Light propagation through black-hole lattices

TL;DR

This study uses numerical relativity and ray-tracing to explore how light propagates in black-hole lattice cosmologies, comparing the luminosity distance-redshift relation to standard models and revealing unique dependencies on lattice parameters.

Contribution

It introduces a new numerical approach with ray-tracing to analyze light propagation in black-hole lattice universes, comparing results with established cosmological models.

Findings

The Empty-Beam Approximation best estimates luminosity distance in these models.

A Friedmann-Lemaître-Robertson-Walker model can match the observable if a time-dependent cosmological constant is used.

The luminosity distance-redshift relation does not approach that of the continuum spacetime as lattice mass-to-spacing ratio decreases.

Abstract

The apparent properties of distant objects encode information about the way the light they emit propagates to an observer, and therefore about the curvature of the underlying spacetime. Measuring the relationship between the redshift $z$ and the luminosity distance $D_{\rm L}$ of a standard candle, for example, yields information on the Universe's matter content. In practice, however, in order to decode this information the observer needs to make an assumption about the functional form of the $D_{\rm L}(z)$ relation; in other words, a cosmological model needs to be assumed. In this work, we use numerical-relativity simulations, equipped with a new ray-tracing module, to numerically obtain this relation for a few black-hole--lattice cosmologies and compare it to the well-known Friedmann-Lema\^itre-Robertson-Walker case, as well as to other relevant cosmologies and to the Empty-Beam Approximation. We find that the latter provides the best estimate of the luminosity distance and formulate a simple argument to account for this agreement. We also find that a Friedmann-Lema\^itre-Robertson-Walker model can reproduce this observable exactly, as long as a time-dependent cosmological constant is included in the fit. Finally, the dependence of these results on the lattice mass-to-spacing ratio $\mu$ is discussed: we discover that, unlike the expansion rate, the $D_{\rm L}(z)$ relation in a black-hole lattice does not tend to that measured in the corresponding continuum spacetime as $\mu \to 0$.