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IMRPhenomHM complex intrinsic-mode assembly

Scope and status

hm.intrinsic.complex-assembly.eq3.v1 composes the source-derived HM frequency, amplitude, and phase components into complex positive-frequency modes. The reference is conditional on supplied PhenomD carrier amplitude and phase, ringdown frequencies, and phase-fit inputs. It checks the assembly equation; it does not claim independently calibrated inputs or whole-model accuracy.

The separate published-calibration waveform specification now composes D fits, remnant and QNM evaluations independently before this assembly. Its explicit published/source-PN option tuple has a separate catalog scope. The conditional tests on this page and earlier inherited-default evidence keep their original scope.

Inputs, conventions, and options

The frequency \(x=Mf\) and all QNM frequencies are positive and dimensionless. The six modes are \((2,2)\), \((2,1)\), \((3,3)\), \((3,2)\), \((4,4)\), and \((4,3)\). The paper variant uses London's printed transition frequencies \(f_0^A=0.018 f_{\ell m}^{\rm RD}/f_{22}^{\rm RD}\) and \(f_0^\varphi=0.014 f_{\ell m}^{\rm RD}/f_{22}^{\rm RD}\). The \((2,1)\), \((3,2)\), and \((4,3)\) ratios use the named Mishra-source 1.5PN options; the \((3,3)\) and \((4,4)\) ratios use their leading-frequency terms. The native phenomd_runtime transition setting and binary-compatible PN defaults are separate variants and are not represented as the paper choices.

The carrier callbacks and calibrated inputs are arguments to the scalar reference. A frontend test may obtain these from its public D and QNM interfaces, but that makes the check conditional: shared errors in those inputs are outside its reach. The 22 component reduces directly to the supplied D carrier. The maintained frontend check covers f_ref=None and f_ref=40 Hz; at finite reference frequency it supplies the frontend's D peak slope as the explicit HM-relative gauge input.

Mathematics and components

London et al., arXiv:1708.00404v4, Methods Eqs. (2)–(9) write the intrinsic Fourier mode as \(\tilde h_{\ell m}=A_{\ell m}\exp(i\varphi_{\ell m})\) and prescribe

\[ A_{\ell m}(x)=|\beta_{\ell m}(x)|\,A_{22}(g_A(x)),\qquad \varphi_{\ell m}(x)=\kappa_b C_{\ell m}(g_\varphi(x)) +\Delta_b[C_{\ell m}]. \]

Here \(g_A\) is Eq. (5), \(g_\varphi\) uses the PhenomD-specific Eq. (8) high branch, and \(\beta\) is Eq. (4) with the retained PN envelopes. In the paper composer, the supplied carrier phase is the input to Eqs. (3), (6)–(8); the public adapter first constructs the London Eq. (9)-corrected D phase. The offsets and inverse slopes enforce the piecewise phase relation. Their separate derivations and gates are recorded in the frequency map, amplitude ratios, and phase composition.

The paper formula uses \(|\beta|\). The native positive-frequency carrier stores the same assembly with an \(\exp(-i\varphi)\) Fourier sign, a named odd-\(m\) body-exchange sign, and optional orbital/reference/time gauges. Those operations are an explicit public adapter, not part of London's Eq. (3). García-Quirós et al., arXiv:2001.10914v2, Sec. II Eqs. (2), (4)–(9) fixes the related positive-frequency mode-pair convention. The SXS convention bridge derives that pair, the frequency-reversing conjugation, and the finite-reference invariant. Angular projection is separately specified in the six-mode projection.

Component ID/version Source Scalar reference Public targets Maintained check
hm.intrinsic.complex-assembly.eq3.v1 London Methods Eqs. (2)–(9); Mishra PN inputs imrphenomhm_intrinsic_modes Torch/JAX IMRPhenomHM unprojected modes test_imrphenomhm_intrinsic_modes_math_reference.py

Implementation mapping

src/diffgw_reference/imrphenomhm_intrinsic_modes.py composes independent scalar reference components without importing Torch or JAX. The maintained test compares complex unprojected mode values from both public frontends under explicitly selected options. The separate projection gate compares mode-to-polarization coefficients and summed plus/cross values. Neither reference is a second implementation of D amplitude or the fitted HM QNMs.

Evidence and limitations

This is a source-to-composition and conditional frontend-agreement gate, not an NR accuracy result. The run receipt records the tested domain, tolerances, dependencies, and untested gauges. London's literal Eq. (4) ratio has a \(0/0\) limit for equal-mass equal-spin odd modes; the native zero convention is an adapter and needs a separate check. The inherited PhenomD fits, remnant and QNM calibration, and the project's N61 QNM refit retain their own provenance and domain limits. The subsequent convention-bridge receipt extends the public complex check to finite f_ref; it does not change the original run's scope.

Reproduce the checks

With both frontends and reference dependencies installed, run JAX_ENABLE_X64=1 JAX_PLATFORMS=cpu python -m pytest -q tests/test_imrphenomhm_intrinsic_modes_math_reference.py --reference-device=cpu -ra. Strict math-reference CI discovers this file. Review the exact variant and supplied-input contract before treating a pass as evidence for another option tuple.