Spin effects in the phasing formula of eccentric compact binary inspirals up to the third post-Newtonian order

Abstract

Compact binary sources that emit gravitational waves (GW) are expected to be both spinning and on eccentric orbits. No closed-form expression for the phasing of GWs are available to date that contain information from both spin and eccentricity. The introduction of eccentricity can slow waveform generation, often requiring slower numerical methods governing its evolution. However, closed-form expressions for the waveform phase can be obtained when eccentricity is treated as a small parameter, enabling quick waveform generation. In this paper, closed-form expressions for the GW phasing in the form of Taylor approximants up to the eighth power in initial eccentricity $(e_0)$ are obtained while also including aligned spins up to the third post-Newtonian order. The phasing is obtained in both time and frequency domains. The fully analytical approximant (TaylorT2) is also resummed for usage in scenarios where initial eccentricities are as high as 0.5. The frequency domain approximant (TaylorF2) based on Stationary Phase approximation is compared with an existing model (TaylorF2Ecc) to assess the importance of the newly computed eccentric/spinning terms. The findings indicate that for eccentricities $\gtrsim 0.15$ (defined at 10 Hz) and small spins $(\sim 0.2)$, the mismatches can be higher than 1%. This leads to an overall loss in signal-to-noise ratio and lower detection efficiency of GWs coming from eccentric spinning compact binary inspirals if the combined effects of eccentricity and aligned spins are neglected in the waveforms.