Searching for periodic sources with LIGO. II: Hierarchical searches

The detection of quasi-periodic sources of gravitational waves requires the accumulation of signal to noise over long observation times. This represents the most difficult data analysis problem facing experimenters with detectors such as those at LIGO. If not removed, Earth-motion induced Doppler modulations and intrinsic variations of the gravitational-wave frequency make the signals impossible to detect. These effects can be corrected (removed) using a parametrized model for the frequency evolution. In a previous paper, we introduced such a model and computed the number of independent parameter space points for which corrections must be applied to the data stream in a coherent search. Since this number increases with the observation time, the sensitivity of a search for continuous gravitational-wave signals is computationally bound when data analysis proceeds at a similar rate to data acquisition. In this paper, we extend the formalism developed by Brady et al. [Phys. Rev. D 57, 2101 (1998)], and we compute the number of independent corrections ${N}_{p}(\ensuremath{\Delta}T,N)$ required for incoherent search strategies. These strategies rely on the method of stacked power spectra---a demodulated time series is divided into N segments of length $\ensuremath{\Delta}T,$ each segment is Fourier transformed, a power spectrum is computed, and the N spectra are summed up. This method is incoherent; phase information is lost from segment to segment. Nevertheless, power from a signal with fixed frequency (in the corrected time series) is accumulated in a single frequency bin, and amplitude signal to noise accumulates as $\ensuremath{\sim}{N}^{1/4}$ (assuming the segment length $\ensuremath{\Delta}T$ is held fixed). For fixed available computing power, there are optimal values for N and $\ensuremath{\Delta}T$ which maximize the sensitivity of a search in which data analysis takes a total time $N\ensuremath{\Delta}T.$ We estimate that the optimal sensitivity of an all-sky search that uses incoherent stacks is a factor of $2\ensuremath{-}4$ better than achieved using coherent Fourier transforms, assuming the same available computing power; incoherent methods are computationally efficient at exploring large parameter spaces. We also consider a two-stage hierarchical search in which candidate events from a search using short data segments are followed up in a search using longer data segments. This hierarchical strategy yields a further $20\ensuremath{-}60%$ improvement in sensitivity in all-sky (or directed) searches for old (\ensuremath{\geqslant}1000 yr) slow (\ensuremath{\leqslant}200 Hz) pulsars, and for young (\ensuremath{\geqslant}40 yr) fast (\ensuremath{\leqslant}1000 Hz) pulsars. Assuming enhanced LIGO detectors (LIGO-II) and ${10}^{12}$ flops of effective computing power, we examine the sensitivity to sources in three specialized classes. A limited area search for pulsars in the Galactic core would detect objects with gravitational ellipticities of $\ensuremath{\epsilon}\ensuremath{\gtrsim}5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ at 200 Hz; such limits provide information about the strength of the crust in neutron stars. Gravitational waves emitted by unstable r-modes of newborn neutron stars would be detected out to distances of $\ensuremath{\sim}8$ Mpc, if the r-modes saturate at a dimensionless amplitude of order unity and an optical supernova provides the position of the source on the sky. In searches targeting low-mass x-ray binary systems (in which accretion-driven spin up is balanced by gravitational-wave spin down), it is important to use information from electromagnetic observations to determine the orbital parameters as accurately as possible. An estimate of the difficulty of these searches suggests that objects with x-ray fluxes exceeding $2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8} {\mathrm{erg}\mathrm{}\mathrm{cm}}^{\ensuremath{-}2}{\mathrm{s}}^{\ensuremath{-}1}$ would be detected using the enhanced interferometers in their broadband configuration. This puts Sco X-1 on the verge of detectability in a broadband search; the amplitude signal to noise would be increased by a factor of order $\ensuremath{\sim}5\ensuremath{-}10$ by operating the interferometer in a signal-recycled, narrow-band configuration. Further work is needed to determine the optimal search strategy when limited information is available about the frequency evolution of a source in a targeted search.