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Cosmological peculiar velocities in general relativity?

Christos G. Tsagas

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.

Cosmological peculiar velocities in general relativity?

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 for peculiar velocities, in contrast to the slower Newtonian/quasi-Newtonian scaling of . 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.

Paper Structure

This paper contains 10 sections, 17 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Relativistic vs quasi-Newtonian study
  3. The initial setup
  4. The quasi-Newtonian approach
  5. The relativistic approach
  6. Reviewing the criticisms
  7. Bulk flows and the deceleration parameter
  8. Unsuspecting vs informed bulk-flow observers
  9. Reviewing the criticisms
  10. Conclusions and discussion

Figures (1)

  • Figure 1: (a) Observers ($O_1$, $O_2$) inside a bulk-flow ($D$), moving with 4-velocity $\tilde{u}_a$ and having mean bulk velocity $v_a$ relative to their CMB counterparts (with 4-velocity $u_a$). (b) The profile of the deceleration parameter measured in the bulk-flow frame, "contaminated" by the bulk-flows local contraction (see Eq. (\ref{['tq3']})). The Einstein-de Sitter limit ($\tilde{q}\rightarrow q=0.5$) is recovered at sufficiently high redshifts, where $\lambda\gg\lambda_T$. Inside the transition length ($\lambda_T$), $\tilde{q}$ appears to turn negative and even to cross the phantom divide ($\tilde{q}=-1$) at $\lambda=\lambda_T/\sqrt{3}$.