  We generalize tensor-scalar theories of gravitation by the introduction of an
abnormally weighting type of energy. This theory of tensor-scalar anomalous
gravity is based on a relaxation of the weak equivalence principle that is now
restricted to ordinary visible matter only. As a consequence, the convergence
mechanism toward general relativity is modified and produces naturally cosmic
acceleration as an inescapable gravitational feedback induced by the
mass-variation of some invisible sector. The cosmological implications of this
new theoretical framework are studied. From the Hubble diagram cosmological
test \textit{alone}, this theory provides an estimation of the amount of
baryons and dark matter in the Universe that is consistent with the independent
cosmological tests of Cosmic Microwave Background (CMB) and Big Bang
Nucleosynthesis (BBN). Cosmic coincidence is naturally achieved from a equally
natural assumption on the amplitude of the scalar coupling strength. Finally,
from the adequacy to supernovae data, we derive a new intriguing relation
between the space-time dependences of the gravitational coupling and the dark
matter mass, providing an example of crucial constraint on microphysics from
cosmology. This glimpses at an enticing new symmetry between the visible and
invisible sectors, namely that the scalar charges of visible and invisible
matter are exactly opposite.
