Adiabatic and post-adiabatic approaches to extreme mass ratio inspiral

Abstract

Extreme mass ratio inspirals (EMRIs) show a strong separation of timescales, with the time characterizing inspiral, T i , much longer than any time To characterizing orbital motions. The ratio of these timescales (which is essentially an EMRI's mass ratio) can be regarded as a parameter that controls a perturbative expansion. Here we describe the value and limitations of an "adiabatic" description of these binaries, which uses only the leading terms arising from such a two-timescale expansion. An adiabatic approach breaks down when orbits evolve through resonances, with important dynamical and observational consequences. We describe the shortfalls of an approach that only includes the adiabatic contributions to EMRI evolution, and outline what must be done to evolve these systems through resonance and to improve our ability to model EMRI systems more generally.Keywords: Black holes; black hole perturbation theory; gravitational waves. Motivation: The large-mass ratio limit of the two-body problem and extreme mass ratio inspiralsBinary systems in which one body is much more massive than the other can be analyzed perturbatively. We can describe such a binary as an exact black hole solution of general relativity (corresponding to the larger member of the binary) plus a correction due to the smaller body. Because the perturbation equations are much simpler to solve than the complete equations of general relativity, this turns out to be a limit that can be modeled very accurately and precisely. At least two major science goals drive studies of large mass ratio systems. First, these binaries represent a limit of the two-body problem that can be solved with high precision. As such, the study of these binaries provides important input to programs to solve the two-body problem of general relativity more generally, such as numerical relativity and the effective one-body approach 1-3 . Second, astrophysical extreme mass ratio inspirals (EMRIs) are expected to be important sources for space-based GW detectors such as eLISA 4 and DECIGO 5 . In this article, we will focus on the role of EMRIs as sources of gravitational waves (GWs). Such binaries are created when multibody interactions scatter stellar mass compact objects onto a strong-field, relativistic orbit of the black hole in a galaxy's center. Further evolution is then driven by GW emission. If the black hole has a mass of around 10 5 − 10 7 M , then these are targets for a detector like eLISA. The GWs that they generate can be heard out to z ∼ 0.5 − 1; we expect dozens to hundreds of events over a space-based detector's mission lifetime 6 . Measuring the GWs from these events will provide precision data on the characteristics of the large black hole, on the small body's orbit, and on the mass of the small bodyin short, a precision probe of the astrophysical population of galactic center black

Prospects for fundamental physics with LISA

“…The perturbation is organized in powers of the EMRI mass ratio µ/M . A phase counting argument [310] suggests that measurement templates will require us to go to at least second order in this expansion, a research program that is just getting underway.…”

Self-force and radiation reaction in general relativity

“…Such calculations are much less computationally expensive than local GSF calculations, and can be done in the convenient framework of Teukolsky's perturbation formalism, working with Ψ 0 or Ψ 4 instead of the full metric perturbation. This approach has been taken by Hughes and collaborators [152,153] in order to compute the adiabatic evolution of (certain types of) EMRIs, circumventing the need for an explicit GSF calculation.…”

Black hole perturbation theory and gravitational self-force

“…If one computes the "in" and "up" mode functions with normalisation such that transmission coefficients are unity, s R in,trans mω = 1 = s R up,trans mω , then the gravitational wave strain can be determined directly from ψ 4 using Eq. ( 77) to give (96) where the weighting coefficient −2 C up mω is to be evaluated in the limit r → ∞. Similarly, the time averaged flux of energy carried by gravitational waves 9 passing through infinity and the horizon can be computed from the "in" and "up" normalization coefficients [95],…”

“…Self-accelerated orbits are often described with a multiscale (or two-timescale) expansion [160,88,129,150,157,96,25,126]. This is essentially equivalent to the averaging transformation described above.…”

“…Starting from seminal work in Refs. [74,128], two groups of authors have developed practical implementations of this scheme [95,175,176,98,59,75,96,101].…”