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.