Effect of inhomogeneities on high precision measurements of cosmological distances

Abstract

We study effects of inhomogeneities on distance measures in an exact relativistic Swiss-cheese model of the universe, focusing on the distance modulus. The model has LCDM background dynamics, and the `holes' are non-symmetric structures described by the Szekeres metric. The Szekeres exact solution of Einstein's equations, which is inhomogeneous and anisotropic, allows us to capture potentially relevant effects on light propagation due to nontrivial evolution of structures in an exact framework. Light beams traversing a single Szekeres structure in different ways can experience either magnification or demagnification, depending on the particular path, and we explore a small additional effect due time evolution of the structures. We study the probability distributions of $\Delta\mu=\mu_{LCDM}-\mu_{SC}$ for sources at different redshifts in various Swiss-cheese constructions, where the light beams travel through a large number of randomly oriented holes with random impact parameters. We find for $\Delta\mu$ the dispersions $0.004\le \sigma_{\Delta\mu} \le 0.008$ mag for sources with redshifts $1.0\le z \le 1.5$, which are smaller than the intrinsic dispersion of, for example, magnitudes of type Ia supernovae. The shapes of the distributions we obtain for our Swiss-cheese constructions are peculiar in the sense that they are not consistently skewed toward the demagnification side, as they are in analyses of lensing in cosmological simulations, reflecting a limitation of these constructions. This could be the result of requiring the continuity of Einstein's equations throughout the overall spacetime patchwork, which imposes the condition that compensating overdense shells must accompany the underdense void regions in the holes. The possibility to explore other uses of these constructions that could circumvent this limitation and lead to different statistics remains open. (Abridged)