IMRPhenomT dominant merger: supplied-input specification v1

Scope and status

This specification implements the updated IMRPhenomTHM paper, Eqs. (28)–(32). It applies to the shared aligned (2,2) carrier in T, THM, TP, and TPHM. The earlier T/TP paper uses different collocation nodes and merger ansätze and is not substituted here.

The executable scalar source is phenomt_merger.py, version phenomt-merger/v1. It imports no production carrier, fit, or amplitude module. Its numerical solves use mpmath; an optional SymPy entry point exposes the exact symbolic ansätze and their derivatives for audit.

Inputs, conventions, and options

In total-mass time u, let x=asinh(alpha u), where alpha=2 pi f_damp and omega_RD=2 pi f_ring are in inverse total-mass units. The paper's merger ansatz is

Mathematics and components

Frequency and phase

omega(u)=omega_RD [1-sum_{k=0}^4 a_k x^k].

The five supplied conditions are omega at the inspiral cut, an interior point, and the peak u=0, plus d omega/du at the cut and peak. The scalar source solves all five coefficients at arbitrary precision. Its analytic derivative is the derivative of that basis; its phase difference is the high-precision quadrature of omega, with no native phase primitive.

For the current dominant carrier, u_cut=-26.982976386771437/eta and u_mid=-5/(eta*0.95^8). The former is an author-carrier join specialization; the latter is the actual dominant interior node, distinct from the paper's generic -25 M description. The peak slope is supplied by the joined ringdown branch. Neither QNM rate nor fitted interior/peak target is inferred from a waveform comparison.

Amplitude

With tau=u-u_peak, the paper's amplitude basis is

H(u)=b0+b1 tau^2+b2 sech(2 alpha tau)^(1/7)+b3 sech(alpha tau).

The four conditions are signed H at the inspiral cut -150 M, interior -25 M, and peak 0, plus dH/du at -150 M. The scalar source solves the four coefficients at arbitrary precision and provides the analytic slope. The native strain uses |H|; a signed zero can therefore make the observed amplitude nondifferentiable and produce a mode-phase jump. The checked finite cases compare magnitudes after solving the signed system.

Implementation mapping

The PN cutoff value and slope, fitted midpoint and peak, and damping rate are supplied inputs. The cutoff slope is computed from the existing PN amplitude and frequency derivatives, so the gate does not independently validate their PN coefficients. The current native amplitude stores merger coefficients in [b0,b3,b2,b1] order; the reference follows the paper's [b0,b1,b2,b3] order.

Evidence and limitations

test_phenomt_merger_math_reference.py checks the symbolic forms, scalar collocation identities, and derivatives, then compares coefficients, frequency, slope, integrated phase, and amplitude at off-node times in Torch and JAX FP64. It covers three mass/spin cases, including mass ratio 10. The len receipt records the execution. This is independent equation and join evidence with supplied inputs; it is not a reconstruction of the calibrated target surfaces, QNM/remnant fits, higher-mode merger, or whole-waveform accuracy.

The pinned MIT author phenomxpy asset remains the source for unprinted calibrated targets. THM Sec. III identifies the fit form and training sources but does not print all target values or surface coefficients. Berti's Kerr QNM data and Jiménez-Forteza's remnant fits offer separate primary lineage checks; they do not replace the particular one-dimensional fits used in this carrier. The existing provenance ledger and THIRD_PARTY_NOTICES.md retain the MIT adaptation attribution.

Reproduce the checks

The len receipt records the environment and commands for the scalar and native checks.