Recently there have been suggestions that the Type Ia supernova data can be explained using only general relativity and cold dark matter with no dark energy. In Swiss cheese models of the Universe, the standard Friedmann-Robertson-Walker picture is m
odified by the introduction of mass compensating spherical inhomogeneities, typically described by the Lemaitre-Tolman-Bondi metric. If these inhomogeneities correspond to underdense cores surrounded by mass-compensating overdense shells, then they can modify the luminosity distance-redshift relation in a way that can mimic accelerated expansion. It has been argued that this effect could be large enough to explain the supernova data without introducing dark energy or modified gravity. We show that the large apparent acceleration seen in some models can be explained in terms of standard weak field gravitational lensing together with insufficient randomization of void locations. The underdense regions focus the light less than the homogeneous background, thus dimming supernovae in a way that can mimic the effects of acceleration. With insufficient randomization of the spatial location of the voids and of the lines of sight, coherent defocusing can lead to anomalously large demagnification effects. We show that a proper randomization of the voids and lines of sight reduces the effect to the point that it can no longer explain the supernova data.
The fitting of the observed redshifts and magnitudes of type Ia supernovae to what we would see in homogeneous cosmological models has led to constraints on cosmological parameters. However, in doing such fits it is assumed that the sampled supernova
e are moving with the Hubble flow, i.e. that their peculiar velocities are zero. In reality, peculiar velocities will modify supernova data in a way that can impact best-fit cosmological parameters. We theoretically quantify this effect in the nonlinear regime with a Monte-Carlo analysis, using data from semi-analytic galaxy catalogs that are built from the Millennium N-body simulation. We find scaling relations for the errors in best-fit parameters resulting solely from peculiar velocities, as a function of the total number of sources in a supernova survey N and its maximum redshift z_max. For low redshift surveys, we find that these errors can be of the same order of magnitude as the errors due to an intrinsic magnitude scatter of 0.1 mag. For a survey with N=2000 and z_max=1.7, we estimate that the expected peculiar velocity-induced errors in the best-fit cosmological constant density and equation of state can be sigma_Lambda~0.009 and sigma_w~0.01, respectively, which are subdominant to the errors due to the intrinsic scatter. We further find that throwing away supernova data below a redshift z~0.01-0.02 can reduce the combined error, due to peculiar velocities and the intrinsic scatter, but by only about 10%.
We calculate the systematic inhomogeneity-induced correction to the cosmological constant that one would infer from an analysis of the luminosities and redshifts of Type Ia supernovae, assuming a homogeneous universe. The calculation entails a post-N
ewtonian expansion within the framework of second order perturbation theory, wherein we consider the effects of subhorizon density perturbations in a flat, dust dominated universe. Within this formalism, we calculate luminosity distances and redshifts along the past light cone of an observer. The resulting luminosity distance-redshift relation is fit to that of a homogeneous model in order to deduce the best-fit cosmological constant density Omega_Lambda. We find that the luminosity distance-redshift relation is indeed modified, by a small fraction of order 10^{-5}. When fitting this perturbed relation to that of a homogeneous universe, we find that the inferred cosmological constant can be surprisingly large, depending on the range of redshifts sampled. For a sample of supernovae extending from z=0.02 out to z=0.15, we find that Omega_Lambda=0.004. The value of Omega_Lambda has a large variance, and its magnitude tends to get larger for smaller redshifts, implying that precision measurements from nearby supernova data will require taking this effect into account. However, we find that this effect is likely too small to explain the observed value of Omega_Lambda=0.7. There have been previous claims of much larger backreaction effects. By contrast to those calculations, our work is directly related to how observers deduce cosmological parameters from astronomical data.
