A template bank for gravitational waveforms from coalescing binary black holes: non-spinning binaries

TL;DR

This work addresses detecting gravitational waves from coalescing non-spinning binary black holes by coherently modeling the inspiral, merger, and ring-down stages using NR informed hybrids. It builds a two parameter phenomenological template bank in the Fourier domain parameterized by the physical binary parameters $M$ and $\eta$, derived by interpolating from an initial 10D bank based on NR-PN hybrids. The authors demonstrate high effectualness and faithfulness (overlaps $>0.99$) across Initial LIGO, Virgo, and Advanced LIGO, and show substantially improved sensitivity over searches that treat the stages separately. The framework enables dense bank construction without exhaustive NR simulations and lays out a path toward enhanced detection capabilities for intermediate-mass black holes, with future directions including longer NR waveforms, higher harmonics, and extensions to spinning binaries.

Abstract

Gravitational waveforms from the inspiral and ring-down stages of the binary black hole coalescences can be modelled accurately by approximation/perturbation techniques in general relativity. Recent progress in numerical relativity has enabled us to model also the non-perturbative merger phase of the binary black-hole coalescence problem. This enables us to \emph{coherently} search for all three stages of the coalescence of non-spinning 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 non-spinning binary black holes that follow quasi-circular 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 \emph{effectual} in detecting the signals from black hole coalescences, but also \emph{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 non-spinning black hole waveforms produced by numerical relativity.

Figures (14)

• Figure 1: Construction of the phenomenological template bank: (i) mapping physical signals (solid curve) into a sub-manifold (dashed curve, with example templates marked by dots) of a larger-dimensional template bank (curved surface), (ii) 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 (iii) 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/red), the 'best-matched' 3.5PN waveforms (dashed/black), and the hybrid waveforms (thin/green) 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 from produced by the Jena group. In each case, the matching region is $-750 \leq t/M \leq -550$ and we plot the real part of the complex strain (the '+' polarization).
• Figure 3: Fourier domain magnitude (left) and phase (right) 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 $\varphi_0$ (see Eq.(\ref{['eq:phenWavePhase']})) 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 Section \ref{['sec:Matching']}). 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.