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 it is more directly applicable to our
observations. Unlike previous studies, the averaging is exact,
non-perturbative, and includes all possible non-linear effects. The
inhomogeneous universes are represented by Sweese-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. Despite the well known argument for photon
flux conservation, the average distance modulus correction at low redshifts is
not zero due to the peculiar velocities. A formula for the maximum possible
average correction as a function of redshift is derived and shown to be in
excellent agreement with the numerical results. The actual average correction
calculated in random and simple cubic void lattices is severely damped below
the predicted maximal average. That is traced to cancelations between the
corrections coming from the fronts and backs of different voids at the same
redshift from the observer. The calculated correction at low redshifts allows
one to readily predict the redshift at which the averaged fluctuation in the
Hubble diagram is below a required precision and suggests a method to extract
the background Hubble constant from low redshift data without the need to
correct for peculiar velocities.
