Introduction

1 Introduction

In this review we consider the problem of detection of deterministic gravitational-wave signals in the noise of a detector and the question of estimation of their parameters. The examples of deterministic signals are gravitational waves from rotating neutron stars, coalescing compact binaries, and supernova explosions. The case of detection of stochastic gravitational-wave signals in the noise of a detector is reviewed in [5 ]. A very powerful method to detect a signal in noise that is optimal by several criteria consists of correlating the data with the template that is matched to the expected signal. This matched-filtering technique is a special case of the maximum likelihood detection method. In this review we describe the theoretical foundation of the method and we show how it can be applied to the case of a very general deterministic gravitational-wave signal buried in a stationary and Gaussian noise.

Early gravitational-wave data analysis was concerned with the detection of bursts originating from supernova explosions [99 ]. It involved analysis of the coincidences among the detectors [52 ]. With the growing interest in laser interferometric gravitational-wave detectors that are broadband it was realized that sources other than supernovae can also be detectable [92 ] and that they can provide a wealth of astrophysical information [85 , 59 ]. For example the analytic form of the gravitational-wave signal from a binary system is known in terms of a few parameters to a good approximation. Consequently one can detect such a signal by correlating the data with the waveform of the signal and maximizing the correlation with respect to the parameters of the waveform. Using this method one can pick up a weak signal from the noise by building a large signal-to-noise ratio over a wide bandwidth of the detector [92 ]. This observation has led to a rapid development of the theory of gravitational-wave data analysis. It became clear that the detectability of sources is determined by optimal signal-to-noise ratio, Equation (24 ), which is the power spectrum of the signal divided by the power spectrum of the noise integrated over the bandwidth of the detector.

An important landmark was a workshop entitled Gravitational Wave Data Analysis held in Dyffryn House and Gardens, St. Nicholas near Cardiff, in July 1987 [86 ]. The meeting acquainted physicists interested in analyzing gravitational-wave data with the basics of the statistical theory of signal detection and its application to detection of gravitational-wave sources. As a result of subsequent studies the Fisher information matrix was introduced to the theory of the analysis of gravitational-wave data [40 , 58 ]. The diagonal elements of the Fisher matrix give lower bounds on the variances of the estimators of the parameters of the signal and can be used to assess the quality of astrophysical information that can be obtained from detections of gravitational-wave signals [32 , 57 , 18 ]. It was also realized that application of matched-filtering to some sources, notably to continuous sources originating from neutron stars, will require extraordinary large computing resources. This gave a further stimulus to the development of optimal and efficient algorithms and data analysis methods [87 ].

A very important development was the work by Cutler et al. [31 ] where it was realized that for the case of coalescing binaries matched filtering was sensitive to very small post-Newtonian effects of the waveform. Thus these effects can be detected. This leads to a much better verification of Einstein’s theory of relativity and provides a wealth of astrophysical information that would make a laser interferometric gravitational-wave detector a true astronomical observatory complementary to those utilizing the electromagnetic spectrum. As further developments of the theory methods were introduced to calculate the quality of suboptimal filters [9 ], to calculate the number of templates to do a search using matched-filtering [74 ], to determine the accuracy of templates required [24 ], and to calculate the false alarm probability and thresholds [50 ]. An important point is the reduction of the number of parameters that one needs to search for in order to detect a signal. Namely estimators of a certain type of parameters, called extrinsic parameters, can be found in a closed analytic form and consequently eliminated from the search. Thus a computationally intensive search needs only be performed over a reduced set of intrinsic parameters [58 , 50 , 60 ].

Techniques reviewed in this paper have been used in the data analysis of prototypes of gravitational-wave detectors [73 , 71 , 6 ] and in the data analysis of presently working gravitational-wave detectors [90 , 15 , 3 , 2 , 1 ].

We use units such that the velocity of light $c = 1$ .