  It is known that the four-dimensional effective field theory arising from
heterotic string theory is general relativity with both a Chern-Simons and
Gauss-Bonnet term. We study the propagation of gravitational waves in this
combination of Chern-Simons and Gauss-Bonnet gravity, both of which have an
associated scalar field, the axion and the dilaton respectively, that are
kinetically coupled. We review how the combination of dynamical Chern-Simons
and Gauss-Bonnet gravities can arise from string theory as corrections to
general relativity and show how the gravitational wave waveform is modified in
such a theory. We compare our results to a novel framework recently introduced
for parametrizing the parity-violating sector (Chern-Simons), and use that to
guide our construction of a similar parametrization for the parity-conserving
(Gauss-Bonnet) sector. In general, we find that the contributions from the
parity-violating and parity-conserving sectors are similar. Moreover, the
kinetic coupling between the axion and dilaton introduces an extra contribution
to the parity-violating sector of the gravitational waves. Using our
parametrization, we are able to comment on initial constraints for the theory
parameters, including the time variations of the axion and dilaton.
