Volume expansion of a Swiss-cheese universe

In order to investigate the effect of inhomogeneities on the volume expansion of the universe, we study a modified Swiss-cheese universe model. Since this model is an exact solution of Einstein's equations, we can get insight into the nonlinear dynamics of an inhomogeneous universe from it. We find that inhomogeneities make the volume expansion slower than that of the background Einstein--de Sitter universe when they can be regarded as small fluctuations in the background universe. This result is consistent with the previous studies based on the second order perturbation analysis. On the other hand, if the inhomogeneities cannot be treated as small perturbations, the volume expansion of the universe depends on the type of fluctuations. Although the volume expansion rate approaches the background value asymptotically, the volume itself can be finally arbitrarily smaller than the background one and can be larger than that of the background, but there is an upper bound on it.