Luminosity distance in Swiss cheese cosmology with randomized voids. II. Magnification probability distributions

We study the fluctuations in luminosity distances due to gravitational lensing by large scale (greater than or similar to 35 Mpc) structures, specifically voids and sheets. We use a simplified Swiss cheese model consisting of a Lambda CDM Friedman-Robertson-Walker background in which a number of randomly distributed nonoverlapping spherical regions are replaced by mass-compensating comoving voids, each with a uniform density interior and a thin shell of matter on the surface. We compute the distribution of magnitude shifts using a variant of the method of Holz and Wald [2], which includes the effect of lensing shear. The standard deviation of this distribution is similar to 0.027 magnitudes and the mean is similar to 0.003 magnitudes for voids of radius 35 Mpc, sources at redshift z(s) = 1.0, with the voids chosen so that 90% of the mass is on the shell today. The standard deviation varies from 0.005 to 0.06 magnitudes as we vary the void size, source redshift, and fraction of mass on the shells today. If the shell walls are given a finite thickness of similar to 1 Mpc, the standard deviation is reduced to similar to 0.013 magnitudes. This standard deviation due to voids is a factor similar to 3 smaller than that due to galaxy scale structures. We summarize our results in terms of a fitting formula that is accurate to similar to 20%, and also build a simplified analytic model that reproduces our results to within similar to 30%. Our model also allows us to explore the domain of validity of weak-lensing theory for voids. We find that for 35 Mpc voids, corrections to the dispersion due to lens-lens coupling are of order similar to 4%, and corrections due to shear are similar to 3%. Finally, we estimate the bias due to source-lens clustering in our model to be negligible.