Bayesian approach to the follow-up of candidate gravitational wave signals

Ground-based gravitational wave laser interferometers (LIGO, GEO-600, Virgo, and Tama-300) have now reached high sensitivity and duty cycle. We present a Bayesian evidence-based approach to the search for gravitational waves, in particular aimed at the follow-up of candidate events generated by the analysis pipeline. We introduce and demonstrate an efficient method to compute the evidence and odds ratio between different models, and illustrate this approach using the specific case of the gravitational wave signal generated during the inspiral phase of binary systems, modeled at the leading quadrupole Newtonian order, in synthetic noise. We show that the method is effective in detecting signals at the detection threshold and it is robust against (some types of) instrumental artifacts. The computational efficiency of this method makes it scalable to the analysis of all the triggers generated by the analysis pipelines to search for coalescing binaries in surveys with ground-based interferometers, and to a whole variety of signal waveforms, characterized by a larger number of parameters.

Figure 1

The distribution of ${log}_{10}$ Bayes factors recovered from running the algorithm on 100 different Gaussian and stationary noise realizations, each of length 13 800 complex samples as described in the text. The distribution in values arises from random fluctuations of the noise, which is contrasted with Fig. 2, where the inherent uncertainty in the algorithm is shown. In both cases, the variation in Bayes factors found in data sets with no signal is extremely small compared to the change in Bayes factor that the presence of a signal produces, which is shown in Fig. 3 for a range of signal-to-noise ratios.

Figure 2

The distribution of ${log}_{10}$ Bayes factors recovered from running the algorithm 200 times on the same data set which contained no signal. The noise is modeled as a Gaussian and stationary random process. The distribution of results is caused by the probabilistic nature of the algorithm, and the range of this distribution can be reduced to have an arbitrarily small range at the expense of increased computation time. In this example and throughout the rest of the paper, the distribution width is chosen to be the same order as the uncertainty caused by different noise realizations, shown in Fig. 1.

Figure 3

The mean recovered ${log}_{10}$ Bayes factors from adding an inspiral signal of increasing signal-to-noise ratio, shown on the $x$-axis, to Gaussian and stationary noise. The algorithm was run on each data set (identical noise realization) 20 times: each point on the plot corresponds to the mean of the Bayes factor. Note that the Bayes factor grows exponentially with increasing SNR, such that it is an extremely sensitive indication of the presence of a signal. The signal and data parameters were chosen as described in the text.

Figure 4

The Bayes factors $B_{\mathrm{SN}}$ for inspiral model vs noise model in the presence of an injected ringdown to simulate such events in instrumental noise. Shown here are the results from two ringdowns, with decay times $\tau =1\mathrm{s}$ (dashed line) and 10 s (solid line), and varying signal-to-noise ratios plotted on the $x$ axis. From these results it is clear that the algorithm is robust against interference from this source, as only the $\tau =10\mathrm{s}$ caused an increase in the Bayes factor, and then only at very high SNR.

Figure 5

The amplitude spectra of the Gaussian (upper black) and Poisson (lower gray) contributions to the data used in Sec. 3e.

Figure 6

The results from injecting a Poissonian component with distribution $P(0.1)$ into a segment of Gaussian noise (and no GW signal) and running the algorithm 100 times. The presence of Poisson noise does not increase the Bayes factor for an inspiral signal, as the Poissonian noise does not match the template for this waveform. The results of plot should be compared with those obtained on pure Gaussian and stationary noise reported in Fig. 2.

Figure 7

The results from adding GW signals of varying signal-to-noise ratio onto the combined noise data from a Poissonian and Gaussian distribution, as described in the text. The resulting Bayes factors are lowered in comparison to Fig. 3.