Template-based analyses of baryon acoustic oscillations (BAO) infer cosmology from the compressed scaling parameters α∥\alpha_{\parallel} and α⊥\alpha_{\perp}, which are interpreted using the sound-horizon scale rdr_d computed from the standard analytic integral rdintr_d^{\rm int} over the early-universe sound speed. Asensio-Rivera, Schöneberg, Gil-Marín and Verde (JCAP 07(2026)046) showed that rdintr_d^{\rm int} differs from the effective sound-horizon scale rdobsr_d^{\rm obs} actually imprinted in and extracted from the clustered matter field, inducing a systematic bias in cosmological inference that becomes a non-negligible fraction of the statistical uncertainty (reaching the conventional 1/51/5-of-statistical-error significance threshold) for parameter excursions of order ∣ΔΩm∣≃0.03|\Delta\Omega_m| \simeq 0.03–0.050.05 and ∣ΔNeff∣≃0.3|\Delta N_{\rm eff}| \simeq 0.3–0.40.4 at DESI Year-5 (Stage-IV) precision, and proposed corrections (per-point bias computation, posterior importance sampling, and a second-order Taylor expansion of the bias). The outstanding problem is the definitive catalogue-level validation of this mismatch and its corrections under full analysis realism — BAO density-field reconstruction, survey mask and geometry, full covariance, and catalogue-level systematics — together with quantification of residual higher-order terms, coverage across tracers, survey specifications and multi-probe likelihoods, the interplay with de-wiggling and broadband parametrizations, and integration into MCMC likelihood pipelines. No published follow-up addresses this validation core; the problem remains open.

Background

Baryon acoustic oscillations (BAO) are oscillations in the density of the primordial photon–baryon plasma, imprinted before recombination as a preferred comoving length scale in the distribution of matter and galaxies. That length — the sound horizon at the baryon-drag epoch, rdr_d — is computed in cosmological theory as an integral over the sound speed and expansion rate of the early universe, rdint=∫zd∞cs(z)/H(z) dzr_d^{\rm int} = \int_{z_d}^{\infty} c_s(z)/H(z)\,dz, and acts as a standard ruler: comparing the observed BAO feature in galaxy clustering to rdr_d measures cosmological distances. Modern surveys such as DESI (the Dark Energy Spectroscopic Instrument) and Euclid do not fit full cosmological models to the clustering data directly. Instead, a template-based analysis compresses the data into the scaling (or dilation) parameters α∥\alpha_{\parallel} and α⊥\alpha_{\perp} — ratios of the inferred radial and transverse BAO scales to their values in an assumed fiducial cosmology — which are then interpreted in terms of cosmological parameters in a separate interpretation step. That interpretation conventionally uses the ratio of integral sound horizons, rdint/rdint,fidr_d^{\rm int}/r_d^{{\rm int, fid}}, in place of the ratio of the scales actually extracted from the data, rdobs/rdobs,fidr_d^{\rm obs}/r_d^{{\rm obs, fid}}. Because the BAO oscillation frequency in the matter power spectrum drifts with wavenumber (the wiggles are not exactly sin⁡(krd)\sin(k r_d)), different extraction estimators — peak positions, BAO-only fits, full power-spectrum or correlation-function fits — return slightly different effective sound horizons, so rdint≢rdobsr_d^{\rm int} \not\equiv r_d^{\rm obs}: the standard ruler is not perfectly standard. This estimator–integral mismatch induces a systematic bias, Δα/α≃Δrd/rd\Delta\alpha/\alpha \simeq \Delta r_d/r_d, in the recovered BAO scaling parameters and hence in downstream cosmological parameters. Thepsuriya and Lewis (JCAP 01(2015)034) first quantified the effect analytically (~0.15% for standard parameter variations); Carter, Beutler, Percival, DeRose, Wechsler and Zhao (MNRAS 494(2):2076–2089, 2020) studied it with N-body-simulation-based pipelines including BAO reconstruction (the procedure that partially undoes nonlinear gravitational smearing of the BAO feature) and estimated effects up to ~0.4–0.5%; Sanz-Wuhl, Gil-Marín, Cuesta and Verde (JCAP 05(2024)116) introduced a template-sound-horizon correction rdTr_d^{T} in BAO analyses of non-flat cosmologies. Asensio-Rivera, Schöneberg, Gil-Marín and Verde (JCAP 07(2026)046) extended the analysis to a broad class of cosmologies, including models that modify pre-recombination physics (parametrized for example by NeffN_{\rm eff}, the effective number of relativistic species), quantified the bias for DESI-like surveys, and showed it becomes a significant fraction of the statistical uncertainty for excursions of order ∣ΔΩm∣≃0.03|\Delta\Omega_m| \simeq 0.03–0.050.05 and ∣ΔNeff∣≃0.3|\Delta N_{\rm eff}| \simeq 0.3–0.40.4 (using the survey-community convention that a systematic is significant when it reaches 1/51/5 of the statistical error) — relevant for DESI Year-5 and other Stage-IV dark-energy surveys such as Euclid, though not for DESI Year-1 precision. They also demonstrated that nonlinear modeling and broadband marginalization in the style of the effective field theory (EFT) of large-scale structure — marginalizing over nuisance parameters describing the smooth, non-oscillatory ('broadband') part of the power spectrum, separated from the BAO wiggles by a de-wiggling algorithm — do not absorb the mismatch, and proposed three corrections: computing the bias at each point of an MCMC (Markov Chain Monte Carlo) parameter exploration, importance-sampling the posterior with the per-point correction, and applying a second-order Taylor expansion of the bias in {Ωb+cdmh2,Neff,Ωbh2}\{\Omega_{b+{\rm cdm}}h^2, N_{\rm eff}, \Omega_b h^2\}, which reduce the systematic by at least ~90% (typically ~96%) within the explored parameter ranges. However, their analysis omits BAO reconstruction, uses a simplified Gaussian covariance for a cubic survey volume rather than the real survey mask and covariance matrix, and excludes catalogue-level observational systematics, leaving the catalogue-level status of the mismatch and of these corrections unestablished.

Problem Statement

Establish, by definitive catalogue-level validation, whether the mismatch between the sound horizon computed from the standard analytic integral (rdintr_d^{\rm int}) and the effective sound horizon extracted from the BAO feature in clustered matter (rdobsr_d^{\rm obs}) biases Stage-IV (DESI Year-5, Euclid) cosmological inference under full analysis realism, and validate corrections that render it harmless. Concretely, using mock (simulated) galaxy catalogues at non-fiducial cosmologies processed through a realistic BAO analysis pipeline that includes density-field reconstruction, the survey mask and geometry, the full covariance matrix, and catalogue-level observational systematics: (1) quantify the mismatch-induced bias in the BAO scaling parameters α∥\alpha_{\parallel}, α⊥\alpha_{\perp} (and in downstream cosmological parameters such as Ωm\Omega_m and NeffN_{\rm eff}) across the relevant parameter excursions — including compensated models (excursions in which relatively large deviations in several parameters combine to keep α∥\alpha_{\parallel} and α⊥\alpha_{\perp} approximately constant over a redshift range, so that BAO data alone cannot exclude them) and the regimes ∣ΔΩm∣∼0.03|\Delta\Omega_m| \sim 0.03–0.050.05 and ∣ΔNeff∣∼0.3|\Delta N_{\rm eff}| \sim 0.3–0.40.4 where the source analysis found the bias reaching the conventional 1/51/5-of-statistical-error significance threshold — and determine whether reconstruction and catalogue effects qualitatively change the estimator–integral mismatch found in the simplified analysis; (2) validate, in this realistic setting, the proposed corrections (per-MCMC-point bias computation, posterior importance sampling, and the second-order Taylor expansion of the bias in {Ωb+cdmh2,Neff,Ωbh2}\{\Omega_{b+{\rm cdm}}h^2, N_{\rm eff}, \Omega_b h^2\}), quantifying residual higher-order terms for larger multi-parameter excursions and their numerical fidelity across survey specifications, tracer samples (the galaxy or quasar populations used to trace the underlying matter density), and combined multi-probe likelihoods — the survey specifications, tracers, likelihood combinations, de-wiggling algorithms and broadband parametrizations delimiting the domain in which the validated correction must hold — and mapping the interplay with different de-wiggling algorithms and broadband parametrizations and their priors; and (3) demonstrate practical integration of the validated correction into an MCMC/likelihood pipeline at DESI/Euclid-scale computational cost, so that the mismatch can be corrected — or bounded in an explicit systematic error budget — in Stage-IV BAO cosmological inference. These three components jointly serve one objective, namely making the sound-horizon interpretation step of BAO analysis safe at Stage-IV precision: (1) establishes the size and character of the systematic under realistic conditions, (2) establishes that the corrections remove it to within quantified residuals, and (3) establishes that the correction is usable in the analyses where it is needed.

References

  1. Asensio-Rivera, Schöneberg, Gil-Marín, Verde, 'The BAO scale – how standard is the standard ruler?', JCAP 07(2026)046, DOI 10.1088/1475-7516/2026/07/046, arXiv:2603.03443, https://arxiv.org/abs/2603.03443
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