Gravitational waves in a flat radiation-matter universe including anisotropic stress

We present novel analytical solutions for linear-order gravitational waves or tensor perturbations in a flat Friedmann-Robertson-Walker universe containing two perfect fluids, radiation and pressureless dust, and allowing for neutrino anisotropic stress. One of the results is applicable to any sub-horizon gravitational wave in such a universe. Another result is applicable to gravitational waves of primordial origin, for example produced during inflation, and works both before and after they cross the horizon. These results improve on analytical approximations previously set out in the literature. Comparison with numerical solutions shows that both these approximations are accurate to within $1\%$, or better, for a wide range of wave-numbers relevant for cosmology.