  We parametrize a large class of corrections to general relativity which
include a long-ranged gravitational scalar field as a dynamical degree of
freedom in two ways: parametrizing the structure of the correction to the
action, and parametrizing the scalar hair (multipole structure) that compact
objects and black holes attain. The presence of this scalar hair violates the
no-hair theorems present in general relativity, which leads to several
important effects. The effects we consider are (i) the interaction between an
isolated body and an external scalar field, (ii) the scalar multipole-multipole
interaction between two bodies in a compact binary, (iii) the additional
pericenter precession of a binary, (iv) the scalar radiation from a binary, and
(v) the modification to the gravitational wave phase from a binary. We apply
this framework to example theories including Einstein-dilaton-Gauss-Bonnet
gravity and dynamical Chern-Simons gravity, and estimate the size of the
effects. Finally, we estimate the bounds that can be placed on parameters of
the theories from the precession of pulsar binaries and from gravitational
waves.
