TaylorF2 adiabatic quadrupolar tides
| Definition field | Catalog value |
|---|---|
| Model ID | taylorf2-adiabatic-quadrupolar-tides-6.5pn/v1 |
| Reference composition | taylorf2-adiabatic-quadrupolar-tides-6.5pn/v1 |
| Scope | Adiabatic quadrupolar gravitoelectric TaylorF2 tidal phase corrections through 6.5PN added to aligned-spin TaylorF2. |
| Domain and modes | frequency; (2, 2) |
| Native backends | torch, jax |
| Scoped verification claims | 0 |
| Option | Default | Native choices | Reference-only choices |
|---|---|---|---|
spin_order |
7 |
-1, 0, 1, 2, 3, 4, 5, 6, 7, 8 |
None |
phasing_convention |
coalescence |
coalescence, reference_frequency |
None |
phenom_d_calibration |
false |
false, true |
None |
Scope and status
Specification: taylorf2-adiabatic-quadrupolar-tides-6.5pn/v1.
The independent reference kernel
adds the 5PN and 6PN electric quadrupole tidal phase and its 6.5PN tail to the
unchanged nonspinning and aligned-spin
specifications. Each inherited component keeps its original ID and version.
The generated coefficient table displays the
component coefficients and effective tidal parameters. Their independent
effective-parameter coefficient forms appear below.
This is a restricted-amplitude, circular, adiabatic inspiral specification. It excludes magnetic and octupolar tides, dynamical tides and resonances, nonlinear tides, internal tidal dissipation, spin-tidal couplings, tidal amplitude corrections, merger, cutoffs, and tapers. The inherited spin sector retains black-hole spin-induced multipoles: adding these tidal terms does not produce a complete spinning neutron-star model. No equation-of-state relation between masses, spins, and deformabilities is imposed.
Inputs, conventions, and options
Coordinates, units, and phase
Write \(M=m_1+m_2\), \(\delta=(m_1-m_2)/M\),
\(X_{1,2}=(1\pm\delta)/2\), and \(\eta=(1-\delta^2)/4\).
The inputs named lambda1 and lambda2 are the dimensionless individual
deformabilities \(\Lambda_i\), not the papers' dimensionful polarizabilities.
In geometric units, \(\lambda_i/M^5=X_i^5\Lambda_i\).
The numerical reference requires finite \(\Lambda_i\geq0\), \(|\delta|<1\), \(|\chi_i|\leq1\), and positive frequency, mass in seconds, distance in seconds, and reference frequency when supplied. Inclination and phase angles are in radians; time shifts are in seconds. The frontend boundary converts solar masses and Mpc using the independently audited unit constants. These domains define the expression and input checks; they do not establish physical accuracy throughout the domain.
Frontend option contract
The mathematical API selects tidal_terms=("T10", "T12", "T13") and
spin_terms= independently. Both default to their full reference term sets.
The frontend adapter
frontend_terms
records integer tidal_order and the presence of each deformability:
tidal_order |
Both deformabilities supplied | Either is None |
|---|---|---|
| 0 through 9 | None | None |
| 10, 11 | T10 |
None |
| 12 (default) | T10, T12 |
None |
| 13 or greater | T10, T12, T13 |
None |
| negative | T10, T12, T13 |
None |
Both frontends default to tidal_order=12 in their coefficient, phase, and
public waveform APIs. Neither argument may be omitted to retain a one-body
tide: supply a numerical zero for the nondeformable body. None disables the
entire tidal contribution even when the other body has nonzero deformability.
The mathematical reference instead requires explicit numerical inputs.
Spin selection keeps the existing documented profiles:
public waveforms default to spin_order=7, low-level phase to spin_order=5,
and all reference spin terms require spin_order=8. The reference's default
full composition therefore differs from the frontend defaults in two explicit
selectors. Tests use the corresponding adapter for each sector.
Mathematics and components
Tidal phase and coefficients
For \(v=(\pi M_{\rm sec} f)^{1/3}\), the added intrinsic phase is
The canonical component form is
The equivalent effective-parameter form is
The two effective parameters are defined in Wade et al., 1402.5156v1, Eqs. (5)–(6) and displayed in the generated table. Their ordered-mass square root becomes signed \(\delta\), extending the same component polynomials to either body labelling. Under simultaneous label exchange, \(\widetilde\Lambda\) is even and \(\delta\widetilde\Lambda\) is odd, so the phase is invariant and smooth at equal mass. Equal masses with equal deformability give \(\widetilde\Lambda=\Lambda\) and \(\delta\widetilde\Lambda=0\).
Both existing phase conventions and the \(e^{-i\Psi}\) carrier are retained. The coalescence convention adds \(I_{\rm tidal}(f)\); the finite-reference convention adds \(I_{\rm tidal}(f)-I_{\rm tidal}(f_{\rm ref})\). The Newtonian amplitude and polarization prefactors are inherited unchanged. Canonical derivatives hold every other input fixed, including both individual deformabilities; they do not differentiate an unprovided equation of state.
Direct paper audit
The audit read the pinned PDF equations directly, independently of frontend outputs and generated fixtures. Wade's Eq. (A26), with \(\lambda_i/M^5=X_i^5\Lambda_i\), gives the displayed component polynomials; exact symbolic expansion agrees with Eqs. (4)–(6). Wade et al., 1402.5156v1, Eq. (A26)
As a separate normalization check, Damour–Nagar–Villain's per-body phase uses \(\kappa_i=3(1-X_i)X_i^4\Lambda_i\). Substitution in Eqs. (B10)–(B11) reproduces the same 5PN/6PN coefficients. Equation (B12) supplies the relative factor \(-\pi v^3\), giving the positive effective \(T_{13}\) above for arbitrary mass ratio. The retained terms do not depend on the unknown higher order quantities elsewhere in that paper. Damour–Nagar–Villain, 1203.4352v1, Eqs. (B8)–(B14)
Seven independently transcribed exact identities checked the two papers' 5PN/6PN forms, tail normalization/sign, and signed-label transformations; no coefficient disagreement was found. The maintained tests additionally compare the kernel's component and effective forms exactly. The papers use different printed Fourier exponential signs (Wade A23, Damour–Nagar–Villain 15); this specification takes their phase polynomials and explicitly keeps the previously audited project carrier and phase gauges.
Implementation mapping
Component traceability
The generated map records every new and inherited component, paper location, reference function, both frontend functions, and maintained check names. Tuple positions are explicit and checked against frontend source. The inherited spin-tail derivation also remains in force. Structural tests resolve links without importing production frontends and detect stale rendered option/component tables.
New components: taylorf2-adiabatic-quadrupolar-tides-6.5pn/v1.
| Component ID | Evidence | Reference expression | Frontends | Checks |
|---|---|---|---|---|
tf2.tidal65.T10.v1 |
wade2014, Eq. A26 | tidal_corrections; T10 |
torch.phase_coefficients[9]; jax.phase_coefficients[9] |
T31, T32, T33, T34 |
tf2.tidal65.T12.v1 |
wade2014, Eq. A26 | tidal_corrections; T12 |
torch.phase_coefficients[10]; jax.phase_coefficients[10] |
T31, T32, T33, T34 |
tf2.tidal65.T13.v1 |
damour2012, Eq. 6, B8, B9, B10, B12, B14 | tidal_corrections; T13 |
torch.phase_coefficients[11]; jax.phase_coefficients[11] |
T31, T32, T33, T34 |
tf2.tidal65.lambda_tilde.v1 |
wade2014, Eq. 5 | tidal_parameters; lambda_tilde; frontend tuple[0] |
torch.tidal_combinations; jax.tidal_combinations |
T35, T34, T36, T37 |
tf2.tidal65.delta_lambda_tilde.v1 |
wade2014, Eq. 6 | tidal_parameters; delta_lambda_tilde; frontend tuple[1] |
torch.tidal_combinations; jax.tidal_combinations |
T35, T34, T36, T37 |
tf2.tidal65.intrinsic_phase.v1 |
Project convention taylorf2-adiabatic-quadrupolar-tides-6.5pn/v1 |
phase_expr; inherited phase plus selected tides |
torch.evaluate_phase; jax.evaluate_phase |
T38, T39, T40, T41, T42, T37, T33, T34 |
tf2.tidal65.coalescence_phase.v1 |
Project convention taylorf2-adiabatic-quadrupolar-tides-6.5pn/v1 |
waveform_expr; phase; reference_frequency=False |
torch.evaluate_phase; jax.evaluate_phase |
T38, T39, T40, T41, T42, T37, T33, T34 |
tf2.tidal65.reference_phase.v1 |
Project convention taylorf2-adiabatic-quadrupolar-tides-6.5pn/v1 |
waveform_expr; phase; reference_frequency=True |
torch.evaluate_phase; jax.evaluate_phase |
T38, T39, T40, T41, T42, T37, T33, T34, T36, T43, T44 |
tf2.tidal65.polarizations.v1 |
Project convention taylorf2-adiabatic-quadrupolar-tides-6.5pn/v1 |
waveform_expr; h_plus, h_cross; both phase conventions |
torch.evaluate; jax.evaluate |
T38, T39, T40, T41, T42, T37, T33, T34, T36, T43, T44 |
Unchanged inherited components: taylorf2-nonspinning-3.5pn/v1.
Unchanged inherited components: taylorf2-aligned-bh-spin-3.5pn/v1.
Additional reference expressions:
| Expression | Evidence | Reference function | Checks |
|---|---|---|---|
| Effective-parameter coefficients: T10, T12, T13 | wade2014, Eq. 4, 5, 6, A26; damour2012, Eq. B12 | effective_tidal_corrections |
T34, T31 |
| Isolated additive tidal phase: 3/(128 eta) times selected v powers | wade2014, Eq. A26; damour2012, Eq. 6, B8, B9, B10, B11, B12, B14 | tidal_phase_expr |
T34, T38, T40 |
Check labels are local to this table; function names identify maintained tests.
Evidence and limitations
The maintained LAL binary check also compares both frontends with LAL at tidal orders 10, 12, and 13 for nonspinning equal- and unequal-mass binaries. It does not qualify spinning tidal LAL parity: the tested LAL call has an additional spin-dependent response to nonzero deformabilities even when its tidal PN selector is zero.
Numerical qualification
On 22 September 2026, the complete required-reference gate passed 576/576 tests, with no skips, expected failures, or deselection. This includes 82 tidal numerical/symbolic tests and 86 tidal traceability tests, alongside the 408 existing reference checks. The run used CPU execution on macOS 26.6 ARM64, Python 3.13.7, SymPy 1.14.0, mpmath 1.3.0, Torch 2.14.0, and JAX 0.11.2 with JAX x64 enabled. The tidal comparisons qualify FP64; GPU and FP32 tidal execution remain unqualified. All 17 pre-existing reference source, test, fixture, and coefficient-table files remained byte-for-byte unchanged.
Maintained FP64 checks compare both frontends against high-precision scalar evaluation for effective parameters, individual tidal coefficients, unwrapped phase, polarizations, and derivatives with respect to mass, deformability, and frequency. They also exercise selector boundaries, missing inputs, zero and one-body tides, body exchange, equal mass, seeded systems, shared and per-binary frequency grids, and both phase conventions. Symbolic derivatives in canonical coordinates are transformed explicitly to frontend coordinates.
The seed 20260924 supplies ten systems alternating component masses in
\([1,2.2]M_\odot\) with one-body-tide systems whose first mass is
\([3,12]M_\odot\) and second is \([1,2.2]M_\odot\). Sampled deformabilities
lie in \([0,2000]\), with lambda1=0 in the latter systems;
\(\chi_1\in[-0.5,0.5]\), \(\chi_2\in[-0.1,0.1]\), and both \(v\) and
\(v_{\rm ref}\) lie in \([0.12,0.30]\). These are engineering checks of
the specified composition, not an astrophysical population or neutron-star
equation of state. Fixed cases include \(f=f_{\rm ref}\) and adjacent FP64
frequencies. A separate 100/160-digit convergence comparison uses
\(f=f_{\rm ref}+10^{-22}\).
| Compared quantity | FP64 budget |
|---|---|
| Tidal coefficients | Relative \(2\times10^{-14}\), absolute \(2\times10^{-13}\) |
| Effective parameters | Relative \(2\times10^{-14}\), absolute \(2\times10^{-12}\) |
| Amplitude | Relative \(5\times10^{-14}\) |
| Unwrapped phase | Relative \(5\times10^{-14}\), absolute \(2\times10^{-10}\) rad |
| Each complex polarization | \(\lvert h-h_{\rm ref}\rvert/A<3\times10^{-10}\) |
| Frequency derivative of phase | Relative \(2\times10^{-13}\), absolute \(2\times10^{-12}\) |
| Scaled parameter Jacobian | Relative and absolute \(2\times10^{-9}\) |
For the parameter Jacobian, amplitude and polarization rows are divided by the reference amplitude; the phase row is unscaled. Columns for \((m_1,m_2,\Lambda_1,\Lambda_2)\) are multiplied by \((1,1,1000,1000)\), so the absolute budget per unit deformability is \(2\times10^{-12}\) in these normalized coordinates. This fixed scale remains defined at zero tide. All budgets are engineering tolerances, not interval bounds or physical error estimates.
Compilation checks use JAX JIT and Torch full-graph aot_eager; the latter
qualifies graph capture and AOT execution, not Inductor code generation.
Cases include a shared one-dimensional grid, explicit per-binary (B,1) and
(B,Nf) grids, vector reference frequencies, and compiled vectorized Jacobians.
The exact cases and assertions remain in the maintained test module.
The tests do not qualify spinning/tidal FP32 evaluation, GPU
execution unless explicitly run there, or physical waveform error against
numerical relativity.
Frozen high-precision values and their precision-convergence check are generated from this kernel. They detect unintended changes but are not independent scientific evidence. Exact identities, source review, and frontend comparisons provide different checks; none makes this reference infallible.
Reproduce the checks
From the repository root after installing .[reference,torch,jax,dev]:
python -m diffgw_reference.tidal_artifacts --check
python -m diffgw_reference.tidal_traceability --check
JAX_ENABLE_X64=1 JAX_PLATFORMS=cpu python -m pytest \
tests/test_waveform_definitions.py tests/test_*math_reference*.py \
tests/test_nsbh_*reference.py tests/test_emri_*reference.py \
tests/test_xp_msa_remnant_boundary_policy.py \
tests/test_xphm_rotation_reference.py tests/test_xphm_nnlo_clean_room.py \
tests/test_spintaylor_reference_angle.py \
--require-math-reference --reference-device=cpu -ra
The complete required-mode run fails on missing dependencies or reference
modules, skipped or expected-failure cases, and deselection. A focused local
run of tests/test_taylorf2_tidal_math_reference.py and
tests/test_tidal_math_reference_traceability.py is useful during development;
it does not replace the complete CPU qualification.
A source, convention, or equation change that changes mathematical meaning requires a new specification/component version, an explained difference, reviewed artifact regeneration, and updated traceability and coverage. This extension leaves prior nonspinning and aligned-spin equations, APIs, and artifacts unchanged.