IMRPhenomHM public intrinsic phase composition

Scope and status

hm.phase.public-composition.eq3-14.v1 checks how the public D-carrier HM generator composes previously checked frequency-map, phase-offset, and ringdown-correction components. This is a bounded executable research component for the five non-(2,2) positive-frequency modes. It is conditional on supplied PhenomD phase, fit coefficients, and HM/D QNM frequencies; it is not a full intrinsic-mode or waveform qualification.

Inputs, conventions, and options

The input is dimensionless positive \(x=Mf\). The tested modes are \((2,1)\), \((3,3)\), \((3,2)\), \((4,4)\), and \((4,3)\). The carrier callback supplies the public PhenomD phase \(P_D(y)\) with \(f_{\rm ref}=\varnothing\), \(\phi_c=t_c=0\). The tested systems have \(m_1>m_2\), so the native signed odd-\(m\) amplitude factor is positive and the unprojected public mode has unit phasor \(\exp[-i\Phi_{\ell m}(x)]\). Other phase origins and amplitude sign branches are separate gates.

Both public outer-map choices are frozen: phenomd_runtime places the amplitude/phase cutoffs at \(0.014/0.018\) times \(f_{\ell m}^{\rm RD}/f_{22}^{\rm RD}\); london_2018 uses the paper's \(0.018/0.014\) pair. These are named options, not interchangeable gauges. The D carrier, \(\eta\), \(\alpha_4\), \(\alpha_5\), HM QNM pairs and D QNM pair are supplied from each frontend. The scalar composer neither imports nor derives those functions or fits.

Mathematics and components

London et al., arXiv:1708.00404v4, Methods Eqs. (3), (5)–(9) map the quadrupole phase through a three-branch frequency map, its inverse slopes \(\kappa_b\), and continuity offsets \(\Delta_b\). Their Eq. (9) changes the merger-ringdown arctangent of Khan et al., arXiv:1508.07253v2, Eq. (14). The existing frequency-map, phase-offset, and ringdown-phase specifications derive the three ingredients separately. This composition fixes their order:

\[ C_{\ell m}(y)=P_D(y)+\delta_{\ell m}^{C^1}(y),\qquad \Phi_{\ell m}(x)=\kappa_b C_{\ell m}(g_b(x)) +\Delta_b[C_{\ell m}]. \]

Here \(g_b\) is London's PhenomD-specific Eq. (8) high branch with the Eq. (5) lower branches, and the offsets are evaluated from the corrected carrier at both joins. The native \(\delta^{C^1}\) uses the D-anchored QNM adaptation and tangent subtraction documented in the ringdown specification; the literal printed London–Khan term is recovered only when their 22 QNM pairs coincide. London's Eq. (7) gives phase offsets; its modified merger-ringdown arctangent is Eq. (9). The stable component ID's eq3-14 suffix records the inherited PhenomD Eq. (14), not a London Eq. (14).

The reciprocal map slopes imply \(C^1\) phase across \(x=f_0^\phi\) and \(x=f_{\ell m}^{\rm RD}\) when \(C_{\ell m}\) is differentiable. The separate D attachment \(J=f_{22}^{\rm RD,D}/2\) is sampled after mapping back to the mode frequency, to check the full public composition at that third join.

Component ID/version	Sources	Variant	Status	Scalar reference	Torch/JAX public target	Maintained check
hm.phase.public-composition.eq3-14.v1	London Methods Eqs. (3), (5)–(9); Khan Eq. (14)	D carrier; both named outer-map settings	executable, conditional	imrphenomhm_phase_composition.composed_phase	IMRPhenomHM(..., return_unprojected_modes=True)	test_imrphenomhm_phase_composition_math_reference.py

Implementation mapping

src/diffgw_reference/imrphenomhm_phase_composition.py composes the two independent mpmath references; it imports no native frontend. The maintained test obtains supplied D phase values at all needed mapped frequencies and join points, then compares the resulting unit phasor with public src/torchwave/approximants/imrphenom_hm.py and src/jaxwave/approximants/imrphenom_hm.py unprojected modes. This reaches the actual public assembly instead of comparing another helper against itself. A nonzero mode amplitude is required to define its phase.

Evidence and limitations

The len CPU-FP64 receipt records values and one-sided slopes across all three joins for two unequal-mass aligned-spin systems, all five non-22 modes, both map settings, and both frontends. It fixes the public phase sign and the zero reference/time gauge. The check does not independently qualify D phase fits, \(\alpha\) surfaces, remnant/QNM fits, amplitude sign/zero behavior, \(f_{\rm ref}\) or time/orbital phase gauges, negative frequencies, projection, whole-mode binary agreement, or NR physical accuracy. In particular, a passing composition check cannot promote the whole IMRPhenomHM family.

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_phase_composition_math_reference.py --reference-device=cpu -ra. Strict math-reference CI discovers this file. Changes to the source equation, sign/gauge adapter, or named map option require a versioned review.