  Corrections to general relativity that introduce long-ranged scalar fields
which are non-minimally coupled to curvature typically predict that neutron
stars possess a non-trivial scalar field profile. An observer far from a star
is most sensitive to the spherically-symmetric piece of this profile that
decays linearly with the inverse of the distance, the so-called scalar charge,
which is related to the emission of dipolar radiation from compact binaries.
The presence of dipolar radiation has the potential to very strongly constrain
extended theories of gravity. These facts may lead people to believe that
gravitational theories with long-ranged scalar fields have already been
constrained strongly from binary pulsar observations. Here we challenge this
"lore" by investigating the decoupling limit of Gauss-Bonnet gravity as an
example, in which the scalar field couples linearly to the Gauss-Bonnet density
in the action. We prove a theorem that neutron stars in this theory cannot
possess a scalar charge. Thus Gauss-Bonnet gravity evades the strong binary
pulsar constraints on dipole radiation. We discuss the astrophysical systems
yielding the best constraints on Gauss-Bonnet gravity and related quadratic
gravity theories. To achieve this we explicitly compute the scalar charge for
slowly-rotating neutron stars in quadratic gravity theories. In generic case,
either neutron star-binary or neutron star-black hole systems can be used to
constrain the theory, but Gauss-Bonnet gravity evades the neutron star-binary
constraints. However, black holes in Gauss-Bonnet gravity do anchor scalar
charge. The best constraints on Gauss-Bonnet gravity will thus come from black
hole observations, for example via pulsar-black hole binaries. We estimate
these constraints to be ten times better than the current estimated bound, and
also include estimated constraints on generic quadratic gravity theories.
