The Hubble tension is inherently multidimensional and bears important implications for parameters beyond H0. We discuss the key role of the matter density parameter Omegam and the physical cold dark matter density omegac. We argue that once Omegam and the physical baryon density omegab are calibrated through Baryon Acoustic Oscillations BAO andor Type Ia Supernovae SNeIa for Omegam and via Big Bang Nucleosynthesis for omegab any model raising H0 requires raising omegac and under minimal assumptions also the clustering parameter S8. We explicitly verify that this behaviour holds when analyzing recent BAO and SNeIa data. We argue that a calibration of Omegam as reliable and modelindependent as possible should be a priority in the Hubble tension discussion and an interesting possibility in this sense could be represented by galaxy cluster gas mass fraction measurements.
['Astrophysics - Cosmology and Nongalactic Astrophysics', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Phenomenology', 'High Energy Physics - Theory']
Multidimensionality of the Hubble tension the roles of Omegam and omegac
{'Multidimensionality of the Hubble tension: the roles of Ω m and ω c': 'Davide Pedrotti, 1, 2, α Jun-Qian Jiang, 3, 1, β Luis A. Escamilla, 4, γ  \nSimony Santos da Costa, 1, 2, δ and Sunny Vagnozzi 1, 2, ϵ  \n1  \nDepartment of Physics, University of Trento, Via Sommarive 14, 38123 Povo (TN), Italy 2 Trento Institute for Fundamental Physics and Applications (TIFPA)-INFN, Via Sommarive 14, 38123 Povo (TN), Italy 3 School of Physical Sciences, University of Chinese Academy of Sciences, No.19(A) Yuquan Road, Beijing 100049, China 4 School of Mathematics and Statistics, University of Sheffield, Hounsfield Road, Sheffield S3 7RH, United Kingdom  \n(Dated: December 12, 2024)  \nThe Hubble tension is inherently multidimensional, and bears important implications for parameters beyond H 0 . We discuss the key role of the matter density parameter Ω m and the physical cold dark matter density ω c . We argue that once Ω m and the physical baryon density ω b are calibrated, through Baryon Acoustic Oscillations (BAO) and/or Type Ia Supernovae (SNeIa) for Ω m , and via Big Bang Nucleosynthesis for ω b , any model raising H 0 requires raising ω c and, under minimal assumptions, also the clustering parameter S 8 . We explicitly verify that this behaviour holds when analyzing recent BAO and SNeIa data. We argue that a calibration of Ω m as reliable and modelindependent as possible should be a priority in the Hubble tension discussion, and an interesting possibility in this sense could be represented by galaxy cluster gas mass fraction measurements.', 'I. INTRODUCTION': "The ability of the simple, six-parameter ΛCDM cosmological model to account for a wide range of cosmological observations remains the main reason for its continued success. Yet, with the increase in precision of cosmological observations over the past decade, this success has been challenged by various tensions whose significance keeps growing: an example is the mismatch between the value of the Hubble constant H 0 = (67 . 36 ± 0 . 54) km / s / Mpc inferred from Cosmic Microwave Background (CMB) data within ΛCDM [1], and a number of direct measurements of the same quantity, including but not limited to the SH0ES Cepheid-calibrated Type Ia Supernovae (SNeIa) local distance ladder measurement of H 0 = (73 . 04 ± 1 . 04) km / s / Mpc [2], see e.g. Refs. [312] for reviews. Explanations based on systematics (e.g. Refs. [13-17]) are growing ever more challenging to defend, which is why serious consideration has been given to the possibility that the Hubble tension calls for new physics beyond ΛCDM (with no claims as to completeness, see Refs. [18-121] for examples).  \nNevertheless, focusing on H 0 can only reveal part of the story and is, at best, misleading. A key role in the Hubble tension is in fact played by Baryon Acoustic Oscillation (BAO) measurements: once the sound horizon at baryon drag r d is calibrated to the ΛCDM value - either from the CMB or through the Big Bang Nucleosynthesis (BBN) determination of the physical baryon density ω b - BAO can be combined with uncalibrated SNeIa measurements to build an inverse distance ladder . This yields a value of  \nH 0 compatible with the 'low' Planck ΛCDM one, yet in principle independent of any CMB data. Another way of seeing this is to note that the distance-redshift diagrams of SH0ES-calibrated SNeIa, and BAO calibrated through the ΛCDM value of r d , are mutually completely inconsistent despite probing the same 0 . 1 ≲ z ≲ 2 redshift range (see e.g. Fig. 1 of Ref. [122], Fig. 1 of Ref. [123], as well as Refs. [124-128]). Given that BAO are sensitive to the combination H 0 r d , it becomes clear that solving the Hubble tension while not altering the SNeIa absolute magnitude M B necessarily requires a decrease in r d . This, in turn, calls for new physics operating before recombination [129-138]. 1  \nHowever, even looking at H 0 and r d is still not the end of the story, as models that aim to solve the Hubble tension are inevitably subject to several other constraints. Lowering r d , therefore, is just a part of what a successful model should do, and various other cosmological quantities such as the age of the Universe t U [142-146], the physical matter density ω m [143, 147, 148], and the fractional matter density parameter Ω m [143, 149-152] appear to play a key role in the Hubble tension discourse. It has therefore become increasingly clear that the Hubble tension is inherently a multi-dimensional problem, as recently emphasized by Ref. [122], where the denomination 'cosmic calibration tension' was suggested.  \nOur work fits in this context by further exploring the role played by two important cosmological parameters: the fractional matter density parameter Ω m , and the physical dark matter (DM) density parameter ω c = Ω c h 2 , with h the dimensionless Hubble constant. Assuming both Ω m and the physical baryon density ω b are somehow calibrated, one can easily show that an increase in H 0 must be accompanied by an increase in ω c . Under minimal assumptions concerning the primordial power  \nspectrum, this inevitably leads to a potentially problematic increase in the clustering parameter S 8 [122]. It so happens that Ω m and ω b can in fact be calibrated in a fairly robust way, through a combination of BAO and uncalibrated SNeIa in the former case, and BBN in the latter. In the rest of our work we discuss these points in more detail, explicitly demonstrating on real data how an increase in H 0 is accompanied by an increase in ω c (and S 8 ), while studying the impact of the Ω m calibration and arguing that a high-fidelity, as model-independent as possible calibration of Ω m , should be a major priority in the quest towards solving the Hubble tension.  \nThe rest of this paper is then organized as follows. We begin by discussing in more detail the role of Ω m and ω c in the context of the Hubble tension in Sec. II. In Sec. III we discuss our datasets and methodology. We present our results in Sec. IV, and critically discuss them in Sec. V. Finally, in Sec. VI we draw concluding remarks.", 'II. MULTIDIMENSIONALITY OF THE HUBBLE TENSION': "The important role of Ω m and ω c in the Hubble tension can most easily be understood by expressing the fractional matter density parameter as follows:  \nΩ m = Ω b h 2 +Ω c h 2 +Ω ν h 2 h 2 = ω b + ω c + ω ν h 2 . (1)  \nwhere Ω m , Ω b , Ω c , and Ω ν are the matter, baryon, cold DM, and (massive) neutrino (fractional) density parameters, h ≡ H 0 / (100 km / s / Mpc) is the reduced Hubble parameter, and ω m , ω b , ω c , and ω ν are the matter, baryon, cold DM, and (massive) neutrino physical density parameters respectively. Note that in writing Eq. (1) we are explicitly assuming a spatially flat Universe. In what follows, we will treat the sum of the neutrino masses ∑ m ν as known and fixed to 0 . 06 eV, so that ω ν = ∑ m ν / 94 . 13 eV ≲ 0 . 0015 is also known and fixed. 2 Let us now assume that both Ω m and ω b can be calibrated (by BAO+SNeIa and BBN respectively, as we shall discuss shortly), and can therefore be considered approximately constant within uncertainties. From Eq. (1) we then see that an increase in h is necessarily accompanied by an increase in ω c :  \nω c = Ω m h 2 -( ω b + ω ν ) . (2)  \nFor small variations δω c and δh , we can therefore expecting the following to hold:  \nδω c ω m ≈ 2 δh h , (3)  \nwhich can be rewritten as follows:  \nδω c ω c = δω c ω m ω m ω c ≈ 2 ( 1 + ω b + ω ν ω c ) δh h . (4)  \nFor reference, taking the best-fit cosmological parameters inferred from a fit to the Planck 2018 TTTEEE+lowE+lensing likelihoods [1], for which ω b = 0 . 022, ω c = 0 . 120, and ω m = 0 . 143, we find that Eq. (4) reduces to the following:  \nδω c ω c ≈ 2 . 38 δh h , (5)  \nwhich shows that the fractional increase in the physical DM density ω c must be larger than twice the fractional increase in the Hubble constant H 0 .  \nWe remark once more that the earlier arguments hinge upon calibrations for Ω m and ω b . We now discuss in more detail how this is achieved. To set the stage, in what follows we work within a spatially flat Friedmann-LemaˆıtreRobertson-Walker (FLRW) metric, and assume that the Etherington distance-duality relation (DDR) [158] holds, in agreement with recent data [159-163]. Within our assumptions, the transverse comoving distance D M is given by the following:  \nD M ( z ) = ∫ z 0 dz ' H ( z ' ) = 1 H 0 ∫ z 0 dz ' E ( z ' ) , (6)  \nwhere we denote by E ( z ) ≡ H ( z ) /H 0 the unnormalized expansion rate. Transverse, line-of-sight, and isotropic (volume-averaged) BAO measurements at an effective redshift z eff are then sensitive to the transverse angular scale θ d , redshift span δz d , and isotropic angular scale θ v , which are given by the following:  \nθ d ( z eff ) = r d D M ( z eff ) = r d H 0 ∫ z eff 0 dz ' /E ( z ' ) , (7)  \nδz d ( z eff ) = r d D H ( z eff ) = r d H ( z eff ) = r d H 0 E ( z eff ) , (8)  \nθ v ( z eff ) = r d D V ( z eff ) = r d [ z eff D 2 M ( z eff ) D H ( z eff )] 1 / 3 , (9)  \nwhere the sound horizon at baryon drag is determined by the following integral:  \nr d = ∫ ∞ z d dz c s ( z ) H ( z ) , (10)  \nwith c s ( z ) being the speed of sound of the photon-baryon plasma, and z d ≈ 1060 denoting the redshift of the drag epoch when baryons are released from the photon drag.  \nA single BAO measurement (at a single effective redshift) of one among θ d , δz d , or θ v is unable to disentangle the effects of r d H 0 and Ω m , which are completely degenerate between each other. However, this degeneracy can be partially broken if the BAO angular scale is measured over a sufficiently wide range of effective redshifts (at present, 0 . 1 ≲ z eff ≲ 2 . 5). The reason is that the slope  \ndescribing the r d H 0 -Ω m correlation slowly changes with z eff (see e.g. Fig. 2 of Ref. [150]), reflecting how the importance of the dark energy contribution relative to the matter one decreases with increasing z eff . This is not sufficient to lead to a very constraining inference of Ω m at the current level of precision (which is why earlier we stated that the degeneracy is only partially broken), but illustrates where the sensitivity of BAO measurements to Ω m arises from.  \nOn the other hand, uncalibrated SNeIa constitute an excellent probe of Ω m . Neglecting the usual stretch and color corrections, which do not alter our subsequent discussion, we recall that the observed SNeIa light-curve B-band rest-frame peak magnitude m B is given by:  \nm B = M B -5 log 10 [ H 0 c ] +5log 10 [ H 0 D L ( z ) c ] +25 , (11)  \nwhere M B denotes the SNeIa absolute magnitude in the same band and is treated as a nuisance parameter. In the absence of any knowledge about M B , we refer to the SNeIa measurements as being uncalibrated. These then probe the uncalibrated luminosity distance H 0 D L ( z ) which, assuming that the Etherington DDR holds, is given by the following:  \nH 0 D L ( z eff ) = H 0 (1 + z eff ) D M ( z ) = (1 + z eff ) ∫ z eff 0 dz E ( z ) . (12)  \nFrom Eq. (12) it is clear that uncalibrated SNeIa can be used as relative distance indicators to constrain the shape of the late-time expansion rate E ( z ) regardless of its overall amplitude. Within the minimal ΛCDM model, and neglecting the radiation component which is completely subdominant at late times, the only free parameter that enters into E ( z ) ≈ √ Ω m (1 + z ) 3 +(1 -Ω m ) is Ω m , which can therefore be determined from uncalibrated SNeIa measurements. This determination can be further improved by combining SNeIa with BAO measurements, whose sensitivity to Ω m we discussed earlier.  \nFor what concerns the physical baryon density ω b , this is tightly constrained by BBN considerations on the abundance of light elements, especially deuterium. This is because ω b controls (among others) the neutron-toproton density at the time of BBN, a parameter to which the yield of light elements is particularly sensitive. It is worth noting that the CMB is also sensitive to ω b through its impact on the relative height of odd and even acoustic peaks. The signature of ω b in CMB data is therefore quite clean, making its determination relatively stable across different models, while still being in part modeldependent. Nevertheless, in support of the BBN determination of ω b , it is worth noting that this is in excellent agreement with the ΛCDM-based CMB determination of the same parameter. Moreover, it has been shown in Ref. [164] that ω b can be inferred in eight independent ways from CMB data, reflecting eight independent ways  \nin which the physical baryon density affects the CMB power spectra, and with all eight determinations being in agreement between each other. Therefore, while the earlier calibration of Ω m is somewhat model-dependent (it implicitly depends on the assumed late-time background expansion), the calibration of ω b can be considered extremely robust and model-independent.  \nWe note that the increase in ω c implied by Eqs. (3-5) has important implications for the clustering amplitude, as quantified by the parameter S 8 ≡ σ 8 √ Ω m / 0 . 3, where σ 8 is the present-day linear theory amplitude of matter fluctuations averaged in spheres of radius 8 h Mpc -1 . Under the assumption that the primordial power spectrum of scalar fluctuations remains ΛCDM-like, an increase in ω c while maintaining ω b fixed as per BBN considerations (and the physical radiation density ω r being fixed by the measured CMB temperature monopole) directly leads to an increase in ω m , and therefore to the onset of matter domination occurring earlier. This, in turn, causes a larger amplitude of fluctuations in the matter power spectrum, as well as a larger net growth of matter perturbations, both of which result in a larger value of σ 8 . Therefore, an increase in ω c indirectly leads to an increase in σ 8 . Moreover, if Ω m is fixed (note that this fixes the large-scale asymptote of the matter power spectrum, see e.g. Fig. 4.6 of Ref. [165]), the small-scale linear matter power spectrum P L scales with H 0 as follows:  \nP L ∼ T 2 ( k ) H 4 0 ∼ k 4 eq H 4 0 ∼ a -2 eq ∼ H 4 0 , (13)  \nwhere T ( k ) denotes the transfer function, k eq is the equality wavenumber, and a eq is the equality scale factor. This necessarily increases S 8 ∼ H 2 0 as well, worsening the discrepancy between values of S 8 determined by weak lensing [166, 167], redshift space distortions [168-170], and cluster counts [171], all of which fall somewhat lower compared to the Planck (TTTEEE+lowE+lensing) determination of S 8 = 0 . 832 ± 0 . 013 [1]. 3 Analogously to Eq. (3), we can expect the following:  \nδS 8 S 8 ≈ 2 δh h . (14)", 'III. DATASETS AND METHODOLOGY': "Our discussions so far have been purely theoretical. In what follows, we will test on real data the validity of our argument that H 0 , ω c , and S 8 increase hand-in-hand.  \nWe make use of the following datasets and priors:  \n- · BAO measurements from the Baryon Oscillation Spectroscopic Survey (BOSS) and extended BOSS (eBOSS) survey programs of the Sloan Digital Sky Survey (SDSS). In particular, the BAO measurements we use come from the Main Galaxy Sample (MGS) at z eff = 0 . 15 [174]; the BOSS galaxy samples at z eff = 0 . 38, 0 . 51 [175]; the eBOSS LRG sample at z eff = 0 . 70 [176]; the eBOSS ELG sample at z eff = 0 . 85 [177]; the eBOSS QSO sample at z eff = 1 . 48 [178]; the eBOSS Lyα sample and the cross-correlation between the Lyα and QSO samples, both of them at z eff = 2 . 33 [179]. We refer to this dataset as BAO .\n- · The PantheonPlus SNeIa catalog [180], consisting of 1701 light curves for 1550 unique SNeIa. We only use SNeIa in the redshift range 0 . 01 < z < 2 . 26, and refer to this dataset as PP .\n- · The Pantheon Type Ia Supernovae (SNeIa) catalog [181], which precedes the PantheonPlus one and consists of 1048 SNeIa within the redshift range 0 . 01 < z < 2 . 26. We refer to this dataset as P .\n- · A Gaussian prior on the physical baryon density ω b = 0 . 02233 ± 0 . 00036, determined from BBN considerations in light of an improved determination of the deuterium burning rate from the LUNA experiment [182], and which we refer to as BBN .\n- · A Gaussian prior on the Hubble constant H 0 = (73 . 04 ± 1 . 04) km / s / Mpc as determined by the SH0ES team through a Cepheid-calibrated SNeIa distance ladder [2], and which we refer to as SH0ES .  \nWe consider various combinations of the above datasets, following a rationale that will be discussed later. Note that we purposely choose not to include CMB data (not even in the form of distance priors), unlike the related Ref. [122]. The reason is that we want our results to depend only on constraints arising from the late-time expansion history as much as possible, whereas including CMB data would introduce an inevitable dependence on the perturbation dynamics of the assumed early-time model, a model-dependence which we are seeking to avoid.  \nFor what concerns the underlying model, we assume that this is ΛCDM. We note, in addition, that the aforementioned datasets are inherently background ones. Therefore, rather than adopting the usual 6-dimensional parameter basis { θ s , ω b , ω c , A s , n s , τ } , it makes more sense to work with the 3-dimensional parameter basis { ω b , Ω m , h } , since A s , n s , and τ play no role in what follows (we recall that θ s and H 0 can be exchanged one for the other). We note that in principle the late-time expansion history is fully specified by Ω m and h alone, since it is really only the sum of ω b + ω c + ω ν = ω m = Ω m h 2 , rather than the two individual components alone, which matters at the background level. However, we treat ω b  \nas an additional free parameter since the key point of our work is to study the relation between H 0 and ω c . This obviously requires disentangling the ω b and ω c contributions to ω m , which is achieved through the BBN prior to ω b as we will discuss shortly.  \nWe sample the posterior distributions of the three cosmological parameters using Monte Carlo Markov Chain (MCMC) methods, adopting the cosmological MCMC sampler SimpleMC 4 . The code can be used to perform parameter estimation against datasets for which only the background expansion history matters, which is precisely the case for the measurements used here. We set wide, flat priors on all three cosmological parameters, verifying a posteriori that our posteriors are not affected by the choice of lower and upper prior boundaries. We assess the convergence of our MCMC chains using the GelmanRubin R -1 parameter [184], with R -1 < 0 . 01 required for our chains to be considered converged. Our chains are subsequently analyzed via the GetDist package [185].  \nAssuming ΛCDM as the underlying cosmological model, we explicitly investigate on real data the connection between h and ω c discussed earlier, see Eq. (2). We do so by first calibrating Ω m by using BAO and/or SNeIa measurements, as discussed in Sec. II. Once this is achieved, we calibrate the resulting distance ladder on opposite ends, resulting in H 0 being pushed to either end of the 'Hubble tension range', 0 . 67 ≲ h ≲ 0 . 74. On the one hand, we can use the BBN prior on ω b to infer r d (assuming ΛCDM), in turn calibrating BAO measurements from the early Universe side (this calibration is eventually transferred to SNeIa measurements in the same range, if they are included). The value of H 0 inferred from this inverse distance ladder is expected to fall on the low side of the Hubble tension range, i.e. closer to the Planck value [133, 186]. On the other hand, the SH0ES prior on H 0 can be used to directly calibrate either or both BAO and SNeIa measurements from the local Universe side, naturally resulting in a value of H 0 falling on the high side of the Hubble tension range, i.e. closer to the SH0ES value.  \nAs the two calibrations discussed above move H 0 across the Hubble tension range, we can expect ω c to increase/decrease as H 0 does the same. Two clarifications are in order before moving on. Firstly, as we will discuss in more detail in Sec. IV, at times we will consider dataset combinations including SNeIa but not BAO, yet we will still include the BBN prior on ω b . In these cases, the role of the BBN prior is to disentangle the ω b and ω c contributions to ω m = ω b + ω c + ω ν , which are otherwise completely degenerate as far as SNeIa data is concerned. Note that when combined with BAO measurements, the BBN prior still plays this role of disentangling ω b and ω c , in addition of course to calibrating r d .  \nFinally we note that, while the value of H 0 inferred from the BAO+BBN combination inherently depends on the assumed early Universe model, required to infer r d from ω b via Eq. (10), the relation between H 0 and ω c is largely independent of the assumed early Universe physics. In fact, such a relation is implied not by the assumed earlyUniverse physics, but by the constraints on Ω m imposed from the late-time expansion history (see also Ref. [122] for further discussions on this point).  \nIn addition, we also study the connection between h and S 8 discussed earlier, see Eq. (14). To do so, we compute S 8 (obviously treated as a derived parameter) for each point in our MCMC chains, using the Boltzmann solver CAMB [187]. As already emphasized earlier, this calculation requires explicit knowledge of the primordial power spectrum of scalar fluctuations, which we assume remains ΛCDM-like (see also Ref. [122] where a similar assumption was made). We assume that the amplitude of the primordial power spectrum is fixed by the value of A s inferred by a fit to the Planck 2018 TTTEEE+lowE+lensing likelihoods. Nevertheless, insofar as we only care about shifts in S 8 (in response to shifts in H 0 ) rather than the reference value thereof, as is the case in this work, we do not expect our subsequent results to depend on the assumed value of A s .  \nFinally, for purely illustrative purposes, we further quantify the relation between ω c , S 8 , and H 0 [see Eqs. (4,14)] by fitting linear relations to the relative variations of both ω c and S 8 ( δω c /ω c and δS 8 /S 8 ) as a function of the relative variation of H 0 , δh/h :  \nδω c ω c = α δh h , δS 8 S 8 = β δh h . (15)  \nFor each dataset combination, we determine the bestfit coefficients α and β from a least-squares fit, minimizing the sum of the squares of the residuals, ∑ ( αδh/h -δω c /ω c ) 2 and ∑ ( βδh/h -δS 8 /S 8 ) 2 respectively. For each set of chains, denoting by x a given parameter among { h, ω c , S 8 } , and by ¯ x its mean value (computed across the chains), we estimate δx/x as δx/x = 2( x -¯ x ) / ( x + ¯ x ).", 'IV. RESULTS': 'We now present the results obtained using the methods and datasets discussed in Sec. III. Constraints on cosmological parameters of interest, as well as the coefficients α and β presented in Eq. (15), are shown in Tab. I. The following four subsections are each devoted to the comparison between the results obtained adopting two specific combinations of likelihoods discussed previously, each of which will make the mutual correlations between ω c , S 8 , and H 0 more or less clear, while allowing us to underscore the importance of a reliable Ω m calibration.', 'A. BBN+PP+BAO vs BBN+PP+BAO+SH0ES': 'We begin by comparing the results obtained from the BBN + PP + BAO versus BBN + PP + BAO + SH0ES dataset combinations. A visual summary of our results is given in the corner plot of Fig. 1, where we show 2D joint and 1D marginalized posterior probability distributions for Ω m , ω c , h , and S 8 .  \nThe rationale behind this first comparison is that the BAO + PP combination is able to calibrate Ω m (as we discussed in Sec. II) in a consistent way across both dataset combinations. Indeed, for both dataset combinations we find Ω m = 0 . 314 ± 0 . 013, see Tab. I. On the other hand, as discussed in Sec. III, calibrating the BAO + PP dataset combination with either the BBN prior on ω b or the SH0ES prior on H 0 pushes H 0 to opposite ends, allowing us to explore how ω c and S 8 change in response to these different calibrations. We note that from the BBN + PP + BAO versus BBN + PP + BAO + SH0ES dataset combinations we infer H 0 = (70 . 6 ± 2 . 2) km / s / Mpc and H 0 = (72 . 0 ± 1 . 7) km / s / Mpc respectively.  \nAs we can clearly see in the triangular plot of Fig. 1, the expected mutual correlations between h , ω c , and S 8 are clearly present, although the shift in ω c and S 8 as the calibration is changed is not strong (less than 1 σ in both cases). We find that ω c increases from 0 . 134 ± 0 . 013 to 0 . 141 ± 0 . 011, whereas S 8 increases from 0 . 846 ± 0 . 071 to 0 . 882 ± 0 . 062. The reason for these relatively small shifts is due to the fact that the shift in H 0 itself is not large. As one would expect, adding BBN + BAO (which in itself favors lower values of H 0 ) to the PP + SH0ES combination, which on its own would favor values of H 0 closer to 74 km / s / Mpc, brings this value down to 72 . 0 km / s / Mpc. For the BBN + PP + BAO dataset combination our bestfit values of α and β are α = 2 . 54 and β = 2 . 06, whereas for the BBN + PP + BAO + SH0ES combination we find α = 2 . 49 and β = 1 . 98, both aligning relatively well with the analytical arguments presented earlier (represented by the dashed black lines in Fig. 1).', 'B. BBN+PP+BAO vs BBN+PP+SH0ES': 'We now move on to comparing the results obtained from the BBN + PP + BAO versus BBN + PP + SH0ES dataset combinations. In practice, compared to the earlier discussion in Sec. IV A, we have removed the BAO dataset from the second combination. A visual summary of our results is given in the corner plot of Fig. 2.  \nIn this case, we are no longer adopting the same dataset to calibrate Ω m (earlier constrained in both cases by the BAO + PP combination), which is calibrated by BAO + PP on one side, and by PP on the other. The resulting calibration of Ω m is broadly consistent, but displays some differences. In particular, from the BBN + PP + BAO combination we find Ω m = 0 . 314 ± 0 . 013, as previously discussed in Sec. IV A, whereas from  \nh  \nFIG. 2. As in Fig. 1, but focusing on the BBN + PP + BAO (red contours and curves) versus BBN + PP + SH0ES (blue contours and curves) dataset combinations.  \n<!-- image -->  \nm  \nc  \n8  \nFIG. 1. Triangular plot showing 2D joint and 1D marginalized posterior probability distributions for the fractional matter density parameter Ω m , the physical cold dark matter density ω c , the reduced Hubble constant h , and the clustering parameter S 8 , in light of the BBN + PP + BAO (red contours and curves) and BBN + PP + BAO + SH0ES (blue contours and curves) dataset combinations. We clearly see that h , ω c , and S 8 increase hand-in-hand. The dashed lines shows the linearized theoretical estimates for the relationships between δω c /ω c and δh/h given by Eq. (5), and between δS 8 /S 8 and δh/h given by Eq. (14).  \nthe BBN + PP + SH0ES combination we find the ≈ 0 . 8 σ larger value of Ω m = 0 . 331 ± 0 . 018, as can be clearly appreciated in Fig. 2. This is not unexpected, as the fact that the PantheonPlus SNeIa sample appears to prefer slightly larger values of Ω m ∼ 0 . 33 compared to its predecessors Pantheon and JLA is well known and documented in the literature [152].  \nComparing these two dataset combinations we find that the expected mutual correlations between h , ω c , and S 8 are again clearly present, but are accentuated compared to the earlier results of Sec. IV A. The reason is two-fold. On the one hand, from entirely analytical arguments [see Eq. (1)], one expects that if the calibration of Ω m is not constant, larger values thereof (as in the BBN + PP + SH0ES combination) will correlate with larger values of ω c and S 8 , which is precisely what we are observing. In addition, removing the BAO dataset from the BBN + PP + SH0ES combination allows the latter to push H 0 towards 74 km / s / Mpc as expected based on our earlier discussions. Indeed, we find H 0 = (73 . 9 ± 2 . 5) km / s / Mpc, completely in line with what one would expect.  \nThe above increase in H 0 leads to substantial increases in both ω c and S 8 . In fact, we find that ω c increases by  \nm  \n1 . 3 σ from 0 . 134 ± 0 . 013 to 0 . 159 ± 0 . 015: we remark that the presence of the BBN prior in the BBN + PP + SH0ES dataset combination plays a crucial role in determining the latter value, given that PP + SH0ES on their own would only be sensitive to the sum of ω b and ω c (more precisely, PP is sensitive to Ω m , so adding the SH0ES information on H 0 naturally determines ω m ), whereas the BBN prior allows to disentangle the baryonic contribution from the DM one. Similarly to ω c , we observe that S 8 increases from 0 . 846 ± 0 . 071 to 0 . 980 ± 0 . 084. In the case of S 8 , although the central value clearly increases to extremely high values, it would be misleading to quantify the significance of the resulting tension with either CMB or weak lensing observations. In fact, we note that the uncertainty on S 8 is of order 0 . 07-0 . 08, i.e. a factor of ≈ 67 larger than that of S 8 = 0 . 832 ± 0 . 013 as determined by Planck , which would result in a tension formally of low significance ( ≲ 2 σ ) in spite of the very high central value. The reason for these large uncertainties is to be sought in the fact that our analysis includes neither CMB nor weak lensing data, both of which are crucial to reduce the uncertainty on S 8 . Nevertheless, even in this way we are able to observe a significant upwards shift in S 8 , in line with the analytical expectations laid out previously. Finally, the best-fit values for α and β we determine for the BBN + PP + SH0ES combination are α = 2 . 27 and β = 1 . 73.  \nh  \n<!-- image -->  \nc  \n8', 'C. BBN+BAO vs BBN+PP+SH0ES': 'Driven by the previous results, we now choose to compare two dataset combinations which exacerbate the tension between the respective determinations of H 0 , in order to check whether a larger shift is observed in both ω c and S 8 . Specifically, we compare the BBN + BAO versus BBN + PP + SH0ES dataset combinations: compared to the earlier discussion in Sec. IV B, we have removed the PP dataset from the first combination. A visual summary of our results is given in the corner plot of Fig. 3.  \nIn this case, analogously to Sec. IV C, we are no longer adopting the same dataset to calibrate Ω m , which is calibrated by the BAO dataset on one side, and the PP dataset on the other. In terms of the resulting value of Ω m , these two datasets display the largest difference among all the ones we discussed. In particular, from the BBN + BAO combination we find Ω m = 0 . 292 ± 0 . 019, whereas from the BBN + PP + SH0ES combination we find the ≈ 1 . 5 σ larger value of Ω m = 0 . 331 ± 0 . 018 discussed previously in Sec. IV B.  \nComparing these two dataset combinations leads to the largest differences between the inferred values of H 0 , for which we find H 0 = (69 . 9 ± 2 . 2) km / s / Mpc from BBN + BAO , and H 0 = (73 . 9 ± 2 . 5) km / s / Mpc from BBN + PP + SH0ES as discussed earlier. Again in line with expectations, we find correspondingly large shifts in both ω c , which increases by 1 . 9 σ from 0 . 121 ± 0 . 014 to 0 . 159 ± 0 . 015, as well as S 8 which increases from 0 . 760 ± 0 . 084 to 0 . 980 ± 0 . 084 (again, we refrain from quoting the level of significance of the S 8 tension due to the large uncertainties).  \nThe significant shifts in ω c and S 8 reported above, while going precisely in the direction expected from our analytical considerations, are however exacerbated by the inconsistent calibration of Ω m . This is very clear from Fig. 3, which shows that the larger value of Ω m in the BBN + PP + SH0ES case is partially responsible for driving ω c and S 8 towards even larger values, as one can expect from the analytical argument presented in Eq. (1). These findings underscore the capital importance of a reliable calibration of Ω m , as even 1 . 5 σ shifts in this parameter are sufficient to drive significant shifts in ω c and S 8 . Finally, for the BAO + BBN combination we find the best-fit values α = 2 . 70 and β = 2 . 27.', 'D. BBN+BAO vs BBN+P+SH0ES': 'In light of all the previous considerations, the final scenario we study is one where Ω m is still calibrated by two different probes, but in a way which is more consistent than the ones considered in Sec. IV B and Sec. IV C. In both cases, we saw that the use of the PantheonPlus SNeIa catalog was responsible for the higher value of Ω m ∼ 0 . 33 inferred in the respective dataset combination. However, it is known that preceding Pantheon SNeIa catalog preferred lower values of Ω m com-  \nFIG. 3. As in Fig. 1, but focusing on the BBN + BAO (red contours and curves) versus BBN + PP + SH0ES (blue contours and curves) dataset combinations.  \n<!-- image -->  \nc  \n8  \npared to PantheonPlus . For this reason, we now choose to compare the BBN + BAO versus BBN + P + SH0ES dataset combinations: compared to the earlier discussion in Sec. IV C, we have therefore replaced the PP dataset in the second combination with the P one. A visual summary of our results is given in the corner plot of Fig. 4. In some sense, this comparison is the best compromise among all the ones we have considered so far. In fact, such a comparison allows for a highly consistent calibration of Ω m , while allowing us to explore the opposite ends of the Hubble tension range precisely because the calibration of Ω m is achieved via different datasets (BAO and SNeIa), which nonetheless are consistent with each other as far as Ω m is concerned.  \nAs expected, we find that Ω m is consistently calibrated across the two dataset combination. From BBN + BAO we infer Ω m = 0 . 292 ± 0 . 019 as already discussed earlier, whereas from BBN + P + SH0ES we find Ω m = 0 . 301 ± 0 . 022, which is consistent within 0 . 3 σ . At the same time, the BBN calibration on one side and the SH0ES calibration on the other push H 0 to opposite ends of the Hubble tension range, with H 0 = (69 . 9 ± 2 . 2) km / s / Mpc from the BBN + BAO dataset combination as already discussed earlier, and H 0 = (73 . 8 ± 2 . 4) km / s / Mpc from the BBN + P + SH0ES one. In line with our expectations, we find that ω c increases by 1 σ from 0 . 121 ± 0 . 014 to 0 . 142 ± 0 . 016, whereas S 8 which increases from 0 . 760 ± 0 . 084 to 0 . 871 ± 0 . 094. The shifts are smaller compared to those reported in Sec. IV C, and the reason is precisely because Ω m is consistently calibrated across the two datasets, albeit using different probes. Such a consistent calibration  \nremoves any spurious shift in ω c and S 8 resulting from an increase in the underlying value of Ω m , underscoring once more the importance of a reliable calibration of the latter. We will return to this point shortly. Finally, for the BAO + P + SH0ES combination we find the best-fit values α = 2 . 33 and β = 1 . 84.  \nFIG. 4. As in Fig. 1, but focusing on the BBN + BAO (red contours and curves) versus BBN + P + SH0ES (blue contours and curves) dataset combinations.  \n<!-- image -->  \nm  \nc  \n8', 'V. DISCUSSION': "The results presented previously fully confirm on real data the expected correlations between H 0 , ω c , and S 8 which, we recall, increase hand-in-hand. While we have specifically assumed ΛCDM at early times, the increase in ω c and S 8 as H 0 increases does not depend on the specific model assumed for the early Universe, since it is solely a consequence of constraints on Ω m imposed from the late-time expansion history 5 . In fact, repeating our analysis in a early Universe-agnostic manner, treating r d as a free parameter and eventually imposing some prior thereon to calibrate the BAO measurements from the early Universe side, would lead to essentially the same conclusions. Therefore, we can expect that any earlytime modification to ΛCDM aiming to solve the Hubble  \ntension will necessarily have to be accompanied by an increase in ω c and, if the primordial power spectrum is ΛCDM-like, in S 8 as well. We also note that we have explicitly assumed that ΛCDM holds at late times, as is usually done when studying early-time models of new physics - we return to this point the end of the Section.  \nOne point worthy of notice is that the increase in ω c is actually welcome from the perspective of early-time models. As explicitly argued by one of us in Ref. [188], many such models, especially those which increase the pre-recombination expansion rate, inevitably lead to an enhanced early integrated Sachs-Wolfe (eISW) effect, as a consequence of their contribution to the decay of gravitational potentials. While some of these models possess ingredients which can allow them to balance the extra eISW power (see e.g. Refs. [189, 190]), most of the others do so precisely through an increase in ω c . The reason is that such an increase anticipates the onset of matter domination, therefore reducing the period over which gravitational potentials decay and drive the eISW effect. In some sense, this increase in ω c kills two birds with one stone: it allows early-time models to fit constraints on Ω m imposed from the late-time expansion history, and brings the eISW power back to a level which is in good agreement with CMB data.  \nA very important point which is underscored by our analysis is the key role played by a (consistent or not) Ω m calibration. As highlighted especially in Sec. IV B and Sec. IV C, an inconsistent calibration of Ω m enhances the shifts in ω c discussed in this work, which in turn exacerbates the S 8 discrepancy. This issue is actually highly relevant at present, given the mild disagreement between the values of Ω m inferred from SNeIa catalogs: as we have discussed earlier, the Pantheon SNeIa dataset indicated Ω m ∼ 0 . 3, which increased to Ω m ∼ 0 . 33 in the PantheonPlus sample. The more recent DES-Y5 and Union3 SNeIa samples instead indicate Ω m ∼ 0 . 35 [191] and Ω m ∼ 0 . 36 [192] respectively, see also Refs. [152, 193] (note that these figures are obtained within ΛCDM, and potentially increase to much larger values when allowing for an evolving dark energy component): we therefore expect that adopting these SNeIa samples would exacerbate the ω c and S 8 shifts we have observed. Finally, while we defer a more detailed exploration of this point elsewhere, we note that these shifts in the values of Ω m from different samples may be connected to recent discussions on redshift-evolution of inferred cosmological parameters as different redshift are probed (see e.g. Refs. [194-208] for more details).  \nThe above discussions reinforce the urgent need, in the cosmology community, for a calibration of Ω m which is as reliable and model-independent as possible. Concerning this last point, we indeed note that the value of Ω m inferred from standard late-time cosmological probes (e.g. BAO, SNeIa, cosmic chronometers, and so on) is inherently dependent on the assumed (late-time) cosmological model, and will in generally change if one changes the dark energy dynamics. One interesting possibility to-  \nTABLE I. 68% credible intervals on the fractional matter density parameter Ω m , the physical cold dark matter density ω c , the Hubble constant H 0 , the clustering parameter S 8 , and the parameters α and β introduced in Eq. (15), in light of various dataset combinations, all of which are discussed in Sec. IV.  \ns a model-independent determination of Ω m could come from measurements of the gas mass fraction f gas in relaxed, massive galaxy clusters, where ≈ 92-95% of the baryons are hot and emit strongly in X-rays. This method relies only on the assumption that the matter content of rich galaxy clusters provides a fair sample of the matter content of the Universe or, in practice, that the cluster potential is able to retain all the matter within the comoving virial radius from which it formed: under these assumptions, the observed ratio of baryonic to total mass should be equal to Ω b / Ω m , independently of any assumed cosmological model, and such a method has indeed been used over the past decades to infer Ω m [209-213]. 6 While the f gas method is not as widely used in the cosmological community as other probes, we believe that its minimal sensitivity to cosmological model assumptions and relatively high level of maturity compared to other less used probes should make it a very important player in the quest towards solving the Hubble tension, in the interest of calibrating Ω m as model-independently as possible. Another interesting possibility towards modelindependently inferring Ω m (with Ω b known) is to use the internal properties of individual galaxies, such as stellar mass, stellar metallicity, and maximum circular velocity [214-218]. Nevertheless, this very interesting method, which demonstrates a potential very tight link between cosmological parameters and the astrophysics of galaxies, has yet to reach the level of maturity of all the other probes discussed so far.  \nIn line with the previous comments, we note that our results do depend on the assumed late-time model, which here we have taken to be ΛCDM, as is usually done when studying early-time models of new physics. In line with the recent findings of Ref. [122] (see also Ref. [148]), we can therefore conclude that successful early-time new physics must be able to reduce r d while either reducing Ω m or increasing ω m (a related result was obtained in  \nRef. [148], which however relied on a completely different argument, exploiting the different r d -H 0 degeneracy directions for BAO versus geometrical information from the CMB). In this sense, we agree with Ref. [122] that, when combined with early-time new physics which reduces r d , the role of late-time new physics can be that of helping relax the constraints on either or both Ω m and S 8 , for instance by allowing for more freedom in the dark energy sector. We note that achieving a successful early-plus-late combination is still not a trivial feat, as not all early-time modifications follow the ideal degeneracy directions. 7 However, as also emphasized in Ref. [122] (see also Ref. [152]), this task is potentially made much easier by the recent DESI BAO data which, in order to be as conservative as possible, we have not adopted here. 8 Moreover, we remark that 'dark scattering' models which feature pure momentum exchange, and do not alter the background to linear order in perturbations, can be particularly promising in terms of relaxing the constraints on σ 8 and, potentially, Ω m (see e.g. Refs. [231-250]). We believe it may be worth exploring these models in combination with successful early-time modifications, as also suggested in Ref. [251]. 9  \nWe close with a few remarks. Firstly, our work provides an unified explanation for the H 0 -S 8 correlations frequently observed in the literature on the Hubble tension, but typically explained on a model-by-model basis. We note that another unified explanation for these correlations was provided in Ref. [148]: however, the latter  \nrelied on a completely different argument compared to ours, and more specifically was based on the different slopes of the r d -H 0 correlation in BAO and (geometrical) CMB data. Our explanation for this correlation is instead based on the (arguably simpler) Eqs. (1,2): to the best of our knowledge, it is the first time that this argument is being clearly presented in the literature. Moreover, while for simplicity we have assumed the ΛCDM model at late times, we remark that our argument is completely general and holds for any model where Eqs. (1,2) are valid, while Ω m and ω b are consistently calibrated. A notable exception could be one where the spatial curvature of the Universe is non-zero, and therefore the contribution of the spatial curvature parameter Ω K should be included in the previous equalities. Indeed, our entire discussion, especially the analytical arguments in Sec. II, have explicitly assumed a spatially flat Universe. Relaxing this assumption is likely to relax all our results, given the well-known degeneracies between Ω m and Ω K , and we plan to explore this in future work.", 'VI. CONCLUSIONS': 'In spite of its name, it is now very clear that the Hubble tension has important implications for other cosmological quantities as well, and that just focusing on H 0 is misleading. The sound horizon at baryon drag r d is another important quantity at play but, even then, looking at H 0 and r d alone obscures part of the story. In this work, we have demonstrated that Ω m and ω c play a particularly important role in this context.  \nWe have argued that, if both Ω m and ω b are calibrated (respectively by BAO and/or uncalibrated SNeIa, and by BBN considerations), then an increase in H 0 must necessarily be accompanied by an increase in ω c . Under the assumption that the primordial power spectrum of scalar fluctuations remains ΛCDM-like, an increase in ω c , via the associated increase in ω m (since ω b is calibrated), implies an increase in the clustering parameter S 8 , worsening the mild S 8 discrepancy. These shifts are not a consequence of the effects of new physics at early times, but follow solely from late-time expansion history constraints. Therefore, successful early-time new physics models must be able to accommodate these changes (which, in the case of ω c , are actually welcome as they help reduce the excess early ISW power typically associated to these models). As emphasized in the recent Ref. [122] (see also Refs. [251, 270]), additional late-time new physics may help in this sense by weakening the constraints on Ω m and/or σ 8 . Although realizing a successful combination of early-plus-late new physics remains non-trivial, our work and Ref. [122] (as well as the earlier attempt in Ref. [219]) emphasize the key role played by Ω m in building such a combination. It is worth noting that, if the recent DESI BAO data are taken at face value, this task becomes somewhat easier.  \nOur work has also underscored the crucial importance  \nof a consistent Ω m calibration. At present, different latetime probes of Ω m are in mild disagreement between each other, with recent SNeIa catalogs pushing towards increasingly high values of Ω m : focusing only on the Pantheon versus PantheonPlus samples, we have explicitly demonstrated the effects of a different Ω m calibration in exacerbating the aforementioned shifts in ω c and S 8 . Moving forward, we therefore believe that obtaining a reliable calibration of Ω m which is as model-independent as possible should be a key priority in the cosmology community (see also Ref. [152]), in order to nail down one of the key actors in the Hubble tension play. We have argued that gas mass fraction measurements from galaxy clusters could be a promising probe in this sense, and we encourage the broader cosmology community to explore these and other less widely used probes. As we have officially entered the era of Stage IV cosmology and a solution to the Hubble tension continues to elude us, we note that the shifts in cosmological parameters which we have discussed (and which we have argued unavoidably accompany successful resolutions) inevitably leave their signatures in other cosmological observables, which current and upcoming cosmological surveys will hopefully be poised to detect [271-275].', 'NOTE ADDED': 'While this work was nearing completion, we became aware of the related Ref. [122] which explores similar aspects to our work, also pointing out the importance of Ω m and ω c . We note that, despite the differences in methodology, our results are in good agreement.', 'ACKNOWLEDGMENTS': "S.V. acknowledges helpful discussions with Vivian Poulin, which originally inspired this work, and with Vittorio Ghirardini on gas mass fraction measurements. D.P., S.S.C., and S.V. acknowledge support from the Istituto Nazionale di Fisica Nucleare (INFN) through the Commissione Scientifica Nazionale 4 (CSN4) Iniziativa Specifica 'Quantum Fields in Gravity, Cosmology and Black Holes' (FLAG). J.-Q.J. acknowledges support from the Joint PhD Training program of the University of Chinese Academy of Sciences. S.S.C. acknowledges support from the Fondazione Cassa di Risparmio di Trento e Rovereto (CARITRO Foundation) through a Caritro Fellowship (project 'Inflation and dark sector physics in light of next-generation cosmological surveys'). S.V. acknowledges support from the University of Trento and the Provincia Autonoma di Trento (PAT, Autonomous Province of Trento) through the UniTrento Internal Call for Research 2023 grant 'Searching for Dark Energy off the beaten track' (DARKTRACK, grant agreement no. E63C22000500003). This publication is based upon work from the COST Action CA21136 'Addressing observa-  \ntional tensions in cosmology with systematics and fun-  \n- [1] N. Aghanim et al. (Planck), Astron. Astrophys. 641 , A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO].\n- [2] A. G. Riess et al. , Astrophys. J. Lett. 934 , L7 (2022), arXiv:2112.04510 [astro-ph.CO].\n- [3] L. Verde, T. Treu, and A. G. Riess, Nature Astron. 3 , 891 (2019), arXiv:1907.10625 [astro-ph.CO].\n- [4] E. Di Valentino et al. , Astropart. Phys. 131 , 102605 (2021), arXiv:2008.11284 [astro-ph.CO].\n- [5] E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, Class. Quant. Grav. 38 , 153001 (2021), arXiv:2103.01183 [astro-ph.CO].\n- [6] L. Perivolaropoulos and F. Skara, New Astron. Rev. 95 , 101659 (2022), arXiv:2105.05208 [astro-ph.CO].\n- [7] N. Schoneberg, G. Franco Abell'an, A. P'erez S'anchez, S. J. Witte, V. Poulin, and J. Lesgourgues, Phys. Rept. 984 , 1 (2022), arXiv:2107.10291 [astro-ph.CO].\n- [8] P. Shah, P. Lemos, and O. Lahav, Astron. Astrophys. Rev. 29 , 9 (2021), arXiv:2109.01161 [astro-ph.CO].\n- [9] E. Abdalla et al. , JHEAp 34 , 49 (2022), arXiv:2203.06142 [astro-ph.CO].\n- [10] E. Di Valentino, Universe 8 , 399 (2022).\n- [11] J.-P. Hu and F.-Y. Wang, Universe 9 , 94 (2023), arXiv:2302.05709 [astro-ph.CO].\n- [12] L. Verde, N. Schoneberg, and H. Gil-Mar'ın, (2023), arXiv:2311.13305 [astro-ph.CO].\n- [13] G. Efstathiou, (2020), arXiv:2007.10716 [astro-ph.CO].\n- [14] E. Mortsell, A. Goobar, J. Johansson, and S. Dhawan, Astrophys. J. 933 , 212 (2022), arXiv:2105.11461 [astroph.CO].\n- [15] E. Mortsell, A. Goobar, J. Johansson, and S. Dhawan, Astrophys. J. 935 , 58 (2022), arXiv:2106.09400 [astroph.CO].\n- [16] R. Wojtak and J. Hjorth, Mon. Not. Roy. Astron. Soc. 515 , 2790 (2022), arXiv:2206.08160 [astro-ph.CO].\n- [17] W. L. Freedman, B. F. Madore, I. S. Jang, T. J. Hoyt, A. J. Lee, and K. A. Owens, (2024), arXiv:2408.06153 [astro-ph.CO].\n- [18] L. A. Anchordoqui, V. Barger, H. Goldberg, X. Huang, D. Marfatia, L. H. M. da Silva, and T. J. Weiler, Phys. Rev. D 92 , 061301 (2015), [Erratum: Phys.Rev.D 94, 069901 (2016)], arXiv:1506.08788 [hep-ph].\n- [19] T. Karwal and M. Kamionkowski, Phys. Rev. D 94 , 103523 (2016), arXiv:1608.01309 [astro-ph.CO].\n- [20] M.-M. Zhao, D.-Z. He, J.-F. Zhang, and X. Zhang, Phys. Rev. D 96 , 043520 (2017), arXiv:1703.08456 [astro-ph.CO].\n- [21] M. Benetti, L. L. Graef, and J. S. Alcaniz, JCAP 07 , 066 (2018), arXiv:1712.00677 [astro-ph.CO].\n- [22] E. Mortsell and S. Dhawan, JCAP 09 , 025 (2018), arXiv:1801.07260 [astro-ph.CO].\n- [23] S. Vagnozzi, S. Dhawan, M. Gerbino, K. Freese, A. Goobar, and O. Mena, Phys. Rev. D 98 , 083501 (2018), arXiv:1801.08553 [astro-ph.CO].\n- [24] S. Kumar, R. C. Nunes, and S. K. Yadav, Phys. Rev. D 98 , 043521 (2018), arXiv:1803.10229 [astro-ph.CO].  \ndamental physics' (CosmoVerse), supported by COST (European Cooperation in Science and Technology).  \n- [25] W. Yang, S. Pan, E. Di Valentino, R. C. Nunes, S. Vagnozzi, and D. F. Mota, JCAP 09 , 019 (2018), arXiv:1805.08252 [astro-ph.CO].\n- [26] A. Banihashemi, N. Khosravi, and A. H. Shirazi, Phys. Rev. D 101 , 123521 (2020), arXiv:1808.02472 [astroph.CO].\n- [27] R.-Y. Guo, J.-F. Zhang, and X. Zhang, JCAP 02 , 054 (2019), arXiv:1809.02340 [astro-ph.CO].\n- [28] L. L. Graef, M. Benetti, and J. S. Alcaniz, Phys. Rev. D 99 , 043519 (2019), arXiv:1809.04501 [astro-ph.CO].\n- [29] A. Banihashemi, N. Khosravi, and A. H. Shirazi, Phys. Rev. D 99 , 083509 (2019), arXiv:1810.11007 [astroph.CO].\n- [30] P. Agrawal, F.-Y. Cyr-Racine, D. Pinner, and L. Randall, Phys. Dark Univ. 42 , 101347 (2023), arXiv:1904.01016 [astro-ph.CO].\n- [31] X. Li and A. Shafieloo, Astrophys. J. Lett. 883 , L3 (2019), arXiv:1906.08275 [astro-ph.CO].\n- [32] W. Yang, S. Pan, S. Vagnozzi, E. Di Valentino, D. F. Mota, and S. Capozziello, JCAP 11 , 044 (2019), arXiv:1907.05344 [astro-ph.CO].\n- [33] S. Vagnozzi, Phys. Rev. D 102 , 023518 (2020), arXiv:1907.07569 [astro-ph.CO].\n- [34] L. Visinelli, S. Vagnozzi, and U. Danielsson, Symmetry 11 , 1035 (2019), arXiv:1907.07953 [astro-ph.CO].\n- [35] E. Di Valentino, A. Melchiorri, O. Mena, and S. Vagnozzi, Phys. Dark Univ. 30 , 100666 (2020), arXiv:1908.04281 [astro-ph.CO].\n- [36] M. Escudero and S. J. Witte, Eur. Phys. J. C 80 , 294 (2020), arXiv:1909.04044 [astro-ph.CO].\n- [37] E. Di Valentino, A. Melchiorri, O. Mena, and S. Vagnozzi, Phys. Rev. D 101 , 063502 (2020), arXiv:1910.09853 [astro-ph.CO].\n- [38] F. Niedermann and M. S. Sloth, Phys. Rev. D 103 , L041303 (2021), arXiv:1910.10739 [astro-ph.CO].\n- [39] J. Sakstein and M. Trodden, Phys. Rev. Lett. 124 , 161301 (2020), arXiv:1911.11760 [astro-ph.CO].\n- [40] G. Ye and Y.-S. Piao, Phys. Rev. D 101 , 083507 (2020), arXiv:2001.02451 [astro-ph.CO].\n- [41] N. B. Hogg, M. Bruni, R. Crittenden, M. Martinelli, and S. Peirone, Phys. Dark Univ. 29 , 100583 (2020), arXiv:2002.10449 [astro-ph.CO].\n- [42] G. Ballesteros, A. Notari, and F. Rompineve, JCAP 11 , 024 (2020), arXiv:2004.05049 [astro-ph.CO].\n- [43] G. Alestas, L. Kazantzidis, and L. Perivolaropoulos, Phys. Rev. D 101 , 123516 (2020), arXiv:2004.08363 [astro-ph.CO].\n- [44] K. Jedamzik and L. Pogosian, Phys. Rev. Lett. 125 , 181302 (2020), arXiv:2004.09487 [astro-ph.CO].\n- [45] M. Ballardini, M. Braglia, F. Finelli, D. Paoletti, A. A. Starobinsky, and C. Umilt'a, JCAP 10 , 044 (2020), arXiv:2004.14349 [astro-ph.CO].\n- [46] E. Di Valentino, S. Gariazzo, O. Mena, and S. Vagnozzi, JCAP 07 , 045 (2020), arXiv:2005.02062 [astro-ph.CO].\n- [47] A. Banerjee, H. Cai, L. Heisenberg, E. O. Colg'ain, M. M. Sheikh-Jabbari, and T. Yang, Phys. Rev. D 103 , L081305 (2021), arXiv:2006.00244 [astro-ph.CO].\n- [48] F. Niedermann and M. S. Sloth, Phys. Rev. D 102 , 063527 (2020), arXiv:2006.06686 [astro-ph.CO].\n- [49] M. Gonzalez, M. P. Hertzberg, and F. Rompineve, JCAP 10 , 028 (2020), arXiv:2006.13959 [astro-ph.CO].\n- [50] M. Braglia, M. Ballardini, F. Finelli, and K. Koyama, Phys. Rev. D 103 , 043528 (2021), arXiv:2011.12934 [astro-ph.CO].\n- [51] S. Roy Choudhury, S. Hannestad, and T. Tram, JCAP 03 , 084 (2021), arXiv:2012.07519 [astro-ph.CO].\n- [52] T. Brinckmann, J. H. Chang, and M. LoVerde, Phys. Rev. D 104 , 063523 (2021), arXiv:2012.11830 [astroph.CO].\n- [53] G. Alestas, L. Kazantzidis, and L. Perivolaropoulos, Phys. Rev. D 103 , 083517 (2021), arXiv:2012.13932 [astro-ph.CO].\n- [54] L.-Y. Gao, Z.-W. Zhao, S.-S. Xue, and X. Zhang, JCAP 07 , 005 (2021), arXiv:2101.10714 [astro-ph.CO].\n- [55] G. Alestas and L. Perivolaropoulos, Mon. Not. Roy. Astron. Soc. 504 , 3956 (2021), arXiv:2103.04045 [astroph.CO].\n- [56] T. Karwal, M. Raveri, B. Jain, J. Khoury, and M. Trodden, Phys. Rev. D 105 , 063535 (2022), arXiv:2106.13290 [astro-ph.CO].\n- [57] F.-Y. Cyr-Racine, F. Ge, and L. Knox, Phys. Rev. Lett. 128 , 201301 (2022), arXiv:2107.13000 [astro-ph.CO].\n- [58] O. Akarsu, S. Kumar, E. Ozulker, and J. A. Vazquez, Phys. Rev. D 104 , 123512 (2021), arXiv:2108.09239 [astro-ph.CO].\n- [59] F. Niedermann and M. S. Sloth, Phys. Lett. B 835 , 137555 (2022), arXiv:2112.00759 [hep-ph].\n- [60] E. N. Saridakis, W. Yang, S. Pan, F. K. Anagnostopoulos, and S. Basilakos, Nucl. Phys. B 986 , 116042 (2023), arXiv:2112.08330 [astro-ph.CO].\n- [61] A. A. Sen, S. A. Adil, and S. Sen, Mon. Not. Roy. Astron. Soc. 518 , 1098 (2022), arXiv:2112.10641 [astroph.CO].\n- [62] L. Herold, E. G. M. Ferreira, and E. Komatsu, Astrophys. J. Lett. 929 , L16 (2022), arXiv:2112.12140 [astroph.CO].\n- [63] S. D. Odintsov and V. K. Oikonomou, EPL 137 , 39001 (2022), arXiv:2201.07647 [gr-qc].\n- [64] L. Heisenberg, H. Villarrubia-Rojo, and J. Zosso, Phys. Dark Univ. 39 , 101163 (2023), arXiv:2201.11623 [astroph.CO].\n- [65] L. Heisenberg, H. Villarrubia-Rojo, and J. Zosso, Phys. Rev. D 106 , 043503 (2022), arXiv:2202.01202 [astroph.CO].\n- [66] R. K. Sharma, K. L. Pandey, and S. Das, Astrophys. J. 934 , 113 (2022), arXiv:2202.01749 [astro-ph.CO].\n- [67] X. Ren, S.-F. Yan, Y. Zhao, Y.-F. Cai, and E. N. Saridakis, Astrophys. J. 932 , 131 (2022), arXiv:2203.01926 [astro-ph.CO].\n- [68] R. C. Nunes, S. Vagnozzi, S. Kumar, E. Di Valentino, and O. Mena, Phys. Rev. D 105 , 123506 (2022), arXiv:2203.08093 [astro-ph.CO].\n- [69] S. Nojiri, S. D. Odintsov, and V. K. Oikonomou, Nucl. Phys. B 980 , 115850 (2022), arXiv:2205.11681 [gr-qc].\n- [70] N. Schoneberg and G. Franco Abell'an, JCAP 12 , 001 (2022), arXiv:2206.11276 [astro-ph.CO].\n- [71] M. Joseph, D. Aloni, M. Schmaltz, E. N. Sivarajan, and N. Weiner, Phys. Rev. D 108 , 023520 (2023), arXiv:2207.03500 [astro-ph.CO].  \n- [72] A. G'omez-Valent, Z. Zheng, L. Amendola, C. Wetterich, and V. Pettorino, Phys. Rev. D 106 , 103522 (2022), arXiv:2207.14487 [astro-ph.CO].\n- [73] H. Moshafi, H. Firouzjahi, and A. Talebian, Astrophys. J. 940 , 121 (2022), arXiv:2208.05583 [astro-ph.CO].\n- [74] S. D. Odintsov and V. K. Oikonomou, EPL 139 , 59003 (2022), arXiv:2208.07972 [gr-qc].\n- [75] S. Banerjee, M. Petronikolou, and E. N. Saridakis, Phys. Rev. D 108 , 024012 (2023), arXiv:2209.02426 [grqc].\n- [76] P. D. Alvarez, B. Koch, C. Laporte, and A. Rincon, Phys. Dark Univ. 45 , 101531 (2024), arXiv:2210.11853 [gr-qc].\n- [77] F. Ge, F.-Y. Cyr-Racine, and L. Knox, Phys. Rev. D 107 , 023517 (2023), arXiv:2210.16335 [astro-ph.CO].\n- [78] O. Akarsu, S. Kumar, E. Ozulker, J. A. Vazquez, and A. Yadav, Phys. Rev. D 108 , 023513 (2023), arXiv:2211.05742 [astro-ph.CO].\n- [79] M. R. Gangopadhyay, S. K. J. Pacif, M. Sami, and M. K. Sharma, Universe 9 , 83 (2023), arXiv:2211.12041 [gr-qc].\n- [80] T. Schiavone, G. Montani, and F. Bombacigno, Mon. Not. Roy. Astron. Soc. 522 , L72 (2023), arXiv:2211.16737 [gr-qc].\n- [81] L.-Y. Gao, S.-S. Xue, and X. Zhang, Chin. Phys. C 48 , 051001 (2024), arXiv:2212.13146 [astro-ph.CO].\n- [82] T. Brinckmann, J. H. Chang, P. Du, and M. LoVerde, Phys. Rev. D 107 , 123517 (2023), arXiv:2212.13264 [astro-ph.CO].\n- [83] M. Khodadi and M. Schreck, Phys. Dark Univ. 39 , 101170 (2023), arXiv:2301.03883 [gr-qc].\n- [84] S. Dahmani, A. Bouali, I. El Bojaddaini, A. Errahmani, and T. Ouali, Phys. Dark Univ. 42 , 101266 (2023), arXiv:2301.04200 [astro-ph.CO].\n- [85] I. Ben-Dayan and U. Kumar, JCAP 12 , 047 (2023), arXiv:2302.00067 [astro-ph.CO].\n- [86] J. de Cruz Perez and J. Sola Peracaula, Phys. Dark Univ. 43 , 101406 (2024), arXiv:2302.04807 [astroph.CO].\n- [87] M. Ballardini, A. G. Ferrari, and F. Finelli, JCAP 04 , 029 (2023), arXiv:2302.05291 [astro-ph.CO].\n- [88] Y.-H. Yao, J.-C. Wang, and X.-H. Meng, Phys. Rev. D 109 , 063502 (2024), arXiv:2303.00961 [astro-ph.CO].\n- [89] M. R. Gangopadhyay, M. Sami, and M. K. Sharma, Phys. Rev. D 108 , 103526 (2023), arXiv:2303.07301 [astro-ph.CO].\n- [90] Y. Zhai, W. Giar'e, C. van de Bruck, E. Di Valentino, O. Mena, and R. C. Nunes, JCAP 07 , 032 (2023), arXiv:2303.08201 [astro-ph.CO].\n- [91] J. Sola Peracaula, A. Gomez-Valent, J. de Cruz Perez, and C. Moreno-Pulido, Universe 9 , 262 (2023), arXiv:2304.11157 [astro-ph.CO].\n- [92] A. G'omez-Valent, N. E. Mavromatos, and J. Sol'a Peracaula, Class. Quant. Grav. 41 , 015026 (2024), arXiv:2305.15774 [gr-qc].\n- [93] Ruchika, H. Rathore, S. Roy Choudhury, and V. Rentala, JCAP 06 , 056 (2024), arXiv:2306.05450 [astro-ph.CO].\n- [94] S. A. Adil, O. Akarsu, E. Di Valentino, R. C. Nunes, E. Ozulker, A. A. Sen, and E. Specogna, Phys. Rev. D 109 , 023527 (2024), arXiv:2306.08046 [astro-ph.CO].\n- [95] E. Frion, D. Camarena, L. Giani, T. Miranda, D. Bertacca, V. Marra, and O. F. Piattella, Open J. As-\n- trophys. 7 , 17 (2023), arXiv:2307.06320 [astro-ph.CO].\n- [96] O. Akarsu, E. Di Valentino, S. Kumar, R. C. Nunes, J. A. Vazquez, and A. Yadav, (2023), arXiv:2307.10899 [astro-ph.CO].\n- [97] L. A. Escamilla, W. Giar'e, E. Di Valentino, R. C. Nunes, and S. Vagnozzi, JCAP 05 , 091 (2024), arXiv:2307.14802 [astro-ph.CO].\n- [98] M. Petronikolou and E. N. Saridakis, Universe 9 , 397 (2023), arXiv:2308.16044 [gr-qc].\n- [99] R. K. Sharma, S. Das, and V. Poulin, Phys. Rev. D 109 , 043530 (2024), arXiv:2309.00401 [astro-ph.CO].\n- [100] I. Ben-Dayan and U. Kumar, Eur. Phys. J. C 84 , 167 (2024), arXiv:2310.03092 [astro-ph.CO].\n- [101] O. F. Ramadan, T. Karwal, and J. Sakstein, Phys. Rev. D 109 , 063525 (2024), arXiv:2309.08082 [astro-ph.CO].\n- [102] C. Fu and S.-J. Wang, Phys. Rev. D 109 , L041304 (2024), arXiv:2310.12932 [astro-ph.CO].\n- [103] G. Efstathiou, E. Rosenberg, and V. Poulin, Phys. Rev. Lett. 132 , 221002 (2024), arXiv:2311.00524 [astroph.CO].\n- [104] G. Montani, N. Carlevaro, and M. G. Dainotti, Phys. Dark Univ. 44 , 101486 (2024), arXiv:2311.04822 [gr-qc].\n- [105] R. Lazkoz, V. Salzano, L. Fernandez-Jambrina, and M. Bouhmadi-L'opez, Phys. Dark Univ. 45 , 101511 (2024), arXiv:2311.10526 [astro-ph.CO].\n- [106] M. Forconi, W. Giar'e, O. Mena, Ruchika, E. Di Valentino, A. Melchiorri, and R. C. Nunes, JCAP 05 , 097 (2024), arXiv:2312.11074 [astro-ph.CO].\n- [107] M. Sebastianutti, N. B. Hogg, and M. Bruni, Phys. Dark Univ. 46 , 101546 (2024), arXiv:2312.14123 [astroph.CO].\n- [108] D. Benisty, S. Pan, D. Staicova, E. Di Valentino, and R. C. Nunes, (2024), 10.1051/0004-6361/202449883, arXiv:2403.00056 [astro-ph.CO].\n- [109] C. Stahl, B. Famaey, R. Ibata, O. Hahn, N. Martinet, and T. Montandon, (2024), arXiv:2404.03244 [astroph.CO].\n- [110] R. Shah, P. Mukherjee, and S. Pal, (2024), arXiv:2404.06396 [astro-ph.CO].\n- [111] W. Giar'e, M. A. Sabogal, R. C. Nunes, and E. Di Valentino, (2024), arXiv:2404.15232 [astroph.CO].\n- [112] G. Montani, N. Carlevaro, L. A. Escamilla, and E. Di Valentino, (2024), arXiv:2404.15977 [gr-qc].\n- [113] R. T. Co, N. Fernandez, A. Ghalsasi, K. Harigaya, and J. Shelton, (2024), arXiv:2405.12268 [hep-ph].\n- [114] O. Akarsu, A. De Felice, E. Di Valentino, S. Kumar, R. C. Nunes, E. Ozulker, J. A. Vazquez, and A. Yadav, (2024), arXiv:2406.07526 [astro-ph.CO].\n- [115] A. Yadav, S. Kumar, C. Kibris, and O. Akarsu, (2024), arXiv:2406.18496 [astro-ph.CO].\n- [116] K. Nozari, S. Saghafi, and M. Hajebrahimi, Phys. Dark Univ. 46 , 101571 (2024), arXiv:2407.01961 [gr-qc].\n- [117] S. Dwivedi and M. Hog˚as, (2024), arXiv:2407.04322 [astro-ph.CO].\n- [118] G. Montani, N. Carlevaro, and M. De Angelis, Entropy 26 , 662 (2024), arXiv:2407.12409 [gr-qc].\n- [119] L. A. Escamilla, D. Fiorucci, G. Montani, and E. Di Valentino, (2024), arXiv:2408.04354 [astroph.CO].\n- [120] L. Perivolaropoulos, (2024), arXiv:2408.11031 [astroph.CO].\n- [121] Y.-H. Yao, J.-Q. Liu, Z.-Q. Huang, J.-C. Wang, and Y. Su, (2024), arXiv:2409.04678 [astro-ph.CO].  \n- [122] V. Poulin, T. L. Smith, R. Calder'on, and T. Simon, (2024), arXiv:2407.18292 [astro-ph.CO].\n- [123] D. Bousis and L. Perivolaropoulos, (2024), arXiv:2405.07039 [astro-ph.CO].\n- [124] M. Raveri, Phys. Rev. D 101 , 083524 (2020), arXiv:1902.01366 [astro-ph.CO].\n- [125] G. Benevento, W. Hu, and M. Raveri, Phys. Rev. D 101 , 103517 (2020), arXiv:2002.11707 [astro-ph.CO].\n- [126] D. Camarena and V. Marra, Mon. Not. Roy. Astron. Soc. 504 , 5164 (2021), arXiv:2101.08641 [astro-ph.CO].\n- [127] D. Camarena and V. Marra, (2023), arXiv:2307.02434 [astro-ph.CO].\n- [128] M. Raveri, (2023), arXiv:2309.06795 [astro-ph.CO].\n- [129] J. L. Bernal, L. Verde, and A. G. Riess, JCAP 10 , 019 (2016), arXiv:1607.05617 [astro-ph.CO].\n- [130] G. E. Addison, D. J. Watts, C. L. Bennett, M. Halpern, G. Hinshaw, and J. L. Weiland, Astrophys. J. 853 , 119 (2018), arXiv:1707.06547 [astro-ph.CO].\n- [131] P. Lemos, E. Lee, G. Efstathiou, and S. Gratton, Mon. Not. Roy. Astron. Soc. 483 , 4803 (2019), arXiv:1806.06781 [astro-ph.CO].\n- [132] K. Aylor, M. Joy, L. Knox, M. Millea, S. Raghunathan, and W. L. K. Wu, Astrophys. J. 874 , 4 (2019), arXiv:1811.00537 [astro-ph.CO].\n- [133] N. Schoneberg, J. Lesgourgues, and D. C. Hooper, JCAP 10 , 029 (2019), arXiv:1907.11594 [astro-ph.CO].\n- [134] L. Knox and M. Millea, Phys. Rev. D 101 , 043533 (2020), arXiv:1908.03663 [astro-ph.CO].\n- [135] N. Arendse et al. , Astron. Astrophys. 639 , A57 (2020), arXiv:1909.07986 [astro-ph.CO].\n- [136] G. Efstathiou, Mon. Not. Roy. Astron. Soc. 505 , 3866 (2021), arXiv:2103.08723 [astro-ph.CO].\n- [137] R.-G. Cai, Z.-K. Guo, S.-J. Wang, W.-W. Yu, and Y. Zhou, Phys. Rev. D 105 , L021301 (2022), arXiv:2107.13286 [astro-ph.CO].\n- [138] R. E. Keeley and A. Shafieloo, Phys. Rev. Lett. 131 , 111002 (2023), arXiv:2206.08440 [astro-ph.CO].\n- [139] O. Akarsu, E. O. Colgain, E. Ozulker, S. Thakur, and L. Yin, Phys. Rev. D 107 , 123526 (2023), arXiv:2207.10609 [astro-ph.CO].\n- [140] I. Tutusaus, M. Kunz, and L. Favre, (2023), arXiv:2311.16862 [astro-ph.CO].\n- [141] L. A. Escamilla, E. Ozulker, O. Akarsu, E. Di Valentino, and J. A. V'azquez, (2024), arXiv:2408.12516 [astroph.CO].\n- [142] R. Jimenez, A. Cimatti, L. Verde, M. Moresco, and B. Wandelt, JCAP 03 , 043 (2019), arXiv:1902.07081 [astro-ph.CO].\n- [143] J. L. Bernal, L. Verde, R. Jimenez, M. Kamionkowski, D. Valcin, and B. D. Wandelt, Phys. Rev. D 103 , 103533 (2021), arXiv:2102.05066 [astro-ph.CO].\n- [144] S. Vagnozzi, F. Pacucci, and A. Loeb, JHEAp 36 , 27 (2022), arXiv:2105.10421 [astro-ph.CO].\n- [145] J.-J. Wei and F. Melia, Astrophys. J. 928 , 165 (2022), arXiv:2202.07865 [astro-ph.CO].\n- [146] S. Capozziello, G. Sarracino, and A. D. A. M. Spallicci, Phys. Dark Univ. 40 , 101201 (2023), arXiv:2302.13671 [astro-ph.CO].\n- [147] L. Pogosian, G.-B. Zhao, and K. Jedamzik, Astrophys. J. Lett. 904 , L17 (2020), arXiv:2009.08455 [astroph.CO].\n- [148] K. Jedamzik, L. Pogosian, and G.-B. Zhao, Commun. in Phys. 4 , 123 (2021), arXiv:2010.04158 [astro-ph.CO].\n- [149] W. Lin, K. J. Mack, and L. Hou, Astrophys. J. Lett. 904 , L22 (2020), arXiv:1910.02978 [astro-ph.CO].\n- [150] W. Lin, X. Chen, and K. J. Mack, Astrophys. J. 920 , 159 (2021), arXiv:2102.05701 [astro-ph.CO].\n- [151] Z. Sakr, Phys. Rev. D 108 , 083519 (2023), arXiv:2305.02846 [astro-ph.CO].\n- [152] M. Baryakhtar, O. Simon, and Z. J. Weiner, (2024), arXiv:2405.10358 [astro-ph.CO].\n- [153] S. Vagnozzi, E. Giusarma, O. Mena, K. Freese, M. Gerbino, S. Ho, and M. Lattanzi, Phys. Rev. D 96 , 123503 (2017), arXiv:1701.08172 [astro-ph.CO].\n- [154] D. Wang, O. Mena, E. Di Valentino, and S. Gariazzo, (2024), arXiv:2405.03368 [astro-ph.CO].\n- [155] D. Naredo-Tuero, M. Escudero, E. Fern'andez-Mart'ınez, X. Marcano, and V. Poulin, (2024), arXiv:2407.13831 [astro-ph.CO].\n- [156] G.-H. Du, P.-J. Wu, T.-N. Li, and X. Zhang, (2024), arXiv:2407.15640 [astro-ph.CO].\n- [157] J.-Q. Jiang, W. Giar'e, S. Gariazzo, M. G. Dainotti, E. Di Valentino, O. Mena, D. Pedrotti, S. S. da Costa, and S. Vagnozzi, (2024), arXiv:2407.18047 [astro-ph.CO].\n- [158] I. M. H. Etherington, Philosophical Magazine 15 , 761 (1933).\n- [159] S. Santos da Costa, V. C. Busti, and R. F. L. Holanda, JCAP 10 , 061 (2015), arXiv:1506.00145 [astro-ph.CO].\n- [160] N. B. Hogg, M. Martinelli, and S. Nesseris, JCAP 12 , 019 (2020), arXiv:2007.14335 [astro-ph.CO].\n- [161] F. Renzi, N. B. Hogg, and W. Giar'e, Mon. Not. Roy. Astron. Soc. 513 , 4004 (2022), arXiv:2112.05701 [astroph.CO].\n- [162] T. Liu, S. Cao, S. Ma, Y. Liu, C. Zheng, and J. Wang, Phys. Lett. B 838 , 137687 (2023), arXiv:2301.02997 [astro-ph.CO].\n- [163] J.-Z. Qi, Y.-F. Jiang, W.-T. Hou, and X. Zhang, (2024), arXiv:2407.07336 [astro-ph.CO].\n- [164] P. Motloch, Phys. Rev. D 101 , 123509 (2020), arXiv:2004.11351 [astro-ph.CO].\n- [165] S. Vagnozzi, (2019), arXiv:1907.08010 [astro-ph.CO].\n- [166] M. A. Troxel et al. (DES), Phys. Rev. D 98 , 043528 (2018), arXiv:1708.01538 [astro-ph.CO].\n- [167] C. Heymans et al. , Astron. Astrophys. 646 , A140 (2021), arXiv:2007.15632 [astro-ph.CO].\n- [168] G. Efstathiou and P. Lemos, Mon. Not. Roy. Astron. Soc. 476 , 151 (2018), arXiv:1707.00483 [astro-ph.CO].\n- [169] D. Benisty, Phys. Dark Univ. 31 , 100766 (2021), arXiv:2005.03751 [astro-ph.CO].\n- [170] R. C. Nunes and S. Vagnozzi, Mon. Not. Roy. Astron. Soc. 505 , 5427 (2021), arXiv:2106.01208 [astro-ph.CO].\n- [171] Z. Sakr, S. Ilic, and A. Blanchard, Astron. Astrophys. 666 , A34 (2022), arXiv:2112.14171 [astro-ph.CO].\n- [172] T. M. C. Abbott et al. (Kilo-Degree Survey, DES), Open J. Astrophys. 6 , 2305.17173 (2023), arXiv:2305.17173 [astro-ph.CO].\n- [173] V. Ghirardini et al. , (2024), arXiv:2402.08458 [astroph.CO].\n- [174] A. J. Ross et al. , Mon. Not. Roy. Astron. Soc. 449 , 835 (2015), arXiv:1409.3242 [astro-ph.CO].\n- [175] S. Alam et al. (BOSS), Mon. Not. Roy. Astron. Soc. 470 , 2617 (2017), arXiv:1607.03155 [astro-ph.CO].\n- [176] Z. Zhai et al. (eBOSS), Astrophys. J. 848 , 76 (2017), arXiv:1607.05383 [astro-ph.CO].\n- [177] A. Tamone et al. (eBOSS), Mon. Not. Roy. Astron. Soc. 499 , 5527 (2020), arXiv:2007.09009 [astro-ph.CO].  \n- [178] J. Hou et al. (eBOSS), Mon. Not. Roy. Astron. Soc. 500 , 1201 (2020), arXiv:2007.08998 [astro-ph.CO].\n- [179] V. de Sainte Agathe et al. (eBOSS), Astron. Astrophys. 629 , A85 (2019), arXiv:1904.03400 [astro-ph.CO].\n- [180] D. Brout et al. , Astrophys. J. 938 , 110 (2022), arXiv:2202.04077 [astro-ph.CO].\n- [181] D. M. Scolnic et al. (Pan-STARRS1), Astrophys. J. 859 , 101 (2018), arXiv:1710.00845 [astro-ph.CO].\n- [182] V. Mossa et al. , Nature 587 , 210 (2020).\n- [183] E. Aubourg et al. (BOSS), Phys. Rev. D 92 , 123516 (2015), arXiv:1411.1074 [astro-ph.CO].\n- [184] A. Gelman and D. B. Rubin, Statist. Sci. 7 , 457 (1992).\n- [185] A. Lewis, (2019), arXiv:1910.13970 [astro-ph.IM].\n- [186] N. Schoneberg, L. Verde, H. Gil-Mar'ın, and S. Brieden, JCAP 11 , 039 (2022), arXiv:2209.14330 [astro-ph.CO].\n- [187] A. Lewis, A. Challinor, and A. Lasenby, Astrophys. J. 538 , 473 (2000), arXiv:astro-ph/9911177.\n- [188] S. Vagnozzi, Phys. Rev. D 104 , 063524 (2021), arXiv:2105.10425 [astro-ph.CO].\n- [189] F. Niedermann and M. S. Sloth, Phys. Rev. D 103 , 103537 (2021), arXiv:2009.00006 [astro-ph.CO].\n- [190] M. Escudero and S. J. Witte, Eur. Phys. J. C 81 , 515 (2021), arXiv:2103.03249 [hep-ph].  \n[191] T. M. C. Abbott  \net al.  \n(DES), (2024), arXiv:2401.02929  \n[astro-ph.CO].  \n- [192] D. Rubin et al. , (2023), arXiv:2311.12098 [astroph.CO].\n- [193] E. O. Colg'ain, S. Pourojaghi, and M. M. SheikhJabbari, (2024), arXiv:2406.06389 [astro-ph.CO].\n- [194] C. Krishnan, E. O. Colg'ain, Ruchika, A. A. Sen, M. M. Sheikh-Jabbari, and T. Yang, Phys. Rev. D 102 , 103525 (2020), arXiv:2002.06044 [astro-ph.CO].\n- [195] C. Krishnan, E. O. Colg'ain, M. M. Sheikh-Jabbari, and T. Yang, Phys. Rev. D 103 , 103509 (2021), arXiv:2011.02858 [astro-ph.CO].\n- [196] M. G. Dainotti, B. De Simone, T. Schiavone, G. Montani, E. Rinaldi, and G. Lambiase, Astrophys. J. 912 , 150 (2021), arXiv:2103.02117 [astro-ph.CO].\n- [197] M. G. Dainotti, B. De Simone, T. Schiavone, G. Montani, E. Rinaldi, G. Lambiase, M. Bogdan, and S. Ugale, Galaxies 10 , 24 (2022), arXiv:2201.09848 [astro-ph.CO].\n- [198] E. O. Colg'ain, M. M. Sheikh-Jabbari, R. Solomon, G. Bargiacchi, S. Capozziello, M. G. Dainotti, and D. Stojkovic, Phys. Rev. D 106 , L041301 (2022), arXiv:2203.10558 [astro-ph.CO].\n- [199] M. G. Dainotti, G. Sarracino, and S. Capozziello, Publ. Astron. Soc. Jap. 74 , 1095 (2022), arXiv:2206.07479 [astro-ph.CO].\n- [200] E. O. Colg'ain, M. M. Sheikh-Jabbari, R. Solomon, M. G. Dainotti, and D. Stojkovic, Phys. Dark Univ. 44 , 101464 (2024), arXiv:2206.11447 [astro-ph.CO].\n- [201] E. O. Colg'ain, M. M. Sheikh-Jabbari, and R. Solomon, Phys. Dark Univ. 40 , 101216 (2023), arXiv:2211.02129 [astro-ph.CO].\n- [202] X. D. Jia, J. P. Hu, and F. Y. Wang, Astron. Astrophys. 674 , A45 (2023), arXiv:2212.00238 [astro-ph.CO].\n- [203] M. Malekjani, R. M. Conville, E. O. Colg'ain, S. Pourojaghi, and M. M. Sheikh-Jabbari, Eur. Phys. J. C 84 , 317 (2024), arXiv:2301.12725 [astro-ph.CO].\n- [204] S. A. Adil, O. Akarsu, M. Malekjani, E. O. Colg'ain, S. Pourojaghi, A. A. Sen, and M. M. SheikhJabbari, Mon. Not. Roy. Astron. Soc. 528 , L20 (2023), arXiv:2303.06928 [astro-ph.CO].\n- [205] O. Akarsu, E. O. Colg'ain, A. A. Sen, and M. M. SheikhJabbari, Universe 10 , 305 (2024), arXiv:2402.04767 [astro-ph.CO].\n- [206] E. O. Colg'ain, M. G. Dainotti, S. Capozziello, S. Pourojaghi, M. M. Sheikh-Jabbari, and D. Stojkovic, (2024), arXiv:2404.08633 [astro-ph.CO].\n- [207] E. O. Colg'ain, M. M. Sheikh-Jabbari, and L. Yin, (2024), arXiv:2405.19953 [astro-ph.CO].\n- [208] X. D. Jia, J. P. Hu, and F. Y. Wang, (2024), arXiv:2406.02019 [astro-ph.CO].\n- [209] S. W. Allen, R. W. Schmidt, and A. C. Fabian, Mon. Not. Roy. Astron. Soc. 334 , L11 (2002), arXiv:astroph/0205007.\n- [210] S. Ettori, P. Tozzi, and P. Rosati, Astron. Astrophys. 398 , 879 (2003), arXiv:astro-ph/0211335.\n- [211] S. Ettori, A. Morandi, P. Tozzi, I. Balestra, S. Borgani, P. Rosati, L. Lovisari, and F. Terenziani, Astron. Astrophys. 501 , 61 (2009), arXiv:0904.2740 [astro-ph.CO].\n- [212] A. B. Mantz, S. W. Allen, R. G. Morris, D. A. Rapetti, D. E. Applegate, P. L. Kelly, A. von der Linden, and R. W. Schmidt, Mon. Not. Roy. Astron. Soc. 440 , 2077 (2014), arXiv:1402.6212 [astro-ph.CO].\n- [213] A. B. Mantz et al. (SPT), Mon. Not. Roy. Astron. Soc. 510 , 131 (2021), arXiv:2111.09343 [astro-ph.CO].\n- [214] F. Villaescusa-Navarro et al. , Astrophys. J. 929 , 132 (2022), arXiv:2201.02202 [astro-ph.CO].\n- [215] N. Echeverri et al. (CAMELS), Astrophys. J. 954 , 125 (2023), arXiv:2304.06084 [astro-ph.CO].\n- [216] C. Chawak, F. Villaescusa-Navarro, N. E. Rojas, Y. Ni, C. Hahn, and D. Angles-Alcazar, Astrophys. J. 969 , 105 (2024), arXiv:2309.12048 [astro-ph.CO].\n- [217] C. Hahn, F. Villaescusa-Navarro, P. Melchior, and R. Teyssier, Astrophys. J. Lett. 966 , L18 (2024), arXiv:2310.08634 [astro-ph.CO].\n- [218] B. Y. Wang and A. Pisani, Astrophys. J. 970 , L32 (2024), arXiv:2405.04447 [astro-ph.CO].\n- [219] Y. Toda, W. Giar'e, E. Ozulker, E. Di Valentino, and S. Vagnozzi, Phys. Dark Univ. 46 , 101676 (2024), arXiv:2407.01173 [astro-ph.CO].\n- [220] N. Schoneberg and L. Vacher, (2024), arXiv:2407.16845 [astro-ph.CO].\n- [221] I. J. Allali, M. P. Hertzberg, and F. Rompineve, Phys. Rev. D 104 , L081303 (2021), arXiv:2104.12798 [astroph.CO].\n- [222] G. Ye, J. Zhang, and Y.-S. Piao, Phys. Lett. B 839 , 137770 (2023), arXiv:2107.13391 [astro-ph.CO].\n- [223] L. A. Anchordoqui, E. Di Valentino, S. Pan, and W. Yang, JHEAp 32 , 28 (2021), arXiv:2107.13932 [astro-ph.CO].\n- [224] N. Khosravi and M. Farhang, Phys. Rev. D 105 , 063505 (2022), arXiv:2109.10725 [astro-ph.CO].\n- [225] H. Wang and Y.-S. Piao, Phys. Lett. B 832 , 137244 (2022), arXiv:2201.07079 [astro-ph.CO].\n- [226] L. A. Anchordoqui, V. Barger, D. Marfatia, and J. F. Soriano, Phys. Rev. D 105 , 103512 (2022), arXiv:2203.04818 [astro-ph.CO].\n- [227] A. Reeves, L. Herold, S. Vagnozzi, B. D. Sherwin, and E. G. M. Ferreira, Mon. Not. Roy. Astron. Soc. 520 , 3688 (2023), arXiv:2207.01501 [astro-ph.CO].\n- [228] S. S. da Costa, D. R. da Silva, A. S. de Jesus, N. PintoNeto, and F. S. Queiroz, JCAP 04 , 035 (2024), arXiv:2311.07420 [astro-ph.CO].  \n- [229] H. Wang and Y.-S. Piao, (2024), arXiv:2404.18579 [astro-ph.CO].\n- [230] J.-Q. Jiang, D. Pedrotti, S. S. da Costa, and S. Vagnozzi, (2024), arXiv:2408.02365 [astro-ph.CO].\n- [231] F. Simpson, Phys. Rev. D 82 , 083505 (2010), arXiv:1007.1034 [astro-ph.CO].\n- [232] M. Escudero, O. Mena, A. C. Vincent, R. J. Wilkinson, and C. Bœhm, JCAP 09 , 034 (2015), arXiv:1505.06735 [astro-ph.CO].\n- [233] T. S. Koivisto, E. N. Saridakis, and N. Tamanini, JCAP 09 , 047 (2015), arXiv:1505.07556 [astro-ph.CO].\n- [234] A. Pourtsidou and T. Tram, Phys. Rev. D 94 , 043518 (2016), arXiv:1604.04222 [astro-ph.CO].\n- [235] S. Kumar and R. C. Nunes, Eur. Phys. J. C 77 , 734 (2017), arXiv:1709.02384 [astro-ph.CO].\n- [236] V. Gluscevic and K. K. Boddy, Phys. Rev. Lett. 121 , 081301 (2018), arXiv:1712.07133 [astro-ph.CO].\n- [237] K. K. Boddy and V. Gluscevic, Phys. Rev. D 98 , 083510 (2018), arXiv:1801.08609 [astro-ph.CO].\n- [238] K. K. Boddy, V. Gluscevic, V. Poulin, E. D. Kovetz, M. Kamionkowski, and R. Barkana, Phys. Rev. D 98 , 123506 (2018), arXiv:1808.00001 [astro-ph.CO].\n- [239] M. Asghari, J. Beltr'an Jim'enez, S. Khosravi, and D. F. Mota, JCAP 04 , 042 (2019), arXiv:1902.05532 [astroph.CO].\n- [240] S. Vagnozzi, L. Visinelli, O. Mena, and D. F. Mota, Mon. Not. Roy. Astron. Soc. 493 , 1139 (2020), arXiv:1911.12374 [gr-qc].\n- [241] J. Beltr'an Jim'enez, D. Bettoni, D. Figueruelo, and F. A. Teppa Pannia, JCAP 08 , 020 (2020), arXiv:2004.14661 [astro-ph.CO].\n- [242] N. Becker, D. C. Hooper, F. Kahlhoefer, J. Lesgourgues, and N. Schoneberg, JCAP 02 , 019 (2021), arXiv:2010.04074 [astro-ph.CO].\n- [243] D. Figueruelo et al. , JCAP 07 , 022 (2021), arXiv:2103.01571 [astro-ph.CO].\n- [244] S. Vagnozzi, L. Visinelli, P. Brax, A.-C. Davis, and J. Sakstein, Phys. Rev. D 104 , 063023 (2021), arXiv:2103.15834 [hep-ph].\n- [245] J. Beltr'an Jim'enez, D. Bettoni, D. Figueruelo, F. A. Teppa Pannia, and S. Tsujikawa, Phys. Rev. D 104 , 103503 (2021), arXiv:2106.11222 [astro-ph.CO].\n- [246] M. S. Linton, R. Crittenden, and A. Pourtsidou, JCAP 08 , 075 (2022), arXiv:2107.03235 [astro-ph.CO].\n- [247] K. K. Rogers, C. Dvorkin, and H. V. Peiris, Phys. Rev. Lett. 128 , 171301 (2022), arXiv:2111.10386 [astroph.CO].\n- [248] F. Ferlito, S. Vagnozzi, D. F. Mota, and M. Baldi, Mon. Not. Roy. Astron. Soc. 512 , 1885 (2022), arXiv:2201.04528 [astro-ph.CO].\n- [249] V. Poulin, J. L. Bernal, E. D. Kovetz, and M. Kamionkowski, Phys. Rev. D 107 , 123538 (2023), arXiv:2209.06217 [astro-ph.CO].\n- [250] J. Beltr'an Jim'enez, E. Di Dio, and D. Figueruelo, JCAP 11 , 088 (2023), arXiv:2212.08617 [astro-ph.CO].\n- [251] S. Vagnozzi, Universe 9 , 393 (2023), arXiv:2308.16628 [astro-ph.CO].\n- [252] H. Desmond, B. Jain, and J. Sakstein, Phys. Rev. D 100 , 043537 (2019), [Erratum: Phys.Rev.D 101, 069904 (2020), Erratum: Phys.Rev.D 101, 129901 (2020)], arXiv:1907.03778 [astro-ph.CO].\n- [253] Q. Ding, T. Nakama, and Y. Wang, Sci. China Phys. Mech. Astron. 63 , 290403 (2020), arXiv:1912.12600 [astro-ph.CO].\n- [254] H. Desmond and J. Sakstein, Phys. Rev. D 102 , 023007 (2020), arXiv:2003.12876 [astro-ph.CO].\n- [255] R.-G. Cai, J.-F. Ding, Z.-K. Guo, S.-J. Wang, and W.-W. Yu, Phys. Rev. D 103 , 123539 (2021), arXiv:2012.08292 [astro-ph.CO].\n- [256] R.-G. Cai, Z.-K. Guo, L. Li, S.-J. Wang, and W.-W. Yu, Phys. Rev. D 103 , 121302 (2021), arXiv:2102.02020 [astro-ph.CO].\n- [257] V. Marra and L. Perivolaropoulos, Phys. Rev. D 104 , L021303 (2021), arXiv:2102.06012 [astro-ph.CO].\n- [258] C. Krishnan, R. Mohayaee, E. O. Colg'ain, M. M. Sheikh-Jabbari, and L. Yin, Class. Quant. Grav. 38 , 184001 (2021), arXiv:2105.09790 [astro-ph.CO].\n- [259] L. Perivolaropoulos and F. Skara, Phys. Rev. D 104 , 123511 (2021), arXiv:2109.04406 [astro-ph.CO].\n- [260] S. Castello, M. Hog˚as, and E. Mortsell, JCAP 07 , 003 (2022), [Erratum: JCAP 09, E01 (2022)], arXiv:2110.04226 [astro-ph.CO].\n- [261] D. Camarena, V. Marra, Z. Sakr, and C. Clarkson, Class. Quant. Grav. 39 , 184001 (2022), arXiv:2205.05422 [astro-ph.CO].\n- [262] L. Perivolaropoulos and F. Skara, Universe 8 , 502 (2022), arXiv:2208.11169 [astro-ph.CO].\n- [263] V. K. Oikonomou, P. Tsyba, and O. Razina, Universe 8 , 484 (2022), arXiv:2209.04669 [gr-qc].  \n- [264] M. Hog˚as and E. Mortsell, Phys. Rev. D 108 , 024007 (2023), arXiv:2303.12827 [astro-ph.CO].\n- [265] M. Hog˚as and E. Mortsell, Phys. Rev. D 108 , 124050 (2023), arXiv:2309.01744 [astro-ph.CO].\n- [266] L. Giani, C. Howlett, K. Said, T. Davis, and S. Vagnozzi, JCAP 01 , 071 (2024), arXiv:2311.00215 [astro-ph.CO].\n- [267] S. Mazurenko, I. Banik, P. Kroupa, and M. Haslbauer, Mon. Not. Roy. Astron. Soc. 527 , 4388 (2024), arXiv:2311.17988 [astro-ph.CO].\n- [268] L. Huang, S.-J. Wang, and W.-W. Yu, (2024), arXiv:2401.14170 [astro-ph.CO].\n- [269] Ruchika, L. Perivolaropoulos, and A. Melchiorri, (2024), arXiv:2408.03875 [astro-ph.CO].\n- [270] S. J. Clark, K. Vattis, J. Fan, and S. M. Koushiappas, Phys. Rev. D 107 , 083527 (2023), arXiv:2110.09562 [astro-ph.CO].\n- [271] L. Amendola et al. , Living Rev. Rel. 21 , 2 (2018), arXiv:1606.00180 [astro-ph.CO].\n- [272] K. N. Abazajian et al. (CMB-S4), (2016), arXiv:1610.02743 [astro-ph.CO].\n- [273] A. Aghamousa et al. (DESI), (2016), arXiv:1611.00036 [astro-ph.IM].\n- [274] P. Ade et al. (Simons Observatory), JCAP 02 , 056 (2019), arXiv:1808.07445 [astro-ph.CO].\n- [275] M. H. Abitbol et al. (Simons Observatory), Bull. Am. Astron. Soc. 51 , 147 (2019), arXiv:1907.08284 [astroph.IM]."}