We study spherically symmetric solutions of the cubic covariant Galileon
model in curved spacetime in presence of a matter source, in the test scalar
field approximation. We show that a cosmological time evolution of the Galileon
field gives rise to an induced matter-scalar coupling, due to the
Galileon-graviton kinetic braiding, therefore the solution for the Galileon
field is non trivial even if the bare matter-scalar coupling constant is set to
zero. The local solution crucially depends on the asymptotic boundary
conditions, and in particular, Minkowski and de Sitter asymptotics correspond
to different branches of the solution. We study the stability of these
solutions, namely, the well-posedness of the Cauchy problem and the positivity
of energy for scalar and tensor perturbations, by diagonalizing the kinetic
terms of the spin-2 and spin-0 degrees of freedom. In addition, we find that in
presence of a cosmological time evolution of the Galileon field, its kinetic
mixing with the graviton leads to a friction force, resulting to efficient
damping of scalar perturbations within matter.Comment: 20 pages, no figure, RevTeX4 format; v2: minor changes reflecting the
published version in PR