Bias and scatter in the Hubble diagram from cosmological large-scale structure

An important part of cosmological model fitting relies on correlating distance indicators of objects (for example, type Ia supernovae) with their redshift, often illustrated on a Hubble diagram. Comparing the observed correlation with a homogeneous model is one of the key pieces of evidence for dark energy. The presence of cosmic structures introduces a bias and scatter, mainly due to gravitational lensing and peculiar velocities but also due to smaller nonlinear relativistic contributions that are more difficult to account for. For the first time we perform ray tracing onto halos in a relativistic N-body simulation. Our simulation is the largest that takes into account all leading-order corrections from general relativity in the evolution of structure, and we present a novel methodology for working out the nonlinear projection of that structure onto the observer’s past light cone. We show that the mean of the bias in the Hubble diagram is indeed as small as expected from perturbation theory. However, the distribution of sources is significantly skewed with a very long tail of highly magnified objects, and we illustrate that the bias of cosmological parameters strongly depends on the function of distance which we consider.

Figure 1

The top panel shows the luminosity distance $D_L$ vs. redshift $z$ for $\sim {10}^7$ halos within a $450{\mathrm{deg}}^2$ field of view (the color gradient illustrates the logarithmic point density) in relation to the luminosity distance ${\overline{D}}_L(z)$ of the homogeneous $\mathrm{\Lambda }\mathrm{CDM}$ model at the same redshift. The bottom panel shows the bias of the source average for various distance functions.

Figure 2

Histograms of the ratio $D_L/{\overline{D}}_L(z)$ for four different redshift bins.

Figure 3

Constraints on ${\mathrm{\Omega }}_m$ and ${\mathrm{\Omega }}_K$ from fits to various functions of the luminosity distance (smaller contours at the top, with $D_L^2$ being the leftmost contour, and then following the order of the legend until $1/D_L^2$ on the right). When binning in redshift, the bias is reduced at the price of an increase in error bars (larger contours at the bottom for $\mu$ on the left and $1/D_L^2$ at the right—the other functions of the luminosity distance would line up as for the smaller contours). The use of $1/D_L^2$ as distance measure combined with redshift binning is able to remove the bias from the lensing of supernovae.