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Livro de Cosmologia

TRANSCRIPT

  • Lecture notes of a course presented in the framework of the3ie`me cycle de physique de Suisse romande

    Inflationary cosmology

    EPFL, March 2006

    Julien LesgourguesLAPTH, Annecy-Le-Vieux, France

  • Contents

    1 Introduction 5

    1.1 Basic recalls on the standard cosmological model . . . . . . . . . . . . . . . . . . . . . . . 51.2 Motivations for inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    1.2.1 Flatness problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Horizon problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.3 Origin of perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.4 Monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    1.3 Quick overview of scalar field inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

    2 Inflation with a single scalar field 14

    2.1 Slow-roll conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Background evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Perturbations and gauge freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Quantization and semi-classical transition . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    2.4.1 Basic recalls on quantization of a free scalar field in flat space-time . . . . . . . . . 172.4.2 Definition of the mode function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.3 Free field in curved space-time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.4 Quantum to semi-classical transition . . . . . . . . . . . . . . . . . . . . . . . . . . 19

    2.5 Tensor perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5.1 General equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5.2 Solution during De-Sitter stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.3 Long-wavelength solution during and after inflation . . . . . . . . . . . . . . . . . . 222.5.4 Primordial spectrum of tensor perturbations . . . . . . . . . . . . . . . . . . . . . 23

    2.6 Scalar perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6.1 General equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6.2 Solution during Quasi-De-Sitter stage . . . . . . . . . . . . . . . . . . . . . . . . . 242.6.3 Long-wavelength solution during and after inflation . . . . . . . . . . . . . . . . . . 252.6.4 Primordial spectrum of scalar perturbations . . . . . . . . . . . . . . . . . . . . . . 26

    2.7 Computing smooth spectra using a slow-roll expansion . . . . . . . . . . . . . . . . . . . . 272.8 Computing Broken Scale Invariance spectra with analytical/numerical methods . . . . . . 29

    3 Current constraints from observations 31

    3.1 Relating the primordial spectra to CMB and LSS observables . . . . . . . . . . . . . . . . 313.2 Overall agreement between observations and inflationary predictions . . . . . . . . . . . . 34

    3.2.1 Gaussianity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.2 Scale-invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.3 Adiabaticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.4 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

    3.3 Recent constraints on slow-roll parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4 Implications for various types of inflaton potentials . . . . . . . . . . . . . . . . . . . . . . 38

    3.4.1 Practical method for constraining potentials . . . . . . . . . . . . . . . . . . . . . . 383.4.2 Monomial potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4.3 New inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4.4 Hybrid inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

    4 Connection with high-energy physics 45

    4.1 Fine-tuning issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Global Supersymmetry models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3 Supergravity models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.4 Compatibility with string theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.5 State of the art of inflationary model building . . . . . . . . . . . . . . . . . . . . . . . . . 514.6 Prospects for observations and theoretical developments . . . . . . . . . . . . . . . . . . . 52

    3

  • 1 Introduction

    1.1 Basic recalls on the standard cosmological model

    In the standard cosmological model and at the level of background quantities (i.e., averaging over spatialfluctuations), the universe is described by the Friedmann-Lematre-Robertson-Walker metric

    ds2 = dt2 a(t)2[

    dr2

    1 k r2 + r2(d2 + sin2 d2)

    ](1)

    where k is a constant related to the spatial curvature (positive for closed models, negative for open ones,zero for a flat universe) and t is the proper time measured by a free-falling observer, that we will callcosmological time. Throughout this course, we adopt units such that c = h = 1.

    The Einstein equation relates curvature to matter. In the case of a homogeneous, isotropic universe,described by the Friedmann metric, the Einstein equation yields the Friedmann equation, which relatesthe total energy density to the space-time curvature: on the one hand, the spatial curvature radiusRk a |k|1/2, and on the other hand the Hubble radius RH H1 = a/a. The Friedmann equationreads

    3

    R2H 3R2k

    = 8G (2)

    where G is the Newton constant. The Friedmann equation is more commonly written as(a

    a

    )2+

    k

    a2=

    8G3

    . (3)

    The Einstein equations also lead, through Bianchi identities, to the energy conservation equation

    = 3 aa(+ p) (4)

    which applies to the total cosmological density and pressure p (actually, this conservation equation canbe derived by variation of the action for each individual homogeneous component in the universe).

    Non-relativistic matter has vanishing pressure and gets diluted according to m a3, while ultra-relativistic matter has pr =

    13r and follows r a4. These dilution laws can be derived more intuitively

    by considering a comoving sphere of fixed comobile radius. The number of particles (ultra-relativistic ornon-relativistic) is conserved inside the sphere (although non-relativistic particles are still in the comobilecoordinate frame, while ultra-relativistic particles flow in and out). The individual energy E m of anon-relativistic particle is independent of the expansion. Instead the energy of a photon is inverselyproportional to its wavelength and is redshifted like a1. The volume of the sphere scales like a3.Altogether these arguments lead to the above dilution laws. The curvature of the universe contributes tothe expansion in the same way as an effective curvature density

    effk 3

    8Gk

    a2(5)

    which scales like a2. Finally, the vacuum energy can never dilute, v = 0 and pv = v. This is validfor the energy of a quantum scalar field in its fundamental state, as well as for a classical field in a stateof equilibrium (its energy density density v is then given by the scalar potential V () at the equilibriumpoint). The vacuum energy is formally equivalent to a cosmological constant which can be added tothe Einstein equation without altering the covariance of the theory,

    G + = 8GT , (6)

    with the identification v = pv = /(8G). The critical density is defined for any given value of theHubble parameter H = a/a as the total energy density that would be present in the universe if the spatialcurvature was null,

    H2 =8G3

    crit . (7)

    With such a definition, the Friedmann equation reads

    crit effk = tot . (8)

    5

  • The contribution of spatial curvature to the expansion of the universe is parametrized by

    k effk

    crit= k

    (aH)2=

    R2HR2k

    . (9)

    Whenever |k| 1, the universe can be seen as effectively flat.

    Let us summarize very briefly how the cosmological model was built step by step:

    Einstein first proposed a solution for a static universe with RH = 0, based on a non-zero cosmo-logical constant, which was proved later to be unstable. The idea of a static (or even stationary)universe was then abandoned in favor of a nearly homogeneous, isotropic, expanding universe, corre-sponding in first approximation to the Friedmann-Lematre-Robertson-Walker metric. This picturewas sustained by the discovery of the homogeneous expansion by Hubble in 1929.

    the minimal assumption concerning the composition of the universe is that its energy density isdominated by the constituent of visible objects like stars, planets and inter-galactic gas: namely,non-relativistic matter. This is the Cold Big Bang scenario, in which the Friedmann equation

    H2 =8G3

    m a3 (10)

    describes the expansion between some initial singularity (a 0) and now, caused by non-relativistic,pressureless matter (cold matter) at a rate a t2/3. Gamow, Zeldovitch, Peebles and othersworked on scenarios for Nucleosynthesis in the Cold Big Bang scenario, and concluded that it wasruled out by the fact that the universe contains a significant amount of hydrogen.

    the next level of complexity is to assume that a radiation component (ultra-relativistic