On constraining the initial orbital eccentricity of inspiral-dominated gravitational-wave events with TaylorF2Ecck

Technical Summary: Constraining Initial Orbital Eccentricity in Inspiral-Dominated GW Events with TaylorF2Ecck

Problem Statement
The LIGO-Virgo-KAGRA (LVK) consortium has detected inspiral-dominated gravitational-wave (GW) events, such as the binary neutron star (BNS) mergers GW170817 and GW190425. These events spend significantly more time in the detector frequency window than binary black hole (BBH) mergers, making them ideal for constraining residual orbital eccentricity ($e_0$). While standard waveform models like TaylorF2 assume quasi-circular orbits, astrophysical formation channels (e.g., dynamical capture in dense clusters) predict non-negligible eccentricity. Previous analyses using the TaylorF2Ecc approximant constrained $e_0$ but omitted periastron advance effects, which enter compact binary dynamics at the 1PN order. Furthermore, existing eccentric models often lack consistency in the order of Post-Newtonian (PN) accuracy between phase and amplitude contributions, potentially introducing systematic biases in parameter estimation.

Methodology
The authors introduce TaylorF2Ecck, a new frequency-domain inspiral approximant implemented within the LALSuite framework. This model analytically describes GWs from non-spinning compact binaries in eccentric orbits. Key technical features include:

• PN Accuracy: The model consistently incorporates orbital, periastron advance, and GW emission effects fully up to 3PN order in the Fourier phase.
• Eccentricity Order: It includes $O(e_0^2)$ contributions to the 3PN-accurate phase and $O(e_0)$ corrections to the quadrupolar-order amplitudes.
• Harmonics: Unlike the dominant-mode-only TaylorF2Ecc, TaylorF2Ecck includes multiple harmonics $(j, n)$, specifically $(1,0), (1,-2), (2,-2),$ and $(3,-2)$, to account for sidebands and mode splitting.
• Comparative Models: The study employs a restricted version of TaylorF2Ecck (denoted TaylorF2Ecch) which omits periastron advance effects to isolate their impact. It also utilizes standard TaylorF2Ecc and quasi-circular TaylorF2 variants with Fourier phases truncated at 3PN, 3.5PN, 4PN, and 4.5PN orders to probe systematic biases.
• Analysis Framework: Bayesian parameter estimation was performed using the Bilby software with nested sampling. The analysis focused on GW170817 and GW190425, utilizing official LVK data releases and BayesWave power spectral densities. The minimum frequency was set to 20 Hz to balance computational cost and eccentricity sensitivity.

Key Contributions

1. Implementation of TaylorF2Ecck: The paper provides a computationally efficient, fully analytic template family that consistently includes leading-order eccentricity contributions up to 3PN order, including periastron advance and multi-harmonic effects.
2. Sanity Checks and Distinguishability: Through match calculations and injection-recovery tests, the authors established that TaylorF2Ecck becomes distinguishable from the quasi-circular TaylorF2 model at $e_0 \gtrsim 0.01$ and from models lacking periastron advance (TaylorF2Ecch) at $e_0 \gtrsim 0.06$ (for GW170817-like parameters).
3. PN Order Sensitivity: The study identified a significant systematic bias in the mass-ratio ($q$) posterior when comparing 3PN-accurate eccentric models against models incorporating 3.5PN-order circular Fourier phase corrections. This highlights the necessity of extending eccentric contributions to at least 3.5PN order for reliable parameter estimation.

Results

• Eccentricity Constraints: For both GW170817 and GW190425, the posterior distributions for initial eccentricity $e_0$ (evaluated at 20 Hz) rail toward zero.
  • GW170817: $e_0 \leq 0.016$ (90% credible interval) with a uniform prior.
  • GW190425: $e_0 \leq 0.023$ (90% credible interval) with a uniform prior.
• Model Preference: Bayes factor comparisons ($|\Delta \log_{10} B| < 1$) show no strong evidence favoring the eccentric TaylorF2Ecck model over the quasi-circular TaylorF2 model, nor over the TaylorF2Ecc model (which lacks periastron advance). This indicates that at the inferred negligible eccentricities, the inclusion of periastron advance and extra harmonics does not significantly alter the results.
• Systematic Biases: While the eccentricity constraints are robust, the study found a substantial shift in the mass-ratio posterior when comparing 3PN models to 3.5PN circular models. This suggests that the known correlation between mass and eccentricity makes the inclusion of higher-order (3.5PN) circular phase corrections critical, even when eccentricity is negligible.
• GW200105 Comparison: A supplementary analysis of the NSBH event GW200105 yielded eccentricity estimates consistent with previous studies ($e_0 \approx 0.12$), though Bayes factors did not strongly favor the eccentric model over the quasi-circular one using the current TaylorF2Ecck implementation.

Significance and Claims
The paper claims that TaylorF2Ecck provides a robust, analytically consistent framework for analyzing low-eccentricity inspiral events. The primary significance lies in the rigorous constraint of initial eccentricity for GW170817 and GW190425, confirming they are consistent with field-formed binaries (negligible eccentricity) rather than dynamical formation channels.

Crucially, the authors argue that while current constraints on $e_0$ are robust, the observed bias in mass-ratio posteriors underscores a methodological imperative: eccentric waveform models must incorporate initial eccentricity contributions up to at least 3.5PN order to avoid systematic errors in parameter estimation. The paper concludes that future work should focus on extending the PN accuracy of eccentric terms to 3.5PN and incorporating spin effects, merger, and ringdown to fully exploit the potential of eccentric GW searches.