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IMRPhenomXHM (3,2) spheroidal ringdown alignment

Scope and status

xhm.phase.32.spheroidal-alignment.v1 specifies the algebra that fixes the linear time and constant phase offsets of a supplied spheroidal (3,2) ringdown phase relative to a supplied (2,2) carrier. It is conditional on QNM frequencies, shape coefficients, carrier values, and two fitted alignment targets. It does not qualify those fitted surfaces, the Eq. (48) coefficient calibration, exceptional cuts, or the full mode.

Freeze CPU FP64 value tolerances rtol=2e-13, atol=5e-12 and complete-input Jacobian tolerances rtol=2e-12, atol=5e-10 before native comparison. Use 70-digit scalar arithmetic and require both Torch and JAX JIT to pass.

Inputs, conventions, and options

All frequencies are positive dimensionless Mf. The ordered input row is (r22,d22,r32,d32,a2,a4,a_lambda,base_time,carrier_slope,carrier_phase,delta_t,delta_phi). The carrier slope is evaluated at f0=r22+d22; carrier phase at f1=r22. delta_t and delta_phi are supplied fitted offsets, with the signs printed in Eqs. (49)–(50). base_time is the unaligned Eq. (48) linear coefficient; the result must be independent of it.

Row r22,d22,r32,d32,a2,a4,a_lambda,base_time,carrier_slope,carrier_phase,delta_t,delta_phi
1 .083,.013,.091,.014,.20,.0001,.30,8,11,-.4,2,.15
2 .104,.016,.115,.018,.13,-.00008,-.24,4,-7,1.2,-1,-.3
3 .072,.010,.078,.012,-.11,.00004,.41,-3,5,.2,.6,-.1

The arXiv:2001.10914v2 source archive is identified by SHA-256 cc05e7dbff7bf0c42a10b7da1f4840eefc55b252168d6bfb085155a7bdc5318e. Its anc/ParspaceFits/IMRPhenomXHM_PhaseFits.nb (SHA-256 48cc263ab9f2837220cd11fbfd99205562b8863af077074f46d730e35382108b) identifies the reference frequencies, and its PhaseFits/32/deltadPhiS.m and deltaPhiS.m hold the respective fit records (SHA-256 07107d0436b316792e0b2cd343b8cce3efa2a96276a26304f6ecbdfb7c975455 and e92a6657458934bea6bb8466b97c7cd90a45d57d98eff9b546e7d64804fac772). The arXiv license grants arXiv a non-exclusive distribution license; it does not state that the ancillary fit records can be redistributed under this repository's MIT license. No fit coefficients are embedded in this specification or reference.

Mathematics and components

García-Quirós et al., arXiv:2001.10914v2, Eqs. (46)–(50) define the ringdown phase shape and the two relative offsets. The author-published phase-fit notebook fixes f0=r22+d22 for Eq. (49) and f1=r22 for Eq. (50). For an unaligned phase P(f;t,c) from Eq. (47), set

t_final = t_base + [carrier_slope + delta_t - P'(f0;t_base,0)]
c_final = carrier_phase + delta_phi - P(f1;t_final,0).

Then P'(f0;t_final,c_final)-carrier_slope=delta_t and P(f1;t_final,c_final)-carrier_phase=delta_phi. These are algebraic identities for supplied inputs, not evidence for the fitted input values.

The same notebook explicitly defines four Eq. (48) derivative targets at Table 9 nodes r22, r32-1.5*d32, r32-0.5*d32, and r32+0.5*d32. The .m archive contains dphiS_p1.m, dphiS_p4.m, and an alternate dphiS_fring32.m, but no separate p2/p3 .m records. The files 3Dfit_p1_lm_32.m through p4 are intermediate-phase, not Eq. (48), targets. Their names must not be used as ringdown fit provenance.

Implementation mapping

diffgw_reference.imrphenomxhm_transition_mixing holds the independent scalar alignment. Both torchwave and jaxwave approximants/imrphenom_xhm.py call their own alignment helper from the production (3,2) ringdown branch. Neither frontend imports the scalar reference. The current production coefficient calibration uses targets at (r22,r32,r32+0.5*d32) and fixes a2_32=a2_22: it is a three-target alternate calibration, not the paper's four-target Eq. (48) system.

Evidence and limitations

The maintained test compares both final offsets and their complete input Jacobians on the frozen three-row grid, and checks both defining residuals. This qualifies only the alignment algebra for supplied inputs. The author-published notebook supplies paper-era fit expressions, but no redistribution terms establishing MIT compatibility or version map to a specific later XHM release. Do not silently replace the production fit or claim it implements Eq. (48).

The paper's Eq. (36)/Table 2 specify default (3,2) cuts and mention high-mass/high-spin adjustments. Neither the paper text nor this fit notebook gives a complete trigger and replacement frequency for that exception. The amplitude-intermediate veto table is a different rule and cannot fill this gap. The exceptional cut remains unqualified.

Reproduce the checks

JAX_ENABLE_X64=1 PYTHONPATH=src:tests python -m pytest -q \
  tests/test_xhm_32_spheroidal_alignment_math_reference.py

The test needs mpmath, numpy, torch, jax, and pytest; no ancillary fit assets are imported. Changing the paper-era frequencies, alignment convention, or selected calibration requires a new component version.