HM signed-mode projection and phase transport

Scope and status

imrphenomhm-signed-projection/v1, component hm.conventions.signed-pair-projection.v1, extends the one-sided projection and Gaussian SPA convention reference without replacing either specification. It derives the exact Fourier transform of a signed TD mode pair, including its opposite-frequency contribution. It is a research component, not a production convention or physical qualification claim. Numerical execution status is recorded in the validation receipt.

Inputs, conventions, and options

Use \(h=h_+-ih_\times\), \(\mathcal F_-[h](f)=\int h(t)e^{-2\pi ift}dt\), and \(h_{\ell,-m}=p\overline{h_{\ell m}}\), where \(p=(-1)^\ell\) and \(m>0\). These are García-Quirós et al., arXiv:2001.10914v2, Sec. II.1, Eqs. (2), (4)–(6). The supported pairs have positive labels 22, 21, 33, 32, 44 and 43. Physical observer azimuth is minus native phi0, as in the existing projection adapter. Harmonic normalization and the Wigner construction are inherited from hm.projection.london-eq1.v1.

Define \(P(f)=\mathcal F_-[\overline{h_{\ell m}}](f)\) and \(Q(f)=\mathcal F_-[h_{\ell m}](f)=\overline{P(-f)}\). Both are retained. A finite damped quadratic chirp has nonzero spectral leakage; one-sided support must not be assumed in an exact transform test.

Mathematics and components

Exact signed-pair projection

Evaluate \(Y_\pm={}_{-2}Y_{\ell,\pm m}\) at native phi0, and reuse

\[D_+=\frac{Y_++p\overline{Y_-}}2,\qquad D_\times=-\frac{i}{2}(Y_+-p\overline{Y_-}).\]

At physical azimuth -phi0, the TD strain of the pair is \(H=\overline{Y_+}h_{\ell m}+p\overline{Y_-}\overline{h_{\ell m}}\). Taking \(h_+=\operatorname{Re}H\) and \(h_\times=-\operatorname{Im}H\) before Fourier transformation gives the exact expressions

\[\widetilde h_+=D_+P+\overline{D_+}Q,\qquad \widetilde h_\times=D_\times P+\overline{D_\times}Q.\]

Conjugation at negative frequency follows directly from \(Q(f)=\overline{P(-f)}\). Setting \(Q(f)=0\) at positive frequency recovers the native assembly and the paper's Eqs. (8)–(9), after its \(\widetilde h_{\ell,-m}=pP\) carrier conversion. This reduction verifies the assembly, not the accuracy of discarding leakage in physical data.

Transport of the fixed SPA phase

For a unit phase \(q_m\), a consistent TD transformation is

\[h_{\ell m}'=\overline{q_m}h_{\ell m},\qquad h_{\ell,-m}'=q_mh_{\ell,-m}.\]

It preserves equatorial symmetry and gives \(P'=q_mP\) and \(Q'=\overline{q_m}Q\). Consequently

\[\widetilde h_A'=D_Aq_mP+\overline{D_Aq_m}Q, \qquad A\in\{+,\times\}.\]

The source-derived SXS-basis candidate is \(q_m=\exp[i(m/2-1)K]\), \(K=3\pi/4\), with one common orbital branch. It leaves both \(m=2\) modes unchanged. Applying \(q_m\) to both \(P\) and \(Q\) would generally violate real-polarization conjugate symmetry. The SPA derivation fixes this candidate from the leading PN coefficient phases and the common 22 phase alignment; it is not fitted to NR residuals.

Why this is not generally a polarization rotation

The identity \(q_m=e^{-iK}e^{imK/2}\) separates an orbital shift from a common analytic carrier phase. In the one-sided approximation, the latter multiplies both Fourier polarizations by the same complex scalar. A physical basis change instead mixes the two real polarizations through a real rotation matrix. At edge-on, a 22 signal has \(\widetilde h_\times=0\): a nonreal common Fourier factor cannot be reproduced by a real matrix while keeping the cross channel zero. Circular polarization at a pole hides this distinction. This counterexample concerns the common factor in the decomposition; the complete \(q_2\) equals one.

Equivalently, after removing the orbital shift, the TD transformation has opposite phases for opposite signs of \(m\). It is not a uniform rotation of the complex strain. The conversion therefore defines a distinct carrier convention; the improved primitive phase residual alone cannot identify it as a harmless observer-basis change or a complete waveform repair.

Implementation mapping

src/diffgw_reference/imrphenomhm_signed_projection.py composes the independent Wigner projection and Gaussian transform references. The maintained checks are in tests/test_imrphenomhm_signed_projection_math_reference.py. Native Torch/JAX waveform evaluation is unchanged. Their existing test_imrphenomhm_projection_math_reference.py gate separately checks the one-sided coefficients against public per-mode and complete polarizations.

Evidence and limitations

The checks cover exact Fourier conjugacy, TD signed-pair symmetry, direct quadrature of the real TD polarizations, all six modes, nonzero azimuth, both hemispheres, and conjugate phase transport. A controlled Gaussian sequence tests suppression of the omitted term. This is evidence about the specified analytic chirps and projection algebra; it does not establish a bound on NR leakage or waveform accuracy.

This component alone does not qualify a production waveform. The subsequent native convention extension applies its phase factor under an explicit option and separately checks the complete composition; full-family qualification remains open. The remaining frequency-dependent phase and amplitude errors in the seen-NR diagnostic remain separate from this convention question.

Reproduce the checks

On len, with the optional reference and both frontend dependencies installed:

PYTHONPATH=src JAX_ENABLE_X64=1 JAX_PLATFORMS=cpu python -m pytest -q \
  tests/test_imrphenomhm_signed_projection_math_reference.py \
  tests/test_imrphenomhm_spa_conventions_math_reference.py \
  tests/test_imrphenomhm_projection_math_reference.py

The new filename joins the strict mathematical reference inventory automatically.