IMRPhenomT dominant ringdown frequency and phase

Scope and status

T-RD-22-FREQ-PHASE-C4ZERO-v1 covers the aligned-spin (2,2) ringdown angular frequency, its phase increment, and frequency slope for supplied peak frequency, QNM frequency, and shape parameters. It checks the actual Torch and JAX carrier evaluation on three binary states. The status is qualified for the peak-matched sign branch, with the printed source inconsistency below retained as an explicit qualification boundary. Neither the parameter-space calibration nor the full waveform is qualified here.

Inputs, conventions, and options

Time u=(t-t_peak22)/M is dimensionless and u>=0; the (2,2) amplitude peak is u=0. The supplied inputs are dimensionless angular frequencies omega_peak=omega_22(0) and omega_RD, a positive inverse-time shape parameter c2, and positive dimensionless c3. The checked states have omega_RD>omega_peak; this is the increasing-from-below branch. The paper sets c4=0 and treats c2 and c3 as calibration outputs. Native final_spin overrides change omega_RD but are outside this gate.

Mathematics and components

Estellés et al., IMRPhenomTHM, arXiv:2012.11923v1, Sec. II.3, Eqs. (24)–(27), and Sec. II.4, Eqs. (28), (31b), (31e) defines the ringdown ansatz and peak matching. The printed equations have an internal sign inconsistency: Eq. (25) writes omega−omega_RD=+c1*c2*c3*exp(−c2*u)/(1+c3*exp(−c2*u)) when c4=0, while Eq. (27a) sets c1=(1+c3)*(omega_RD−omega_peak)/(c2*c3). Inserting u=0 into the printed Eq. (25) gives omega(0)=2*omega_RD−omega_peak, not omega_peak. Eqs. (28), (31b) require the ringdown and merger frequencies to agree at the peak. Thus this specification infers the minus sign in Eq. (25) while preserving Eq. (27a) and the stated join:

\[ c_1=\frac{(1+c_3)(\omega_{\rm RD}-\omega_{\rm peak})}{c_2c_3},\qquad E(u)=e^{-c_2u},\qquad \omega(u)=\omega_{\rm RD}-\frac{c_1c_2c_3 E(u)}{1+c_3E(u)}. \]

Integrating d phi/du=omega with phi(0)=0, and differentiating the frequency, gives

\[ \Delta\phi(u)=\omega_{\rm RD}u+c_1\log\!\frac{1+c_3E(u)}{1+c_3},\qquad \omega'(u)=\frac{c_1c_2^2c_3E(u)}{(1+c_3E(u))^2}. \]

These formulas yield omega(0)=omega_peak, Delta phi(0)=0, positive slope, and omega(u)→omega_RD as u→∞. The sign choice is an algebraic resolution of contradictory printed statements, not a claim that the paper literally prints this branch or that an erratum establishes it. One could instead reverse Eq. (27a)'s c1 sign and retain Eq. (25)'s plus; that gives the same observable omega(u) but a different coefficient convention. The source does not establish which printed line carries the typo, so the check explicitly adopts the native Eq. (27a) convention.

Component ID	Source	Reference	Native production path	Check
T-RD-22-FREQ-PHASE-C4ZERO-v1	THM Eqs. (25), (27a), (28), (31b,e), with documented sign inference	diffgw_reference/phenomt_ringdown_frequency.py	both IMRPhenomT._coeff and _evaluate; _domega_full	test_phenomt_ringdown_frequency_math_reference.py

Implementation mapping

The scalar mpmath reference imports no native implementation or fit. The gate extracts supplied omega_peak, c2, c3, and omega_RD from each frontend's production _coeff state; checks its peak-derived c1; then compares production-called _evaluate frequency and phase increments and _domega_full slope at six ringdown times. It subtracts the native constant phase at u=0, so only the covered phase increment is compared. The native QNM and calibration providers are input providers in this test, not independent source evidence for their coefficients.

Evidence and limitations

The scalar reference separately checks the peak identity, its analytic phase against a high-precision integral, and slope against numerical differentiation. The frontend gate uses FP64 CPU and tolerances rtol=3e-12, atol=1e-13 for frequency, slope and c1, and atol=2e-12 for phase. The execution receipt records the tested versions and result. This confirms only the supplied input ringdown algebra on the three declared states. It does not establish the printed Eq. (25) literally, QNM/remnant or c2,c3,omega_peak fit provenance, native override behavior, amplitude, merger/inspiral joins, other modes, or physical agreement with NR.

Reproduce the checks

On len, run JAX_ENABLE_X64=1 JAX_PLATFORMS=cpu PYTHONPATH=src:tests python -m pytest -q tests/test_phenomt_ringdown_frequency_math_reference.py --reference-device=cpu -ra. The strict math-reference CI gate should collect the file without skips.