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

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.

💡 Research Summary

This paper presents a comprehensive validation framework for assessing the fidelity of local‑universe peculiar velocity field reconstructions using direct distance tracers such as Tully‑Fisher, Fundamental Plane, and Type Ia supernovae. The authors argue that peculiar velocities are a crucial probe of the matter distribution, growth of structure, and gravity, and that accurate velocity models are indispensable for contemporary cosmological analyses (e.g., SH0ES H₀ determinations and standard‑sirens distance inference).

The framework combines four complementary statistical diagnostics: (i) Bayesian evidence, which quantifies the overall likelihood of a reconstruction given the distance‑tracer data; (ii) residual redshift (Δz) distributions, which test how well the model reproduces observed redshifts after accounting for the Hubble flow; (iii) a velocity‑scaling parameter β (or β⋆ for the Carrick et al. model), allowing the overall amplitude of the reconstructed field to float and thereby absorbing uncertainties in the growth rate fσ₈ and galaxy bias; and (iv) an external bulk‑flow term that captures large‑scale motions entering the surveyed volume from beyond its boundaries. By jointly fitting these quantities, the framework simultaneously corrects for homogeneous and inhomogeneous Malmquist biases, selection effects, and any systematic mismatches between model and data.

The authors apply the framework to five representative reconstructions:

1. Carrick et al. (2015, C15) – a linear‑theory reconstruction based on the 2M++ redshift catalogue, with a free β⋆ that rescales the velocity field.
2. Lilow et al. (2024, L24) – a machine‑learning approach that trains a U‑Net auto‑encoder on Quijote N‑body simulations and then predicts density and velocity from the 2MRS survey.
3. CSiBORG1 – constrained N‑body simulations built from BORG‑derived initial conditions (Jasche & Lavaux 2019) using Planck‑2014 cosmology and a 10‑step particle‑mesh integrator.
4. CSiBORG2 – an updated version of CSiBORG with newer BORG initial conditions (Stopyra et al. 2024), Planck‑2020 cosmology, and a 20‑step COLA integrator, yielding higher mass resolution and better cluster mass recovery.
5. Sorce (2018, S18) – constrained realisations that use only radial peculiar velocities from the CosmicFlows‑2 catalogue, applying Wiener filtering and reverse Zel’dovich reconstruction.

All models are evaluated against the same set of distance tracers (2M++ and CF2), with careful treatment of selection functions and Malmquist bias. The Bayesian evidence strongly favours the BORG‑based reconstructions (both CSiBORG1 and CSiBORG2), which achieve the highest likelihoods and require negligible external bulk‑flow corrections. The Lilow et al. model performs well on intermediate scales (∼100 h⁻¹ Mpc) but still needs a modest β adjustment and bulk‑flow term to match the data. The linear C15 reconstruction needs a substantial amplitude rescaling (β⋆≈0.43) and exhibits larger residuals, indicating that linear theory underestimates large‑scale flows. The Sorce constrained realisations, while internally consistent, lack the ability to capture flows sourced outside the CF2 volume, resulting in a significant bulk‑flow term.

Beyond model comparison, the authors exploit the linear C15 reconstruction to compute the cross‑correlation between the reconstructed density field and the observed peculiar velocities. This yields an estimate of the growth factor fσ₈, which they translate into the commonly used parameter S₈ = σ₈(Ω_m/0.3)^0.5, obtaining S₈ = 0.793 ± 0.035. This value is in excellent agreement with Planck CMB results and recent weak‑lensing surveys, but it is in tension with several earlier peculiar‑velocity analyses that reported lower S₈ values. The authors interpret this discrepancy as evidence that non‑linear, high‑resolution reconstructions (such as BORG) provide a more faithful representation of the true velocity field.

In the discussion, the implications for cosmological inference are highlighted. Accurate peculiar‑velocity models reduce systematic uncertainties in H₀ measurements from supernovae and in distance estimates from gravitational‑wave standard sirens, where the local velocity field can dominate the error budget for nearby events. The authors recommend adopting BORG‑based reconstructions as the default velocity model for such analyses, while using their validation framework to fine‑tune β and bulk‑flow parameters for any specific data set.

The paper concludes with four key take‑aways: (1) a robust, multi‑metric validation framework enables fair comparison of diverse velocity reconstructions; (2) non‑linear, constrained N‑body reconstructions (CSiBORG) outperform linear or machine‑learning approaches in matching direct distance data; (3) the derived S₈ value supports the concordance ΛCDM picture and alleviates some reported tensions in peculiar‑velocity literature; and (4) future work should incorporate larger, deeper distance‑tracer samples (e.g., upcoming Tully‑Fisher surveys, JWST supernovae) and higher‑resolution simulations to further tighten constraints on growth and gravity.