The reason peculiar velocities grow faster in general relativity than in Newtonian gravity

TL;DR

The paper addresses the discrepancy between Newtonian predictions and observed large-scale bulk flows by performing a covariant comparison of Newtonian, quasi-Newtonian, and fully relativistic treatments of linear peculiar velocities. It shows that in general relativity the peculiar flux contributes to gravity and that the 4-acceleration acts as a driving force, leading to a minimal relativistic growth of $v ∝ t$ (and up to $v ∝ t^{4/3}$ when structure formation is included), faster than the Newtonian $v ∝ t^{1/3}$. An effective Newtonian mapping with a divergence-free term in Poisson’s equation demonstrates how some relativistic results can be recovered in a Newtonian framework, though this mapping is not unique and serves as an illustrative bridge. The findings imply that relativistic effects could naturally accommodate faster and deeper bulk flows within the standard $\Lambda$CDM model and provide a gauge-invariant, covariant basis to test gravity in the post-recombination universe, guiding both analytic and numerical studies of bulk flows.

Abstract

An increasing number of surveys has been reporting large-scale peculiar motions with sizes and speeds in excess of those allowed by the concordance cosmological model. These are the so called bulk flows, the presence of which has come to be treated as a problem for the $Λ$CDM paradigm. However, the limits of the $Λ$CDM model are based on Newtonian studies, which predict the mediocre $v\propto t^{1/3}$ growth-rate for the peculiar-velocity field ($v$). Recently, a few fully relativistic treatments have appeared in the literature, arguing for a much stronger velocity growth that could explain the reported fast and deep bulk flows. What separates the Newtonian from the relativistic studies is the gravitational input of the peculiar flux, namely of the kinetic energy triggered by the moving matter. The latter has no direct gravitational contribution in Newtonian theory, but it does so in general relativity. This drastically changes the driving agent of the peculiar-velocity field and boosts its linear growth. The aim of this work is to directly compare the two treatments, as well as identify and discuss the reasons for their different results. In the process, we also demonstrate how one could recover the relativistic growth-rate from a Newtonian setup by selectively including certain (typically ignored) source-free terms into the Poisson equation. This way, we provide a unified covariant comparison of the Newtonian, the quasi-Newtonian and the fully relativistic studies.