Particle Dark Matter in the Solar System Annika H. G. Peter A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Department of Physics June, 2008 c Copyright 2008 by Annika H. G. Peter. All rights reserved. Abstract Particle dark matter in the Galactic halo may be bound to the solar system either by elastic scattering through weak interactions with nucleons in the Sun (weak scattering) or by gravitational interactions with the planets, mainly Jupiter (gravitational capture). In this thesis, I simulate weak scattering, gravitational capture, and the subsequent evolution of the bound orbits to determine the distribution of bound dark matter at the position of the Earth. Previous work on this subject suggested that the event rate in dark matter detection experiments due to bound particles could be of order the event rate of halo particles in direct detection experiments, and several orders of magnitude higher for neutrinos arising from dark matter annihilation in the Earth. I use direct integration of orbits in a simplified solar system consisting of the Sun and Jupiter. I follow bound orbits until the particles are either rescattered in the Sun onto orbits that no longer intersect the Earth, ejected from the solar system, or reach the lifetime of the solar system tSS = 4.5 Gyr. Since many aspects of the particle orbits pose severe problems for traditional orbit integration methods, I develop a novel integration scheme for this problem, which has only small and oscillatory energy errors even for highly eccentric orbits over very long times. Using the bound dark matter distribution functions I generate from the simulations, I show that bound dark matter has a small effect on direct detection event rates, and that it will be almost impossible to detect neutrinos from dark matter annihilation in the Earth with the new generation of km3- scale neutrino telescopes. I also show how the distribution functions and resulting direct detection and neutrino telescope event rates can be scaled to other particle masses and elastic scattering cross sections. iii Acknowledgements There are many people who have guided my interests and academic path through life, and many other who have kept me sane during graduate school. I’d like to thank a few of those people here. First and foremost, I would like to thank my parents, Lyle and Beate Peter, for provid- ing an example and a home life that encouraged learning and discovery. Thanks to their support, my siblings and I are each following our passions and we know that we can do anything we want as long as we put our noses to the grindstone. I would also like to thank my parents for their advice and cheerleading (“it’s too late to quit now”, “Augen zu und durch” ) when I was discouraged and tired of working on this thesis, as well as celebrating the many milestones along the way. I have also benefited from the guidance and love of my grandparents, Bernie and Eleanor Peter, and Dr. Hans and Else Vossloh, as well as our dear great-aunts, Anni Vossloh and Reinhild Vossloh. Each has lived a remarkable life, and each is an inspiration to me. In particular, I would like to recognize Tante Reinhild for leading by example and by encouraging my studies by sending books on women in science. The Shoreline School District has many excellent teachers, but has been especially suc- cessful in staffing its Highly Capable Program. I would like to thank the district and all my teachers for their dedication to and belief in their students. I am grateful to the Early Entrance Program at the University of Washington for giving me (and many others) with a solid foundation for college, providing a safe and nurturing place on campus, and allowing nerds from all over Puget Sound to become lifelong friends. I would like to thank the Ronald Geballe, Maren Halvorsen, Bill Horder, Kate Noble, Ellen iv Mosolf, Nancy Robinson, Nancy Sisko, and Kay Soberg for their support during my years as a TSer and EEPer. There are many people in the Physics-Astronomy Building at the University of Wash- ington who have put me on the path to graduate school. The SNO group provided me with my first glimpse of how physics works outside of the classroom. The Astronomy Depart- ment was especially good at fostering a community that was inclusive of undergraduates. In particular, Julianne Dalcanton, Chris Stubbs, and Paula Szkody were instrumental in inspiring undergrads, and encouraging us to consider a careers in astrophysics. I am grateful to George Fuller and UC San Diego and Andreas Zezas at the CfA for mentoring me during my NSF REU summers. In graduate school, I have found homes in Gravity Group and the Department of As- trophysical Sciences, not just my home department. I would like to thank Paul Steinhardt and Suzanne Staggs for advising me on projects during my first years, and for bringing me into the Gravity Group fold. My advisor, Scott Tremaine, is a model of prudence, patience, and productivity. I have learned a lot about how to do science from him. I am indebted to Alice Shapley for cheerleading and guiding me through both an observational project and post-doc applications. My thesis work was heavily computational, and I could not have completed the work without the facilities at Princeton or the people who maintain the computers. I’d like to that Steve Huston, Leigh Koven, and Bill Wichser in the Department of Astrophysical Sci- ences, Neelesh Arora and Vinod Gupta in the Physics Department, and the dedicated staff at the Computer Science and Engineering Support Group (CSES) for keeping the comput- ers (Hydra, Feynman, Della, Orangena) up and running. And now, on to the question of sanity. Graduate school turned out to be much harder and quite different than I imagined as an undergraduate. My friends from Seattle have been a source of strength and valued outside observers through the past five years. My siblings (Lukas, Elias, and Cara) keep me laughing with jokes of questionable taste and photos and videos of Milo engaging in ridiculous feline antics. I have made a lot of treasured friends v at Princeton. I would especially like to thank Eric Dahl for getting people to socialize. I would like to thank Janice and Khali Hester and Amin Nasr for putting up with my dismal apartment-cleaning skills for two years. My officemates, Amol Upadhye and Beth Reid, and my across-the-wall neighbors Kevin Huffenberger and Nikhil Padmanabhan, always guaranteed a good time in the office. The (astro)physics women are a fantastic bunch. In a department that is > 90% male, it is wonderful to have such a cool group of women surrounding me. I have acquired a husband, Chris Hirata, and another family (the Hiratas and Lefevres), thanks to my association with the Physics Department at Princeton. Even though I have been known to complain, on occasion, about the state of New Jer- sey, there are institutions here that sustained me and which I miss now that I have moved. Crown of India has fed me well (no other chicken tikka masala compares!), and the Wawa has satisfied every junk food craving (especially for Haribo Goldb¨aren). Chris and I have spent a lot of time drinking coffee and reading books at Market Fair, and whiling away happy evenings with friends in the dbar. Finally, I would like to thank my husband Chris for emotional support, being an invalu- able sounding board, and the many trips to the Wawa and Starbucks on my behalf. vi Contents Abstract iii Acknowledgements iv Contents vii List of Figures x List of Tables xxii 1 Introduction 1 1.1 EvidenceforDarkMatter . 1 1.1.1 AstrophysicalEvidence . 2 1.1.2 Tantalizing Hints from Particle Physics . ..... 20 1.2 Various Probes of Dark Matter Properties . ...... 23 1.2.1 (Extra)Galactic Methods . 23 1.2.2 SolarSystemMethods .......................... 28 1.3 The Local Dark Matter Distribution Function . ....... 46 1.3.1 The Halo Dark Matter Distribution Function at the Solar Circle . 47 1.3.2 Motion of the Earth in the Halo . 57 1.4 Corrections to the Distribution Function: Effects Within the Solar System . 62 1.4.1 WeakScattering ............................. 63 1.4.2 Gravitational Scattering . 69 vii 1.5 ThisApproach .................................. 71 2 Initial Conditions 74 2.1 ParticleandAstrophysicsInput. ..... 74 2.1.1 TheDarkMatterCandidate. 74 2.1.2 The Halo Distribution Function . 80 2.1.3 TheSun.................................. 82 2.1.4 TheSolarSystem............................. 82 2.2 WeakScatteringintheSun . 83 2.3 Gravitational Scattering . 88 2.4 SimulationSpecifics .............................. 94 3 Numerical Methods 96 3.1 TheIntegrator .................................. 96 3.1.1 The Adaptive Time Step Symplectic Integrator . ..... 97 3.1.2 ErrorsAlongtheOrbitalPath . 101 3.1.3 Special Cases: Strong Perturbations to the Keplerian Potential . 109 3.1.4 CoordinateChoice ............................ 117 3.2 SubsequentScatteringintheSun . 119 3.3 SimulationSpecifics .............................. 122 3.3.1 Weak Scattering Simulations . 122 3.3.2 Gravitational Scattering Simulations . 128 3.3.3 Computing ................................ 131 4 From Simulations to Distribution Functions 132 4.1 Distribution Function Estimators . 132 4.2 Estimating Distribution Functions in Practice . .......... 137 4.3 The Distribution Function in the Earth . 138 viii 5 Results 141 5.1 DistributionFunctions . 142 5.1.1 Weak Scatter Simulations . 142 5.1.2 Gravitational Capture Simulations . 187 5.1.3 Summary of the Gravitationally Captured WIMPs . 199 5.2 Impact on Direct Detection Experiments . 200 5.3 Impact on Neutrino Telescope Event Rates . 211 5.3.1 AnAsideonNeutrinosfromtheSun . 225 6 Conclusion 229 6.1 Summary ..................................... 229 6.2 FutureDirections................................ 231 6.2.1 PossibleSpeed-Ups...........................