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

Reconstructing the metric of the local Universe from number counts observations