Extreme mass ratio inspirals, from orbits to forecasts
A black hole of ten solar masses circling one a million times heavier falls inward so slowly that it completes hundreds of thousands of orbits first. Start with what those orbits look like, count the wave cycles they leave, recompute that count from a published table, then read what LISA forecasts say and do not say.
What hundreds of thousands of orbits can reveal about a black hole
A small black hole circling a giant one swings its orbit forward every lap. Over many laps those swings could map the giant’s mass, spin and shape.
02 · L1 · ChapterHow orbits precess around a spinning black hole and where they stop being stable
Derive circular-orbit frequencies around a Kerr black hole, find the innermost stable circular orbit and compute how far the periapsis advances per lap.
03 · L2 · ChapterHow energy loss sets the cycles left before the last stable orbit
Turn gravitational-wave energy loss into an inspiral time and a count of wave cycles, check it against a published flux table and see what it leaves out.
04 · L3 · ChapterWhy the wave from an eccentric orbit is a comb of lines
Compute an eccentric Kerr orbit’s radial and azimuthal frequencies, see why its quadrupole wave is a comb of lines and check the sum against Peters–Mathews.
05 · L3 · MethodRecompute an EMRI frequency evolution from a published flux table
Run one Python script that turns the Fujita–Tagoshi flux table into wave cycles and frequency against time, then test it with three deliberate changes.
06 · L4 · Paper guideReading the EMRI literature on forecasts, self-force, resonances and multipole tests
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.
Primary sources and what we read in each
- Babak et al. (2017), Science with the space-based interferometer LISA. V. Extreme mass-ratio inspirals, Phys. Rev. D 95, 103012
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.
- Flanagan & Hinderer (2012), Transient resonances in the inspirals of point particles into black holes, Phys. Rev. Lett. 109, 071102
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.
- Barack & Pound (2019), Self-force and radiation reaction in general relativity, Rep. Prog. Phys., doi:10.1088/1361-6633/aae552
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.
- Fujita & Tagoshi (2004), New numerical methods to evaluate homogeneous solutions of the Teukolsky equation, Prog. Theor. Phys. 112, 415
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.
- Glampedakis & Kennefick (2002), Zoom and whirl: eccentric equatorial orbits around spinning black holes and their evolution under gravitational radiation reaction, Phys. Rev. D 66, 044002
arXiv version 1. Section II, eqs. 1–24: equatorial geodesic constants in terms of p and e, the quartic for L − aE, and the separatrix condition; Appendix A. We did not use the printed integrands for the periods; ours were derived separately and checked numerically.
- Abramowicz & Kluźniak (2004), Epicyclic frequencies derived from the effective potential: simple and practical formulae, arXiv:astro-ph/0411709, Astrophys. Space Sci. doi:10.1007/s10509-005-1173-z
arXiv version 1. Eqs. 20–34: circular-orbit angular velocity, energy and angular momentum in a stationary axisymmetric metric, and the radial and vertical epicyclic frequencies from the effective potential.
- Tagoshi & Sasaki (1994), Post-Newtonian expansion of gravitational waves from a particle in circular orbit around a Schwarzschild black hole, Prog. Theor. Phys. 92, 745
arXiv version 1. Eq. (43): the energy flux to infinity relative to the quadrupole value, through fourth post-Newtonian order; used only as an independent weak-field check of the flux table.
- Brito, Cardoso & Pani, Superradiance, Lecture Notes in Physics 971 (2020), arXiv:1501.06570 version 8
arXiv version 8, Section 6.7, eq. 6.29 only: the energy flux into the horizon of a spinning hole changes sign when the orbital frequency exceeds the horizon frequency.