Estimation of parameters of gravitational wave signals from coalescing binaries

Abstract

With the ongoing construction of several large and medium scale laser interferometric gravitational wave antennas around the globe, the problem of the detection of gravitational waves has acquired great impetus. Since gravitational wave signals from astrophysical sources are expected to be weak, despite the state of the art technology being employed, the development of optimal signal extraction techniques and the consequent accurate determination of the parameters of the signal is of major importance. Coalescing binary systems are one of the most promising sources of gravitational waves. One reason is that such sources are easier to model and thus one can design detection strategies tuned to such signals. A lot of attention has been devoted in the literature studying such techniques and most of the work has revolved around matched filtering and maximum likelihood estimation. In a previous work, Monte Carlo simulations were carried out of the detection process using matched filtering for the initial LIGO-VIRGO configuration for the first post-Newtonian corrected coalescing binary waveform. We had compared the results of our simulations with available estimates obtained from covariance matrix considerations of the errors in the determination of the parameters. Our results showed that the covariance matrix underestimates, by over a factor of two, the actual errors in the estimation of parameters even when the signal-to-noise ratio (SNR) is as high as 10. Sources having SNR higher than 10 are expected to be few and hence this issue is of major concern. In this paper we probe the question as to why the Monte Carlo simulations give such high errors as opposed to those obtained via the covariance matrix. We present a computationally viable statistical model of the distribution of the maximum likelihood estimates (MLEs) of the parameters. This model reproduces the essential features of the Monte Carlo simulations, thereby explaining the large root mean square errors in the estimates obtained in numerical experiments. The chief reason for the large errors seems to be the fact that the probability distribution of the estimated parameters is multimodal. Though only the central peak (corresponding to the actual values of the parameters) is dominant, the subsidary peaks occur "far" away thereby contributing to large variances. We suggest that the variance or the standard deviation of an estimated parameter may not provide the best measure of the error, for the kind of situation we encounter here. We therefore propose another criterion by which the MLE should be judged. In order to illustrate the model we have considered the Newtonian as well as the first post-Newtonian corrected waveform. We have assumed Gaussian noise with a power spectrum typical of the LIGO-VIRGO type of detectors. The model we have used, however, is quite general, and robust, and will be relevant to many other parameter estimation problems.