Expanding universe with nonlinear gravitational waves

TL;DR

This paper numerically tests Isaacson's formula for high frequency, low amplitude gravitational waves, finding it holds within 3% deviation even when wave length is comparable to the Hubble scale.

Contribution

The study provides a numerical solution of nonlinear standing gravitational waves in vacuum Einstein equations to validate Isaacson's formula in a cosmological setting.

Findings

Deviation from Isaacson's formula is at most 3% without averaging.

Deviation drops below 0.1% with temporal averaging.

The formula holds even when wave length is comparable to the Hubble scale.

Abstract

We test the validity of Isaacson's formula which states that high frequency and low amplitude gravitational waves behave as a radiation fluid on average. For this purpose, we numerically construct a solution of the vacuum Einstein equations which contains nonlinear standing gravitational waves. The solution is constructed in a cubic box with periodic boundary conditions. The time evolution is solved in a gauge in which the trace of the extrinsic curvature $K$ of the time slice becomes spatially uniform. Then, the Hubble expansion rate $H$ is defined by $H=-K/3$ and compared with the effective scale factor $L$ defined by the proper volume, area and length of the cubic box. We find that, even when the wave length of the gravitational waves is comparable to the Hubble scale, the deviation from Isaacson's formula $H\propto L^{-2}$ is at most 3\% without taking a temporal average and is below 0.1\% with a temporal average.