Master's Thesis in Theoretical Physics - Universiteit Utrecht

5.2 Quantization of the nonminimally coupled inflaton fieldWe consider again the action for a real scalar field that is nonminimally coupled to gravity.For convenience, we will give the explicit action again∫S =d 4 x −g(− 1 2 gµν (∂ µ φ)(∂ ν φ) − V (φ) − 1 )2 ξRφ2 . (5.44)The field equation for φ iswheredV (φ)−□φ + ξRφ + = 0, (5.45)dφ□φ = 1 −g∂ µ −gg µν ∂ ν φ. (5.46)In quantum field theory, our field φ is a quantum field. The main contribution is the classicalhomogeneous inflaton field, but on top of that there are small quantum fluctuations. Wetherefore consider the following fieldφ(t,x) = φ 0 (t) + δφ(t,x), (5.47)where the expectation value of the field is the classical inflaton field φ 0 , i.e 〈φ(t,x)〉 = φ 0 (t),whereas the fluctuations satisfy 〈δφ(t,x)〉 = 0. We expand our action to quadratic order influctuations and derive the field equations for the classical inflaton field and the quantumfluctuations. First of all, the expansion of the action (5.44) to linear order in fluctuationsvanishes, since this is precisely what we do when we want to find the classical field equationfor the background field φ 0 . If we use the quartic potential V (φ) = 1 4 λφ4 from Eq. (4.15) andthe FLRW metric (B.11) the classical field equation is¨φ 0 + 3H ˙φ0 + ξRφ 0 + λφ 3 0= 0. (5.48)The classical equation of motion (5.48) determines the dynamics of the inflaton field andis responsible for inflation, which we have discussed extensively in section 4.1. The Lagrangiandensity for the quantum fluctuations is thenL Q = [−g − 1 2 gµν (∂ µ δφ)(∂ ν δφ) − 1 (3λφ22 0+ ξR ) ]δφ 2 + O(δφ 3 ), (5.49)where I have used the index Q to indicate that this Lagrangian contains quantum fluctuations.We see that the classical inflaton field acts as a mass term for the quantumfluctuations, so I will define a mass termm 2 ≡ 3λφ 2 0 . (5.50)Again with the substitution of the FLRW metric we find the field equation for the fluctuationsδ ¨φ + 3Hδ ˙φ − ∇ 2 δφ + ξRδφ + m 2 δφ = 0. (5.51)The quantum fluctuations, described by Eq. (5.51), are the origin of density perturbations,whose spectrum we can measure from the CMBR. Also, quantum fluctuations can influencethe dynamics of fermions, which we will discuss in chapter 6. For now, let us continue withthe quantization of our inflaton field.