Large-Scale surveys for continuous gravitational waves: from data preparation to multi-stage hierarchical follow-ups

The gravitational wave event GW150914 was the first direct detection of gravitational waves roughly 100 years after their prediction by Albert Einstein. The detection was a breakthrough, opening another channel to observe the Universe. Since then over 90 detections of merging compact objects have been made, most of them coalescences of binary black holes of different masses. There have been two black hole-neutron star, and two binary neutron-star mergers. Another breakthrough was the first binary neutron-star merger, GW170817, associated with a slew of electromagnetic observations, including a gamma-ray burst 1.7s after the merger. Compact binary coalescence events are cataclysmic events in which multiple solar masses are emitted in gravitational waves in ~seconds. Still, their gravitational wave detection requires sophisticated measuring devices: kilometer-scale laser interferometers. Another not yet detected form of gravitational radiation are continuous gravitational waves from e.g., but not limited to, fast-spinning neutron stars nonaxisymmetric relatively to their rotational axis. The gravitational wave amplitude on Earth is orders of magnitude weaker than the compact binary coalescence events, but, in the case of the nonaxisymmetric neutron star, is emitted as long as the neutron star is spinning and sustaining the deformation, which may be months to years. The gravitational wave is mostly emitted at twice the rotational frequency, with a possible frequency evolution (spin-down) due to the energy emitted by gravitational waves, as well as other braking mechanisms. This nearly monochromatic continuous wave is received by observers on Earth Doppler modulated by Earth's orbit and spin. Although the waveform is seemingly simple, the detection problem for signals from unknown sources is very challenging. The all-sky search for unknown neutron stars in our galaxy detailed in this work used the volunteer distributed computing project Einstein@Home and the ATLAS supercomputer for several months, taking tens of thousands of total CPU-time years to complete. In this work I describe the full-scale data analysis procedure, including data preparation, search set-up optimization and post-processing of search results, whose design and implementation is the core of my doctoral research work. I also present a number of observational results that demonstrate the real-world application of the methodologies that I designed.Das Gravitationswellenereignis GW150914 war der erste direkte Nachweis von Gravitationswellen rund 100 Jahre nach deren Vorhersage durch Albert Einstein. Die Entdeckung war ein Durchbruch und eröffnete einen weiteren Kanal zur Beobachtung des Universums. Seitdem wurden über 90 weitere verschmelzende kompakte Objekte entdeckt, die meisten binäre schwarze Löcher unterschiedlicher Masse, aber auch zweimal verschmelzende Schwarze Löcher mit Neutronensternen und zwei Verschmelzungen von binären Neutronensternen. Ein weiterer Durchbruch war die Beobachtung der ersten Verschmelzung zweier Neutronensterne, GW170817, die mit einer Reihe von elektromagnetischen Beobachtungen einherging, darunter ein Gammastrahlenausbruch 1.7s nach der Verschmelzung. Bei der Verschmelzung kompakter Objekte handelt es sich um kataklysmische Ereignisse, bei denen innerhalb von ~Sekunden mehrere Sonnenmassen in Form von Gravitationswellen ausgestoßen werden. Ihr Nachweis erfordert jedoch hochentwickelte Messgeräte: Laserinterferometer im Kilometermaßstab. Eine weitere, noch nicht nachgewiesene Form der Gravitationsstrahlung sind kontinuierliche Gravitationswellen, die z.B., aber nicht nur, von schnell rotierenden Neutronensternen ausgehen, die relativ zu ihrer Rotationsachse nicht achsensymmetrisch sind. Die Amplitude der kontinuierlichen Gravitationswellen auf der Erde ist um Größenordnungen schwächer als die der verschmelzenden kompakten Objekte, wird aber im Fall des nicht achsensymmetrischen Neutronensterns so lange abgestrahlt, wie der Neutronenstern rotiert und die Deformation aufrechterhält, was Monate bis Jahre sein können. Die Gravitationswelle wird meist mit der doppelten Rotationsfrequenz ausgestrahlt, wobei eine Frequenzentwicklung (Spin-down) aufgrund der von Gravitationswellen ausgesandten Energie, sowie anderer Bremsmechanismen möglich ist. Diese nahezu monochromatische, kontinuierliche Welle wird von einem Beobachter auf der Erde Doppler-moduliert durch die Erdumlaufbahn und die Erddrehung empfangen. Obwohl die Wellenform scheinbar einfach ist, ist das Problem des Nachweises von Signalen aus unbekannten Quellen eine große Herausforderung. Die in dieser Arbeit beschriebene Suche nach unbekannten Neutronensternen in unserer Galaxie über den kompletten Himmel verwendete über mehrere Monate hinweg das Volunteer-Computing-Projekt Einstein@Home und den ATLAS-Supercomputer und benötigte insgesamt Zehntausende von Jahren an Rechenzeit. In dieser Arbeit beschreibe ich das vollständige Datenanalyseverfahren einschließlich der Datenvorbereitung, der Optimierung der Suchparameter und der Nachbearbeitung der Suchergebnisse, dessen Entwurf und Implementierung das Kernstück meiner Doktorarbeit darstellt. Außerdem stelle ich eine Reihe von Beobachtungsergebnissen vor, welche die praktische Anwendung der von mir entwickelten Methoden demonstrieren

Loosely coherent search in LIGO O1 data for continuous gravitational waves from Terzan 5 and the galactic center

We report results of a search for continuous gravitational waves from a region covering the globular cluster Terzan 5 and the galactic center. Continuous gravitational waves are expected from fast-spinning, slightly non-axisymmetric isolated neutron stars as well as more exotic objects. The regions that we target are believed to be unusually abundant in neutron stars. We use a new loosely coherent search method that allows to reach unprecedented levels of sensitivity for this type of search. The search covers the frequency band 475-1500 Hz and frequency time derivatives in the range of [-3e-8, +1e-9] Hz/s, which is a parameter range not explored before with the depth reached by this search. As to be expected with only a few months of data from the same observing run, it is very difficult to make a confident detection of a continuous signal over such a large parameter space. A list of parameter space points that passed all the thresholds of this search is provided. We follow-up the most significant outlier on the newly released O2 data and cannot confirm it. We provide upper limits on the gravitational wave strength of signals as a function of signal frequency

Tile-Based Two-Dimensional Phase Unwrapping for Digital Holography Using a Modular Framework

A variety of physical and biomedical imaging techniques, such as digital holography, interferometric synthetic aperture radar (InSAR), or magnetic resonance imaging (MRI) enable measurement of the phase of a physical quantity additionally to its amplitude. However, the phase can commonly only be measured modulo 2π, as a so called wrapped phase map. Phase unwrapping is the process of obtaining the underlying physical phase map from the wrapped phase. Tile-based phase unwrapping algorithms operate by first tessellating the phase map, then unwrapping individual tiles, and finally merging them to a continuous phase map. They can be implemented computationally efficiently and are robust to noise. However, they are prone to failure in the presence of phase residues or erroneous unwraps of single tiles. We tried to overcome these shortcomings by creating novel tile unwrapping and merging algorithms as well as creating a framework that allows to combine them in modular fashion. To increase the robustness of the tile unwrapping step, we implemented a model-based algorithm that makes efficient use of linear algebra to unwrap individual tiles. Furthermore, we adapted an established pixel-based unwrapping algorithm to create a quality guided tile merger. These original algorithms as well as previously existing ones were implemented in a modular phase unwrapping C++ framework. By examining different combinations of unwrapping and merging algorithms we compared our method to existing approaches. We could show that the appropriate choice of unwrapping and merging algorithms can significantly improve the unwrapped result in the presence of phase residues and noise. Beyond that, our modular framework allows for efficient design and test of new tile-based phase unwrapping algorithms. The software developed in this study is freely available

Tile-based phase unwrapping schematic.

<p>A wrapped phase image is tessellated into rectangular subregions called tiles. First these tiles are unwrapped individually and second they are merged to a continuous surface.</p

Unwrap of a synthetic phase map with noise and residues.

<p><b>A:</b> Wrapped phase distribution with a noise level of <i>σ</i> = 0.3. The residue is indicated by an arrow and the a close-up of the surrounding area is shown in an inset. <b>B:</b> Pixel-based SRNCP unwrap. <b>C:</b> Tilebased unwrap with <i>N</i>_<i>τ</i> = 40 × 40 tiles using Strand’s tile unwrapper (<i>N</i>_<i>ρ</i> = 20) and Strand’s unidirectional merger. <b>D:</b> Unwrapped phase using the proposed MLSQU tile unwrapper with <i>P</i> = 2, <i>N</i>_<i>ρ</i> = 40 and the unidirectional tile merger. Tile count is <i>N</i>_<i>τ</i> = 20 × 20, resulting in double the tile size compared to C. <b>E:</b> Same as D, but using the proposed tile-based <i>τ</i>SRNCP algorithm for merging. <b>F:</b> Unwrapped phase using MLSQU tile unwrapper with <i>P</i> = 6, <i>N</i>_<i>ρ</i> = 60 and the tile-based <i>τ</i>SRNCP merger. The image was tessellated into a 2 × 2 grid. All images have dimensions of 600 × 600 pixels and the same color scale as <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143186#pone.0143186.g002" target="_blank">Fig 2</a>.</p

Phase unwrap of ZMTH3 canine adenoma cells captured with digital holography.

<p><b>A:</b> Wrapped phase map. <b>B:</b> Pixel-based unwrap with the SRNCP algorithm. <b>C:</b> Unwrap using the proposed approach by tessellating the image into 8 × 8 tiles and unwrapping individual tiles with the model-based approach using polynomial order <i>P</i> = 4. Tiles were merged with the unidirectional merger. A magnification of a region with phase residues is shown. The phase maps consist of 1400 × 1400 pixels. Scale bar is 15 <i>μ</i><i>m</i>.</p

Comparison of phase unwrap with different algorithms for various noise levels.

<p><b>A,B,C:</b> Wrapped phase distributions for different noise levels. <b>D,E,F:</b> Tile-based unwrap with a tessellation into 20 × 20 tiles. Tile unwrapping was performed with the model-based least squares unwrapper (MLSQU) using <i>P</i> = 2, <i>N</i>_<i>ρ</i> = 40. Merging was performed with the tile-based <i>τ</i>SRNCP merger. <b>G,H,I:</b> Tile-based unwrap with a tessellation into 40 × 40 tiles. Tile unwrapping was done with Strand’s unwrapper (<i>N</i>_<i>r</i><i>ho</i> = 20) and a unidirectional merger. This corresponds to Strand’s original algorithm. <b>J,K,L:</b> Unwrapped phase map using the pixel-based SRNCP algorithm. For a full resolution graphic see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143186#pone.0143186.s001" target="_blank">S1 Fig</a>. Algorithms proposed in this paper are in black boxes.</p

Deviation of the unwrapped phase from a noisy ground truth for different algorithms.

<p>The mean deviation of 25 phase images to the ground truth with a white Gaussian noise of specified <i>σ</i> is shown. (▿) The proposed algorithm using a model-based tile unwrapper with polynomial base <i>P</i> = 2 and <i>N</i>_<i>τ</i> = 20 × 20 tiles and <i>N</i>_<i>ρ</i> = 40 steps for minimization. Merging was performed using our <i>τ</i>SRNCP merger. (•) The pixel-based SRNCP algorithm. At <i>σ</i> = 1.0 the deviation is outside the range of the plot with a value of 7.7 radians. (■) Strand’s original algorithm using a tessellation into <i>N</i>_<i>τ</i> = 40 × 40 tiles and <i>N</i>_<i>ρ</i> = 20. The plots for (✦) and (△) show the results of using Strand’s unwrapper with a tile size containing more than one phase wrap using <i>N</i>_<i>ρ</i> = 40. In this case, the unwrapping of those tile fails and the deviation is analyzed for two different merging algorithms: (✦) the unidirectional merger and (△) the <i>τ</i>SRNCP merger. Algorithms proposed in this paper are in black boxes.</p