Abstract
Recent development of the observational cosmology indicates the necessity of general relativistic treatment of our universe. In this paper we present approach to the construction of a realistic inhomogeneous universe starting from the first principle of general relativity. The approach does not assume that the density contrast is small and gives a method using a spatial averaging to calculate the effect that local inhomogeneities determine the global geometry. It clarifies the conditions under which the homogeneous and isotropic geometry appears in the global scale as the result of the averaging. Our approach allows one to calculate the back reaction by local inhomogeneities on the cosmic expansion and we express explicitly the deviation from Friedman-Lemaitre-Robertson-Walker (FLRW) geometry in terms of local inhomogeneities. We discuss also the condition for the applicability of the Newtonian approximation in the cosmological situation. As an application of the present formalism, the propagation of light rays in such an inhomogeneous universe is considered and the distance-redshift relation is derived which would play a fundamental role in the observational cosmology. On the basis of the derived relation we discuss the validity of the Dyer-Roeder distance.
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