Testing Models of Cosmological Inflation Christopher Michael Graham Thesis submitted for the degree of Doctor of Philosophy School of Mathematics and Statistics University of Newcastle upon Tyne Newcastle upon Tyne United Kingdom April 2010 Dedicated to the memory of Norah Mander. Acknowledgements This research was made possible through financial support from the Science and Technology Facilities Council and the School of Mathematics and Statistics. I’d like to say a big thank you to my fellow postgraduate students, companions on my PhD journey. From my 1st year, the M416 Merz Court crew, in particular Dave Ryan, who’s singing brightened many a gloomy afternoon and Sam James for helpful physics discussions and as a team leader in all things extra-curricular. More recently thanks to Andrew Baggaley, Stuart Cockburn (and his supervisor Nick Proukakis) for many helpful chats on thermalisation, and all of the other postgrad- uate students with whom I shared offices, borrowed books and stationary or conversed with over lunch. A big thank you to my second supervisor Andrew Fletcher who has always been available for useful discussions about my work, and to Anvar Shukurov and David Toms who stepped in to help with my viva arrangements in the last few months. Thank you to my external examiners also, Arjun Berera and Ed Copeland. A huge thank you to Antonia who endured the strains of living with me, and was always there to support me, and to my parents, my sister Catherine and my extended family. To my friends outside of academia too, particularly Andy and Darren who helped me juggle my life in recent months. And finally this thesis would not have been possible without my supervisor Ian Moss. Without Ian’s guidance, enthusiasm and seemingly infinite knowledge of the subject there would still be no end in sight and it is to him that I am most grateful. Abstract Over the past two decades cosmological inflation has become an essential ingredient to any model of the early universe. Measurements of temperature fluctuations observed in the Cosmic Microwave Background are in remarkable agreement with the predictions of inflationary models, and for the first time these results have allowed these models to be intensively tested against observational data. This thesis considers the building and testing of inflationary models which can be consistent with the Cosmic Microwave Background data. In particular the consistency of warm inflation is tested, a model where density fluctuations arise from thermal, rather than quantum vacuum fluctuations, and the possibility of distinguishing this model from standard inflationary theories is considered. The decay channels for the production of low-mass particles during standard and warm inflation are analysed to yield new results for the particle production rates. The subsequent thermalisation of particles is investigated numerically for both cases. It is concluded that in some situations the radiation produced during inflation can remain in thermal equilibrium, allowing modified inflaton equations parameterised by a single temperature-dependent friction coefficient. New results are obtained for the power spectrum of fluctuations at the end of inflation for several models of warm inflation, results that will be very important when testing temperature-dependent models against Cosmic Microwave Background data. The primordial power spectrum of fluctuations alone is not enough to distinguish between different models of inflation. The bispectrum statistic of non-Gaussianity is calculated for temperature-independent warm inflation, and for comparison the curva- ton model, to conclude that forthcoming Cosmic Microwave Background satellite data should be able to detect and distinguish any non-Gaussian signal for warm inflation from other models through its signal strength and the shape of the bispectrum. Preface This thesis is based partially upon the following papers: ”Testing Models of Inflation with CMB non-Gaussianity” • Ian G. Moss, Chris M. Graham JCAP 0711:004,2007 ”Particle Production and Reheating in the Inflationary Universe” • Ian G. Moss, Chris M. Graham Phys.Rev.D78:123526,2008 ”Density Fluctuations from Warm Inflation” • Chris M. Graham, Ian G. Moss JCAP07(2009)013 i Contents 1 Introduction 1 2 Cosmological inflation 4 2.1 Motivation for inflation . 4 2.2 Inflationarydynamics ............................ 9 2.3 Theslow-rollapproximation . 12 2.4 Reheating ................................... 13 2.5 Inflationarymodels.............................. 13 2.6 Warminflation ................................ 15 3 The source of density fluctuations 17 3.1 Comovingscales ............................... 18 3.2 Vacuum fluctuations as the source of density fluctuations ........ 18 3.3 Thermal fluctuations as the source of density fluctuations ........ 21 3.3.1 Fluctuations in the case Γ = Γ (φ) ................. 22 3.3.2 Fluctuations in the case Γ = Γ (T, φ)................ 25 3.4 Numerical simulation of coupled fluctuations . ...... 27 3.4.1 Numericalapproach . 27 3.4.2 Dependency on r and c ....................... 29 3.4.3 Dampingeffects ........................... 30 4 Reheating the universe 34 4.1 Particleproductionmethod . 35 4.1.1 Someusefuldefinitions. 35 ii Contents 4.1.2 Theparticlenumberdensity. 36 4.1.3 Theparticleproductionrate . 37 4.2 Particle production calculations . ..... 39 4.2.1 Oscillating fields by the scattering matrix method . ..... 40 4.2.2 Oscillating fields by the general method . 41 4.2.3 Particle production in the adiabatic approximation . ....... 43 4.2.4 Particle production in the 2-Stage decay model . 44 4.3 The thermalisation of particles . 46 4.3.1 Intheoscillatingphase . 49 4.3.2 Intheslow-rollphase . 50 4.3.3 Thermalisation with the full Boltzmann collision integral . 52 5 The cosmic microwave background 57 5.1 Evolution of perturbations after inflation . ....... 57 5.2 MeasurementoftheCMB . .. .. .. .. .. .. .. 59 5.3 CMBdataanalysis.............................. 60 5.4 Theangularpowerspectrum . 60 5.5 Power spectrum predictions using numerical code . ....... 63 6 Testing models of inflation with CMB non-Gaussianity 66 6.1 Quantifyingthenon-Gaussianity . 67 6.2 TheCMBbispectrum ............................ 67 6.2.1 Thecurvatonmodel ......................... 69 6.2.2 Warminflation ............................ 74 6.3 Numerical calculation of the bispectrum . ..... 74 6.4 Distinguishing between inflationary models . ...... 77 6.5 Non-Gaussianity tests on CMB data . 80 7 Conclusion 85 A Numerical methods 87 A.1 Numerical simulation of the stochastic fluctuations . ......... 87 iii Contents A.2 ModificationstoCMBFAST. 89 A.3 CalculatingtheFishermatrix . 92 B Reduction of the Collision Integral 95 C Bispectrum calculations 99 C.1 General expression for the angle-averaged bispectrum . ......... 99 C.2 Warm inflation angle-averaged bispectrum . 100 iv Chapter 1 Introduction The discovery and subsequent precise measurements of the Cosmic Microwave Back- ground (CMB) in the past two decades have transformed the subject that is Cosmo- logical Inflation. The early moments of the universe have long been studied, but it is with the measurement of the CMB that the earliest of these, the exponential expansion of the universe known as inflation at some 10−42 seconds old, can be studied in great depth. Indeed, the very title of this thesis, ’testing models of inflation’, is now a reality. It is very well known that towards the end of the Hot Big Bang, a very long time after inflation, photons and electrons in a plasma universe decouple, leaving photons to travel freely through space in the transparent universe that we see today. It is these photons released at the decoupling time, which after billions of years are observable as the microwaves of the CMB. While this is overwhelming evidence for the Hot Big Bang, even more astounding is that the imprint of inflation is left in the CMB in the form of small scale temperature anisotropies. At the onset of inflation, the universe is small enough that quantum vacuum fluctu- ations are important. Inflationary models predict that these quantum fluctuations are the seed for density fluctuations which are not only visible in the CMB temperature spectrum, but also in the distribution of large scale structure in the universe. With this snapshot of the universe at the decoupling time, measurements of the CMB are an extremely powerful tool for determining the physics of the early universe, both during inflation and beyond. Indeed, it would not be bold to say that the precise measurements of the COBE satellite in 1991 and WMAP satellite in 2001-2007 have transformed the subject of Cosmology. This thesis considers the building and testing of inflationary models which can be consistent with the CMB data. There are at present many many models that can be considered consistent and the quest now is to find a definitive model of inflation, a task 1 Chapter 1. Introduction that will require analysis after analysis of each model, and ultimately even more precise measurement of the CMB. In particular, this thesis considers the consistency of a model known as warm infla- tion, and the possibility of distinguishing this model from standard (cold) inflationary theories. In warm inflation, radiation is produced alongside the expansion of the uni- verse, which solves very neatly one, perhaps undesirable, problem of standard inflation- ary models, that the universe cools as it expands, leaving it far below the temperatures required for the Hot Big Bang to continue. In standard theories this is solved by having a separate period of reheating at