Two-timescale evolution of extreme-mass-ratio inspirals: Waveform generation scheme for quasicircular orbits in Schwarzschild spacetime

Abstract

Extreme-mass-ratio inspirals, in which a stellar-mass compact object spirals into a supermassive black hole in a galactic core, are expected to be key sources for LISA. Modelling these systems with sufficient accuracy for LISA science requires going to second (or post-adiabatic) order in gravitational self-force theory. Here we present a practical two-timescale framework for achieving this and generating post-adiabatic waveforms. The framework comprises a set of frequency-domain field equations that apply on the fast, orbital timescale, together with a set of ordinary differential equations that determine the evolution on the slow, inspiral timescale. Our analysis is restricted to the special case of quasicircular orbits around a Schwarzschild black hole, but its general structure carries over to the realistic case of generic (inclined and eccentric) orbits in Kerr spacetime. In our restricted context, we also develop a tool that will be useful in all cases: a formulation of the frequency-domain field equations using hyperboloidal slicing, which significantly improves the behavior of the sources near the boundaries. We give special attention to the slow evolution of the central black hole, examining its impact on both the two-timescale evolution and the earlier self-consistent evolution scheme.

Second-order gravitational self-force in a highly regular gauge

“…dτ [11][12][13]. Hence, to correctly track the orbital frequencies to within a relative error much smaller than , we must compute all of f α 1 and the dissipative piece of f α 2 .…”

“…The literature on second-order self-force, going back to Ref. [46], has focused on solving the reduced problem via a puncture scheme [13,20,36,[39][40][41]47] (see also Refs. [9,[48][49][50][51][52]).…”

“…µν near the worldline. That method, which is used in the only extant second-order implementation [13,20], involves performing two-dimensional numerical integrations of the four-dimensional h P1 µν on a grid of r BL values around r o . Such a procedure will be the overwhelming computational expense in any second-order calculations using current methods.…”

“…As discussed in Refs. [11,13], to avoid other, similar errors, in general we should also allow the coefficients h n µν to depend on -and time-dependent external physical parameters. Examples of such parameters are perturbations to the large black hole's mass and spin in a binary.…”

“…where T µν 1 is the stress-energy of a point mass on the accelerated curve γ, and the O 2 term is a spatially noncompact source proportional to the acceleration. 13 Given these subtleties, there are at least two paths we can follow (in Sec. V, we discuss a third path that one of us followed in previous papers).…”