Cosmological peculiar velocities in general relativity?

TL;DR

This paper critically compares Newtonian and relativistic approaches to cosmological peculiar velocities, highlighting limitations of quasi-Newtonian methods and confirming the need for a relativistic treatment.

Contribution

It provides a detailed analysis contrasting Newtonian and relativistic frameworks, clarifies misconceptions, and identifies inconsistencies in recent literature.

Findings

Relativistic analyses confirm limitations of quasi-Newtonian methods.

Quasi-Newtonian approach yields Newtonian-like equations unsuitable for large scales.

The paper clarifies how relativistic treatment resolves issues in peculiar velocity studies.

Abstract

Cosmological peculiar velocities have traditionally been studied within the framework of Newtonian theory. Around the turn of the century, a few quasi-Newtonian analyses appeared in the literature, but led to equations and results identical to those of the purely Newtonian approach [1]. More recently, a series of studies introduced a relativistic treatment of the peculiar-velocity problem, criticising the quasi-Newtonian approach as effectively Newtonian in nature [2]. These works also reported a linear growth-rate of $v \propto t$ for peculiar velocities, in contrast to the slower Newtonian/quasi-Newtonian scaling of $v \propto t^{1/3}$. In a manuscript uploaded to the archives a few days ago [3], the authors defended their earlier quasi-Newtonian work and criticised the more recent relativistic treatments. However, the limitations of the quasi-Newtonian approach are not a new concern, but they have been noted at least since [4]. There, it was clearly stated that the quasi-Newtonian approximation leads to Newtonian-like equations and results, and readers were cautioned against applying it to large-scale cosmological studies. Given that one of the authors of [3] was also a coauthor of [4], the self-contradiction is evident. The relativistic analyses have, in fact, confirmed the concerns of [4], clarified the underlying issues and shown how they can be resolved. Motivated by [3], we present a critical comparison of the two approaches and in the process identify several internal inconsistencies in that manuscript.