Reading the EMRI literature on forecasts, self-force, resonances and multipole tests

How each statement is labelled

Published theory result: a derivation or calculation in a refereed paper, which later work can check without new data. Model prediction: a number that depends on an astrophysical population model, a waveform approximation or a statistical approximation. Open problem: something the authors themselves say is not yet solved. Our calculation: a number computed for this course, with its inputs published.

The versions below are the arXiv versions we read. A refereed version can differ in wording or numbers, so quote section and figure numbers together with the version.

Babak et al. 2017: how many EMRIs, and how well measured

Read arXiv version 2, stamped 11 April 2018. Start with Table I, which lists twelve population models, M1 to M12, and their detection rates. The models vary the black hole mass function, spin, cusp erosion, the black hole mass–velocity dispersion relation, the number of plunges per EMRI and the compact object mass. The authors find that these choices change the intrinsic rate by up to three orders of magnitude.

Section IV explains the waveforms. The analytic kludge starts from a Keplerian orbit and adds precession and inspiral with post-Newtonian rules. It is run in two versions, stopped at the Schwarzschild ISCO, AKS, or at the Kerr ISCO, AKK. Figure 5 checks them for one case, a prograde circular equatorial inspiral into a hole of spin 0.99: the AKS and AKK detection horizons bracket the one computed from Teukolsky fluxes.

Figures 11 and 12 give the parameter errors as distributions over detected sources, from the Fisher matrix. Fractional errors on the redshifted masses, spin and plunge eccentricity are typically $10^{-6}$ to $10^{-4}$, the distance about 10%, the sky area a few square degrees. All of these are model predictions. The Fisher matrix assumes a high signal-to-noise ratio and a waveform that matches the signal, and the analytic kludge is the waveform.

Reading: Babak et al. (2017)

The multipole test in Figure 13

In general relativity a vacuum black hole’s quadrupole moment is fixed by its mass and spin, $Q_K=-a^2M^3$. Babak et al. add a free deviation and ask how well it can be measured. Their parameter is $\delta Q=(Q-Q_K)/M^3$, chosen to be independent of redshift. It is a phenomenological knob, not a prediction of any specific alternative theory.

Figure 13 shows the $1\sigma$ error on that knob over the twelve models: percent level or better, and smaller with AKK because a modified quadrupole matters most in the late inspiral. That is a model prediction, resting on the same waveform and Fisher assumptions as Figures 11 and 12.

Reading: Babak et al. (2017)

Barack and Pound 2019: why the self-force is needed

Read arXiv version 2, stamped 19 November 2018. Section 5.2 sets up the two-timescale expansion. Eq. (72) gives the slow change of the orbital constants as a series in the mass ratio $\eta$; eq. (74) gives the phase as $\eta^{-1}$ times a leading term plus a subleading term.

The key statement follows eq. (74). The leading phase needs only the orbit-averaged dissipative part of the first-order self-force, the adiabatic approximation. The subleading term is of order one and needs the full first-order force plus the averaged dissipative part of the second-order force, the post-adiabatic approximation. This is a published theory result.

Section 6.1, eqs. (77) and (78), gives the balance laws that make the adiabatic order computable from fluxes alone, which is the approach of the second chapter. It notes that the horizon flux can be negative for some Kerr orbits.

Reading: Barack & Pound (2019)

Flanagan and Hinderer 2012: transient resonances

Read arXiv version 3, stamped 11 July 2012. The abstract states the claim: when the ratio of the polar and radial frequencies passes a low-order rational number, the adiabatic approximation breaks down briefly, and the orbital phase shifts by a few tens of cycles for mass ratios near $10^{-6}$.

The order-of-magnitude section explains the scaling. A resonance of order $v$ lasts a time of order $1/\sqrt{v\varepsilon}$ and leaves a phase error of the same order, which grows as the mass ratio shrinks. The same section notes that resonances need orbits that are both eccentric and inclined around a spinning hole. Equatorial orbits, like those in this course’s Lab, never cross one.

Barack and Pound, Section 5.3, add that the resonant phase makes the outcome sensitive to the orbit’s initial state. Accurate modelling through resonances is an open problem in that review.

Reading: Flanagan & Hinderer (2012) · Barack & Pound (2019)

A claim-by-claim checklist

Use the table as a template: one claim per row, where it is stated, and what kind of statement it is.

Reading: Babak et al. (2017) · Barack & Pound (2019) · Flanagan & Hinderer (2012) · Fujita & Tagoshi (2004)

What this guide does not cover

We read the 2019 self-force review and not the second-order self-force waveforms published after it, so this guide makes no claim about their accuracy. We did not reopen Babak et al.’s rate calculation or Fisher matrices; the numbers above are quoted from their figures and text.