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Traces of Exotic Physics in Cosmology
WILLS, DANIELLE,ELIZABETH,STANBR
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Traces of Exotic Physics in
Cosmology
Danielle Elizabeth Wills
A Thesis presented for the degree of
Doctor of Philosophy
Centre for Particle Theory
Department of Mathematical Sciences
University of Durham
England
March 2015
Dedicated to
Shiva and Shakti
Traces of Exotic Physics in Cosmology
Danielle Elizabeth Wills
Submitted for the degree of Doctor of Philosophy
March 2015
Abstract
In recent years it has become increasingly clear that our universe is far more
intricate than we might ever have imagined. While theoretical formulations of the
fundamental aspects of Nature have, for many years, hinted at its vast and elusive
complexity, suggesting that our known world is but a tiny facet of the greater re-
ality in which it is embedded, it has only been within the last several decades that
observations have really begun to confirm this. Indeed, while deep-field surveys
of the universe have uncovered myriads of galaxies, constituting an untold number
of gravitationally bound microcosms such as ours, precision cosmological measure-
ments have revealed that all of this luminous baryonic matter is a near negligible
fraction of the total energy and matter in the universe. The vast majority of our
cosmos is a dark universe, comprised of some kind of invisible substances or dark
fluids that only interact gravitationally with visible matter.
Even among the objects that are visible to us, there are many mysterious entities
which are predicted by theory and which may or may not as yet have been glimpsed
in the cosmos. In the first part of this thesis we will study the interactions between
two such entities, namely cosmic strings and rotating black holes. In the latter part,
we will turn to the invisible sector and explore whether or not the dark phenomena
in the universe could in fact be the shadows of fundamental objects moving in higher
dimensions beyond our own.
Declaration
The work in this thesis is based on research carried out at the Centre for Particle
Theory, Department of Mathematical Sciences, Durham University, England. No
part of this thesis has been submitted elsewhere for any other degree or qualification
and it is all my own work unless referenced to the contrary in the text.
Chapters 3, 4 and 5 are based on collaborative research papers. The work in
Chapter3wasdoneincollaborationwithRuthGregoryandDavidKubizn¯a´k,thatin
Chapter 4 with Ruth Gregory, David Kubizn¯´ak, Robert Mann and Peter Gustainis,
and finally that in Chapter 5 with Ivonne Zavala and Tomi Koivisto.
Copyright (cid:13)c 2015 by Danielle Elizabeth Wills.
“The copyright of this thesis rests with the author. No quotations from it should be
published without the author’s prior written consent and information derived from
it should be acknowledged”.
iv
Acknowledgements
First of all I would like to extend my deepest thanks to my supervisor Ruth Gre-
gory, who has inspired, supported and guided me in my studies, provided fantastic
and numerous opportunities for me, and most importantly, has really been a friend
to me during my time as her student. Thanks to Ruth, my Ph.D life has been
full to the brim of diverse research, stimulating research trips, interesting and in-
depth discussions, and memorable adventures around the world such as hiking in
the Grand Canyon and cross-country skiing in snowy Canada. Both intellectually
and personally, I have grown so much as a result of having Ruth as a supervisor.
In addition, I am most grateful to Paul Sutcliffe for being my second supervisor,
and for always being so available to me when I have needed advice or guidance.
I would like to thank Ivonne Zavala for introducing me to the world of theoretical
physics as a Masters student, and for setting the standard so high as my first re-
search supervisor. I have benefited so much from Ivonne’s breath of knowledge and
expertise as a theoretical physicist, and have so much enjoyed spending time with
her and getting to know her in an informal context. As with Ruth, she is definitely
a friend to me.
I would further like to thank Hans Peter Nilles and Stefan Fo¨rste for their guid-
anceandsupportduringmyMSc,andtomyresearchgroupinBonnfortwofantastic
years.
Thank you to Ivonne along with all my collaborators, David Kubizn¯a´k, Robert
Mann, Peter Gustainis and Tomi Koivisto, for all their hard work that went into
obtaining the results that are presented in this thesis. I feel very privileged to
compile together into my own thesis the research and ideas that belong to all of us.
In addition, I am deeply grateful to Gianmassimo Tasinato and Simon Ross for their
v
vi
careful reading and critiquing of this manuscript.
So many people at Durham have made my time as a student so memorable.
Thank you to my office mates David Tapp and Paul Jennings, who made the office
such an animated and interesting place to be, where there was never a dull moment.
ThankyoualsotoSarahChadburn, PhillippBurda, BenWithers, SimonGentleand
Jonathan Pearson for memorable times in Durham and in Canada at the Perimeter
Institute. Thank you also to all my fellow CPT and maths students and to my
Durham friends outside of academic life, most notably Natasha Cooper, for adding
so many wonderful things to my Ph.D experience.
My family have always provided me with an enormous amount of support and
encouragement. I am truly grateful to Liz Evans, John Wills, Natalie Wills, Justin
Wills, and Hayley Wills, and later on to Julia Wills and James Laidlaw, for the
enumerable ways in which you have all enhanced my life. Liz Evans in particular
has shared and fed my passion for uncovering the subtleties of Nature and I thank
her for many stimulating philosophical discussions, for the exchanging of ideas and
books, and for guiding me to the right path.
Finally, with enormous gratitude and love, I thank my fianc´e Mike Pandher, for
looking with me from the same place into the vastness of the universe.
Contents
Abstract iii
Declaration iv
Acknowledgements v
Preface 1
1 Theories of Gravity 3
1.1 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Lagrangian formulation . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 Black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.3 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2 Quantum gravity in higher dimensions . . . . . . . . . . . . . . . . . 31
1.2.1 A history of extra dimensions . . . . . . . . . . . . . . . . . . 32
1.2.2 Superstring theory . . . . . . . . . . . . . . . . . . . . . . . . 38
1.3 Scalar-tensor theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2 Cosmic Strings 50
2.1 Symmetry breaking in the universe . . . . . . . . . . . . . . . . . . . 51
2.2 Topological defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.3 Abelian Higgs model . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.4 The Nielsen–Olesen vortex . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5 Strings and black holes . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.5.1 Vortex in a Schwarzschild spacetime . . . . . . . . . . . . . . 66
2.5.2 Flux expulsion in Reissner-Nordstrom . . . . . . . . . . . . . . 70
vii
Contents viii
3 Rotating Black Hole Hair 75
3.1 Higgs hair for the Kerr black hole . . . . . . . . . . . . . . . . . . . . 76
3.1.1 Approximate solution . . . . . . . . . . . . . . . . . . . . . . . 78
3.1.2 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . 80
3.2 Extremal Kerr black holes . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2.1 Near horizon expansion . . . . . . . . . . . . . . . . . . . . . . 87
3.2.2 Flux penetration and expulsion . . . . . . . . . . . . . . . . . 89
3.3 Backreaction of the vortex on the black hole . . . . . . . . . . . . . . 91
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4 Vortex Hair for AdS Black Holes 101
4.1 Vortices in AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2 Vortices in Kerr-AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.2.1 Approximate solution . . . . . . . . . . . . . . . . . . . . . . . 106
4.2.2 Extremal black holes . . . . . . . . . . . . . . . . . . . . . . . 108
4.2.3 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . 115
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5 Dark D-brane Cosmology 122
5.1 The general set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.1.1 Disformal coupling from moving D-branes . . . . . . . . . . . 126
5.1.2 Disformally Interacting Massive Particles (DIMPs) . . . . . . 131
5.2 Disformal Dark D-brane Cosmology . . . . . . . . . . . . . . . . . . . 134
5.2.1 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.2.2 Phase space analysis . . . . . . . . . . . . . . . . . . . . . . . 140
5.2.3 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . 150
5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6 Concluding Remarks 159
Bibliography 165
List of Figures
1.1 A visualisation of the large-scale structure from the Millennium-XXL
Simulation, on a scale of a few hundred Mpc. The bright regions
are very dense clusters of galaxies which form where the filaments
intersect each other. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2 Coincidence problem: Ω and Ω , the fractional energy of the cosmo-
Λ X
logical constant and the fractional energy of the cosmological fluid,
are of the same order of magnitude only during a tiny window of the
cosmological evolution, yet we happen to measure them to be right
within that window today. . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1 Potential for the Higgs field: When the Higgs field sits at φ = 0, the
unstable vacuum state respects the U(1) symmetry of the potential,
as rotations about the z-axis do not change the location of the Higgs
field. Once the Higgs field rolls down into the true vacuum, the
symmetry is broken, as rotations about the z-axis now “move” the
Higgs field around the circle. . . . . . . . . . . . . . . . . . . . . . . . 54
2.2 Formation of vortices: The Kibble mechanism for the simple cases of
n = 0 and n = 1. If around a closed loop the phase of the Higgs
does not vary (n = 0), then one can contract that loop to a point
by defining a phase at that point. On the other hand, for n = 1 the
phase is not definable at the central point. This means that the Higgs
field must vanish at that point, which then corresponds to a vortex. . 56
2.3 Intercommutation of strings: Two strings cross and exchange end-
points, leading to the formation of a loop and two smaller strings. . . 57
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