Production and Evaporation of Higher Dimensional Black Holes Marco Oliveira Pena Sampaio Churchill College A dissertation submitted to the University of Cambridge for the degree of Doctor of Philosophy March 2010 Production and Evaporation of Higher Dimensional Black Holes Marco Oliveira Pena Sampaio Abstract This thesis is a study of the theory and phenomenology of trans-Planckian black holes, in TeV gravity extra-dimensional theories. The introduction starts with the motivation for this beyond the Standard Model scenario (chapter 1), a summary of the theoretical tools to formulate the theory, and a summary of the best bounds from experiment (chapter 2). In chapter 3, after setting up some notation and describing well known solutions in 4+ n-dimensional general relativity, we construct an approximate effective background for a brane charged rotating higher-dimensional black hole. This is achieved by solving Maxwell’s equations perturbatively on the brane to obtain the electromagnetic field. A brief study of the effect of rotation on the absorption of classical particles is also provided. Chapter 4 is a review of methods to model black hole production focusing on the trapped surface method. A model for the mass and angular momentum loss into gravita- tional radiation is described. A detailed study of the effects of particle mass and charge, for fermions and scalars on the effective brane charged background, is presented in chapters 5 and 6. After coupling the fields to the background, the separated radial wave equations for both perturbations are obtained (chapter 5) and they are integrated using a detailed numerical method as well as analytic approximations (chapter 6). Similarly, a method is described to obtain high accuracy angular functions based on series expansions. We conclude the theoretical study by evaluating the Hawking spectra for various combinations of spin, mass, charge and rotation parameters, and discuss them comparatively. The last part of the thesis is on the implementation of the theoretical results in the new CHARYBDIS2 Monte Carlo simulation of black hole production and decay (chapter 7), and on the analysis of the phenomenological consequences (chapter 8). The main new features implemented in CHARYBDIS2 are: a full treatment of the spin-down phase using the angular and energy distributions of the associated Hawking radiation; an improved model for energy and angular momentum loss in the production process, and a wider range of options for the Planck-scale termination of the decay. The main conclusions of this thesis and an outlook on future directions are summarised in the final chapter. iii Declaration This dissertation is the result of my own work, except where explicit reference is made to the work of others, and has not been submitted for another qualification to this or any other university. The approximate charged rotating background constructed in chapter 3 was published in [1]. The original study of charge and mass effects developed in chapters 5 and 6 was published in [1, 2]. The event generator described in chapter 7 was released in [3] and published in [4] with part of the phenomenological study of chapter 8. This thesis does not exceed the 60,000 word limit prescribed by the Degree Committee for Physics and Chemistry. Marco Oliveira Pena Sampaio v Acknowledgements I start by thanking my supervisor Prof. Bryan Webber for all his support, guidance, patience and positive thinking throughout my PhD. I have learnt a lot about doing re- search, and the process of building independent ideas. I must thank him in particular for the incredible continuous feedback provided during the preparation of this document. I am very grateful for the pleasure of having collaborated with: Marc Casals, Sam Dolan, Jonathan Gaunt, and especially Prof. M. Andrew Parker and James Frost. I must single out James for his dedication and support to the CHARYBDIS2 project, especially for his good-humoured and friendly character, positiveness and loyalty. I have been lucky to be surrounded by many other stimulating researchers in the Cambridge high energy physics group and the Cambridge Supersymmetry working group. I have learned from them through discussions about many interesting areas of particle physics. In particular, I thank my near office mates Deirdre Black, Steve Kom, Are Raklev and the lineage of PhD students Jenni Smillie, Seyi Latunde-Dada, Andreas Pa- paefstathiou, Jo Gaunt, Lucian Harland-Lang and Eleni Vryonidou. With them I had the most satisfying discussions, and shared problems, hopes and successes. A special note must go to a collaborator from a different project, and friend, Carlos Herdeiro for his wise advice and positiveness during some important periods of my PhD. This project was funded by Funda¸c˜ao para a Ciˆencia e Tecnologia (FCT) - Portu- gal, grant SFRH/BD/23052/2005 co-financed by POPH/FSE. I am grateful to Churchill College for providing not only academic resources, but most importantly, a home. I thank Bryan, James and Jo for proof-reading the document. There are so many people outside physics who made these years the best. I especially want to single out Martin Huarte-Espinosa, Johann von Kirchbach, Kasia Gilewicz, Frida Weierud, Carlos Guedes, Chris Edge and Raquel Ribeiro for being my Cambridge family. I was also lucky to spend many hours with the Cambridge Volleyball friends, who kept me physically healthy and cheerful. In Portugal, I thank my friends from Neiva, Viana do Castelo and Porto for always receiving me as if I had never left. I am most grateful of all to my two siblings and my parents. They believe and support everything I do, and they are the constants in my life. vii Contents 1 Introduction 1 1.1 Particle physics and the Standard Model . ..... 1 1.2 Gravity ..................................... 6 1.3 Comparingcouplings .............................. 7 1.4 Hierarchies and other unexplained properties . ......... 9 1.5 Extradimensions ................................ 11 2 Theories with extra dimensions 13 2.1 Generalformulation.. .. .. 14 2.2 TheADDscenario ............................... 17 2.3 The Randall-Sundrum scenario . .. 18 2.4 Further brane constructions and other scenarios . ......... 20 2.5 Experimental bounds on extra dimensions . ..... 21 2.5.1 Laboratorybounds . .. .. 21 2.5.2 Astrophysicsandcosmology . 23 2.5.3 Summaryofthebounds . .. .. 24 3 Strong gravity I: Charged rotating black holes 27 3.1 Black holes in general relativity . ..... 28 3.1.1 Definitionsandproperties . 29 3.1.2 Some solutions in four dimensions . .. 30 3.1.3 Exact solutions in higher dimensions . ... 35 3.2 Construction of a brane charged background . ...... 38 3.2.1 The brane Maxwell field as a perturbation . .. 40 3.2.2 Commentsonbackreaction. 42 3.2.3 Systems of units and orders of magnitude . ... 45 3.3 Geodesics and the geometrical cross section . ........ 47 3.3.1 Thecriticalimpactparameter . 48 3.3.2 Perturbative and numerical minimisation . ..... 54 4 Strong gravity II: Models for black hole production 61 4.1 Settinguptheinitialstate . ... 62 4.2 Gravitationalcollapse. ... 63 ix x CONTENTS 4.3 Trapped surface and other analytic bounds . ...... 64 4.3.1 Theoretical studies of black hole production . ...... 64 4.3.2 The model for CHARYBDIS2 ....................... 65 4.4 Latest developments in numerical relativity . ......... 69 5 Black hole decay and Hawking radiation 71 5.1 Perturbation theory and approximate decoupling . ........ 71 5.2 Hawkingradiation ............................... 75 5.3 Perturbations of a brane charged black hole . ...... 78 5.3.1 Wave equations I: Coupling the background . ... 78 5.3.2 Wave equations II: Separability . .. 80 5.3.3 Higherspins............................... 82 5.3.4 Decomposition of spheroidal waves into plane waves . ....... 84 6 Analytic and numerical study of perturbations 89 6.1 Theangularequations ............................. 89 6.2 The radial equations I: Analytic methods . ...... 94 6.2.1 Nearhorizonequation . 94 6.2.2 Far field solution and low energy matching . ... 96 6.2.3 High energy approximation based on WKB arguments . ... 98 6.3 The radial equations II: Numerical methods . ......100 6.3.1 Nearhorizonexpansions . .102 6.3.2 Farfieldexpansions. .102 6.4 Numericalresults ................................ 104 6.4.1 Theeffectofparticlemass . .105 6.4.2 The effect of black hole charge on neutral particles . ......108 6.4.3 Theeffectofparticlecharge . 109 6.4.4 Theeffectofblackholerotation . 113 6.4.5 Interplay between rotation and charge effects . ......119 6.5 High energy absorption cross sections . ......121 6.6 Conclusions ...................................124 7 CHARYBDIS2 127 7.1 Production ...................................129 7.1.1 Adding the intrinsic spin of the colliding particles . .........132 7.2 Evaporation...................................133 7.2.1 Back-reactionandspin-down. 135 7.3 Remnants ....................................140 7.3.1 Termination of the black hole decay . 140 7.3.2 Fixed-multiplicity decay model . 141 7.3.3 Variable-multiplicity decay model . .142 7.3.4 Boilingmodel ..............................143 7.3.5 Stableremnantmodel . .143 CONTENTS xi 7.3.6 Straight-to-remnantoption. 144 7.4 Programstructureandusage. 144 7.4.1 Generalstructure . .. .. .145 7.4.2 Initialisation...............................147 7.4.3 Eventgeneration . .. .. .148 7.4.4 Termination...............................151 7.5 CHARYBDIS2 andothergenerators . .151 8 Phenomenological study 153 8.1 Production ...................................154 8.2 Evaporation...................................156