A parametrized test of general relativity for inspiralling eccentric binaries in LISA
The space-based detector Laser Interferometer Space Antenna (LISA) will observe inspiralling black hole binaries in the mHz band, many of which may retain orbital eccentricity. In contrast to quasicircular binaries, eccentric systems radiate through multiple orbital harmonics, while relativistic periastron precession introduces a secular phase structure in the waveform. We exploit this structure to develop a parametrized test of general relativity (GR) for eccentric bound orbits. We construct a frequency-domain eccentric waveform model in which a phenomenological deviation parameter $\delta\alpha$ modifies the GR prediction for the conservative azimuthal-to-radial frequency ratio, with $\delta\alpha=0$ corresponding to GR. We consider two parametrizations of this deformation. In the first, $\delta\alpha$ modifies only the explicit precession-dependent sideband structures of the waveform. In the second, the secular precession phase is assigned to the dominant lower-order angular carriers and retained in resummed form. The latter produces a substantially stronger response because the dominant waveform components coherently accumulate the modified phase. We implement both models within \textsc{lisabeta} and perform a Bayesian analysis including the time- and frequency-dependent LISA response and associated time-delay-interferometry observables. We find that LISA can place stringent constraints on the parametrized deviation. For a binary with chirp mass $3000 M_{\odot}$, initial eccentricity $e_0=0.5$ observed for four years at an SNR of $50$, the second model yields a $90\%$ credible bound of $|\delta\alpha|\lesssim 10^{-4}$. Increasing eccentricity further sharpens the constraints by introducing additional harmonic structure and reducing degeneracies among the binary parameters. The framework developed here is general and can be extended to more complete eccentric waveform models.