The magnitude-redshift relation in a realistic inhomogeneous universe
  The light rays from a source are subject to a local inhomogeneous geometry
generated by inhomogeneous matter distribution as well as the existence of
collapsed objects. In this paper we investigate the effect of inhomogeneities
and the existence of collapsed objects on the propagation of light rays and
evaluate changes in the magnitude-redshift relation from the standard
relationship found in a homogeneous FRW universe. We give the expression of the
correlation function and the variance for the perturbation of apparent
magnitude, and calculate it numerically by using the non-linear matter power
spectrum. We use the lognormal probability distribution function for the
density contrast and spherical collapse model to truncate the power spectrum in
order to estimate the blocking effect by collapsed objects. We find that the
uncertainties in $\Omega_m$ is $\sim 0.02$, and that of $w$ is $\sim 0.04$. We
also discuss a possible method to extract these effects from real data which
contains intrinsic ambiguities associated with the absolute magnitude.
