IMRPhenomHM (2,1) 1.5PN amplitude ratio
Scope and status
hm.amp.beta21.1p5pn.eq4.v1 is the (2,1) amplitude factor in
London et al., arXiv:1708.00404v4, Eqs. (3)–(4) and the paragraph after
Eq. (4). London retains terms through
relative 1.5PN for \(|m|=\ell-1\) modes. Mishra et al.,
arXiv:1601.05588v1, Eqs. (11), (12a), (12b)
print the corresponding complete SPA polynomial through \(V_1^4\).
This source-first version froze the equation and gate before the scalar,
production correction, and public checks. The sampled CPU FP64 gate passed
on len; see the validation receipt.
The printed equation is available through the public
hm_21_pn_variant="mishra_eq12b" option. The default
hm_21_pn_variant="binary_compatible" preserves the established HM binary
behavior; the source gate does not assert that the printed equation is the
binary model. This is not a whole-HM qualification.
Inputs, conventions, and options
Let \(x=Mf_{21}>0\), \(g(x)>0\) be the supplied (2,2) amplitude-map frequency, \(m_1\ge m_2>0\), \(\delta=(m_1-m_2)/(m_1+m_2)\), and \(\eta=m_1m_2/(m_1+m_2)^2=(1-\delta^2)/4\). Aligned component spins obey \(|\chi_i|\le1\), with \(\chi_s=(\chi_1+\chi_2)/2\) and \(\chi_a=(\chi_1-\chi_2)/2\), as in Mishra Eq. (7). Eq. (3) uses the magnitude of the complex ratio. At \(\delta=\chi_a=0\), the (2,1) polynomial vanishes identically and literal Eq. (4) is \(0/0\); this scalar component excludes that point. Equal masses with unequal spins are valid.
The public hm_mapping_transitions setting chooses phenomd_runtime
(default) or london_2018 outer amplitude-map cutoff. The D carrier,
QNM frequencies, phase, distance, and projection remain supplied by the
public assembly. The mishra_eq12b option requires that D carrier.
The two Eq. (5) joins are \(f_0^A\) and \(f_{21}^{\rm RD}\);
one-sided slopes are used there.
Mathematics and components
Set \(V_m(y)=(2\pi y/m)^{1/3}\) and \(v_y=V_1(y)\). Mishra Eq. (12b) gives \(\hat H_{21}(y)=-(\sqrt2/3)P(v_y)\) through \(v_y^4\), where
Mishra Eqs. (11), (12a) give the full SPA envelopes \(\mathcal H_{21}(y)=-(\sqrt2/3)V_1(y)^{-7/2}P(v_y)\) and \(\mathcal H_{22}(y)=-V_2(y)^{-7/2}\) at the retained order. Their \(V_m^{-7/2}\) factors cancel between the two ratios in London Eq. (4):
The nonspinning leading reduction is \((\sqrt2/3)\delta[\pi g(x)]^{1/3}\), but it is not the prescribed 1.5PN public factor. Eq. (5) supplies \(g'=2\) below \(f_0^A\), \(g'=(f_{22}^{\rm RD}-2f_0^A)/(f_{21}^{\rm RD}-f_0^A)\) between joins, and \(g'=1\) above ringdown. For \(B\ne0\), its one-sided slope is \(\operatorname{Re}(\overline B\,B')/|B|\), using the corresponding \(g'\). At \(\delta=\chi_a=0\), the public mode takes the finite zero-mode branch; the literal scalar Eq. (4) quotient is undefined there. No limiting amplitude ratio independent of the path toward this point is claimed.
| Component | Source | Scalar | Native production | Gate |
|---|---|---|---|---|
hm.amp.beta21.1p5pn.eq4.v1 |
London Eqs. (3)–(4); Mishra Eqs. (11), (12a), (12b) | diffgw_reference.imrphenomhm_amplitude_ratio |
Torch/JAX public IMRPhenomHM D-carrier (2,1) with hm_21_pn_variant="mishra_eq12b" |
test_imrphenomhm_amplitude_ratio_math_reference.py |
Implementation mapping
An independent SymPy reduction must begin with both complete SPA
envelopes, and separately reduce the printed \(v^4\) spin coefficients.
An mpmath scalar evaluates \(P\), \(B\), \(|B|\), the Eq. (5) composition, and
the analytic one-sided magnitude slopes. It imports no native helper or
calibration data. The public test divides the unprojected Torch/JAX (2,1)
mode magnitude with hm_21_pn_variant="mishra_eq12b" by the public PhenomD (2,2) carrier magnitude at the
independently mapped frequency. This checks the production-called ratio,
including its complex polynomial, through Eq. (3). It does not use the
older public h_hat_multipole helper as a substitute for production.
Evidence and limitations
Frozen len CPU FP64 grid: \((m_1,m_2,\chi_1,\chi_2)\) equal to
\((35,20,0.2,-0.1)\), \((35,20,0,0)\), and \((30,30,0.2,-0.1)\), masses in
solar units, distance 100 Mpc, and both outer-map choices. Sample below,
at, between, and above both joins. Require public quotient versus mpmath
relative error \(\le10^{-10}\) with absolute floor \(10^{-12}\). At each join,
compare one-sided secants with \(h=10^{-7}\) in geometric frequency to
analytic slopes, relative error \(\le5\times10^{-5}\) and absolute floor
\(5\times10^{-8}\). Check exact SymPy Eq. (4) reduction, the printed \(v^4\)
spin coefficients, a hand-reduced nonspinning scalar, equal-mass unequal-spin
nonzero behavior, and rejection of the \(0/0\) equal-mass equal-spin point.
Capture the pre-correction public failure before changing production.
The receipt
records that failure, the source-option implementation, and the passing
source/public and neighboring HM checks. The strict binary oracle separately
qualifies the default binary_compatible option.
Passing this gate establishes only the sampled (2,1) 1.5PN magnitude factor. Higher PN orders, independent QNM and D-amplitude correctness, odd-mode phase and parity, projection, LAL agreement, 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_math_reference.py
--reference-device=cpu -ra. The linked receipt records the exact checkout,
environment, command, pre-correction failure, final result, and skips.
A changed source equation, truncation, map convention,
or tolerance requires a new component version and reviewed evidence.