The position of the baryon acoustic oscillation (BAO) standard ruler, as measured in the Lyman-alpha forest auto-correlation and its cross-correlation with quasars at redshift z≈2z\approx2–44, is shifted by nonlinear gravitational evolution and gas physics at the sub-percent level. Three dedicated studies — field-level approximate simulations, a one-loop effective-field-theory fit to hydrodynamical simulations, and FGPA mocks painted on NN-body particles — disagree on both the magnitude and the sign of this shift, and existing hydrodynamical simulations lack the volume to measure it directly at the required precision. The open problem, posed as the foremost unresolved issue in the DESI DR2 Lyman-alpha BAO analysis, is to determine the sign and magnitude of the nonlinear Lyman-alpha BAO shift (radial, transverse, auto- and cross-correlation) with quantified robustness to intergalactic-medium modeling choices such as helium reionization and ultraviolet-background fluctuations, thereby enabling the conversion of DESI's conservative systematic allowance into a validated correction or a reduced uncertainty.

Background

Baryon acoustic oscillations (BAO) are the frozen acoustic waves of the primordial photon-baryon plasma: they imprint a preferred comoving separation of roughly 150 Mpc (the sound horizon at drag epoch, rdr_d) on the matter distribution, making the BAO peak a standard ruler for measuring cosmic distances. The Dark Energy Spectroscopic Instrument (DESI) measures this ruler in the Lyman-alpha forest — the suite of hydrogen absorption lines blueward of the Lyman-alpha emission line in quasar spectra, produced by neutral hydrogen in the intergalactic medium (IGM) along the line of sight — which traces structure at redshifts z≈2z\approx2–44, the highest-redshift rung of the BAO distance ladder. DESI measures the BAO scale in the forest auto-correlation and in its cross-correlation with quasars, reporting an isotropic dilation parameter α\alpha and, anisotropically, the transverse and line-of-sight ratios α⊥=DM/rd\alpha_\perp=D_M/r_d and α∥=DH/rd\alpha_\parallel=D_H/r_d (the comoving angular-diameter distance and Hubble distance in units of the sound horizon). The ratio of these two scales, read from the anisotropic distortion of the intrinsically isotropic BAO feature, is the Alcock-Paczyński measurement — a purely geometric constraint on the expansion history between the survey redshift and today that does not depend on the absolute ruler size. Nonlinear structure formation shifts the BAO peak position from its linear-theory location — an effect that is well understood at the roughly 0.5% level for matter and low-redshift galaxy clustering, but which for the Lyman-alpha forest depends additionally on the complex thermal and ionization physics of the IGM and on how the forest traces density. The forest is a biased tracer, described in effective-field-theory (EFT) language by a set of bias coefficients multiplying operators built from the density and velocity fields; the fluctuating Gunn-Peterson approximation (FGPA) — the assumption that optical depth traces gas density with a power-law temperature-density relation — underlies several fast simulation approaches. Three studies quantified the nonlinear Lyman-alpha BAO shift before DESI DR2: (i) Sinigaglia et al. used field-level simulations based on the FGPA applied to Lagrangian-perturbation-theory density fields in 1 Gpc/h1\,\mathrm{Gpc}/h boxes and found a negative isotropic shift of about 1%; (ii) de Belsunce et al. fitted a one-loop EFT of the Lyman-alpha forest to the ACCEL2^2 hydrodynamical simulations and predicted shifts of −0.2%-0.2\% radially and −0.3%-0.3\% transversally in the auto-correlation; (iii) Hadzhiyska et al. painted the FGPA on the AbacusSummit NN-body suite with linear control variates (a variance-reduction technique that subtracts a correlated, low-noise estimator) and measured a positive radial shift of +0.35%+0.35\% at 3σ3\sigma in the auto-correlation — where the transverse shift is negligible — together with a negative transverse shift of about −0.25%-0.25\% at 2σ2\sigma in the cross-correlation; this is a radial sign opposite to the EFT result, with each study detecting its shift only at roughly 3σ3\sigma. Because these results disagree in both magnitude and sign, the DESI DR2 Lyman-alpha BAO analysis (DESI Collaboration, Phys. Rev. D 112, 083514 (2025)) did not correct its data; instead it added a conservative systematic allowance (about 0.3% in the isotropic parameter) in quadrature to the statistical error. The same paper notes that the EFT study did not quantify how simulation modeling choices — the treatment of helium reionization, fluctuations in the spatially varying ultraviolet ionizing background, and other small-scale astrophysics — affect the fitted bias parameters and hence the inferred shift, and that available hydrodynamical simulations lack the volume to measure the shift directly at comparable precision. A related but distinguishable systematic — the interaction between quasar redshift errors and forest continuum estimation — was partially mitigated with mock studies and close-pair removal but is acknowledged as needing further work with more realistic mocks.

Problem Statement

Determine the nonlinear shift of the BAO peak position induced by structure formation and Lyman-alpha tracer bias at the few-tenths-of-a-percent precision level relevant to DESI-era surveys, separately for the Lyman-alpha forest auto-correlation and the quasar-Lyman-alpha cross-correlation, and separately in the radial (line-of-sight) and transverse directions — resolving the current disagreement between approximate-simulation, EFT-based, and NN-body-painted predictions, which extends to the sign of the radial shift. The determination must come either from direct measurement in hydrodynamical (or hydrodynamically calibrated) simulations of sufficient volume, or from an EFT-based theoretical prediction validated on such simulations, and must quantify how intergalactic-medium and astrophysical modeling choices — the treatment of helium reionization, fluctuations in the ultraviolet ionizing background, and other small-scale astrophysics — propagate through the Lyman-alpha bias parameters into the inferred shift, so that the result is demonstrably robust rather than pipeline-specific. If the shift is thereby confirmed at a level that matters for the distance measurements, apply the validated prediction to real Lyman-alpha data (e.g., the DESI DR2 or later samples) to convert the current conservative systematic allowance into a precise correction or a demonstrably reduced uncertainty; this data-level application, explicitly deferred as future work by the source paper, is an optional extension rather than a required deliverable — the required deliverable is complete once the shift has been determined, its robustness to intergalactic-medium modeling choices quantified, and the origin of the disagreement among the approximate-simulation, EFT-based, and NN-body-painted predictions (including the sign of the radial shift) identified.

References

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