Parametrized post-Einsteinian gravitational waveforms in various modified theories of gravity

Despite the tremendous success of general relativity so far, modified theories of gravity have received increased attention lately, motivated from both theoretical and observational aspects. In particular, gravitational wave observations opened new possibilities for testing the viability of such theories in the dynamical and strong-field regime. One could test each modified theory of gravity against observed data one at a time, though perhaps a more efficient approach would be to first probe gravity in a theoryagnostic way and map such information to that on specific theories afterward. One example of such mode-lindependent tests of gravity with gravitational waves is the parametrized post-Einsteinian formalism, in which one introduces generic parameters in the amplitude and phase that capture non-Einsteinian effects. In this paper, we derive gravitational waveforms from inspiraling compact binaries in various modified theories of gravity that violate at least one fundamental pillar in general relativity, such as the strong equivalence principle, Lorentz and parity invariance, and commutativity of spacetime. We achieve this by first deriving relations between corrections to the waveform amplitude/phase and those to the frequency evolution and Kepler's third law, since the latter two have already been (or can easily be) derived in several example modified theories of gravity. In particular, such an analysis allows us to derive corrections to the waveform amplitude, which extends many of previous works that focused on deriving phase corrections only. Moreover, we derive modified gravitational waveforms in theories with a varying gravitational constant. In particular, we extend the previous work by introducing two different gravitational constants (the conservative one entering in the binding energy and the dissipative one entering in the gravitational wave luminosity) and allowing masses of binary constituents to also vary with time. We also correct some errors in the previous literature. Our results can be used to improve current analyses of testing general relativity with available gravitational wave data as well as to achieve new projected constraints on various modified theories of gravity with future gravitational wave observations.