NRTidalv2 Pv2 positive-22 projection boundary

Scope and status

Specification nrtidalv2-pv2-positive-22-projection/v1 covers the positive-frequency projection of a supplied aligned-frame (2,2) mode into complex plus/cross polarizations. It is an executable, conditional component of the Pv2 construction, not an IMRPhenomPv2_NRTidalv2 waveform. The prior aligned-frame gate covers the signed finite-size term. Dietrich et al. (2019), Sec. III.4 places that term into the aligned carrier before the precession rotation.

Inputs, conventions, and options

The input is a complex positive-frequency h^L_{2,-2}(f), with supplied Euler angles alpha, beta, gamma, line-of-sight polar angle theta_JN, and polarization-basis angle zeta, in radians. All are evaluated pointwise on the same grid. The spin-weighted harmonics have azimuth zero. The output is (h_plus,h_cross) in the inverse-rotated polarization basis. This gate uses arbitrary normalized mode values and nondegenerate angles. It does not infer the angle dynamics. The dimensional conversion from Eq. (24)'s absolute tidal amplitude to h^L_{2,-2} is covered by a separate normalization gate.

Mathematics and components

For the supplied mode H, let Y_m = sqrt(5/(4 pi)) d^2_{m,2}(theta_JN) and A_{m,m'} = exp(-i m alpha) d^2_{m,m'}(beta) Y_m. The positive-frequency polarization pair is

\[ p=\tfrac12 H e^{2i\gamma}\sum_{m=-2}^{2} (A_{m,-2}+A^*_{m,2}),\qquad c=\tfrac i2 H e^{2i\gamma}\sum_{m=-2}^{2} (A_{m,-2}-A^*_{m,2}). \]

The returned pair is (cos(2 zeta) p + sin(2 zeta) c, cos(2 zeta) c - sin(2 zeta) p). This is Pratten et al. (2020), Eqs. (11)–(13), (53), and the inverse of Eqs. (93)–(94). The reference evaluates Eq. (53)'s factorial sum at 80 decimal digits; the native implementations evaluate their own tensor expressions. Because this projection is linear in H, an aligned-frame correction D must obey P(H+D)=P(H)+P(D).

Component ID/version	Source	Variant	Status	Scalar reference	Torch	JAX	Check
nrtidalv2-pv2-positive-22-projection/v1	Pratten (2020), Eqs. (11)–(13), (53), (93)–(94)	Supplied positive-frequency (2,-2) mode	Conditional executable	nrtidalv2_pv2_projection.project_positive_22	project_twisted_positive_modes	phenompv2_waveform._project	test_nrtidalv2_pv2_projection_math_reference.py

Implementation mapping

src/diffgw_reference/nrtidalv2_pv2_projection.py has no native imports. The native Torch and JAX projectors are used by their existing Pv2 BBH waveforms. Neither frontend currently provides the tidal Pv2 carrier assembly or its public BNS dispatch. This test calls the projector boundary directly, with no LAL source or calibration.

Evidence and limitations

The frozen cases are three complex H values, three independent complex perturbations D, and nonzero frequency-dependent alpha, beta, gamma, theta_JN, and zeta, as recorded in the maintained test. Torch and JAX must each match both complex polarizations from the scalar reference for H, D, and H+D to 1e-12 after normalization by maximum input-mode magnitude. Each backend must also satisfy the pre-rotation linearity identity to 1e-12. CPU FP64 is required; there is no phase/time alignment, fitting, or output rescaling. The execution receipt records the run.

This operator gate does not establish the source-frame angle map, the IMRPhenomD-to-Pv2 BBH carrier normalization, the native tidal Pv2 assembly, a pinned-binary match, or physical validation. Eq. (24)'s supplied-mode normalization is tested separately in the linked gate. The historical paper-D and binary_compatible variants are unchanged.

Reproduce the checks

With mpmath, Torch, JAX, and pytest available, run CPU FP64:

PYTHONPATH=src:tests JAX_ENABLE_X64=1 python -m pytest -q \
  tests/test_nrtidalv2_pv2_projection_math_reference.py

Changing the sign, mode, basis, or angle convention requires a new spec version and a new independent reference comparison.