Observables in a lattice Universe: the cosmological fitting problem

We explore observables in a lattice Universe described by a recently found solution to Einstein field equations. This solution models a regular lattice of evenly distributed objects of equal masses. This inhomogeneous solution is perturbative, and, up to second order in a small parameter, it expands at a rate exactly equal to the one expected in a dust dominated Friedmann-Lemaitre-Robertson-Walker (FLRW) model with the equivalent, smoothed energy density. Therefore, the kinematics of both cosmologies are identical up to the order of perturbation studied. Looking at the behaviour of the redshift and angular distance, we find a condition on the compactness of the objects at the centre of each cell under which corrections to the FLRW observables remain small, i.e. of order of a few percents at most. Nevertheless, we show that, if this condition is violated, i.e. if the objects are too compact, our perturbative scheme breaks down as far as the calculations of observables are concerned, even though the kinematics of the lattice remains identical to its FLRW counterpart (at the perturbative order considered). This may be an indication of an actual fitting problem, i.e. a situation in which the FLRW model obtained from lightcone observables does not correspond to the FLRW model obtained by smoothing the spatial distribution of matter. Fully non-perturbative treatments of the observables will be necessary to answer that question.