TESTING GENERAL RELATIVITY WITH GRAVITATIONAL WAVES by Laura Mae Sampson A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics MONTANA STATE UNIVERSITY Bozeman, Montana March 2014 COPYRIGHT by Laura Mae Sampson 2014 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License iii ACKNOWLEDGEMENTS Thank you to my advisor, Neil Cornish, to the rest of my committee members, in particular Nico Yunes, and to all of my professors in the physics department. A special thank you also goes to Margaret Jarrett, who made sure that all of the paperwork hoops involved in graduate school were as painless as possible. iv TABLE OF CONTENTS 1. INTRODUCTION . .1 2. GRAVITATIONAL WAVES IN GENERAL RELATIVITY . .3 2.1. Linearized Gravity . .3 2.2. Gravitational Wave Detectors . .8 2.2.1. Interferometric Detectors . .9 2.2.2. Pulsar Timing Array . 11 2.3. Gravitational Wave Sources . 12 3. NON-GW TESTS OF GR . 15 3.1. Perihelion Precession of Mercury . 15 3.2. Gravitational Lensing . 19 3.3. Shapiro Time Delay . 21 3.4. Testing the Equivalence Principle . 23 3.5. Constraints from Binary Pulsars . 25 4. NON-GR THEORIES OF GRAVITY . 29 4.1. Theoretical Properties . 29 4.2. Scalar-Tensor Theories . 30 4.3. Massive Graviton Theories . 32 4.4. Modified Quadratic Gravity . 34 5. PARAMETERIZING ALTERNATIVES TO GENERAL RELATIVITY . 37 5.1. Parameterized post-Newtonian . 43 5.2. Parameterized post-Keplerian . 48 5.3. Parameterized post-Einsteinian . 53 5.4. ppN-ppE correspondence . 56 5.5. ppN-ppK Correspondence . 62 5.6. ppE-ppK Correspondence . 66 5.7. Current Constraints . 70 5.8. Conclusion . 72 6. BAYESIAN PROBABILITY THEORY AND MARKOV METHODS . 74 6.1. Bayesian Inference . 74 6.2. Computational Techniques . 75 6.2.1. Differential Evolution . 79 6.2.2. Langevin Proposals . 81 6.3. Calculating Bayes Factors . 83 6.3.1. Thermodynamic Integration . 85 6.3.2. Volume Tessellation Algorithm . 88 v TABLE OF CONTENTS { CONTINUED 6.3.3. Reversible Jump MCMC . 89 6.3.4. Savage-Dickey Density Ratio . 91 6.3.5. Comparison of Techniques . 92 6.4. Fitting Factor . 100 7. TESTING GR USING PPE TEMPLATES . 108 7.1. Analysis Framework . 112 7.1.1. Instrument Response . 112 7.1.2. Likelihood Function. 113 7.2. Approximate Bounds and Comparison with Pulsar Bounds . 116 7.3. Noise Modeling and Signal Strength . 123 7.4. Full Bayesian Bounds and Model Selection . 128 7.5. Parameter Biases . 134 7.6. More Complicated Waveforms . 136 7.7. Optimal Model Selection . 147 7.7.1. Overfitting . 148 7.7.2. Inclusion of Spin . 150 7.7.3. Parsimonious Fitting . 151 7.8. Detecting GWs from Scalar-Tensor Gravity . 156 7.9. Inspiral-Merger-Ringdown Signals . 164 7.9.1. Extracting with Inspiral-only ppE Templates . 165 7.9.2. Extracting with Inspiral-Merger-Ringdown ppE Templates . 171 8. SUMMARY . 179 REFERENCES CITED . 184 vi LIST OF TABLES Table Page 1. The ten ppN parameters, as well as their physical significance, and their value in GR and in semi-conservative theories, in which energy and momentum are conserved. 46 2. The current experimental constraints on the four ppN parameters that we will consider in this thesis, along with the effect used to ¯ measure that constraint [144]. We use the definitions βppN “ βppN ´ 1; γ¯ppN “ γppN ´ 1 ............................................ 48 3. Uncertainty in measured values for PSR J0737-3039A. [86] . 53 4. The values that the ppE parameters take on in various non-GR theories [165]. 56 5. Values of ppN parameters for a selection of alternative theories. These expressions are in the large ! limit. [144, 28, 79, 60] . 59 6. ppK parameters expressed as combinations of the ppN parameters. 64 7. The current constraints that can be placed on βppE from Solar System and binary pulsar tests. These values are only for b “ ´3.. 70 8. Source parameters for sources used in Fig. 33 (top), Fig. 34 (middle) and Figs. 25, 26, and 35 (bottom). All sources had pα; cos δ; φL; cos θLq “ p1:0; 0:77; 1:76; ´0:43q.......................... 142 9. Number of useful cycles from the different injected ppE terms - Fig 33 and Fig 34. 146 10. Source parameters for Figures 37 and 38. The βb values listed are for a particular case - the ratio between different βb values was kept constant for each injected signal. The ratio for sub- critical was ˆ1:0, critical was ˆ30, and super-critical was ˆ100. All sources had pα; cos δ; φL; cos θLq “ p1:42; 0:87; 2:5; 0:43q............... 152 11. Number of useful cycles from the different injected ppE terms - Fig. 37 .......................................................... 155 vii LIST OF TABLES { CONTINUED Table Page 12. ISCO frequency and percentage of total SNR before the ISCO for systems of different total mass. 170 viii LIST OF FIGURES Figure Page 1. The effect of the two different polarizations of GWs on a ring of test particles. The top panel shows the plus polarization, and the bottom panel shows the cross. [75] . .9 2. Cartoon of a Michelson interferometer, like the LIGO and VIRGO detectors. A beam of light is split in two and sent down both arms of the detector. Each half of the beam bounces off of the test mass in one arm, and the beam is recombined at the photodetector. Changes in the length of the two arms will re- sults in changes of the light intensity at the photodetector, which allows us to use these interferometers to detect GWs. [36] . 10 3. The perihelion precession of a planet around the Sun - greatly exaggerated so as to be visible. (Copyright 2010, Professor Ken- neth R. Lang, Tufts University ) . 16 4. Light from a distant star being bent by the Sun as it travels to Earth. [144] . 20 5. The measured orbital period for the Hulse-Taylor pulsar (points) plotted with the GR prediction (line). The agreement between the measured values and the theoretical predictions is very pre- cise. [138] . 27 6. The measured value of parameters that characterize possible de- viations from GR in binary pulsar orbits. Each parameter is a function of the two masses. If all of the measured curves overlap in the same region of the mass1- mass2 plane, the data is con- sistent with GR. As we can see, the curves do, in fact, overlap. [85].............................................................. 27 7. Limits that can be placed on the ppE strength parameter, βppE, using the known limits.