Template Bank Research Articles

Quasinormal ringing of Kerr black holes. II. Excitation by particles falling radially with arbitrary energy

The analytical understanding of quasinormal mode ringing requires an accurate knowledge of the Green's function describing the response of the black hole to external perturbations. We carry out a comprehensive study of quasinormal mode excitation for Kerr black holes. Relying on the formalism developed by Mano, Suzuki, and Takasugi, we improve and extend previous calculations of the quasinormal mode residues in the complex frequency plane (``excitation factors ${B}_{q}$''). Using these results we compute the ``excitation coefficients'' ${C}_{q}$ (essentially the mode amplitudes) in the special case where the source of the perturbations is a particle falling into the black hole along the symmetry axis. We compare this calculation with numerical integrations of the perturbation equations, and we show quantitatively how the addition of higher overtones improves the agreement with the numerical waveforms. Our results should find applications in models of the ringdown stage and in the construction of semianalytical template banks for gravitational-wave detectors, especially for binaries with large mass ratios and/or fast-spinning black holes.

Template mode hierarchies for binary black hole mergers

Matched filtering is a popular data analysis framework used to search for gravitational wave signals emitted by compact object binaries. The templates used in matched filtering searches are constructed predominantly from the quadrupolar mode since this is the energetically most dominant mode. However, for highly precessing binaries or binaries with moderately large mass ratios, quadrupole templates lose sensitivity for a significant fraction of source orientations. We investigate how the inclusion of higher modes in the templates alleviates this loss of sensitivity and thus increases the prospects for detecting gravitational waves. Specifically, we use numerical relativity waveforms from the late inspiral and coalescence of binary black holes to identify mode hierarchies from which one can construct templates that cover the entire sky of binary orientations. The ordering in these hierarchies depends on the characteristics of the binary system and the mode strengths. The proposed hierarchies could assist in deciding which modes to add to template banks according to their ability to maximize sky coverage.

Balancing act of template bank construction: Inspiral waveform template banks for gravitational-wave detectors and optimizations at fixed computational cost

Gravitational-wave searches for signals from inspiraling compact binaries have relied on matched filtering banks of waveforms (called template banks) to try to extract the signal waveforms from the detector data. These template banks have been constructed using four main considerations, the region of parameter space of interest, the sensitivity of the detector, the matched filtering bandwidth, and the sensitivity one is willing to lose due to the granularity of template placement, the latter of which is governed by the minimal match. In this work we describe how the choice of the lower frequency cutoff, the lower end of the matched filter frequency band, can be optimized for detection. We also show how the minimal match can be optimally chosen in the case of limited computational resources. These techniques are applied to searches for binary neutron star signals that have been previously performed when analyzing Initial LIGO and Virgo data and will be performed analyzing Advanced LIGO and Advanced Virgo data using the expected detector sensitivity. By following the algorithms put forward here, the volume sensitivity of these searches is predicted to improve without increasing the computational cost of performing the search.

Template banks to search for low-mass binary black holes in advanced gravitational-wave detectors

Coalescing binary black holes (BBHs) are among the most likely sources for the Laser Interferometer Gravitational-Wave Observatory (LIGO) and its international partners Virgo and KAGRA. Optimal searches for BBHs require accurate waveforms for the signal model and effectual template banks that cover the mass space of interest. We investigate the ability of the second-order post-Newtonian TaylorF2 hexagonal template placement metric to construct an effectual template bank, if the template waveforms used are effective one-body waveforms tuned to numerical relativity (EOBNRv2). We find that by combining the existing TaylorF2 placement metric with EOBNRv2 waveforms, we can construct an effectual search for BBHs with component masses in the range $3{M}_{\ensuremath{\bigodot}}\ensuremath{\le}{m}_{1}$, ${m}_{2}\ensuremath{\le}25{M}_{\ensuremath{\bigodot}}$. We also show that the (computationally less expensive) TaylorF2 post-Newtonian waveforms can be used in place of EOBNRv2 waveforms when $M\ensuremath{\lesssim}11.4{M}_{\ensuremath{\bigodot}}$. Finally, we investigate the effect of modes other than the dominant $l=m=2$ mode in BBH searches. We find that for systems with $({m}_{1}/{m}_{2})\ensuremath{\le}1.68$ or inclination angle $\ensuremath{\iota}\ensuremath{\le}0.31$ or $\ensuremath{\iota}\ensuremath{\ge}2.68$ radians, there is no significant loss in the total possible signal-to-noise ratio due to neglecting modes other than $l=m=2$ in the template waveforms. For a source population uniformly distributed in spacial volume, over the entire sampled region of the component-mass space, the loss in detection rate (averaged over a uniform distribution of inclination angle and sky-location/polarization angles) remains below $\ensuremath{\sim}11%$. For binaries with high mass ratios and $0.31\ensuremath{\le}\ensuremath{\iota}\ensuremath{\le}2.68$, including higher-order modes could increase the signal-to-noise ratio by as much as 8% in Advanced LIGO. Our results can be used to construct matched-filter searches in Advanced LIGO and Advanced Virgo.

Interpolation in waveform space: Enhancing the accuracy of gravitational waveform families using numerical relativity

Matched filtering for the identification of compact object mergers in gravitational wave antenna data involves the comparison of the data stream to a bank of template gravitational waveforms. Typically the template bank is constructed from phenomenological waveform models, since these can be evaluated for an arbitrary choice of physical parameters. Recently it has been proposed that singular value decomposition (SVD) can be used to reduce the number of templates required for detection. As we show here, another benefit of SVD is its removal of biases from the phenomenological templates along with a corresponding improvement in their ability to represent waveform signals obtained from numerical relativity (NR) simulations. Using these ideas, we present a method that calibrates a reduced SVD basis of phenomenological waveforms against NR waveforms in order to construct a new waveform approximant with improved accuracy and faithfulness compared to the original phenomenological model. The new waveform family is given numerically through the interpolation of the projection coefficients of NR waveforms expanded onto the reduced basis and provides a generalized scheme for enhancing phenomenological models.

MultidetectorF-statistic metric for short-duration nonprecessing inspiral gravitational-wave signals

We derive explicit expressions for the multi-detector F-statistic metric applied to short-duration non-precessing inspiral signals. This is required for template bank production associated with coherent searches for short-duration non-precessing inspiral signals in gravitational-wave data from a network of detectors. We compare the metric's performance with explicit overlap calculations for all relevant dimensions of parameter space and find the metric accurately predicts the loss of detection statistic above overlaps of 95%. We also show the effect that neglecting the variations of the detector response functions has on the metric.

Gravitational waves from spinning eccentric binaries**This paper is dedicated to the memory of our colleague and friend Péter Csizmadia a young physicist, computer expert and one of the best Hungarian mountaineers who disappeared in China’s Sichuan near the Ren Zhong Feng peak of the Himalayas on 23 Oct. 2009. We started to develop CBwaves jointly with Péter a couple of

This paper is to introduce a new software called CBwaves which provides a fast and accurate computational tool to determine the gravitational waveforms yielded by generic spinning binaries of neutron stars and/or black holes on eccentric orbits. This is done within the post-Newtonian (PN) framework by integrating the equations of motion and the spin precession equations, while the radiation field is determined by a simultaneous evaluation of the analytic waveforms. In applying CBwaves various physically interesting scenarios have been investigated. In particular, we have studied the appropriateness of the adiabatic approximation, and justified that the energy balance relation is indeed insensitive to the specific form of the applied radiation reaction term. By studying eccentric binary systems, it is demonstrated that circular template banks are very ineffective in identifying binaries even if they possess tiny residual orbital eccentricity, thus confirming a similar result obtained by Brown and Zimmerman (2010 Phys. Rev. D 81 024007). In addition, by investigating the validity of the energy balance relation we show that, contrary to the general expectations, the PN approximation should not be applied once the PN parameter gets beyond the critical value ∼0.08 − 0.1. Finally, by studying the early phase of the gravitational waves emitted by strongly eccentric binary systems—which could be formed e.g. in various many-body interactions in the galactic halo—we have found that they possess very specific characteristics which may be used to identify these type of binary systems.

Detecting binary neutron star systems with spin in advanced gravitational-wave detectors

The detection of gravitational waves from binary neutron stars is a major goal of the gravitational-wave observatories Advanced LIGO and Advanced Virgo. Previous searches for binary neutron stars with LIGO and Virgo neglected the component stars' angular momentum (spin). We demonstrate that neglecting spin in matched-filter searches causes advanced detectors to lose more than 3% of the possible signal-to-noise ratio for 59% (6%) of sources, assuming that neutron star dimensionless spins, $c\mathbf{J}/GM^2$, are uniformly distributed with magnitudes between 0 and 0.4 (0.05) and that the neutron stars have isotropically distributed spin orientations. We present a new method for constructing template banks for gravitational wave searches for systems with spin. We present a new metric in a parameter space in which the template placement metric is globally flat. This new method can create template banks of signals with non-zero spins that are (anti-)aligned with the orbital angular momentum. We show that this search loses more than 3% of the maximium signal-to-noise for only 9% (0.2%) of BNS sources with dimensionless spins between 0 and 0.4 (0.05) and isotropic spin orientations. Use of this template bank will prevent selection bias in gravitational-wave searches and allow a more accurate exploration of the distribution of spins in binary neutron stars.

Summed parallel infinite impulse response filters for low-latency detection of chirping gravitational waves

With the upgrade of current gravitational wave detectors, the first detection of gravitational wave signals is expected to occur in the next decade. Low-latency gravitational wave triggers will be necessary to make fast follow-up electromagnetic observations of events related to their source, e.g., prompt optical emission associated with short gamma-ray bursts. In this paper we present a new time-domain low-latency algorithm for identifying the presence of gravitational waves produced by compact binary coalescence events in noisy detector data. Our method calculates the signal to noise ratio from the summation of a bank of parallel infinite impulse response (IIR) filters. We show that our summed parallel infinite impulse response (SPIIR) method can retrieve the signal to noise ratio to greater than 99% of that produced from the optimal matched filter. We emphasise the benefits of the SPIIR method for advanced detectors, which will require larger template banks.

Performance of a Chirplet-based analysis for gravitational-waves from binary black-hole mergers

The gravitational wave (GW) signature of a binary black hole (BBH) coalescence is characterized by rapid frequency evolution in the late inspiral and merger phases. For a system with total mass larger than 100 M☉, ground based GW detectors are sensitive to the merger phase, and the in-band whitened waveform is a short-duration transient lasting about 10-30 ms. For a symmetric mass system with total mass between 10 and 100 M☉, the detector is sensitive instead to the inspiral phase and the in-band signal has a longer duration, between 30 ms -3 s. Omega is a search algorithm for GW bursts that, with the assumption of locally stationary frequency evolution, uses sine-Gaussian wavelets as a template bank to decompose interferometer strain data. The local stationarity of sine-Gaussian waveforms induces a performance loss for the detection of lower mass BBH signatures, due to the mismatch between template and signal. We present the performance of a modified version of the Omega algorithm, Chirplet Omega, which allows a linear variation of frequency, to target BBH coalescences. The use of Chirplet-like templates enhances the measured signal-to-noise ratio due to less mismatch between template and data, and increases the detectability of lower mass BBH coalescences. We present the results of a performance study of Chirplet Omega in colored Gaussian noise at initial LIGO sensitivity.

Accessibility of the gravitational-wave background due to binary coalescences to second and third generation gravitational-wave detectors

Compact binary coalescences, such as binary neutron stars or black holes, are among the most promising candidate sources for the current and future terrestrial gravitational-wave detectors. While such sources are best searched using matched template techniques and chirp template banks, integrating chirp signals from binaries over the entire universe also leads to a gravitational-wave background (GWB). In this paper we systematically scan the parameter space for the binary coalescence GWB models, taking into account uncertainties in the star formation rate and in the delay time between the formation and coalescence of the binary, and we compare the computed GWB to the expected sensitivities of the second and third generation gravitational-wave detector networks. We find that second generation detectors are likely to detect the binary coalescence GWB, while the third generation detectors will probe most of the available parameter space. The binary coalescence GWB will, in fact, be a foreground for the third generation detectors, potentially masking the GWB background due to cosmological sources. Accessing the cosmological GWB with third generation detectors will therefore require identification and subtraction of all inspiral signals from all binaries in the detectors' frequency band.

Towards low-latency real-time detection of gravitational waves from compact binary coalescences in the era of advanced detectors

Electromagnetic (EM) follow-up observations of gravitational wave (GW) events will help shed light on the nature of the sources, and more can be learned if the EM follow-ups can start as soon as the GW event becomes observable. In this paper, we propose a computationally efficient time-domain algorithm capable of detecting gravitational waves (GWs) from coalescing binaries of compact objects with nearly zero time delay. In case when the signal is strong enough, our algorithm also has the flexibility to trigger EM observation before the merger. The key to the efficiency of our algorithm arises from the use of chains of so-called Infinite Impulse Response (IIR) filters, which filter time-series data recursively. Computational cost is further reduced by a template interpolation technique that requires filtering to be done only for a much coarser template bank than otherwise required to sufficiently recover optimal signal-to-noise ratio. Towards future detectors with sensitivity extending to lower frequencies, our algorithm's computational cost is shown to increase rather insignificantly compared to the conventional time-domain correlation method. Moreover, at latencies of less than hundreds to thousands of seconds, this method is expected to be computationally more efficient than the straightforward frequency-domain method.

Reduced basis representations of multi-mode black hole ringdown gravitational waves

We construct compact and high-accuracy reduced basis (RB) representations of single and multiple quasinormal modes (QNMs). The RB method determines a hierarchical and relatively small set of the most relevant waveforms. We find that the exponential convergence of the method allows for a dramatic compression of template banks used for ringdown searches. Compressing a catalog with a minimal match MM = 0.99, we find that the selected RB waveforms are able to represent any QNM, including those not in the original bank, with extremely high accuracy, typically less than 10−13. We then extend our studies to two-mode QNMs. Inclusion of a second mode is expected to help with detection, and might make it possible to infer details of the progenitor of the final black hole. We find that the number of RB waveforms needed to represent any two-mode ringdown waveform with the above high accuracy is smaller than the number of metric-based, one-mode templates with MM = 0.99. For unconstrained two modes, which would allow for consistency tests of general relativity, our high accuracy RB has around 104 fewer waveforms than the number of metric-based templates for MM = 0.99. The number of RB elements grows only linearly with the number of multipole modes versus exponentially with the standard approach, resulting in very compact representations even for many multiple modes. The results of this paper open the possibility of searches of multi-mode ringdown gravitational waves.

Estimating the sensitivity of wide-parameter-space searches for gravitational-wave pulsars

This paper presents an in-depth study of how to estimate the sensitivity of searches for gravitational-wave pulsars -- rapidly-rotating neutron stars which emit quasi-sinusoidal gravitational waves. It is particularly concerned with searches over a wide range of possible source parameters, such as searches over the entire sky and broad frequency bands. Traditional approaches to estimating the sensitivity of such searches use either computationally-expensive Monte Carlo simulations, or analytic methods which sacrifice accuracy by making an unphysical assumption about the population of sources being searched for. This paper develops a new, analytic method of estimating search sensitivity which does not rely upon this unphysical assumption. Unlike previous analytic methods, the new method accurately predicts the sensitivity obtained using Monte Carlo simulations, while avoiding their computational expense. The change in estimated sensitivity due to properties of the search template bank, and the geographic configuration of the gravitational wave detector network, are also investigated.

Improvement of the performance of a classical matched filter by an independent component analysis preprocessing

Current gravitational wave searches for compact binaries coalescence are done using a bank of templates (matched filters) on each running detector. Given a network of interferometers, we propose to use a denoising strategy based on an independent component analysis which considers two interferometers at a time and then to use a standard matched filter on the processed data. We show that this method allows to lower the level of noise and increases the signal-to-nose ratio at the output of the matched filter.

Efficiently enclosing the compact binary parameter space by singular-value decomposition

Gravitational-wave searches for the merger of compact binaries use matched-filtering as the method of detecting signals and estimating parameters. Such searches construct a fine mesh of filters covering a signal parameter space at high density. Previously it has been shown that singular value decomposition can reduce the effective number of filters required to search the data. Here we study how the basis provided by the singular value decomposition changes dimension as a function of template bank density. We will demonstrate that it is sufficient to use the basis provided by the singular value decomposition of a low density bank to accurately reconstruct arbitrary points within the boundaries of the template bank. Since this technique is purely numerical it may have applications to interpolating the space of numerical relativity waveforms.

Banks of templates for directed searches of gravitational waves from spinning neutron stars

We construct efficient banks of templates suitable for directed searches of almost monochromatic gravitational waves originating from spinning nuetron stars in our Galaxy in data being collected by currently operating interferometric detectors. We thus assume that the position of the gravitational-wave source in the sky is known, but we do not assume that the wave's frequency and its derivatives are a priori known. In the construction we employ simplified model of the signal with constant amplitude and phase which is a polynomial function of time. All our template banks enable usage of the fast Fourier transform algorithm in the computation of the maximum-likelihood F-statistic for nodes of the grids defining the bank. We study and employ the dependence of the grid's construction on the choice of the position of the observational interval with respect to the origin of time axis. We also study the usage of the fast Fourier transform algorithms with non-standard frequency resolutions achieved by zero padding or folding the data. In the case of the gravitational-wave signal with one spindown parameter included we have found grids with covering thicknesses which are only 0.1%--16% larger than the thickness of the optimal two-dimensional hexagonal covering.

Accuracy and effectualness of closed-form, frequency-domain waveforms for nonspinning black hole binaries

The coalescences of binary black hole (BBH) systems, here taken to be non-spinning, are among the most promising sources for gravitational wave (GW) ground-based detectors, such as LIGO and Virgo. To detect the GW signals emitted by BBHs, and measure the parameters of the source, one needs to have in hand a bank of GW templates that are both effectual (for detection), and accurate (for measurement). We study the effectualness and the accuracy of the two types of parametrized banks of templates that are directly defined in the frequency-domain by means of closed-form expressions, namely 'post-Newtonian' (PN) and 'phenomenological' models. In absence of knowledge of the exact waveforms, our study assumes as fiducial, target waveforms the ones generated by the most accurate version of the effective one body (EOB) formalism. We find that, for initial GW detectors the use, at each point of parameter space, of the best closed-form template (among PN and phenomenological models) leads to an effectualness >97% over the entire mass range and >99% in an important fraction of parameter space; however, when considering advanced detectors, both of the closed-form frequency-domain models fail to be effectual enough in significant domains of the two-dimensional [total mass and mass ratio] parameter space. Moreover, we find that, both for initial and advanced detectors, the two closed-form frequency-domain models fail to satisfy the minimal required accuracy standard in a very large domain of the two-dimensional parameter space. In addition, a side result of our study is the determination, as a function of the mass ratio, of the maximum frequency at which a frequency-domain PN waveform can be 'joined' onto a NR-calibrated EOB waveform without undue loss of accuracy.

Analytic Modeling of Tidal Effects in the Relativistic Inspiral of Binary Neutron Stars

To detect the gravitational-wave (GW) signal from binary neutron stars and extract information about the equation of state of matter at nuclear density, it is necessary to match the signal with a bank of accurate templates. We present the two longest (to date) general-relativistic simulations of equal-mass binary neutron stars with different compactnesses, C=0.12 and C=0.14, and compare them with a tidal extension of the effective-one-body (EOB) model. The typical numerical phasing errors over the ≃22 GW cycles are Δϕ≃±0.24 rad. By calibrating only one parameter (representing a higher-order amplification of tidal effects), the EOB model can reproduce, within the numerical error, the two numerical waveforms essentially up to the merger. By contrast, the third post-Newtonian Taylor-T4 approximant with leading-order tidal corrections dephases with respect to the numerical waveforms by several radians.

Detection of GW bursts with chirplet-like template families

Gravitational wave (GW) burst detection algorithms typically rely on the hypothesis that the burst signal is ‘locally stationary’, that is with slow variations of its frequency. Under this assumption, the signal can be decomposed into a small number of wavelets with constant frequency. This justifies the use of a family of sine-Gaussian wavelets in the Omega pipeline, one of the algorithms used in LIGO–Virgo burst searches. However, there are plausible scenarios where the burst frequency evolves rapidly, such as in the merger phase of a binary black-hole and/or neutron-star coalescence. In those cases, the local stationarity of sine Gaussians induces performance losses, due to the mismatch between the template and the actual signal. We propose an extension of the Omega pipeline based on chirplet-like templates. Chirplets incorporate an additional parameter, the chirp rate, to control the frequency variation. In this paper, we show that the Omega pipeline can easily be extended to include a chirplet template bank. We illustrate the method on a simulated data set, with a family of phenomenological binary black-hole coalescence waveforms embedded into Gaussian LIGO/Virgo-like noise. Chirplet-like templates result in an enhancement of the measured signal-to-noise ratio.