Template bank for gravitational waveforms from coalescing binary black holes: Nonspinning binaries

Gravitational waveforms from the inspiral and ring-down stages of the binary black-hole coalescences can be modeled accurately by approximation/perturbation techniques in general relativity. Recent progress in numerical relativity has enabled us to model also the nonperturbative merger phase of the binary black-hole coalescence problem. This enables us to coherently search for all three stages of the coalescence of nonspinning binary black holes using a single template bank. Taking our motivation from these results, we propose a family of template waveforms which can model the inspiral, merger, and ring-down stages of the coalescence of nonspinning binary black holes that follow quasicircular inspiral. This two-dimensional template family is explicitly parametrized by the physical parameters of the binary. We show that the template family is not only effectual in detecting the signals from black-hole coalescences, but also faithful in estimating the parameters of the binary. We compare the sensitivity of a search (in the context of different ground-based interferometers) using all three stages of the black-hole coalescence with other template-based searches which look for individual stages separately. We find that the proposed search is significantly more sensitive than other template-based searches for a substantial mass range, potentially bringing about remarkable improvement in the event rate of ground-based interferometers. As part of this work, we also prescribe a general procedure to construct interpolated template banks using nonspinning black-hole waveforms produced by numerical relativity.

Figure 1

Construction of the phenomenological template bank: (a) mapping physical signals (solid curve) into a submanifold (dashed curve, with example templates marked by dots) of a larger-dimensional template bank (curved surface), (b) obtaining a lower-dimensional phenomenological bank with the same number of parameters as physical parameters, through interpolation (solid curve on the curved surface, with example templates marked by triangles), and c) Estimating the bias of the lower-dimensional interpolated bank by mapping physical signals into the bank (with images of example signals marked by dots).

Figure 2

NR waveforms (thick lines), the best-matched 3.5PN waveforms (dashed lines), and the hybrid waveforms (thin lines) from three binary systems. The top panel corresponds to $\eta =0.25$ NR waveform produced by the AEI-CCT group. The second, third, and fourth panels, respectively, correspond to $\eta =0.25$, 0.22, and 0.19 NR waveforms produced by the Jena group. In each case, the matching region is $-750\le t/M\le -550$ and we plot the real part of the complex strain (the “$+$” polarization).

Figure 3

Fourier domain magnitude (top) and phase (bottom) of the (normalized) hybrid waveforms. The constant phase term and the term linear in time (and frequency) have already been subtracted from the phase. Symmetric mass ratio $\eta$ of each waveform is shown in the legends. These waveforms are constructed by matching 3.5PN waveforms with the long NR waveforms produced by the Jena group.

Figure 4

Fitting factors of the hybrid waveforms with the phenomenological waveform family. Horizontal axis shows the symmetric mass ratio of the binary. Fitting factors are calculated assuming a white noise spectrum, and hence are independent of the mass of the binary.

Figure 5

Hybrid waveforms (solid lines) in the frequency domain, and the best-matched phenomenological waveforms (dashed lines). The left panel shows the Fourier domain magnitude, while the right one shows the phase. In the hybrid waveforms, the constant phase term and the term linear in time (and frequency) have already been subtracted from the phase. In the phenomenological waveforms, $t_0$ and ${\phi }_0$ [see Eq. (4.16)] have been chosen to minimize the phase difference between the hybrid and phenomenological waveforms. These waveforms correspond to a binary with $\eta =0.25$, and are constructed from the “short” NR waveforms produced by the Jena group (see Sec. 4b). The “dip” in the left panel at $Mf\simeq 2\times {10}^{-2}$ is due to the small eccentricity present in the first few cycles of the NR waveform. All waveforms are normalized assuming a flat noise spectral density.

Figure 6

Best-matched amplitude parameters ${\mathit{\alpha }}_{\max }$ in terms of the physical parameters of the binary (assuming white noise spectrum). The horizontal axis shows the symmetric mass-ratio of the binary. Quadratic polynomial fits ${\mathit{\alpha }}_{\mathrm{int}}$ to the data points are also shown.

Figure 7

Best-matched phase parameters ${\mathit{\psi }}_{\max }$ in terms of the physical parameters of the binary (assuming white noise spectrum). The horizontal axis shows the symmetric mass-ratio of the binary. Quadratic polynomial fits ${\mathit{\psi }}_{\mathrm{int}}$ to the data points are also shown.

Figure 8

Fitting factor of the (two-dimensional) template family computed using three different noise spectra. The panel in the left correspond to the Initial LIGO noise PSD, the one in the middle to the Virgo noise PSD, and the one in the right to the Advanced LIGO noise PSD. The horizontal axis represents the symmetric mass ratio $\eta$ of the binary and the legends display the total mass $M$ (in units of $M_{\odot }$).

Figure 9

Same as in Fig. 8, except that the plots show the faithfulness of the (two-dimensional) template family.

Figure 10

Bias in the estimation of $M$. Horizontal axis reports the total mass $M$ (in units of $M_{\odot }$) and vertical axis reports the symmetric mass ratio $\eta$ of the binary. Shading (colors online) in the plot corresponds to the percentage bias, $|\Delta M|/M\times 100$. The left panel corresponds to the Initial LIGO noise PSD, the middle panel to the Virgo noise PSD, and the right panel to the Advanced LIGO noise PSD.

Figure 11

Same as in Fig. 10, except that the plots show the percentage bias $|\Delta \eta |/\eta \times 100$ in the estimation of $\eta$.

Figure 12

Same as in Fig. 10, except that the plots show the percentage bias $|\Delta M_{\mathrm{c}}|/M_{\mathrm{c}}\times 100$ in the estimation of $M_{\mathrm{c}}=M{\eta }^{3/5}$.

Figure 13

Fitting factor of the (two-dimensional) template family with “more accurate” hybrid waveforms (see Sec. 4e). The overlaps are computed using three different noise spectra. The panel in the left correspond to the Initial LIGO PSD, the one in the middle to the Virgo noise PSD, and the one in the right to the Advanced LIGO noise PSD. The horizontal axis represents the total mass $M$ (in units of $M_{\odot }$) and the legends display the symmetric mass ratio $\eta$ of the binary.

Figure 14

Distance to optimally located and oriented equal-mass binaries which can produce an optimal SNR of 8 at the Initial LIGO (left plot), Virgo (middle plot), and Advanced LIGO (right plot) noise spectra. Horizontal axis reports the total mass of the binary (in units of $M_{\odot }$) and vertical axis reports the distance in Mpc. In each plot, the thin solid line corresponds to a search using standard PN templates truncated at ISCO, the dotted-dashed line to a search using effective-one-body waveform templates truncated at the light ring, the dashed line to a search using ring-down templates, and the thick solid line to a search using the template family proposed in this paper. The ring-down horizon distance is computed assuming that $ϵ=0.7\%$ of the black-hole mass is radiated in the ring-down stage, while the Kerr parameter $a=0.69$ is known from the numerical simulation. Since the value of $ϵ$ has some amount of uncertainty in it, we have also included the shaded region in the plot corresponding to $0.18\%\le ϵ\le 2.7\%$.