Classical back reaction of low-frequency cosmic gravitational radiation

We study in a Brill-Hartle type of approximation the back reaction of a superposition of linear gravitational waves on its own mean gravitational field up to second order in the wave amplitudes. The background field is taken as a spatially flat Einstein–de Sitter geometry. In order to follow inflationary scenarios, the wavelengths are allowed to exceed the temporary Hubble distance. As in optical coherence theory, the wave amplitudes are considered as random variables, which form a homogeneous and isotropic stochastic process, sharing the symmetries of the background metric. A segregation of the field equations into equations for the wave amplitudes and equations for the background field is performed by averaging the field equations and interpreting the averaging process as a stochastic (ensemble) average. The spectral densities satisfy a system of ordinary differential equations. The effective stress-energy tensor for the random gravity waves is calculated in terms of correlation functions and covers subhorizon as well as superhorizon modes, where superhorizon modes give in many cases negative contributions to energy density and pressure. We discuss solutions of the second-order equations including pure gravitational radiation universes.