IMRPhenomHM six-mode projection component

Scope and status

hm.projection.london-eq1.v1 is a bounded, research-only projection gate for the six aligned-spin IMRPhenomHM modes. It checks the angular conversion of public positive-frequency complex modes into complex plus/cross polarizations. The intrinsic mode carrier, QNM inputs, LAL agreement, and physical accuracy are outside this component.

Inputs, conventions, and options

The modes are (2,2), (2,1), (3,3), (3,2), (4,4), (4,3) at positive Fourier frequency. Inclination is in [0,π]; native phi0 is an observer azimuth, phic is the orbital phase, and tc is in seconds. The frontends use the positive-frequency time factor exp(-2πiftc). This gate uses FP64 and the default PhenomD carrier. It treats each public return_unprojected_modes value as an input, so no intrinsic mode formula is assumed.

London et al., arXiv:1708.00404v4, Eq. (1) write the time-domain strain as h = h₊ − i h× = Σ hₗₘ {}₋₂Yₗₘ. For aligned, nonprecessing binaries the equatorial-reflection relation is hₗ,₋ₘ(t)=(-1)^ℓ h*ₗₘ(t). London Eq. (1) alone does not specify a Fourier kernel. García-Quirós et al., arXiv:2001.10914v2, Sec. II, Eqs. (2), (4)–(9) explicitly use exp(-2πift): at positive f, their carrier is Nₗₘ(f)=FT[hₗ,₋ₘ](f). The existing HM public carrier has the equivalent positive-m adapter Pₗₘ=(-1)^ℓ Nₗₘ, with the HM azimuth argument opposite the physical azimuth used in those equations. This identifies the convention being checked; the gate does not infer it from London's Eq. (1) alone.

Mathematics and components

Construct the harmonics independently from the Wigner small-d matrix:

Yₗₘ(ι,φ) = {}₋₂Yₗₘ = (-1)^m sqrt((2ℓ+1)/(4π)) d^ℓ_{m,2}(ι) exp(imφ).

The normalization gives Y₂₂=sqrt(5/(64π))(1+cos ι)² exp(2iφ). Write p=(-1)^ℓ and evaluate both harmonics at the native phi0. The coefficients multiplying each public unprojected mode are

D₊ = (Yₗₘ + p Y*ₗ,₋ₘ)/2, D× = -i(Yₗₘ - p Y*ₗ,₋ₘ)/2.

Thus h₊,ₗₘ=D₊ Pₗₘ and h×,ₗₘ=D× Pₗₘ; the full polarizations are the sum over the requested modes. Equivalently, García-Quirós's negative-m carrier formula at physical azimuth −phi0 gives these expressions after Nₗₘ=p Pₗₘ. This equality fixes both the odd-ℓ sign and the cross sign. The mode's orbital phase is exp(im phic) and the common time shift is exp(-2πiftc). At face-on only the two m=2 modes, (2,2) and (3,2), survive in this six-mode set; edge-on and generic angles exercise both members of each pair.

Implementation mapping

The independent SymPy Wigner-d evaluator lives in src/diffgw_reference/imrphenomhm_projection.py. The public Torch and JAX IMRPhenomHM implementations return unprojected modes and polarizations; their private harmonic helpers are not inputs to the reference gate. tests/test_imrphenomhm_projection_math_reference.py compares each mode and the six-mode sum at source angles and nonzero phase/time shifts.

Evidence and limitations

The focused CPU FP64 gate covers six per-mode complex plus/cross contributions and their sum at face-on, edge-on, and two generic inclinations with nonzero azimuth, orbital phase, and time shift. It separately checks the m=2 pole selection, odd-ℓ orientation, azimuth/phase/time covariance, body-exchange (-1)^m parity, and the equal-mass/equal-spin odd-mode zero. The reference builds all harmonics with SymPy Wigner matrices and evaluates them with 50-digit mpmath; it imports no Torch/JAX projection helper. The focused len validation passed 13/13 strict math-reference cases on both frontends, with no skips.

This component is conditional on the public positive-frequency carrier adapter above. It checks source harmonic normalization, the six angular coefficients, and their assembly with live carriers. It cannot qualify the carrier amplitude, phase, fit inputs, negative-frequency reconstruction, detector response, or a complete waveform. No whole-model verification claim is implied by passing this gate.

The separate signed-pair reference retains the opposite-frequency contribution and derives real-polarization reconstruction and conjugate phase transport. It extends this conditional projection contract without changing its version or promoting its intrinsic-carrier claims.

An independent public-binary polarization holdout compares complete plus/cross from both frontends with pinned LALSuite at three unseen generic orientations and phase/time settings. That sampled binary gate does not promote this source-projection component to a whole-model claim.

Reproduce the checks

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