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

Abstract

We study the fluctuations in luminosity distances due to gravitational lensing by large scale ( 35 Mpc) structures, specifically voids and sheets. We use a simplified "Swiss cheese" model consisting of a ΛCDM Friedman-Robertson-Walker background in which a number of randomly distributed non-overlapping 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 & Wald (1998), which includes the effect of lensing shear. The standard deviation of this distribution is ∼ 0.027 magnitudes and the mean is ∼ 0.003 magnitudes for voids of radius 35 Mpc, sources at redshift zs = 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 ∼ 1 Mpc, the standard deviation is reduced to ∼ 0.013 magnitudes. This standard deviation due to voids is a factor ∼ 3 smaller than that due to galaxy scale structures. We summarize our results in terms of a fitting formula that is accurate to ∼ 20%, and also build a simplified analytic model that reproduces our results to within ∼ 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 ∼ 4%, and corrections to due shear are ∼ 3%. Finally, we estimate the bias due to source-lens clustering in our model to be negligible.

Interpretation of the Hubble diagram in a nonhomogeneous universe

“…The optical properties of such models have been extensively studied (see Refs. [54][55][56][57][58][59][60][61][62]) to finally conclude that the average luminosityredshift relation remains unchanged with respect to the purely homogeneous case, contrary to the early results of Refs. [52,53].…”

“…It has been proven in Refs. [58,61,62] that the global effect averages to zero when many sources are considered. Hence, LTB holes introduce an additional dispersion to the Hubble diagram, but no statistically significant bias.…”

“…VI D, such a claim was proven to be inaccurate in Refs. [57,58,62], because it relies on observations along a peculiar line of sight. When many random directions of observation are taken into account, the mean magnification goes back to 1.…”

Redshift and distance in the inhomogeneous universe

“…While the null shear depends on the Weyl focusing, the image distortions γ can be estimated from the derivatives of the gravitational potential [65,13,44]. Both approaches lead to a similar conclusion: the large scale structure of the universe, i.e.…”

The theory of stochastic cosmological lensing

“…The most common examples are Swiss-cheese models [23,24], where inhomogeneities are introduced within a background FL spacetime by inserting spherical patches of another exact solution of Einstein's equation. Recent analyses generally exploit the Lemaître-Tolman-Bondi (LTB) [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] or Szekeres [40][41][42][43] geometries as interior solutions, which aim at describing large-scale inhomogeneities such as superclusters or cosmic voids (see also Refs. [44,45]).…”