Cosmology Meets Condensed Matter Mark N. Brook Thesis submitted to the University of Nottingham for the degree of Doctor of Philosophy. July 2010 The Feynman Problem-Solving Algorithm: 1. Write down the problem 2. Think very hard 3. Write down the answer – R. P. Feynman att. to M. Gell-Mann Supervisor: Prof. Peter Coles Examiners: Prof. Ed Copeland Prof. Ray Rivers Abstract This thesis is concerned with the interface of cosmology and condensed matter. Although at either end of the scale spectrum, the two disciplines have more in common than one might think. Condensed matter theorists and high-energy field theorists study, usually independently, phenomena embedded in the structure of a quantum field theory. It would appear at first glance that these phenomena are disjoint, and this has often led to the two fields developing their own procedures and strategies, and adopting their own nomenclature. We will look at some concepts that have helped bridge the gap between the two sub- jects, enabling progress in both, before incorporating condensed matter techniques to our own cosmological model. By considering ideas from cosmological high-energy field theory, we then critically examine other models of astrophysical condensed mat- ter phenomena. In Chapter 1, we introduce the current cosmological paradigm, and present a somewhat historical overview of the interplay between cosmology and condensed matter. Many concepts are introduced here that later chapters will follow up on, and we give some examples in which condensed matter physics has had a very real effect on informing cosmology. We also reflect on the most recent incarnations of the condensed matter / cosmology interplay, and the future of these developments. Chapter 2 presents the Einstein-Klein-Gordon system of equations and their non-relat- ivistic and nonlinear counterparts, the Schr¨odinger-Poisson, and nonlinear Schr¨odinger (Gross Pitaevskii)-Poisson systems. We give a more technical overview of the various applications of these systems of equations, as well as discussing the role and interpre- tation of condensates in the field of cosmology. Abstract iii In Chapter 3 we discuss more qualitatively the fluid-mechanical methods used in a wave-mechanical approach to structure formation, and in formulations of condensed matter models. Taking a lead from the condensed matter side, we look at some of the details of the Gross-Pitaevskii equation, particularly with regard to quantum vortices, and then put this quantum-mechanical system into a cosmological environment by coupling it to the Poisson equation, in an effort to pin down some of the parameters that may be consistent with the existence of vortices in a cosmological Bose-Einstein condensate. In Chapter 4 we turn to high-energy field theory and elucidate further some of the re- lationships with condensed matter physics that are present. We also critically examine a Bose-Einstein dark matter model in light of these considerations. Chapter 5 rounds off with a discussion and suggestions for further work based upon models we have discussed, as well as some ideas for models that have not yet been mentioned. An appendix discusses techniques for moving from the relativistic Einstein-Klein- Gordon equations to the Schr¨odinger-Poisson system. Acknowledgements Gratitude is extended to those who have conspired to keep my life interesting. Includ- ing, but not limited, to: David, Will, Edo, Jensen, Ryan, Sarah, Thea, Dan, Nic, Ben, Karen, Seb, Charlotte, Jon, Hari, Tony, Abigail, Ben, Dave, Matt, Bruce, Holger, Kiera, Martin, Bethan, Ed, Mike, Peter, Meghan, Kate, Kathleen, Omar, Paul, Nicky, Dan, Rob, Evelyn, Amanda, Beth, David, Lisa, Dolf, Emma, Steph, Dan, Mike, Ruth, Amanda, Ian, Malgorzata, Joe, Peter, Tom, Jesse, Amie, Jen, Shannon, Troy, Fred, Tiana, Charles, Justine, Joe, Richard, Andrew, Philip, Frazer, Anita, David, Ray, Sophie, BJ, Kyle, Carl, Chris, June, Simon, Sean, Gemma, Iris, David, Mike, Christine, Rowena, Janine, Emma, Gustavo, Emmaline, Dale, Jon, Katrina, David, Peter, Aman, Jonathan, Kate, Chris, Kev, Boris, Magda, Mark, Dave, Carly, Lizzy, Belinka, Elisa, Judith, Daniele, Jes´us, Dan, Maryam, Marco, Natasha, Giordano, Riccardo, Marion, Lynne, Jennie, Danny, Phillip, Harvey, Anne, Tim, Duncan, Mitsuo, Graeme, Marianne, Bec, Karima, Marco, Kate, Julie, Ben, Martyn, Clare, Michelle, Nara, Nat, Phil, Fiona, Caitlin, Alberto, Ned, Karen, Alessandra, Rachael, Alkistis, Becca, Tom, Claire, Gary, Martin, Arianna, Yara, Olivia, Philip. Contents Abstract ii Acknowledgements iv List of Figures and Tables viii 1 Cosmology and Condensed Matter 1 1.1 Cosmology ............................... 1 1.1.1 TheBirthofGeneralRelativity. 1 1.1.2 InflationandDarkEnergy . 13 1.1.3 The Traditional Approach to Structure Formation . ... 27 1.1.4 DarkMatter........................... 30 1.2 CosmologyMeetsCondensedMatter. 38 1.2.1 The‘Higgs-Anderson’mechanism . 39 1.2.2 Kibble-Zurek mechanism - Condensed Matter again informs Cosmology ........................... 41 1.3 TheFutureofMulti-DisciplinaryResearch . .... 45 1.4 Discussion................................ 48 2 Technical Background 49 2.1 BeyondColdDarkMatter. 50 2.2 A Wave-Mechanical Approach to Structure Formation . ..... 52 2.3 QuantumMechanicalDarkMatter . 55 2.3.1 DarkMatterasaBose-EinsteinCondensate . 56 2.4 ScalarFieldDarkMatter . 59 2.4.1 Scalar Fieldsand Bose-EinsteinCondensation . ... 63 2.5 ExoticObjects.............................. 66 2.5.1 BosonStars ........................... 66 2.5.2 Oscillatons ........................... 68 2.6 PenroseandtheQuantumStateReductionProblem . ... 69 CONTENTS vi 2.7 Discussion................................ 71 3 The nonlinear Schrodinger¨ Equation in Condensed Matter and Cosmol- ogy 73 3.1 TheBose-EinsteinCondensate . 74 3.2 TheGross-PitaevskiiEquation . 77 3.2.1 TheMadelungTransformation . 82 3.2.2 Vortices ............................. 83 3.2.3 ApproximationstotheDensityProfile . 86 3.3 The Wave-Mechanical Approach to Structure Formation . ..... 88 3.4 Gravitational Stability of Vortices in Bose-Einstein Condensate Dark Matter.................................. 90 3.4.1 Vortices in Gravitationally Coupled Bose-Einstein Condensates 92 3.4.2 Vortex Stability in Gravitationally Coupled Bose-Einstein Condensates........................... 93 3.4.3 BoundsonParameters . 95 3.4.4 Results ............................. 99 3.5 Discussion................................ 100 4 Relations to Field Theory 105 4.1 ComparisonofRelatedModels . 106 4.1.1 TheAbelian-HiggsModel . 106 4.1.2 TheGoldstoneModel. 110 4.1.3 TheLandau-GinzburgModel . 111 4.1.4 TheGross-PitaevskiiEquation . 113 4.2 The Relationship between Bose-Einstein Condensation and Sponta- neousSymmetryBreaking, and the Role of Temperature . 115 4.3 TheAxion................................ 120 4.4 The Domain Wall Problem in a Model of Bose-Einstein Condensate DarkMatter ............................... 123 4.4.1 TheModel ........................... 124 4.4.2 Existence of Domain Walls in the Silverman-Mallett Model. 125 4.4.3 Cosmological Constraints on the Existence of Domain Walls . 125 4.4.4 Mass and Symmetry-Breaking Parameters in the Silverman- MallettModel.......................... 127 4.4.5 FinalRemarks.......................... 129 4.5 Discussion................................ 130 CONTENTS vii 5 Conclusions and Further Work 133 A From Einstein-Klein-Gordon to Schrodinger-Poisson¨ 138 Bibliography 142 List of Figures 1.1 Hubble’s original data [23], compared with that of more recent surveys (see Kirshner [24] for image and data references). Note the difference inscale. ................................. 7 1.2 Comparison of Λ CDM models. From Choudhury and Padmanabhan [99]. Observational data points are obtained from the ‘Gold’ sample of Riess et al. [100] from the High-z Supernova Search. .. 24 1.3 Equation of state. From the Supernova Cosmology Project [89]. ... 25 1.4 Temperature anisotropies from WMAP 5 year data [101]. ..... 26 1.5 PowerspectrumfromWMAP5yeardata[102]. 27 3.1 Numeric solution to eqn. (3.59) (blue), the Pad`eapproximation, eqn. - (3.62) (red), and the scaled approximation used in this analysis, eqn. - (3.61)(green)............................... 87 3.2 Velocity Profiles for vG (green, dot dash) and vω (blue, dash). Density profile plotted schematically for comparison (red, solid). ....... 95 3.3 Allowedregionin(V0, m) parameter space, for a healing length of a0 = 1 1016 m ( 1parsec). ........................ 100 × ∼ 3.4 Allowed regions in (V0, m) parameter space, with n = 10. Healing lengthsaslabelled. .. .. .. .. .. .. .. .. 101 List of Tables 1.1 WMAP5YearParameters. 26 3.1 VortexEnergyperUnitLength.. 103 Chapter 1 Cosmology and Condensed Matter 1.1 Cosmology 1.1.1 The Birth of General Relativity The Einstein centenary in 2005 was a chance to celebrate the significant achievements made in cosmology in particular, and physics in general, during the hundred years since Einstein’s annus miribilis. This Latin phrase has also been linked to Newton’s own miracle year in 1666, three hundred and thirty nine years previously, and one might begin to wonder when another might be due. The history of progress in cosmology is interesting in its own right. A revolution in science is rarely as drastic or cataclysmic as the image invoked. The history of progress, as reported, does seem to come in fits and starts, but it is by more of an evolutionary process that the standard paradigms are formed. Often, many incorrect suggestions or interpretations are made, with a theory becoming established within