CONSTRAINING THE NEUTRON STAR EQUATION OF STATE WITH ASTROPHYSICAL OBSERVABLES by Carolyn A. Raithel Copyright © Carolyn A. Raithel 2020 A Dissertation Submitted to the Faculty of the DEPARTMENT OF ASTRONOMY In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY WITH A MAJOR IN ASTRONOMY AND ASTROPHYSICS In the Graduate College THE UNIVERSITY OF ARIZONA 2020 3 ACKNOWLEDGEMENTS Looking back over the last five years, this dissertation would not have been possible with the support of many people. First and foremost, I would like to thank my advisor, Feryal Ozel,¨ from whom I have learned so much { about not only the science I want to do, but about the type of scientist I want to be. I am grateful as well for the support and mentorship of Dimitrios Psaltis and Vasileios Paschalidis { I have so enjoyed working with and learning from you both. To Joel Weisberg, my undergraduate research advisor who first got me started on this journey and who has continued to support me throughout, thank you. I believe a scientist is shaped by the mentors she has early in her career, and I am grateful to have had so many excellent ones. I am deeply thankful for my friends, near and far, who have supported me, encouraged me, and helped preserve my sanity over the last five years. To our astronomy crafting group Lia, Ekta, Samantha, and Allie; to my office mates David, Gabrielle, Tyler, Kaushik, and Michi; to Sarah, Marina, Tanner, Charlie, and Lina{ thank you. I will be forever grateful to my family for their continual support { espe- cially my parents, Don and Kathy, who instilled in me a love for learning from a very young age and who have encouraged me ever since. Finally, I would like to thank my husband, Nathan, who has been there from the start, when we were just two undergraduates studying pulsars in our first summer of research. You have encouraged me daily, made me laugh when I didn't think it was possible, helped me think through more than one statistical pathology, and been my rock through it all. Thank you. 4 DEDICATION For my parents 5 TABLE OF CONTENTS LIST OF FIGURES . .9 LIST OF TABLES . 12 ABSTRACT . 13 CHAPTER 1 Introduction . 15 1.1 Overview . 15 1.2 Predicting neutron star observables from theoretical EOS . 17 1.3 Observations of neutron star properties . 18 1.3.1 Masses . 19 1.3.2 Radii . 22 1.3.3 Moment of inertia . 25 1.3.4 Tidal deformability . 26 1.4 Constraining the EOS from neutron star observables . 28 1.5 Dynamical phenomena and the finite-temperature EOS . 30 1.6 Outline of this work . 31 CHAPTER 2 An Optimal Parametrization of the EOS . 34 2.1 Optimizing the parametric EOS . 35 2.2 From EOS to observables . 37 2.3 Generating mock equations of state . 39 2.3.1 The mock EOS . 41 2.3.2 Determining the goodness of the parametric representation . 44 2.3.3 Results of parametrization of mock EOS . 44 2.4 Adding phase transitions to the mock EOS . 47 2.5 Application of the parametrization to physically-motivated EOS . 49 2.6 Conclusions . 52 CHAPTER 3 Bayesian Inference of EOS Pressures . 54 3.1 Bayesian inference of EOS pressures . 55 3.1.1 Priors on the pressures . 56 3.1.2 Regularizers . 57 3.2 Testing the Bayesian inference with mock data . 58 3.3 Biases due to marginalization . 64 3.4 Conclusions . 69 6 TABLE OF CONTENTS { Continued CHAPTER 4 Model-Independent Mapping from a Moment of Inertia Measure- ment to the Neutron Star Radius . 72 4.1 The moment of inertia in the double pulsar system . 73 4.2 Neutron star moments of inertia for various EOS . 74 4.3 Absolute bounds on the moment of inertia . 77 CHAPTER 5 Gravitational Wave Events as a Direct Probe of the Neutron Star Radius . 83 5.1 Gravitational waves and the EOS . 84 5.2 Properties of GW170817 . 86 5.2.1 Updated analysis of GW170817 . 87 5.3 Effective tidal deformability for GW170817 . 88 5.4 Effective tidal deformability in the Newtonian limit . 90 5.5 Comparison to existing radius constraints . 93 5.6 Bayesian inference of the radius . 97 5.7 Black hole-neutron star mergers . 100 5.8 Conclusions . 100 CHAPTER 6 Connecting Gravitational Wave Events to Fundamental Nuclear Parameters . 102 6.1 Motivations for studying the nuclear symmetry energy . 102 6.2 Nuclear expansion of the equation of state . 105 6.3 Polytropic approximation . 107 6.4 Relating the tidal deformability to the symmetry energy . 113 6.5 Constraints on higher-order nuclear parameters . 119 6.6 Conclusions . 122 CHAPTER 7 Optimized Statistical Approach for Comparing Multi-Messenger Neutron Star Data . 124 7.1 Overview of multi-messenger data for neutron stars . 124 7.2 Motivation and past work . 127 7.3 Defining Bayesian priors . 131 7.4 Transformation functions . 134 7.4.1 From the nuclear symmetry energy to the neutron star radius 135 7.4.2 From tidal deformability to the neutron star radius . 135 7.4.3 Summary of transformations . 136 7.5 Example application to gravitational wave data . 138 7.6 Composite constraints on the neutron star radius . 141 7.7 Conclusions . 146 7 TABLE OF CONTENTS { Continued CHAPTER 8 The Birth Mass Distribution of Neutron Stars and Black Holes 147 8.1 Previous studies of the compact object mass distribution . 148 8.2 Pre-SN stellar evolution and explosion . 152 8.3 Observations of remnant masses . 156 8.4 Black hole mass distribution . 159 8.4.1 Comparing the simulated and observed black hole mass dis- tributions . 164 8.5 Neutron star mass distribution . 167 8.5.1 Comparing the simulated and observed pulsar mass distributions171 8.6 Missing high-mass remnants . 173 8.6.1 High-mass black holes . 174 8.6.2 Missing high-mass neutron stars . 177 8.6.3 Effect of binary evolution . 178 8.7 Conclusions . 179 CHAPTER 9 Finite-Temperature Extension for Cold Neutron Star EOS . 181 9.1 Dynamical phenomena and the finite-temperature EOS . 181 9.2 Overview of finite-temperature EOS . 186 9.3 Generic model of a finite temperature EOS . 191 9.4 Derivation of the cold symmetry energy in the Fermi Gas limit . 194 9.5 Thermal contribution to the energy . 200 9.5.1 M ∗-approximation . 204 9.5.2 Performance of the M ∗-approximation of thermal effects at fixed Yp .............................. 209 9.5.3 M ∗-approximation for non-RMF models . 212 9.6 Putting it all together . 214 9.7 Complete model: Comparison of realistic EOS at arbitrary Yp and T . 218 9.8 Conclusions . 221 CHAPTER 10 Future Prospects . 223 10.1 Upcoming new data . 224 10.1.1 X-ray observations . 224 10.1.2 Radio pulsar observations . 225 10.1.3 Gravitational waves . 227 10.2 New simulations of binary neutron star mergers . 231 10.3 Conclusions . 232 CHAPTER 11 Appendices . 234 A Linear parametrization of the EOS . 234 8 TABLE OF CONTENTS { Continued B Relationship between the β-equilibrium proton fraction and the sym- metry energy for n-p-e matter . 237 C Calculation of the sound speed . 239 REFERENCES . 243 9 LIST OF FIGURES 1.1 Mass-radius relations for realistic EOS . 18 1.2 Mass distribution of neutron stars and black holes. 20 1.3 Current sample of radius measurements . 25 2.1 Piecewise polytrope schematic . 36 2.2 Distribution of polytropic indices for realistic EOS . 39 2.3 Grid of polytropic, mock EOS . 42 2.4 Mass-radius curves for sample of mock EOS . 43 2.5 Distribution of errors in radii and the maximum mass introduced by various parametrizations, for the mock EOS . 45 2.6 Distribution of errors in the moment of inertia introduced by various parametrizations, for the mock EOS . 46 2.7 Distribution of errors in radii and the maximum mass introduced by various parametrizations, for mock EOS with a first-order phase transition . 48 2.8 Distribution of errors introduced by the optimal parametrization for 42 realistic EOS . 50 2.9 Four realistic EOS and their mass-radius relations, for both the full EOS and our optimal parametrization of the EOS . 51 3.1 Distribution of second derivatives in pressure for realistic EOS . 58 3.2 EOS inference for example mass-radius data with various priors . 59 3.3 Individual mass-radius curves for inferred EOS . 60 3.4 Correlations between the five inferred pressures . 61 3.5 EOS inference from example mass-radius data with a Gaussian regu- larizer . 65 3.6 Distribution of errors in inferred EOS for many example datasets . 66 3.7 Bias introduced by two-dimensional histograms of mass-radius relations 67 3.8 Fidelity of recovered radii for inferred EOS . 69 4.1 Moments of inertia and radii for realistic EOS . 77 4.2.