IMRPhenomHM (4,3) 1.5PN amplitude ratio
Scope and status
hm.amp.beta43.1p5pn.eq4.v1 is the restricted (4,3) amplitude factor
from London et al., arXiv:1708.00404v4, Eqs. (3)–(4) and the paragraph
following Eq. (4). London keeps terms
through relative 1.5PN for \(|m|=\ell-1\) modes. Mishra et al.,
arXiv:1601.05588v1, Eqs. (7), (11), (12a), (12g)
print the SPA polynomial used here. This source-first version freezes the
equation and validation gate before native changes. The printed formula is
selected by hm_43_pn_variant="mishra_eq12g"; the
"binary_compatible" default retains the established native carrier.
This is a component claim, not whole-HM qualification.
Inputs, conventions, and options
Let \(x=Mf_{43}>0\) and \(g(x)>0\) be the supplied (2,2) amplitude-map
frequency in geometric units. Require \(m_1\ge m_2>0\),
\(\delta=(m_1-m_2)/(m_1+m_2)\),
\(\eta=m_1m_2/(m_1+m_2)^2=(1-\delta^2)/4\),
\(\chi_s=(\chi_1+\chi_2)/2\), and \(\chi_a=(\chi_1-\chi_2)/2\) with
\(|\chi_i|\le1\). The aligned spins are the projections on the Newtonian
orbital angular momentum in Mishra Eq. (7). London Eq. (3) takes the
magnitude of the complex Eq. (4) ratio. The option requires the public
PhenomD (2,2) carrier. hm_mapping_transitions selects the default
phenomd_runtime or source london_2018 outer-map cutoff.
At \(\delta=\chi_a=0\), the (4,3) SPA polynomial is identically zero and literal Eq. (4) is \(0/0\); the scalar ratio excludes that point. Equal mass with unequal spins is valid. The public zero-mode branch is checked separately. This component neither supplies QNM frequencies nor validates the PhenomD amplitude, phase, distance, or projection.
Mathematics and components
Set \(V_m(y)=(2\pi y/m)^{1/3}\) and
At the retained order Mishra Eqs. (11), (12a), (12g) give \(\mathcal H_{43}(y)=-(3/4)\sqrt{3/35}\,V_3(y)^{-7/2}P_{43}(V_3(y))\) and \(\mathcal H_{22}(y)=-V_2(y)^{-7/2}\). In London Eq. (4), the SPA \(V_m^{-7/2}\) factors cancel between the two ratios, yielding
The nonspinning reduction is \((3/4)\sqrt{3/35}\,\delta(1-2\eta)[\pi g(x)]\); it does not replace the printed spin term. The Eq. (5) amplitude map has \(g'=2/3\) below \(f_0^A\), an affine slope between \(f_0^A\) and \(f_{43}^{\rm RD}\), and \(g'=1\) above ringdown. At a nonzero ratio, \(d|B|/dx=\operatorname{Re}(\overline B B')/|B|\), with the appropriate one-sided map slope at each join.
| Component | Source | Scalar | Native public option | Gate |
|---|---|---|---|---|
hm.amp.beta43.1p5pn.eq4.v1 |
London Eqs. (3)–(5); Mishra Eqs. (7), (11), (12a), (12g) | diffgw_reference.imrphenomhm_amplitude_ratio_43 |
Torch/JAX IMRPhenomHM, D carrier, hm_43_pn_variant="mishra_eq12g" |
test_imrphenomhm_amplitude_ratio_43_math_reference.py |
Implementation mapping
Independent SymPy reduction starts with both complete SPA envelopes and
checks the printed \(v^4\) coefficient. The mpmath scalar evaluates the
unreduced polynomial, the Eq. (4) magnitude, and analytic one-sided Eq. (5)
slopes without native imports or calibration data. The public test removes
the separately evaluated PhenomD (2,2) carrier magnitude at \(g(x)\) from
the unprojected (4,3) mode and compares the remaining factor with mpmath.
It does not substitute the public h_hat_multipole helper for production.
Evidence and limitations
Frozen len CPU FP64 cases are \((m_1,m_2,\chi_1,\chi_2)\) equal to
\((35,20,0.2,-0.1)\), \((35,20,0,0)\), and \((30,30,0.2,-0.1)\), in solar
masses; distance is 100 Mpc. Both map settings sample below, at, between,
and above \(f_0^A\) and \(f_{43}^{\rm RD}\). The public quotient must agree
with mpmath to rtol=1e-10, atol=1e-12. One-sided secants at geometric
step \(10^{-7}\) use rtol=5e-5, atol=5e-8. Check the exact symbolic
reduction, a hand-reduced nonspinning scalar, equal-mass unequal-spin
nonzero behavior, and rejection of the scalar \(0/0\) point. Capture the
pre-change public failure. The default must equal an explicit
binary_compatible request exactly; the paper option must be observable.
These checks establish sampled source and implementation agreement for the (4,3) amplitude factor only. QNM and D-carrier correctness, phase, projection, same-model LAL parity, physical accuracy, and the complete HM waveform remain separate. No LAL code, constants, or binary output enters this gate.
Reproduce the checks
On len with the repository's reference, Torch, JAX, and test
dependencies, run JAX_ENABLE_X64=1 JAX_PLATFORMS=cpu OMP_NUM_THREADS=1
python -m pytest -q tests/test_imrphenomhm_amplitude_ratio_43_math_reference.py
--reference-device=cpu -ra. The validation receipt records the exact
checkout, environment, red/green commands, results, and skips.
A changed source equation, truncation, map convention, or tolerance
requires a new component version and reviewed evidence.