The Pennsylvania State University The Graduate School Department of Physics BAYESIAN ANALYSIS ON GRAVITATIONAL WAVES AND EXOPLANETS A Dissertation in Physics by Xihao Deng c 2015 Xihao Deng Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2015 ii The dissertation of Xihao Deng was read and approved1 by the following: Lee Samuel Finn Professor of Physics Dissertation Advisor Chair of Committee Chad Hanna Assistant Professor of Physics Stephane Coutu Professor of Physics Aleksander Wolszczan Professor of Astronomy and Astrophysics Nitin Samarth Professor of Physics Head of the Department of Physics 1Signatures on file in the Graduate School. iii Abstract Attempts to detect gravitational waves using a pulsar timing array (PTA), i.e., a collection of pulsars in our Galaxy, have become more organized over the last several years. PTAs act to detect gravitational waves generated from very distant sources by observing the small and correlated effect the waves have on pulse arrival times at the Earth. In this thesis, I present advanced Bayesian analysis methods that can be used to search for gravitational waves in pulsar timing data. These methods were also applied to analyze a set of radial velocity (RV) data collected by the Hobby- Eberly Telescope on observing a K0 giant star. They confirmed the presence of two Jupiter mass planets around a K0 giant star and also characterized the stellar p-mode oscillation. The first part of the thesis investigates the effect of wavefront curvature on a pulsar’s response to a gravitational wave. In it we show that we can assume the gravitational wave phasefront is planar across the array only if the source luminosity distance 2πL2/λ, where L is the pulsar ≫ distance to the Earth ( kpc) and λ is the radiation wavelength ( pc) in the PTA waveband. ∼ ∼ Correspondingly, for a point gravitational wave source closer than 100 Mpc, we should take into ∼ account the effect of wavefront curvature across the pulsar-Earth line of sight, which depends on the luminosity distance to the source, when evaluating the pulsar timing response. As a consequence, if a PTA can detect a gravitational wave from a source closer than 100 Mpc, the effects of ∼ wavefront curvature on the response allows us to determine the source luminosity distance. This technique is very similar to the use of pulsar timing parallax to measure the distances to nearby pulsars, because they both try to measure the phasefront curvature of a wave passing through a long baseline (pulsar-Earth distance and Sun-Earth distance) to determine the wave source distance. iv The second and third parts of the thesis propose a new analysis method based on Bayesian nonparametric regression to search for gravitational wave bursts and a gravitational wave back- ground in PTA data. Unlike the conventional Bayesian analysis that introduces a signal model with a fixed number of parameters, Bayesian nonparametric regression sets constraints on the function space that may be reasonably thought to characterize the range of gravitational wave signals. For example, focus attention on the detection of a gravitational wave burst, by which we mean a signal that begins and ends over the course of an observational epoch. The burst may result from a source that we know how to model - e.g., a near-unity mass ratio black hole binary system - or it may be the result of a process, which we have not imagined and, so, have no model for. Similarly, a gravitational wave background resulting from a superposition of a number of weak sources may be difficult to characterize if the number of weak sources is sufficiently large that none can be individually resolved, but not so large that their superposition leads to a reasonably Gaussian distribution. Correspondingly, the Bayesian nonparametric regression method may be very useful to help search for gravitational wave bursts and a gravitational wave background in the pulsar timing data. By testing this new method on simulated data sets, it is found that we can use it to detect gravitational wave bursts and a gravitational wave background, and we can also characterize their important physical parameters such as the burst durations and the amplitude of the background even if their signal-to-noise ratios are low. The fourth part develops Bayesian analysis methods that can be used to detect gravitational waves generated from circular-orbit supermassive black hole binaries with a pulsar timing array. PTA response to such gravitational waves can be modeled as the difference between two sinusoidal terms — the one with a coherent phase among different pulsars called “Earth term” and the other one with incoherent phases among different pulsars called “pulsar term”. For gravitational waves from slowly evolving binaries, the two terms in the PTA response model have the same frequency. v Previous methods aimed at detecting gravitational waves from circular-orbit binaries ignored pulsar terms in data analysis since those terms were considered to be negligible when averaging over all the pulsars. However, it is found that we can incorporate the contributions of pulsar terms into data analysis in the case of slowly evolving binaries by treating the incoherent phases in pulsar terms as unknown parameters to be marginalized. By testing the new method on simulated data sets, the improvement, compared to previous analyses, is equivalent to halving the strength of timing noise associated with each pulsar. The final part of this thesis applies Bayesian analysis to search for the evidence of a planetary system around the K0 giant star HD 102103 detected by the Penn State-Torun planet group at the Hobby Eberly Telescope. It analyzes 116 observations of the star’s radial velocity. However, the stellar p-mode oscillation also contributes to the radial velocity data, challenging the search for the planets around the star. The Bayesian method models the stellar oscillation effect and the potential exoplanet signal together, simultaneously inferring their parameters from the data. Consequently, the method removes the ambiguities of the presence of two Jupiter mass planets around the K0 giant and as a bonus, it also characterizes the strength and the frequency of the stellar oscillation. vi Table of Contents List of Tables ............................................ xii List of Figures ........................................... xiii Acknowledgments ......................................... xv Chapter 1. Introduction ...................................... 1 1.1 Overviewofpulsartiming . .... 1 1.1.1 Observatory clock correction . .... 3 1.1.2 Atmospheric propagation delay . .... 3 1.1.3 Roemerdelay................................. 3 1.1.4 General relativistic corrections . ....... 4 1.1.5 Dispersionmeasure. .. .. .. .. .. .. .. .. 5 1.1.6 Excess vacuum propagation delay due to secular motion ......... 6 1.1.7 Orbitalmotion ................................ 6 1.1.8 Pulsar spin, spin down and timing residuals . ....... 7 1.2 Overview of gravitational waves . ....... 8 1.2.1 Gravitational wave fundamentals . ..... 8 1.2.2 Gravitational wave detection with a pulsar timing array......... 11 1.2.3 Gravitational wave sources . 13 1.3 Dataanalysis .................................... 13 1.3.1 Overview of Bayesian data analysis . ..... 14 1.3.2 Detection ................................... 14 vii 1.3.2.1 Introduction ............................ 15 1.3.2.2 Deviance Information Criterion . 17 1.3.2.3 Comparison of DIC and Bayes factor . 20 1.3.3 Computationalmethod . 21 1.3.3.1 Markov Chain Monte Carlo . 21 1.3.3.2 Convergence diagnostics . 22 1.4 Application of gravitational wave data analysis methods to detect exoplanetary systems ........................................ 24 1.4.1 Overview of exoplanet detection . 24 1.4.2 Radial velocity planet detection . ...... 24 1.4.3 Detecting exoplanets around giant stars . ....... 26 1.5 Outline ........................................ 26 Chapter 2. Pulsar Timing Array Observations of Gravitational Wave Source Timing Parallax 28 2.1 Introduction.................................... 28 2.2 Pulsar timing residuals from spherically-fronted gravitational waves . 29 2.2.1 Timingresiduals ............................... 29 2.2.2 Gravitational waves from a compact source . ...... 30 2.2.3 Responsefunction .............................. 31 2.3 Discussion...................................... 33 2.3.1 Plane wave approximation timing residuals . ....... 35 2.3.2 Correction to plane wave approximation . ...... 38 2.3.3 Gravitational wave source distance and location on sky ......... 39 2.4 Conclusion...................................... 42 viii Chapter 3. Searching for Gravitational Wave Bursts via Bayesian Nonparametric Data Anal- ysis with Pulsar Timing Arrays ........................... 45 3.1 Introduction.................................... 45 3.2 Bayesian nonparametric methodology . ........ 47 3.2.1 Framework of Bayesian nonparametric analysis . ........ 47 3.2.1.1 LikelihoodfunctionΛ . 47 3.2.1.2 Prior probability density q .................... 48 3.2.1.3 Bayesian nonparametric inference . 50 3.2.2 Comparison with Bayesian parametric analysis . ........ 52 3.3 Bayesian nonparametric analysis on gravitational wave bursts with pulsar timing arrays ......................................... 53 3.3.1 Properties of the pulsar timing response to the passage of a gravitational waveburst .................................. 54 3.3.2 The choice of prior probability distribution . ......... 55 3.3.2.1 Priors of τ(+)