IMRPhenomXHM (3,2) transition and mixing reference

Scope and status

This is a paper-equation reference for the (3,2) intermediate phase, ringdown join, and spherical–spheroidal projection. It does not provide the calibration fits, QNM frequencies, or complex mixing-coefficient tables needed to reproduce a particular LALSuite release. Its component IDs are xhm.phase.32.intermediate-ringdown-join.v1 and xhm.mixing.32.spherical-spheroidal.v1.

For the intermediate-phase gate, the acceptance thresholds were fixed before native comparison: Table-7 node absolute error 2e-14; six coefficients relative error 5e-7 with absolute floor 1e-10; phase and first derivative absolute error 2e-7; second derivative at the join absolute error 2e-5; and target-gradient relative error 2e-4 with absolute floor 1e-9.

Inputs, conventions, and options

Frequency is dimensionless Mf; a complex mode is written h = A exp(-i φ). The four intermediate nodes and phase-derivative targets, join frequency, QNM ring/damping frequencies, and complex mixing coefficients are caller inputs. The mixed mode must be nonzero where its phase derivative is evaluated. No release-specific fit or mixing option is selected. The default four intermediate nodes additionally take 0<eta<=1/4, the (3,2) inspiral phase cut, and the (2,2) ring/damping frequencies. Native checks use CPU float64 and the scalar reference uses 70 decimal digits.

Mathematics and components

The source is García-Quirós et al., arXiv:2001.10914v2, Eqs. (38), (40), (46)–(48), Table 7, and (A1)–(A3). Some native comments and the earlier version of this document used section-style equation labels; these numbers refer to the pinned v2 paper.

Equation (38) gives

\[ \phi'_{32,\mathrm{int}}(f)= a_\lambda\frac{f_\mathrm{damp}}{f_\mathrm{damp}^{2}+(f-f_\mathrm{ring})^{2}} +\sum_{k=0}^{4}a_k f^{-k}. \]

The six coefficients solve a linear system: four prescribed derivative values at intermediate collocation frequencies, then the spherical mixed ringdown phase's first and second derivatives at the join frequency. A final additive phase offset matches its value there. These conditions make the phase, first derivative, and second derivative continuous at the join; the intermediate coefficients are not themselves calibrated parameters.

For the (3,2) default Table-7 nodes, let s=[1+0.001(0.25/eta-1)] f_ins^32 and e=f_ring^22-0.5 f_damp^22. The four collocation frequencies are s, [(sqrt(3)+2)s-(sqrt(3)-2)e]/4, (e+3s)/4, and (e+s)/2. The ringdown join f_RD^32 is a separate input. The sixth equation is the frequency derivative of Eq. (38), evaluated at that join; the Lorentzian derivative is -2 d (f-r)/[d²+(f-r)²]² and each inverse-power term contributes -k a_k/f^(k+1).

The spheroidal (3,2) ringdown phase primitive and derivative are transcribed from Eqs. (46)–(47), including the arctangent primitive of the Lorentzian. The scalar reference checks the derivative analytically. The four ringdown collocation values in Eq. (48) remain caller data.

Appendix A gives the two-mode basis map. Eliminating the spheroidal (2,2) mode yields

\[ h_{32}=\frac{\alpha_{322}}{\alpha_{222}}h_{22} +\left(\alpha_{332}-\frac{\alpha_{322}\alpha_{232}}{\alpha_{222}}\right)h^S_{32}. \]

Component ID	Source	Scalar reference	Torch/JAX
`xhm.phase.32.intermediate-ringdown-join.v1`	Eqs. (38), (40), (46)–(48), Table 7	`default_intermediate_32_nodes`, `intermediate_phase_coefficients`, `intermediate_phase`	`_intermediate_32_phase_nodes`, `_intermediate_32_phase_coefficients`, phase/derivative evaluators
`xhm.mixing.32.spherical-spheroidal.v1`	App. A, Eqs. (A1)–(A3)	`spherical_32_from_spheroidal`, `mixed_phase_derivatives`	`_spherical_32_from_spheroidal`

Implementation mapping

The independent scalar implementation is imrphenomxhm_transition_mixing.py. It takes all four complex α values as inputs and evaluates the mixed phase derivatives by logarithmic differentiation. Both production imrphenom_xhm.py frontends route their (3,2) collocation solve through the named helper. The real/imaginary Cartesian Torch mixing path takes a real h32_s amplitude.

Evidence and limitations

test_xhm_transition_mixing_math_reference.py checks the projection identity, phase derivatives, four intermediate collocation values, value/first/second-derivative join, and the same solve and projection in Torch and JAX. The production (3,2) branch uses named helpers for the tested collocation solve. The complementary test_xhm_32_intermediate_phase_math_reference.py gates the printed nodes, coefficients, integrated phase and derivative, C2 join, Torch/JAX batching, JAX JIT, and target gradients against independent scalar central differences. Test inputs are synthetic and do not assert release-calibration accuracy.

The frozen XHM/XPHM release qualification still fails its catalog and holdout waveform gates. The release-122022 calibration surfaces and mixing tables have not been proven identical to the paper-era companion assets; this component must not be described as full waveform qualification.

Reproduce the checks

On len, with the project on PYTHONPATH, run python -m pytest -q tests/test_xhm_transition_mixing_math_reference.py. The Eq. (40) completion also runs with JAX_ENABLE_X64=1 PYTHONPATH=src:tests python -m pytest -q tests/test_xhm_32_intermediate_phase_math_reference.py.