Large-Scale Tides in General Relativity

TL;DR

This paper develops a relativistic framework for understanding large-scale tidal effects in cosmology, extending the 'separate universe' concept to anisotropic perturbations and providing practical equations for second-order structure growth.

Contribution

It introduces a closed set of relativistic equations for nonlinear anisotropic perturbations using the conformal Fermi frame, extending the 'separate universe' paradigm beyond isotropic cases.

Findings

Tidal effects are encoded by the Weyl tensor, distinct from Bianchi I anisotropy.

The equations match second-order density evolution but not third order.

The framework simplifies relativistic structure growth calculations at second order.

Abstract

Density perturbations in cosmology, i.e. spherically symmetric adiabatic perturbations of a Friedmann-Lema\^itre-Robertson-Walker (FLRW) spacetime, are locally exactly equivalent to a different FLRW solution, as long as their wavelength is much larger than the sound horizon of all fluid components. This fact is known as the "separate universe" paradigm. However, no such relation is known for anisotropic adiabatic perturbations, which correspond to an FLRW spacetime with large-scale tidal fields. Here, we provide a closed, fully relativistic set of evolutionary equations for the nonlinear evolution of such modes, based on the conformal Fermi (CFC) frame. We show explicitly that the tidal effects are encoded by the Weyl tensor, and are hence entirely different from an anisotropic Bianchi I spacetime, where the anisotropy is sourced by the Ricci tensor. In order to close the system, certain higher derivative terms have to be dropped. We show that this approximation is equivalent to the local tidal approximation of Hui and Bertschinger (1995). We also show that this very simple set of equations matches the exact evolution of the density field at second order, but fails at third and higher order. This provides a useful, easy-to-use framework for computing the fully relativistic growth of structure at second order.