Computing inspirals in Kerr in the adiabatic regime. I. The scalar case

TL;DR

This paper develops a comprehensive method for computing scalar field inspirals in Kerr spacetime under the adiabatic approximation, providing explicit formulas for waveform generation and constants of motion evolution, aiding gravitational wave data analysis.

Contribution

It offers a complete, self-contained computational framework for scalar radiation reaction in Kerr spacetime, including explicit formulas for constants of motion evolution, based on Mino's approach.

Findings

Derived an expression for the time-averaged change of the Carter constant.

Verified that circular orbits remain circular under radiation reaction.

Presented explicit formulas suitable for numerical implementation.

Abstract

A key source for LISA will be the inspiral of compact objects into supermassive black holes. Recently Mino has shown that in the adiabatic limit, gravitational waveforms for these sources can be computed using for the radiation reaction force the gradient of one half the difference between the retarded and advanced metric perturbations. Using post-Newtonian expansions, we argue that the resulting waveforms should be sufficiently accurate for signal detection with LISA. Data-analysis templates will require higher accuracy, going beyond adiabaticity; this remains a significant challenge. We describe an explicit computational procedure for obtaining waveforms based on Mino's result, for the case of a point particle coupled to a scalar field. We derive an expression for the time-averaged time derivative of the Carter constant, and verify that the expression correctly predicts that circular orbits remain circular while evolving via radiation reaction. The derivation uses detailed properties of mode expansions, Green's functions and bound geodesic orbits in the Kerr spacetime, which we review in detail. This paper is about three quarters review and one quarter new material. The intent is to give a complete and self-contained treatment of scalar radiation reaction in the Kerr spacetime, in a single unified notation, starting with the Kerr metric and ending with formulae for the time evolution of all three constants of the motion that are sufficiently explicit to be used immediately in a numerical code.