Luminosity distance in Swiss cheese cosmology with randomized voids and
  galaxy halos
  We study the fluctuations in luminosity distance due to gravitational lensing
produced both by galaxy halos and large scale voids. Voids are represented via
a "Swiss cheese" model consisting of a \LambdaCDM Friedman-Robertson-Walker
background in which a number of randomly distributed, spherical regions of
comoving radius 35 Mpc are removed. A fraction of the removed mass is then
placed on the shells of the spheres, in the form of randomly located halos,
modeled with Navarro-Frenk-White profiles. The remaining mass is placed in the
interior of the spheres, either smoothly distributed, or as randomly located
halos. We compute the distribution of magnitude shifts using a variant of the
method of Holz & Wald (1998), which includes the effect of lensing shear. In
the two models we consider, the standard deviation of this distribution is
0.065 and 0.072 magnitudes and the mean is -0.0010 and -0.0013 magnitudes, for
voids of radius 35 Mpc, sources at redshift 1.5, with the voids chosen so that
90% of the mass is on the shell today. The standard deviation due to voids and
halos is a factor ~ 3 larger than that due to 35 Mpc voids alone with a 1 Mpc
shell thickness which we studied in our previous work. To a good approximation,
the variance of the distribution depends only on the mean column depth and
concentration of halos and on the fraction of the mass density that is in the
form of halos (as opposed to smoothly distributed): it is independent of how
the halos are distributed in space. We derive an approximate analytic formula
for the variance that agrees with our numerical results to \lesssim 20% out to
z\simeq 1.5.
