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.
