Why reducing the cosmic sound horizon alone cannot fully resolve the Hubble tension

The standard ΛCDM model, comprising baryons, cold dark matter (CDM), and dark energy (Λ), accurately describes many cosmological observations. However, it predicts a Hubble constant (H₀) of 67.36 ± 0.54 km/s/Mpc based on Planck satellite CMB data, significantly lower than direct local measurements like the SHOES collaboration's 73.5 ± 1.4 km/s/Mpc. This 4.2σ discrepancy, the "Hubble tension," has motivated numerous ΛCDM modifications. The precise measurement of acoustic peaks in CMB anisotropy spectra determines the angular size of the sound horizon at recombination (θ = rs/D(z)), where rs is the comoving distance sound waves traveled to recombination and D(z) is the comoving distance to the last scattering surface. In ΛCDM, D(z) depends on Ωmh² and h (H₀/100 km/s/Mpc). A higher locally measured H₀ implies a larger θ unless other factors in the equation are modified. Models addressing this tension either modify late-time expansion (e.g., dynamical dark energy) or reduce rs (early-time modifications). Late-time modifications struggle due to BAO and supernovae data consistency with constant dark energy density. Early-time solutions aim to reduce rs, either by increasing Hubble expansion around recombination or altering the recombination rate. Examples include early dark energy, extra radiation (neutrinos or dark sector), dark energy-dark matter interactions, primordial magnetic fields, non-standard recombination, or varying fundamental constants. This research investigates whether solely changing rs can resolve the Hubble tension.

Numerous papers propose modifications to the ΛCDM model to address the Hubble tension. Late-time modifications, often using phenomenological parameterizations, generally fall short of completely resolving the tension because BAO and supernova data are consistent with a constant dark energy density. More flexible parameterizations, allowing non-monotonically evolving dark energy, are proposed but can imply instabilities in simple dark energy and modified gravity theories. Early-time modifications, focused on reducing the sound horizon at recombination (rs), involve various mechanisms such as early dark energy, extra radiation (from neutrinos or other sources), dark energy-dark matter interactions, and modifications to the recombination process itself. These models, while aiming to alleviate the Hubble tension, often introduce new parameters and complexities. The existing literature lacks a comprehensive model-independent analysis of the feasibility of resolving the tension by solely modifying rs.

The research employs a model-independent approach, treating the sound horizon at recombination (rs) as a free parameter. The relationship between rs and the sound horizon at the drag epoch (ra), where photon drag on baryons becomes insignificant (ra ≈ 1.02rs), remains largely unchanged across various models. The authors use two key equations: one relating the angular size of the sound horizon in the CMB (θ = rs/D(z*)) and another for BAO (θBAO(zobs) = ra/D(zobs)), where D(z) is the comoving distance and zobs is the observed redshift. These equations define lines in the ra-H₀ plane. The study analyzes the slopes of these lines derived from CMB and BAO observations at various redshifts (e.g., z = 0.5 and z = 1.5) for different values of Ωmh². The analysis considers the marginalized constraints from BAO observations (combining data from various surveys like eBOSS, 6dF, and SDSS) and the ΛCDM constraints from Planck. Additionally, it incorporates constraints on σ8 (matter clustering amplitude) from weak lensing surveys (DES and KiDS). The authors investigate the parameter space to determine if a reduction in rs alone can reconcile CMB, BAO, and SHOES H₀ measurements while remaining consistent with weak lensing data. The analysis includes a compilation of rs, H₀, Ωmh², and σ8 values from various models in the literature, aiming to identify consistent trends.

The key finding is that solely reducing rs cannot fully resolve the Hubble tension without creating inconsistencies with either BAO or weak lensing data. The different slopes of the ra-H₀ degeneracy lines for CMB and BAO observations at different redshifts prevent simultaneous agreement with both datasets when only rs is modified. Attempts to reconcile CMB, BAO, and SHOES H₀ measurements by decreasing rs necessitate a higher Ωmh². However, this increased Ωmh² creates tension with weak lensing surveys like DES and KiDS, which already show some tension with the standard ΛCDM model. The analysis of various models from the literature confirms this trend: models with low Ωmh² fail to achieve a sufficiently high H₀ or are inconsistent with BAO, while models with high Ωmh² are in tension with DES and KiDS. The maximum achievable H₀ while maintaining reasonable agreement with BAO and DES/KiDS is around 70 km/s/Mpc, falling short of the locally measured values. The CMB provides more information than just the acoustic peak positions; modifying rs to resolve the Hubble tension often worsens the fit to other CMB features. Therefore, a full resolution requires either multiple modifications to ΛCDM or the identification of systematic effects in the datasets.

The results highlight a fundamental limitation of models solely aiming to reduce the sound horizon to resolve the Hubble tension. The inherent difference in the slopes of the ra-H₀ degeneracy lines from CMB and BAO prevents a simultaneous solution. This model-independent demonstration underscores the need for more complex solutions involving multiple modifications of ΛCDM or revisiting systematic errors in observational data. The finding does not invalidate early-time solutions but emphasizes their limitations when solely focusing on rs reduction. The tension between high Ωmh² (required for high H₀ with reduced rs) and weak lensing data is a significant challenge. While some models attempt to circumvent this, they often introduce additional parameters and theoretical complexities, potentially diminishing their explanatory power.

This study demonstrates that reducing the cosmic sound horizon alone is insufficient to resolve the Hubble tension while maintaining consistency with other cosmological data. The different slopes of the rs-H₀ degeneracy lines for CMB and BAO preclude a full resolution through this mechanism. Resolving the Hubble tension requires either multiple modifications to the ΛCDM model or the discovery of systematic errors in the datasets. Future research should focus on exploring alternative models that incorporate multiple modifications or carefully scrutinize systematic effects in the CMB, BAO, and weak lensing datasets.

The analysis assumes that the expansion of the universe after recombination is well-described by the ΛCDM model. Deviations from this assumption in specific models could influence the results. The analysis also considers only a limited set of models from the existing literature; other early-time solutions may exist that evade these constraints. Finally, the uncertainties in BAO and weak lensing measurements could affect the conclusions.