LISA data analysis : The monochromatic binary detection and initial guess problems

We consider the detection and initial guess problems for the LISA gravitational wave detector. The detection problem is the problem of how to determine if there is a signal present in instrumental data and how to identify it. Because of the Doppler and planeprecession spreading of the spectral power of the LISA signal, the usual power spectrum approach to detection will have difficulty identifying sources. A better method must be found. The initial guess problem involves how to generate a priori values for the parameters of a parameter-estimation problem that are close enough to the final values for a linear leastsquares estimator to converge to the correct result. A useful approach to simultaneously solving the detection and initial guess problems for LISA is to divide the sky into many pixels and to demodulate the Doppler spreading for each set of pixel coordinates. The demodulated power spectra may then be searched for spectral features. We demonstrate that the procedure works well as a first step in the search for gravitational waves from monochromatic binaries. I. The detection problem The signal coming down from a spaceborne gravitational wave detector will contain signals from many close compact binaries, but, to look at the signal, one would see nothing but noise. This is because the instrument noise, integrated over the bandwidth of the detector, far exceeds the strength of any individual binary signal. This situation is common in data analysis, and the usual procedure for detecting weak coherent signals in the presence of large incoherent noise is to generate the power spectrum of the signal. Since the binary star signal is monochromatic, its power will be concentrated in a single frequency bin of the power spectrum, while the noise is spread over all frequencies. The signal will then be seen as a single spike rising above the neighboring noise spectrum. However, the usefulness of this power spectrum method for signal detection is limited for the spaceborne gravitational wave detectors, due to the motion of the detector around the sun and to the change in the orientation of the detector. As a result of these motions, the gravitational wave signal from a particular source will be strongly phase modulated, spreading the total power from the source over many frequency bins. It is therefore quite possible that none of the frequency components of the signal will be seen above the noise. To illustrate, we show in Fig. 1 the power spectrum of the gravitational wave signal from a binary star, where the signal frequency is 0.01 Hz and the amplitude is 3.5 × 10. The signal has been sampled at 10 s intervals for one year. This is the power spectrum that would be seen if the detector were at rest in the solar system. The power spectrum as seen in a moving detector is shown in Fig. 2. To generate this figure, the detector has been assumed to be the LISA detector [1], in which three free-flying spacecraft move on heliocentric trajectories in such a way as to remain at roughly constant distances from each other and to remain in a plane inclined by 60◦ to the plane of the ecliptic, the plane precessing about the ecliptic pole once per year. The spreading of the power over about 100 frequency bins can be clearly seen in the figure. If a noise spectrum, characteristic of the proposed LISA instrument noise, is added to the signal, the power spectrum in Fig. 3 results. As may be seen, the spreading of the binary signal power over many frequency bins has produced a power spectrum that cannot now be seen above the noise. One of the goals of this paper is to show how this “detection problem” may be solved. II. The initial guess problem The ultimate goal of gravitational wave detectors is to use the gravitational wave signals to determine the astrophysical properties of the gravitational wave source. Papers by Cutler and Vecchio [2,3] and by Moore and Hellings [4] have addressed the problem of what information is available in the signal and how well the parameters of the source may be determined from a signal with a particular signal-to-noise ratio. The data analysis technique to be used to extract the information from the signal is called linear least-squares parameter estimation. This technique accounts for the possible correlation of the parameters with each other and gives the best-fit values of the parameters and their formal standard deviation, with the assumption of a background of Gaussian noise. However, the “linear” in “linear least-squares” reminds us that this procedure will only work in a totally linear problem or in the limit that the problem may be linearized by writing it in terms of small quantities, the differences between the final best-fit values for the parameters and some a priori values. The signal from a monochromatic binary is characterized by seven independent param-