IMRPhenomHM–SXS mode conventions

Scope and status

hm.conventions.sxs-bridge.v1 is a standalone, source-derived convention bridge for the six positive-frequency IMRPhenomHM carriers and the SXS strain modes. Its SymPy/mpmath reference fixes Fourier signs, the negative-\(m\) pair, orbital/time transformations, and a phase statistic that separates gauges from mode-relative discrepancies. The finite-\(f_\mathrm{ref}\) public HM adapter is checked conditionally against both frontends. This is a convention and implementation gate, not a fit to NR or a physical qualification of the six-mode waveform.

This v1 bridge retains the historical hm_reference_phase="legacy" adapter. The separately versioned orbital reference-phase extension omits its additional finite-reference factor. Existing bridge evidence is not silently reassigned to that new option.

Inputs, conventions, and options

Use positive \(f\) in Hz and \(m>0\). SXS defines \(h=h_+-ih_\times=\sum_{\ell m}h_{\ell m}\,{}_{-2}Y_{\ell m}\). For a Fourier sign \(s=\pm1\), define \(\mathcal F_s[x](f)=\int x(t)e^{s2\pi ift}\,dt\). The SXS holdout uses \(s=-1\) and the local positive carrier \(P_{\ell m}(f)=\mathcal F_-[\overline{h_{\ell m}}](f)\). The sampled transform checks also cover \(s=+1\) to expose an accidental sign assumption. The public HM comparison covers \((2,2),(2,1),(3,3),(3,2),(4,4),(4,3)\), the asymmetric aligned-spin binary \((35,20)M_\odot\), \((\chi_1,\chi_2)=(0.2,-0.1)\), \(\phi_c=0.37\), \(t_c=0.008\,\)s, and \(f_\mathrm{ref}\in\{\mathrm{None},40\,\mathrm{Hz}\}\). Only the CPU-FP64 public check evaluates frontend values.

Mathematics and components

García-Quirós et al., Sec. II Eqs. (2), (4)–(9) give \(h_{\ell,-m}(t)=(-1)^\ell\overline{h_{\ell m}(t)}\) and the negative-\(m\) positive-frequency carrier \(N_{\ell m}=\mathcal F_-[h_{\ell,-m}]\). Therefore

\[ N_{\ell m}(f)=(-1)^\ell\overline{\mathcal F_-[h_{\ell m}](-f)}, \qquad P_{\ell m}(f)=(-1)^\ell N_{\ell m}(f). \]

The frequency reversal in the first equality is essential. The second identity maps the paper's mode pair to the SXS holdout target, independent of an empirical phase alignment. London et al., Eq. (1) and the 2019 SXS catalog, Eq. (52) and its sign discussion fix the strain expansion and the catalog's current \(h_+-ih_\times\) sign; the third catalog describes the retroactive convention update. A common strain sign \(c=-1\) multiplies every \(P_{\ell m}\) by \(c\); it does not select an arbitrary per-mode phase.

With \(\mathcal F_-\), a delay \(t\mapsto t-t_c\) multiplies \(P_{\ell m}\) by \(e^{-2\pi ift_c}\). García-Quirós et al., Eq. (14) give the physical rotation \(h_{\ell m}\mapsto e^{im\delta}h_{\ell m}\), so \(P_{\ell m}\mapsto e^{-im\delta}P_{\ell m}\). The native public phase \(\phi_c\) acts as \(e^{im\phi_c}\) on \(P\), hence its rotation angle in that equation is \(\delta=-\phi_c\).

The public HM adapter also supplies \(B=(d\Psi_D/d(Mf))_\mathrm{peak}M_\mathrm{seconds}f_\mathrm{ref}\) and \(e^{i(m/2-1)B}\), with \(B=0\) when \(f_\mathrm{ref}=\mathrm{None}\). This \(B\) term documents the existing native adapter, not a factor deduced from SXS. The scalar reference takes \(B\) as an input and never reimplements the calibrated D slope.

For nonzero carriers, let \(u_{\ell m}=P_{\ell m}/|P_{\ell m}|\) and \(T_{\ell m}(f)=u_{\ell m}(f)^2\overline{u_{22}(2f/m)}^{\,m}\). The orbital rotation and delay cancel exactly; the finite-reference adapter rotates \(T\) by \(e^{i(m-2)B}\). A common strain factor \(c\) rotates it by \(c^{2-m}\), giving \((-1)^m\) for a catalog sign flip. This statistic does not erase frequency-dependent mode-shape errors.

Component ID/version	Source and rule	Reference	Public targets	Maintained check
`hm.conventions.sxs-bridge.v1`	London Eq. (1); García-Quirós Sec. II Eqs. (2), (4)–(9), (14); SXS 2019 Eq. (52): pair, Fourier, rotation and strain sign	`imrphenomhm_sxs_conventions`	Torch/JAX HM positive carriers through the finite-reference adapter	`test_imrphenomhm_sxs_conventions_math_reference.py`; `test_imrphenomhm_intrinsic_modes_math_reference.py`

Implementation mapping

src/diffgw_reference/imrphenomhm_sxs_conventions.py contains the symbolic gauge expression, scalar phase factor, mode-pair conversion, and doubled relative invariant. It imports neither Torch nor JAX. The sampled chirp test computes both Fourier integrals directly, including \(f\leftrightarrow-f\). The public six-mode test composes the independent HM scalar reference with frontend-supplied D amplitude/phase, remnant/QNM inputs, and the D peak slope, then compares full complex values with Torch and JAX. The native implementations do not call the reference. The paper's mode-pair convention, native positive carrier, and SXS time-domain modes remain separately named.

Evidence and limitations

The run receipt records the exact checks. The algebra and synthetic chirp test establish the stated identities for either Fourier sign; the frontend test establishes conditional agreement for two transition selections and the sampled input. Neither evaluates an independent PhenomD slope or HM calibration fit.

The diagnostic phase \(-e^{i\pi/4}\) seen in the earlier 3631 receipt is not introduced as a production correction. A quarter-cycle stationary phase term appears in García-Quirós et al., Eqs. (22)–(24), but that later model does not prove a constant HM-to-SXS conversion for the current implementation. The frozen 2265, 3631, and 3630 projection failures remain red. A future physical qualification needs a separately predeclared NR gate after any source-justified model change.

The full-carrier reference-frequency audit includes the D carrier's own phase subtraction. After an orbital compensation, it leaves the common analytic phase \(e^{-iB}\), generally not a physical polarization rotation. Removing the explicit HM-relative factor restores orbital covariance in the diagnostic while preserving \((2,2)\); this is not yet a production change or a physical qualification result.

Reproduce the checks

With SymPy, mpmath, NumPy, Torch, JAX and CPU FP64 available, run JAX_ENABLE_X64=1 JAX_PLATFORMS=cpu python -m pytest -q tests/test_imrphenomhm_sxs_conventions_math_reference.py tests/test_imrphenomhm_intrinsic_modes_math_reference.py --reference-device=cpu -ra. Both files are discovered by strict math-reference CI. A changed Fourier sign, phase gauge, or mode-pair convention requires a new component version, new source review, and refreshed tests and receipt; the NR protocols are versioned separately.