All-sky search algorithms for monochromatic signals in resonant bar gravitational wave detector data

1 I N T RO D U C T I O N It is generally believed that the most intense gravitational waves (GWs) arriving at the Earth from remote sources in the Universe correspond to very short duration (,1 ms) bursts, generated in the explosion of a supernova (Thorne 1987), or in gamma-ray bursters (Roland, Frossati & Teyssier 1994). Since their very first origins, cylindrical bar GWantennas have been applied to the detection of this sort of event (Weber 1969), and the more modern cryogenic bars have also been used for this purpose, with considerably enhanced sensitivities (Astone et al. 1991; Astone et al. 1993; Hamilton et al. 1994): the long decay times of the oscillations of the bar make it well suited for the measurement of impulsive, short duration signals (Gibbons & Hawking 1971; Astone, Bonifazi & Pallottino 1990). It so happens, however, that some cylindrical GW antennas have been in continuous operation for many consecutive months, even years. This is the case, for example, with the Explorer detector, owned and operated by the Ricerche Onde Gravitazionali (ROG) group in Rome (Italy) and installed within the CERN premises in Geneva, Switzerland (Astone et al. 1993). Long-term operation naturally provides the appropriate background for a search of monochromatic signals in the detector data, as the requisite long integration times become available. Monochromatic signals are most probably generated by the rotation of asymmetric stars, such as a pulsar or a neutron star. The intensity of the GWs depends strongly on the amount of asymmetry of the source, and this is in turn dependent on its equation of state (Bonazzola & Gourgoulhon 1996). Reasonably optimistic upper bounds on typical star parameters give an extremely weak signal estimation of h , 10 (Thorne 1987), which must be seen against a noisy background. Clearly, long integration times are required to reveal this kind of signal. A systematic search for such a signal must face a practical difficulty which derives from the fact that the signal is received in the antenna Doppler-shifted as a result of the daily and yearly motions of the Earth – in addition to possible internal motions within the source if it is, e.g., in a binary system. Fourier analysis of long stretches of data results in high-frequency resolution (Kay 1990), thence in signal spread across several spectrum bins if it is Doppler shifted. This can naturally cause a significant reduction in the post-filter signal-to-noise ratio. The problem is easily overcome if the source position in the sky is known (or assumed) ahead of time by means of suitable corrections based on ephemeris calculations. Analyses of this type exist in the literature: traces of a pulsar in the centre of the supernova SN1987A were sought in four days of data generated by the 30-m Garching interferometer in 1989 March (Niebauer et al. 1993), and Frasca & La Posta studied almost four years of data generated by the room-temperature bar detector GEOGRAV in search of periodic signals from the Galactic Centre and the Large Magellanic Cloud (Frasca & La Posta 1991). More recently, Mauceli (Mauceli 1997) has looked for monochromatic GW signals coming Mon. Not. R. Astron. Soc. 301, 729–744 (1998) q 1998 RAS * E-mail: [email protected] (JAL) from the region of Tuc 47 and from the Galactic Centre in three months of data generated by the cryogenic detector ALLEGRO at Louisiana State University. A different strategy must of course be used for an all-sky search. The philosophy of the procedure put forward by Frasca & La Posta (Frasca & La Posta 1991) consists of the construction of a large bank of spectra, taken over shorter stretches of data such that the frequency resolution in each individual spectrum be sufficiently low that daily Doppler-shifted signals fit in a single spectral bin. Suitable comparison and averaging are thereafter applied to the spectra in order to draw conclusions about the intensity and/or bounds on signals. Astone et al. (Astone et al. 1997a; Astone 1998) have looked at one year (1991) of data taken by the above-mentioned Explorer detector in order to perform an all-sky search for monochromatic sources of GWs. Their method is based upon local maxima identification in a bank of spectra, followed by close up analyses of frequency peaks, looking for evidence of Doppler-shift patterns across the duration of the entire data set. In this paper we design and develop algorithms for the analysis of data generated by a resonant bar detector of GWs, in search of monochromatic signals within the sensitive frequency band of the system. We are also interested in an all-sky search, but adopt a different point of view. Rather than scanning a bank of spectra, we propose to use a matched filter technique to estimate both frequency and phase of candidate signals. We then set a threshold, using the Neyman–Pearson criterion, to select those events which have a given probability of crossing it as a consequence of pure random noise fluctuations. We have tested our methods in simulations with real Explorer detector data from 1991, and have seen that they perform very satisfactorily. We plan to apply our methods to the massive processing of long stretches of data from the same antenna in a future paper, in order to provide complementary analyses to the procedures and methods already reported in Astone et al. (1997b) and Astone (1998). The article is structured as follows. In Section 2 we present a few technical generalities and set the basic conventions of notation. Section 3 is devoted to a detailed study of a situation in which the signal has a frequency exactly equal to one of those in the discrete Fourier spectrum of the data (Lobo & Montero 1997); this corresponds to an idealized situation the consideration of which is methodologically useful, as it allows us to determine the phase of the signal, and to investigate the statistical properties of the filter output; it also characterizes the main guidelines for the more realistic study in subsequent sections. In Section 4 the method is illustrated with a signal artificially added to real detector data, which includes the estimation of the noise spectral density in the presence of such a signal. In Section 5 we address the real case, in which the signal frequency no longer exactly matches any of the discrete samples, so that it leaks across neighbouring spectrum bins (Lobo & Montero 1998), and also assess the statistical properties of the filter output (Montero 1998). Finally, in Section 6 we apply the method again to real data with an external control signal added, and show that it works satisfactorily. The paper closes with a summary of conclusions and future prospects. 2 L I N E A R DATA F I LT E R I N G We begin with a review of some fundamental concepts of linear data processing, fixing also the basic notation which we will be using throughout this article. In the general case, let uðnÞ (n 1⁄4 0; . . . ;N 1 1) be the discrete set of samples which constitute our experimental data. A linear filter consists of a discrete set of numbers gðn; miÞ depending on several parameters, mi, which acts on the experimental data as yðmiÞ 1⁄4 XN11 n1⁄40 gðn; miÞuðnÞ; ð1Þ producing what we shall call the filter output. It is usually assumed that uðnÞ is the sum of two different contributions: on the one hand is the signal, xðnÞ, the presence of which we want to assess, represented by a deterministic function, and on the other hand the noise rðnÞ, a stochastic process, uðnÞ 1⁄4 xðnÞ þ rðnÞ: ð2Þ For any choice of parameters, it is appropiate to ask for the filter response both to the signal, yx, and to the noise, yr, the latter also being a stochastic process. The ratio of the mean square values of these quantities is called in the literature the output signal-to-noise ratio (SN), r ; yx