Separate Universes beyond General Relativity

TL;DR

This paper develops geometric criteria to validate the separate universe ansatz, linking local volume expansion and curvature in long-wavelength perturbations to a local Friedmann-Robertson-Walker cosmology, with applications in testing modified gravity.

Contribution

It introduces metric-based conditions for the separate universe ansatz, enabling identification of local expansion and curvature in long-wavelength perturbations, applicable even in nonlinear modified gravity regimes.

Findings

Lapse perturbation must be much smaller than curvature potential.

Effective stress energy must comove with freely falling observers.

Matching expansion histories tests consistency in simulations.

Abstract

We establish purely geometric or metric-based criteria for the validity of the separate universe ansatz, under which the evolution of small-scale observables in a long-wavelength perturbation is indistinguishable from a separate Friedmann-Robertson-Walker cosmology in their angle average. In order to be able to identify the local volume expansion and curvature in a long-wavelength perturbation with those of the separate universe, we show that the lapse perturbation must be much smaller in amplitude than the curvature potential on a time slicing that comoves with the Einstein tensor. Interpreting the Einstein tensor as an effective stress energy tensor, the condition is that the effective stress energy comoves with freely falling synchronous observers who establish the local expansion, so that the local curvature is conserved. By matching the expansion history of these synchronous observers in cosmological simulations, one can establish and test consistency relations even in the nonlinear regime of modified gravity theories.