IMRPhenomNSBH low-tidal ringdown width

Scope and status

Specification nsbh.amplitude.low-lambda-width/v1 freezes the \(0\leq\Lambda\leq1\) branch of Thompson et al. (2020), Eq. (29), with Eq. (28). It is a conditional component gate with supplied PhenomC \(\delta_2\), final-state ringdown frequency \(f_{\rm RD}\), and quality factor \(Q\). It does not qualify the \(\Lambda>1\) branch, the Eq. (26) premerger correction, the Lorentzian carrier, or the complete NSBH waveform.

Inputs, conventions, and options

\(\Lambda\) is dimensionless and belongs to the closed interval \([0,1]\); the equality case selects this branch. The supplied \(\delta_2\geq10^{-4}\) is dimensionless, consistent with the paper's stated positivity guard. The supplied \(f_{\rm RD}>0\) is in Hz and \(Q>0\) is dimensionless. This gate does not evaluate their upstream fits or alter production input guards.

Mathematics and components

The paper prints \(b_0=0.81248\) and

\[ \delta'_2=\delta_2-2(\delta_2-b_0)\Lambda +(\delta_2-b_0)\Lambda^2, \qquad \sigma=\delta'_2\frac{f_{\rm RD}}{Q}. \]

The equivalent form \(\delta'_2=b_0+(\delta_2-b_0)(1-\Lambda)^2\) establishes \(\delta'_2(0)=\delta_2\), \(\delta'_2(1)=b_0\), positivity, and zero first \(\Lambda\) derivative at the branch endpoint. For independently supplied inputs, the exact partials are

\[ \partial_\Lambda\delta'_2=2(\delta_2-b_0)(\Lambda-1),\quad \partial_{\delta_2}\delta'_2=(1-\Lambda)^2,\quad \partial_\Lambda\sigma=\frac{f_{\rm RD}}Q \partial_\Lambda\delta'_2,\quad \partial_{\delta_2}\sigma=\frac{f_{\rm RD}}Q(1-\Lambda)^2,\quad \partial_{f_{\rm RD}}\sigma=\frac{\delta'_2}Q,\quad \partial_Q\sigma=-\frac{\delta'_2f_{\rm RD}}{Q^2}. \]

These derivatives are conditional partials, not derivatives through the physical remnant or PhenomC providers.

Implementation mapping

diffgw_reference.nsbh_low_lambda_width will evaluate both printed expressions independently in high-precision scalar arithmetic. A production-called delta2_prime_small_lambda helper in each native backend will replace only the inline low-\(\Lambda\) polynomial; the existing Eq. (28) width multiplication and high-\(\Lambda\) selection remain unchanged. Tests will compose the helper with Eq. (28) and compare values and automatic derivatives with the scalar reference. No LAL source or binary-derived coefficients enter the reference.

Evidence and limitations

The grid and limits are frozen before numerical native comparison: \(\delta_2\in\{10^{-4},0.5,0.81248,2\}\), \(\Lambda\in\{0,2^{-20},1/8,1/2,7/8,1-2^{-20},1\}\), \(f_{\rm RD}\in\{100,600\}\) Hz, and \(Q\in\{2,8\}\). This includes the BBH limit, both sides of the low-branch interior, and the closed endpoint; it deliberately excludes \(\Lambda>1\). Independent 70- and 90-digit scalar values and partials must differ by at most \(10^{-30}\) absolutely. Torch/JAX CPU FP64 values must agree with the 70-digit scalar to \(2\times10^{-13}\) absolute for \(\delta'_2\) and \(\sigma\), and \(2\times10^{-12}\) absolute for the six partials listed above. Check batched shape and the \(\Lambda=0,1\) identities. These limits must not be retuned after comparison.

The variant audit records that the printed \(\Lambda>1\) Eq. (29) window sign differs from the native compatibility variant; this gate makes no claim about that branch. The Eq. (26) polynomial/baseline ambiguity, low/no-torus joining center, upstream remnant extrapolation, full amplitude, and whole waveform remain separate qualification gates.

Reproduce the checks

On len CPU with FP64 enabled, run JAX_ENABLE_X64=1 PYTHONPATH=src python -m pytest -q tests/test_nsbh_low_lambda_width_math_reference.py. Record package versions, numerical maxima, neighboring tests, and any environment limitation in a validation receipt before claiming a pass.