Waveform of gravitational waves in the ghost-free parity-violating
  gravities
  Gravitational waves (GWs) provide an excellent opportunity to test the
gravity in the strong gravitational fields. In this article, we calculate the
waveform of GWs, produced by the coalescence of compact binaries, in an
extension of the Chern-Simons gravity by including higher derivatives of the
coupling scalar field. By comparing the two circular polarization modes, we
find the effects of amplitude birefringence and velocity birefringence of GWs
in their propagation caused by the parity violation in gravity, which are
explicitly presented in the GW waveforms by the amplitude and phase
modifications respectively. Combining the two modes, we obtain the GW waveforms
in the Fourier domain, and find that the deviations from those in General
Relativity are dominated by effects of velocity birefringence of GWs. In
addition, we also map the effects of the parity violation on the waveform onto
the parameterized post-Einsteinian (PPE) framework and identify explicitly the
PPE parameters.
