Robust sound-horizon-free measurement of $H_0$ from galaxy clustering at future-survey precision: validity of the rescaling methodology, the role of equality-scale information, and the power to constrain early-Universe solutions of the Hubble tension

The DESI analysis of Zaborowski et al. (2025) measured the Hubble constant to sub-3% precision from the full-shape galaxy power spectra of the first DESI data release while deliberately removing all information from the sound horizon scale, instead anchoring the measurement primarily to the matter-radiation equality scale; the sound horizon was marginalized by a heuristic rescaling of the baryon acoustic oscillation wiggles by a free parameter $q_{\rm BAO}$. The authors themselves flag the central unresolved issues: the rescaling is a heuristic whose robustness at future-survey precision and across beyond-$\Lambda$CDM models is unquantified; the analyzed scales exclude the turnover of the power spectrum at $k_{\rm eq}\simeq0.015\,h\,{\rm Mpc}^{-1}$, so the balance between equality-scale and BAO-derived information is unknown; several modeling approximations (fixed neutrino mass, approximate Jeffreys prior, Taylor-series emulator) are only partially quantified relative to the achieved sub-3% precision; a mild $q_{\rm BAO}<1$ preference is undiagnosed; and the ability of such measurements to arbitrate early dark energy and other pre-recombination solutions of the Hubble tension is unestablished. The open problem is to turn this heuristic measurement into a validated, model-robust sound-horizon-free determination of $H_0$ at the precision of completed DESI and Euclid-class surveys, with a quantified account of what it can and cannot constrain about early-Universe physics.

Background

The Hubble tension is the roughly $5\sigma$ discrepancy between the local distance-ladder value of the Hubble constant, $H_0\approx73\,{\rm km\,s^{-1}\,Mpc^{-1}}$ measured by the SH0ES team (Supernovae $H_0$ for the Equation of State, a Hubble Space Telescope distance-ladder program), and the value $H_0\approx67\,{\rm km\,s^{-1}\,Mpc^{-1}}$ inferred from the cosmic microwave background (CMB) within the standard $\Lambda$CDM cosmological model. Most early-Universe determinations of $H_0$ rely on the sound horizon $r_{\rm d}$ — the comoving distance sound waves traveled in the photon-baryon plasma before recombination — as a ``standard ruler'' that calibrates the baryon acoustic oscillation (BAO) feature in the galaxy power spectrum and the CMB acoustic peak scale. Consequently, most proposed solutions of the tension (for example early dark energy, EDE, a period of extra energy density before recombination that shrinks $r_{\rm d}$) modify precisely this scale, which makes sound-horizon-based measurements self-adjusting and blunts their power to test such models.

This motivated a line of work measuring $H_0$ from the shape of the galaxy power spectrum with the sound horizon information removed. Philcox, Sherwin, Farren and Baxter (2021) showed that a full-shape analysis of the BOSS galaxy power spectrum — fitting complete model spectra, computed within the effective field theory of large-scale structure (EFTofLSS, a perturbative treatment of biased tracers valid on large scales), rather than compressed BAO parameters — with an uninformative prior on the baryon density yields an $H_0$ constraint independent of $r_{\rm d}$, driven instead by the matter-radiation equality scale $k_{\rm eq}$ (the wavenumber of the horizon at matter-radiation equality, which sets the turnover scale of the power spectrum and modulates its broadband shape). Farren, Philcox and Sherwin (2022) improved this with a heuristic rescaling procedure: the BAO wiggles are shifted by a free parameter $q_{\rm BAO}$ and marginalized over, so the absolute sound horizon is left uncalibrated while the informative baryon-density prior is retained; they also forecast performance for a Euclid-like survey and tested early-dark-energy cosmologies. Philcox et al. (2022) combined this with CMB lensing and uncalibrated type Ia supernovae (SNe Ia, whose relative distances constrain $\Omega_{\rm m}$ without an absolute calibration) to reach 3.6% precision, and Brieden, Gil-Marín and Verde (2023) compared this ``shape'' route to the traditional BAO route. Independently, Smith, Poulin and Simon (2023) warned that such $r_{\rm d}$-free constraints depend strongly on the assumed model and priors (e.g. CMB-derived priors on $n_s$ and $A_s$), so $\Lambda$CDM-based analyses cannot by themselves arbitrate new physics.

The source paper (DESI Collaboration: Zaborowski et al. 2025, arXiv:2411.16677) applied this machinery to the first DESI (Dark Energy Spectroscopic Instrument) data release: joint full-shape EFTofLSS analysis of luminous red galaxies, emission line galaxies, quasars and the bright galaxy sample in six redshift bins, for wavenumbers $k\in[0.02,0.20]\,h\,{\rm Mpc}^{-1}$, with the sound horizon marginalized via the $q_{\rm BAO}$ rescaling, combined with Planck+ACT CMB lensing, uncalibrated SNe Ia (DESY5, Pantheon+, Union3), and Alcock-Paczyński information from the DESI Lyman-$\alpha$ forest (forest-like absorption by intergalactic neutral hydrogen along quasar sightlines; the Alcock-Paczyński effect is the apparent distortion of clustering statistics when a wrong fiducial cosmology is used to convert redshifts to positions). The tightest sound-horizon-free results, $H_0=66.7^{+1.7}_{-1.9}$, $67.9^{+1.9}_{-2.1}$, and $67.8^{+2.0}_{-2.2}\,{\rm km\,s^{-1}\,Mpc^{-1}}$, remain in 2.2–3.0$\sigma$ tension with SH0ES while being consistent with sound-horizon-based early-time measurements, disfavoring only the more extreme (ACT-favored) EDE models. The paper is explicit that the $k$-range analyzed excludes the expected turnover scale $k_{\rm eq}\sim0.015\,h\,{\rm Mpc}^{-1}$, that the sum of neutrino masses is fixed to $0.06\,{\rm eV}$, that an approximate Jeffreys prior (a prior designed to be invariant under reparameterization, implemented here via profile maximization) is used for nuisance parameters, that a fourth-order Taylor-series emulator with subsequent importance sampling (reweighting Monte Carlo chains by the ratio of exact to emulated posteriors) accelerates the likelihood evaluation, that a mild preference for $q_{\rm BAO}<1$ (up to $2\sigma$) is observed but remains undiagnosed, and that ``uncalibrated, unnormalized'' BAO information is left for future work.

Problem Statement

The overarching objective is to convert the heuristic, sub-3% sound-horizon-free measurement of $H_0$ by Zaborowski et al. (2025) into a validated and model-robust determination of $H_0$ from large-scale-structure clustering at the precision of upcoming surveys (completed DESI, Euclid-class volumes), and to establish what such a measurement implies for early-Universe solutions of the Hubble tension. This requires, as jointly necessary components of one objective: (1) validating or replacing the heuristic $q_{\rm BAO}$ rescaling procedure — quantifying any residual sound-horizon information it leaves in the likelihood at future-survey precision, and its behavior across a wide class of beyond-$\Lambda$CDM models (including early dark energy, extra relativistic species, and modified recombination); (2) determining whether including very large scales ($k<0.02\,h\,{\rm Mpc}^{-1}$, containing the power-spectrum turnover at $k_{\rm eq}\simeq0.015\,h\,{\rm Mpc}^{-1}$) materially changes the balance of equality-scale versus BAO-derived information for sound-horizon-free $H_0$ constraints at that precision; (3) quantifying the residual impact on $H_0$ of the modeling approximations used for tractability — the fixed neutrino mass sum $\sum m_\nu=0.06\,{\rm eV}$, the approximate Jeffreys-prior treatment of nuisance parameters via profile maximization, the Taylor-series emulator with importance sampling, and the dependence on externally applied priors (e.g. $n_s$, $A_s$, $\sum m_\nu$) that may smuggle in $\Lambda$CDM assumptions — at the level relevant to the targeted future-survey precision; (4) establishing whether the mild preference for $q_{\rm BAO}<1$ observed in DESI data is a statistical fluctuation, a modeling residual, or evidence of beyond-$\Lambda$CDM physics; and (5) using the sound-horizon-marginalized pipeline to map systematically which early-Universe models of the Hubble tension (including less-extreme early dark energy) such a measurement can and cannot constrain. The problem is complete when a sound-horizon-free $H_0$ constraint at future-survey precision (attained either on released future data or through forecasts and simulations of completed-DESI/Euclid-class volumes, as in the Euclid-like forecasts of the cited lineage) is delivered jointly with the five components identified above: a validation or replacement of the heuristic $q_{\rm BAO}$ rescaling that quantifies any residual sound-horizon information it leaves at that precision and its behavior across beyond-$\Lambda$CDM models; a quantified account of the equality-scale versus BAO information balance, including whether the turnover scales at $k<0.02\,h\,{\rm Mpc}^{-1}$ materially change it; a quantification of the residual impact on $H_0$ of the modeling approximations used for tractability at that precision; a determination of whether the mild preference for $q_{\rm BAO}<1$ observed in DESI data is a statistical fluctuation, a modeling residual, or evidence of beyond-$\Lambda$CDM physics; and an explicit, evidence-backed statement of the constraining power over early-Universe model space. A partial delivery of this set does not resolve the problem.

References

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