  Extreme-mass-ratio inspirals, in which a stellar-mass compact object spirals
into a supermassive black hole, are prime candidates for detection with
space-borne milliHertz gravitational wave detectors, similar to the Laser
Interferometer Space Antenna. The gravitational waves generated during such
inspirals encode information about the background in which the small object is
moving, providing a tracer of the spacetime geometry and a probe of
strong-field physics. In this paper, we construct approximate,
"analytic-kludge" waveforms for such inspirals with parameterized
post-Einsteinian corrections that allow for generic, model-independent
deformations of the supermassive black hole background away from the Kerr
metric. These approximate waveforms include all of the qualitative features of
true waveforms for generic inspirals, including orbital eccentricity and
relativistic precession. The deformations of the Kerr metric are modeled using
a recently proposed, modified gravity bumpy metric, which parametrically
deforms the Kerr spacetime while ensuring that three approximate constants of
the motion remain for geodesic orbits: a conserved energy, azimuthal angular
momentum and Carter constant. The deformations represent modified gravity
effects and have been analytically mapped to several modified gravity black
hole solutions in four dimensions. In the analytic kludge waveforms, the
conservative motion is modeled by a post-Newtonian expansion of the geodesic
equations in the deformed spacetimes, which in turn induce modifications to the
radiation-reaction force. These analytic-kludge waveforms serve as a first step
toward complete and model-independent tests of General Relativity with extreme
mass-ratio inspirals.
