Parametrized post-Einsteinian framework for precessing binaries

In general relativity, isolated black holes obey the no-hair theorems, which fix the multipolar structure of their exterior spacetime. However, in modified gravity, or when the compact objects are not black holes, the exterior spacetime may have a different multipolar structure. When two black holes are in a binary, this multipolar structure determines the morphology of the dynamics of orbital and spin precession. In turn, the precession dynamics imprint onto the gravitational waves emitted by an inspiraling compact binary through specific amplitude and phase modulations. The detection and characterization of these amplitude and phase modulations can therefore lead to improved constraints on fundamental physics with gravitational waves. Recently, analytic precessing waveforms were calculated in two scenarios: (i) dynamical Chern-Simons gravity, where the no-hair theorems are violated, and (ii) deformed compact objects with generic mass quadrupole moments. In this work, we use these two examples to propose an extension of the parametrized post-Einsteinian (ppE) framework to include precession effects. The new framework contains 2 � ppE parameters ( � ( � ′ , � ) ppE , � ( � ′ , � ) ppE ) for the waveform phase, and 2 � ppE parameters ( � ( � ′ , � ) ppE , � ( � ′ , � ) ppE ) for the waveform amplitudes. The number of ppE corrections � corresponds to the minimum number of harmonics necessary to achieve a given likelihood threshold when comparing the truncated ppE waveform with the exact one, and ( � ′ , � ) corresponds to the harmonic numbers of the harmonics containing ppE parameters. We show explicitly how these ppE parameters map to the specific example waveforms discussed above. The proposed ppE framework can serve as a basis for future tests of general relativity with gravitational waves from precessing binaries.