Master's Thesis in Theoretical Physics - Universiteit Utrecht

Since the spatial background is homogeneous according to the cosmological principle, ourscalar field also has to be spatially homogeneous, i.e φ(x) = φ(t). This allows us to identifyby using Eq. (2.3)ρ = 1 2 ˙φ 2 + V (φ)p = 1 2 ˙φ 2 − V (φ). (2.35)Now we use conservation of the energy momentum tensor, ∇ µ T µν = 0, which lead us to theenergy conservation law Eq. (2.13). Using the expressions for ρ and p for a real homogeneousscalar field, we finddV (φ)¨φ + 3H ˙φ + = 0. (2.36)dφNote that we could just as well have derived the equation of motion for the field φ from theaction (2.33). This would give usdV (φ)−□φ + = 0, (2.37)dφwhere□φ = 1 −g∂ µ −gg µν ∂ ν φ. (2.38)Using the flat FLRW metric with g µν = (−1, a 2 , a 2 , a 2 ), we recover Eq. (2.36). Since we canderive the two equations (2.11) and (2.12) from the action (2.29) and we can express thesetwo equations in terms of ρ and φ, we find that the equation of motion relates these twoequations. Thus we have three equations: two Einstein equations derived by varying theaction with respect to the metric g µν and one equation of motion for φ. In chapter 4 we willactually solve the field equations for a specific matter action, and we will use the fact thatonly two of these equations are independent. To be precise, we will solve the equation foronly H and the field φ because we will be able to eliminate Ḣ from these equations.