Searching for periodic sources with LIGO

We investigate the computational requirements for all-sky, all-frequency searches for gravitational waves from spinning neutron stars, using archived data from interferometric gravitational wave detectors such as LIGO. These sources are expected to be weak, so the optimal strategy involves coherent accumulation of signal-to-noise using Fourier transforms of long stretches of data (months to years). Earth-motion-induced Doppler shifts, and intrinsic pulsar spindown, will reduce the narrow-band signal-to-noise by spreading power across many frequency bins; therefore, it is necessary to correct for these effects before performing the Fourier transform. The corrections can be implemented by a parametrized model, in which one does a search over a discrete set of parameter values (points in the parameter space of corrections). We define a metric on this parameter space, which can be used to determine the optimal spacing between points in a search; the metric is used to compute the number of independent parameter-space points Np that must be searched, as a function of observation time T. This method accounts automatically for correlations between the spindown and Doppler corrections. The number Np(T) depends on the maximum gravitational wave frequency and the minimum spindown age τ=f/ḟ that the search can detect. The signal-to-noise ratio required, in order to have 99% confidence of a detection, also depends on Np(T). We find that for an all-sky, all-frequency search lasting T=107 s, this detection threshold is hc≈(4–5)h3/yr, where h3/yr is the corresponding 99% confidence threshold if one knows in advance the pulsar position and spin period. We define a coherent search, over some data stream of length T, to be one where we apply a correction, followed by a fast Fourier transform of the data, for every independent point in the parameter space. Given realistic limits on computing power, and assuming that data analysis proceeds at the same rate as data acquisition (e.g., 10 days of data gets analyzed in ∼10 days), we can place limitations on how much data can be searched coherently. In an all-sky search for pulsars having gravity-wave frequencies f~1000 yr, one can coherently search ∼18 days of data on a teraflops computer. In contrast, a teraflops computer can only perform a ∼0.8-day coherent search for pulsars with frequencies f<~1 kHz and spindown ages as low as 40 yr. In addition to all-sky searches we consider coherent directed searches, where one knows in advance the source position but not the period. (Nearby supernova remnants and the galactic center are obvious places to look.) We show that for such a search, one gains a factor of ∼10 in observation time over the case of an all-sky search, given a 1 Tflops computer. The enormous computational burden involved in coherent searches indicates the need for alternative data analysis strategies. As an example we briefly discuss the implementation of a simple hierarchical search in the last section of the paper. Further work is required to determine the optimal approach