Why reducing the cosmic sound horizon alone can not fully resolve the Hubble tension

The mismatch between the locally measured expansion rate of the universe and the one inferred from the cosmic microwave background measurements by Planck in the context of the standard $\Lambda$CDM, known as the Hubble tension, has become one of the most pressing problems in cosmology. A large number of amendments to the $\Lambda$CDM model have been proposed in order to solve this tension. Many of them introduce new physics, such as early dark energy, modifications of the standard model neutrino sector, extra radiation, primordial magnetic fields or varying fundamental constants, with the aim of reducing the sound horizon at recombination $r_{\star}$. We demonstrate here that any model which only reduces $r_{\star}$ can never fully resolve the Hubble tension while remaining consistent with other cosmological datasets. We show explicitly that models which achieve a higher Hubble constant with lower values of matter density $\Omega_m h^2$ run into tension with the observations of baryon acoustic oscillations, while models with larger $\Omega_mh^2$ develop tension with galaxy weak lensing data.

Decades of progress in observational and theoretical cosmology have led to the consensus that our universe is well described by a flat Friedman-Robertson-Lemaitre metric and is currently comprised of around 5% baryons, 25% cold dark matter (CDM), and 70% dark energy in its simplest form -the cosmological constant Λ. Although this ΛCDM model fits many observations exquisitely well, its prediction for the present day cosmic expansion rate, H 0 = 67.36 ± 0.54 km/s/Mpc [1], based on precise cosmic microwave background (CMB) radiation observations by the Planck satellite, do not compare well with direct measurements of the Hubble constant. In particular, the Supernovae H0 for the Equation of State (SH0ES) collaboration [2], using Cepheid calibrated supernovae Type Ia, finds a much higher value of H 0 = 73.5 ± 1.4 km/s/Mpc. This 4.2σ disagreement, known as the "Hubble tension", has spurred much interest in modifications of the ΛCDM model capable of resolving it (cf. [3] for a comprehensive list of references). Several other determinations of H 0 , using different methods, are also in some degree of tension with Planck, such as the Megamaser Cosmology Project [4] finding 73.9 ± 3.0 km/s/Mpc or H0LiCOW [5] finding 73.3 +1.7 −1.8 km/s/Mpc. It is worth noting that a somewhat lower value of 69.8 ± 2.5 km/s/Mpc was obtained using an alternative method for calibrating SNIa [6].
Among the most precisely measured quantities in cosmology are the locations of the acoustic peaks in the CMB temperature and polarization anisotropy spectra. They determine the angular size of the sound horizon at recombination, with an accuracy of 0.03% [1]. The sound horizon r is the comoving distance a sound wave could travel from the beginning of the universe to recombination, a standard ruler in any given model, and D(z ) is the comoving distance from a present day observer to the last scattering surface, i.e., to the epoch of recombination. D(z ) is determined by the redshift-dependent expansion rate H(z) = h(z) × 100 km/s/Mpc which, in the flat ΛCDM model, depends only on two parameters (see Appendix A for details): Ω m h 2 and h, where Ω m is the fractional matter energy density today and h = h(0) = H 0 /100 km/s/Mpc. Thus, given r and an estimate of Ω m h 2 , one can infer h from the measurement of θ . Using the Planck best fit values of Ω m h 2 = 0.143 ± 0.001 and r = 144.44 ± 0.27 Mpc, obtained within the ΛCDM model [1], yields a Hubble constant significantly lower than the more direct local measurements.
If the value of the Hubble constant was the one measured locally, i.e., h ≈ 0.735, it would yield a much larger value of θ unless something else in Eq. (1) was modified to preserve the observed CMB acoustic peak positions. There are two broad classes of models attempting to resolve this tension by introducing new physics. One introduces modifications at late times (i.e., lower redshifts), e.g., by introducing a dynamical dark energy or new interactions among the dark components that alter the Hubble expansion to make it approach a higher value today, while still preserving the integrated distance D in Eq. (1). In the second class of models, the new physics aims to reduce the numerator in Eq. (1), i.e., modify the sound horizon at recombination.
Late time modifications based on simple phenomenological parameterizations tend to fall short of fully resolving the tension [7]. This is largely because the baryon acoustic oscillation (BAO) and supernovae (SN) data, probing the expansion in the 0 z 1 range, are generally consistent with a constant dark energy density. One can accommodate a higher value of H 0 by making pa-rameterizations more flexible, as e.g., in [8,9], that allow for a non-monotonically evolving effective dark energy fluid. Such non-monotonicity tends to imply instabilities within the context of simple dark energy and modified gravity theories [10] but can, in principle, be accommodated within the general Horndeski class of scalar-tensor theories [11].
Early-time solutions aim to reduce r with essentially two possibilities: (i) a coincidental increase of the Hubble expansion around recombination or (ii) new physics that alters the rate of recombination. Proposals in class (i) include the presence of early dark energy [12][13][14][15][16][17], extra radiation in either neutrinos [18][19][20][21] or some other dark sector [22][23][24][25][26][27], and dark energy-dark matter interactions [28]. Proposals in class (ii) include primordial magnetic fields [29], non-standard recombination [30], or varying fundamental constants [31,32]. In this work we show that any early-time solution which only changes r can never fully resolve the Hubble tension without being in significant tension with either the weak lensing (WL) surveys [33,34] or BAO [35] observations. The acoustic peaks, prominently seen in the CMB anisotropy spectra, are also seen as BAO peaks in the galaxy power spectra and carry the imprint of a slightly different, albeit intimately related, standard ruler -the sound horizon at the "cosmic drag" epoch (or the epoch of baryon decoupling), r d , when the photon drag on baryons becomes unimportant. As the latter takes place at a slightly lower redshift than recombination, we have r d ≈ 1.02r with the proportionality factor being essentially the same in all proposed modified recombination scenarios. More importantly for our discussion, the BAO feature corresponds to the angular size of the standard ruler at z z , i.e., in the range 0 z 2.5 accessible by galaxy redshift surveys. For the BAO feature measured using galaxy correlations in the transverse direction to the line of sight, the observable is where z obs is the redshift at which a given BAO measurement is made. For simplicity, we do not discuss the line of sight and the "isotropic" BAO measurements [36] here, but our arguments apply to them as well. It is well known that BAO measurements at multiple redshifts provide a constraint on r d h and Ω m . In any particular model, r (and r d ) is a derived quantity that depends on Ω m h 2 , the baryon density and other parameters. However, in this work, for the purpose of illustrating trends that are common to all models, we treat r as an independent parameter and assume that no new physics affects the evolution of the universe after recombination.
Without going into specific models, we now consider modifications of ΛCDM which decrease r , treating the latter as a free parameter. The relation between r and r d in different models that reduce the sound horizon is largely the same as the one in ΛCDM, hence we fix it at r d = 1.0184r based on the Planck best fit ΛCDM value. For a given Ω m h 2 , Eq. (1) defines a line in the r d -H 0 plane and, since Eqs. (1) and (2) are the same in essence, a BAO measurement at each different redshift also defines a respective line in the r d -H 0 plane. However, the significant difference between z and z obs results in different slopes of the respective r d (h) lines (see Appendix B for details), as illustrated in Fig. 1. The latter shows the r d (h) lines from two different BAO observations, one at redshift z = 0.5 and another at z = 1.5, at Ω m h 2 fixed to the Planck best fit ΛCDM value of 0.143, and the analogous lines defined by the CMB acoustic scale plotted for three values of Ω m h 2 : 0.143, 0.155 and 0.167. Both lines correspond to transverse BAO measurements. Slopes derived from the line of sight and isotropic BAO at the same redshift would be different, but the trend with increasing redshift is the same. The lines are derived from the central observational values and do not account for the uncertainties in θ BAO ⊥ and θ (although the uncertainty in θ is so tiny that it would be difficult to see by eye on this plot). As anticipated, the slope of the r d (h) lines becomes steeper with increased redshift. Also shown in Fig. 1 are the marginalized 68% and 95% confidence levels (CL) derived from the combination of all presently available BAO observations in a recombinationmodel-independent way, namely, while treating r d as an independent parameter (see [38] and Appendix B for details). The red contours show the ΛCDM based constraint from Planck, in good agreement with BAO at H 0 ≈ 67 km/s/Mpc, but in tension with the SH0ES value shown with the grey band. In order to reconcile Planck with SH0ES solely by reducing r d , one would have to move along one of the CMB lines. Doing it along the line at Ω m h 2 = 0.143 would quickly move the values of r d and H 0 out of the purple band, creating a tension with BAO. Full consistency between the observed CMB peaks, BAO and the SH0ES Hubble constant could only be achieved at a higher value of Ω m h 2 ≈ 0.167 1 . However, unless one supplements the reduction in r d by yet another modification of the model, such high values of Ω m h 2 would cause tension with galaxy WL surveys such as the Dark Energy Survey (DES) [33] and the Kilo-Degree Survey (KiDS) [34], which we illustrate next.
DES and KiDS derived strong constraints on the quantity The values of S 8 and Ω m obtained by DES and KiDS are already in slight tension with the Planck best fit ΛCDM model, and the tension between KiDS and Planck is notably stronger than that between DES and Planck. Increasing the matter density aggravates this tensiona trend that can be seen in Fig. 2. The figure shows the 68% and 95% CL joint constraints on S 8 -Ω m by DES supplemented by the Pantheon SN sample [39] (which helps by providing an independent constraint on Ω m ), along with those by Planck within the ΛCDM model. The purple contours (Model 2) correspond to the model that can simultaneously fit BAO and CMB acoustic peaks at Ω m h 2 = 0.155, i.e., the model defined by the overlap between the BAO band and the θ (2) (blue dashed) line in Fig. 1. The green contours (Model 3) are derived from the model with Ω m h 2 = 0.167 corresponding to the overlap region between the θ (3) (green dotted) line and the BAO and SH0ES bands in Fig. 1 (see Appendix C for details). The figure shows that when attempting to find a full resolution of the Hubble tension, with CMB, BAO and SH0ES in agreement with each other, one exacerbates the tension with DES and KiDS. 1 The dependence of the CMB r d (h) lines on Ωmh 2 may appear contradictory to the Ωmh 2 dependence shown in Fig. 1 of a wellknow paper by Knox and Millea [40]. There, increasing Ωmh 2 moves the CMB best fit (r d , h) point in a direction orthogonal to where our CMB lines move. The reason for the difference is that their r d is a derived parameter obtained from the standard recombination model and, hence, depends on Ωmh 2 . In our derivation of the CMB lines, on the other hand, the Ωmh 2 dependence only appears in D(z ) and D(z obs ). We note that there is much more information in the CMB than just the positions of the acoustic peaks. It is generally not trivial to introduce new physics that reduces r and r d without also worsening the fit to other features of the temperature and polarization spectra [40]. Our argument is that, even if one managed to solve the Hubble tension by reducing r while maintaining a perfect fit to all CMB data, one would still necessarily run into problems with either the BAO or WL.
Surveying the abundant literature of the proposed early-time solutions to the Hubble tension, one finds that the above trends are always confirmed. Fig. 3 shows the best fit values of r d h, H 0 and S 8 in models from Refs. [13,14,18,23,24,[28][29][30]32]. Note that there are other proposed early-time solutions to the Hubble tension. Fig. 3 only shows the models for which explicit estimates of H 0 , Ω m h 2 , S 8 , and possibly r d h were provided. One can see that, except for the model represented by the red dot at the very right of the plot, corresponding to the strongly interacting neutrino model of [18], solutions requiring low Ω m h 2 are in tension with BAO, whereas solutions with higher Ω m h 2 are in tension with DES and KiDS. This latter tension was previously observed and extensively discussed in the context of the early dark energy models [41][42][43][44][45][46]. As we have shown in this paper, it is part of a broader problem faced by all proposals aimed at reducing the Hubble tension in which the main change amounts to a reduction of r d .
In most of the models represented in Fig. 3, the effect of introducing new physics only amounts to a reduction in r d . We note that, in any specific model of a reduced  [1]. With the exception of the red dot, corresponding to the model from [18] with multiple modifications of ΛCDM fit to Planck temperature anisotropy data only, there is a consistent trend: models with low Ωmh 2 either fail to achieve a sufficiently high H0 or are in tension with baryonic acoustic oscillations (BAO), and models with high values of Ωmh 2 run into tension with DES or KiDS. . r d , the best fit values of other cosmological parameters also change, which can affect the quality of the fit to various datasets. However, such changes, e.g. in the best fit value of the spectral index n s which affects S 8 , tend to be small for the models studied in the literature and have a minor impact compared to the effect of reducing r d , which is a pre-requisite for reconciling CMB with SH0ES. As we have argued, this will necessarily limit their ability to address the Hubble tension while staying consistent with the large scale structure data. Resolving the Hubble tension by new early-time physics without creating other observational tensions requires more than just a reduction of the sound horizon. This is exemplified by the interacting dark matter-dark radiation model [25] and the neutrino model [18] proposed as solutions.
Here, extra tensions are avoided by supplementing the reduction in the sound horizon due to extra radiation by additional exotic physics: dark matter-dark radiation interactions in the first case and neutrino self-interactions and non-negligible neutrino masses in the second case. Consequently, with so many parameters, the posteriori probabilities for cosmological parameters are highly inflated over those for ΛCDM. It is not clear how theoretically appealing such scenarios are, and the model in [18] seems to be disfavoured by the CMB polarization data.
In conclusion, we have argued that any model which tries to reconcile the CMB inferred value of H 0 with that measured by SH0ES by only reducing the sound horizon automatically runs into tension with either the BAO or the galaxy WL data. While we do not expect our findings to be surprising for the majority of the community, the novelty of our result is in isolating and clearly stating the essence of the problem -that the slopes of the r -H 0 degeneracy lines for BAO and CMB are vastly different, thus making it impossible to reconcile CMB with SH0ES by reducing r without violating BAO. We believe this very simple fact has not been stated before in this context in a model-independent way. With just a reduction of r , the highest value of the Hubble constants one can get, while remaining in a reasonable agreement with BAO and DES/KiDS, is around 70 km/s/Mpc. Thus, a full resolution of the Hubble tension will require either multiple modifications of the ΛCDM model or discovering systematic effects in one or more of the datasets.

ACKNOWLEDGMENTS
We thank Eiichiro Komatsu and Joulien Lesgourgues for helpful comments on the draft of the paper and Kanhaiya Pandy and Toyokazu Sekiguchi for kindly providing us with data of their models. We gratefully acknowledge using CosmoMC [47] and GetDist [48]. This research was enabled in part by support provided by West-Grid (www.westgrid.ca) and Compute Canada Calcul Canada (www.computecanada.ca). L.P. (A5) and a completely analogous equation for BAO. It is important to realize that the derivative is very different for CMB and BAO due to the vast difference in redshifts at which the standard ruler is observed, z ≈ 1100 for CMB vs z obs ∼ 1 for BAO, resulting in different values of the integral in Eq. (A5). This results in different slopes of the respective r d (h) lines. Note that the slopes of the r d (h) lines differ for the transverse, parallel and volume averaged BAO measured at the same redshift. While important for constraining cosmological parameters [50], these differences are small compared to that caused by the big difference between the BAO and CMB redshifts.