  Scalar-tensor theories of gravity modify General Relativity by introducing a
scalar field that couples non-minimally to the metric tensor, while satisfying
the weak-equivalence principle. These theories are interesting because they
have the potential to simultaneously suppress modifications to Einstein's
theory on Solar System scales, while introducing large deviations in the strong
field of neutron stars. Scalar-tensor theories can be classified through the
choice of conformal factor, a scalar that regulates the coupling between matter
and the metric in the Einstein frame. The class defined by a Gaussian conformal
factor with negative exponent has been studied the most because it leads to
spontaneous scalarization (i.e. the sudden activation of the scalar field in
neutron stars), which consequently leads to large deviations from General
Relativity in the strong field. This class, however, has recently been shown to
be in conflict with Solar System observations when accounting for the
cosmological evolution of the scalar field. We study whether this remains the
case when the exponent of the conformal factor is positive, as well as in
another class of theories defined by a hyperbolic conformal factor. We find
that in both of these scalar-tensor theories, Solar System tests are passed
only in a very small subset of parameter space, for a large set of initial
conditions compatible with Big Bang Nucleosynthesis. However, while we find
that it is possible for neutron stars to scalarize, one must carefully select
the coupling parameter to do so, and even then, the scalar charge is typically
two orders of magnitude smaller than in the negative exponent case. Our study
suggests that future work on scalar-tensor gravity, for example in the context
of tests of General Relativity with gravitational waves from neutron star
binaries, should be carried out within the positive coupling parameter class.
