Cosmological peculiar velocities in general relativity

TL;DR

This paper clarifies that the covariant approach to cosmological peculiar velocities aligns with standard perturbation theory and does not predict anomalously strong growth or mimic accelerated expansion.

Contribution

It demonstrates that the covariant 1+3 approach, when applied consistently, reproduces standard results and corrects recent claims of stronger velocity growth.

Findings

The covariant approach matches standard perturbation results.

Inconsistent treatments led to false claims of stronger growth.

Standard relativistic treatment does not imply accelerated expansion.

Abstract

We reconsider the late-time evolution of galaxy peculiar velocities in the 1+3 covariant approach to cosmological perturbation theory. It has recently been claimed that this approach predicts substantially stronger growth of peculiar velocities than standard metric-based perturbation theory -- on the grounds that the covariant treatment is fully relativistic whereas standard treatments are effectively Newtonian. We show that this is not the case. When the covariant equations are applied consistently, the $1+3$ approach reproduces exactly the standard perturbative result for peculiar-velocity growth. The stronger growth laws claimed in recent work arise from an inconsistent treatment of the coupled covariant system, in which terms constrained by the field equations are treated as if they were independent sources. Further claims are made that the stronger bulk flows can mimic accelerated expansion in a dust universe. We argue that these claims rest on a confusion between the kinematics of an arbitrarily chosen observer congruence and the physical expansion of the matter congruence traced by galaxies. We conclude that the standard treatment of peculiar velocities is correct and fully relativistic~-- and does not lead to anomalous bulk flows or to apparent accelerated expansion.