Reconstructing the Large-Scale Velocity Field from DESI DR1 with Bayesian Inference

Survey boundaries and nonlinear dynamics complicate reconstruction of large-scale matter-density and peculiar-velocity fields from galaxy redshift surveys. We develop and compare two reconstruction methods within a common Bayesian framework, based on linear perturbation theory and the Zel'dovich approximation, and apply them to the DESI Data Release 1 Bright Galaxy Survey. We infer the initial density field using a Gaussian prior and likelihood, with the Zel'dovich model displacing particles from a uniform Lagrangian grid. Both yield maximum a posteriori (MAP) fields; only the linear posterior is sampled. Our main findings are: \textit{(i)} Both reconstructions yield similar large-scale density structures and coherent velocity patterns in the survey interior, beyond roughly \(50\,h^{-1}\,\mathrm{Mpc}\) from the footprint boundaries. \textit{(ii)} Comparison with Cosmicflows-4 groups, predominantly from the SDSS Fundamental Plane sample in the DESI overlap, shows close agreement between mean observed radial velocities and MAP predictions when binned by the latter, despite substantial individual scatter. \textit{(iii)} Both yield similar reflex-dipole profiles, defined as mean shell velocities relative to those at randomly sampled centres, out to \(\sim225\,h^{-1}\,\mathrm{Mpc}\). The profiles depend weakly on the prior normalization \(σ_8\), but lie below the fiducial linear \(Λ\)CDM reference at \(r<50\,h^{-1}\,\mathrm{Mpc}\). Sparse sampling limits the largest radii. \textit{(iv)} In an exploratory BAO test, shifting galaxies and randoms to a target redshift \(z=8\) using reconstructed displacements sharpens the acoustic feature in both cases, with a slightly more pronounced peak for Zel'dovich. Survey geometry and galaxy bias remain the principal limitations.