Gravitational waves from a compact star in a circular, inspiral orbit, in the equatorial plane of a massive, spinning black hole, as observed by LISA

Results are presented from high-precision computations of the orbital evolution and emitted gravitational waves for a stellar-mass object spiraling into a massive black hole in a slowly shrinking, circular, equatorial orbit. The focus of these computations is inspiral near the innermost stable circular orbit (isco)—more particularly, on orbits for which the angular velocity $\Omega$ is $0.03\lesssim \Omega /{\Omega }_{\mathrm{isco}}<~\mathrm{1.0.}$ The computations are based on the Teuksolsky-Sasaki-Nakamura formalism, and the results are tabulated in a set of functions that are of order unity and represent relativistic corrections to low-orbital-velocity formulas. These tables can form a foundation for future design studies for the LISA space-based gravitational-wave mission. A first survey of applications to LISA is presented: Signal to noise ratios $S/N$ are computed and graphed as functions of the time-evolving gravitational-wave frequency for the lowest three harmonics of the orbital period, and for various representative values of the hole’s mass M and spin a and the inspiraling object’s mass $\mu ,$ with the distance to Earth chosen to be $r_o=1\mathrm{}\hspace{0.5em}\mathrm{Gpc}.$ These $S/N’\mathrm{s}$ show a very strong dependence on the black-hole spin, as well as on M and μ. Graphs are presented showing the range of the $\{M,a,\mu \}$ parameter space, for which $S/N>\mathrm{}10\mathrm{}$ at $r_0=1\mathrm{}\hspace{0.5em}\mathrm{Gpc}$ during the last year of inspiral. The hole’s spin a has a factor of $\sim 10$ influence on the range of M (at fixed μ) for which $S/N>\mathrm{}10,$ and the presence or absence of a white-dwarf–binary background has a factor of $\sim 3$ influence. A comparison with predicted event rates shows strong promise for detecting these waves, but not beyond about 1 Gpc if the inspiraling object is a white dwarf or neutron star. This argues for a modest lowering of LISA’s noise floor. A brief discussion is given of the prospects for extracting information from the observed waves.