Mimicking the cosmological constant for more than one observable with large scale inhomogeneities

Assuming the definition of the inversion problem (IP) as the exact matching of the terms in the low redshift expansion of cosmological observables calculated for different cosmological models, we solve the IP for $D_L(z)$ and the redshift spherical shell mass density $mn(z)$ for a central observer in a LTB space without cosmological constant and a generic $\Lambda CDM $ model. We show that the solution of the IP is unique, corresponds to a matter density profile which is not smooth at the center and that the same conclusions can be reached expanding self-consistently to any order all the relevant quantities. On contrary to the case of a single observable inversion problem, it is impossible to solve the IP (LTB vs. $\Lambda$CDM) for both $mn(z)$ and $D_L(z)$ while setting one the two functions $k(r)$ or $t^b(r)$ to zero, even allowing not smooth matter profiles. Our conclusions are general, since they are exclusively based on comparing directly physical observables in redshift space, and don't depend on any special ansatz or restriction for the functions defining a LTB model.