The Velocity Field Olympics: Assessing velocity field reconstructions with direct distance tracers

TL;DR

This work develops a Bayesian framework to validate local-Universe velocity reconstructions against direct distance tracers, reconciling density/velocity fields with Tully\u2013Fisher and Type Ia SN measurements. By jointly calibrating flow models and distance indicators and computing model evidences, the study systematically compares six reconstructions, finding the non-linear CSIBORG2 implementation to be the most consistent with data across multiple tracers. The analysis yields a robust $S_8$ constraint, $S_8 = 0.793\pm0.035$, compatible with weak lensing and Planck but in tension with some peculiar-velocity studies, and highlights $\sigma_v$ as a strong diagnostic of model fit. The approach provides a practical framework for selecting velocity-field models in cosmological analyses of SNs and gravitational waves, and it underscores the promise of non-linear, forward-modelled reconstructions for local-Scale cosmology.

Abstract

The peculiar velocity field of the local Universe provides direct insights into its matter distribution and the underlying theory of gravity, and is essential in cosmological analyses for modelling deviations from the Hubble flow. Numerous methods have been developed to reconstruct the density and velocity fields at $z \lesssim 0.05$, typically constrained by redshift-space galaxy positions or by direct distance tracers such as the Tully-Fisher relation, the fundamental plane, or Type Ia supernovae. We introduce a validation framework to evaluate the accuracy of these reconstructions against catalogues of direct distance tracers. Our framework assesses the goodness-of-fit of each reconstruction using Bayesian evidence, residual redshift discrepancies, velocity scaling, and the need for external bulk flows. Applying this framework to a suite of reconstructions -- including those derived from the Bayesian Origin Reconstruction from Galaxies (BORG) algorithm and from linear theory -- we find that the non-linear BORG reconstruction consistently outperforms others. We highlight the utility of such a comparative approach for supernova or gravitational wave cosmological studies, where selecting an optimal peculiar velocity model is essential. Additionally, we present calibrated bulk flow curves predicted by the reconstructions and perform a density--velocity cross-correlation using a linear theory reconstruction to constrain the growth factor, yielding $S_8 = 0.793 \pm 0.035$. The result is in good agreement with both weak lensing and Planck, but is in strong disagreement with some peculiar velocity studies.

Figures (16)

• Figure 1: Slices of the density field in the $\mathrm{SGX}$-$\mathrm{SGY}$ plane from $-155~h^{-1}\,\mathrm{Mpc}$ to $155~h^{-1}\,\mathrm{Mpc}$ in supergalactic coordinates at $z = 0$ for the local Universe reconstructions used in this work (\ref{['tab:reconstructions']}). Redder (bluer) colours correspond to overdensities (underdensities). The density fields are presented without additional smoothing. For the fields in the bottom row we resimulated the IC at higher resolution. The CSIBORG suites and Courtois2023 are averaged over $20$ posterior samples. The black cross marks the origin, indicating the approximate position of the Local Group.
• Figure 2: Examples of the reconstructed velocities along the LOS to the Virgo cluster, whose approximate distance is indicated by the dashed vertical line. For reconstructions providing multiple samples of the velocity field, a shaded region between 16th and 84th percentiles is shown. For Carrick_2015 we assume the fiducial value of $\beta^\star = 0.43$.
• Figure 3: The distribution of observed redshifts converted to the CMB frame ($z_{\rm obs}$) for the peculiar velocity samples used in this work. With the exception of the Foundation sample, the majority lie within $z_{\rm obs} \lesssim 0.05$, while the 2MTF sample is constrained to $z_{\rm obs} \lesssim 0.03$ due to its magnitude limit. For visual clarity, the samples are arbitrarily separated into three rows, sharing the $x$-scale but each with its own $y$-scale.
• Figure 4: Differences in logarithmic evidences $\mathcal{Z}$ from our flow validation model for various local Universe reconstructions (shown on the $x$-axis, see \ref{['tab:reconstructions']}), compared against peculiar velocity samples (individual panels, see \ref{['tab:pv_samples']}). Higher bars indicate a preferred model, and a bar of zero height indicates the reference (least successful) model. The logarithmic evidences are normalised with respect to the reference model as only relative differences are meaningful. Solid bars show evidences using the highest available resolution for each model, while hatched bars show evidences when all fields are smoothed to the resolution of $7.8~h^{-1}\,\mathrm{Mpc}$, twice that of Courtois2023. Overall, CSiBORG2 is the preferred model while the CosmicFlows-based reconstructions (Sorce_2018 and Courtois2023) are disfavoured. Upon smoothing, the reconstruction of Lilow2024 becomes marginally preferred.
• Figure 5: Differences in logarithmic evidence ($\Delta \log_{10} \mathcal{Z}$, with higher values indicating a better goodness-of-fit) and inferred $\sigma_v$ (scatter between observed and predicted redshifts) as functions of smoothing scale $R_{\rm smooth}$ for the joint SN and TFR dataset against Carrick_2015. The $\sigma_v$ axis is inverted. Both the goodness-of-fit, and $\sigma_v$ quickly degrade with increased smoothing, indicating that $\sigma_v$ is an effective goodness-of-fit indicator.