IMRPhenomNSBH compactness and disruption root

Scope and status

Specification nsbh-compactness-disruption-root/v1 covers only the neutron-star compactness fit, the largest positive relativistic tidal root, and the tidal radius/frequency derived from that root. It does not qualify the NSBH amplitude branches, remnant or ringdown fits, phase gauge, or full waveform. The independent paper gate below is separate from any comparison with a public LALSuite waveform binary.

Inputs, conventions, and options

The black hole is body 1, the nonspinning neutron star is body 2, \(q=m_1/m_2\), \(\chi\) is the aligned black-hole spin, and \(\Lambda\) is the neutron-star dimensionless tidal deformability. CPU FP64 is required for both native backends. The frozen Cartesian gate is \(q\in\{1,2,5,10,15\}\), \(\chi\in\{-0.5,0,0.5\}\), and \(\Lambda\in\{0,1,10,300,800,2600,5000\}\). The compactness and root use all 105 cases. Radius and frequency use the 60 cases with \(\Lambda\ge300\) and \(m_2=1.4M_\odot\). The paper's calibration domain is \(1\le q\le15\), \(|\chi|\le0.5\), and \(0\le\Lambda\le5000\). The frequency gate requires a positive, finite denominator in Eq. (8); all 60 frozen cases must satisfy that precondition.

Mathematics and components

Thompson et al. (2020), Appendix B Eq. (32) define \(C(\Lambda)\) by the cubic below for \(\Lambda\le1\) and the logarithmic fit for \(\Lambda>1\), with \((a_0,a_1,a_2)=(0.360,-0.0355,0.000705)\):

\[ C(\Lambda)=\begin{cases} \tfrac12+(3a_0-a_1-\tfrac32)\Lambda^2+ (a_1-2a_0+1)\Lambda^3,&\Lambda\le1,\\ a_0+a_1\ln\Lambda+a_2(\ln\Lambda)^2,&\Lambda>1. \end{cases} \]

With \(\mu=qC\), Appendix A Eq. (10) defines \(\xi\) as the largest positive real root of

\[ P(\xi)=\xi^5-3\mu\xi^4+2\chi\sqrt{\mu^3\xi^7} -3q\xi^2+6q\mu\xi-3q\mu^2\chi^2=0. \]

The independent scalar reference uses \(y=\sqrt\xi\) and brackets the rightmost crossing of the resulting degree-ten polynomial from a Cauchy upper bound, then bisects at high precision. This does not share the native fixed-iteration Newton solver. Appendix A Eqs. (8)–(9) give \(\widetilde r_{\rm tide}=\xi m_{2,\rm sec}(1-2C)/C\) and \(f_{\rm tide}=[\pi(\chi m_{1,\rm sec}+ \sqrt{\widetilde r_{\rm tide}^3/m_{1,\rm sec}})]^{-1}\). The positive-frequency native adapter is immaterial on the frozen positive-denominator frequency grid.

Implementation mapping

src/diffgw_reference/nsbh_disruption.py evaluates the published scalar expressions without importing a production backend or LALSuite. The Torch and JAX paths are ns_compactness_from_lambda, solve_xi_tide, and nsbh_frequencies in their respective imrphenom_nsbh.py modules. Tests call these components directly, without an IMRPhenomD carrier or assembled NSBH waveform.

Evidence and limitations

The gate requires maximum absolute compactness error \(\le2\times10^{-13}\), maximum relative root error \(\le2\times10^{-11}\), and normalized native polynomial residual \(\le2\times10^{-12}\). The normalization is one plus the sum of absolute polynomial terms at the returned root. On the 60-case frequency subset, maximum relative radius and frequency errors must each be \(\le2\times10^{-10}\). A separate degree-ten polynomial root enumeration checks that the scalar bracket selected the largest positive root. These thresholds and cases were fixed before any public-binary evaluation.

An assembled LAL NSBH waveform does not expose \(C\), \(\xi\), or \(f_{\rm tide}\) as a public observable, so waveform matching cannot serve as an independent oracle for this component. Binary comparison belongs to a separate composition gate after the remnant, amplitude and phase components have their own references. Numerical agreement here does not validate the underlying NR calibration or imply whole-waveform qualification.

Reproduce the checks

Run JAX_ENABLE_X64=1 PYTHONPATH=src pytest -q tests/test_nsbh_disruption_reference.py with the reference, torch, and jax extras on CPU. The measured result and environment are recorded in the 27 September 2026 validation receipt.