Waveform of gravitational waves in the general parity-violating
  gravities
  As an extension of our previous work [J.Qiao, T.Zhu, W.Zhao & A.Wang,
arXiv:1909.03815], in this article, we calculate the effects of parity
violation on gravitational-wave (GW) waveforms during their propagation in the
most general parity-violating gravities, including Chern-Simons modified
gravity, ghost-free scalar-tensor gravity, symmetric teleparallel equivalence
of GR theory, Ho\v{r}ava-Lifshitz gravity and so on. For this purpose, we
consider the GWs generated by the coalescence of compact binaries and
concentrate on the imprints of the parity violation in the propagation of GWs.
With a unified description of GW in the theories of parity-violating gravity,
we study the effects of velocity and amplitude birefringence on the GW
waveforms. Decomposing the GWs into the circular polarization modes, the two
birefringence effects exactly correspond to the modifications in phase and
amplitude of GW waveforms respectively. We find that, for each circular
polarization mode, the amplitude, phase and velocity of GW can be modified by
both the parity-violating terms and parity-conserving terms in gravity.
Therefore, in order to test the parity symmetry in gravity, we should compare
the difference between two circular polarization modes, rather than measuring
an individual mode. Combining two circular modes, we obtain the GW waveforms in
the Fourier domain, and obtain the deviations from those in General Relativity.
The GW waveforms derived in this paper are also applicable to the theories of
parity-conserving gravity, which have the modified dispersion relations (e.g.
massive gravity, double special relativity theory, extra-dimensional theories,
etc), or/and have the modified friction terms (e.g. nonlocal gravity,
gravitational theory with time-dependent Planck mass, etc).
