NRTidalv2 nonspinning tidal amplitude and merger taper
Scope and status
Specification nrtidalv2-nonspinning-tidal-amplitude-taper/v1 covers
the signed additive Fourier tidal amplitude as an independent Eq. (24)
component, plus the published merger-frequency fit and descending Planck
taper for the dominant mode, zero component spins, CPU FP64, the four frozen
binaries below, and 100 Mpc distance. Dietrich et al. (2019), Table II
assigns the amplitude correction to IMRPhenomPv2_NRTidalv2 but not
IMRPhenomD_NRTidalv2. The explicit D amplitude_variant="paper_2019"
therefore uses tidal phase and taper without Eq. (24); its self-spin
composition follows the printed 2PN+3PN D scope. The default
binary_compatible variant preserves the historical native amplitude
normalization and passes the frozen pinned-LAL catalog and holdout cases, but that normalization
has unresolved paper provenance. The scalar Eq. (24) gate passes separately;
the paper-defined D option differs from the pinned LAL D binary at nonzero tide.
Inputs, conventions, and options
With \(M_{\rm sec}=GM/c^3\), \(D_{\rm sec}=D_L/c\), \(\eta=m_1m_2/M^2\), \(x=(\pi M_{\rm sec}f)^{2/3}\) and \(\kappa_{\rm eff}^{T}=3\widetilde\Lambda/16\), the frozen systems \((m_1,m_2;\Lambda_1,\Lambda_2)\) in solar masses are \((1.35,1.35;400,400)\), \((1.6,1.2;200,800)\), the latter with both bodies exchanged, and \((1.35,1.35;0,0)\). Inclination is 0.4 rad. Amplitude frequencies are 100, 300, 800 and 1200 Hz. Taper frequencies are \(f/f_{\rm mrg}=\) 0.9, 1, 1.025, 1.05, 1.1, 1.15, 1.195, 1.2 and 1.25. The zero-tide case probes the taper independently of Eq. (24).
Mathematics and components
Dietrich et al. (2019), Eqs. (24)–(25) give the amplitude and its multiplication by a Planck taper in the amplitude-carrying Pv2 variant. Table II and the surrounding text specify phase-only tidal extension for D and SEOBNRv4_ROM. Dietrich et al. (2018), Eq. (11) give the merger-frequency fit, substituting \(\kappa_{\rm eff}^{T}\) for \(\kappa_2^{T}\) as the paper states for its LAL implementation. The mass-weighted \(\widetilde\Lambda\) follows Wade et al. (2014), Eq. (5). The signed face-on amplitude is
The square-root normalization is part of Eq. (24). It also follows
independently from the leading term of Dietrich et al. (2019), Eq. (23):
\((27/16)\sqrt{160\pi\eta/27}=9\sqrt{5\pi\eta/24}\), while
\(x^{-7/4}x^5=x^{13/4}\). Thus Eqs. (23) and (24) have the same
low-frequency normalization. The physical Newtonian IMRPhenomD carrier is
\(\sqrt{5\eta/24}\,M_{\rm sec}^{5/6}/(\pi^{2/3}D_{\rm sec}f^{7/6})\)
(Cutler–Flanagan, Eqs. (62)–(63)).
In the same \(x\) variable this is exactly
\(\sqrt{5\pi\eta/24}\,M_{\rm sec}^2 x^{-7/4}/D_{\rm sec}\).
The earlier Kawaguchi et al. (2018), Eq. (2.25)
already has this \(\sqrt{5\pi\eta/24}\) factor. Its full additive amplitude
becomes the numerator of Dietrich Eq. (24) after the latter's explicitly stated
\(D_{\rm eff}\to D_L\) substitution and \(\kappa_{\rm eff}^T=3\widetilde\Lambda/16\);
the independent reference checks this identity symbolically. The first
Dietrich 2019 preprint, Table II
also designates D_NRTidalv2 as phase-only, so the D/Pv2 distinction is not
a later revision of the paper.
The published additive correction therefore has the same physical amplitude
scale as the carrier; the historical binary-compatible native
\(M_{\rm sec}^2/(2D_{\rm sec})\) scale is not Eq. (24) and has unresolved
paper provenance. Its ratio to the paper scale is
\([2\sqrt{5\pi\eta/24}]^{-1}\): 1.236077446 at \(\eta=1/4\) and
1.248886881 at \(\eta=12/49\). A fixed inclination or polarization
projection cannot produce this mass-ratio dependence when inclination is
held fixed. Equation (24) specifies the
absolute additive tidal amplitude \(\widetilde A_T\), rather than a
dimensionless multiplier or the full BNS amplitude. Equation (25) adds it
once to \(\widetilde A_{\rm BBH}\) before tapering in the amplitude-carrying
variant. The paper_2019 D option instead has \(\widetilde A_{\rm BBH}W\).
The decimal fit inputs are used exactly as printed. For \(X_A\) the heavier mass fraction,
\(X_B\) the lighter, the quoted merger fit is
Equation (25) gives \((\widetilde A_{\rm BBH}+\widetilde A_T)W(f)\) for Pv2. The descending Planck window is 1 through \(f_{\rm mrg}\), 0 from \(1.2f_{\rm mrg}\) onward, and for \(z=(f-f_{\rm mrg})/(0.2f_{\rm mrg})\in(0,1)\) is \(W=[1+\exp(-1/z-1/(z-1))]^{-1}\). Inclined \(h_+\) applies the positive projection \((1+\cos^2\iota)/2\) to \(\widetilde A_T\).
Implementation mapping
The independent scalar reference src/diffgw_reference/nrtidalv2_amplitude.py
uses SymPy/mpmath and no native waveform or LAL implementation. Torch/JAX
retain Eq. (24) scalar component functions for direct checks; neither D
variant calls them. The default uses the separately named historical
M^2/(2D) correction, while paper_2019 omits tidal amplitude. The tests
compare each contract with its appropriate independent math or pinned binary.
Evidence and limitations
The paper-math tests in tests/test_nrtidalv2_amplitude_math_reference.py
require merger-frequency relative error \(\le10^{-11}\) and scalar/window
amplitude error \(\le10^{-10}\). The now-green native amplitude checks in
tests/test_nrtidalv2_amplitude_math_reference_native.py
retain the frozen Torch/JAX scalar amplitude relative limit \(10^{-11}\).
The untapered paper_2019 D plus-polarization amplitude ratio must be unity
within the existing absolute limit \(10^{-10}\).
The pinned-binary tests in tests/test_nrtidalv2_amplitude_lal_binary.py
compare the unaligned modulus ratios
\(|h_+^{\rm BNS}|/|h_+^{\rm BBH}|\) with the paper-D taper result using each
backend's own BBH amplitude, then compare the default native and pinned LAL D ratios. Maximum
absolute ratio error must be \(\le5\times10^{-5}\) before and within the
taper, including its endpoints. LALSuite 7.26.15 is required under
DIFFGW_REQUIRE_LAL_BINARY=1. These limits are not retuned on failure.
Disposition: Torch/JAX agree with the independent Eq. (24) scalar
reference while the published D option omits that component and passes its
untapered unity-ratio test. The pinned LALSuite 7.26.15 D binary has a
nonzero-tide amplitude quotient, contrary to Table II and the later
Colleoni et al. (2025), Sec. II A
description of the older D variant. The native merger fit and Planck window
pass the scalar gate. The default binary-compatible variant retains the
historical \(1/2\) normalization and passes the frozen catalog and holdout
cases; its provenance remains unresolved. Its dedicated high-frequency
taper-ratio tests retain strict expected failures for the approximately
25 ppm boundary discrepancy. Comparisons of the paper_2019 option with
the pinned LAL D binary also remain strict expected failures at nonzero tide.
Public binary quotients imply an approximately 25 ppm lower effective
taper boundary for the nonzero-tide cases if their untapered amplitudes
continue to agree inside the window. This is a diagnostic inference, not
an identification of the pinned binary's internal merger calculation.
The separate premerger complex-composition holdout
passes for the default binary_compatible D variant on three new
nonspinning systems through 1200 Hz. It jointly checks the tidal phase and
default amplitude against the pinned public binary and checks the phase
against the independent paper reference. The result does not resolve the
default amplitude's paper provenance or the failed nonzero-tide taper gate.
This gate tests only literature transcription and named-binary agreement for this slice. It does not validate the NR calibration, self-spin, nonzero spins, other modes, other distances/inclinations, FP32/GPU or full-family behavior.
Reproduce the checks
Run PYTHONPATH=src:tests DIFFGW_REQUIRE_LAL_BINARY=1 JAX_ENABLE_X64=1
python -m pytest -q tests/test_nrtidalv2_amplitude_math_reference.py
tests/test_nrtidalv2_amplitude_math_reference_native.py
tests/test_nrtidalv2_amplitude_lal_binary.py on CPU FP64 with the pinned
binary installed. Exact measurements and commands are in the
26 September 2026 receipt.