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.Comment: 18 pages,5 figure
