ESIGMA Inspiral Termination: x_final, the ISCO, and Spin

Split out of under_construction.md (the ESIGMA XLA-compilation and gradient log). This note collects the quantified analysis of the inspiral-termination radius x_final versus the physically-motivated ISCO. Current modeling decision: keep esigmapy’s fixed r = 4M; revisit later.

1. What x_final actually is

x_final is not the ISCO — it is a fixed r = 4M inspiral-termination radius. ESIGMAInspiral sets self.x_final = 1.0 / inspiral_end_radius with inspiral_end_radius = 4.0 (esigma.py:90), i.e. x_final = 0.25. In the PN variable x = (M\omega_{\rm orb})^{2/3} (leading order x ≈ M/r), that is r = 4M. The Schwarzschild test-particle ISCO is r = 6M, i.e. x_ISCO = 1/6 ≈ 0.167 — so the code terminates the inspiral at x = 0.25 > 1/6, deeper than (inside) the ISCO. It is a heuristic end-of-inspiral radius, not an ISCO estimate, and it does not depend on mass ratio or spin.

This mirrors esigmapy’s own numba PN backend, which sets f_gw_isco = 1/(TRANS^{3/2} \pi M) with TRANS = inspiral_end_radius = 4.0 and x_final = (\pi M f_gw_isco)^{2/3} = 1/4 (pn_main.py:529,572-574) — it labels this “ISCO” but it is the same fixed 4M. So the jaxpe adapter faithfully reproduces esigmapy’s PN model, and test_esigma_parity validates that agreement.

2. The physically-motivated ISCO (spin and mass-ratio dependent)

r = 6M holds only for a non-spinning test particle around a Schwarzschild BH.

• Kerr test-particle ISCO (Bardeen–Press–Teukolsky 1972), dimensionless spin χ ∈ [-1,1] (aligned; negative = retrograde): \(r_{\rm ISCO}/M = 3 + Z_2 \mp \sqrt{(3 - Z_1)(3 + Z_1 + 2 Z_2)},\) \(Z_1 = 1 + (1-\chi^2)^{1/3}\left[(1+\chi)^{1/3} + (1-\chi)^{1/3}\right], \quad Z_2 = \sqrt{3\chi^2 + Z_1^2},\) upper sign (−) prograde, lower (+) retrograde. χ=0 → 6M; χ=+1 → 1M; χ=−1 → 9M.

• Finite mass-ratio + aligned spin: esigmapy already ships this as esigmapy.utils.f_ISCO_spin(m1, m2, s1z, s2z) (used by its surrogate backend, not the PN one). It forms an effective spin a_eff = a_tot + \zeta\,\eta\,(s_{1z}+s_{2z}) (a_tot the mass-weighted spin, ζ = 0.41616), evaluates the Kerr ISCO of the remnant (final spin χ_f via an η-dependent fit, then the BPT radius/frequency of that remnant), and returns the Kerr ISCO GW frequency in Hz. η enters through the final-spin/final-mass fit. This is a phenomenological remnant-ISCO transition frequency (Healy–Lousto–Zlochower-style fits), not the strict two-body ISCO; EOB radial-potential or MECO (dE/dx = 0) conditions are the more first-principles finite-η alternatives.

3. Consequence for highly spinning black holes

The fixed x_final = 0.25 cannot track the ISCO, which in the code’s own x variable sweeps x_ISCO ∈ [0.11, 0.51] (r ∈ [1M, 9M]) across aligned spin. Using the Kerr formula with the Kerr orbital frequency at ISCO, M\omega_{\rm ISCO} = 1/(r_{\rm ISCO}^{3/2} + \chi) and x_ISCO = (M\omega_{\rm ISCO})^{2/3}:

χ (aligned)	r_ISCO/M	x_ISCO	vs fixed x_final = 0.25
+0.99	1.45	0.510	stops short (large early cut)
+0.90	2.32	0.370	stops short
+0.70	3.39	0.275	stops slightly short
+0.50	4.23	0.228	runs slightly past
0.00	6.00	0.167	runs past
−0.50	7.56	0.135	runs past
−0.90	8.72	0.117	runs far past ISCO
−0.99	8.97	0.114	runs far past

Two distinct failure modes:

• Prograde, χ ≳ 0.6 → premature truncation. The true ISCO is inside x = 0.25, so the fixed cutoff ends the inspiral early and discards the loudest late-inspiral cycles. The termination frequency is off by (x_ISCO / 0.25)^{3/2}: at χ = +0.9 the real ISCO frequency is 1.8× higher than where the code stops (≈2.9× at χ = +0.99). For PE this lost band biases the likelihood and pushes recovered spin/masses.
• Retrograde and low spin, χ ≲ 0.5 → over-integration past the ISCO. The true ISCO is outside x = 0.25, so the code integrates the adiabatic PN inspiral well past it — for χ = −0.9 down to ~3× past the ISCO frequency — into a regime with no stable circular orbits, where an inspiral-only model is unphysical (this includes the non-spinning case, since 4M is already inside 6M).

No single fixed radius can be right, since the ISCO sweeps 1M → 9M across spin.

PN-validity caveat that limits the upside of simply extending the cutoff. At x = 0.25, v ≈ 0.5c and the PN series is already marginal; for high prograde spin the ISCO sits at x = 0.37–0.51, where the PN expansion (and its large spin-orbit/spin-spin terms) is genuinely unreliable. So a spin-dependent ISCO would properly cut retrograde/low-spin systems at their outer ISCO (removing the unphysical over-integration — a clear win) but would extend prograde systems toward their inner ISCO only as far as PN remains trustworthy.

4. If we later adopt a proper ISCO in jaxpe

Replace the constant with x_final = (m_sec · \pi · f_ISCO_spin(m1, m2, s1z, s2z))^{2/3} (the same x = (M\pi f_{\rm GW})^{2/3} mapping already used for x_init), porting f_ISCO_spin to JAX (pure algebra — differentiable; guard sign/abs at zero spin).

Interaction with the ISCO-clip gradient fix (§17–18 of under_construction.md). Making x_final a function of (η, s1z, s2z) is fully compatible with the jnp.minimum(x, x_final) cap: the cap stays differentiable and would correctly propagate the ISCO’s parameter dependence into the mass/spin gradients (a new, physically-correct contribution, since x_final also enters t_max). It does not reintroduce the frozen-sensitivity bug, which is about the RHS freeze and is orthogonal to the value of x_final.

Decision (deferred). Two consistent choices:

1. Keep r = 4M (current) to remain a faithful, parity-validated reproduction of esigmapy’s PN model.
2. Adopt the η+spin ISCO for better physics — but this changes where the inspiral ends, diverges from esigmapy’s PN reference, and would require updating test_esigma_parity and re-validating against the surrogate/reference. esigmapy’s own PN backend would then also need changing for consistency.

Sources

Bardeen, Press & Teukolsky, ApJ 178, 347 (1972); Kerr ISCO calculator (L. C. Stein, duetosymmetry.com); ISCO parameter review arXiv:1605.04189; PN/self-force ISCO conditions arXiv:1010.2553; esigmapy/utils.py::f_ISCO_spin.