---
title: "Reading the EMRI literature on forecasts, self-force, resonances and multipole tests"
description: "Where to look in three key EMRI papers, which figures and equations carry the claims, and which statements are forecasts, theory results or open problems."
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last_modified: "2026-11-12"
---

[Extreme mass ratio inspirals, from orbits to forecasts](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/)

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

Sandbox Physics · 2026-11-12

No EMRI signal has been observed. Every number in this field is a calculation or a forecast, and the papers mix the two with open problems. This guide takes three primary sources, points to the sections, equations and figures that carry their main claims, and labels each claim by what kind of evidence stands behind it.

## Before you begin

The three chapters of this course, and familiarity with matched filtering and the Fisher information matrix.

## 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.

$$
Q_K=-a^2M^3,\qquad \delta Q\equiv\frac{Q-Q_K}{M^3}
$$

A measured $\delta Q$ consistent with zero would support the Kerr description; a nonzero one would point to new physics or matter near the hole.

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.

| Claim | Where | Kind |
| --- | --- | --- |
| At least a few EMRIs per year are detectable, up to a few thousand | Babak v2, abstract; Table I; conclusions item 1 | Model prediction |
| Intrinsic parameters to $10^{-6}$–$10^{-4}$ | Babak v2, Fig. 11; conclusions item 4 | Model prediction |
| Distance to about 10%, sky to a few square degrees | Babak v2, Fig. 12 | Model prediction |
| Quadrupole deviation to percent level or better | Babak v2, Fig. 13 | Model prediction |
| The post-adiabatic phase term is of order one | Barack–Pound v2, Sec. 5.2, eq. (74) | Published theory result |
| Orbit-averaged energy balance at first order | Barack–Pound v2, Sec. 6.1, eq. (77) | Published theory result |
| Resonances shift the phase by a few tens of cycles at $10^{-6}$ | Flanagan–Hinderer v3, abstract | Published estimate |
| Accurate models through resonances | Barack–Pound v2, Sec. 5.3 | Open problem in the review |
| Quadrupole flux 0.51 to 1.19 times the Teukolsky flux at $r=6$ for spins from $-0.9$ to $0.9$ | This course, from Fujita–Tagoshi Table VII | Our calculation |

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.

## Try it in the Lab

### [See what an equatorial model can and cannot test](https://sandboxphysics.com/labs/emri-spacetime-mapping/)

1. Open the Lab and note that every preset stays in the equatorial plane.
2. Change the spin and watch how the periapsis advance and the last stable orbit both respond to the same number.

The Lab shows equatorial geodesic orbits in the Kerr spacetime and their quadrupole harmonics.

## Exercises and explained answers

Babak et al. report results for two waveform variants. Why does that matter when you quote their error forecasts?

Because the analytic kludge is the only waveform in the Fisher calculation, and the two variants are the authors’ way of bracketing its modelling error. Quote both, or say which one you use. AKK keeps more of the late inspiral, so it gives smaller errors for parameters such as the quadrupole deviation.

A colleague says resonances will spoil the Lab’s cycle count. Is that right?

No. Resonances need orbits that are both eccentric and inclined around a spinning hole, and the Lab’s orbits are equatorial. The count has other limits: it uses the quadrupole flux, circular orbits and the adiabatic order only.

## Primary sources and what we read in each

1. [Babak et al. (2017), Science with the space-based interferometer LISA. V. Extreme mass-ratio inspirals, Phys. Rev. D 95, 103012](https://arxiv.org/abs/1703.09722v2)

   arXiv version 2, stamped 11 April 2018. Abstract; Table I, the twelve population models and their rates; Section IV on the AKS and AKK kludge waveforms; Fig. 5, the detection horizon for a prograde circular equatorial inspiral into a hole of spin 0.99; Figs. 11–13, Fisher-matrix error distributions for intrinsic parameters, distance and sky position, and the quadrupole deviation; the conclusions.

2. [Flanagan & Hinderer (2012), Transient resonances in the inspirals of point particles into black holes, Phys. Rev. Lett. 109, 071102](https://arxiv.org/abs/1009.4923v3)

   arXiv version 3, dated 11 July 2012. Abstract, introduction and the order-of-magnitude estimates with eqs. (1)–(2): resonances occur when the ratio of polar and radial frequencies passes a low-order rational number, only for orbits that are both eccentric and inclined around a spinning hole; the phase shift scales as the inverse square root of the mass ratio, a few tens of cycles near one in a million.

3. [Barack & Pound (2019), Self-force and radiation reaction in general relativity, Rep. Prog. Phys., doi:10.1088/1361-6633/aae552](https://arxiv.org/abs/1805.10385v2)

   arXiv version 2, dated 19 November 2018. Section 5.2 and eqs. (72)–(74): the two-timescale phase expansion and what the adiabatic and post-adiabatic orders need; Section 5.3 on transient resonances; Section 6.1 and eqs. (77)–(78), the orbit-averaged balance laws, including negative horizon fluxes.

4. [Fujita & Tagoshi (2004), New numerical methods to evaluate homogeneous solutions of the Teukolsky equation, Prog. Theor. Phys. 112, 415](https://arxiv.org/abs/gr-qc/0410018v1)

   arXiv version 1. Table VII, the gravitational-wave luminosity through l = 6 for circular equatorial orbits at seven spins; Table III, the mode-by-mode luminosity at r = 6M. The unit normalisation and the reading as flux to infinity are our inference, checked against the quadrupole and post-Newtonian limits.

## Keep reading

Extreme mass ratio inspirals · Reading path

### A small black hole maps a large one

1. [Article **What hundreds of thousands of orbits can reveal about a black hole**](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/what-hundreds-of-thousands-of-orbits-reveal/)
2. [Article **How orbits precess around a spinning black hole and where they stop being stable**](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/geodesics-precession-and-the-last-stable-orbit/)
3. [Article **How energy loss sets the cycles left before the last stable orbit**](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/radiation-reaction-and-the-adiabatic-inspiral/)
4. [Article **Why the wave from an eccentric orbit is a comb of lines**](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/fundamental-frequencies-and-harmonics/)
5. [Article **Recompute an EMRI frequency evolution from a published flux table**](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/recompute-a-frequency-evolution/)
6. [You are here **Reading the EMRI literature on forecasts, self-force, resonances and multipole tests**](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/reading-emri-forecasts-self-force-resonances-multipoles/)