Average luminosity distance in inhomogeneous universes

Using numerical ray tracing, the paper studies how the average distance modulus in an inhomogeneous universe differs from its homogeneous counterpart. The averaging is over all directions from a fixed observer not over all possible observers (cosmic), thus is more directly applicable to our observations. In contrast to previous studies, the averaging is exact, non-perturbative, and includes all non-linear effects. The inhomogeneous universes are represented by Swiss-cheese models containing random and simple cubic lattices of mass-compensated voids. The Earth observer is in the homogeneous cheese which has an Einstein-de Sitter metric. For the first time, the averaging is widened to include the supernovas inside the voids by assuming the probability for supernova emission from any comoving volume is proportional to the rest mass in it. Voids aligned along a certain direction give rise to a distance modulus correction which increases with redshift and is caused by cumulative gravitational lensing. That correction is present even for small voids and depends on their density contrast, not on their radius. Averaging over all directions destroys the cumulative lensing correction even in a non-randomized simple cubic lattice of voids. At low redshifts, the average distance modulus correction does not vanish due to the peculiar velocities, despite the photon flux conservation argument. A formula for the maximal possible average correction as a function of redshift is derived and shown to be in excellent agreement with the numerical results. The formula applies to voids of any size that: (1) have approximately constant densities in their interior and walls; and (2) are not in a deep nonlinear regime. The average correction calculated in random and simple cubic void lattices is severely damped below the predicted maximal one after a single void diameter. That is traced to cancellations between the corrections from the fronts and backs of different voids. The results obtained allow one to readily predict the redshift above which the direction-averaged fluctuation in the Hubble diagram falls below a required precision and suggest a method to extract the background Hubble constant from low redshift data without the need to correct for peculiar velocities.