Effect of inhomogeneities on the expansion rate of the universe

While the expansion rate of a homogeneous isotropic universe is simply proportional to the square-root of the energy density, the expansion rate of an inhomogeneous universe also depends on the nature of the density inhomogeneities. In this paper we calculate to second order in perturbation variables the expansion rate of an inhomogeneous universe and demonstrate corrections to the evolution of the expansion rate. While we find that the mean correction is small, the variance of the correction on the scale of the Hubble radius is sensitive to the physical significance of the unknown spectrum of density perturbations beyond the Hubble radius.