IMRPhenomXHM positive-frequency mode projection

Scope and status

This is a paper-equation gate for the complete five-mode projection of aligned-spin XHM's positive-frequency carriers. It specifies the angular coefficients and the common time/orbital phase factor, treating each intrinsic complex carrier as an input. It does not qualify those carriers, their calibration fits, the precessing XPHM rotation, or a full LAL waveform. Component ID: xhm.positive-frequency.mode-pair-projection.v1.

Inputs, conventions, and options

García-Quirós et al., arXiv:2001.10914v2, Sec. II, Eqs. (2), (4)–(9), defines the Fourier transform with exp(-2πift), the equatorial-reflection symmetry h_{ℓm}(t)=(-1)^ℓ h*_{ℓ,-m}(t), and the positive-frequency carrier H_{ℓm}(f)=\tilde h_{ℓ,-m}(f) for m>0.

The native H_{ℓm} also carries exp[-i(2πft_c-mφ_c)]; the observer azimuth enters the harmonics as exp(±imφ_0). The supported mode set is (2,2), (2,1), (3,3), (3,2), (4,4). The scalar reference constructs the spin-−2 harmonics from Wigner d matrices, normalized by {}_{-2}Y_{22}=sqrt(5/(64π))(1+cos ι)^2 exp(2iφ_0).

Mathematics and components

The paper's two polarization contributions are

\[ \tilde h_{+,\ell m} = \frac{1}{2} \left({}_{-2}Y_{\ell,-m} + (-1)^\ell {}_{-2}Y^*_{\ell m}\right) H_{\ell m}, \qquad \tilde h_{\times,\ell m} = \frac{i}{2} \left({}_{-2}Y_{\ell,-m} - (-1)^\ell {}_{-2}Y^*_{\ell m}\right) H_{\ell m}. \]

Implementation mapping

imrphenomxhm_projection.py implements the two source coefficients without importing either frontend. The shared HM spin-weighted-harmonic helper instead returns coefficients for the positive-m carrier. XHM evaluates that helper at −φ_0; the source's negative-m carrier requires the additional factor (-1)^ℓ in both plus and cross. Thus the existing even-ℓ coefficients stay unchanged, while the (3,3) and (3,2) coefficients change sign in Torch and JAX. The shared HM helpers are unchanged, so HM projections are unaffected. XPHM's scalar aligned-spin fast path delegates to XHM and inherits this correction; its precessing twist-up requests unprojected XHM carriers, which this change leaves untouched.

Evidence and limitations

test_xhm_projection_math_reference.py checks each complex plus/cross contribution and their sum at three viewing angles and three positive frequencies using each frontend's unprojected carrier as the independent projection input. It also checks the source's quadrupole normalization and the common time and mode-dependent orbital phase factor in Torch and JAX FP64. The test does not compare carrier amplitudes or phases to LAL, and it cannot certify the XHM full waveform or XPHM rotation. The frozen XHM/XPHM release gates retain their strict tolerances and failing status pending separate work.

Reproduce the checks

With SymPy, Torch, JAX, and pytest installed, run PYTHONPATH=src JAX_ENABLE_X64=1 python -m pytest -q tests/test_xhm_projection_math_reference.py.