TaylorF2 eccentric phase at order \(e_0^2\)

Scope and status

Component specification ecc.phase-3pn-e2/v1 covers the secular, dominant-harmonic stationary-phase correction through relative 3PN and order \(e_0^2\). A separate ecc.phase-fref-gauge/v1 specifies the native phase origin. The scalar SymPy reference transcribes Moore et al. (2016), Eq. (6.26) without importing either frontend or LAL. This is a component-level result, not qualification of the complete waveform or the optional amplitude factor.

Inputs, conventions, and options

Input masses are in solar masses and both frequencies are positive GW frequencies in Hz. Let \(M=m_1+m_2\), \(\eta=m_1m_2/M^2\), \(v=(\pi GMf/c^3)^{1/3}\), and \(v_0=v(f_{0,\rm ecc})\). The eccentricity \(e_0\) is defined at \(f_{0,\rm ecc}\); it is not defined at the separate waveform phase origin \(f_{\rm ref}\). The paper's leading-eccentricity approximation was estimated to apply to comparable-mass binaries at roughly \(e_0\lesssim0.1\), with a more restrictive extreme-mass-ratio range. The native API accepts a wider input range; that acceptance is not a validity claim. The focused gate uses CPU float64, positive frequencies, and no ISCO cutoff. No numerical table or trained asset enters this component.

The native ecc_order values 0, 2, 3, 4, 5, and 6 retain the Newtonian, 1PN, 1.5PN, 2PN, 2.5PN, and 3PN eccentric terms respectively. Order 6 means relative 3PN, not \(e_0^6\). An explicit f0_ecc takes precedence over f_lower, then a positive f_ref, then the lowest positive grid value, with a 20 Hz fallback on an empty-positive grid. The gate supplies f0_ecc explicitly so this grid-dependent default is outside the paper check.

Mathematics and components

Eq. (6.26) gives the eccentric addition to the SPA phase:

\[ \Delta\Psi_{\rm paper}(f;f_0) = -\frac{2355}{1462}\frac{3}{128\eta v^5} e_0^2\left(\frac{v_0}{v}\right)^{19/3} P_{3\rm PN}(v,v_0,\eta). \]

The independently transcribed polynomial is stored by relative PN order in pn_terms(); its 3PN term includes both \(\ln(16v^2)\) and \(\ln(16v_0^2)\). The project waveform uses the distinct phase-origin adapter

\[ \Delta\Psi_{\rm native}(f) = \Delta\Psi_{\rm paper}(f;f_{0,\rm ecc})- \Delta\Psi_{\rm paper}(f_{\rm ref};f_{0,\rm ecc}). \]

This subtraction is a native convention, not an additional term in Moore's equation. The native positive-frequency polarization convention uses \(\exp[-i\Psi]\), so its eccentric-to-circular complex quotient is \(\exp[-i\Delta\Psi_{\rm native}]\).

Component ID	Source	Variant	Status	Check
`ecc.phase-3pn-e2/v1`	Moore et al. Eq. (6.26)	Paper phase, selectors 0/2/3/4/5/6	Executable	Individual PN orders and increments against high-precision scalar
`ecc.phase-fref-gauge/v1`	Native phase-origin convention	Subtraction at explicit `f_ref`	Executable	Complex quotient of both polarizations with `f_ref` above, below, and equal to `f0_ecc`
`ecc.amplitude-adapter/v0`	Primary derivation unresolved	Optional project factor; default off	Pending	Outside this gate

Implementation mapping

The reference functions pn_terms, phase_expr, evaluate, and anchored_phase live in src/diffgw_reference/taylorf2_eccentric.py. TaylorF2_Eccentric.eccentric_phase_correction in src/torchwave/approximants/taylorf2_eccentric.py and src/jaxwave/approximants/taylorf2_eccentric.py are the production paths. Their waveform evaluators resolve f0_ref and f_ref_circ separately and subtract the eccentric term at the latter. Neither frontend calls the scalar reference. The scalar test compares eccentric/circular waveform quotients; the circular carrier is common to numerator and denominator and is not requalified by this specification.

Evidence and limitations

tests/test_taylorf2_eccentric_math_reference.py checks equal- and unequal-mass systems \((1.4,1.4)\), \((1.7,1.2)\), and \((19,7)M_\odot\), \(e_0=0.02\)–0.08, \(f_0=20\)–37 Hz and sampled positive frequencies through 82 Hz. It compares each of six PN truncations against 80-digit scalar evaluation in both Torch and JAX CPU float64 at relative \(2\times10^{-12}\) and absolute \(2\times10^{-11}\) rad, tests each PN increment at relative \(3\times10^{-9}\), and checks both complex polarization quotients at absolute \(3\times10^{-10}\). The gauge cases put f_ref above, below, and equal to f0_ecc; none fits a phase or time shift. The strict --require-math-reference inventory automatically collects this named test module and rejects skips, expected failures, and deselection in the full suite.

The 2026-09-27 execution receipt records the pinned base, backend environment, and focused test result.

This establishes sampled paper-coefficient agreement and native gauge composition. A separate versioned nonspinning amplitude-off waveform slice has two strict same-model LAL holdouts. This phase component alone does not establish physical accuracy, spin-eccentric couplings, oscillatory eccentric terms, cutoff behavior, GPU/FP32 behavior, or the optional amplitude factor.

Reproduce the checks

With SymPy, mpmath, Torch, JAX, and pytest installed, run:

JAX_ENABLE_X64=1 JAX_PLATFORMS=cpu PYTHONPATH=src:tests \
  python -m pytest -q tests/test_taylorf2_eccentric_math_reference.py \
  tests/test_eccentric.py tests/parity/test_eccentric_parity.py \
  --reference-device=cpu

Run the full strict inventory with the CPU qualification command. A change to the paper coefficients, the f0_ecc meaning, or the native f_ref subtraction requires a component version increment and renewed scalar and backend evidence.