  We show how a nearly massless scalar field conformally and disformally
coupled to matter can affect the dynamics of two bodies in their inspiralling
phase before merging. We discuss both the conservative dynamics, e.g. how the
energy of the bound system is corrected by the conformal and disformal
interactions, and the dissipative part where scalars and gravitons are emitted.
The first disformal correction to the Einstein-Infeld-Hoffmann Lagrangian is
obtained using both the Fokker method relying on the equations of motion and an
Effective Field Theory approach using Feynman diagrams. This leads to a
correction to the energy functional at the 2PN level for eccentric orbits,
which vanishes for circular orbits up to the 7PN order. The dissipative power
from the disformal interaction gives a correction to the monopole and
quadrupole terms in the presence of a conformal coupling. Although this
correction vanishes for circular orbits at leading order, this is not the case
for elliptical orbits allowing us to derive a bound on the disformal coupling
from the time drift of the period for the Hulse-Taylor binary pulsar, which is
slightly stronger than the one from fifth force tests. We conclude that the
prospect of observing disformal effects for inspiralling systems lies in the
accurate monitoring of eccentric trajectories as expected for the LISA
experiment.
