4.8 False alarms, detection threshold and coincident observation

Gravitational-wave event rates in initial interferometers is expected to be rather low: about a few per year. Therefore, one has to set a high threshold, so that the noise-generated false alarms mimicking an event are negligible.

For a detector output sampled at 1 kHz and processed through a large number of filters, say 103, one has ∼ 3 × 1013 instances of noise in a year. If the filtered noise is Gaussian, then the probability P(x ) of observing an amplitude in the range of x to x + dx is

( ) ---1-- −-x2 P (x) dx = √ --- exp 2σ2 dx, (71 ) 2π σ
where σ is the standard deviation. The above probability-distribution function implies that the probability that the noise amplitude is greater than a given threshold η is
∫ ∫ ( ) ∞ 1 ∞ − x2 P(x |x ≥ η) = P (x)dx = √----- exp 2σ2- dx. (72 ) η 2 πσ η
Demanding that no more than one noise-generated false alarm occur in a year’s observation means that P (x|x ≥ η) = 1∕(3 × 1013). Solving this equation for η, one finds that η ≃ 7.5σ in order that false alarms are negligible in a year’s observation. Therefore, a source is detectable only if its amplitude is significantly larger than the effective noise amplitude, i.e., f&tidle;h(f ) ≫ hn(f ).

The reason for accepting only such high-sigma events is that the event rate of a transient source, i.e., a source lasting for a few seconds to minutes, such as a binary inspiral, could be as low as a few per year, and the noise generated false alarms, at low SNRs ∼ 3–4, over a period of a year, tend to be quite large. Setting higher thresholds for detection helps in removing spurious, noise generated events. However, signal enhancement techniques (cf. Section 5) make it possible to detect a signal of relatively low amplitude, provided there are a large number of wave cycles and the shape of the wave is known accurately.

Real detector noise is neither Gaussian nor stationary and therefore the filtered noise cannot be expected to obey these properties either. One of the most challenging problems is how to remove or veto the false alarm generated by a non-Gaussian and/or nonstationary background. There has been some effort to address the issue of non-Gaussianity [125] and nonstationarity [263]; more work is needed in this direction. However, it is expected that the availability of a network of gravitational wave detectors alleviates the problem to some extent. This is because a high amplitude gravitational wave event will be coincidentally observed in several detectors, although not necessarily with the same SNR, while false alarms are, in general, not coincident, as they are normally produced by independent sources located close to the detectors.

We have seen that coincident observations help to reduce the false alarm rate significantly. The rate can be further reduced, and possibly even nullified, by subjecting coincident events to further consistency checks in a detector network consisting of four or more detectors. As discussed in Section 2, each gravitational wave event is characterized by five kinematic (or extrinsic) observables: location of the source with respect to the detector (D, 𝜃,φ ) and the two polarizations (h ,h ) + ×. Each detector in a network measures a single number, say the amplitude of the wave. In addition, in a network of N detectors, there are N − 1 independent time delays in the arrival times of the wave at various detector locations, giving a total of 2N − 1 observables. Thus, the minimum number of detectors needed to reconstruct the wave and its source is N = 3. More than three detectors in a network will have redundant information that will be consistent with the quantities inferred from any three detectors, provided the event is a true coincident event and not a chance coincidence, and most likely a true gravitational wave event. In a detector network consisting of N (≥ 4) detectors, one can perform 2N − 6 consistency checks. Such consistency checks further reduce the number of false alarms.

When the shape of a signal is known, matched filtering is the optimal strategy to pull out a signal buried in Gaussian, stationary noise (see Section 5.1). The presence of high-amplitude transients in the data can render the background nonstationary and non-Gaussian, therefore matched filtering is not necessarily an optimal strategy. However, the knowledge of a signal’s shape, especially when it has a broad bandwidth, can be used beyond matched filtering to construct a 2 χ veto [31Jump To The Next Citation Point] to distinguish between triggers caused by a true signal from those caused by high-amplitude transients or other artifacts. One specific implementation of the χ2 veto compares the expected signal spectrum with the real spectrum to quantify the confidence with which a trigger can be accepted to be caused by a true gravitational wave signal and has been the most powerful method for greatly reducing the false alarm rate. We shall discuss the 2 χ veto in more detail in Section 5.1.

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