TaylorF2_Eccentric nonspinning low-e mathematical specification

Definition field	Catalog value
Model ID	`TaylorF2_Eccentric`
Reference composition	`taylorf2-eccentric-nonspinning-e2-3pn/v1`
Scope	Nonspinning comparable-mass (q <= 2), 0 < e0 <= 0.1, positive-frequency inspiral: restricted quadrupole amplitude and 3PN order-e0² eccentric phase at explicit f0_ecc and f_ref, with amplitude correction off.
Domain and modes	`frequency`; `(2, 2)`
Native backends	`torch`, `jax`
Scoped verification claims	1

Option	Default	Native choices	Reference-only choices
`phasing_convention`	`coalescence`	`coalescence`, `reference_frequency`	None
`ecc_order`	`6`	`0`, `2`, `3`, `4`, `5`, `6`	None
`include_amplitude_correction`	`false`	`false`, `true`	None
`f_isco_cutoff`	`true`	`false`, `true`	None

Scope and status

Specification taylorf2-eccentric-nonspinning-e2-3pn/v1 composes the unchanged nonspinning TaylorF2 3.5PN carrier with the paper-derived 3PN eccentric phase. It covers the positive-frequency, dominant-harmonic, restricted-amplitude inspiral with zero spins, comparable masses \(1\le q=m_1/m_2\le2\), and \(0<e_0\le0.1\). The eccentricity is defined at an explicit \(f_{0,\rm ecc}\). Both native backends and the scalar reference use an explicit, positive \(f_{\rm ref}\) for the project reference-frequency phase convention.

The native API accepts more parameter choices than this versioned slice. Optional eccentric amplitude, aligned-spin terms and their possible spin-eccentric couplings, other phase origins, arbitrary mass ratios, larger eccentricities, and cutoff-boundary behavior remain provisional. This is a bounded mathematical and implementation qualification, not a uniform physical accuracy statement.

Inputs, conventions, and options

Masses are positive solar masses ordered \(m_1\ge m_2\), distance is positive Mpc, \(0\le\iota\le\pi\), and \(t_c,\phi_c\) are finite. The grid frequencies, \(f_{0,\rm ecc}\), and \(f_{\rm ref}\) are positive Hz. The sampled band stays below Schwarzschild \(f_{\rm ISCO}=[6^{3/2}\pi M]^{-1}\), with \(M=G(m_1+m_2)/c^3\). Thus the native default f_isco_cutoff=True has no effect on the qualified samples. The scalar reference is unwindowed; no assertion is made at the hard cutoff or after it.

Select chi1=chi2=0, no tides, ecc_order=6, include_amplitude_correction=False, and phasing_convention="reference_frequency". Supply e0, f0_ecc, and f_ref explicitly. ecc_order=6 selects the relative 3PN eccentric phase at order \(e_0^2\); it does not select order \(e_0^6\). The native API's implicit reference-frequency priority and other PN truncations are described in the phase component specification, outside this composition claim. The strict external cases use CPU float64.

Mathematics and components

Let \(v=(\pi Mf)^{1/3}\), \(v_0=v(f_{0,\rm ecc})\), and \(\eta=m_1m_2/(m_1+m_2)^2\). The circular intrinsic phase \(I_{3.5\rm PN}\) and restricted quadrupole amplitude \(A_0\) are exactly the versioned TaylorF2 base expressions. The eccentric term from Moore et al., Eq. (6.26) is

\[ E(f;f_{0,\rm ecc})=-\frac{2355}{1462}\frac{3e_0^2}{128\eta v^5} \left(\frac{v_0}{v}\right)^{19/3}P_{3\rm PN}(v,v_0,\eta). \]

The complete \(P_{3\rm PN}\) polynomial, including distinct logarithms in \(v\) and \(v_0\), is transcribed and tested in the phase component specification. The composed project phase and polarizations are

\[ \Psi(f)=2\pi ft_c-2\phi_c+\frac{3\pi}{4} +I_{3.5\rm PN}(f)-I_{3.5\rm PN}(f_{\rm ref}) +E(f;f_{0,\rm ecc})-E(f_{\rm ref};f_{0,\rm ecc}), \]

\[ h_+(f)=\tfrac12(1+\cos^2\iota)A_0(f)e^{-i\Psi(f)},\qquad h_\times(f)=-i\cos\iota\,A_0(f)e^{-i\Psi(f)}. \]

The two \(f_{\rm ref}\) subtractions and \(3\pi/4\) offset are DiffGW phase conventions, not terms in Moore's paper equation. The amplitude contains no eccentric factor in this slice. Reused tf2.ns35.*.v1 components retain the base version; tf2.ecc.paper_phase.v1, tf2.ecc.reference_phase.v1, and tf2.ecc.polarizations.v1 identify the extension and composition.

Implementation mapping

src/diffgw_reference/taylorf2_eccentric.py builds waveform_expr() from the independent base waveform_expr() and its own Moore polynomial, then evaluates it at high precision through evaluate_waveform(). It does not import either tensor frontend or LAL. The native Torch and JAX classes add the eccentric phase to their existing TaylorF2 carrier; their definition_id metadata binds this scoped catalog entry. No production waveform coefficients or numerical paths changed for this promotion.

Evidence and limitations

The phase receipt records the direct Eq. (6.26) transcription and strict Torch/JAX CPU float64 component checks. The frozen two-case holdout receipt records nonzero-eccentricity, off-catalog direct LAL TaylorF2Ecc comparisons with different mass ratios, \(e_0\), \(f_{0,\rm ecc}\), and \(f_{\rm ref}\). Both backend polarizations and six detector projections passed the unchanged strict complex-\(L_2\) and mismatch gates. The test uses the existing PyCBC optimized match and named PSDs; no phase or time shift was fitted to the complex-\(L_2\) results. These two points do not establish an error envelope over the stated domain or the accuracy of the underlying approximation.

The native aligned-spin/eccentric combination is measured in an older frozen catalog row but has no independent paper-backed cross-term derivation here. The optional native amplitude multiplier also lacks primary-source provenance. Neither is included in this model version.

Reproduce the checks

With LALSimulation, PyCBC, Torch, JAX, SymPy and mpmath installed:

JAX_ENABLE_X64=1 JAX_PLATFORMS=cpu PYTHONPATH=src:tests \
  python -m pytest -q tests/test_taylorf2_eccentric_holdout.py \
  tests/test_taylorf2_eccentric_math_reference.py \
  tests/test_waveform_definitions.py tests/test_parity_harness_controls.py \
  --reference-device=cpu

python scripts/generate_waveform_library_table.py --check checks the catalog-generated header and waveform-library row. Changing the base phase, Moore polynomial, eccentricity reference, or project phase origin requires a new composition version and renewed scalar and external checks.