Observing complete gravitational wave signals from dynamical capture binaries

We assess the detectability of the gravitational wave signals from highly eccentric compact binaries. We use a simple model for the inspiral, merger, and ringdown of these systems. The model is based on mapping the binary to an effective single black hole system described by a Kerr metric, thereby including certain relativistic effects such as zoom-whirl-type behavior. The resultant geodesics source quadrupolar radiation and, in turn, are evolved under its dissipative effects. At the light ring, we attach a merger model that was previously developed for quasicircular mergers but also performs well for eccentric mergers with little modification. We apply this model to determine the detectability of these sources for initial, Enhanced, and Advanced laser interferometer gravitational-wave observatory across the parameter space of nonspinning close capture compact binaries. We conclude that, should these systems exist in nature, the vast majority will be missed by conventional burst searches or by quasicircular waveform templates in the advanced detector era. Other methods, such as eccentric templates or, more practically, a stacked excess power search, must be developed to avoid losing these sources. These systems would also have been missed frequently in the initial laser interferometer gravitational-wave observatory data analysis. Thus, previous null coincidence results with detected gamma-ray bursts cannot exclude the possibility of coincident gravitational wave signals from eccentric binaries.

Figure 1

Comparison of the $\ell =2$, $m=2$ component of $\overline{h}$ for fly-by waveforms from $4:1$ BH-NS simulations (solid line) and our model with best-fit periapse distance (dashed line). The approximate effective geodesic orbital parameters of the simulated system (left to right, top to bottom) are $(r_p,e)=(8.3,1.0)$, (8.0, 0.8), (5.6, 1.0), and (5.0, 1.0). The fit parameters are given in Table .

Figure 2

Comparison of the $\ell =2$, $m=2$ component of $\overline{h}$ for fly-by waveforms from equal mass BH-BH (top panel) and NS-NS (bottom panel) simulations (solid lines) and our model (dotted lines) with best-fit parameters. The approximate effective geodesic orbital parameters of the simulated systems are $(r_p,e)=(8.7,1.0)$ for both cases. The fit parameters are given in Table . The feature in the waveform after the peak in the NS-NS simulation is from $f$-mode excitation that occurs during the close encounter.

Figure 3

Comparison of the merger GW strain from a $4:1$ BH-NS simulation (solid lines) and the IRS model (dotted lines) with best-fit parameters. The top panel shows a case where the initial BH was nonspinning. The bottom panel shows a case with $a_{\mathrm{BH}}=0.5$, which results in more whirling behavior and tidal disruption of the NS. The best-fit parameters in (7) are $(\kappa ,c)=(0.66,0.28)$ (top) and (0.46, 0.18) (bottom), and the matches are 0.98 and 0.96, respectively. The match is weighted based on the “whitened” waveforms as described in Sec. 3 assuming a total mass of $10M_{\odot }$.

Figure 4

Comparison of merger waveforms from an equal mass BH-BH simulation (top panel) and NS-NS simulation that forms a BH (bottom panel) with the IRS model (dotted lines) with best-fit parameters. The best-fit parameters in (7) are $(\kappa ,c)=(0.31,0.03)$ (top) and (0.36, 0.19) (bottom), and the matches are 0.98 and 0.97, respectively. The match is weighted based on the “whitened” waveforms as described in Sec. 3 assuming a total mass of $20M_{\odot }$ and $2.8M_{\odot }$ for the BH-BH and NS-NS binaries, respectively.

Figure 5

GW strain generated with our model and initial conditions $r_p=8M$ and $e=1$. The top panel shows the entire waveform, while the bottom panel shows a zoomed-in view of the end of the waveform.

Figure 6

Evolution of orbital parameters for a $4:1$ (top panel) and an equal mass ratio (bottom panel) binary. The effective eccentricity is calculated from successive apoapse and periapse distances as $e=(r_a-r_p)/(r_a+r_p)$.

Figure 7

Relative difference in energy and angular momentum lost in a close encounter for the 2.5 and 3.5 PN approximations versus our model. For this comparison the orbits are chosen to initially have zero energy and the same value of angular momentum which, for a geodesic with the same initial conditions, corresponds to the value of $r_p$ indicated on the $x$ axis.

Figure 8

The characteristic strain $h_{\mathrm{c}}$ is shown for the initial (thin dash-dotted line), Enhanced (dash-dotted line), and Advanced LIGO (solid line) detectors, as well as for two example signals at $D_L=1\mathrm{Gpc}$. The first signal corresponds to an orientation-averaged source with $M=100M_{\odot }$, $q=1$, and $r_{\mathrm{p}}=5M$ (dashed line), and the second signal is from a source with $M=10M_{\odot }$, $q=0.1$, and $r_{\mathrm{p}}=10M$ (dotted line). Both signal spectra are smoothed to diminish fluctuations and make the trend more clear. The system with $q=0.1$ has little contribution from the merger, so the repeated burst phase dominates the spectra, with $h_{\mathrm{c}}{\propto }_{\sim }f$, whereas the $q=1$ system signal comes largely from the merger, where $h_{\mathrm{c}}\approx \mathrm{\text{constant}}$ over a small band of frequencies.

Figure 9

Whitened waveforms for a $10M_{\odot }$ (top panel) and a $100M_{\odot }$ (bottom panel) binary with initial $e=1$ and $r_p=5M$, along with the (whitened) best-fit template among sine-Gaussian and ringdown templates.

Figure 10

Contours of horizon distance ($\rho =8$) as a function of mass ratio $q$ and pericenter separation $r_p$ for initial LIGO using an optimal filter (top panel), sine-Gaussian templates (middle panel), and an estimate of a power-stacking search (bottom panel) as described in the text. We fix one component to be a $1.35M_{\odot }$ neutron star and change the total mass with mass ratio accordingly. We include a contour at $D_L=0.77\mathrm{Mpc}$ and another at 3.6 Mpc, to show the region of parameter space where existing LIGO searches would not have seen a gravitational wave counterpart to GRB070201 [74] and GRB051103 [64], respectively.

Figure 11

Contours of horizon distance as a function of rest mass $M$ and mass ratio $q$ for Enhanced LIGO using an optimal filter (top panel), sine-Gaussian templates (middle panel), and ringdown templates (bottom panel) for an initial pericenter separation of $r_p=6M$.

Figure 12

Contours of horizon distance as a function of rest mass $M$ and mass ratio $q$ for Advanced LIGO using an optimal filter (top panel), sine-Gaussian templates (middle panel), and ringdown templates (bottom panel) for an initial pericenter separation of $r_p=6M$.

Figure 13

Contours of detection probability $p\equiv V/V_{\max }$ as a function of rest mass $M$ and mass ratio $q$ for Enhanced LIGO for a source inside the optimal filtering distance horizon, using sine-Gaussian (top panel) and ringdown (bottom panel) templates for an initial pericenter separation of $r_p=6M$.

Figure 14

Contours of detection probability $p\equiv V/V_{\max }$ as a function of rest mass $M$ and mass ratio $q$ for Advanced LIGO for a source inside the optimal filtering distance horizon, using sine-Gaussian (top panel) and ringdown (bottom panel) templates for an initial pericenter separation of $r_p=6M$.

Figure 15

Contours of horizon distance as a function of rest mass $M$ and pericenter separation $r_p$ for Enhanced LIGO using an optimal filter (top panel), sine-Gaussian templates (middle panel), and ringdown templates (bottom panel). The mass ratio is $q=1$.

Figure 16

Contours of horizon distance as a function of rest mass $M$ and pericenter separation $r_p$ for Advanced LIGO using, from top to bottom, an optimal filter, sine-Gaussian templates, ringdown templates, and a power-stacking search. The mass ratio is $q=1$.

Figure 17

Contours of detection probability as a function of rest mass $M$ and pericenter separation $r_p$ for Enhanced LIGO for a source inside the optimal filtering distance horizon, using sine-Gaussian (top panel) and ringdown (bottom panel) templates. The mass ratio is $q=1$.

Figure 18

Contours of detection probability as a function of rest mass $M$ and initial pericenter separation $r_p$ for Advanced LIGO for a source inside the optimal-filtering horizon distance, using sine-Gaussian (top panel) and ringdown (middle panel) templates and a power-stacking search (bottom panel). The mass ratio is $q=1$. Note the different scale in the bottom figure.