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Primordial Magnetic Fields in Cosmology

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Primordial Magnetic Fields in Cosmology Primordial Magnetic Fields in Cosmology by Iain A. Brown THE THESIS IS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF THE DEGREE OF DOCTOR OF PHILOSOPHY OF THE UNIVERSITY OF PORTSMOUTH arXiv:0812.1781v1 [astro-ph] 9 Dec 2008 March, 2006 Copyright c Copyright 2006 by Iain A. Brown. All rights reserved. The copyright of this thesis rests with the Author. Copies (by any means) either in full, or of extracts, may not be made without the prior written consent from the Author. ii Abstract Magnetic fields have been observed in galaxies, clusters of galaxies and probably in superclusters. While mechanisms exist to generate these in the late universe, it is possible that magnetic fields have existed since very early times. A field existing before the formation of the cosmic microwave background will leave imprints from its impact on plasma physics that might soon be observable. This thesis is concerned with investigating methods to predict the form of such imprints. In chapter 2 we review in detail a standard, linearised cosmology based on a Robertson-Walker metric and a universe filled with photons, massless neutrinos, cold dark matter, a cosmological constant, and baryons. We work in a synchronous gauge for ease of implementation in a Boltzmann code, and keep our formalism general. We then consider the statistics of the cosmic microwave background radiation, assuming that only scalar (density) perturbations cause significant impact. Chapter 3 introduces an electromagnetic field of arbitrary size and presents the equations governing the magnetised cosmology, and the structure of the electromagnetic fields, in greater detail than has hitherto been shown. We then invoke a hierarchy of approximations, treating the conductivity of the universe as infinite and thus removing the electric field, and considering the energy density of the magnetic field to be small. We present the resulting system in a computationally useful form and end by reviewing previous studies into the damping scales induced by photon viscosity. Chapter 4 considers the intrinsic statistics of the magnetic stresses. We approach this issue in two ways. Analytical methods are exact but reliant on an underlying Gaussian distribution function for the magnetic field, and simulating fields on a finite grid allows a great freedom in the form of the field but imposes an undesirable granularity and an unphysical infra-red cut-off. We construct the two-point moments in Fourier space, extending and improving the analytical results, some of which we present for the first time. There is excellent agreement between the results from analysis and those from the simulated fields. At the one- and three-point level we find significant intrinsic non-Gaussianities. In chapter 5 we turn to the observable impacts a primordial magnetic field. We briefly review previous studies into constraints from the epoch of nucleosynthesis before turning to consider the cosmic microwave background. We briefly consider the potential impact of an evolving damping scale, which may cause decoherence in the sources. Assuming coherence, which is likely to be accurate for very infra-red fields, the statistics of the source can be mapped onto the cosmic microwave background in an extensible man- ner, modelling the source statistics and the photon evolution entirely seperately. We demonstrate that our approach is valid by reproducing the signals for Gaussian power law fields on the microwave sky. After outlining how we will improve our predictions by employing a Boltzmann code, we show that it is a purely technical matter to extend the method to the three-point function. With detector sensitivity increasing, the non-Gaussianity of the cosmic microwave background will be well constrained in the near future and our work allows us to employ a new probe into the nature of an early-universe magnetic field. iii Preface The workof this thesis was carriedoutat the Institute of Cosmology and Gravitation University of Portsmouth, United Kingdom. The work in this thesis is based on work in collaboration with Robert Crittenden (ICG, Portsmouth). Much of chapter 4 is based on the paper “Non-Gaussianity from Cosmic Magnetic Fields”, I. Brown and R. Crittenden, Phys. Rev. D72 063002 (2005). I hereby declare that this thesis has not been submitted, either in the same or different form, to this or any other university for a degree, and that it represents my own work. Iain Brown iv Acknowledgements The work presented in this thesis could not have been performed without the help and collaboration of my supervisor, Robert Crittenden. One of the chapters of this thesis is based on and expanded from a paper written in collaboration with him. The support of the Institute of Cosmology and Gravitation, and in particular of Prof. Roy Maartens, has also been invaluable, as have been discussions with Kishore Ananda, Chris Clarkson, David Wands, David Parkinson, Marco Bruni, Konstantinos Dimopoulos, Antony Lewis and others too numerable to mention. I must also thank my family for their support as writing the thesis wore on, and the patience and tolerance of my friends, particularly of Mike, Ricki, Laura, Emma, Chris and Vivian. The largest thanks go to Kishore, Richard, Mat and Dan, who offered much appreciated hospitality and sat through innumerable ”discussions” about the thesis and the state of the world, and to Kathryn, who rarely complained about the weight of emails and helped my sanity enormously. Thanks to them all, this was a lot less unpleasant than it might otherwise have been. v Table of Contents Abstract iii Preface iv Acknowledgements v 1 Introduction 1 2 Standard Linearised FLRW Cosmology 6 2.1 TheFLRWGeometry................................. .... 6 2.2 FluidMatter..................................... ..... 12 2.2.1 GenericStress-EnergyTensor . ........ 12 2.2.2 ColdDarkMatter................................ ... 16 2.2.3 Single-FluidUniverses . ....... 16 2.3 BoltzmannFluids ................................. ...... 18 2.3.1 GeneralBoltzmannFluids . ...... 18 2.3.2 ThePhotons .................................... 24 2.3.3 EandBModes.................................... 34 2.3.4 TheLine-of-SightApproach . ....... 34 2.3.5 TheNeutrinos.................................. ... 37 2.3.6 Truncation .................................... 38 2.4 BaryonicMatter.................................. ...... 39 2.4.1 Tight-Coupling ................................ .... 40 2.5 ScalarFieldMatter ............................... ....... 42 2.6 TheEvolutionoftheUniverse . ......... 44 2.6.1 PeriodsintheUniversalEvolution . .......... 44 2.6.2 VacuumDomination–Inflation . ...... 44 2.6.3 RadiationDomination . ..... 45 2.6.4 Matter-RadiationEquality . ........ 46 2.6.5 MatterDomination .............................. .... 47 2.6.6 Recombination–FormationoftheCMB . ....... 48 2.6.7 Late-TimeAcceleration . ...... 48 2.7 TheCosmicMicrowaveBackground . ......... 49 2.7.1 TheSachs-WolfeEquation . ...... 49 2.7.2 StatisticalMeasuresontheSky . ........ 50 2.7.3 Temperature Auto-Correlation - Traditional Approach ............... 51 2.7.4 Angular Power Spectra from the Line-of-Sight Approach.............. 53 2.7.5 TheCMBAngularPowerSpectrum . ..... 55 3 Magnetised Cosmology 59 3.1 TheElectromagneticField . ......... 59 3.1.1 FieldandStress-EnergyTensors . ......... 59 3.1.2 Maxwell Equations and Stress-Energy Conservation . .............. 61 3.1.3 The Electromagnetic Field: Robertson-Walker and BianchiCosmologies . 62 3.2 Baryonic Matter: Cosmological Magnetohydrodynamics . ................. 63 3.2.1 RealSpace ..................................... 63 vi 3.2.2 FourierSpace.................................. ... 66 3.3 DampingofCosmologicalMagneticFields . ............ 69 3.4 Conclusions..................................... ..... 72 4 Statistics of Cosmic Magnetic Sources 73 4.1 TangledMagneticFields . ........ 73 4.1.1 UnderlyingStatisticsoftheFields . ........... 74 4.1.2 RealisationsofMagneticFields . ......... 75 4.2 One-PointMoments................................ ...... 77 4.3 Two-PointMoments ................................ ..... 77 4.4 Three-PointMoments.... .... ... .... .... .... ... .... ....... 81 4.4.1 GeneralConsiderations. ....... 81 4.4.2 ScalarBispectra ............................... .... 82 4.4.3 VectorandTensorCrossBispectra . ......... 83 4.4.4 TensorAuto-Correlation . ....... 85 4.4.5 Results ....................................... 86 4.5 TheLorentzForces ................................ ...... 88 4.6 ADampedCausalField.............................. ...... 89 4.7 DiscussionandConclusions . ......... 90 5 Observable Impacts of a Magnetic Field 92 5.1 Overview ........................................ ... 92 5.2 Nucleosynthesis................................. ....... 93 5.2.1 DirectBounds .................................. 93 5.2.2 IndirectBounds................................ .... 94 5.3 Large-ScaleMagnetisedCosmology . ........... 95 5.3.1 GeneralApproach ............................... ... 95 5.3.2 DampingScales ................................. 96 5.3.3 Non-EqualTimeCorrelations . ....... 98 5.3.4 VectorTransferFunctions . ....... 99 5.3.5 TensorTransferFunctions . ....... 101 5.4 CMBPowerSpectrafromTangledMagneticFields . ............. 102 5.4.1 AnalyticalEstimatesfromtheLiterature . ............. 103 5.4.2 NumericalEstimatesfromtheLiterature. ............ 103 5.4.3 A New ApproachtotheMagnetisedCMB PowerSpectrum . ......... 104 5.4.4 An Outline forGeneratingStatistics fromCMBFast . ............. 108 5.5 TheCMBBispectrum...............................
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