IMRPhenomHM leading (3,3) amplitude ratio
Scope and status
The component hm.amp.beta33.leading.eq4.v1 covers the leading restricted stationary-phase (3,3) rescaling in London et al., arXiv:1708.00404v4, Eqs. (3)–(4), using the independently printed SPA terms in Mishra et al., arXiv:1601.05588v1, Eqs. (11), (12a), (12c). It does not claim the complete HM amplitude, the other modes, or the PhenomD carrier.
Inputs, conventions, and options
Let \(x=M f_{\rm physical}>0\) be the (3,3) Fourier frequency, \(g(x)>0\) the supplied (2,2) amplitude-map frequency in the same geometric units, and \(\delta=(m_1-m_2)/(m_1+m_2)>0\). The mass ordering excludes the equal-mass zero and isolates the paper's \(|\beta|\) without relying on an odd-mode sign convention. Spins may vary: London retains only the leading (3,3) term, so this component contains no explicit spin factor. QNM frequencies, the PhenomD amplitude \(A_{22}\), distance, and phase are supplied by the public assembly, not fitted in this reference. The public hm_mapping_transitions choice changes \(g\) through its amplitude cutoff; both named choices are tested. The historical default is unchanged.
The map's joins are \(f_0^A\) and \(f_{33}^{\rm RD}\). The reference checks one-sided frequency slopes, since the map is continuous but generally not differentiable at either join. At \(x\le0\), or \(\delta=0\), the public quotient used by this gate is outside its domain.
Mathematics and components
Mishra's \(V_m(y)=(2\pi y/m)^{1/3}\) and Eq. (11) give a common SPA prefactor times \(V_m^{-7/2}\hat H_{\ell m}\). At London's restricted order,
Substitute the full SPA amplitudes, including \(V_m^{-7/2}\), into London's Eq. (4). The second ratio cancels the harmonic-frequency factor, leaving
For \(x\le f_0^A\), \(g=2x/3\); between \(f_0^A\) and ringdown, \(g'=(f_{22}^{\rm RD}-2f_0^A/3)/(f_{33}^{\rm RD}-f_0^A)\); above ringdown, \(g'=1\). These are London's Eq. (5) amplitude branches. Eq. (3) uses \(A_{33}(x)\simeq|\beta_{33}(x)|A_{22}(g(x))\). The native odd-mode phase adapter is not part of this positive-\(\delta\) amplitude claim.
| Component | Source | Status | Scalar | Torch/JAX | Gate |
|---|---|---|---|---|---|
| hm.amp.beta33.leading.eq4.v1 | London Eqs. (3)–(4); Mishra Eqs. (11), (12a), (12c) | Executable | diffgw_reference.imrphenomhm_amplitude_ratio | Public D-carrier IMRPhenomHM (3,3) mode | test_imrphenomhm_amplitude_ratio_math_reference.py |
Implementation mapping
The scalar reference reduces the published leading SPA expressions with SymPy and evaluates the reduced formula with mpmath. It imports neither frontend nor a fit table. Torch and JAX production call their own _production_hhat inside the Eq. (4) path, then multiply the carrier amplitude at the mapped frequency. The maintained test calls the public IMRPhenomHM (3,3) mode and separately calls the public PhenomD (2,2) carrier at \(g(x)\). Dividing the unprojected mode magnitude by the carrier magnitude removes the otherwise opaque carrier, testing the production-called ratio rather than an unused helper.
The CPU FP64 validation receipt
records the integrated public-path run on len.
Evidence and limitations
Frozen CPU-FP64 gate: \(m_1=35M_\odot\), \(m_2=20M_\odot\), \(\chi_1=0.2\), \(\chi_2=-0.1\); both named outer-map choices; samples below, at, between, and above the two amplitude-map joins. Require relative ratio-value error \(\le 5\times10^{-11}\) with absolute floor \(5\times10^{-13}\). One-sided secant slopes use a geometric-frequency step \(h=10^{-7}\), with relative error \(\le 2\times10^{-5}\) and absolute floor \(2\times10^{-8}\). An exact symbolic reduction and a hand-reduced scalar example check the independent reference. These bounds test algebra and sampled production composition. They do not validate the QNM fit, 22 carrier, phase, projections, spins in other modes, calibration, or full-waveform accuracy. No LAL code or binary output is used.
Reproduce the checks
Run python -m pytest tests/test_imrphenomhm_amplitude_ratio_math_reference.py --reference-device=cpu -ra with the repository's reference, torch, jax, and test dependencies. The strict reference suite discovers this file by name. A changed source equation, truncation, map convention, or tolerance requires a new component version and reviewed evidence.