Light-cone averages in a Swiss-cheese universe

Abstract

We analyze a toy Swiss-cheese cosmological model to study the averaging problem. In our Swiss-cheese model, the cheese is a spatially flat, matter only, Friedmann-Robertson-Walker solution (i.e., the Einstein-de Sitter model), and the holes are constructed from a Lema\^{\i}tre-Tolman-Bondi solution of Einstein's equations. We study the propagation of photons in the Swiss-cheese model, and find a phenomenological homogeneous model to describe observables. Following a fitting procedure based on light-cone averages, we find that the expansion scalar is unaffected by the inhomogeneities (i.e., the phenomenological homogeneous model is the cheese model). This is because of the spherical symmetry of the model; it is unclear whether the expansion scalar will be affected by nonspherical voids. However, the light-cone average of the density as a function of redshift is affected by inhomogeneities. The effect arises because, as the universe evolves, a photon spends more and more time in the (large) voids than in the (thin) high-density structures. The phenomenological homogeneous model describing the light-cone average of the density is similar to the $\ensuremath{\Lambda}\mathrm{CDM}$ concordance model. It is interesting that, although the sole source in the Swiss-cheese model is matter, the phenomenological homogeneous model behaves as if it has a dark-energy component. Finally, we study how the equation of state of the phenomenological homogeneous model depends on the size of the inhomogeneities, and find that the equation-of-state parameters ${w}_{0}$ and ${w}_{a}$ follow a power-law dependence with a scaling exponent equal to unity. That is, the equation of state depends linearly on the distance the photon travels through voids. We conclude that, within our toy model, the holes must have a present size of about 250 Mpc to be able to mimic the concordance model.