# Reconstructing the metric of the local Universe from number counts observations

## TL;DR

This paper presents a method to reconstruct the local Universe's metric from number counts observations, using an exact spherically symmetric solution to improve accuracy over linear perturbation theory.

## Contribution

It introduces a low red-shift expansion approach for density and energy density, enabling more precise metric reconstruction from observational data.

## Key findings

- The low red-shift expansion outperforms linear perturbation theory.
- The inversion method accurately reconstructs the metric within its validity regime.
- Numerical tests confirm the method's effectiveness for observational data.

## Abstract

Number counts observations available with new surveys such as the Euclid mission will be an important source of information about the metric of the Universe. We compute the low red-shift expansion for the energy density and the density contrast using an exact spherically symmetric solution in presence of a cosmological constant. At low red-shift the expansion is more precise than linear perturbation theory prediction. We then use the local expansion to reconstruct the metric from the monopole of the density contrast. We test the inversion method using numerical calculations and find a good agreement within the regime of validity of the red-shift expansion. The method could be applied to observational data to reconstruct the metric of the local Universe with a level of precision higher than the one achievable using perturbation theory.