Deepest All-Sky Surveys for Continuous Gravitational Waves
Holger J. Pletsch
[This is an invited article from Dr. Holger J. Pletsch who is the recipient of the 2009 GWIC (Gravitational Wave International Committee) Thesis Prize for his PhD thesis “Data Analysis for Continuous Gravitational Waves: Deepest All-Sky Surveys” (PDF). The thesis also received the 2009 Dieter Rampacher Prize of the Max Planck Society in Germany -- awarded to its youngest Ph.D. candidates usually between the ages of 25 and 27 for their outstanding doctoral work. -- 2Physics.com]
Author: Holger J. Pletsch
Affiliation: Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut) and Leibniz Universität Hannover
Besides validating Einstein's theory of General Relativity, direct detection of gravitational waves will also constitute an important new astronomical tool. Prime target sources of continuous gravitational waves (CW) for current Earth-based laser-interferometric detectors as LIGO  are rapidly spinning compact objects, such as neutron stars, with nonaxisymmetric deformations .
Very promising searches are all-sky surveys for prior unknown CW emitters. As most neutron stars are electromagnetically invisible, gravitational-wave observations might allow to reveal completely new populations of neutron stars. Therefore, a CW detection could potentially be extremely helpful for neutron-star astrophysics. Even the null results of today's search efforts, yielding observational upper limits , already constrain the physics of neutron stars.
2Physics articles by past winners of the GWIC Thesis Prize:
Henning Vahlbruch (2008): "Squeezed Light – the first real application starts now"
Keisuke Goda (2007): "Beating the Quantum Limit in Gravitational Wave Detectors"
Yoichi Aso (2006): "Novel Low-Frequency Vibration Isolation Technique for Interferometric Gravitational Wave Detectors"
Rana Adhikari (2003-5)*: "Interferometric Detection of Gravitational Waves : 5 Needed Breakthroughs"
*Note, the gravitational wave thesis prize was started initially by LIGO as a biannual prize, limited to students of the LIGO Scientific Collaboration (LSC). The first award covered the period from 1 July 2003 to 30 June 2005. In 2006, the thesis prize was adopted by GWIC, renamed, converted to an annual prize, and opened to the broader international community.
The expected CW signals are extremely weak, and deeply buried in the detector instrument noise. Thus, to extract these signals sensitive data analysis methods are requisite. A powerful method is coherent matched filtering, where the signal-to-noise ratio (SNR) increases with the square root of observation time. Hence, detection is a matter of observing long enough, to accumulate sufficient SNR.
The CW data analysis is further complicated by the fact that the terrestrial detector location Doppler-modulates the amplitude and phase of the waveform, as the Earth moves relative to the solar system barycenter (SSB). The parameters describing the signal's amplitude variation may be analytically eliminated by maximizing the coherent matched-filtering statistic. The remaining search parameters describing the signal's phase are the source's sky location, frequency and frequency derivatives. The resulting coherent detection statistic is commonly called the F-statistic .
However, what ultimately limits the sensitivity of all-sky surveys for unknown CW sources using the F-statistic is the finite computing power available. Such searches are computationally very expensive, because for maximum sensitivity one must convolve the full data set with many signal waveforms (templates) corresponding to all possible sources. But the number of templates required for a fully coherent F-statistic search increases as a high power of the coherent observation time. For a year of data, the computational cost to search a realistic range of parameter space exceeds the total computing power on Earth [4,5]. Thus a fully coherent search is limited to much shorter observation times.
Searching year-long data sets is accomplished by less costly hierarchical, so-called “semicoherent” methods [6,7]. The data is broken into segments, which are much smaller than one year. Just every segment is analyzed coherently, computing the F-statistic on a coarse grid of templates. Then the F-statistics from all segments (or statistics derived from F) are incoherently combined using a common fine grid of templates. This way phase information between segments is discarded, hence the term “semicoherent”.
A central long-standing problem in these semicoherent methods was the design of, and link between, the coarse and fine grids. Previous methods, while creative and clever, were arbitrary and ad hoc constructions. In most recent work , the optimal solution for the incoherent combination step has been found. The key quantity is the fractional loss, called mismatch, in expected F-statistic for a given signal at a nearby grid point. Locally Taylor-expanding the mismatch (to quadratic order) in the differences of the coordinates defines a positive definite metric. Previous methods considered parameter correlations in the F-statistic to only linear order in coherent integration time, discarding higher orders from the metric.
The F-statistic has strong "global" (large-scale) correlations in the physical coordinates, extending outside a region in which the mismatch is well approximated by the metric. In recent work , an improved understanding of the large-scale correlations in the F-statistic was found. Particularly, for realistic segment durations (a day or longer) it turned out to be also crucial to consider the fractional loss in F to second order in coherent integration time.
Exploiting these large-scale correlations in the coherent detection statistic F has lead to a significantly improved semicoherent search technique for CW signals . This novel method is optimal if the semicoherent detection statistic is taken to be the sum of one coarse-grid F-statistic value from each data segment.
More precisely, the improved understanding of large-scale correlations yields new coordinates on the phase parameter space. In these coordinates the first fully analytical metric for the incoherent combination step is obtained, accurately approximating the mismatch. Hence, the optimal (closest) coarse-grid point from each segment can be determined for any given fine-grid point in the incoherent combination step. So the new method combines the coherent segment results much more efficiently, since previous techniques did not use metric information beyond linear order in coherent integration time.
Fig.1: The Einstein@Home screensaver
The primary application area of this new technique is the volunteer distributed computing project Einstein@Home . Members of the public can sign up their home or office computers (hosts) through the web page, and download a screensaver. When idle, the screensaver displays (Fig.1), and in the background hosts automatically download small chunks of data from the servers, carry out analysis, and report back results. While more than 250k individuals have already contributed, the computational power (0.25 PFlop/s) achieved is in fact competitive with the world's largest supercomputers.
What improvement to expect from the new search technique for Einstein@Home? Via Monte Carlo simulations an implementation of this new method has been compared to the conventional Hough transform technique  that has been previously used on Einstein@Home. To provide a realistic comparison, simulated data covered the same time intervals as the input data of a recent Einstein@Home search run, which employed the conventional Hough technique. Those data, from LIGO’s fifth science run (S5), included 121 data segments of 25-hour duration. The false alarm probabilities were obtained using many simulated data sets with different realizations of stationary Gaussian white noise. To find the detection probabilities, different CW signals with fixed source gravitational-wave amplitude were added. Other source parameters were randomly drawn from uniform distributions.
Fig.2: (right click on the image to see higher resolution version) Performance demonstration of the new search method. Left panel: Receiver operating characteristic curves for fixed source strain amplitude. Right panel: Detection probability as a function of source strain amplitude, at 1% false alarm probability.
The results of this comparison are illustrated in Fig. 2. The right panel of Fig. 2 shows the detection efficiencies for different values of source gravitational-wave amplitude (strain), at a fixed 1% false alarm probability. The new method has been applied in two modes of operation: first, F-statistics were simply summed across segments; second, only ones or zeros were summed (number counts) depending upon whether F exceeds a predefined threshold in a given segment. In both modes of operation, the new technique performs significantly better than the conventional Hough method. For instance, 90% detection probability with the new method (in number-count operation mode) is obtained for a value of source strain amplitude about 6 times smaller as needed by the Hough method (which is also based number counts): thus the "distance reach" of the new technique is about 6 times larger. This increases the number of potentially detectable sources by more than 2 orders of magnitude, since the "visible" spatial volume increases as the cube of the distance, as illustrated in Fig. 3.
Fig.3: Artist’s illustration of increased "visible" spatial volume due to the novel search technique.
The current Einstein@Home search run  in fact deploys this new technique for the first time, analyzing about two years of LIGO’s most sensitive S5 data. The combination of a better search technique, plus more and more sensitive data, greatly increases the chance of making the first gravitational wave detection of a CW source. In the long term, the detection of CW signals will provide new means to discover and locate neutron stars, and will eventually provide unique insights into the nature of matter at high densities.
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 Einstein@Home: http://einstein.phys.uwm.edu/.