Post-circular expansion of eccentric binary inspirals: Fourier-domain waveforms in the stationary phase approximation

We lay the foundations for the construction of analytic expressions for Fourier-domain gravitational waveforms produced by eccentric, inspiraling compact binaries in a post-circular or small-eccentricity approximation. The time-dependent, “plus” and “cross” polarizations are expanded in Bessel functions, which are then self-consistently reexpanded in a power series about zero initial eccentricity to eighth order. The stationary-phase approximation is then employed to obtain explicit analytic expressions for the Fourier transform of the post-circular expanded, time-domain signal. We exemplify this framework by considering Newtonian-accurate waveforms, which in the post-circular scheme give rise to higher harmonics of the orbital phase and to amplitude corrections of the Fourier-domain waveform. Such higher harmonics lead to an effective increase in the inspiral mass reach of a detector as a function of the binary’s eccentricity $e_0$ at the time when the binary enters the detector sensitivity band. Using the largest initial eccentricity allowed by our approximations ($e_0<0.4$), the mass reach is found to be enhanced up to factors of approximately 5 relative to that of circular binaries for Advanced LIGO, LISA, and the proposed Einstein Telescope at a signal-to-noise ratio of ten. A post-Newtonian generalization of the post-circular scheme is also discussed, which holds the promise to provide “ready-to-use” Fourier-domain waveforms for data analysis of eccentric inspirals.

Figure 1

SNR for an equal-mass binary at optimal orientation as a function of total mass for circular binaries and elliptic binaries with initial eccentricity of 0.3. The assumed low-frequency cutoffs for AdvLIGO, ET and LISA are 20, either 1 or 10, and ${10}^{-4}\mathrm{Hz}$, respectively. The sources are at 100 Mpc for AdvLIGO and ET and at 3 Gpc for LISA. The initial eccentricity corresponds to that at the low-frequency cutoff.

Figure 2

Normalized mass reach enhancement, as a function of initial eccentricity. The normalization is given by the value of the mass reach for circular binaries, namely, $M_0=219M_{\odot }$, $M_0=440M_{\odot }(4397M_{\odot })$ and $M_0=4.239\times {10}^7{M}_{\odot }$ for AdvLIGO, ET and LISA, respectively.

Figure 3

Plot of the sine of the phase calculated numerically (solid black line) and expanded in Bessel functions for a system with $e_0=0.99$, following Eq. (2.14), and keeping 3 (dotted red line), 7 (dashed green line), 10 (dotted-dashed blue line) and 15 (dotted-dotted-dashed orange line) terms.

Figure 4

Plot of the absolute value of the fractional relative difference between the sine of the eccentric anomaly, calculated numerically and expanded in Bessel functions with 9 terms, for different eccentricities ($e=0.4$ solid black curve, $e=0.2$ blue dashed curve, and $e=0.1$ red dotted curve). The results are here normalized by the exact numerical solution.

Figure 5

Eccentricity as a function of frequency for different value of the initial eccentricity $e_0$ evaluated at $F_0=20\mathrm{Hz}$. Solid lines correspond to the eccentricity as given by Eq. (3.11) for $e_0=\{0.1,0.3,0.5,0.6,0.7,0.8,0.9\}$, in ascending order. Dotted lines correspond to the approximant of Eq. (3.12) for the same initial eccentricities.

Figure 6

SNR for an equal-mass binary at optimal orientation as a function of total mass in solar mass units for different initial eccentricities. The top figures correspond to the AdvLIGO (left) and ET (right) detectors, while results for the LISA detector are shown in the bottom panel.

Figure 7

Absolute value of the difference in SNR computed up to ${\ell }_{\max }=(2,3,4,5,6,7,8,9)$ and up to ${\ell }_{\max }=10$. We use the AdvLIGO noise curve and an initial eccentricity of $e_0=0.01$, $e_0=0.1$, $e_0=0.3$ and $e_0=0.4$ (top to bottom). The insets zoom on the $y$ axis in the region between 200 and 800 solar masses.

Figure 8

Histograms of the AdvLIGO SNR (see text) for a set of random observer orientations. The left (right) panel corresponds to a $M=100M_{\odot }$ ($M=300M_{\odot }$) system. Systems with initial eccentricity $e_0=(0.1,0.2,0.4)$ are shown in dotted black, dashed blue and solid red, respectively.

Figure 9

Frequency 1 yr prior to merger as a function of total mass for equal-mass binaries with different initial eccentricity: $e_0=0$ (crosses), $e_0=0.1$ (circles), $e_0=0.2$ (squares), and $e_0=0.4$ (diamonds). The quantity $f={10}^{-4}\mathrm{Hz}$ corresponds to the acceleration noise cutoff frequency.