Why reducing the cosmic sound horizon 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 Λ CDM, known as the Hubble tension, has become one of the most pressing problems in cosmology. A large number of amendments to the Λ 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 (cid:63) . We demonstrate here that any model which only reduces r (cid:63) can never fully resolve the Hubble tension while remaining consistent with other cosmological datasets. We show explicitly that models which operate at lower matter density Ω m h 2 run into tension with the observations of baryon acoustic oscillations, while models operating at higher Ω m h 2 develop tension with galaxy weak lensing data.

a constraint of Ω m h 2 = 0.143 ± 0.001 within the ΛCDM model 1 , yielding 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 8 . This is largely because the baryon acoustic oscillation (BAO) and 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 parameterizations more flexible, as e.g., in 9,10 , 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 11 but can, in principle, be accommodated within the general Horndeski class of scalar-tensor theories 12 .
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 [13][14][15][16][17][18] , extra radiation in either neutrinos [19][20][21][22] or some other dark sector 23-28 , and dark energy-dark matter interactions 29 . Proposals in class (ii) include primordial magnetic fields 30 , non-standard recombination 31 , or varying fundamental constants 32, 33 . 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 surveys 34, 35 or BAO 36 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. It is well known that BAO measurements at multiple redshifts provide a constraint on r d h and Ω m .
Without going into specific models, we now consider modifications of ΛCDM which decrease r , treating the latter as a free parameter and taking r d = 1.0184r . For a given Ω m h 2 , Eq. 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 41 . Our argument is that one will generally run into problems even before considering these additional potential complications.
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  In most of the models represented in Fig. 3, the effect of introducing new physics only amounts to a reduction in r d . 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 26 and the neutrino model 19 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.
In conclusion, we have argued that any model which tries to reconcile the CMB inferred

Methods 1 The acoustic scale measurements from the CMB and BAO
The geometric information of the Universe is mapped by the CMB via There are three types of BAO observables corresponding to the three ways of extracting the acoustic scale from galaxy surveys 37 : using correlations in the direction perpendicular to the line of sight, using correlations in the direction parallel to the line of sight, and the angle-averaged or "isotropic" measurement. For the purpose of our discussion, it suffices to consider just the first type, which is the closest to CMB in its essence, but our conclusions apply to all three. In fact, our numerical analysis includes all three types. Namely, let us consider where z obs is the redshift at which a given BAO measurement is made.
As the integrals in the denominators of Eqs.
(3) and (5)  (1 + z) 3 + h 2 /ω m − 1 , where ω m = Ω m h 2 , and an analogous equation for BAO with the replacement (r , θ , z ) → 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. (7). This results in different slopes of the respective r d (h) lines.

Obtaining the S 8 constraints
To derive the Model 2 and Model 3 contours in Fig. 2 Data Availability The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.