ECCENTRIC COMPACT BINARIES: MODELING THE INSPIRAL AND GRAVITATIONAL WAVE EMISSION by Nicholas Peter Loutrel A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics MONTANA STATE UNIVERSITY Bozeman, Montana April 2018 c COPYRIGHT by Nicholas Peter Loutrel 2018 All Rights Reserved ii DEDICATION I dedicate this to my parents, Brian Charles Loutrel & Annette Marie Manzo Loutrel; without their endless love and support, this dissertation would not be possible. iii ACKNOWLEDGEMENTS I would like to thank my Ph.D. advisor, Nicol´asYunes (NY), for providing invaluable advice and helping me become a professional researcher. Further, I would like to thank K. G. Arun, Alejandro Cardenas-Avenda~no,Katerina Chatziioannou, Neil Cornish, Samuel Liebersbach, Sean McWilliam, and Frans Pretorius (FP) for supporting this work through discussions and collaborations. This research was supported by National Science Foundation grants PHY- 1065710 (FP), PHY-1305682 (FP), PHY-1114374 (NY), PHY-1250636 (NY), EAPSI Award No. 1614203, NASA grant NNX11AI49G, under sub-award 00001944 (FP and NY), the Simons Foundation (FP). iv TABLE OF CONTENTS 1. INTRODUCTION ........................................................................................1 2. ECCENTRIC BINARIES IN GENERAL RELATIVITY...............................10 Radiation Reaction in General Relativity ..................................................... 10 Basics................................................................................................. 10 Quasi-Keplerian Parametrization.......................................................... 14 Averaging and Enhancement Factors .................................................... 20 Fourier Decomposition of Multipole Moments............................................... 24 Multipole Moments at Newtonian Order............................................... 24 Mass Quadruple at 1PN Order............................................................. 27 3. HEREDITARY EFFECTS IN ECCENTRIC COMPACT BINARY INSPIRALS................................................................................................32 Energy & Angular Momentum Fluxes: Tail Effects ....................................... 35 Integral Definitions and Orbit-Averages ................................................ 36 Tail Enhancement Factors.................................................................... 39 Tail Fluxes: Resummation of Asymptotic Enhancement Factors .................... 45 Asymptotic Resummation Method for the Enhancement factors............. 45 Superasymptotic Enhancement Factors................................................. 52 Hyperasymptotic Enhancement Factors ................................................ 61 Discussion .................................................................................................. 68 4. ECCENTRIC GRAVITATIONAL WAVE BURSTS IN THE POST- NEWTONIAN FORMALISM......................................................................71 Constructing Burst Models.......................................................................... 74 The Newtonian Burst Model ................................................................ 74 Orbit Evolution ........................................................................... 75 Centroid Mapping........................................................................ 78 Volume Mapping.......................................................................... 81 The Inverse Problem and Degeneracies ................................................. 83 A Simplified Formalism ....................................................................... 84 A Generic PN Formalism ............................................................................ 88 Example PN Burst Models................................................................... 94 Burst Model at 1PN Order ........................................................... 95 Burst Model at 2PN Order ........................................................... 97 Burst Model at 3PN Order ......................................................... 100 Properties of the PN Burst Model ............................................................. 105 v TABLE OF CONTENTS { CONTINUED Accuracy of the Burst Model ............................................................. 105 Pericenter Braking............................................................................. 108 Discussion ................................................................................................ 114 5. PARAMETERIZED POST-EINSTEINIAN FRAMEWORK FOR GRAVITATIONAL WAVE BURSTS ......................................................... 117 Kepler Problem in the ppE Formalism ....................................................... 122 Modeling Beyond GR................................................................................ 126 Size of Tiles ...................................................................................... 127 Mapping Between Tiles...................................................................... 128 A Parameterized-Post Einsteinian Burst Framework............................ 130 Burst Models in Modified Gravity.............................................................. 132 Einstein-Dilaton-Gauss-Bonnet Gravity .............................................. 132 Brans-Dicke Theory of Gravity........................................................... 137 Projected Constraints........................................................................ 141 Discussion ................................................................................................ 145 6. SECULAR GROWTH OF ECCENTRICITY IN THE LATE IN- SPIRAL ................................................................................................... 147 Radiation Reaction Force .......................................................................... 147 Multiple Scale Analysis ............................................................................. 151 Properties of Secular Growth..................................................................... 157 7. SUMMARY.............................................................................................. 163 REFERENCES CITED.................................................................................. 167 APPENDICES .............................................................................................. 182 APPENDIX A : Fourier Coefficients of the Mass Octopole and Current Quadrupole............................................................................ 183 APPENDIX B : Pad´eApproximant Coefficients......................................... 185 APPENDIX C : Post-Newtonian Recursion Relations................................. 187 APPENDIX D : Third Post-Newtonian Quasi-Keplerian Parametrization .... 190 PN Vector Fields ...................................................................................... 191 1PN Amplitude Vector Fields............................................................. 191 Amplitude Vector Fields to 3PN Order............................................... 195 APPENDIX E : Harmonic Decomposition of the Radiation Reac- tion Force........................................................................................... 202 vi LIST OF TABLES Table Page 2 3.1 Leading order dependence on = 1 − et of the integrals in Eq. (3.67) for various positive powers of n. ....................................... 51 4.1 Initial values of the PN expansion parameter x for the set of compact binary systems studied. The values are obtained by requiring the initial GW frequency to be 10Hz. The final column provides an estimate of the semi-major axis of the binary, since ar = M=x + O(1) in PN theory................................... 107 vii LIST OF FIGURES Figure Page 2.1 Diagram of Keplerian elliptical orbits in an effective one body frame. A point mass µ orbits around a central mass m located at the focus of the ellipse. The semi-major axis a, ~ pericenter distance rp, and orbital angular momentum L are constants of the orbit when radiation reaction is neglected. The orbital radius r is the line connecting m and µ, while the true anomaly V is the angle from pericenter, which is located along the x-axis, to r. The eccentric anomaly u is the angle measured from the x-axis to a line whose end points are the center of the ellipse and a point on a circle that is concentric to the ellipse and of radius a, where the end point on the circle is determined by a line parallel to the y-axis that passes through µ. ..................................................... 16 3.1 Accuracy of the uniform asymptotic expansion of the Bessel function Jp(pet) as a function of p and for eccentricity et = 0:9. Going to higher order in the asymptotic expansion causes the series to become more accurate compared to the Bessel function, but at sixth order in 1=p, the relative error achieves a minimum. ....................................................................... 48 3.2 Accuracy of the asymptotic series for the enhancement factor '(et). The relative error to numerical results increases as we go to higher order in the series, but 2 −2 reaches a minimum at O [(1 − et ) ]. The series can thus be optimally truncated at this order, generating the superasymptotic series for '(et). ...................................................... 54 3.3 Comparison of the numerical results for '(et) with its hyperasymptotic series, 'hyper(et), at different orders in