  We investigate the cosmological properties of Galileon models with positive
kinetic terms. We include both conformal and disformal couplings to matter and
focus on constraints on the theory that arise because of these couplings. The
disformal coupling to baryonic matter is extremely constrained by astrophysical
and particle physics effects. The disformal coupling to photons induces a
cosmological variation of the speed of light and therefore distortions of the
Cosmic Microwave Background spectrum which are known to be very small. The
conformal coupling to baryons leads to a variation of particle masses since Big
Bang Nucleosynthesis which is also tightly constrained. We consider the
background cosmology of Galileon models coupled to Cold Dark Matter (CDM),
photons and baryons and impose that the speed of light and particle masses
respect the observational bounds on cosmological time scales. We find that
requiring that the equation of state for the Galileon models must be close to
-1 now restricts severely their parameter space and can only be achieved with a
combination of the conformal and disformal couplings. This leads to large
variations of particle masses and the speed of light which are not compatible
with observations. As a result, we find that cosmological Galileon models are
viable dark energy theories coupled to dark matter but their couplings, both
disformal and conformal, to baryons and photons must be heavily suppressed
making them only sensitive to CDM.
