IMRPhenomXHM non-mixing ringdown phase and C1 join
Scope and status
xhm.phase.ringdown.nonmix-c1.v1 specifies the published ringdown phase
shape and value/slope attachment for the (2,1), (3,3), and (4,4)
intrinsic modes. It takes the phase coefficients, QNM frequencies, join
frequency, and intermediate phase and slope at the join as inputs. The fitted
coefficient surfaces, Eq. (45) rescaling, intermediate fit targets, physical
mode phase gauge, and complete waveform remain separate qualifications.
Before native comparison, the fixed CPU float64 gate uses the three cases and
grids below. Phase and shift tolerances are 5e-13 relative with 2e-11
absolute floor. Derivative and parameter-gradient tolerances are 2e-11
relative with 2e-9 absolute floor. Join value/slope residuals are at most
2e-11 and 2e-9, respectively. The scalar derivative identity must agree
with 70-digit numerical differentiation to 1e-55 absolute.
Inputs, conventions, and options
All frequencies are dimensionless Mf. Require 0 < j <= f <= 0.3,
r>0, and d>0, where j is the ringdown join, r the mode ring
frequency, and d its damping frequency. Supplied real coefficients are
a2, a_lambda; supplied boundary data are p=phi_int(j) and
g=phi_int'(j). The additive phase gauge is fixed only relative to p.
| Mode role | j |
r |
d |
a2 |
a_lambda |
p |
g |
f grid |
|---|---|---|---|---|---|---|---|---|
| 33 | .08 | .095 | .015 | .28 | .13 | 2.7 | 26 | .08, .09, .095, .13, .20 |
| 44 | .10 | .12 | .018 | -.12 | .08 | -1.4 | 18.5 | .10, .11, .12, .16, .25 |
| 21 | .063 | .075 | .012 | .4 | -.07 | .8 | 16.2 | .063, .07, .075, .105, .15 |
The scalar evaluator uses 70-digit mpmath. The native gate uses a three-row CPU float64 batch, Torch autograd, and JAX JIT/grad.
Mathematics and components
García-Quirós et al., arXiv:2001.10914v2, Sec. VI.2.1, Eqs. (43)--(44) and the paragraph following Eq. (45) give, for non-mixing modes,
The paper requires a smooth connection to the intermediate reconstruction;
for its value p and slope g at j, the unique shifts are
Hence phi_RD(j)=p and phi'_RD(j)=g. An independent second derivative is
-2a2 r²/f³ - 2a_lambda d(f-r)/[d²+(f-r)²]². These derivative identities
provide stronger checks than phase-value agreement alone.
Implementation mapping
The scalar reference is
diffgw_reference.imrphenomxhm_nonmix_ringdown_phase. Production Torch
and JAX imrphenom_xhm.py modules call their own
_nonmix_ringdown_phase_shifts and _nonmix_ringdown_phase helpers in
the (2,1), (3,3), and (4,4) paths. Neither frontend imports the
reference.
Evidence and limitations
The gate checks scalar antiderivatives, both C1 conditions, native shifts,
batched phases and frequency/parameter gradients for all three mode roles.
It proves the printed phase shape and join algebra for supplied inputs, not
the fitted inputs or agreement with numerical relativity. The (3,2)
spheroidal phase and mixed spherical reconstruction are outside this slice.
The validation receipt
records the fixed cases and len run.