Systematic corrections to the measured cosmological constant as a result of local inhomogeneity

Abstract

We explicitly calculate the lowest order systematic inhomogeneity-induced corrections 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-Newtonian 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 luminosity distance-redshift relation is then averaged over viewing angles and ensemble averaged, assuming that density fluctuations at a given cosmic time are a homogeneous random process. The resulting relation is fit to that of a homogeneous model containing dust and a cosmological constant, in order to deduce the best-fit cosmological constant density ${\ensuremath{\Omega}}_{\ensuremath{\Lambda}}$. We find that the luminosity distance-redshift relation is indeed modified, even for large sample sizes, but only by a very small fraction, of order ${10}^{\ensuremath{-}5}$ for $z\ensuremath{\sim}0.1$. This lowest order deviation depends on the peculiar velocities of the source and the observer. However, when fitting this perturbed relation to that of a homogeneous universe, via maximizing a likelihood function, 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 ${z}_{\mathrm{min}}=0.02$ out to a limiting redshift ${z}_{\mathrm{max}}=0.15$, we find that ${\ensuremath{\Omega}}_{\ensuremath{\Lambda}}\ensuremath{\approx}0.004$. The value of ${\ensuremath{\Omega}}_{\ensuremath{\Lambda}}$ has a large variance, and its magnitude tends to get progressively larger as the limiting redshift ${z}_{\mathrm{max}}$ gets smaller, implying that precision measurements of ${\ensuremath{\Omega}}_{\ensuremath{\Lambda}}$ from nearby supernova data will require taking this effect into account. This effect has been referred to in the past as the ``fitting problem,'' and more recently as subhorizon ``backreaction.'' We find that it is likely too small to explain the observed value ${\ensuremath{\Omega}}_{\ensuremath{\Lambda}}\ensuremath{\approx}0.7$. There have been previous claims of much larger backreaction effects. In contrast to those calculations, our work is directly related to how observers deduce cosmological parameters from astronomical data.