IMRPhenomHM ringdown phase-shape component

Scope and status

hm.phase.ringdown.eq9-c1.v1 specifies the changed merger-ringdown arctangent shape in London's IMRPhenomHM phase, conditional on supplied PhenomD coefficients and QNM frequencies. It is a bounded executable research component, not a complete phase, mode, or waveform qualification. The printed formula and native D-carrier QNM adaptation have different claims, distinguished below.

Inputs, conventions, and options

All frequencies are positive dimensionless \(Mf\); \(\eta>0\). Supply PhenomD coefficients \(\alpha_4,\alpha_5\), HM QNM pairs \((f_{22}^{\rm RD,H},f_{22}^{\rm damp,H})\) and \((f_{\ell m}^{\rm RD,H},f_{\ell m}^{\rm damp,H})\), and the D carrier pair \((f_{22}^{\rm RD,D},f_{22}^{\rm damp,D})\). H and D identify QNM conventions, not different remnants. This component derives none of these inputs. The public HM assembly uses the correction only for its PhenomD carrier; its optional XAS carrier bypasses the branch.

Mathematics and components

London et al., arXiv:1708.00404v4, Eq. (9) modify the last arctangent of Khan et al., arXiv:1508.07253v2, Eq. (14). With \(R=f_{22}^{\rm RD,H}/f_{\ell m}^{\rm RD,H}\), the changed term is

\[ L_{\ell m}(f)=\frac{\alpha_4}{\eta}R\arctan\!\left(\frac{f-\alpha_5 f_{22}^{\rm RD,H}}{R f_{\ell m}^{\rm damp,H}}\right), \]

and the original D term is

\[ D(f)=\frac{\alpha_4}{\eta}\arctan\!\left(\frac{f-\alpha_5 f_{22}^{\rm RD,D}}{f_{22}^{\rm damp,D}}\right). \]

These form the literal London-minus-Khan delta only when both 22 QNM pairs coincide. The live Torch/JAX code permits distinct pairs. It uses HM ratios \(R\) and \(T=f_{\ell m}^{\rm damp,H}/f_{22}^{\rm damp,H}\), but anchors the absolute center and width to the D carrier:

\[ W=R T f_{22}^{\rm damp,D},\quad C=\alpha_5 f_{22}^{\rm RD,D},\quad \delta(f)=\frac{\alpha_4}{\eta}\left[R\arctan\!\left(\frac{f-C}{W}\right)-\arctan\!\left(\frac{f-C}{f_{22}^{\rm damp,D}}\right)\right]. \]

This is a native carrier-consistency adaptation, not another equation printed by London. It reduces to \(L_{\ell m}-D\) when the 22 pairs coincide. PhenomD begins its MR phase at \(J=f_{22}^{\rm RD,D}/2\) (Khan et al., Sec. V.A). The native correction removes the value and tangent there:

\[ \Delta(f)=\begin{cases}0,&f\leq J,\\ \delta(f)-\delta(J)-(f-J)\delta'(J),&f>J. \end{cases} \]

Thus both value and slope join at zero. This affine attachment is a native choice, not another HM fit. For the 22 control, \(R=T=1\) and \(\Delta\equiv0\). Other terms of Khan's Eq. (14) remain inherited from D. The eq9-c1 component ID already names London's numbered equation; the native _production_eq14_delta_c1 helper names the PhenomD carrier equation.

Component ID/version	Source	Variant	Status	Scalar reference	Torch	JAX	Maintained check
hm.phase.ringdown.eq9-c1.v1	London Eq. (9); Khan Eq. (14), Sec. V.A	supplied alpha/QNM, D-carrier C1	executable, conditional	carrier_delta_c1	_production_eq14_delta_c1	_production_eq14_delta_c1	test_imrphenomhm_ringdown_phase_math_reference.py

Implementation mapping

The independent 70-digit mpmath evaluator is src/diffgw_reference/imrphenomhm_ringdown_phase.py; it imports no frontend, fit, or QNM code. The maintained test calls the live Torch and JAX _production_eq14_delta_c1 helpers, each called by public HM D-carrier assembly. All parameters are supplied, so passing cannot establish their physical or fit provenance.

Evidence and limitations

The len validation receipt records passing checks of the literal paper limit, sampled values on both sides of the join and through the arctangent center, frequency and \(\alpha_4\) gradients, C1 value/slope, and a 22-mode control. Unequal HM/D 22 QNM fixtures expose the native adaptation rather than silently identifying conventions. The checks do not validate PhenomD \(\alpha\) surfaces, remnant/QNM calculations, mode-dependent QNM choice, full phase gauge, mapped-phase attachment, complete modes, or NR/LAL accuracy.

Reproduce the checks

With reference, Torch, JAX, and test extras installed, run JAX_ENABLE_X64=1 JAX_PLATFORMS=cpu python -m pytest -q tests/test_imrphenomhm_ringdown_phase_math_reference.py --reference-device=cpu -ra. Strict mathematical-reference CI discovers this file. A changed equation or convention requires a versioned source/fixture review.