NRTidalv2 self-spin in the published D composition

Scope and status

Specification nrtidalv2-self-spin-d/v1 covers the positive-deformability, aligned-spin, dominant-mode Fourier self-spin component in the printed IMRPhenomD_NRTidalv2 composition. It covers CPU FP64 scalar phase components and Torch/JAX complex quotient agreement on the frozen systems below. It is a conditional equation-level gate: the paper's Fig. 7 example is inconsistent with its printed Eqs. (28)–(29), and this gate does not validate the universal fit against neutron-star data or the complete waveform against a binary oracle.

Inputs, conventions, and options

Dietrich et al. (2019), arXiv:1905.06011v2, Eqs. (26)–(29) and Table II are the primary source. \(m_A,m_B>0\) may be passed in either order; \(X_i=m_i/M\), \(M=m_A+m_B\), \(\nu=X_AX_B\), \(x=(\pi M_{\rm sec}f)^{2/3}\), \(f>0\), and aligned dimensionless spins \(\chi_i\) use the native signed convention. The printed Love–Q relation requires \(\Lambda_i>0\). The gate tests \(\Lambda_i\ge180\); zero and tiny values use native clamps outside this paper reference. The paper D composition retains the 2PN and 3PN EOS self-spin terms, but not the Pv2 3.5PN quadrupole tail/cubic-spin terms (Table II). Its tidal amplitude is omitted. The historical binary_compatible default is preserved.

The existing API calls this choice amplitude_variant. For v2, amplitude_variant="paper_2019" now selects the entire printed D finite-size composition: Eq. (28) quadrupole fit, 2PN+3PN self-spin, and no Eq. (24) tidal amplitude. binary_compatible retains the inherited earlier Love–Q fit, 3.5PN self-spin and historical tidal amplitude. This option name predates the self-spin qualification and remains for compatibility.

Mathematics and components

For each body, let \(\widehat C_Q=C_Q-1\) and \(\widehat C_{Oc}=C_{Oc}-1\). Dietrich Eq. (28) prints

\[ \log C_Q=0.1940+0.09163\ell+0.04812\ell^2 -0.004286\ell^3+0.00012450\ell^4,\qquad \ell=\log\Lambda. \]

Equation (29) prints

\[ \log C_{Oc}=0.003131+2.071q-0.7152q^2+0.2458q^3-0.03309q^4, \qquad q=\log C_Q. \]

Equation (27) gives the per-body coefficients

\[ P_{2,i}=-50\widehat C_{Q,i}X_i^2\chi_i^2,\quad P_{3,i}=\frac5{84}(9407+8218X_i-2016X_i^2) \widehat C_{Q,i}X_i^2\chi_i^2, \]

\[ P_{3.5,i}=10\left[(X_i^2+\frac{308}{3}X_i)\chi_i+ (X_j^2-\frac{89}{3}X_j)\chi_j-40\pi\right] \widehat C_{Q,i}X_i^2\chi_i^2 -440\widehat C_{Oc,i}X_i^3\chi_i^3. \]

The independent scalar reference evaluates each order separately with \(K=3/(128\nu x^{5/2})\): \(S_2=Kx^2\sum_iP_{2,i}\), \(S_3=Kx^3\sum_iP_{3,i}\), and \(S_{3.5}=Kx^{7/2}\sum_iP_{3.5,i}\). nrtidalv2.self-spin.eq26-29.printed/v1 comprises these components. The paper D quotient uses \(\exp[-i(\psi_T+S_2+S_3)]\) before taper; \(S_{3.5}\) is retained only as a separate Eq. (27) component for the published Pv2 scope. At phasing_convention="reference_frequency", the native D adapter instead uses \(S_2(f)+S_3(f)-S_2(f_{ref})-S_3(f_{ref})\); this subtraction is a project phase convention, not Eq. (26).

Implementation mapping

The independent mpmath scalar reference is src/diffgw_reference/nrtidalv2_self_spin.py. Its printed fit and separated Eq. (27) terms are compared with src/torchwave/approximants/imrphenomd_nrtidal.py and src/jaxwave/approximants/imrphenomd_nrtidal.py. The maintained gate tests/test_nrtidalv2_self_spin_math_reference.py also compares the full pre-taper complex BNS/BBH quotient using the separate diffgw_reference.nrtidalv2_phase component. Neither native implementation imports the reference. tests/test_nrtidal_reference.py remains a regression for the historical default and is labelled accordingly.

Evidence and limitations

The frozen cases \((m_1,m_2;\chi_1,\chi_2;\Lambda_1,\Lambda_2)\) are \((1.35,1.35;0.20,0.20;350,350)\), \((1.6,1.2;0.04,-0.03;250,800)\), its exchanged copy, and \((1.45,1.25;0,0;300,650)\). Frequencies are 30, 100, 300 and 800 Hz. The absolute component, unaligned complex quotient, and spin-derivative limits are \(10^{-12}\), \(10^{-10}\), and \(10^{-10}\) respectively. The quotient is tested in coalescence and 30 Hz reference-frequency conventions, without fitting. The execution receipt records the actual result. The separate spinning composition gate joins this self-spin phase with the published tidal phase and merger taper in one complex quotient through the taper boundary.

At \(\Lambda=350\), direct evaluation of the printed Eqs. (28)–(29) gives \((C_Q,C_{Oc})\simeq(5.29659,10.49832)\); the Fig. 7 description instead quotes \((4.30,7.28)\). This discrepancy is recorded by a diagnostic test and is unresolved. It limits source-fidelity interpretation beyond the equations as printed. A passed scalar gate does not certify the EOS-insensitive relation, physical BNS accuracy, zero-\(\Lambda\) behavior, binary agreement, taper, or the full waveform. The historical default remains separately qualified where its existing evidence supports it.

Reproduce the checks

With CPU FP64 Torch, JAX, mpmath and pytest installed, run PYTHONPATH=src:tests python -m pytest -q tests/test_nrtidalv2_self_spin_math_reference.py tests/test_nrtidal_reference.py.