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

The ΛCDM model, featuring a flat FRW metric with about 5% baryons, 25% CDM, and 70% dark energy (Λ), fits many observations but predicts a present-day expansion rate H0 ≈ 67.4 km/s/Mpc from Planck CMB data that is in 4+σ tension with local measurements (e.g., SHOES: H0 ≈ 73.5 km/s/Mpc), with other late-Universe probes often favoring similarly high values. The acoustic peak locations in the CMB determine the angular size of the sound horizon θ = r_s / D(z*), with r_s the sound horizon at recombination and D the comoving distance to last scattering, which in flat ΛCDM depends mainly on Ω_m h^2 and h. To accommodate a higher local H0 while preserving θ, one must alter either late-time expansion (modifying D) or reduce r_s with early-time physics. Simple late-time parameterizations generally fail due to BAO and SN supporting near-constant dark energy at 0 ≲ z ≲ 1. Early-time proposals aim to reduce r_s via a transient increase in H(z) around recombination or modified recombination physics. The core question of this work is whether reducing r_s alone can reconcile CMB-inferred and locally measured H0 without conflicting with other datasets.

Two broad solution classes are discussed: (1) late-time modifications (evolving dark energy, dark sector interactions) that adjust H(z) at low redshift to preserve CMB θ while increasing H0; and (2) early-time changes that reduce r_s by altering pre- or peri-recombination physics. Prior proposals include early dark energy; extra relativistic species or dark sector radiation; dark matter–dark radiation interactions; primordial magnetic fields; non-standard recombination; and varying fundamental constants. Previous analyses show late-time simple parameterizations struggle with BAO and SN, while early-time models often face tensions with either BAO (if Ω_m h^2 is low) or weak lensing (if Ω_m h^2 is high). The paper compiles best-fit parameter trends from multiple early-time model papers, confirming this dichotomy and noting that only models with multiple concurrent new-physics ingredients sometimes evade all tensions, often at the cost of complexity and potential conflicts with detailed CMB data.

The analysis isolates the geometric constraints from the acoustic scales in a model-agnostic way by treating r_s (and closely related r_a, the drag-epoch sound horizon with r_a ≈ 1.0184 r_s) as free parameters, while assuming standard ΛCDM expansion after recombination. Key steps: (1) Consider the CMB acoustic scale θ = r_s / D(z*) and BAO angular ruler θ_BAO(z_obs) = r_a / D(z_obs). For a fixed Ω_m h^2, each relation defines a degeneracy line between r_a (or r_s) and H0 in the r_a–H0 plane. Because CMB (z* ≈ 1100) and BAO (z_obs ≲ 2.5) probe very different redshifts, the slopes of these degeneracy lines differ substantially. (2) Fix r_a = 1.0184 r_s (Planck ΛCDM value, largely unchanged across reduced r_s models) and examine how CMB lines for different Ω_m h^2 intersect with BAO-derived constraints and the SHOES H0. (3) Derive marginalized r_a–H0 constraints from a compilation of BAO datasets (6dF, SDSS DR7, eBOSS DR16 LRG/ELG, QSO, Ly-α) using CosmoMC modified to treat r_a as an independent parameter; marginalize over relevant cosmological parameters (e.g., Ω_m, Ω_Λ, H0, curvature). (4) Compare with Planck ΛCDM constraints and with SHOES H0. (5) Assess weak lensing consistency by deriving S8–Ω_m constraints: use DES Y1 + Pantheon SN (standard CosmoMC), and construct two representative models: Model 2 with Ω_m h^2 ≈ 0.155 (CMB–BAO overlap) and Model 3 with Ω_m h^2 ≈ 0.167 (CMB–BAO–SHOES overlap), adopting Gaussian priors on Ω_m h^2 and h, and fixing A_s and n_s to Planck best-fit ΛCDM values to infer S8 and Ω_m.

- Reducing r_s alone cannot simultaneously fit CMB acoustic peaks, BAO distances, and the high SHOES H0. The fundamental reason is the vastly different slopes of the r_a–H0 degeneracy lines for CMB and BAO, stemming from their very different redshift ranges. - If one moves along the CMB degeneracy line at Ω_m h^2 ≈ 0.143 to reach higher H0, the implied (r_a, H0) exits the BAO-allowed region, creating BAO tension. - Achieving consistency among CMB, BAO, and SHOES requires larger Ω_m h^2 (≈ 0.167). However, such higher matter density increases structure growth and S8, exacerbating tension with galaxy weak-lensing constraints (DES, KiDS). - Representative models: Model 2 (Ω_m h^2 ≈ 0.155) can fit CMB and BAO but not SHOES; Model 3 (Ω_m h^2 ≈ 0.167) can align CMB, BAO, and SHOES but worsens tension with DES/KiDS in S8–Ω_m. - Survey of published early-time solutions shows a consistent trend: models with low Ω_m h^2 either fail to reach H0 ≳ 73 km/s/Mpc or conflict with BAO; models with high Ω_m h^2 face WL tensions. - Even if a model perfectly fits all detailed CMB spectra while reducing r_s, it will still encounter BAO or WL inconsistencies unless additional new physics beyond reducing r_s is invoked. - The highest H0 attainable while remaining in reasonable agreement with BAO and DES/KiDS is around 70 km/s/Mpc.

The analysis directly addresses whether reducing the pre-recombination sound horizon r_s alone can reconcile the H0 discrepancy. Because BAO and CMB measure the same standard ruler at widely separated redshifts, their r_a–H0 degeneracy directions differ. Thus, lowering r_s to raise H0 along the CMB line inevitably clashes with BAO unless Ω_m h^2 is increased. But increasing Ω_m h^2 raises growth, leading to higher S8 that conflicts with weak lensing observations (DES, KiDS). This geometric and growth-based tension persists across a wide range of early-time models surveyed in the literature, indicating that simply shrinking r_s is insufficient. Consequently, a full resolution likely requires multiple concurrent modifications (e.g., extra radiation plus interactions affecting growth) or else identifying systematics in one or more datasets. The work clarifies in a model-independent way the slope-mismatch mechanism that blocks a simple r_s-only fix and contextualizes why many proposed early-time solutions face BAO/WL challenges.

Reducing the cosmic sound horizon alone cannot fully resolve the Hubble tension without violating other cosmological constraints. Due to the different r_a–H0 degeneracy slopes for CMB and BAO, moving to higher H0 by shrinking r_s conflicts with BAO unless Ω_m h^2 is raised, which in turn creates tension with weak-lensing constraints through increased S8. A comprehensive literature survey confirms this trend across many early-time models. With only a reduction in r_s, the maximum H0 compatible with BAO and DES/KiDS is roughly 70 km/s/Mpc. A full resolution likely requires multiple new-physics ingredients affecting both early and late-time observables, or uncovering systematic errors in existing datasets.

- The analysis assumes standard ΛCDM expansion history after recombination (no late-time new physics) and treats r_s (and r_a) as independent, which may not capture models with correlated post-recombination effects. - The fixed relation r_a = 1.0184 r_s (Planck ΛCDM value) is adopted; while largely stable across proposed models, small deviations are not explored. - Illustrative r_a–H0 lines are shown without individual uncertainty bands; full uncertainties are represented by the marginalized BAO and Planck contours. - Weak lensing comparisons for the representative models fix A_s and n_s to Planck best-fit values, which may not reflect all parameter degeneracies in specific extended models. - While MCMC includes multiple BAO observables (transverse, line-of-sight, isotropic), the simplified discussion focuses on the transverse case; however, the main conclusions are stated to be insensitive to this choice.