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 $\Lambda \mathrm{CDM}$ Friedmann-Robertson-Walker background from 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. The halos are assumed to be nonevolving and are modeled with Navarro-Frenk-White profiles of a fixed mass. 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 and Wald [Phys. Rev. D 58, 063501 (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 and the 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 $\sim 3$ larger than that due to 35 Mpc voids alone with a 1 Mpc shell thickness, which we studied in our previous work. We also study the effect of the existence of evacuated voids, by comparing to a model where all the halos are randomly distributed in the interior of the sphere with none on its surface. This does not significantly change the variance but does significantly change the demagnification tail. To a good approximation, the variance of the distribution depends only on the mean column density of halos (halo mass divided by its projected area), the concentration parameter of the halos, and 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 $z\simeq 1.5$, and that can be used to study the dependence on halo parameters.

• Received 16 July 2012

DOI:https://doi.org/10.1103/PhysRevD.88.043004