  Modified theories of gravity that offer viable models for dark energy often
rely on mechanisms that screen their effects in high density environments. From
this perspective, it would appear that, once solar system constraints are
satisfied, these theories would predict a trivial phenomenology for (much
denser) neutron stars. In this work we explore the fact that in scalar-tensor
theories the scalar degree of freedom does not couple to the mass density
alone, but to the trace of the energy-momentum tensor - which can increase and
eventually change sign as density and pressure build up in the core of neutron
stars -, and investigate whether there could be a partial unscreening of the
scalar field inside the most compact stars found in Nature. For this purpose,
we construct neutron star solutions with realistic equations of state in
theories with screening mechanisms and study their stability under radial
perturbations. In particular, we find that stable solutions with a unscreened
core can exist in chameleon models, while for the environmentally-dependent
dilaton model a wealth of new, scalarized equilibrium solutions are found, some
of which can be stable.
