Astroparticle Physics

254 12 Inflationexcitation of this field corresponds to a spin-0 particle. Theenergy density ϱ and pressure P associated with φ are (inunits with ¯h = c = 1) given by [38]ϱ = 1 2 ˙φ 2 + 1 2 (∇φ)2 + V(φ), (12.20)P = 1 2 ˙φ 2 − 1 6 (∇φ)2 − V(φ). (12.21)Higgs fieldThe term V corresponds to a potential which could emergefrom some particle physics theory. A particular form for Vis, for example, predicted for a Higgs field (see below). Fornow V(φ)will be treated as a function which one can choosefreely.Now, for a field φ(x,t), which is almost constant in timeand space such that the ˙φ 2 and (∇φ) 2 terms can be neglectedrelative to V(φ), one then hasϱ ≈−P ≈ V(φ), (12.22)Fig. 12.1Schematic illustration of thepotential V(φ)associated with theHiggs field φ in the StandardModel of particle physicsHiggs potentialFig. 12.2Schematic illustration of thepotential V(φ)first proposed toprovide inflationmodified Higgs potentialand therefore the equation-of-state parameter w is approximately−1. Since this fulfills the relation w0.Now, in constructing a quantum field theory there is afairly wide degree of freedom in writing down the potentialterm V(φ). For example, Fig. 12.1 shows the potential associatedwith the Higgs field. This is a scalar field in the StandardModel of particle physics needed to explain the massesof the known particles.In 1981, Alan Guth [39] proposed that a scalar Higgsfield associated with a Grand Unified Theory could be responsiblefor inflation. The potential of this field should havea dip around φ = 0, as shown in Fig. 12.2. As a consequenceof the local minimum at φ = 0, there should be a classicallystable configuration of the field at this position. In this state,the energy density of the field is given by the height of thepotential at φ = 0. Suppose the field goes into this state at atime t i . The scale factor then follows an exponential expansion,(√ )8πGϱR(t) = R(t i ) exp (t − t i ) , (12.23)3with ϱ = V(0).In a classical field theory, if the field were to settle intothe local minimum at φ = 0, then it would stay there forever.In a quantum-mechanical theory, however, it can tunnel from