# Convergence of Fourier-domain templates for inspiraling eccentric compact binaries

## TL;DR

This paper analyzes the convergence and accuracy of frequency-domain inspiral templates for eccentric black hole binaries, crucial for gravitational wave detection with LISA, focusing on systematic and statistical errors up to 2PN order and eccentricity $e_0^6$.

## Contribution

It evaluates the convergence of eccentric inspiral templates and assesses the impact of neglecting higher-order terms on systematic errors in gravitational wave analysis.

## Key findings

- Eccentric waveforms increase statistical errors due to parameter correlations.
- Including terms of order $e_0^2$ or higher keeps systematic errors negligible.
- Templates converge well up to 2PN order and $e_0^6$ in eccentricity.

## Abstract

The space-based detector LISA may observe gravitational waves from the early inspiral of stellar-mass black hole binaries, some of which could have significant eccentricity. Current gravitational waveform templates are only valid for small orbital velocities (i.e., in a post-Newtonian expansion) and small initial eccentricity $e_0$ ("post-circular" expansion). We conventionally define $e_0$ as the eccentricity corresponding to an orbital frequency of $5 \text{ mHz}$, and we study the convergence properties of frequency-domain inspiral templates that are accurate up to 2PN and order $e_0^6$ in eccentricity. We compute the so-called "unfaithfulness" between the full template and "reduced" templates obtained by dropping some terms in the phasing series; we investigate the conditions under which systematic errors are negligible with respect to statistical errors, and we study the convergence properties of statistical errors. In general, eccentric waveforms lead to larger statistical errors than circular waveforms due to correlations between the parameters, but the error estimates do not change significantly as long as we include terms of order $e_0^2$ or higher in the phasing.