Ready-to-use analytic model for gravitational waves from a hierarchical triple with Kozai-Lidov oscillations

Gravitational waves emitted by inner binaries in hierarchical triple systems are interesting astrophysical candidates for space-based detectors like the Laser Interferometer Space Antenna, LISA. In the presence of a third body, such as a supermassive black hole, an inner binary consisting of intermediate mass black holes can undergo oscillations in eccentricity and inclination angle due to the Kozai-Lidov mechanism. In this work, we construct analytic gravitational waveforms in the Fourier domain, taking into account the Kozai-Lidov effect at Newtonian (leading) order. Using multiple-scale analysis, we make use of the separability of timescales to combine the effects of both Kozai-Lidov oscillations and radiation-reaction. We assume small eccentricity and present analytic solutions to the evolution of the other orbital elements. Our analytic calculation can be systematically extended to higher orders in eccentricity, and can be used to construct inspiral-merger-ringdown models. The imprint on the waveform, due to this combined evolution, is computed under the stationary-phase approximation. We find that the oscillations leave a clear signature on the Fourier amplitude of the waveform while leaving a measurable imprint on the gravitational wave phase, and that our analytic results are consistent with numerics. Further, with our study of the astrophysical parameters of the hierarchical triple, we outline potential source candidates, along with potential implications for gravitational wave data analysis.