IMRPhenomHM (3,2) 1.5PN amplitude ratio
Scope and status
hm.amp.beta32.1p5pn.eq4.v1 is the aligned-spin (3,2) amplitude factor of
London et al., arXiv:1708.00404v4, Eqs. (3)–(4) and the paragraph after
Eq. (4). For \(|m|=\ell-1\), London
retains the spin-dependent 1.5PN expression.
Mishra et al., arXiv:1601.05588v1, Eqs. (7), (11), (12a), (12d)
provide its SPA envelope. This paper equation and the numerical gate are
frozen here before implementation. The existing native default uses only
the leading term; a separately named paper option is required to preserve
the pinned binary-compatible default. This component alone does not qualify
the complete HM waveform.
Inputs, conventions, and options
Let \(x=Mf_{32}>0\) and \(g(x)>0\) be the supplied (2,2) amplitude-map
frequency. The masses obey \(m_1\ge m_2>0\), with
\(\eta=m_1m_2/(m_1+m_2)^2\in(0,1/4]\). Aligned spins obey
\(|\chi_i|\le1\) and \(\chi_s=(\chi_1+\chi_2)/2\). Frequencies are
dimensionless geometric \(Mf\). The public amplitude map has the existing
phenomd_runtime and london_2018 cutoff choices; the printed PN option
will be hm_32_pn_variant="mishra_eq12d", with
"binary_compatible" as the unchanged default. Both paths use the D
carrier. This component compares magnitudes, as London Eq. (3) prescribes.
Mathematics and components
Set \(v_y=V_2(y)=(\pi y)^{1/3}\) and \(P_{32}(v)=(1-3\eta)v^2+4\eta\chi_s v^3\). Mishra Eq. (12d), truncated as London specifies, is \(\hat H_{32}(y)=-(1/3)\sqrt{5/7}\,P_{32}(v_y)\); Eqs. (11) and (12a) give the complete SPA envelopes \(\mathcal H_{32}(y)=V_2(y)^{-7/2}\hat H_{32}(y)\) and \(\mathcal H_{22}(y)=-V_2(y)^{-7/2}\) at the retained order. Since \(m=2\), London's Eq. (4) second quotient is \(\mathcal H_{32}(x)/\mathcal H_{32}(2x/m)=1\) wherever it is defined. The common SPA power cancels in the first quotient, leaving
The leading nonspinning reduction is \((\sqrt{5/7}/3)(1-3\eta)(\pi g)^{2/3}\). For \(P\ne0\), the one-sided derivative at either Eq. (5) map join is \(\operatorname{sgn}(P)(\sqrt{5/7}/3) [2(1-3\eta)v+12\eta\chi_s v^2]v\,g'_{\pm}/(3g)\). At a root of \(P\), the magnitude has a cusp; that point is excluded from the slope gate. Equal masses are valid and generally have nonzero (3,2).
| Component | Source | Scalar | Native production | Gate |
|---|---|---|---|---|
hm.amp.beta32.1p5pn.eq4.v1 |
London Eqs. (3)–(4); Mishra Eqs. (7), (11), (12a), (12d) | diffgw_reference.imrphenomhm_amplitude_ratio |
Public Torch/JAX IMRPhenomHM D-carrier (3,2) with hm_32_pn_variant="mishra_eq12d" |
test_imrphenomhm_amplitude_ratio_math_reference.py |
Implementation mapping
The independent SymPy reference reduces the complete SPA quotient, then
checks the printed \(4\eta\chi_s\) coefficient. An mpmath scalar evaluates
\(P_{32}\), the Eq. (5) composition, and analytic one-sided slopes without
native helpers or fit data. The public gate divides the unprojected (3,2)
mode magnitude by a separately evaluated public PhenomD (2,2) carrier
magnitude at \(g(x)\). It thus exercises the production-called ratio, not the
older public h_hat_multipole helper. Tests check the paper option and exact
default preservation in both frontends.
Evidence and limitations
Use CPU FP64 on len at three predeclared rows:
\((m_1,m_2,\chi_1,\chi_2)=(35,20,0.2,-0.1)\),
\((35,20,0,0)\), and \((30,30,0.2,-0.1)\) (solar masses and dimensionless
spins), distance 100 Mpc, both map choices, and points below, at, between,
and above both Eq. (5) joins. Require public quotient versus mpmath
rtol=1e-10, atol=1e-12; for secants at geometric step \(10^{-7}\),
require analytic one-sided slopes within rtol=5e-5, atol=5e-8.
Capture a pre-change red gate and post-change green gate. Independently
check exact SymPy reduction, the spin coefficient, the nonspinning limit,
and scalar domain. The pinned binary gate separately protects the default.
The public quotient takes the D carrier, amplitude map, and QNM frequencies as supplied inputs. Their correctness, higher PN orders, phase/parity, projection, LAL agreement, physical accuracy, and whole-HM qualification are separate. No LAL source or output enters this equation gate.
Reproduce the checks
On len, with reference, Torch, JAX, and test dependencies installed, 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 -k beta32. The validation receipt records the
exact host, checkout, environment, command, outcomes, and skips. A changed
equation, truncation, map convention, or tolerance requires versioned
review.