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The Pennsylvania State University the Graduate School Eberly College of Science The Pennsylvania State University The Graduate School Eberly College of Science SEARCHING FOR GRAVITATIONAL WAVES FROM NEUTRON STARS A Dissertation in Physics by Ashikuzzaman Idrisy © 2015 Ashikuzzaman Idrisy Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2015 The dissertation of Ashikuzzaman Idrisy was reviewed and approved∗ by the following: Benjamin J. Owen Professor of Physics Dissertation Advisor, Chair of Committee Martin Bojowald Professor of Physics Michael Eracleous Professor of Astronomy Sarah Shandera Assistant Professor of Physics Nitin Samarth Professor of Physics George A. and Margaret M. Downsbrough Department Head ∗Signatures are on file in the Graduate School. ii Abstract In this dissertation we discuss gravitational waves (GWs) and their neutron star (NS) sources. We begin with a general discussion of the motivation for searching for GWs and the indirect experimental evidence of their existence. Then we discuss the various mechanisms through which NS can emit GWs, paying special attention the r-mode oscillations. Finally we end with discussion of GW detection. In Chapter 2 we describe research into the frequencies of r-mode oscillations. Knowing these frequencies can be useful for guiding and interpreting gravita- tional wave and electromagnetic observations. The frequencies of slowly rotating, barotropic, and non-magnetic Newtonian stars are well known, but subject to various corrections. After making simple estimates of the relative strengths of these corrections we conclude that relativistic corrections are the most important. For this reason we extend the formalism of K. H. Lockitch, J. L. Friedman, and N. Andersson [Phys. Rev. D 68, 124010 (2003)], who consider relativistic polytropes, to the case of realistic equations of state. This formulation results in perturbation equations which are solved using a spectral method. We find that for realistic equations of state the r-mode frequency ranges from 1.39–1.57 times the spin frequency of the star when the relativistic compactness parameter (M/R) is varied over the astrophysically motivated interval 0.110–0.310. Following a successful r-mode detection our results can help constrain the high density equation of state. In Chapter 3 we present a technical introduction to the data analysis tools used in GW searches. Starting from the plane-wave solutions derived in Chapter 1 we develop the F-statistic used in the matched filtering technique. This technique relies on coherently integrating the GW detector’s data stream with a theoretically modeled wave signal. The statistic is used to test the null hypothesis that the data contains no signal. In this chapter we also discuss how to generate the parameter space of a GW search so as to cover the largest physical range of parameters, while keeping the search computationally feasible. Finally we discuss the time- domain solar system barycentered resampling algorithm as a way to improve to the computational cost of the analysis. iii In Chapter 4 we discuss a search for GWs from two supernova remnants, G65.7 and G330.2. The searches were conducted using data from the 6th science run of the LIGO detectors. Since the searches were modeled on the Cassiopeia A search paper, Abadie et. al. [Astrophys. J. 722,1504–1513, 2010], we also used the frequency and the first and second derivatives of the frequency as the parameter space of the search. There are two main differences from the previous search. The first is the use of the resampling algorithm, which sped up the calculation of the F-statistic by a factor of 3 and thus allowed for longer stretches of data to be coherently integrated. Being able to integrate more data meant that we could beat the indirect limit on GWs expected from these sources. We used a 51 day integration time for G65.7 and 24 days for G330.2. The second difference is that the analysis pipeline is now more automated. This allows for a more efficient data analysis process. We did not find a credible source of GWs and so we placed upper limits on the gravitational wave strain, ellipticity, and r-mode amplitude of the sources. The best upper-limit for the strain was 3.0 × 10−25, for ellipticity it was 7.0 × 10−6 and for r-mode amplitude it was 2.2 × 10−4. iv Table of Contents List of Figures viii List of Tables ix Acknowledgments x Chapter 1 Introduction 1 1.1 Indirect evidence . 2 1.2 Theory of gravitational waves . 4 1.2.1 Linearized gravity . 6 1.2.2 Generation of gravitational waves . 8 1.3 Neutron stars . 12 1.3.1 Gravitational waves from neutron stars . 14 1.3.2 r-modes . 15 1.4 Detection of gravitational waves . 17 1.4.1 History and Development . 17 1.4.2 Operating Principles . 18 1.4.3 Data Analysis challenge . 20 1.4.4 Detectors of the future . 22 Chapter 2 R-mode frequencies for slowly rotating neutron stars with real- istic equations of state 23 2.1 Introduction . 23 2.1.1 General relativity . 26 2.1.2 Rapid rotation . 27 2.1.3 The crust . 28 2.1.4 Other effects . 28 2.1.5 Summary . 29 v 2.2 Formulation . 29 2.2.1 Equilibrium Solution for non-rotating star . 30 2.2.2 Interpolation Schemes . 31 2.2.3 Equilibrium solution for slowly rotating star . 35 2.2.4 Perturbation equations . 36 2.2.4.1 Perturbations of non-rotating stars . 36 2.2.4.2 Perturbations of slowly rotating stars . 37 2.2.5 Boundary Conditions . 39 2.3 Numerical Solution . 41 2.3.1 Chebyshev-Galerkin Method . 41 2.3.2 Finding κ ............................ 43 2.3.3 The eigenfunctions . 45 2.4 Results . 46 2.4.1 The n = 1 polytrope . 46 2.4.2 Tabulated Equations of State . 47 2.5 Discussion . 51 2.6 Conclusion . 54 Chapter 3 Fundamentals of gravitational wave data analysis 56 3.1 Introduction . 56 3.2 Signal model . 57 3.3 Maximum likelihood detection . 60 3.4 Template spacing . 63 3.4.1 Sphere-covering . 63 3.4.2 Template Spacing Metric . 64 3.4.3 Lattice generation . 66 3.5 Resampling . 67 3.5.1 Implementation . 70 3.5.2 Speed up . 71 3.5.3 Issues . 72 Chapter 4 Using barycentric resampling in gravitational wave searches from two supernova remnants 74 4.1 Introduction . 74 4.2 Data analysis method . 77 4.2.1 Data Prepration . 83 4.2.2 Search . 84 4.2.3 Validation . 85 vi 4.2.4 Upper Limits . 88 4.3 Implementation . 88 4.4 Results . 90 4.5 Discussion . 92 Chapter 5 Conclusion 98 Bibliography 100 vii List of Figures 1.1 Energy loss from Hulse-Taylor binary system . 3 1.2 Regions of influence of gravitational object . 5 1.3 Polarization of gravitational waves . 8 1.4 Digram of gravitational wave source . 10 1.5 Visualization of r-mode oscillatios . 16 1.6 Schematic of LIGO detectors . 18 1.7 GW passing a LIGO interferometer . 19 1.8 LVC interferometer sensitivity . 20 2.1 Convergence of eigenvalue . 45 2.2 Plot of eigenfunctions . 46 2.3 Plot of eigenfunctions . 47 2.4 Eigenvalues for n=1 polytrope . 48 2.5 Eigenvalues for realistic equations of state . 49 2.6 Compare eigenvalue for different equations of state . 50 3.1 Schematic of resampling . 68 4.1 First 12 steps of the pipeline . 78 4.2 Steps 13 to 17 of the pipeline . 79 4.3 Veto bands . 86 4.4 2F distribution . 87 4.5 Schematic of DAG . 90 4.6 Strain amplitude upper limit for G65.7 . 92 4.7 Ellipticity upper limit for G65.7 . 93 4.8 r-mode upper limit for G65.7 . 94 4.9 Strain amplitude upper limit for 330.2 . 94 4.10 Ellipticity upper limit for G330.2 . 95 4.11 r-mode upper limit for G330.2 . 96 viii List of Tables 2.1 We present a list of all the tabulated EoS, for each EoS we show the stable maximum mass, the radius for a 1.4M star, κ for a compactness of .15, the coefficients for the quadratic fit of the κ 2 M M of the form a + b R + c R , and the root mean square error (RMSE) of the EoS data points to the quadratic fit. 51 2.2 This table gives the numerical κ for all tabulated EoS, over the range of compactness values considered in our analysis. These values are plotted in Fig. 2.5. The “-" indicates a compactness that could not be obtained with that EoS. 55 4.1 Search targets and astronomical parameters . 83 4.2 Target objects, and astronomical parameters used in each search along with information about the SFTs used in the searches. 83 ix Acknowledgments First and foremost I would like to thank my research advisor Professor Benjamin Owen, for none of this would be possible without his support. Thank you for taking a chance on a untested second year graduate student. Thank you for teaching me about the stars. Thank you for molding me into a scientist. Thank you for giving me the freedom to explore different career opportunities. Thank you for all the guidance given, when real-life came knocking. Thank you, so very much, for everything. I would like to thank professors Bojowald, Eracleous, and Shandera for being on my committee. Thank you for taking the time to read and edit this thesis and for overseeing my comprehensive examination. I would like to thank Dr. Ra Inta without whom I could not have finished this work. Thank you for taking the time to help me debug code, for teaching me about Bash, vim, Perl, and Python, and for all the practical advice.
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