  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 parameterized
post-Einsteinian~(ppE) framework to include precession effects. The new
framework contains $2n$ ppE parameters $(\mathscr{b}^{\rm ppE}_{(m',n)}, b^{\rm
ppE}_{(m',n)})$ for the waveform phase, and $2n$ ppE parameters
$(\mathscr{a}^{\rm ppE}_{(m',n)}, a^{\rm ppE}_{(m',n)})$ for the waveform
amplitudes. The number of ppE corrections $n$ 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 $(m',n)$ 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.
