The fitting problem in a lattice Universe

TL;DR

This paper constructs an exact second-order lattice solution to Einstein's equations, compares it with FLRW models, and discusses conditions under which observational equivalence holds, highlighting the importance of object compactness.

Contribution

It provides a second-order exact lattice solution to Einstein's equations and analyzes the conditions for observational equivalence with FLRW models.

Findings

The lattice solution is kinematically equivalent to FLRW with same average density.

A criterion on object compactness determines when observables match FLRW predictions.

Non-perturbative analysis is needed if the compactness criterion is not satisfied.

Abstract

We present a regular cubic lattice solution to Einstein field equations that is exact at second order in a small parameter. We show that this solution is kinematically equivalent to the Friedmann-Lema\^itre-Robertson-Walker (FLRW) solution with the same averaged energy density. This allows us to discuss the fitting problem in that framework: are observables along the past lightcone of observers equivalent to those in the analogue FLRW model obtained by smoothing spatially the distribution of matter? We find a criterion on the compacity of the objects that must be satisfied in order for the answer to this question to be positive and given by perturbative arguments. If this criterion is not met, the answer to this question must be addressed fully non perturbatively along the past lightcone, even though the spacetime geometry can be described perturbatively.