A Scalar Field Theory for Dark Matter{Dark Energy Interaction Pedro Miguel Greg´orioCarrilho Disserta¸c~aopara obter o Grau de Mestre em Engenharia F´ısicaTecnol´ogica J´uri Presidente: Professora Maria Teresa Haderer de la Pe~naStadler Orientador: Doutor Jorge Tiago Almeida P´aramos Orientador Externo: Professor Orfeu Bertolami Vogal: Professor Jos´ePedro Mimoso Vogal: Doutor Nuno Miguel Candeias dos Santos September 2012 Acknowledgments I would like to start by thanking my supervisors Orfeu Bertolami and Jorge P´aramosfor their guidance and support throughout the development of this work, for their pacience with my mistakes and for their teachings, not only on physics but on scientific research in general. Secondly I thank my family not only for their support and for the freedom they gave me to choose physics, but also for the sacrificies they made so that I could have a better education. Next I would like to show my appreciation to my collegues and friends for the nice moments we spent together either having fun, learning physics, or both, and also for frequently stimulating me to improve myself. Also I would like to thank my friends who lived with me, for giving me a nice atmosphere to live in and for their pacience when listening to my ramblings about physics, even when they had little clue what I was talking about. Last, but certainly not least, I would like to thank Susana for her constant efforts in making me happy and for her expertise in casting away all the threats to my sanity. i Abstract In this thesis we study the effects of an interaction between dark matter and dark energy. To fulfil this objective, we introduce two scalar fields φ and χ, and endow them with an interaction potential V (φ, χ) = e−λφP (φ, χ), where P (φ, χ) is a polynomial. This improves on standard cosmology, which is instead based on the cosmological constant and non-interacting cold dark matter. In this model, we demonstrate that the relevant features of the present Universe are reproduced for a large range of the bare mass of the dark matter field. We also study modifications of the potential, revealing important implications of the interaction, including the possibility of transient acceleration solutions. The original work presented in this thesis closely follows Ref. [1]. Keywords Dark energy, Dark matter, Scalar field, Interaction model, Transient accelerated expansion. iii Resumo Nesta tese estudam-se os efeitos de uma intera¸c~aoentre mat´eriaescura e energia escura. Para cumprir esse objectivo, s~aointroduzidos dois campos escalares φ e χ, que s~aodotados de um potencial de interac¸c~ao V (φ, χ) = e−λφP (φ, χ), em que P (φ, χ) ´eum polin´omio. Este modelo complementa a cosmologia padr~ao,que ´ebaseada na constante cosmol´ogicae na mat´eriaescura fria sem interac¸c~oes. Neste modelo, demonstra-se que as propriedades relevantes do presente estado do Universo s~aorepro- duzidas para um grande conjunto de valores da massa livre do campo de mat´eriaescura. Estudam-se tamb´emmodifica¸c~oes do potencial, revelando implica¸c~oesimportantes da interac¸c~ao,incluindo a pos- sibilidade de solu¸c~oescom acelera¸c~aotransit´oria. O trabalho original apresentado nesta tese segue a Ref. [1]. Palavras Chave Energia escura, Mat´eriaescura, Campo Escalar, Modelo de intera¸c~ao,Expans~aoacelerada tran- sit´oria. v Contents 1 Introduction 1 1.1 Standard Cosmology......................................1 1.1.1 FLRW Cosmology...................................1 1.1.2 Dark Matter......................................3 1.1.3 The Cosmological Constant and Dark Energy....................4 1.1.4 Inflation.........................................6 1.2 Modifying the Standard Cosmological Model........................6 2 Scalar Field Cosmology7 2.1 General Equations.......................................8 2.2 Scalar Dark Energy.......................................9 2.3 Scalar Dark Matter....................................... 11 3 Dark Matter { Dark Energy Interaction 13 3.1 Interaction Model........................................ 15 3.1.1 Basic equations..................................... 15 3.1.2 Interaction Potential.................................. 15 3.1.3 Average Evolution Equations............................. 16 3.2 Results.............................................. 17 3.2.1 Analytical Considerations............................... 17 3.2.2 Numerical Solutions.................................. 19 4 Conclusions and Future Work 25 Bibliography 27 vii List of Figures 3.1 log ρdm and log ρde for different values of the bare mass m.................. 20 −60 3.2 log ρdm and log ρde for m = 10 and different values of the initial condition χi..... 20 3.3 Transient solution for m = 5:9 × 10−57, λ¯ = 9:5........................ 21 −15 ¯ 3.4 log ρdm and log ρde for m = 10 and different values of λ................. 22 3.5 Transient solution for m = 10−15, λ¯ = 2:8........................... 22 3.6 (λ,¯ m) parameter space and transient line........................... 23 3.7 Effective potential for m = 10−15, λ¯ = 2:8........................... 23 ix Abbreviations CDM Cold Dark Matter FLRW Friedmann-Lem^aitre-Robertson-Walker EOS Equation of State DM Dark Matter CMB Cosmic Microwave Background DE Dark Energy WIMPs Weakly Interacting Massive Particles GCG Generalized Chaplygin Gas VAMP VAriable Mass Particle xi 1 Introduction 1.1 Standard Cosmology The Standard Cosmological Model is currently a widely accepted phenomenological description of the evolution of the Universe. The reasons for that lie in its simplicity and the fact that it reproduces a variety of observational results coming from independent sources. For this last reason it is also called the Concordance Model. It is based on three fundamental ingredients: inflation, Cold Dark Matter (CDM) and the cosmological constant (Λ), as well as on the Standard Model of Particle Physics and General Relativity. In this section of the introduction we review the most relevant results of this Concordance Model as well as some of its limitations. 1.1.1 FLRW Cosmology We begin our review by pointing out a main ingredient of most cosmological models: the cosmolog- ical principle. This principle states that the properties of the Universe are the same for all observers, on sufficiently large scales, which implies the Universe to be homogeneous and isotropic. This in turn constrains the metric tensor gµν to be the Friedmann-Lem^aitre-Robertson-Walker (FLRW) metric, given by the line element: dr2 ds2 = g dxµdxν = −dt2 + a2(t) + r2dΩ2 ; (1.1) µν 1 − kr2 2 where a(t) is the cosmological scale factor, normalized in this work so that at present a(t0) = 1, dΩ is the line element for the 2-sphere and k is a constant proportional to the scalar curvature of the spatial section of space-time: for flat space it equals zero, for a closed one it is positive and it is negative if space is open, i.e. hyperbolic. The dynamics of this space-time are related exclusively with the scale factor a(t), which is so far an arbitrary function. To find the evolution equations one must input the metric (1.1) into Einstein's 1 1. Introduction equations1 [2] 1 R − Rg = T ; (1.2) µν 2 µν µν µν where Rµν is the Ricci tensor, R = g Rµν is the Ricci scalar and Tµν is the energy momentum tensor. Having chosen the metric tensor, Eq. (1.1), the only ingredient yet unknown is Tµν . However, there is very little freedom in that choice due to our assumption of homogeneity and isotropy: the energy momentum tensor reduces to that of a perfect fluid: µ [Tν ] = diag(−ρ(t); p(t); p(t); p(t)) ; (1.3) in which ρ is the energy density and p is the pressure of the fluid. What results from the substitutions of this and the metric in Eq. (1.2) are Friedmann and Raychaudhuri equations, respectively: 1 k H2 = ρ − ; (1.4) 3 a2 a¨ 1 = − (ρ + 3p) ; (1.5) a 6 with H =a=a _ the expansion rate, also called the Hubble parameter. Without further information about the specific fields originating the energy momentum tensor, these equations provide a complete description of the evolution of the Universe. What we mean by this is that the evolution of the energy density, ρ_ + 3H(ρ + p) = 0 ; (1.6) is not independent of the previous equations. It is, nevertheless, quite useful to determine ρ(t) as a function of the scale factor a(t). This is only possible, of course, if there exists an Equation of State (EOS) p = p(ρ) relating the pressure to the energy density. The EOS depends fundamentally on the type of matter one introduces into the theory. However, for cosmological purposes, a simple and important example is a fluid with a linear EOS, that is: p = wρ ; (1.7) in which the EOS parameter w is taken to be constant for the moment. We see immediately from Eq. (1.5) that the sign ofa ¨ is intimately related to the value of this parameter. In particular, the threshold for zero acceleration occurs for w = −1=3, meaning that only a negative pressure, smaller than this value, can accommodate for an accelerated expansion. Beyond that, with this EOS, the solution to Eq. (1.6) is simply −3(1+w) ρ(a) = ρ0a ; (1.8) with ρ0 the energy density at present. The known fields of the Standard Model of Particle Physics can be described cosmologically by fluids with this EOS, separated in two extreme cases: non-relativistic matter, with wm = 0, and relativistic matter or radiation with wr = 1=3. As expected, the energy −3 −4 density varies with the volume a for non-relativistic matter (ρm), or with a for radiation (ρr) due to an additional contribution from the cosmological redshift. Other cases exist, such as ultra- −6 stiff matter, with w = 1 and ρs / a and also the more exotic case with w = −1 that leads to a 1 Note that we use reduced Planck units, i.e.