IMRPhenomXHM ringdown amplitude shape

Scope and status

xhm.amplitude.ringdown.ansatz.v1 specifies only the dimensionless ringdown amplitude ratio in García-Quirós et al., Eq. (42). Its fitted coefficients, QNM frequencies, normalization to physical strain, interval selection, intermediate attachment, and spherical/spheroidal (3,2) mixing are outside this component. This is not a full-mode qualification.

Before native comparison, the fixed CPU float64 gate uses the two cases and frequencies below. Relative value tolerance is 5e-13 with absolute floor 2e-12; frequency and parameter derivative tolerance is 2e-11 relative with absolute floor 2e-10. Scalar analytical derivatives must agree with 70-digit numerical differentiation to 1e-55 absolute.

Inputs, conventions, and options

Use dimensionless positive Fourier frequency f=M f_GW, positive ring frequency r, damping frequency d, and sigma>0. The supplied real parameters are a_lambda != 0 and lambda. The (3,2) inputs refer to the spheroidal ringdown basis. The reference and native gate use:

Case	Mode role	r	d	sigma	a_lambda	lambda	f grid
A	33, fixed Table 8 sigma	.09	.016	1.3	.015	.4	.06, .08, .09, .11, .15
B	21, supplied sigma	.08	.012	.91	-.023	.65	.05, .07, .08, .10, .14

The chosen interior grid has no active native numerical clamp. Each case is also evaluated as a batch item. The scalar reference uses 70-digit mpmath; native checks use Torch/JAX CPU float64 and JAX JIT.

Mathematics and components

García-Quirós et al., arXiv:2001.10914v2, Sec. VI.1, Eq. (42) and Table 8 specify, with w=d sigma,

\[ H(f)=\frac{A_{RD}^{\ell m}}{A_0^{22}} = f^{-1/12}\frac{|a_\lambda|w}{(f-r)^2+w^2} \exp\!\left[-\frac{(f-r)\lambda}{w}\right]. \]

Table 8 fixes sigma=1.3 for (3,3) and 1.33 for (3,2) and (4,4); the (2,1) sigma remains fitted. All are caller-supplied here. An independent derivative identity used by the gate is

\[ \partial_f H=H\left[-\frac{1}{12f} -\frac{2(f-r)}{(f-r)^2+w^2}-\frac{\lambda}{w}\right],\quad \partial_\lambda H=-\frac{f-r}{w}H,\quad \partial_{a_\lambda}H=H/a_\lambda. \]

Implementation mapping

The scalar implementation lives in diffgw_reference.imrphenomxhm_ringdown_amplitude. The production eval_h_rd closures in the Torch and JAX imrphenom_xhm.py modules call their own _ringdown_amplitude_ratio helpers. Native positive-frequency, exponential, and denominator guards remain for numerical robustness; this component checks the published expression only where those guards are idle.

Evidence and limitations

The gate compares 70-digit scalar values with native batched values and analytical frequency/coefficient gradients, including negative a_lambda, and verifies JAX compilation. This establishes the functional shape for supplied parameters. It does not establish the provenance or quality of a_lambda, lambda, the (2,1) sigma, Kerr QNM values, physical amplitude normalization, or the full XHM ringdown and attachment.

Reproduce the checks

The component receipt records the frozen cases and measured results.

JAX_ENABLE_X64=1 PYTHONPATH=src:tests python -m pytest -q \
  tests/test_xhm_ringdown_amplitude_math_reference.py