Some clarifications about Lema\^itre-Tolman models of the Universe used to deal with the dark energy problem

During the past fifteen years, inhomogeneous cosmological models have been put forward to explain the observed dimming of the SNIa luminosity without resorting to dark energy. The simplest models are the spherically symmetric Lema\^itre-Tolman (LT) solutions with a central observer. Their use must be considered as a mere first step towards more sophisticated models. Spherical symmetry is but a mathematical simplification and one must consider spherical symmetric models as exhibiting an energy density smoothed out over angles around us. However, they have been taken at face value by some authors who tried to use them for either irrelevant purposes or to put them to the test as if they were robust models of our Universe. We wish to clarify how these models must be used in cosmology. We first use the results obtained by Iguchi and collaborators to derive the density profiles of the pure growing and decaying mode LT models. We then discuss the relevance of the different test proposals in the light of the interpretation given above. We show that decaying-mode (parabolic) LT models always exhibit an overdensity near their centre and growing-mode (elliptic or hyperbolic) LT models, a void. This is at variance with some statements in the literature. We dismiss all previous proposals merely designed to test the spherical symmetry of the LT models, and we agree that the value of $H_0$ and the measurement of the redshift drift are valid tests of the models. However, we suspect that this last test, which is the best in principle, will be more complicated to implement than usually claimed.