Ready-to-use Fourier domain templates for compact binaries inspiraling along moderately eccentric orbits

We derive analytic expressions that provide the Fourier domain gravitational wave (GW) response function for compact binaries inspiraling along moderately eccentric orbits. These expressions include amplitude corrections to the two GW polarization states that are accurate to the first post-Newtonian (PN) order. Additionally, our fully third post-Newtonian (3PN)-accurate GW phase evolution incorporates eccentricity effects up to sixth order at each PN order. Further, we develop a prescription to incorporate analytically the effects of the 3PN-accurate periastron advance in the GW phase evolution. This is how we provide a ready-to-use and efficient inspiral template family for compact binaries in moderately eccentric orbits. Preliminary GW data analysis explorations suggest that our template family should be required to construct analytic inspiral-merger-ringdown templates to model moderately eccentric compact binary coalescence.

Figure 1

Matches between eccentric waveform models with different orders of eccentricity corrections. We are comparing waveforms that take only leading-order $\mathcal{O}(e_0^2)$ eccentricity corrections into account to those that include eccentricity corrections up to next-to-next-to leading order $\mathcal{O}(e_0^6)$. We consider three configurations of a NS and a BH with masses of $1.4M_{\odot }$ and $10M_{\odot }$, respectively: i.e., NS-NS (blue curve), NS-BH (orange curve), and BH-BH (pink curve) systems. The initial orbital eccentricity $e_0$ refers to the eccentricity of the binary system at 20 Hz. Given the same $e_0$, the effect of higher-order eccentricity corrections on the agreement between signal and template is strongly dependent on the total mass of the compact binary source. The solid black line denotes the threshold $\mathcal{M}=0.97$, associated with the effectualness of a model for GW detection and its faithfulness for source parameter estimation. The inset plot zooms into the region of parameter space where we can expect the effect of higher-order eccentricity corrections to become distinguishable from noise for $\mathrm{SNR}=30$, leading to systematic errors in parameter estimation; the dashed black line represents the indistinguishability criterion.

Figure 2

Matches between eccentric waveform models that include or neglect effects of the periastron advance. We consider the same three configurations of binaries with NS and BH components as in Fig. 1: i.e., NS-NS (blue curve), NS-BH (orange curve), and BH-BH (pink curve) systems. The initial orbital eccentricity $e_0$ is again defined at the lower cutoff frequency 20 Hz. We infer that the significance of periastron advance effects for GW data analysis is rather independent of the total mass of the source. We interpret our results by considering the threshold $\mathcal{M}=0.97$ (represented by the solid black line) below which a waveform model should be considered ineffectual for detection and unfaithful for parameter estimation. In the inset plots, we highlight the parameter space of small eccentricities to probe the importance of systematic errors in parameter estimation due to waveform uncertainties. The dashed black line represents the distinguishable limit for a fiducial GW signal with $\mathrm{SNR}=30$.

Figure 3

Matches between eccentric waveform models with Newtonian and 1PN-accurate amplitudes. We consider compact binary systems with a total mass of $m=m_1+m_2$, with different choices for the mass ratio $q=m_1/m_2$. As expected, the effect of amplitude corrections on waveform faithfulness is largely independent of the orbital eccentricity $e_0$ at 20 Hz. Waveforms with Newtonian amplitudes are faithful representations of amplitude-corrected waveforms only if $q\le 3$ (blue and orange curves); for higher mass ratios $q\ge 5$ (pink, green and purple curves) the match between waveforms with Newtonian and 1PN-accurate amplitudes falls below the threshold of $\mathcal{M}=0.97$ (denoted by the black line) even in the circular limit.