IMRPhenomT dominant ringdown amplitude

Scope and status

T-RD-22-AMP-v1 is a source-level, supplied-input component for the aligned-spin (2,2) ringdown amplitude. Its domain is peak-relative time u=(t-t_peak22)/M >= 0. The only native behavior in scope is the amplitude returned by the public Torch and JAX IMRPhenomT.time_domain carriers and its local time derivative. This is not a whole-waveform or calibration claim.

Inputs, conventions, and options

H_peak=H_22(0)>0 is the dimensionless positive-mode peak amplitude; alpha_1>0 and alpha_2>alpha_1 are QNM damping rates in inverse total-mass time; and real d_3 is the fitted shape input. The reference uses the printed paper branch d_2=(alpha_2-alpha_1)/2. The native carriers accept physical solar masses and seconds, use the same dimensionless internal u, and multiply the mode by their existing distance/inclination projection. Neither this specification nor its tests derives the remnant, QNMs, peak fit, or d_3 fit.

Both carriers also bound d_2 for some parameters. When that branch is active, a separate carrier-input check may supply the actual d_2 to the same ansatz, but its result is not Eq. (27d) fidelity. The test must identify which branch each sampled binary takes. Native defaults and this bound are unchanged.

Mathematics and components

Estellés et al., Time domain phenomenological model of gravitational wave subdominant harmonics for quasi-circular non-precessing binary black hole coalescences, arXiv:2012.11923v1, Sec. II C, Eqs. (26), (27c)–(27e) gives, at the (2,2) peak u=0,

\[ \begin{aligned} d_2&=(\alpha_2-\alpha_1)/2,\\ d_1&=H_{\rm peak}\,\alpha_1\cosh^2(d_3)/d_2,\\ d_4&=H_{\rm peak}-d_1\tanh(d_3),\\ H_{22}^{\rm RD}(u)&=e^{-\alpha_1u} [d_1\tanh(d_2u+d_3)+d_4]. \end{aligned} \]

Direct differentiation gives

\[ H'_{22}(u)=e^{-\alpha_1u}\left[ d_1d_2\operatorname{sech}^2(d_2u+d_3) -\alpha_1\{d_1\tanh(d_2u+d_3)+d_4\}\right]. \]

The algebra requires H(0)=H_peak and H'(0)=0; these are independent reference identities. A local stationary point alone does not establish a global maximum or accuracy against numerical relativity.

Component ID	Source	Scalar reference	Native carrier	Gate
T-RD-22-AMP-v1	THM Eqs. (26), (27c)–(27e)	diffgw_reference.phenomt_ringdown_amplitude	Torch/JAX IMRPhenomT.time_domain via _phenomt_amplitude	test_phenomt_ringdown_amplitude_math_reference.py

Implementation mapping

The independent scalar reference uses mpmath and imports no native waveform, QNM, or fit code. The native state provides the QNM, peak, and shape inputs. A face-on public waveform exposes the positive mode amplitude as hypot(h_plus,h_cross)/(scale*sqrt(5/(4*pi))), where scale=(M*MSUN_SEC)/(distance*MPC_SI/C_SI). The gate differentiates this public modulus with respect to u through t=u*M*MSUN_SEC (equivalently, its seconds derivative times M*MSUN_SEC). The arbitrary carrier phase and reference-frequency phase offset cancel in this modulus.

Evidence and limitations

The maintained check enforces high-precision peak/derivative identities on independent scalar inputs, then compares the public Torch and JAX FP64 CPU amplitudes and derivatives at nonnegative ringdown times. Paper-branch cases separately check d_2=(alpha_2-alpha_1)/2; a bounded d_2 case is a carrier-algebra diagnostic only. The execution receipt records the len CPU result. The QNM/remnant, peak, and shape fits, inspiral/merger joins, other modes, physical waveform accuracy, and whole IMRPhenomT qualification remain outside this evidence.

Reproduce the checks

On len CPU, run JAX_ENABLE_X64=1 JAX_PLATFORMS=cpu PYTHONPATH=src:tests python -m pytest -q tests/test_phenomt_ringdown_amplitude_math_reference.py --reference-device=cpu -ra. The linked execution receipt records the environment, sample coverage, and results.