IMRPhenomXHM non-mixing intermediate phase

Scope and status

xhm.phase.intermediate.nonmix-collocation.v1 is a research-only component for the (2,1), (3,3), and (4,4) intrinsic mode phases. It specifies the printed default collocation geometry and the five-coefficient phase-derivative solve for supplied fitted targets. It does not certify the target fit surfaces, inspiral/ringdown attachment, (3,2) mode, or a complete waveform.

The acceptance thresholds below were fixed before comparing the native frontends: default-node absolute error at most 2e-14, coefficient relative error at most 5e-8 (with absolute floor 1e-10), phase/derivative absolute error at most 2e-8 on the declared dimensionless-frequency fixtures, and autodiff against independent central differences to 2e-5 relative.

Inputs, conventions, and options

Inputs are dimensionless Fourier frequencies Mf > 0, symmetric mass ratio 1e-6 <= eta <= 1/4 for native comparisons, a positive inspiral phase cut f_ins, ring frequency f_ring > f_ins, damping frequency f_damp > 0, and five supplied fitted phase-derivative targets. The default nodes are the paper's [C1,C2,E2,E3,E5] subset. This component leaves the additive phase constant at zero; the full model fixes it at an attachment. The scalar evaluator uses mpmath at 70 decimal digits, while native comparisons use CPU float64. The production node helpers retain the pre-existing eta floor of 1e-6 outside this declared comparison domain; the scalar reference evaluates the printed formula for any positive eta.

Mathematics and components

García-Quirós et al., arXiv:2001.10914v2, Sec. V B, Eqs. (38)--(39), Table 7, and the paragraph following Table 7 give

\[ \phi'_{\ell m}(f)=a_\lambda\frac{d}{d^2+(f-r)^2} +a_0+\frac{a_1}{f}+\frac{a_2}{f^2}+\frac{a_4}{f^4}, \quad (\ell,m)\in\{(2,1),(3,3),(4,4)\}, \]

where r=f_ring, d=f_damp, and a_3=0. The five coefficients solve phi'(f_i)=F_i. With s=beta(eta) f_ins, e=f_ring, and beta=1+0.001(0.25/eta-1), the default nodes are

\[ (f_1,f_2,f_3,f_4,f_6)= \left(s,\frac{(\sqrt3+2)s-(\sqrt3-2)e}{4}, \frac{e+3s}{4},\frac{e+s}{2},\frac{7e+s}{8}\right). \]

Termwise integration, with an unfixed constant, gives a_lambda atan((f-r)/d)+a_0 f+a_1 log(f)-a_2/f-a_4/(3f^3). The independent scalar reference constructs the five linear equations from these expressions and solves them at high precision.

Implementation mapping

The independent reference is diffgw_reference.imrphenomxhm_intermediate_phase. Torch and JAX production call _nonmix_intermediate_phase_nodes, _nonmix_intermediate_phase_coefficients, and the phase and derivative evaluators in their respective imrphenom_xhm.py modules. Neither frontend imports the reference or fitted targets from it.

Evidence and limitations

The gate checks default nodes, coefficient solve, all five interpolation conditions, integrated phase and analytic derivative, native batch values, Torch parameter gradients, and JAX JIT/grad. Targets are deliberately supplied fixtures: agreement proves the printed algebra and native solve for those cases, not the provenance or accuracy of the actual fitted surfaces. The paper's alternate sparse-calibration subset is outside this v1 component. The full XHM and XPHM release gates remain unqualified.

Reproduce the checks

On a float64 Torch/JAX CPU environment:

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