Complete waveform model for compact binaries on eccentric orbits

We present a time domain waveform model that describes the inspiral, merger and ringdown of compact binary systems whose components are nonspinning, and which evolve on orbits with low to moderate eccentricity. The inspiral evolution is described using third-order post-Newtonian equations both for the equations of motion of the binary, and its far-zone radiation field. This latter component also includes instantaneous, tails and tails-of-tails contributions, and a contribution due to nonlinear memory. This framework reduces to the post-Newtonian approximant TaylorT4 at third post-Newtonian order in the zero-eccentricity limit. To improve phase accuracy, we also incorporate higher-order post-Newtonian corrections for the energy flux of quasicircular binaries and gravitational self-force corrections to the binding energy of compact binaries. This enhanced prescription for the inspiral evolution is combined with a fully analytical prescription for the merger-ringdown evolution constructed using a catalog of numerical relativity simulations. We show that this inspiral-merger-ringdown waveform model reproduces the effective-one-body model of Ref. [Y. Pan et al., Phys. Rev. D 89, 061501 (2014).] for quasicircular black hole binaries with mass ratios between 1 to 15 in the zero-eccentricity limit over a wide range of the parameter space under consideration. Using a set of eccentric numerical relativity simulations, not used during calibration, we show that our new eccentric model reproduces the true features of eccentric compact binary coalescence throughout merger. We use this model to show that the gravitational-wave transients GW150914 and GW151226 can be effectively recovered with template banks of quasicircular, spin-aligned waveforms if the eccentricity $e_0$ of these systems when they enter the aLIGO band at a gravitational-wave frequency of 14 Hz satisfies $e_0^{\mathrm{GW}150914}\le 0.15$ and $e_0^{\mathrm{GW}151226}\le 0.1$. We also find that varying the spin combinations of the quasicircular, spin-aligned template waveforms does not improve the recovery of nonspinning, eccentric signals when $e_0\ge 0.1$. This suggests that these two signal manifolds are predominantly orthogonal.

Figure 1

The panels show the eccentricity at the ISCO of black hole binaries prior to the merger event. We assume that the black hole population on each panel has an initial eccentricity $e_0$ at a gravitational-wave frequency $f_{\mathrm{GW}}=15\mathrm{Hz}$. These results have been obtained using PN equations of motion that include conservative and radiative corrections up to 3PN order.

Figure 2

The panels show the difference in the number of GW cycles when we use a waveform model that includes conservative corrections up to 3PN order and radiative corrections up to 2PN or 3PN order. The binary black hole population in each panel has an initial eccentricity $e_0$ at $f_{\mathrm{GW}}=15\mathrm{Hz}$. Note that the discrepancy between the two approximate models becomes very noticeable when $e_0\gtrsim 0.3$ for systems with asymmetric mass ratios.

Figure 3

Left panel: Time evolution of the orbital frequency evolution, $M\mathrm{\Omega }(t)$, of NR simulations and a SEOBNRv2 waveform ($q=1000$) compared with our gIRS model, Eq. (19). The right panel shows that the analytical expressions given by Eqs. (25)–(27) can accurately reproduce the orbital frequency evolution of NR waveforms at mass ratios that were not used for their calibration.

Figure 4

The panels present the merger model introduced in the main text. We present a direct comparison with waveforms used to calibrate the model, i.e., those with mass ratio $q=\{2.5,10\}$, and two additional cases to exhibit the performance of this approach. For reference, $h_{+}=\Re [h_{\text{merger}}]$, where $h_{\text{merger}}$ is given by Eq. (30) in the main text.

Figure 5

The panels present the inspiral (red) and merger-ringdown (blue) evolution of two binary black hole systems [see Eq. (32)]. The top panel presents a BBH system with component masses $(10M_{\odot },10M_{\odot })$ with an initial orbital eccentricity $e_0=0.3$ at a GW frequency $f_{\mathrm{GW}}=15\mathrm{Hz}$. This panel has two insets that show the imprint of eccentricity at low frequencies, and the late-time evolution of this system when the eccentricity has been radiated away. Bottom: BBH system with mass ratio $q=5$, total mass $M=45M_{\odot }$ and $e_0=0.15$ at $f_{\mathrm{GW}}=15\mathrm{Hz}$.

Figure 6

The panels present the time our code takes to generate a waveform, averaged over 15 iterations, for a given set of parameters. We assume that the binary systems have an initial eccentricity $e_0$ at a gravitational-wave frequency of 15 Hz. Left panel: $e_0=0$. The contour lines indicate how fast we can generate IMR waveforms in different regions of the BBH parameter space under consideration. Right panel: Same as left panel but for systems with $e_0=0.4$.

Figure 7

Overlap between the $ax$–model and the approximant TaylorT4 including 3.5PN corrections. We have used the zero-detuned high-power sensitivity configuration for Advanced LIGO and a lower frequency cutoff of 15 Hz.

Figure 8

Overlap between the $ax$–model in the zero-eccentricity limit and the SEOBNRv2 model. Left panel: The overlap is computed from an initial gravitational-wave frequency of 15 Hz using the zero-detuned high-power sensitivity configuration for aLIGO. Right panel: The overlap is computed from an initial gravitational-wave frequency of 25 Hz using the mid aLIGO sensitivity curve described in Ref. [9]. We note that in both cases a large portion of the parameter space under consideration is accurately reproduced by the $ax$–model with $\mathcal{O}\gtrsim 0.95$.

Figure 9

Using the Richardson extrapolation we provide a phase error estimate for each of our highest-resolution eccentric NR simulations. The vertical lines indicate the merger time of each of the BBH systems under consideration.

Figure 10

For an equal-mass BBH system with initial eccentricity $e_0=0.076$ and mean anomaly ${\ell }_0=3.09$ at a gauge-invariant frequency value $x_0=0.074$, we present a direct comparison of the dynamics predicted by our IMR $ax$–model and an eccentric NR simulation. Left panel: Our IMR $ax$ predicts with very good accuracy the orbital frequency evolution throughout late inspiral, merger and ringdown. Right panel: Our IMR $ax$–model can accurately reproduce the true NR features of the amplitude and phase evolution of an equal-mass, eccentric BBH merger.

Figure 11

Same as Fig. 10, but now for a BBH system with mass ratio $q=2$, initial eccentricity $e_0=0.1$, mean anomaly ${\ell }_0=3.11$ and gauge-invariant frequency parameter $x_0=0.076$.

Figure 12

The panels present overlap calculations between IMR $ax$ and SEOBNRv2 waveforms. The IMR $ax$ waveforms are generated for binaries that enter the aLIGO band with eccentricities $e_0\in [0,0.4]$ at $f_{\mathrm{GW}}=14\mathrm{Hz}$. The overlaps are computed from an initial gravitational-wave frequency $f_{\min }=15\mathrm{Hz}$ [see Eq. (33)] using the zero-detuned high-power sensitivity configuration for aLIGO.

Figure 13

The panel shows the coverage of the mass parameter space ($m_1$, $m_2$) using $10^6$ quasicircular template waveforms. We also show the mass distribution of the $8\times 10^3$ simulated eccentric signals or “injections.”

Figure 14

Left panels: Effectualness of a bank of quasicircular, nonspinning SEOBNRv2 templates to recover a population of eccentric, nonspinning signals. Right panels: Recovery of nonspinning eccentric injections with an aligned-spin template bank of SEOBNRv2 waveforms. Each panel indicates the eccentricity $e_0$ at which these systems enter aLIGO band at $f_{\mathrm{GW}}=14\mathrm{Hz}$. The $\mathcal{F}\mathcal{F}$’s are computed using $f_{\min }=15\mathrm{Hz}$ [see Eqs. (33) and (37)], and the zero-detuned high-power sensitivity configuration for aLIGO. The green and black stars represent the GW transients GW150914 and GW151226, respectively.

Figure 15

Same as Fig. 14 but now for compact binary populations with $e_0=\{0.1,0.15\}$. Note that the color bar has been adjusted to the range [0.7, 1] to exhibit additional structure for low $\mathcal{F}\mathcal{F}$ values.

Figure 16

Same as Fig. 14 but now for compact binary populations with $e_0=\{0.2,0.3\}$. Note that the color bar range is [0.5, 1].

Figure 17

Effective spin ${\chi }_{\mathrm{eff}}$ with which eccentric signals are recovered [see Eq. (42) in the main text]. The magnitude of ${\chi }_{\mathrm{eff}}$ indicates that spin-aligned SEOBNRv2 template banks significantly improve the recovery of nonspinning, eccentric waveforms for low-total-mass systems.

Figure 18

Left column: Difference in number of cycles using the pairwise comparison $\mathrm{\Delta }\mathcal{N}=|\mathcal{N}(2.5\mathrm{PN})-\mathcal{N}(2\mathrm{PN})|$. Right column: Pairwise comparison between $\mathrm{\Delta }\mathcal{N}=|\mathcal{N}(3\mathrm{PN})-\mathcal{N}(2.5\mathrm{PN})|$.

Figure 19

Same as Fig. 18, but now for $e_0=\{0.3,0.4\}$.