter (AdS) phase around the recombination period as a solution to the Hubble tension. AdS vacua are theoretically important because they naturally emerge within the string theory framework (for late-time AdS see references [251–253]).

This EDE model with an AdS phase will make the energy injection more efficient without spoiling the fit to CMB data. We have w_DE > −1 when the EDE field rolls down to V < 0. Therefore, ρ_ϕ ∼ a^−3(1+w) redshifts very rapidly. In [250] the potential is modeled as:

$$\begin{equation}V(\phi )=\begin{cases}_{0}{\left(\frac{\phi }{{M}_{\text{Pl}}}\right)}^{4}-{V}_{\text{AdS}}\enspace ,\quad \quad \text{for}\enspace \frac{\phi }{{M}_{\text{Pl}}}{\leqslant}{\left(\frac{{V}_{\text{AdS}}}{{V}_{0}}\right)}^{1/4},\\ 0\enspace ,\quad \quad \quad \quad \quad \quad \quad \quad \quad \text{for}\enspace \frac{\phi }{{M}_{\text{Pl}}}{ >}{\left(\frac{{V}_{\text{AdS}}}{{V}_{0}}\right)}^{1/4},\end{cases}\end{equation} \tag{ 10 }$$

where V_AdS is the depth of the AdS well.

While a constraint on H₀ from Planck data alone is missing, Planck 2018 + CMB lensing + BAO + Pantheon + R19 gives ${H}_{0}=72.6{4}_{-0.64}^{+0.57}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [250] solving the tension within one standard deviation. However, the presence of the R19 Gaussian prior in the analysis makes difficult to assess the consistency between the measurements. An extended model considering the temperature of the CMB T₀ free to vary has been studied in reference [254], finding consistent results.

In reference [253] the graduated dark energy model (gDE) is introduced, inspired by reference [255]. A limiting case of the gDE is a sign-switching cosmological constant, that can be appealing from the string theory perspective. Using the Planck information as a BAO data point at redshift z = 1090, and using also SNIa JLA [256] + BAO + CC measurements, the authors in reference [253] argue that this model is in agreement with the local H₀ measurements. However, a complete and robust data analysis considering the perturbations and the full Planck 2018 data is missing to date.

Acoustic dark energy (ADE) has been proposed in reference [257] to alleviate the Hubble tension. The authors consider a general phenomenological model of perturbations in a dark fluid which becomes important around matter-radiation equality. The presence of ADE impacts on the CMB through the gravitational effects on the acoustic oscillations. More concretely, ADE consists of a perfect dark fluid specified by its background equation of state w_ADE(a) and its rest frame sound speed ${c}_{\mathrm{s}}^{2}$. The ADE equation of state changes around the scale factor a = a_c, ranging from w_ADE = −1 to w_f as:

$$\begin{equation}{w}_{\text{ADE}}(a)=-1+\frac{1+{w}_{\mathrm{f}}}{{[1+{({a}_{\mathrm{c}}/a)}^{3(1+{w}_{\mathrm{f}})/p}]}^{p}},\end{equation} \tag{ 11 }$$

where the index p controls the rapidity of the transition, such that small values lead to sharper transitions. For p = 1, the model described in section 4.1 is obtained. In reference [257], the ADE model with p = 1/2 is analysed by fitting against Planck 2015 + CMB lensing + BAO + Pantheon + R19 data, obtaining H₀ = 70.60 ± 0.85 km s⁻¹ Mpc⁻¹ at 68% CL [257] and reducing the Hubble tension to 1.6σ. An updated analysis without the Gaussian prior on the Hubble constant is presented in reference [258], where the combination of Planck 2018 + CMB lensing + ACT + Pantheon + BAO gives ${H}_{0}=68.5{0}_{-0.93}^{+0.55}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, restoring the tension with R20 at the 3.6σ level.

4.9.1. Exponential acoustic dark energy

Acoustic dark energy in which the equation of state has an exponential dependence on the scale factor (eADE) has been explored in reference [259]:

$$\begin{equation}{w}_{\text{eADE}}(a)=-1+{2}^{1-\frac{{a}_{\mathrm{c}}}{2a}},\end{equation} \tag{ 12 }$$

where a_c corresponds to the critical scale factor at which the eADE fluid becomes dominant. The equation of state evolves from the value w = −1 before the transition to w ≈ 1 at present time. The fractional energy density evolves as

$$\begin{equation}{{\Omega}}_{\text{eADE}}(a)=2{f}_{\mathrm{c}}\enspace \frac{{({c}_{\mathrm{s}}^{2}+1)}^{2}-{({w}_{\text{eADE}}(a)+1)}^{2}}{{({c}_{\mathrm{s}}^{2}+1)}^{2}},\end{equation} \tag{ 13 }$$

where f_c is the fractional contribution of eADE at a_c, and c_s is the sound speed. For this model, a fit to Planck 2018 + CMB lensing + BAO + Pantheon + R19 datasets provides the constraint ${H}_{0}=71.6{5}_{-4.40}^{+1.62}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [259], solving within 1σ the tension with R20. However, the analysis already includes a Gaussian prior on the Hubble constant.

In the framework of inflation (see section 11), it is possible to introduce a class of models that possess an attractor point predicting the value of the scalar spectral tilt n_s and the tensor-to-scalar ratio r, independently from the specific functional form of the inflaton potential V(ϕ) [260–262]. An EDE model can also be extended to include α-attractors, with a potential for the EDE scalar field of the form [263]:

$$\begin{equation}V(\phi )={\Lambda}+{V}_{0}\frac{{(1+\beta )}^{2n}\enspace \mathrm{tanh}{\left(\phi /\sqrt{6\alpha }{M}_{\mathrm{P}\mathrm{l}}\right)}^{2p}}{{\left[1+\beta \enspace \mathrm{tanh}\left(\phi /\sqrt{6\alpha }{M}_{\text{Pl}}\right)\right]}^{2n}},\end{equation} \tag{ 14 }$$

where V₀, p, n, α and β are constants. The shape of the potential, away from the plateau and around its minimum, regulates the shape of the energy injection and it is thus crucial to successfully alleviate the Hubble tension. For the choice p = 2 and n = 4, the scalar field oscillates at the bottom of the potential, making this case more similar to the original EDE proposal [212]. For these values of the model parameters, the analysis of Planck 2018 + CMB lensing + BAO + Pantheon + R19 data gives H₀ = 70.9 ± 1.1 km s⁻¹ Mpc⁻¹ at 68% CL, softening the tension with R20 down to the 1.4σ level [263]. Note, however, that this result already incorporates a Gaussian prior in the Hubble constant.

We first consider a model in which the equation of state for the DE component is independent of redshift and generally differs from the cosmological constant value, w_DE(z) ≡ w₀ ≠ −1. This simple extension of ΛCDM is referred to as the wCDM model, where 'w' stands for the equation of state w₀. Here, equation (16) gives:

$$\begin{equation}f(z)={(1+z)}^{3(1+{w}_{0})}.\end{equation} \tag{ 18 }$$

For the case w₀ = −1, the function f(z) is also independent of redshift and the DE component acts as a cosmological constant of density parameter Ω_DE.

A likelihood analysis with Planck 2018 data for this model assumes a constant equation of state for DE, ${w}_{0}=-1.5{8}_{-0.35}^{+0.16}$ at 68% CL and H₀ > 69.9 km s⁻¹ Mpc⁻¹ at 95% CL [11]. Such a wCDM scenario would therefore solve the H₀ tension within two standard deviations. The Hubble constant is in fact almost unconstrained in the wCDM scenario, due to the geometrical degeneracy between w_DE and H₀. Therefore, this scenario can perfectly accommodate a Hubble constant in agreement with R20, at the price of a phantom-like DE equation of state, i.e. w₀ < −1. Such a result implies that the energy density of DE is increasing over time, so that the scale factor of the Universe would reach infinity in a finite time and the Universe would end in a 'big rip' [268]. In addition, the Hamiltonian of the theory could have vacuum instabilities due to negative kinetic terms. Nevertheless, the reader should keep in mind that despite of these many theoretical problems, there exist models with an effective energy density with a phantom-like equation of state which avoid the aforementioned difficulties, see e.g. references [269–273]. This model is however in tension with additional datasets, and considering Planck 2018 + Pantheon + BAO, the Hubble constant will be H₀ = 68.34 ± 0.82 km s⁻¹ Mpc⁻¹ at 68% CL [11], in 3.2σ tension with R20. Other approaches in the literature that explored the ability of a phantom DE to solve the Hubble tension considered a redshift-binned DE model [274], a wCDM model in which w₀ is fixed to some specific values [275], taking into account previously unconsidered systematic effects affecting the SNIa measurements [276], exploiting the H₀ − w₀ degeneracy [277], reanalysing the BOSS DR12 data using the EFTofLSS formalism [278], considering an extreme combination of Hubble measurements [103], and exploring the epoch that possibly sourced the H₀ tension [231].

We now discuss some models in which the equation of state for DE depends on the redshift. Among such models, we first consider the Chevallier–Polarski–Linder parameterization (CPL) [279, 280]:

$$\begin{equation}{w}_{\text{DE}}(a)={w}_{0}+(1-a){w}_{a},\end{equation} \tag{ 19 }$$

where a is the cosmological scale factor normalized to unity today, w₀ is the DE equation of state today, and w_a describes its evolution with time. We refer to this scenario as the w₀ w_a CDM model. For example, if w_a < 0 (w_a > 0), w_DE(a) becomes more negative (positive) as we look backwards in time. Within the CPL parameterization, Planck 2018 provides the constraints ${w}_{0}=-1.2{1}_{-0.60}^{+0.33}$ and w_a < −0.85 at 68% CL [281], ²² and H₀ > 63 km s⁻¹ Mpc⁻¹ at 95% CL, in agreement with R20 within 2σ. However, when additional datasets are considered, Planck 2018 + Pantheon + BAO gives H₀ = 68.35 ± 0.84 km s⁻¹ Mpc⁻¹ at 68% CL [11], in 3.2σ tension with R20.

Other studies that account how the CPL parameterization of the DE equation of state addresses the Hubble tension explore the w₀ w_a CDM model by changing the pivot redshift [283], taking into account unconsidered systematic effects affecting the SNIa [276], exploiting the degeneracy between H₀ and w_DE [277], considering an extreme combination of Hubble measurements [103], demanding a higher power of polarizations with respect to ΛCDM to be in agreement with R19 [284], or showing how this solution worsens the Ω_m − σ₈ growth tension [285].

In order to identify the optimal extension of the minimal ΛCDM model to alleviate the H₀ tension, leading to a better fit to observations, one can allow to vary more than one well-motivated cosmological parameters simultaneously. In other words, one should try a combination of parameters that can ameliorate the Hubble tension without considering only one specific mechanism. Indeed, many assumptions and simplifications made in the six parameter description of the ΛCDM model may not be fully justified, and perhaps could hide some physical aspects essential in the evolution of the Universe. In a multi-parameter space, the biases introduced by the choice of the model are easily avoided [286–290].

To begin with, the authors consider an 11-parameter space model in which the ΛCDM model is augmented by the running of the scalar spectral index α_s, the total neutrino mass Σm_ν , the effective number of relativistic degrees of freedom N_eff (see section 7 for details), a constant DE equation of state w₀, and the A_lens parameter [17]. In this scenario, a fit of the 11-parameter space model to the Planck 2018 data results in ${H}_{0}=7{3}_{-20}^{+10}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, in agreement with R20 within 1σ. When additional data are considered, Planck 2018 + BAO gives H₀ = 67.9 ± 1.7 km s⁻¹ Mpc⁻¹ at 68% CL [289], in 2.5σ tension with R20, and Planck 2018 + Pantheon gives H₀ = 66.9 ± 2.0 km s⁻¹ Mpc⁻¹ at 68% CL [289], in 2.6σ tension.

The ΛCDM model can be further extended by considering, instead, a dynamical DE equation of state w_DE(z), parameterized by the CPL relation in equation (19). This is the same as considering the 11-parameter space model but with a DE equation of state modeled with the CPL relation. In this 12-parameter space, a fit to the Planck 2018 data gives H₀ = 72 ± 20 km s⁻¹ Mpc⁻¹ at 68% CL [289], in agreement with R20 within 1σ. Again, the results prefer a phantom-like DE at more than three standard deviations. When additional data are considered, Planck 2018 + BAO gives ${H}_{0}=64.{8}_{-2.9}^{+2.5}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [289], in 3σ tension with R20, and Planck 2018 + Pantheon gives H₀ = 66.8 ± 2.1 km s⁻¹ Mpc⁻¹ at 68% CL [289], in 2.6σ tension.

Dynamical DE parameterizations with two free parameters have been extensively studied in the literature, see for instance [279, 280, 291–307]. Apart from the most well known dynamical DE prescribed by the CPL parameterization with two free parameters [279, 280, 291], some other two-parameter parameterizations have recently been confronted with the latest Planck 2018 data in reference [281], namely:

• The JBP parameterization of the DE equation of state proposed by Jassal–Bagla–Padmanabhan [295]:
  $$\begin{equation}{w}_{\text{DE}}(a)={w}_{0}+{w}_{a}\enspace a\enspace (1-a),\end{equation} \tag{ 20 }$$
  that, when analysed in light of Planck 2018 measurements, provides ${H}_{0}=8{5}_{-7}^{+13}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, while Planck 2018 + BAO gives ${H}_{0}=67.{4}_{-2.9}^{+1.9}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL.
• The logarithmic DE equation of state parameterization proposed by Efstathiou [292]:
  $$\begin{equation}{w}_{\text{DE}}(a)={w}_{0}-{w}_{a}\enspace \mathrm{ln}\enspace a,\end{equation} \tag{ 21 }$$
  for which the Planck 2018 data analysis results in a value of the Hubble constant ${H}_{0}=8{3}_{-8}^{+15}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, while Planck 2018 + BAO gives H₀ = 64.8 ± 2.1 km s⁻¹ Mpc⁻¹ at 68% CL.
• The BA parameterization proposed by Barboza and Alcaniz [298]:
  $$\begin{equation}{w}_{\text{DE}}(a)={w}_{0}+{w}_{a}\left(\frac{1-a}{2{a}^{2}-2a+1}.\right),\end{equation} \tag{ 22 }$$
  for which the Planck 2018 data analysis results in a value of the Hubble constant ${H}_{0}=8{3}_{-8}^{+15}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, while Planck 2018 + BAO gives ${H}_{0}=65.{2}_{-2.8}^{+2.2}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL.

All of the parameterizations above are in agreement with R20 within 2σ for Planck 2018 only, while for Planck 2018 + BAO are in tension at 2.5σ, 3.4σ and 3.1σ, respectively.

Compared to the dynamical DE parameterizations with two free parameters, there are only a few dynamical DE parameterizations with one free parameter [308, 309]. However, some recent investigations clearly demonstrate that dynamical DE parameterizations with a single parameter are very effective in alleviating the Hubble tension (see reference [309]).

The models considered in [309] are reported in table 1, together with their results on the value of H₀ are all at 68% CL. Note, that w₀ is the present value of the DE equation of state, that means w₀ = w_DE(a = 1).

Table 1. The models considered in reference [309] and the Hubble constant obtained by analysing the Planck 2015 data and its combination with BAO and JLA.

Hubble constant H0
Model	Equation of state	Planck 2015	+BAO +JLA
(i)	wDE(a) = w0 exp(a − 1)	$7{4}_{-7}^{+11}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$	$69.{2}_{-1.0}^{+1.1}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$
(ii)	wDE(a) = w0 a[1 − log(a)]	$8{1}_{-9}^{+12}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$	$69.{0}_{-1.1}^{+1.0}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$
(iii)	wDE(a) = w0 a exp(1 − a),	$8{4}_{-8}^{+10}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$	69.4 ± 1.0 km s−1 Mpc−1
(iv)	wDE(a) = w0 a[1 + sin(1 − a)]	$84.{3}_{-6.5}^{+9.9}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$	$70.0{7}_{-0.94}^{+0.91}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$
(v)	wDE(a) = w0 a[1 + arcsin(1 − a)]	$8{3}_{-7}^{+12}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$	$69.{6}_{-1.2}^{+1.0}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$

All the models considered in reference [309] are in agreement with R20 within 1σ for Planck 2015 only, always at the price of a phantom DE equation of state today, and within 2.6σ for Planck 2015 + BAO + JLA. However, a re-analysis with the most recent Planck 2018 dataset is still missing in the literature.

Another possibility to solve the Hubble constant problem relies on metastable DE models, where the DE energy density can decay or increase depending only on its intrinsic nature and not on the external parameters [310–314]. In the simplest model of metastable DE, the DE energy density evolves as [310, 311]:

$$\begin{equation}\frac{\mathrm{d}{\rho }_{\text{DE}}}{\mathrm{d}t}=-{\Gamma}{\rho }_{\text{DE}},\end{equation} \tag{ 23 }$$

where Γ is a constant decay rate and t denotes cosmic time. The equation of state obtained from equation (23) is:

$$\begin{equation}{w}_{\text{DE}}=-1-\frac{1}{3H}\frac{\mathrm{d}\enspace \mathrm{ln}\enspace {\rho }_{\text{DE}}}{\mathrm{d}t}=-1+\frac{{\Gamma}}{3H}.\end{equation} \tag{ 24 }$$

The fit against Pantheon + BAO data provides ${H}_{0}=75.0{1}_{-5.80}^{+4.71}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL. However, when CMB distance priors from Planck 2018 are included, the Hubble tension is restored at more than 3σ [311].

Reference [314] performs an analysis of this metastable DE model against Planck 2018 (Planck 2018 + BAO + DES + R19) data which leads to a value of the Hubble constant ${H}_{0}=69.{3}_{-3.5}^{+5.9}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (H₀ = 71.94 ± 1.08 km s⁻¹ Mpc⁻¹) at 68% CL, solving the tension with R20 within 1σ. Notice that the alleviation of the tension for Planck 2018 alone is mainly due to the large error bars in H₀.

If a phantom crossing model is accounted for, the Hubble tension can be solved within one standard deviation, without spoiling the agreement with the BAO data [315]. If the DE density is Taylor-expanded around an extremum at scale factor a = a_m as:

$$\begin{equation}{\rho }_{\text{DE}}(a)={\rho }_{0}+{\rho }_{2}{(a-{a}_{\mathrm{m}})}^{2}+{\rho }_{3}{(a-{a}_{\mathrm{m}})}^{3}={\rho }_{0}[1+\alpha {(a-{a}_{\mathrm{m}})}^{2}+\beta {(a-{a}_{\mathrm{m}})}^{3}],\end{equation} \tag{ 25 }$$

where ρ₀, ρ₂, ρ₃ are constants and α ≡ ρ₂/ρ₀, β ≡ ρ₃/ρ₀, the DE equation of state results in:

$$\begin{equation}{w}_{\text{DE}}(a)=-1-\frac{a\enspace [2\alpha (a-{a}_{\mathrm{m}})+3\beta {(a-{a}_{\mathrm{m}})}^{2}]}{3[1+\alpha {(a-{a}_{\mathrm{m}})}^{2}+\beta {(a-{a}_{\mathrm{m}})}^{3}]}.\end{equation} \tag{ 26 }$$

For this particular parameterization, an analysis to Planck 2018 + BAO measurements results in a Hubble constant value of ${H}_{0}=71.{0}_{-3.8}^{+2.9}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [315]. A full dataset combination of Planck 2018 + CMB lensing + BAO + R19 + Pantheon gives instead H₀ = 70.25 ± 0.78 km s⁻¹ Mpc⁻¹ at 68% CL [315], in agreement with R20 at 2σ.

Another possibility for solving the Hubble tension is to consider a late DE transition, in which the equation of state for DE sharply changes from the cosmological constant value w_DE = −1 to a phantom-like value w_DE < −1 at redshift $z\sim \mathcal{O}(0.1)$ [60, 316–318]. Such a transition is referred to as a 'hockey stick' because of the shape of the equation of state w_DE(z).

Starting from the prediction for the Hubble constant ${\tilde {H}}_{0}$ in ΛCDM, a late DE transition leads to the actual Hubble constant ${H}_{0}=(1+\delta ){\tilde {H}}_{0}$ where δ is the fractional change in the Hubble constant. To model this, one considers a DE energy density content ρ_DE(z) that transitions from the cosmological constant value ρ_Λ = Ω_Λ ρ_crit,0 to a phantom-like fluid at redshift z_t. The transition is modulated by a smooth step function f(z) as:

$$\begin{equation}{\rho }_{\text{DE}}(z)={\rho }_{{\Lambda}}\enspace \left[1+f(z)\right];\end{equation} \tag{ 27 }$$

$$\begin{equation}f(z)=\frac{2\delta }{{{\Omega}}_{{\Lambda}}}\frac{S(z)}{S(0)};\end{equation} \tag{ 28 }$$

$$\begin{equation}S(z)=\frac{1}{2}\left[1-\mathrm{tanh}\left(\frac{z-{z}_{\mathrm{t}}}{{\Delta}z}\right)\right],\end{equation} \tag{ 29 }$$

where Δz is the duration of the transition. For z ≫ z_t, the expansion history is indistinguishable from the ΛCDM scenario. In reference [317] it has been shown that the combination Planck 2018 + CMB lensing + BAO + Pantheon + R19 provides H₀ = 72.5 ± 1.85 km s⁻¹ Mpc⁻¹ at 68% CL, solving the Hubble tension within one standard deviation. However, notice that this result incorporates already a Gaussian prior on the Hubble constant corresponding to R19.

In reference [318], a sudden change in the DE equation of state by a quantity Δw is considered as a possible solution to the Hubble tension. The equation of state is modeled as:

$$\begin{equation}{w}_{\text{DE}}(z)=-1+{\Delta}w\enspace {\Theta}({z}_{\mathrm{t}}-z),\end{equation} \tag{ 30 }$$

where Θ is the Heaviside step function.

The possibility that a late phantom transition ever occurred has been recently challenged in reference [319], where it has been shown that a 'hockey stick' DE cannot solve the Hubble crisis because the SNIa absolute magnitude M_B considered to obtain R19 is inconsistent with the M_B necessary to fit SNIa, BAO and CMB data.

However, if a corresponding transition for the SNIa absolute magnitude M is accounted for, as in reference [318]:

$$\begin{equation}M(z)={M}_{\mathrm{C}}+{\Delta}M\enspace {\Theta}({z}_{\mathrm{t}}-z),\end{equation} \tag{ 31 }$$

then the late phantom transition approach is again a viable possibility to address the Hubble tension. However, a full CMB data analysis is currently missing.

The running vacuum model was proposed in references [320, 321] to solve the 'coincidence problem' by using a quantum field theory approach in cosmology, where the vacuum energy density can be derived from a general renormalization group equation whose beta-function takes the form of an adiabatic expansion in powers of the Hubble rate and its time derivatives (see also the explanations in references [322–324] and the analysis in references [325–327]). Therefore, in this model the cosmological constant is assumed to be an affine power-law function of the Hubble rate, Λ = Λ(H). The story of the running vacuum model and related ideas can be found in the reviews [328, 329], while the extensions for a curved spacetime and for a string Universe are carried out in references [330, 331], respectively. Another extension of the ΛCDM model that accounts for this parameterizations are the dynamical quasi-vacuum models (wDVMs), in which the Hubble tension is reduced because of the phantom-like behavior of DE [332]. The analysis of Planck 2015 + CMB lensing 2015 + BAO + R16 provides, indeed, H₀ = 70.95 ± 1.46 km s⁻¹ Mpc⁻¹ at 68% CL [332], solving the Hubble tension at 1.1σ. However, in this result the R16 Gaussian prior on the Hubble constant and the BAO data are both considered.

Another extension named as RRVM of type-II, where the vacuum dynamics is not caused by an interaction between the vacuum and matter sectors, but by the running of the gravitational coupling G, has been studied in reference [333], where Planck 2018 + Pantheon + DES + BAO + RSD + CC + a prior on H₀ from [68] gives ${H}_{0}=70.9{3}_{-0.87}^{+0.93}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, in agreement at 1.4σ with R20, but already including a prior on the Hubble constant.

When a parametric model where a transition in the DE equation of state is accounted for, in order to be consistent with H₀ ∼ 73 km s⁻¹ Mpc⁻¹, the DE component is not yet present until redshifts around z = 2, but its energy density has instead a rapid change between z = 0.5 and z = 2 [334]. This result has been obtained with a model-independent Gaussian regression analysis process using Planck 2015, BAO, Pantheon and R16 data, but a complete analysis with perturbations and the full Planck 2018 data is absent.

In reference [335] the authors assume that the Universe follows a ΛCDM cosmology at higher redshifts (z ⩾ 4), in agreement with the Planck measurements, and reanalyse the low redshift cosmological data in order to reconstruct a Hubble rate H(z) which is in full agreement with R16. Once the energy density for the DE component is computed as a function of redshift without assuming a specific model, they find a local minimum of the DE energy density with a negative value. While this scenario could be ascribed to a negative cosmological constant plus an evolving DE component, the model considered deserves further investigations since these findings compromise its stability. A model which comprises a negative cosmological constant plus a time-evolving quintessence field has been considered in references [251, 252] and tested against BAO surveys and the Pantheon SNIa data, however, a test against the full Planck dataset is still missing.

Bulk viscous models have been proposed to alleviate the H₀ tension. A bulk viscous fluid is characterized by its energy density ρ and a pressure term p which comprises two components, the first being the conventional pressure term p_con = w₀ ρ, where w₀ is a constant equation of state, and the second one being a viscosity component ${p}_{\text{vis}}=-\xi (t){{u}^{\mu }}_{;\mu }$ that depends on the coefficient of bulk viscosity ξ(t) > 0 and on the four-velocity of the fluid u^μ [336]. Therefore, the effective pressure term p takes the form $p={w}_{0}\rho -\xi (\rho ){{u}^{\mu }}_{;\mu }$.

The bulk viscosity can play an effective role in describing the evolution of the Universe in its early and late phases [337] (see the review in reference [338] for more details). For any bulk viscous fluid as described above, its evolution in the FLRW Universe is given by:

$$\begin{equation}\frac{\mathrm{d}\rho }{\mathrm{d}t}+3H(1+{w}_{0})\rho =9{H}^{2}\xi (t),\end{equation} \tag{ 32 }$$

where H has been defined in equation (17). In general, two different kinds of bulk viscous models are considered: one where DE has a viscous nature [339, 340] but matter has an independent evolution, or alternatively, a unified bulk viscous model in which DM and DE cannot be distinguished [341]. Both scenarios can alleviate the H₀ tension.

Considering that DE has a viscous nature, where the viscosity coefficient is proportional to the Hubble parameter ξ(t) = η₀ H, as introduced in reference [339], the authors of reference [340] have found that for this model, the combination of Planck 2018 CMB distance priors + Pantheon + BAO + BBN + CC results in H₀ = 69.3 ± 1.7 km s⁻¹ Mpc⁻¹ at 68% CL, reducing the Hubble tension to the 1.9σ level.

A model in which the bulk viscosity is proportional to the energy density and inversely proportional to the Hubble parameter, $\xi (t)={\eta }_{0}\enspace \sqrt{{\rho }_{\text{DE}}}/H$, has been introduced in references [342, 343] and considered in light of the Hubble tension in reference [340]. An analysis that fits the Planck 2018 CMB distance priors + Pantheon + BAO + BBN + CC data provides H₀ = 69.3 ± 1.7 km s⁻¹ Mpc⁻¹ at 68% CL, thus reducing the tension with R20 down to 1.9σ [340].

The bulk viscosity can be a thermodynamic function $\xi (t)={\eta }_{0}\enspace {\rho }_{\text{DE}}^{\nu }$, as introduced in reference [344]. A fit to the combination of Planck 2018 CMB distance priors + Pantheon + BAO + BBN + CC data on this model gives H₀ = 69.2 ± 1.7 km s⁻¹ Mpc⁻¹ at 68% CL [340], alleviating the Hubble tension down to 1.9σ as in the two previous models. Note that a CMB-only analysis for all these models is currently missing.

In reference [341] the authors have investigated a unified cosmic scenario endowed with a bulk viscosity in which the bulk viscosity coefficient follows a general law ξ(t) = αρ^m (see also reference [344]). We see that in the scenario where w₀ = 0 and m is a free parameter, Planck 2015 + Pantheon leads to H₀ = 68.0 ± 1.1 km s⁻¹ Mpc⁻¹ at 68% CL [341] and is hence in disagreement with R20 at 3.1σ. For the case of a free w₀ with m = 0, instead, Planck 2015 + Pantheon leads to ${H}_{0}=70.{2}_{-1.9}^{+1.6}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [341], solving the tension at 1.4σ. Lastly, for the case in which both w₀ and m are free parameters, Planck 2015 + Pantheon gives ${H}_{0}=68.{0}_{-2.4}^{+2.7}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [341], alleviating the tension with R20 at 1.7σ.

Models in which a viscous inhomogeneous fluid describes the content of the late Universe are adopted in reference [345]. In the models studied, the pressure of the single fluid considered is a function of both the Hubble rate and density, with parameters that are fixed through a Bayesian learning method over measured values of H(z) for z ≲ 2.5. For the models considered, this method yields H₀ = 73.4 ± 0.1 km s⁻¹ Mpc⁻¹ and H₀ = 73.52 ± 0.15 km s⁻¹ Mpc⁻¹ at 68% CL [345], respectively. Note however that an analysis that uses Planck 2018 data is still missing.

An interesting DE candidate that was proposed following the holographic principle is the holographic dark energy (HDE) [346–348]. The model was extensively studied for its ability to explain the late-time cosmic acceleration (see the review of reference [349]). In this model, the DE equation of state is given by:

$$\begin{equation}{w}_{\text{DE}}(z)=-\frac{1}{3}-\frac{2}{3c}\sqrt{{{\Omega}}_{\text{DE}}(z)},\end{equation} \tag{ 33 }$$

where c is a dimensionless parameter.

In reference [350], the authors argue that the HDE model can alleviate the tension with the local measurements of H₀. A fit to the Planck 2015 + CMB lensing + BAO + JLA + R16 data for HDE returns ${H}_{0}=69.6{7}_{-0.94}^{+0.95}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [350], which alleviates the tension with R20 down to the 2.2σ level. The fit within the extended model HDE $+{N}_{\text{eff}}+{m}_{\nu ,\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{l}\mathrm{e}}^{\text{eff}}$ (in which a massive sterile neutrino is included) to the same dataset gives H₀ = 70.70 ± 1.10 km s⁻¹ Mpc⁻¹ at 68% CL [350], alleviating the tension with R20 at 1.5σ.

An updated analysis for HDE using Planck 2018 data can be found in reference [351], where Planck 2018 + BAO + R19 gives H₀ = 73.12 ± 1.14 km s⁻¹ Mpc⁻¹ at 68% CL, in agreement with R20. Reference [351] also considers the effect of including Pantheon data, finding that this inclusion shifts the value of H₀ to a lower value. In fact, reference [351] further uncovered that both the subsets of the Pantheon data with z > 0.2 and z < 0.2 prefer a higher value of H₀, but they have a large negative correlation in between which has not yet been fully understood. Reference [351] also considered an analysis that includes Pantheon data, finding that this inclusion shifts the value of H₀ to a lower value. In fact, the analysis in reference [352] considers Planck 2018 + BAO + Pantheon and obtains H₀ = 67.94 ± 0.80 km s⁻¹ Mpc⁻¹ at 68% CL [352], at 3.5σ tension with R20 once the Pantheon data are included.

In this context one may be interested in the stability of the de Sitter state in the DE models following a holographic approach [353]. As argued in reference [353], unlike in the ΛCDM model where the de Sitter state is assumed to be stable in the distance future, the DE model following a holographic approach could alleviate the Hubble constant tension leading to an unstable de Sitter state in the distance future. This instability is actually responsible for a turning point [354] which seems crucial in capturing the H₀ tension quantitatively and providing a common ground with the Swampland conjectures.

5.13.1. Tsallis holographic dark energy

String theory is a potential candidate for a UV-complete theory. A large number of string vacua are expected, therefore providing a consistent low-energy EFT limit [357]. These well-behaved solutions that populate the 'landscape' are conjectured to be surrounded by a 'swampland' of semi-classical EFTs for which a consistent theory of quantum gravity does not exist [358]. Various recipes have been conjectures in the attempt to understand the conditions under which a given EFT does not lie in the swampland, such as the weak-gravity conjecture [359] and a set of swampland conjectures [360–363]. In particular, two of these swampland conjectures constrain the excursion range Δϕ of a scalar field ϕ in field space as well as the logarithmic gradient of the scalar field potential V(ϕ). The first 'distance' conjecture avoids that a tower of light states emerges when a scalar field moves by a distance Δϕ ⩾ O(1) (in Planck units) [364–366]. The second of these swampland criteria establishes that a scalar field potential V arising from a consistent quantum theory of gravity should satisfy |V_ϕ | ⩾ cV, where $c\sim \mathcal{O}(1)$ (in Planck units) is a positive constant and V_ϕ = dV(ϕ)/dϕ (see references [367, 368] for the implications of the swampland conjecture in cosmology).

Scalar field models obeying the swampland conjectures have recently gained considerable attention in relation with the proposed solutions to the Hubble tension. In fact, one could construct physically viable scalar field models that could explain the DE effects at late time and satisfy the swampland criteria [369–372].

In reference [370], it is found that a scalar field model, satisfying the swampland criteria with a fixed value of c, can alleviate the Hubble tension. For example, an analysis using Planck 2015 + CMB lensing + BAO + Pantheon + R19 fixing c = 0.1 gives ${H}_{0}=69.0{6}_{-0.73}^{+0.66}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, reducing the tension down to 2.8σ level (notice that R19 is already included in the analysis).

The authors in reference [373] use low-redshift measurements of H₀ to fit a polynomial expansion of the Hubble rate H(z) and test the different features against ΛCDM to alleviate the Hubble tension (see also reference [374]). In particular, the functions for H(z) in reference [373] possess a turning point at a critical redshift z_c = z_c(Ω_m), where Ω_m is the fractional matter density today. Unfortunately, a full data analysis for this model is missing to date.

A consequence of the swampland criteria within string theory is to consider a quintessence field instead of a cosmological constant. In reference [375], this possibility is investigated for solving the Hubble tension, concluding that quintessence models always prefer a lower Hubble constant value than that obtained within the standard ΛCDM. The addition of an exponential coupling to the DM sector does not change this result.

In reference [376], a quintessence field which transits from a matter-like to a cosmological constant-like behavior between recombination and the present time has been proposed to alleviate the Hubble tension. The authors model a transition with the effective DE equation of state:

$$\begin{equation}{w}_{\text{DE}}(a)=\frac{{w}_{\phi 0}}{1+{\left(\frac{a}{{a}_{\text{tr}}}\right)}^{-\frac{2}{{\Delta}}}},\end{equation} \tag{ 34 }$$

where a_tr is the scale factor of the transition and Δ defines its duration. They conclude that Planck 2015 data exclude this model as a possible solution of the Hubble tension, since the best fit value of the Hubble constant is H₀ = 67.20 ± 0.64 km s⁻¹ Mpc⁻¹ at 68% CL [376], at 4.3σ tension with R20.

A similar observation was found in the context of a minimally coupled slowly or moderately rolling quintessence field with a smooth potential [377]. The authors of reference [377] considered the curvature parameter in the analysis and found that the H₀ tension in such models remains at more than 3σ.

In reference [378] a braneworld scenario, introduced in [379], has been proposed to increase the Hubble constant estimate. In this model the observable Universe is situated in a four-dimensional brane embedded in a fifth dimension, the 'bulk'. The braneworld DE has an equation of state phantom-like, and the accelerated expansion of the Universe is therefore a consequence of this modification of gravity. Using a combination of Planck 2015 CMB distance priors, Union 2.1 SNIa and BAO in reference [378] the authors find for this scenario H₀ = 70.75 ± 1.30 km s⁻¹ Mpc⁻¹ at 68% CL, solving the Hubble tension within 1.4σ. A full 2018 CMB data analysis is however missing.

In reference [380], a frame dependent, although scale invariant, DE theory was proposed to alleviate the Hubble tension. In this late time model, the Hubble constant can take extremely large values comparable with the local measurements of H₀, offering at the same time an excellent fit to the CMB spectra. The model has some interesting implications, however it needs to be robustly investigated by means of a full data analysis.

In reference [381] a chameleon field (see also references [382–390]) has been proposed to alleviate the Hubble tension. In this paper, the possibility that a matter overdensity, coupled to the chameleon DE, can increase the Hubble constant locally, introducing the cosmic inhomogeneity in the Hubble expansion rate at late-time, is taken into consideration. A full data analysis is however missing, but it does not go against the No-Go theorem of general chameleon [389].

A famous possibility is a phenomenologically emergent dark energy (PEDE) model, in which a redshift-dependent DE component emerges at late times. In this model, firstly introduced in reference [391], the fractional DE energy density has the following parameterization:

$$\begin{equation}{\tilde {{\Omega}}}_{\text{PEDE}}(z)\equiv \frac{{\rho }_{\text{DE}}}{{\rho }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t},0}}={{\Omega}}_{\text{PEDE,0}}\left[1-\mathrm{tanh}\enspace ({\mathrm{log}}_{10}(1+z))\right],\end{equation} \tag{ 35 }$$

where Ω_PEDE,0 = Ω_PEDE(z = 0) and ${\rho }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t},0}=3{H}_{0}^{2}{M}_{\text{Pl}}^{2}$ is the present critical energy density. The fluid described by equation (35) has a phantom-like equation of state that asymptotically approaches the cosmological constant value w_Λ = −1 as time proceeds [391]:

$$\begin{equation}{w}_{\text{PEDE}}(z)=-1-\frac{1}{3\enspace \mathrm{ln}\enspace 10}\enspace \left[1+\mathrm{tanh}\left({\mathrm{log}}_{10}(1+z)\right)\right].\end{equation} \tag{ 36 }$$

Using the Planck 2015 CMB distance priors + Pantheon + BAO + Ly-α data, reference [391] finds ${H}_{0}=71.0{2}_{-1.37}^{+1.45}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, solving the Hubble tension at 1.1σ. Considering a full CMB analysis for this scenario, Planck 2015 alone gives instead ${H}_{0}=72.5{8}_{-0.80}^{+0.79}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [392], solving the Hubble tension within 1σ, and Planck 2015 + BAO gives ${H}_{0}=71.5{5}_{-0.57}^{+0.55}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, in agreement with R20 at 1.2σ. This result is in agreement with reference [393], where CC measurements are considered. The very same model has been updated in reference [394], which finds ${H}_{0}=72.3{5}_{-0.79}^{+0.78}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL for the Planck 2018 data, and H₀ = 72.16 ± 0.44 km s⁻¹ Mpc⁻¹ at 68% CL for Planck 2018 + CMB lensing + BAO + Pantheon + DES + R19, confirming the agreement with R20 within one standard deviation. However, in reference [395] it has been argued that, while at the background level the flat-PEDE model fits the data as well as the ΛCDM scenario, at the perturbation level the PEDE model cannot fit the observational data in cluster scales compared to the ΛCDM. Extensions of this model considering neutrinos or a non-zero curvature of the Universe can be found in references [394–396].

6.1.1. Generalized emergent dark energy

A generalization of the PEDE model, including one more degree of freedom Δ, known as generalized emergent dark energy (GEDE) can be found in reference [397]. In the GEDE model, the evolution for the DE density is written as [397]:

$$\begin{equation}{\tilde {{\Omega}}}_{\text{GEDE}}(z)\equiv \frac{{\rho }_{\text{DE}}}{{\rho }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t},0}}=\enspace {{\Omega}}_{\text{GEDE,0}}\enspace \frac{1-\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left({\Delta}\enspace {\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\left(\frac{1+z}{1+{z}_{\mathrm{t}}}\right)\right)}{1+\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left({\Delta}\enspace {\mathrm{l}\mathrm{o}\mathrm{g}}_{10}(1+{z}_{\mathrm{t}})\right)},\end{equation} \tag{ 37 }$$

where the redshift z_t marks the transition at which the densities in DE and matter equate, ${\tilde {{\Omega}}}_{\text{GEDE}}({z}_{\mathrm{t}})\enspace =\enspace {{\Omega}}_{\mathrm{m}}{(1+{z}_{\mathrm{t}})}^{3}$. The redshift z_t is thus not a free parameter. For Δ = 0 this model recovers the ΛCDM scenario, while for Δ = 1 and z_t = 0, the PEDE model is recovered. The GEDE equation of state is:

$$\begin{equation}{w}_{\text{GEDE}}(z)=-1-\frac{{\Delta}}{3\enspace \mathrm{ln}\enspace 10}\enspace \left(1+\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left[{\Delta}\enspace {\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\left(\frac{1+z}{1+{z}_{\mathrm{t}}}\right)\right]\right).\end{equation} \tag{ 38 }$$

The analysis of Planck 2018 data at the background level for this GEDE scenario provides ${H}_{0}=66.7{6}_{-0.76}^{+2.9}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [397], reducing the Hubble tension down to the 2σ level. However, this result is mostly driven by a volume effect, due to the increased volume of the parameter space. This result is in agreement with the results of reference [393], where CC data are considered. An updated result of this scenario is presented in reference [398] considering also the effects of the DE perturbations where for Planck 2018, ${H}_{0}=8{5}_{-6}^{+12}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL.

6.1.2. Modified emergent dark energy

Another generalization of the PEDE model which includes one additional degree of freedom α is the modified emergent dark energy (MEDE), in which the DE equation of state can be written as [399]:

$$\begin{equation}{w}_{\text{MEDE}}(z)=-1-\frac{\alpha }{3\enspace \mathrm{ln}\enspace 10}\enspace \left(1+\mathrm{tanh}\left[\alpha \enspace {\mathrm{log}}_{10}\left(1+z\right)\right]\right).\end{equation} \tag{ 39 }$$

If α = 0, the model reduces to the ΛCDM scenario, while for α = 1 the PEDE model is recovered. A fit to Planck 2018 + BAO data within the MEDE model provides H₀ = 68.4 ± 1.5 km s⁻¹ Mpc⁻¹ at 68% CL and reduces the Hubble tension to 2.4σ [399].

The vacuum metamorphosis (VM) model is a cosmological scenario which is physically motivated by quantum gravitational effects, where a gravitational phase transition occurs at late times [269–271]. The phase transition is induced when the Ricci scalar curvature R is of the order of the mass squared of the field m², after which R is frozen. The value of m² determines the matter density today Ω_m, and therefore the VM model has the same number of free parameters than the flat ΛCDM scenario.

It has been found that this specific model can be an excellent candidate to solve the H₀ tension [400]. In more detail, the expansion rate above and below the phase transition reads as [400]:

$$\begin{equation}\frac{{H}^{2}}{{H}_{0}^{2}}=\begin{cases}{{\Omega}}_{\mathrm{m}}{(1+z)}^{3} + {{\Omega}}_{\mathrm{r}}{(1+z)}^{4} + M\left\{1 - {\left[3{\left(\frac{4}{3{{\Omega}}_{\mathrm{m}}}\right)}^{4}M{(1-M)}^{3}\right]}^{-1}\right\},\quad & \text{for}\enspace z{ >}{z}_{\mathrm{t}}^{\text{ph}};\\ (1-M){(1+z)}^{4}+M,\quad & \text{for}\enspace z{\leqslant}{z}_{\mathrm{t}}^{\text{ph}},\end{cases}\end{equation} \tag{ 40 }$$

where $M={m}^{2}/(12{H}_{0}^{2})$ and the phase transition occurs at the redshift

$$\begin{equation}{z}_{\mathrm{t}}^{\text{ph}}=-1+\frac{3{{\Omega}}_{\mathrm{m}}}{4(1-M)}.\end{equation} \tag{ 41 }$$

The effective DE equation of state is [400]:

$$\begin{equation}{w}_{\text{VMDE}}(z)=-1-\frac{1}{3}\frac{3{{\Omega}}_{\mathrm{m}}{(1+z)}^{3}-4(1-M){(1+z)}^{4}}{M+(1-M){(1+z)}^{4}-{{\Omega}}_{\mathrm{m}}{(1+z)}^{3}},\end{equation} \tag{ 42 }$$

below the phase transition, while w_VMDE(z) = −1 above the phase transition.

For the VM scenario, Planck 2015 gives H₀ = 78.61 ± 0.38 km s⁻¹ Mpc⁻¹ at 68% CL [400], reducing the tension at 3.9σ. An updated analysis is performed in reference [401], where a fit to the Planck 2018 data gives H₀ = 81.1 ± 2.1 km s⁻¹ Mpc⁻¹ at 68% CL, reducing the tension at 3.1σ, and an analysis to Planck 2018 + BAO + R19 gives H₀ = 75.22 ± 0.60 km s⁻¹ Mpc⁻¹ at 68% CL, in agreement at 1.4σ with R20. An extension of the model considering a non-zero curvature of the Universe can be found in reference [401]. The extension considering the neutrino sector, instead, explored in reference [402], provides, for Planck 2018, ${H}_{0}=78.{0}_{-2.6}^{+3.8}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, alleviating the tension at 1.7σ, and, for Planck 2018 + BAO + R19, H₀ = 74.60 ± 0.97 km s⁻¹ Mpc⁻¹ at 68% CL, in agreement with R20 within 1σ.

6.2.1. Elaborated vacuum metamorphosis

While in the original VM, M is not a free parameter, but fixed by:

$$\begin{equation}{{\Omega}}_{\mathrm{m}}=\frac{4}{3}{\left[3M{(1-M)}^{3}\right]}^{1/4}\enspace ,\end{equation} \tag{ 43 }$$

a scenario where the model has one more free parameter M, can also be regarded as a possible cosmological scenario. For the VM scenario Planck 2015 gives ${H}_{0}=71.{6}_{-5.1}^{+2.8}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [400], solving the Hubble tension within 1σ. This result is confirmed by the updated analysis performed in reference [401], where Planck 2018 gives ${H}_{0}=76.{7}_{-2.6}^{+3.9}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, and Planck 2018 + BAO + R19 ${H}_{0}=73.6{3}_{-0.48}^{+0.33}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, in agreement within 1σ with R20. An extension of this model, considering a curvature component, can be found in reference [401]. The extension considering the neutrino sector, instead, explored in reference [402], provides, for Planck 2018, ${H}_{0}=75.{2}_{-2.4}^{+1.6}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, and, for Planck 2018 + BAO + R19, H₀ = 73.19 ± 0.85 km s⁻¹ Mpc⁻¹ at 68% CL, both in agreement with R20 within 1σ.

A possibility for having extra relativistic degrees of freedom at recombination is the presence of additional sterile neutrinos [416–419], since they are not forbidden by any fundamental symmetry within the standard model of elementary particles. They can contribute to N_eff in a fractional way, if not fully thermalized, and can still be in agreement with the CMB data, while a thermalized sterile neutrino with ${\Delta}{N}_{\text{eff}}={N}_{\text{eff}}-{N}_{\text{eff}}^{\text{SM}}=1$ is ruled out at about 6σ.

Light sterile neutrinos are strongly motivated by neutrino short baseline oscillation anomalies. In fact, there is a 6.1σ indication for an electron–neutrino appearance, when combining together the MiniBooNE [420] and the liquid scintillator neutrino detector (LSND) [421] data, even if this result has been challenged by STEREO [422] and PROSPECT [423]. These anomalous datasets can be explained with one sterile neutrino [424–428] with ΔN_eff ≈ 1, in strong contradiction with cosmological constraints. A possibility is the presence of some non-standard interactions [429–431], low-temperature reheating [432], or other special mechanisms. Nevertheless, in reference [433], the authors showed that the combination of the Planck 2015 CMB distance priors + BAO + Pantheon + BBN + R16 gives N_eff ≈ 4 and ${H}_{0}=73.6{4}_{-2.68}^{+2.61}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, perfectly in agreement with R20. A full Planck 2018 analysis is however in contradiction with this result.

In reference [434], instead, the authors showed that additional radiation produced just before the BBN by an unstable sterile neutrino with a mass of the order of tens of MeV can alleviate the Hubble tension by increasing N_eff in the right amount. Also, in reference [435], it is shown that, in the presence of a large lepton asymmetry, sterile neutrinos with the masses in the (150–450) MeV range can increase N_eff by 0.2–0.4 and reduce the Hubble tension.

A large lepton number asymmetry ξ in one or more neutrino species contributes to ΔN_eff in the following way, accounting for the thermal distributions of two light mass eigenstates [436–439]:

$$\begin{equation}{\Delta}{N}_{\text{eff}}=\frac{15}{7}\sum\limits _{i=1,2}{\left(\frac{{\xi }_{i}}{\pi }\right)}^{2}\left[2+{\left(\frac{{\xi }_{i}}{\pi }\right)}^{2}\right].\end{equation} \tag{ 45 }$$

Given the role of the extra dark radiation in solving the Hubble tension, in reference [440] it has been investigated the possibility that a primordial lepton asymmetry can alleviate the tension between the CMB and R16. The combination of Planck 2015 + BICEP2 & Keck array (BKP) [441] results in H₀ = 67.71 ± 0.95 km s⁻¹ Mpc⁻¹ at 68% CL [440], showing still a disagreement at 3.4σ with R20.

The QCD axion [442, 443] is a hypothetical particle that emerges from the Peccei–Quinn solution to the strong CP problem [444, 445] (see reference [446] for a review). These particles may be copiously produced in the early Universe through either non-thermal mechanisms comprising CDM or thermal mechanisms which lead to a population of relativistic axions that would contribute to N_eff [413, 447–455].

The production of thermal axions might proceed through various mechanisms. Since the QCD axion couples to the gluon through a model-independent interaction, thermal axions are produced via scatterings off gluons for the decoupling temperature T_D ≳ 1 GeV [456], and through the scattering off pions and nucleons at lower temperatures, regardless of the QCD axion model (see reference [457] for recent updates). Since the coupling of the QCD axion with other SM particles is model-dependent, other production mechanisms via scattering off photons [458] and SM leptons [459] or quarks [460] might also arise. The energy density of radiation in the late Universe leads to a deviation in the effective number of relativistic degrees of freedom given by:

$$\begin{equation}{\Delta}{N}_{\text{eff}}=\frac{8}{7}{\left(\frac{11}{4}\right)}^{\frac{4}{3}}\frac{{\rho }_{\mathrm{a}}}{{\rho }_{\gamma }},\end{equation} \tag{ 46 }$$

where the axion energy density is found in terms of the current axion number density n_a and the internal degrees of freedom of the relativistic gas of bosons g, and it reads as:

$$\begin{equation}{\rho }_{\mathrm{a}}=\frac{{\pi }^{2}}{30}{\left(\frac{{\pi }^{2}{n}_{\mathrm{a}}}{\zeta (3)g}\right)}^{\frac{4}{3}}.\end{equation} \tag{ 47 }$$

The assessment of a model in which hot axions are produced from the coupling with muons leads to an alleviation of the Hubble tension at 3σ level [459]. In this ΛCDM + N_eff scenario, the fit to Planck 2018 + BAO data results in a value for the Hubble constant corresponding to ${H}_{0}=68.{0}_{-1.1}^{+2.3}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 95% CL.

Cosmological scenarios in which it is present a decaying DM component [461] could be alternative proposals to reduce the Hubble constant tension. For instance, a DM fluid could decay into invisible massless particles after recombination while still avoiding photon overproduction [462]. This model has been explored in references [463–465] making use of different combinations of the Planck 2015 data with other cosmological probes. In reference [466], a model of CDM decaying before recombination is inspected and it is shown to not to be a satisfactory solution to the Hubble tension. Nevertheless, the best case scenario to alleviate the Hubble constant occurs when the DM particles decay exclusively into dark radiation.

A cosmological model where a fraction of the DM density decays into dark radiation increasing ΔN_eff, as proposed by reference [467], has been considered as a solution for the Hubble tension by many authors. Such a decaying scenario in terms of the background equations reads as:

$$\begin{equation}{\dot {\rho }}_{\text{DM}}+3\mathcal{H}{\rho }_{\text{DM}}=-Q;\end{equation} \tag{ 48 }$$

$$\begin{equation}{\dot {\rho }}_{\text{DR}}+4\mathcal{H}{\rho }_{\text{DR}}=Q,\end{equation} \tag{ 49 }$$

where here and in the following, a dot stands for a differentiation with respect to conformal time, and the source term Q = Γρ_DM, depends on a constant decay rate Γ (see references [468–470] for different functional forms of Γ). Within this scenario, the authors of reference [471] have found from the analysis of Planck 2015 + R18 a value for the Hubble constant ${H}_{0}=70.{6}_{-1.3}^{+1.1}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, reducing the Hubble tension down to the 1.5σ level. Notice however that this result may be biased due to the fact that it already includes a R18 prior on the Hubble constant.

If we consider two different regimes, i.e. one for long-lived decaying CDM (with a lifetime longer than the epoch corresponding to recombination) and another one for short-lived decaying CDM particles, for which the mass–energy density decreases significantly well before recombination, the latter will leave a strong imprint on the CMB. In reference [472], Planck 2015 + BAO gives ${H}_{0}=67.9{3}_{-0.63}^{+0.53}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL for long-lived decaying CDM, in disagreement with R20 at more than 3.8σ, while ${H}_{0}=68.3{7}_{-0.89}^{+0.61}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL for short-lived decaying CDM, in disagreement with R20 at 3.5σ. An update is presented in reference [473], where Planck 2018 provides H₀ = 67.7 ± 1.2 km s⁻¹ Mpc⁻¹ (${H}_{0}=67.{8}_{-1.5}^{+1.4}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$) at 95% CL for long (short)-lived decaying CDM particles, in disagreement with R20 at more than 3.7σ (3.6σ), and Planck 2018 + BAO instead lead to ${H}_{0}=67.{7}_{-0.9}^{+1.0}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (${H}_{0}=68.{6}_{-1.4}^{+1.2}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$) at 95% CL, in disagreement with R20 at more than 3.9σ (3.2σ). If, instead, DM is composed of decaying warm DM particles, it has been shown in reference [474] that for a DM particle mass m = 40 eV, Planck 2018 + CMB lensing (Planck 2018 + CMB lensing + BAO + R19) provides the constraint ${H}_{0}=69.0{5}_{-0.95}^{+0.66}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (${H}_{0}=70.2{0}_{-0.94}^{+0.79}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$) at 68% CL, reducing the Hubble tension down to the 2.8σ (2σ) significance level.

7.4.1. Self-interacting dark matter

7.4.2. Two-body dark matter decays

Different from the two previous cases is the model presented in reference [480] and analysed in reference [481], where a parent particle decays into two daughter particles, one massless and one massive, with the form ψ → χ + γ. This decay is well-known within the context of super weakly interacting massive particles (super WIMPs) [482]. This decaying DM model has therefore two free parameters: the fraction of the energy of the parent particle which is transferred to the massless particle, and the lifetime τ = 1/Γ, where Γ is the decay rate. Assuming there are no decays prior the recombination period, the energy densities of the parent particle ρ₀ and of the massless daughter particle ρ₁ evolve as:

$$\begin{equation}\frac{\mathrm{d}{\rho }_{0}}{\mathrm{d}t}+3H{\rho }_{0}=-{\Gamma}{\rho }_{0};\end{equation} \tag{ 50 }$$

$$\begin{equation}\frac{\mathrm{d}{\rho }_{1}}{\mathrm{d}t}+4H{\rho }_{1}={\epsilon}\enspace {\Gamma}{\rho }_{0};\end{equation} \tag{ 51 }$$

$$\begin{equation}3{H}^{2}(a)={\kappa }^{2}\sum\limits _{i}\enspace {\rho }_{i}(a),\end{equation} \tag{ 52 }$$

where, referring to the massive daughter particle with the subscript 2, the total energy density is:

$$\begin{equation}\sum\limits _{i}\enspace {\rho }_{i}(a)={\rho }_{0}(a)+{\rho }_{1}(a)+{\rho }_{2}(a)+{\rho }_{\mathrm{r}}(a)+{\rho }_{\nu }(a)+{\rho }_{\mathrm{b}}(a)+{\rho }_{{\Lambda}}.\end{equation} \tag{ 53 }$$

In this scenario, an analysis with Planck 2015 + CMB lensing + R18 + BAO provides ${H}_{0}=7{0}_{-3}^{+4}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, in agreement with R20 within 1σ [481]. However, since both the R18 prior on the Hubble constant and the BAO data are present in the joint analysis, so it is difficult to assess how well the model can solve the Hubble tension for the CMB dataset alone. ²⁴ An updated CMB result is nevertheless present in reference [484], where taking into account the late-Universe decaying DM effects like ISW and lensing, an analysis with Planck 2018 + CMB lensing data results in ${H}_{0}=67.3{1}_{-0.56}^{+0.53}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 95% CL, consistent with the ΛCDM value and in disagreement with R20 at 4.2σ, excluding at a high significance the preferred region of the earlier analysis carried out in reference [481].

7.4.3. Light gravitino scenarios

In reference [485] the authors study the keV gravitino DM model arguing that this could reduce the Hubble tension at around the 3σ level. The bino, the superpartner of the U(1) weak hypercharge gauge field, can have a late decay into a gravitino nearly relativistic in the early Universe, increasing the radiation density by:

$$\begin{equation}{\rho }_{\mathrm{R}}^{\text{extra}}=f{\times}{\rho }_{3/2}{\times}({\gamma }_{3/2}-1),\end{equation} \tag{ 54 }$$

where γ_3/2 will be the boost factor of the gravitino from the bino decay, and f will be the fraction of the non-thermal gravitino density in the total gravitino production. Therefore, the gravitino contributes to the effective number of relativistic degrees of freedom ΔN_eff as [486]:

$$\begin{equation}{\Delta}{N}_{\text{eff}}=f{\times}\left({\gamma }_{3/2}-1\right)/0.16\enspace .\end{equation} \tag{ 55 }$$

A similar contribution to ΔN_eff is expected from a light DM candidate suggested in reference [487] to solve both the lithium problem and reconcile the Hubble tension. Another particle physics model to address the Hubble tension is presented in reference [488], where the authors consider a gravitino as decaying DM and a quintessence DE axion, nevertheless a full Planck 2018 analysis is however still missing.

7.4.4. Decaying Z'

7.4.5. Dynamical dark matter

7.4.6. Degenerate decaying fermion dark matter

Neutrinos interacting with DM have been investigated in the literature [494–503], because DM annihilations into neutrinos can mimic an increase in the value of the dark radiation N_eff [504, 505], and therefore solve the Hubble tension. In reference [506], it has been shown that varying N_eff, Planck 2015 gives ${H}_{0}=66.{8}_{-1.9}^{+1.8}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, reducing the tension with R20 down to the 2.9σ level.

7.5.1. Neutrino–Majoron interactions

7.5.2. FIMPs decay into neutrinos

An interacting dark radiation component increasing ΔN_eff has been proposed to alleviate the Hubble tension in reference [517]. In this model, the energy density in relativistic particles is [518]:

$$\begin{equation}{\rho }_{\mathrm{r}}={\rho }_{\gamma }\enspace \left[1+\frac{7}{8}{\left(\frac{4}{11}\right)}^{4/3}\left({N}_{\text{eff}}+{N}_{\text{fl}}\right)\right],\end{equation} \tag{ 57 }$$

where N_fl is the interacting counterpart of the dark radiation component. If N_eff = 3.046 and N_fl is free to vary, a fit to Planck 2018 + CMB lensing (Planck 2018 + CMB lensing + BAO + R19) provides ${H}_{0}=69.1{4}_{-1.26}^{+0.77}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (${H}_{0}=70.6{4}_{-1.00}^{+0.93}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$) at 68% CL [517], alleviating the tension with R20 down to the 2.7σ (1.6σ) level.

The possibility of a DM interacting with massless relics from the dark sector, i.e. dark radiation [519–531], has been proposed to solve the Hubble tension. One example is given by the ETHOS formalism [532], in which it is assumed that a single DM species interacts with a relativistic component via 2-to-2 scattering processes of the form DM + DR ↔ DM + DR, with a comoving interaction rate that depends on temperature as Γ_DR–DM ∝ Tⁿ . For the case n = 0, corresponding to a class of non-abelian DM models, the analysis of Planck 2015 + BAO datasets gives ${H}_{0}=68.0{4}_{-0.60}^{+0.50}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [533], in disagreement with R20 at 3.7σ. An update of this work is presented in reference [534], where Planck 2018 + BAO results in H₀ = 68.73 ± 0.96 km s⁻¹ Mpc⁻¹ at 68% CL, reducing the Hubble tension with R20 at 2.8σ.

A scattering process that allows three particles to annihilate into two has been proposed as a possible DM scenario in reference [535]. Dark matter 'cannibalism' is a generic feature arising in any hidden sector in which a mass gap exists between two species.

This model has been considered for solving the Hubble tension, because it can increase the dark radiation component in the Universe [536]. The thermal history can be divided into three different phases:

• The cannibal dark matter paradigm, made of a real scalar field ϕ, behaves as a radiation fluid, indistinguishable from an extra contribution to N_eff, while its temperature is above the ϕ mass.
• A cannibalistic phase happens at a given scale factor, when the ϕ-fluid cools below the mass of the particles: the interaction 3 → 2 starts processing mass into temperature and the temperature drops logarithmically.
• The 3 → 2 interactions decouple and the temperature drops as in the ordinary non-relativistic matter case.

Due to its strongly exothermic nature, cannibal DM acts as a warm DM component for a long period and turns non-relativistic at later times than CDM. In the simplest scenario, this strongly suppresses structure growth, so cannibal DM cannot be all of the DM [537]. In the analysis of reference [536], only ∼1% of the DM is considered as cannibalistic. As in many of the previous scenarios, a quantitative assessment of the ability of this model to solve the Hubble constant tension cannot be performed, since a full Planck 2018 analysis for these models is absent in the literature.

A class of models that takes advantage of both the ΔN_eff and the EDE models, improving on their downsides, has been proposed in reference [538]. The authors study the decaying ultralight scalar model, which does not suffer from the EDE fine-tuning. In this model, the dark sector contains an ultralight scalar field ϕ of mass m ≳ H_eq ∼ 10⁻²⁸ eV, where H_eq is the Hubble rate at matter-radiation equality, which resonantly decays into an abelian gauge field A_μ when the axion field starts to oscillate at m ∼ H. Mimicking the perturbative preheating stage after inflation, see e.g. references [539, 540], an effective description of the model in terms of coupled fluid equations is:

$$\begin{equation}\frac{\mathrm{d}{\rho }_{\phi }}{\mathrm{d}t}+3H(1+{w}_{\phi }){\rho }_{\phi }=-{\Gamma}(t){\rho }_{\phi };\end{equation} \tag{ 58 }$$

$$\begin{equation}\frac{\mathrm{d}{\rho }_{\mathrm{A}}}{\mathrm{d}t}+4H{\rho }_{\mathrm{A}}={\Gamma}(t){\rho }_{\phi },\end{equation} \tag{ 59 }$$

where Γ(t) is a time-dependent decay rate and the equation of state for the ultralight scalar field transitions from w_ϕ = −1 to w_ϕ = 0 at m ∼ H. A combined analysis to Planck 2018 + CMB lensing + BAO + Pantheon + R19 data gives ${H}_{0}=69.{9}_{-0.86}^{+0.84}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [538], reducing the H₀ tension at 2.2 standard deviations. However, a Gaussian prior for the Hubble constant coming from the Sh0ES measurement is also included in the analysis.

In reference [541] the possibility that an extra radiation density could be due to extra vector fields, different from the visible photon field, has been considered for solving the Hubble tension. These vector fields, called dark photon fields, must interact very weakly with the visible matter and should have a small mass (see references [542, 543]), contributing as DM. In reference [544] this model has been studied considering the BBN observations concluding that even if the addition of three dark massive vector fields can help to soften the Hubble tension, the mechanism cannot resolve it completely.

Recently a strong interest for primordial black holes (PBHs) as a possible DM component of our Universe (see e.g. references [545–548]) has developed. In particular, in reference [545] the implications of the Hawking evaporation of light PBHs have been studied, showing that those can affect either N_eff or w_DE, depending on their precise mass, and potentially alleviate the Hubble tension. The authors of reference [545] find that Planck 2018 CMB distance priors (Planck 2018 CMB distance priors + BAO + Pantheon + H(z)) give for this model H₀ = 65.37 ± 1.92 km s⁻¹ Mpc⁻¹ (H₀ = 67.09 ± 1.76 km s⁻¹ Mpc⁻¹) at 68% CL, reducing the tension at 3.4σ (2.8σ). However, they also speculate that an ultra-light PBH, decaying around the neutrino decoupling period, could raise H₀ = 70.49 ± 1.34 km s⁻¹ Mpc⁻¹ at 68% CL for Planck 2018 CMB shift, solving the Hubble tension with R20 at 1.4σ.

In reference [549] a new scenario for the formation of PBHs within the dark sector is proposed. This scenario predicts a modification to N_eff by:

$$\begin{equation}{\Delta}{N}_{\text{eff}}=14{\left(\frac{{g}_{1}}{{g}_{{\ast}}({T}_{\mathrm{D}})}\right)}^{4/3},\end{equation} \tag{ 60 }$$

i.e. by about 0.1–0.2, which could potentially alleviate the Hubble tension. Unfortunately, a full data analysis for this model is missing and therefore it is not possible to fully quantify its effectiveness in solving the H₀ tension.

The physics beyond the SM could contain a sector that is conformally invariant in the infra-red region and classically scale-invariant in the ultra-violet limit [550, 551], referred to as the 'unparticle' and the Banks–Zaks phases [552]. Unparticles behave like radiation at high energies, increasing ΔN_eff and therefore reducing the Hubble tension due to their correlation [553]. In addition, unparticles may act as a cosmological constant at low energies mimicking the standard ΛCDM model. A full data analysis for this model is however required to quantitatively assess its ability to resolve the Hubble tension.

Along with the early and late time solutions, a generalized cosmological scenario in which the DM and the DE interact with each other in a non-gravitational way received massive attention in the literature. These are known as interacting dark energy (IDE) or coupled dark energy (CDE) models [554]. The possibility of an interaction was thought to deal with the cosmological constant problem [555]. Later on, it was argued that an interaction between the dark fluids, namely, DM and DE, can be used to provide a possible solution to the cosmic coincidence problem [556–562]. Additionally, an interaction in the dark sector can explain the phantom DE regime without any scalar field having a negative kinetic term [556, 563–567]. For those reasons, IDE scenarios have been substantially investigated in the literature, see e.g. references [568–631] (see also two review articles [632, 633] in this context for a comprehensive reading) with some interesting consequences.

The IDE models, as examined by several investigators over the last couple of years, can play an effective role to alleviate/solve the Hubble constant tension. In this section, we therefore revisit different IDE models which significantly increase the H₀ value. The basic framework of this theory is the coupling between DM and DE characterizing the energy flow between these dark sectors. Due to such a coupling between these dark fluids, the continuity equations can be written as:

$$\begin{equation}{\dot {\rho }}_{\text{DM}}+3\mathcal{H}{\rho }_{\text{DM}}=a\enspace Q;\end{equation} \tag{ 61 }$$

$$\begin{equation}{\dot {\rho }}_{\text{DE}}+3\mathcal{H}(1+{w}_{\text{DE}}){\rho }_{\text{DE}}=-a\enspace Q,\end{equation} \tag{ 62 }$$

where the dot corresponds to the derivative with respect to conformal time τ, a is the scale factor, $\mathcal{H}\equiv \mathrm{d}\enspace \mathrm{ln}\enspace a/\mathrm{d}\tau$ is the conformal expansion rate of the Universe, and ρ_DM, ρ_DE are respectively the energy density of DM and DE. The quantity Q denotes the interaction rate/interaction function/coupling function which characterizes the energy or/and momentum flow between these dark fluids. In the following, we classify the models based on the functional form of Q.

8.1.1. Interacting vacuum energy

The simplest interacting scenario is the case where vacuum energy characterized by the equation of state w_DE ≡ p_DE/ρ_DE = −1 interacts with DM, known as the interacting vacuum scenario (IVS). Consistent observational evidences indicate that IVS can solve the H₀ tension [634–641] in an exceptional way, even if this result is mostly driven by the existing parameter degeneracies [642].

A possibility is to assume that the rate of the interaction Q between the two dark components is proportional to the DE density ρ_DE as:

$$\begin{equation}Q=\xi \enspace \mathcal{H}\enspace {\rho }_{\text{DE}},\end{equation} \tag{ 63 }$$

where ξ is a dimensionless parameter that quantifies the coupling between DM and DE. An analysis that accounts for Planck 2015 TT power spectra data (Planck 2015 TT + R16 + KiDS-450 [13]) results in ${H}_{0}=72.{2}_{-5.0}^{+3.5}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (H₀ = 73.6 ± 1.6 km s⁻¹ Mpc⁻¹) at 68% CL [639], solving the Hubble tension within 1σ.

Repeating the analysis in an extended scenario where the parameters in the neutrino sector are also allowed to vary, and using the Planck 2015 + JLA + BAO datasets yields the Hubble constant H₀ = 68.2 ± 1.4 km s⁻¹ Mpc⁻¹ at 68% CL [635], alleviating the Hubble tension at 2.6σ. For this latter case, results obtained by fitting Planck observations alone are absent in the literature.

An update to this scenario that considers a flux of energy from DM to the DE components is presented in reference [637], where it is shown that a fit against Planck 2018 (Planck 2018 + BAO + R19) data gives ${H}_{0}=72.{8}_{-1.5}^{+3.0}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (H₀ = 71.7 ± 1.1 km s⁻¹ Mpc⁻¹) at 68% CL, resolving completely the Hubble tension within 1σ. This is in agreement with the findings in references [103, 631, 640] and reference [636], where instead the interaction term differs by a factor of three, and for which an assessment against Planck 2018 (Planck 2018 + BAO) data gives ${H}_{0}=70.{8}_{-2.5}^{+4.3}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL (${H}_{0}=68.{8}_{-1.5}^{+1.3}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL), solving the tension within 1σ (at 2.4σ). In references [631, 636], an extended model in which the neutrino sector is also allowed to vary has been considered.

The case in which there is a transition between an interacting and a non-interacting scenario is instead analysed in reference [638], where it was shown that the Hubble constant resulting from Planck 2015 data cannot solve the tension with R19.

8.1.2. Coupled scalar field

We now discuss the CDE scenario, in which DM interacts via a dark force mediated by a new scalar field ϕ, which in turn drives cosmic acceleration [554, 643–646]. In this model, the coupling in the dark sector is described by the Lagrangian term:

$$\begin{equation}\mathcal{L}=-\frac{1}{2}{\partial }^{\mu }\phi {\partial }_{\mu }\phi -V(\phi )-m(\phi )\bar{\psi }\psi +{\mathcal{L}}_{\text{kin}}[\psi ],\end{equation} \tag{ 64 }$$

in which the mass of DM field ψ, m(ϕ), is a function of the scalar field ϕ, V(ϕ) is a self-coupling potential, and the last term describes the DM kinetic term. This model has been assessed against WMAP [647], Planck 2013 [584], and Planck 2015 [57] measurements, by employing a Peebles–Ratra potential [648]:

$$\begin{equation}V(\phi )={V}_{0}\enspace {\phi }^{-\alpha },\end{equation} \tag{ 65 }$$

where V₀ and α > 0 are constants. Recently, the model has been reconsidered in light of the Planck 2018 data (Planck 2018 + BAO + Pantheon + CC + R19 + H0LiCOW), obtaining the result for the Hubble constant ${H}_{0}=67.7{4}_{-0.66}^{+0.57}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (${H}_{0}=68.7{9}_{-0.40}^{+0.35}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$) at 68% CL [649], a value that is in tension with R20 at 3.9σ (3.4σ).

In reference [650] a quintessence model with a Yukawa interaction between DE and DM has been explored, finding that Planck 2015 gives ${H}_{0}=66.8{9}_{-2.3}^{+2.1}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, at 2.5σ tension with R20.

Recently, in reference [630] the authors have explored the scalar field interacting scenario proposed in reference [651]. Using a combination of CC + BAO + high redshift HII Galaxy measurements (HIIG) + JLA, the authors find ${H}_{0}=69.{9}_{-1.02}^{+0.46}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [630], alleviating the tension with R20 at 2.4σ.

8.1.3. IDE with a constant DE equation of state

8.1.4. IDE with variable DE equation of state

8.1.5. IVS and IDE with variable coupling

In most of the IDE models, the coupling parameter of the interaction model is assumed to be constant. However, the most general case is only realized when a time varying coupling parameter is considered, as there is no theoretical principle that can exclude this possibility. In references [662, 663], the two following coupling functions were considered:

$$\begin{equation}\text{Model}\;\text{A:}\quad Q=3\xi (a)H{\rho }_{\text{DE}};\end{equation} \tag{ 70 }$$

$$\begin{equation}\text{Model}\;\text{B:}\quad Q=3\xi (a)H\frac{{\rho }_{\text{DM}}\enspace {\rho }_{\text{DE}}}{{\rho }_{\text{DM}}+{\rho }_{\text{DE}}},\end{equation} \tag{ 71 }$$

where ξ(a) is the time dependent and dimensionless coupling parameter having the form

$$\begin{equation}\xi (a)={\xi }_{0}+{\xi }_{a}\enspace (1-a).\end{equation} \tag{ 72 }$$

The above interaction models together with the variable coupling function were investigated for w_DE = −1 (IVS), in reference [662], and for a constant value of w_DE ≠ −1 in reference [663] (IDE).

In reference [662], for model A (B), Planck 2015 alone estimates H₀ = 69.2 ± 5 km s⁻¹ Mpc⁻¹ ($68.{3}_{-6.2}^{+6}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$) at 95% CL. Consequently, due to the large error bars, the H₀ tension with R20 is solved at 1.5σ level for both models.

In reference [663], models A and B are tested for a constant value of w_DE ≠ −1. For model A, when a quintessence regime w_DE > −1 is assumed, Planck 2018 (Planck 2018 + BAO) gives ${H}_{0}=70.{2}_{-3.1}^{+4.1}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL (${H}_{0}=68.{4}_{-2.5}^{+2.7}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 95% CL). Due to the very large error bars, the H₀ tension with R20 is alleviated within 1σ (2.6σ). For model B and w_DE < −1, a fit to Planck 2018 (Planck 2018 + BAO) data provides H₀ > 73 km s⁻¹ Mpc⁻¹ at 95% CL (${H}_{0}=69.{4}_{-2.3}^{+2.6}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 95% CL). Notice that the tension is solved within 2σ (at 2.1σ).

8.1.6. IDE with sign-changing interaction

8.1.7. Anisotropic stress in IDE

An IDE scenario where the anisotropic stress of the large scale inhomogeneities is also considered has been explored in reference [670]. In this model, the conservation equations are:

$$\begin{equation}{\dot {\rho }}_{\text{DM}}+3\mathcal{H}{\rho }_{\text{DM}}=-aQ;\end{equation} \tag{ 73 }$$

$$\begin{equation}{\dot {\rho }}_{\text{DE}}+3\mathcal{H}(1+{w}_{\text{DE}}){\rho }_{\text{DE}}=aQ,\end{equation} \tag{ 74 }$$

and the interaction function Q reads:

$$\begin{equation}Q=3(1+{w}_{\text{DE}})\enspace \mathcal{H}\enspace {\rho }_{\text{DE}}.\end{equation} \tag{ 75 }$$

A fit of this model against Planck 2015 (Planck 2015 + BAO + R16 + CFHTLenS [671]) data gives ${H}_{0}=68.{6}_{-5.8}^{+4.1}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (${H}_{0}=70.{3}_{-1.3}^{+1.1}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$) at 68% CL [670], solving the tension with R20 at 1.1σ (1.7σ). The updated analysis of this scenario using Planck 2018 measurements is absent from the literature to date.

8.1.8. Interaction in the anisotropic Universe

8.1.9. Metastable interacting dark energy

A model of metastable interacting DE was studied in references [310, 311, 314]. The conservation equations for DE–DM in this scenario follow:

$$\begin{equation}{\dot {\rho }}_{\text{DE}}=-\enspace {\Gamma}{\rho }_{\text{DE}};\end{equation} \tag{ 76 }$$

$$\begin{equation}{\dot {\rho }}_{\text{DM}}+3\mathcal{H}{\rho }_{\text{DM}}={\Gamma}{\rho }_{\text{DE}},\end{equation} \tag{ 77 }$$

where Γ is a constant (for Γ < 0 DE density increases, while for Γ > 0 DE density decreases). Notice that it is an interacting DE–DM scenario with Q = Γρ_DE.

For the combination Pantheon + BAO data the model results in ${H}_{0}=71.{8}_{-4.6}^{+4.7}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL, showing that the Hubble tension is solved within 1σ [311]. However, in presence of the CMB distance priors from Planck 2018, the Hubble tension is restored at more than 3σ [311].

When a full analysis with Planck 2018 (Planck 2018 + BAO + DES + R19) is performed with this model, an increase of DE density (Γ < 0) is supported by the data together with a larger value of the Hubble constant ${H}_{0}=70.{3}_{-2.0}^{+3.3}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (${H}_{0}=69.1{2}_{-0.45}^{+0.46}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$) at 68% CL [314], solving the tension with R20 within 1σ (within 2.9σ).

8.1.10. Quantum field cosmology

The IDE model proposed in reference [673], that relies on the Einstein–Cartan gravitational theory and considers the Universe in the scale invariant ultra-violet fixed point of the theory (referred to as quantum field cosmology), has been investigated as a possible solution to the Hubble tension. In this model, Newton's gravitational constant G_N and the cosmological constant Λ possess a mild dependence on redshift, and vary according to the scaling laws:

$$\begin{equation}\frac{{G}_{\mathrm{N}}}{{G}_{0}}={(1+z)}^{-{\delta }_{G}},\quad \text{and}\quad \frac{{\Lambda}}{{{\Lambda}}_{0}}={(1+z)}^{{\delta }_{{\Lambda}}},\end{equation} \tag{ 78 }$$

with δ_G, δ_Λ ≪ 1, and approaching to an ultraviolet fixed point of G₀ and Λ₀ where the classical Einstein theory is realized.

This picture is extended in reference [674] to include the matter and radiation energy densities:

$$\begin{equation}\frac{{G}_{\mathrm{N}}}{{G}_{0}}{\rho }_{\mathrm{m},\mathrm{r}}={\rho }_{\mathrm{m},\mathrm{r}}^{0}{(1+z)}^{3(1+{w}_{\mathrm{m},\mathrm{r}})-{\delta }_{G}};\end{equation} \tag{ 79 }$$

$$\begin{equation}\frac{{G}_{\mathrm{N}}}{{G}_{0}}{\rho }_{{\Lambda}}={\rho }_{{\Lambda}}^{0}{(1+z)}^{{\delta }_{{\Lambda}}},\end{equation} \tag{ 80 }$$

where ${\rho }_{\mathrm{m}}^{0}$, ${\rho }_{\mathrm{r}}^{0}$, and ${\rho }_{{\Lambda}}^{0}$ are the present-day values of ρ_m, ρ_r, and ρ_Λ, and w_m, w_r are the equations of state for matter and radiation, respectively. When this model is analysed using the Planck 2018 distance priors + BAO + Pantheon, the Hubble constant obtained is ${H}_{0}=66.8{7}_{-1.61}^{+1.57}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [674], in disagreement with R20 at 3.2σ.

8.1.11. Interacting quintom dark energy

A quintom model [675–679], i.e. a DE model with two scalar fields where one of them has canonical kinetic energy and the second one a negative kinetic energy term, modified to include an interaction between DM and DE, can be considered a possible alternative to reconcile the Hubble constant tension, as argued in reference [680]. The addition of this extra component X with negative density, in the Friedmann equation, will leave unaltered the Planck's constraints on the matter and DE densities, but will match the requirements for solving the Hubble tension, acting as a phantom field, while the second scalar field will be quintessence. A full data analysis is however missing.

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In the standard ΛCDM model, DM is assumed to be collisionless. Therefore, a possible extension is a DM interacting with the other components of the Universe. This process can help in reconciling the Hubble constant tension. We already explored the possibility of self-interacting DM, decaying DM, and DM interacting with neutrinos and DE in the previous sections, therefore we shall restrict ourselves in the following to the remaining cases exclusively.

8.2.1. DM–photon coupling

A non-minimal coupling between photons and DM [681–684] has also been considered to ameliorate the Hubble tension.

The coupling between the DM fluid and photons can be described by:

$$\begin{equation}{\dot {\rho }}_{\text{DM}}+3\mathcal{H}{\rho }_{\text{DM}}=-Q;\end{equation} \tag{ 81 }$$

$$\begin{equation}{\dot {\rho }}_{\gamma }+4\mathcal{H}{\rho }_{\gamma }=Q,\end{equation} \tag{ 82 }$$

where $Q={{\Gamma}}_{\gamma }\mathcal{H}{\rho }_{\text{DM}}$. For this scenario, where the neutrino sector is free to vary, Planck 2015 TT + CMB lensing + BAO gives H₀ = 71.9 ± 4.0 km s⁻¹ Mpc⁻¹ at 68% CL [685], solving the H₀ tension within 1σ. However, this result has been obtained fitting the CMB temperature power spectrum only.

An extension of this model has been investigated in reference [686], considering a CPL parameterization for the DE equation of state, obtaining H₀ = 67.4 ± 3.9 km s⁻¹ Mpc⁻¹ at 68% CL for Planck 2015 + BAO, and alleviating the tension with R20 at 1.4σ .

An updated analysis is instead presented in reference [534], where Planck 2018 + BAO gives H₀ = 67.70 ± 0.43 km s⁻¹ Mpc⁻¹ at 68% CL, showing a disagreement with R20 at 4σ.

8.2.2. DM–baryon coupling

Contrary to the search for DM, for which realistic particle models motivate the search in direct detection experiments, a laboratory search of DE is conceptually complicated to start with, since the nature of DE is not clear. For example, DE could be due to a theory of gravity beyond GR, or it could be a manifestation of new fields. In the latter case, it is not even clear what the associated mass scale should be; for the case of a light scalar field, for example, we expect a field of a mass of the order of the Hubble constant [223, 227].

Surprisingly, the interaction between DE and baryon could proceed through a large Thompson cross section $\sim \mathcal{O}(\mathrm{b})$, with negligible impact on the CMB or structure formation [696, 697]. If instead a time-varying cross section is invoked, it is possible to have detectable signatures of an elastic interaction between baryons and DE. In this latter case, an analysis that accounts for the Planck 2018 + CMB lensing + BAO + JLA + CFHTLensS + Planck SZ datasets finds ${H}_{0}=67.6{5}_{-0.64}^{+0.80}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [698], and thus in disagreement with R20 at 3.7σ.

The physics of neutrinos is one of the appealing topics in modern cosmology. The neutrinos may in principle interact with each other or with other cosmic sectors, see for instance reference [699]. The possibility of an interacting neutrino sector has been explored recently to reconcile the Hubble constant tension. While the possibility of a DM sector interacting with neutrinos has been already discussed in section 7.5 in light of a contribution to ΔN_eff, here we shall restrict ourselves to previously unexplored models.

8.4.1. Self-interacting neutrinos

8.4.2. Self-interacting sterile neutrino model

8.4.3. Dark neutrino interactions

The generalized Chaplygin gas model (gcg) is characterized by the equation of state:

$$\begin{equation}{p}_{\text{gcg}}=-\frac{A}{{({\rho }_{\text{gcg}})}^{\alpha }},\end{equation} \tag{ 83 }$$

where A and α are two real constants, and p_gcg and ρ_gcg are, respectively, the pressure and energy density of this fluid. For this model, Planck 2015 estimates ${H}_{0}=71.{0}_{-3.7}^{+1.7}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [744], and this solves the tension with R20 at 1σ. The model needs to be updated with the final CMB data from Planck 2018.

In reference [745], we find a new type of a unified model based on field theory grounds. In this model, the explicit relation between the pressure p_u and the energy density ρ_u is [745–747]:

$$\begin{equation}{p}_{\mathrm{u}}=-{\rho }_{\mathrm{u}}+{\rho }_{\mathrm{u}}\enspace \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left(\frac{\mu \pi {\rho }_{\mathrm{u},0}}{{\rho }_{\mathrm{u}}}\right),\end{equation} \tag{ 84 }$$

where sinc(θ) = sin θ/θ, μ ≠ 0 is a dimensionless quantity, and ρ_u,0 is the energy density of the unified dark fluid today.

A fit to Planck 2015 data alone to this model results in ${H}_{0}=77.3{3}_{-0.73}^{+0.71}$ at 68% CL [747], alleviating the tension with R20 at 2.8σ. However, an analysis with the new Planck 2018 data is absent in the literature.

In references [748, 749] the Λ(t)CDM model has been considered to address the Hubble constant tension. The authors of references [748, 749] analyse a class of interacting models behaving as a gcg at the background level, i.e. like CDM at early times and a cosmological constant in the asymptotic future. The explicit expression of Λ(t) as considered in both the works has a Hubble dependence as:

$$\begin{equation}{\Lambda}(t)=\sigma {H}^{-2\alpha },\end{equation} \tag{ 85 }$$

where α > −1 is the interaction parameter and $\sigma =3(1-{{\Omega}}_{\mathrm{m}}){H}_{0}^{2(1+\alpha )}$. This is an example of a unified dark sector model where the Hubble rate follows

$$\begin{equation}\frac{H(z)}{{H}_{0}}=\sqrt{{\left[(1-{{\Omega}}_{\mathrm{m}})+{{\Omega}}_{\mathrm{m}}{(1+z)}^{3(1+\alpha )}\right]}^{\frac{1}{(1+\alpha )}}+{{\Omega}}_{\mathrm{r}}{(1+z)}^{4}},\end{equation} \tag{ 86 }$$

which recovers the standard ΛCDM model for α = 0.

For the above model of Λ(t)CDM, Planck 2015 TT + JLA + BBN + R19 estimates ${H}_{0}=69.1{2}_{-3.7}^{+1.7}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [748], and this solves the tension with R20 at 2.4σ, including already a Gaussian prior on H₀. An updated analysis is presented in reference [749], where Planck 2018 + CMB lensing + JLA + BBN + R19 gives H₀ = 70.73 ± 1.02 km s⁻¹ Mpc⁻¹ at 68% CL [749], solving the Hubble tension at 1.5σ, but always including a Gaussian prior.

In reference [750] the possibility that a cosmological constant, describing both the accelerated expansion of the Universe and the dynamics of Galaxy groups and clusters, could solve the Hubble tension is taken into account. This theory is called Λ-gravity and is considered in the modified weak-field limit of GR. In this context it is possible to have a local Hubble constant of a local flow and a global one [751], as a consequence of the common nature of DE and DM, solving naturally the Hubble constant problem.

Einstein theory of gravitation can be recovered from the principle of least action, once the Einstein–Hilbert action is introduced as

$$\begin{equation}\mathcal{S}=\frac{1}{2{\kappa }^{2}}\int {\mathrm{d}}^{4}x\sqrt{-g}\enspace \mathcal{R}+{\mathcal{S}}_{\mathrm{m}},\end{equation} \tag{ 87 }$$

where g is the determinant of the metric tensor, $\mathcal{R}$ is the Ricci scalar, κ² = 8πG_N, and ${\mathcal{S}}_{\mathrm{m}}$ is the action describing any matter fields appearing in the theory.

The simplest generalization of Einstein gravity is the $f(\mathcal{R})$ gravity, in which the Ricci scalar appears in the action in a generic function $f(\mathcal{R})$:

$$\begin{equation}\mathcal{S}=\frac{1}{2{\kappa }^{2}}\int {\mathrm{d}}^{4}x\sqrt{-g}\enspace f(\mathcal{R})+{\mathcal{S}}_{\mathrm{m}}.\end{equation} \tag{ 88 }$$

The modified gravity theory described in equation (88) has been widely investigated over the past years, considering various choices of the function $f(\mathcal{R})$. In this context, of particular interest is the Hu–Sawicki $f(\mathcal{R})$ model [796],

$$\begin{equation}f(\mathcal{R})=\mathcal{R}-{m}^{2}\frac{{c}_{1}{(\mathcal{R}/{m}^{2})}^{n}}{{c}_{2}{(\mathcal{R}/{m}^{2})}^{n}+1},\end{equation} \tag{ 89 }$$

where c₁, c₂ are constants, n > 0 is an index, and ${m}^{2}={H}_{0}^{2}{{\Omega}}_{\mathrm{m}}$. In this class of models, an accelerating phase can be achieved without introducing a cosmological constant while satisfying both galactic and solar-system constraints.

Recently, the Hu–Sawicki model has been tested in light of the H₀ tension with different conclusions, depending on the cosmological datasets considered. The authors of reference [797] study the Hu–Sawicki model for n = 1 using the geometrical data. Using the CC + Pantheon datasets, their best estimate for the Hubble constant is H₀ = 69.5 ± 2.0 km s⁻¹ Mpc⁻¹ at 68% CL [797], which alleviates the tension with R20 at 1.5σ. In reference [798], the author performed the analyses exploiting Planck 2018 data in combination with other cosmological probes, leaving the index n free to vary and also considering some specific values of this parameter. For example, for n = 1 Planck 2018 + CMB lensing (Planck 2018 + CMB lensing + RSD + BAO + Pantheon + CC) gives H₀ = 67.58 ± 0.64 km s⁻¹ Mpc⁻¹ (H₀ = 67.86 ± 0.42 km s⁻¹ Mpc⁻¹) at 68% CL [798], i.e. the H₀ tension is not alleviated within this specific $f(\mathcal{R})$ gravity model. Recently, further investigations aimed at testing whether the H₀ tension can be solved within the $f(\mathcal{R})$ theory have been performed in references [799, 800].

The theory of Einstein–Cartan is an extension of GR that describes gravity in spacetime metrics with a connection that has both torsion and curvature [801]. In the framework of Einstein–Cartan theory, GR is a limit which is formulated based on Levi-Civita connections, for which the spacetime metric is torsion-free and has a possible non-zero curvature. A different limit, in which the spacetime connection has a non-zero torsion tensor ${{\mathcal{T}}^{\lambda }}_{\mu \nu }$ and zero curvature (Weitzenböck connection) is teleparallel gravity, see e.g. references [802, 803]. A torsion scalar $\mathcal{T}$ can be constructed by contractions of the torsion tensor [804].

Models based on a modification of teleparallel gravity might lead to a successful alternative to inflationary models, resulting in an accelerated expansion rate without introducing an inflaton field [805–807]. ²⁶ These models are characterized by the inclusion in the action of an arbitrary function $f(\mathcal{T})$ of the torsion scalar:

$$\begin{equation}\mathcal{S}=\frac{1}{2{\kappa }^{2}}\int {\mathrm{d}}^{4}x\sqrt{-g}\enspace f(\mathcal{T})+{\mathcal{S}}_{\mathrm{m}}.\end{equation} \tag{ 90 }$$

An $f(\mathcal{T})$ model can be studied in a EFT framework, in which the action describing perturbations is expanded around a time-dependent background [809]. In this case, the first Hubble equation is modified as

$$\begin{equation}3{H}^{2}={\kappa }^{2}\rho +\frac{1}{2}\left[\mathcal{T}-f(\mathcal{T})+2\mathcal{T}{f}_{\mathcal{T}}\right],\end{equation} \tag{ 91 }$$

where ${f}_{\mathcal{T}}=\mathrm{d}f/\mathrm{d}\mathcal{T}$.

A simple parameterization for teleparallel gravity is the power-law model [810]:

$$\begin{equation}f(\mathcal{T})=-\mathcal{T}+\alpha {\mathcal{T}}^{b},\end{equation} \tag{ 92 }$$

where the torsion scalar, in the mostly plus sign convention for the metric signature, is $\mathcal{T}=6{H}^{2}$, and α, b are constants. In this model, the GR metric for ΛCDM is recovered for b = 0 and α = −2Λ. The model described in equation (92) may be able to alleviate the Hubble tension [811]: a fit to Planck 2015 + BAO data yields ${H}_{0}=72.{4}_{-4.1}^{+3.3}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [812], in agreement with R20 within 1σ, even in presence of the BAO measurements. An alternative analysis, based on Gaussian processes and H(z) data, is presented in reference [813], where the tension is also efficiently alleviated. In reference [797], instead, CC + Pantheon gives H₀ = 69.1 ± 1.9 km s⁻¹ Mpc⁻¹ at 68% CL, in tension at 1.8σ with R20. An updated analysis is presented in reference [814], where Planck 2018 for this scenario gives H₀ = 66.51 ± 3.65 km s⁻¹ Mpc⁻¹ at 68% CL, where the H₀ value is shifted towards a lower mean value but with a larger error, alleviating therefore the Hubble tension (1.7σ).

In order to attain a small variation of the gravitational coupling, reference [815] adopts a $f(\mathcal{T})$ model with an exponential form:

$$\begin{equation}f(\mathcal{T})=-\mathcal{T}+\frac{1-{{\Omega}}_{\mathrm{m}}}{(1+p){\mathrm{e}}^{-p}-1}{\mathcal{T}}_{0}\enspace \left(1-{\mathrm{e}}^{-p\sqrt{\mathcal{T}/{\mathcal{T}}_{0}}}\right),\end{equation} \tag{ 93 }$$

where ${\mathcal{T}}_{0}=6{H}_{0}^{2}$ and p > 0. The prefactor in equation (93) is obtained by evaluating equation (91) at present time. Note, that ΛCDM is recovered in the limit p → +∞. For this scenario, a fit to Planck 2018 data gives H₀ = 67.11 ± 0.56 km s⁻¹ Mpc⁻¹ at 68% CL, value in disagreement with R20 at the level of 4.4σ [814].

A different $f(\mathcal{T})$ model with an exponential form has been explored in reference [816]:

$$\begin{equation}f(\mathcal{T})=-\mathcal{T}+\frac{1-{{\Omega}}_{\mathrm{m}}}{(1+2q){\mathrm{e}}^{-q}-1}{\mathcal{T}}_{0}\enspace \left(1-{\mathrm{e}}^{-q\mathcal{T}/{\mathcal{T}}_{0}}\right),\end{equation} \tag{ 94 }$$

where q is a parameters. For this scenario, a fit to the Planck 2018 data yields the result H₀ = 67.12 ± 0.56 km s⁻¹ Mpc⁻¹ at 68% CL [814], value that shows a 4.4σ disagreement with R20.

Another $f(\mathcal{T})$ parameterization with an exponential form [808]:

$$\begin{equation}f(\mathcal{T})=-\mathcal{T}\enspace {\text{e}}^{\beta ({\mathcal{T}}_{0}/\mathcal{T})},\end{equation} \tag{ 95 }$$

where β is found from solving 1–2β = Ω_me^−β , has been explored in reference [817] in relation with the Hubble tension. A fit to Pantheon + R20 + BBN + BAO gives at the background level H₀ = 70.7 ± 1.3 km s⁻¹ Mpc⁻¹ at 68% CL [817], alleviating the Hubble tension at 1.4σ. This estimate, however, already includes a Gaussian prior on the Hubble constant. A full CMB analysis including perturbations has been performed in reference [818], where Planck 2018 (Planck 2018 + CMB lensing + BAO) gives H₀ = 72.03 ± 0.70 km s⁻¹ Mpc⁻¹ (H₀ = 71.49 ± 0.47 km s⁻¹ Mpc⁻¹) at 68% CL, solving the Hubble tension within 1σ (at 1.2σ) without the introduction of extra free parameters.

Finally, reference [819] presents constraints on teleparallel gravity and its $f(\mathcal{T})$ extensions using Gaussian processes, and reference [820] reconstructs the free function of $f(\mathcal{T})$ gravity in a model-independent manner using different datasets and relieving the Hubble tension.

An extension of the $f(\mathcal{T})$ scenario is the $f(\mathcal{T},\mathcal{B})$ gravity theory where, along with the torsion $\mathcal{T}$, the boundary term $\mathcal{B}=2{\nabla }_{\mu }{\mathcal{T}}_{\nu }^{\nu \mu }$ is also included [821]. Recently, some specific models of $f(\mathcal{T},\mathcal{B})$ gravity were examined with the observational data in reference [822], where the authors argued there that the H₀ tension can be weakened in this context. In particular, the authors of reference [822] investigated two different models:

$$\begin{equation}f(\mathcal{T},\mathcal{B})={b}_{0}{\mathcal{B}}^{k}+{t}_{0}{\mathcal{T}}^{\mathrm{m}}\hspace{25.0pt}(\text{Power}-\text{law}\;\text{model});\end{equation} \tag{ 96 }$$

$$\begin{equation}f(\mathcal{T},\mathcal{B})={f}_{0}{\mathcal{B}}^{k}\enspace {\mathcal{T}}^{\mathrm{m}}\hspace{25.0pt}(\text{Mixed}\;\text{power}-\text{law}\;\text{model}),\end{equation} \tag{ 97 }$$

where b₀, t₀, k, m and f₀ are all arbitrary constants. A fit to BAO + Pantheon + CC datasets gives H₀ = 67.74 ± 1.1 km s⁻¹ Mpc⁻¹ at 68% CL [822] for the power-law model, reducing the H₀ tension with R20 down to the 3.2σ level, and ${H}_{0}=67.8{6}_{-1.1}^{+1.2}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [822] for the mixed power-law model, reducing the tension down to the 3σ level. In this context, an analysis with the full CMB data is missing.

All models discussed so far assume ∇_α g_μν = 0, which is a condition that assures that angles and lengths are preserved under parallel transport. This assumption is dropped in extensions of GR that include non-Riemannian spacetime metrics, introducing a non-zero non-metricity tensor ${\mathcal{Q}}_{\alpha \mu \nu }={\nabla }_{\alpha }{g}_{\mu \nu }$ (see e.g. references [823–825]). In this framework, the action for a model that includes an arbitrary function $f(\mathcal{Q})$ on a torsion- and curvature-free geometry is:

$$\begin{equation}\mathcal{S}=\frac{1}{2{\kappa }^{2}}\int {\mathrm{d}}^{4}x\sqrt{-g}\enspace f(\mathcal{Q})+{S}_{\mathrm{m}},\end{equation} \tag{ 98 }$$

where the non-metricity scalar $\mathcal{Q}$ is defined as $\mathcal{Q}=-{\mathcal{Q}}_{\alpha \mu \nu }{P}^{\alpha \mu \nu }$, in terms of the non-metricity conjugate:

$$\begin{equation}{P}_{\mu \nu }^{\alpha }=-\frac{1}{2}\left(\frac{1}{2}{{\mathcal{Q}}^{\alpha }}_{\mu \nu }-{{\mathcal{Q}}_{(\mu \nu )}}^{\alpha }\right)+\frac{1}{4}\left({\mathcal{Q}}^{\alpha }-{\tilde {Q}}^{\alpha }\right){g}_{\mu \nu }-\frac{1}{4}{\delta }_{\left(\right.\mu }^{\alpha }{\mathcal{Q}}_{\nu \left.\right)}.\end{equation} \tag{ 99 }$$

Here, ${\mathcal{Q}}_{\alpha }={g}^{\mu \nu }{\mathcal{Q}}_{\alpha \mu \nu }$ and ${\tilde {\mathcal{Q}}}_{\alpha }={g}^{\mu \nu }{\mathcal{Q}}_{\mu \alpha \nu }$ are the two independent traces of the non-metricity tensor, and round brackets mean a symmetrisation over the indices.

In reference [825], the $f(\mathcal{Q})$ modified gravity model is tested against the Pantheon sample using a cosmographic approach, in which the parameterization of $f(\mathcal{Q})$ involves an increase in the numbers of derivatives in the theory. More specifically, the authors consider three cosmographic $f(\mathcal{Q})$ models, namely M1, M2, and M3. The estimated values of H₀ are higher than R20, since H₀ = 79.5 ± 2.5 km s⁻¹ Mpc⁻¹ at 68% CL [825] for M1, H₀ = 79.2 ± 3.1 km s⁻¹ Mpc⁻¹ at 68% CL [825] for M2, and H₀ = 79.5 ± 2.6 km s⁻¹ Mpc⁻¹ at 68% CL [825] for M3. The tension with R20 is reduced down to the 2.3σ, 1.8σ and 2.2σ levels, respectively. However, a full analysis with Planck CMB data is missing for this theoretical framework.

The replacement of Newton's gravitational constant G_N with a coupling that varies with cosmic time, G_N(t), has been proposed for the first time by Brans & Dicke (BD) [826]. In the BD theory, Newton's constant is promoted to a dynamical field that depends on the spacetime coordinates. In more detail, the action describing the BD theory depends on the BD field Φ as

$$\begin{equation}\mathcal{S}=\frac{1}{2{\kappa }^{2}}\int {\mathrm{d}}^{4}x\sqrt{-g}\enspace \left[{\Phi}\mathcal{R}-\frac{\omega }{{\Phi}}{g}^{\mu \nu }{\partial }_{\mu }{\Phi}{\partial }_{\nu }{\Phi}-2V({\Phi})\right]+{\mathcal{S}}_{\mathrm{m}},\end{equation} \tag{ 100 }$$

where ω is a new parameter in the theory and κ depends on the value of G_N measured today. It can be shown that the GR limit in equation (87) is recovered for ω → +∞.

The Jordan [827], Brans & Dicke (JBD) gravity has been extensively studied in the literature (see references [386, 828–848]) and can possibly embed the running vacuum model [849, 850] (see also section 5.9). Models of JBD gravity where the Hubble tension is alleviated have also been discussed, as reviewed below.

10.5.1. BD-ΛCDM

In reference [851], a BD cosmology with an additional cosmological constant term (the BD-ΛCDM model) is considered in light of easing the Hubble tension. In this case the action reads

$$\begin{equation}\mathcal{S}=\int {\mathrm{d}}^{4}x\sqrt{-g}\enspace \left[\frac{1}{2{\kappa }^{2}}\left({\Phi}\mathcal{R}-\frac{\omega }{{\Phi}}{g}^{\mu \nu }{\partial }_{\mu }{\Phi}{\partial }_{\nu }{\Phi}\right)-{\rho }_{{\Lambda}}\right]+{\mathcal{S}}_{\mathrm{m}}.\end{equation} \tag{ 101 }$$

For this scenario, Planck 2015 + CMB lensing + BAO + RSD + KiDS-450 + R19 gives H₀ = 72.0 ± 1.0 km s⁻¹ Mpc⁻¹ at 68% CL, solving the tension within 1σ. Nevertheless, these results include a Gaussian prior for the Hubble constant.

An update with new data has been performed in reference [852], where Planck 2018 + Pantheon + BAO + RSD + CC + H₀ from [68] gives ${H}_{0}=69.8{5}_{-0.85}^{+0.81}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [852], alleviating the tension down to 2.2σ. However, a prior on the Hubble constant is already included in the analysis.

A result without this prior, and also without CMB polarization measurements, provides instead ${H}_{0}=68.8{6}_{-1.24}^{+1.15}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [852], for Planck 2018 TT + Pantheon + RSD + BAO + CC, reducing the tension down to the 2.6σ level.

The JBD theory can be reformulated to include the equivalent formulation of induced gravity (IG) in a scalar–tensor model [853, 854]:

$$\begin{equation}\mathcal{S}=\frac{1}{2{\kappa }^{2}}\int {\mathrm{d}}^{4}x\sqrt{-g}\enspace \left[F(\sigma )\mathcal{R}-{g}^{\mu \nu }{\partial }_{\mu }\sigma {\partial }_{\nu }\sigma -2V(\sigma )-2{\Lambda}\right]+{\mathcal{S}}_{\mathrm{m}},\end{equation} \tag{ 102 }$$

where σ is the scalar field in units of M_Pl which is responsible for generating Newton's gravitational constant G_N through the spontaneous breaking of scale invariance and moving in a potential V(σ), while $F(\sigma )={N}_{\text{Pl}}^{2}+\xi {\sigma }^{2}$ where N_Pl is a parameter and ξ > 0 is the coupling to the Ricci scalar. The conformal coupling case is ξ = −1/6 and N_Pl = 0 (ξ = 0 and N_Pl = 1) for IG (GR).

In reference [855], an extended JBD is considered to alleviate the Hubble tension, assuming an effectively massless scalar field σ with a potential V ∝ F². A fit to Planck 2015 TT + CMB lensing + BAO data gives ${H}_{0}=69.{4}_{-0.9}^{+0.7}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [855], reducing the tension with R20 at 2.5σ. An update of this model is performed in reference [856], where Planck 2018 + CMB lensing (Planck 2018 + CMB lensing + BAO + R19) gives ${H}_{0}=69.{6}_{-1.7}^{+0.8}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (H₀ = 70.06 ± 0.81 km s⁻¹ Mpc⁻¹) at 68% CL [856], alleviating the Hubble tension at 2.4σ (2.1σ).

For the conformal coupling model, a fit to Planck 2015 TT + CMB lensing + BAO data gives ${H}_{0}=69.1{9}_{-0.93}^{+0.77}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [855], in disagreement with R20 at 2.7σ. An update of this model is performed in reference [856], where Planck 2018 + CMB lensing (Planck 2018 + CMB lensing + BAO + R19) gives ${H}_{0}=69.{0}_{-1.2}^{+0.7}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (${H}_{0}=69.6{4}_{-0.73}^{+0.65}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$) at 68% CL [856], with the Hubble tension still in disagreement at 2.8σ (2.4σ).

An extension of the previous scenario is studied in reference [857], where the non-minimal coupling of the scalar field to the Ricci scalar is:

$$\begin{equation}F(\sigma )=1+\xi {\sigma }^{n}.\end{equation} \tag{ 103 }$$

Unfortunately, all the cases considered in the context of this model are in disagreement with R20 at more than 3σ.

A similar scenario, where a variation of the Newton's gravitational constant G_N between the early and the late Universe is accounted for, in the context of a scalar field model which is non-minimally and quadratically coupled to gravity, is considered in reference [858]. The H₀ value for this model using Planck 2018 + BAO is estimated to be ${H}_{0}=68.2{4}_{-0.79}^{+0.5}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [858] and the disagreement with R20 is at the level of 3.5σ.

10.6.1. Early modified gravity

10.6.2. Screened fifth forces

The Über gravity model is a fixed point in the space of the gravity models obtained from varying the Ricci scalar [869]. This model mimics the Einstein–Hilbert theory in the high-curvature regime, while in the low-curvature regime it predicts a sharp transition at a model-dependent Ricci scale R₀. The cosmological model embedded in this theory, the ÜΛCDM model, is characterised by a density-dependent transition between ΛCDM and a phase in which the Ricci scalar is constant [870].

This scenario has been proposed to alleviate the Hubble tension in reference [871]. The combined analysis of Planck 2015 TT + R16 + BAO for this model estimates ${H}_{0}=70.{6}_{-1.3}^{+1.1}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [871], in agreement with R20 at 1.5σ. However, this result relies exclusively on Planck temperature data at high multipoles, on a Gaussian prior on H₀, as measured by R16, and on BAO measurements. In reference [871] it is shown that for Planck 2015 alone the constraints are largely relaxed, and therefore a possible agreement with R20 would be possible within one standard deviation. An updated analysis for this scenario has been performed in reference [59], and Planck 2018 (Planck 2018 + CMB lensing + R19) measurements provide a value of ${H}_{0}=70.{7}_{-2.6}^{+1.4}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at (H₀ = 73.2 ± 1.3 km s⁻¹ Mpc⁻¹) at 68% CL [59], in agreement with R20 within 1.3σ (1σ).

The covariant Galileon model is a theory of modified gravity in which the accelerated expansion rate of the Universe is driven by a scalar field φ, whose Lagrangian is invariant under the Galilean shift symmetry by a constant vector b_μ , ∂_μ φ → ∂_μ φ + b_μ [872, 873]. One aspect of this model is that the background component of the Galileon field ϕ is described by a 'tracker' evolution, Hdϕ/dt ∝ ξ, where ξ is a constant [874]. Once the Galileon field has reached the tracker solution, its energy density contributes appreciably to the total energy budget, reaching the present value

$$\begin{equation}{{\Omega}}_{\phi 0}=\frac{{c}_{2}}{6}{\xi }^{2}-2{c}_{3}{\xi }^{3}+{c}_{4}\frac{15}{2}{\xi }^{4}+{c}_{5}\frac{7}{3}{\xi }^{5}.\end{equation} \tag{ 104 }$$

Depending on the highest exponent for ξ, we refer either to the cubic (c₄ = c₅ = 0), quartic (c₅ = 0), or quintic Galileon model.

In reference [875] the Galileon gravity scenario has been proposed to solve the Hubble tension. When the total neutrino mass is allowed to vary in addition to the standard parameters, the combination Planck 2015 TT + CMB lensing + BAO leads to the values H₀ = 71.6 ± 2.1 km s⁻¹ Mpc⁻¹, H₀ = 72.4 ± 2.0 km s⁻¹ Mpc⁻¹ and H₀ = 72.3 ± 2.1 km s⁻¹ Mpc⁻¹ for the several possible Galileon scenarios (cubic, quartic and quintic, respectively), all with 95% CL errors. The Hubble tension is therefore reduced in these cases within 1σ, even if the BAO observations are also included. Updated results with Planck 2018 are not yet available.

A similar scenario has been analysed in reference [876], where the authors studied a generalized cubic covariant Galileon scenario. Planck 2015 TT (Planck 2015 TT + BAO + RSD + JLA) gives in this case ${H}_{0}=7{2}_{-5}^{+8}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (H₀ = 68.4 ± 0.9 km s⁻¹ Mpc⁻¹) at 95% CL [876], solving the Hubble tension within 1σ (3.4σ). Once the Planck high-ℓ polarization is included, the Hubble tension is however restored.

In reference [877], instead, it has been argued that the problem can be overcome in the enhanced early gravity model, i.e. an exponentially coupled cubic Galileon scenario, where Planck 2018 + BAO gives H₀ = 68.7 ± 1.5 km s⁻¹ Mpc⁻¹ at 68% CL, relaxing the Hubble tension down to 2.3σ level.

Within the subclass of generalized Proca interactions [878–880], the authors of reference [881] focus on the cubic Galileon scenario, based on a vector field for the solution of the Hubble constant tension, mainly due to the phantom-like behaviour of DE. Using a combination of Planck 2018 + BAO + Pantheon + R19, they find H₀ = 70.1 ± 0.76 km s⁻¹ Mpc⁻¹ at 68% CL [881], alleviating the Hubble tension at 2.4σ, but including a Gaussian prior on H₀. The same dataset combination without R19 restores the tension above 3σ.

Finally, in reference [882] a Galileon ghost condensate model has been studied to alleviate the Hubble constant tension. For this scenario Planck 2015 TT (Planck 2015 TT + BAO + RSD + JLA) gives ${H}_{0}=69.{3}_{-3.0}^{+3.6}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (H₀ = 68.1 ± 1.1 km s⁻¹ Mpc⁻¹) at 95% CL [882], solving the Hubble tension within 1.8σ (3.6σ).

The introduction of quantum gravity effects in the Einstein–Hilbert action leads to the presence of non-local effects that typically signal the presence of quantum properties corresponding to the local fundamental action of gravity, including the effect of quantum fluctuations [883, 884]. Among the possible nonlocal gravity models there is the RR scenario, in which the Einstein–Hilbert action in equation (87) is modified as [885]

$$\begin{equation}\mathcal{S}=\frac{1}{2{\kappa }^{2}}\int {\mathrm{d}}^{4}x\sqrt{-g}\enspace \left[\mathcal{R}-\frac{{m}^{2}}{6}\mathcal{R}\frac{1}{{\square }^{2}}\mathcal{R}\right]+{\mathcal{S}}_{\mathrm{m}},\end{equation} \tag{ 105 }$$

where m is a new mass parameter. In in this scenario, the nonlocal term acts as an effective DE with a phantom equation of state.

The RR model is relevant for solving the Hubble tension [886]. For this particular scenario, the combination of Planck 2015 + BAO + JLA results in ${H}_{0}=69.4{9}_{-0.80}^{+0.79}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [886], reducing the Hubble constant tension down to 2.5σ.

Updated results for nonlocal gravity models have been performed in reference [887]. While the RR model does not satisfy the lunar laser ranging constraints, the RT model [888] works better and for Planck 2018 + Pantheon + BAO the minimal case gives ${H}_{0}=68.7{4}_{-0.51}^{+0.59}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [887], reducing the tension with R20 at 3.2σ.

Another gravitational theory having close resemblance to the Einstein gravity is the unimodular gravity. The unimodular gravity is obtained by adding the unimodular condition [889] to the Einstein-gravity action through the Lagrange multiplier λ [890].

The possibility that this gravitational theory could help in resolving of the H₀ tension has been recently discussed in reference [890]. The H₀ tension can be alleviated by allowing for a non-gravitational interaction between the DM and the DE fluids within this gravitational context. Considering four different interaction rates between the DM and DE, namely, sudden transfer model (model 1), anomalous decay of the matter density (model 2), barotropic model (model 3) and continuous spontaneous localization (model 4), the best estimations of the Hubble constant for the combined dataset including Planck 2018 CMB distance priors + Pantheon + R19 + H0LiCOW, are, respectively, ${H}_{0}=73.{4}_{-0.6}^{+1.5}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [890] (model 1), ${H}_{0}=73.{2}_{-0.9}^{+1.4}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [890] (model 2), H₀ = 70 ± 1 km s⁻¹ Mpc⁻¹ at 68% CL [890] (model 3) and H₀ = 72 ± 1 km s⁻¹ Mpc⁻¹ at 68% CL [890] (model 4). For model 1, model 2 and model 4, the H₀ values are in agreement with R20 within 1σ, while for model 3 the H₀ tension with R20 is reduced to 2σ. However, when only CMB data are considered, the value of H₀ is unconstrained, and therefore the Gaussian priors on the Hubble constant are essential in the analysis to constrain it. A complete data analysis to Planck 2018 observations is missing.

A cosmological model with a scale—dependent scenario of gravity has been proposed in reference [891] to potentially alleviate the Hubble tension. Unfortunately, the data analysis is missing.

A cosmological theory where the cosmological constant term Λ of the standard ΛCDM scenario is replaced by a free function V(ϕ), without introducing any extra physical degrees of freedom, was proposed in reference [892]. The 'V' of VCDM therefore stands for the free function V(ϕ). The authors of reference [893] studied a specific model in this context finding that the H₀ tension can be alleviated. Planck 2018 + BAO + Pantheon gives ${H}_{0}=71.7{3}_{-0.29}^{+0.58}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 95% CL [893], alleviating the tension with R20 at 1.1σ.

In the single-field inflation model, there exists a degeneracy between the spectral index of the primordial scalar power spectrum, n_s, and N_eff, see e.g. reference [965]. Therefore, in principle, it is possible to build a model in which the interplay between the inflationary mechanism and the presence of additional dark radiation may alleviate the Hubble tension, due to the strong correlation between ΔN_eff and H₀. In the following, we shall discuss the inflationary models that embed additional dark radiation.

The authors of reference [966] re-examine various inflationary models in light of the presence of additional dark radiation. They study the large-field inflation scenario with a potential V(ϕ) ∝ ϕ² [967], the natural inflation model [968, 969], the Starobinsky model [970], and the power-law inflation paradigm (PLI) [971], in which the potential is given in terms of an amplitude M and an index α > 0 by: ²⁹

$$\begin{equation}V(\phi )={M}^{4}\enspace \mathrm{exp}\left(-\alpha \frac{\phi }{{M}_{\text{Pl}}}\right).\end{equation} \tag{ 106 }$$

In the ΛCDM, the PLI model is excluded since it predicts a value of the tensor-to-scalar ratio r = 8(1 − n_s) that lies above the limit from current observations [972]. However, when including an extra component of dark radiation, the relatively large value of n_s predicted within PLI, turns into an opportunity to address the Hubble tension since, using Planck 2015 TT + CMB lensing datasets, the Hubble constant results in H₀ = 73.6 ± 0.95 km s⁻¹ Mpc⁻¹ at 68% CL [966], in agreement with R20 within 1σ (see also reference [973] for similar results). For the very same scenario, the addition of the CMB polarization data and a lower value for the optical depth τ, as preferred by the new Planck 2018 power spectra, restores the Hubble constant tension above 3 standard deviations [974]. The same results are confirmed even when an origin of the Universe from the quantum landscape multiverse is considered (see reference [975]).

A possibility for solving the Hubble tension is to change the PPS. In reference [976] it has been shown that band-limited features in the PPS cannot resolve the Hubble tension.

In reference [977], instead, the shape of the PPS is reconstructed by implementing a modified Richardson–Lucy algorithm (MRL), and assuming the fitting of the Planck 2015 TT data, H₀ from R18 and S₈ from the cosmic shear data. This reconstructed model allows the data to be perfectly in agreement with the measured R20 Hubble constant. The reconstructed form of the PPS will have a suppression of power at large scales and sharp fluctuations at wave numbers larger than 0.02 Mpc⁻¹.

A generalization is performed in reference [978], where a class of PPS, that continuously deforms between the best-fit power-law and the MRL-reconstructed PPS, is parameterized. This interpolation is called 'deformation model', and the Hubble constant is degenerate with the new degree of freedom in the PPS. Using Planck 2018 TT, the Hubble constant is H₀ = 70.2 ± 1.2 km s⁻¹ Mpc⁻¹ at 68% CL [978], solving the tension with R20 at 1.7σ. However, it is unclear whether this result holds once polarization data are included in the analysis.

In reference [979] the authors show that the quintessential inflation, coming from the Lorentzian distribution introduced in references [980, 981], agrees with the recent observations and is in agreement with R20. In particular they find for Planck 2018 CMB distance priors + BAO + Pantheon + CC + R19, H₀ = 71.75 ± 0.89 km s⁻¹ Mpc⁻¹ at 68% CL [979].

Because of the existing degeneracy between n_s, N_eff, and H₀, in reference [982] the authors pointed out that in light of the Hubble tension a Harrison–Zel'dovich [983–985] PPS n_s = 1 is not ruled out by the data.

A modified effective electron rest mass m_e during the cosmological recombination era [988] could provide a mechanism to reduce the Hubble constant tension. In reference [989] it has been shown that Planck 2018 + BAO gives H₀ = 69.1 ± 1.2 km s⁻¹ Mpc⁻¹ at 68% CL [989] for a varying m_e, lowering the H₀ tension down to the 2.3σ level. The concordance model results in a larger electron rest mass m_e = (1.0078 ± 0.0067)m_e,0 at 68% CL [989].

In reference [990], an explicit model showing how the recombination history of the Universe can be modified has been proposed, in which a time varying electron mass m_e plays a key role. Specifically, a time varying electron mass can shift the recombination epoch z_* and the drag epoch z_d from the baseline model without affecting the CMB power spectra. Thus, considering the varying electron mass within the Ω_k ΛCDM model, the best estimated value of the Hubble constant for Planck 2018 + BAO + Pantheon is ${H}_{0}=72.{3}_{-2.8}^{+2.7}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [990]. This reconciles the tension with R20 at less than 1σ.

The authors of reference [991] present a simple model in which a light axion is coupled to the Higgs field. In this scenario, the Higgs vacuum expectation value in the early Universe is larger than its measured value, thus modifying the electron rest mass and possibly alleviating the Hubble tension (see section 12.1). The largest estimate presented in reference [991] for the Hubble constant is achieved with an analysis a posteriori of the results obtained in [989] for Planck 2018 + BAO, with m_e free to vary. Assuming a model with non-linear BBN, the analysis gives H₀ = 69.24 ± 0.68 km s⁻¹ Mpc⁻¹ at 68% CL [991], alleviating the tension with R20 at 2.6σ. A complete full CMB data analysis is however missing.

Additional small-scale, mildly non-linear inhomogeneities in the baryon density changing the recombination history could be a possible route to alleviate the H₀ tension [992]. These might be caused by the evolution of PMFs prior to recombination.

Using the model proposed in reference [993] and analysed in reference [994], the combination of Planck 2018 + CMB lensing + R19 + H0LiCOW + MCP gives H₀ = 71.03 ± 0.74 km s⁻¹ Mpc⁻¹ at 68% CL [992], reducing the Hubble tension at 1.4σ. However, Gaussian priors on the Hubble constant are already included in this analysis, inducing a possible bias in the result.

Another possibility is to have a weaker impact on recombination, with only a tiny fraction of the total volume in high density regions. Planck + CMB lensing + R19 + H0LiCOW + MCP results in a value of H₀ = 69.81 ± 0.62 km s⁻¹ Mpc⁻¹ at 68% CL [992], alleviating the Hubble tension down to the 2.4σ level. Nevertheless, the Gaussian priors are also already included in this analysis, as in the previous case, which may bias the result.

The double-Λ cold dark matter (Λ ΛCDM) scenario is inspired by the Ising model, a classic model of critical phenomena describing the phase transition from para-magnet to ferro-magnet at Curie temperature. This cosmological scenario assumes a cosmological constant with two values before a transition redshift and with a single value afterwards. In reference [995] it has been shown that, with this phase transition in the dark sector, the Hubble constant tension can be solved. Considering a χ² analysis, and the combination of Planck 2015 TT + BAO + R19, the Hubble constant is H₀ = 72.8 ± 1.6 km s⁻¹ Mpc⁻¹ at 68% CL [995]. The H₀ tension with R20 is therefore solved within 1σ, but including already a Gaussian prior on H₀.

In the Ginzburg–Landau theory of DE a phase transition happens, causing a spontaneous symmetry breaking, in the Landau approximation. Considering a χ² analysis, and the combination of Planck 2015 CMB distance priors on the angular size of horizon at decoupling and Ω_m h² + BAO + R18 + quasars H(z) data, the Hubble constant is H₀ = 71.89 ± 0.93 km s⁻¹ Mpc⁻¹ at 68% CL [996]. While the H₀ tension with R20 is therefore solved within 1σ, this result relies on a non full CMB data analysis and includes already a Gaussian prior on H₀.

Based on the physics of the critical phenomena, a DE model named as critically emergent DE was recently proposed in reference [997]. The evolution of the DE in this model takes the form:

$$\begin{equation}{{\Omega}}_{\text{DE}}(z)=(1-{{\Omega}}_{\mathrm{m}}-{{\Omega}}_{\mathrm{r}})\enspace \sqrt{\frac{{z}_{\mathrm{c}}-z}{{z}_{\mathrm{c}}}},\end{equation} \tag{ 107 }$$

where z_c is the transition redshift from which DE starts to emerge, and the corresponding DE equation of state is

$$\begin{equation}{w}_{\text{DE}}(z)=-1-\frac{1+z}{6({z}_{\mathrm{c}}-z)}.\end{equation} \tag{ 108 }$$

For this model, a fit to Planck 2018 data results in a value of ${H}_{0}=70.{0}_{-2.7}^{+1.2}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [997], which solves the tension with R20 within 1.8σ.

Inhomogeneities in the density distribution could lead to a modification of the expansion rate over some finite region of spacetime; a domain average density in the locally observed region would lead to a modification of the estimate for the local Hubble parameter from the value inferred from using the global background energy density, see section 3.1 for the relevant literature.

In reference [998] the possibility that the Hubble tension could be solved within the general relativistic framework of perturbation theory in an inhomogeneous Universe is investigated. The authors find that the crucial point is the first-order effect due to inhomogeneities at linear order in perturbation theory.

In reference [999] a simple anisotropic correction to the standard ΛCDM model by replacing the spatially flat FLRW metric with the Bianchi type-I metric has been investigated. Adopting a compilation of 36 H(z) measurements from CC, BAO signal in both Galaxy and Ly-α forest distributions, the authors estimate H₀ = 70.4 ± 1.7 km s⁻¹ Mpc⁻¹ at 68% CL [999] in the anisotropic ΛCDM scenario, in combination with the Planck 2015 distance prior, which is in agreement with both the CMB and the R20 values within 2σ. A full analysis considering CMB data is still missing, but the scenario is highly promising.

The inclusion of the backreaction from inhomogeneities in the cosmological expansion within GR has been proposed in references [181–184], see also section 3.1. This scheme is generally referred to in the literature as the 'Buchert equations'.

In reference [1000] a class of 'scaling solutions' satisfying this scalar averaging scheme has been proposed to solve the Hubble tension, while at the same time being in agreement with a slightly positively curved Universe as measured by Planck [11, 18, 19]. In fact, in this generic average model, there is a dynamical curvature, i.e. curvature and structure in the matter distribution are dynamically coupled within GR and without the necessity of introducing a DE component. A full data analysis is however missing for this interesting possibility.

In reference [1001], a GR fluid simulation of the LSS that includes a nonlinear evolution of structures leads to a negative emerging spatial curvature and to a value of the Hubble constant H₀ = 72.5 ± 2.1 km s⁻¹ Mpc⁻¹. Instead, neglecting the inhomogeneities, the same simulation finds a lower value H₀ = 68.1 ± 2.0 km s⁻¹ Mpc⁻¹, in agreement with the findings in reference [1000]. Instead, an independent analysis following a fully inhomogeneous, anisotropic relativistic simulation finds that these effects alone are not sufficient to reconcile the discrepancy represented by the Hubble tension [1002].

In reference [1003] the possibility of varying the CMB monopole temperature T₀, typically fixed when considering the CMB data, is explored to solve the Hubble tension. This is in fact fixed because of the extremely good precision of measurements of T₀ from the cosmic background explorer (COBE) far infrared absolute spectrophotometer (FIRAS) data, molecular lines, and balloon-borne experiments, but should in principle be considered as an extra free cosmological parameter to be varied in the Bayesian analysis [1004]. Using Planck 2018 + CMB lensing (Planck 2018 + CMB lensing + BAO), reference [1003] finds ${H}_{0}=55.{3}_{-4.9}^{+1.7}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (${H}_{0}=67.9{2}_{-0.51}^{+0.49}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$) at 68% CL, increasing the tension with R20 at several standard deviations (3.8σ) when varying T₀. Therefore, even if a strong anti-correlation is present between T₀ and H₀, this is not enough for solving the Hubble tension because the CMB data prefer the wrong direction.

14.4.1. Open and hotter Universe

In reference [1007] it has been assumed that a non-Gaussian covariance, due to possibly non-Gaussian primordial fluctuations, can be extracted from a four-point correlation function. This non-Gaussian covariance can be modeled through two additional degrees of freedom describing the trispectrum in the theoretical CMB angular power spectrum, and the resulting model has been named as super-ΛCDM. The combination of Planck 2015 TT + τ prior (Planck 2015 TT + τ prior + R19 + Pantheon) gives ${H}_{0}=68.{4}_{-2.3}^{+2.5}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (H₀ = 69.9 ± 1.7 km s⁻¹ Mpc⁻¹) at 95% CL [1007], reducing the Hubble tension at 2.7σ (2.1σ). It should be checked if this result holds after the inclusion in the fit of the Planck 2018 polarization data.

In reference [1008] it has been studied how the Heisenberg principle can affect the reliability of cosmological measurements. The authors ascribe the Hubble tension as the effect due to the indetermination associated to the comparison of kinematical versus dynamical measurements. They conclude that the uncertainty on a possible photon mass not accounted for, can be the reason for the H₀ disagreement.

Another possible way to alleviate the Hubble tension, considering an effective energy flux from the matter sector into DE has been proposed in reference [1009]. This scenario results naturally from a combination of unimodular gravity and an energy diffusion process. While the two simple models proposed in this study (one of them assuming a quick transfer of energy from the matter density to the cosmological constant sector, and a second one in which a diffusion process decreases anomalously the matter density) may be able to solve the Hubble tension. A complete data analysis is absent in the literature.

In reference [1010] the extrapolation of physics of the quantum vacuum [1011], a theory well-tested in atomic, molecular and optical physics, has been proposed to solve the Hubble tension. In this model the vacuum energy is time-dependent because of the Casimir forces, and therefore Λ varies with the cosmic expansion, allowing a larger value for H₀.

In reference [1012], the author argues that the inclusion of the surface forces of the homogeneous and isotropic Universe from the Euler Cauchy stress principle can explain the present accelerating expansion of the Universe without any DE fluid. The model was constrained using a joint analysis of Hubble parameter measurements and the Pantheon sample. The resulting value of the Hubble constant is enhanced (${H}_{0}=74.6{3}_{-2.7}^{+3.2}\enspace \mathrm{k}\mathrm{m}\enspace {\mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ at 68% CL [1012]) solving perfectly the tension with R20. A full analysis involving Planck 2018 data is missing.

In reference [1013], the author considers alleviating the Hubble tension within the Milne model [1014]. However, the model needs to be fitted with the observational data in order to be more conclusive in this direction.

In reference [1015] a running of H₀ as a function of redshift is proposed to alleviate the Hubble tension, although a full data analysis is missing. A similar idea has been explored in reference [1016] to analyse the Pantheon SNIa data.

In reference [1017] a rapid transition in the value of the relative effective gravitational constant is proposed to explain the lower luminosity of local SNIa and solve the H₀ crisis. In particular, the authors assume that there is a transition of this luminosity at z_t = 0.01, with a 10% higher luminosity at z > 0.01, due to a gravitational transition, and they argue that this is a defined testable assumption which would fully resolve both the H₀ and the S₈ tensions, in addition to provide an equally good fit to BAO, SNIa and Planck. A full data analysis is however missing.

In reference [1018] it has been argued that CMB maps show 'causal horizons' where cosmological parameters within each horizon can differ significantly, because those regions of the Universe have never been in causal contact. Within these causal horizons (see also reference [1019]) H₀ takes values which differ up to 20%, and therefore, if similar 'causal horizons' are present in the local Universe, variations between the local and high-z measures of the Hubble constant are expected.

In reference [1020] a Milgromian dynamics has been proposed as a possibility to solve the Hubble tension. Assuming a cosmological MOND model extended with the presence of sterile neutrinos with mass 11 eV/c², it has been shown that the Keenan–Barger–Cowie void [1021] can arise, despite being highly unexpected within the ΛCDM framework, and can naturally resolve the Hubble tension.

In reference [1022] a model of charged DM has been explored to solve the Hubble tension. In this scenario, the DM is charged under a dark non-linear electromagnetic force which features a screening of the K-mouflage type [1023]. The idea is that the expansion of different shells is modified by the presence of the electric repulsion, and therefore the H₀ value measured locally (inner shells) can be larger for the expansion rate due to the electric interaction with respect to the outer shells (see also reference [1024]). A full data analysis is however missing for this proposal.