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

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 ${\Omega }_{\Lambda }$. 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 ${10}^{-5}$ for $z\sim 0.1$. 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 $z_{\min }=0.02$ out to a limiting redshift $z_{\max }=0.15$, we find that ${\Omega }_{\Lambda }\approx 0.004$. The value of ${\Omega }_{\Lambda }$ has a large variance, and its magnitude tends to get progressively larger as the limiting redshift $z_{\max }$ gets smaller, implying that precision measurements of ${\Omega }_{\Lambda }$ 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 ${\Omega }_{\Lambda }\approx 0.7$. 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.

Figure 1

The best-fit cosmological constant density ${\Omega }_{\Lambda }$ plotted as a function of the maximum redshift $z_{\max }$, for the choices $z_{\min }=0.01$, 0.02, and 0.03. The horizontal dash-dotted line shows the actual model value ${\Omega }_{\Lambda }=0$.

Figure 2

The function $1+f(r)$ plotted versus $k_{eq}r$, where $k_{eq}$ is the wave number of the dominant perturbation mode.

Figure 3

The relative size of the perturbation $\log [|\Delta D_L(z)|/D_{L(0)}(z)]$ plotted versus $\log (z)$, assuming that the dominant perturbation wavelength is ${10}^3$ times smaller than the Hubble scale: $k_{eq}/H_0={10}^3$.