Master's Thesis in Theoretical Physics - Universiteit Utrecht

In general, we can define the so-called slow-roll parameters ɛ and η byɛ = 1 2 M2 p( V′) 2(3.17)V ′′Vη = M 2 pV , (3.18)where V ′ ≡ dV /dφ and the Planck mass was defined in Eq. (3.1). The slow-roll conditionsare thenɛ ≪ 1, η ≪ 1. (3.19)The qualitative argument for these slow-roll conditions is the following: in order for inflationto take place the potential energy must dominate the kinetic energy. This means thatin a chaotic inflationary scenario (see Fig. 3.1) the scalar field should not roll down thepotential well ’too fast’. In order to achieve this, the slope of the potential well should not betoo steep. Since the slow-roll parameters in Eq. (3.17) and (3.18) contain derivatives of thepotential, they therefore tell us something about the steepness of the slope. The slow-rollconditions (3.19) are such that the slope is not too steep.Note the two different definitions for ɛ in Eq. (3.17) and (2.10). One can show that theseare equal in the slow-roll approximation. Take the time derivative of Eq. (3.15) to get anequation for Ḣ, eliminate the ˙φ term by substituting Eq. (3.16) and divide by H 2 . You willrecover Eq. (3.17).In a simple inflationary scenario with a real scalar field in a quartic potential V (φ) = λφ 4 , wecan easily see that the slow-roll conditions (3.19) are satisfied when the fields φ ∼ O(10M P ).In chaotic inflationary scenarios this is a typical initial value for the field φ.3.6 Constraints on inflationary modelsSo far we have only considered characteristics of general inflationary models. We havehowever not yet discussed constraints on inflationary models. Although inflation is a hypotheticalera that happened in the very early universe, we fortunately can constrain inflationarymodels. These constraints follow from cosmological perturbation theory, see forexample Refs. [9][10][11]. The idea is that in the early universe there are small quantumfluctuations of the scalar field and metric. During inflation some of these fluctuations arefrozen in and inflation actually amplifies the vacuum fluctuations. The scalar metric perturbationsare responsible for the large scale structure formation in the universe and leavetheir imprint on the spectrum of the CMBR. This effect can be measured and constrains theinflationary model. Let us sketch in a few steps how this works.Consider a simple free scalar field theory with field equation−□φ = ¨φ + 3H ˙φ −∇ 2a 2 φ = 0.Suppose now that the spatial derivative term is large compared to the damping term withH, so we can neglect the second term. We now see that we have a wave equation with planewave solutions. If we see the field φ as a small quantum fluctuation and Fourier transformthe field, we see that for sub-Hubble wavelengths with aH ≪ k the Fourier-transformedfield φ k obeys a harmonic oscillator equation and fluctuates.If the spatial derivative terms are small such that we can neglect the third term, we findthat the term 3H ˙φ leads to damping of φ with a characteristic decay rate proportional toH. This means that ˙φ ∝ Hφ, such that we find that H 2 ≫ k2a 2 in this limit. These are