Peculiar velocity fields from analytic solutions of General Relativity

TL;DR

This work develops a non-perturbative relativistic framework to model peculiar velocity fields in cosmology by employing exact solutions of Einstein's equations sourced by irrotational, shear-free fluids with nonzero energy flux. Interpreting the energy flux as the nonrelativistic limit of a Lorentz boost between non-comoving fluids, the authors connect these solutions to cosmological perturbation theory while avoiding perturbative constraints. A conformally flat, nearly FLRW, spherically symmetric example demonstrates that peculiar velocities of order a few thousand km/s can arise relative to the CMB frame, with redshift relations consistent with observations and a controllable perturbative expansion around a closed FLRW background. The results suggest that richer, more general solutions within this class could model time- and space-varying 3D velocity fields and be tested against peculiar velocity surveys and CMB dipole measurements, paving the way for further exploration of realistic, non-perturbative cosmologies.

Abstract

Peculiar velocities are analyzed through cosmological perturbations in the Newtonian longitudinal gauge characterized by irrotational shear-free congruences in an Eulerian frame. We show that non-trivial peculiar velocity fields can be generated through Lorentzian boosts in the non-relativistic limit, where the Eulerian frame is obtained from analytic solutions of Einstein's equations sourced by an irrotational shear-free fluid with nonzero energy flux. This approach provides a physically viable interpretation of these analytic solutions, which (in general) admit no isometries, thus allowing, in principle, for modeling time and space varying 3-dimensional fields of peculiar velocities that can be contrasted with observational data on our local cosmography. As a ``proof of concept'' we examine the peculiar velocities of varying dark matter and dark energy perfect fluids with respect to the CMB frame using a simple, spherically symmetric particular solution. The resulting peculiar velocities are qualitatively compatible with observational data on the CMB dipole.

Figures (7)

• Figure 1: Peculiar velocities, $v_{\tiny{\hbox{pec}}}$, as a function of $(a,r)$. Notice the peculiar velocity vanishes at $a=0$ for all values of $r$ and also vanishes at the symmetry centers $r=0,\pi$. Refer to main text for discussion.
• Figure 2: Peculiar velocities, $v_{\tiny{\hbox{pec}}}$, from Scenario 1 (continuous curves) and Scenario 2 (dashed curves) as functions of redshift $z=1/a-1$ for various values of $r$, corresponding to dark matter (Scenario 1) and dark energy (scenario 2). Notice that $v_{\tiny{\hbox{pec}}}$ is slightly larger in Scenario 2.
• Figure 3: $\Delta^{(\rho)}$ and $|\Delta^{(p)}|$ as functions of $a$. Continuous curves depict $\epsilon_0=-0.005$, while dashed curves show $\epsilon_0=-0.00001$. See main text for details.
• Figure 4: Ratio $w=p^{(DE)}/\rho^{(DE)}$. Top: $w$ as a function of $a$. Bottom: $w$ as a function of $z$ for various values of $r$. Details are discussed in the text.
• Figure 5: Top: Ingoing null geodesic, highlighted curve shows the observer at $r=0$. and $a=1$, Bottom: Redshift $z$ and $r$ as functions of the comoving distance $D$. Refer to main text for further discussion.