Master's Thesis in Theoretical Physics - Universiteit Utrecht

Finally, we can also vary the action with respect to the field φ to obtain the equation ofmotion for φ. The equation of motion for φ is¨φ + 3H ˙φ + ξRφ +dV (φ)dφ= 0. (4.9)The Ricci scalar R can be expressed in terms of H 2 and Ḣ (see appendix, Eq. (B.17)). Wenow want to express the Ricci scalar in terms of φ and ˙φ alone. We first substitute Eq. (4.9)into Eq. (4.8) and obtainḢ =1(M 2 p − ξφ 2 )which allows us to writeThen R isḢ =1(M 2 p − ξ(1 − 6ξ)φ 2 )R = 6(Ḣ + 2H 2 )=6(M 2 p − ξ(1 − 6ξ)φ 2 )==6(M 2 p − ξ(1 − 6ξ)φ 2 )1(M 2 p − ξ(1 − 6ξ)φ 2 )[− 1 2 ˙φ 2 + ξ ˙φ2 − 4ξHφ ˙φ − 6ξ 2 (Ḣ + 2H 2 )φ 2 − ξφ[− 1 2 ˙φ 2 + ξ ˙φ2 − 4ξHφ ˙φ − 12ξ 2 H 2 φ 2 − ξφ[− 1 2 ˙φ 2 + ξ ˙φ2 − 4ξHφ ˙φ − 12ξ 2 H 2 φ 2 − ξφ[− 1 2 ˙φ 2 + ξ ˙φ2 − 4ξHφ ˙φ + 2(M2[− ˙φ2 + 6ξ ˙φ2 + 4V (φ) − 6ξφdV (φ)dφdV (φ)dφdV (φ)dφdV (φ)],].]+ 2H 2dφ]P − ξφ2 )H 2 dV (φ)− ξφdφ]. (4.10)Note that for a quartic potential V (φ) = 1 4 λφ4 the Ricci scalar vanishes for the special valueξ = 1 6. This is the conformal coupling value of ξ in 4 dimensions. The vanishing of R suggeststhat a conformally coupled massless scalar field behaves as in flat space, i.e. it does not feelthe expansion of the universe. In fact, one can reduce the field equation for φ in Eq. (4.9)for R = 0 to the field equation for a scalar field in flat space by a conformal rescaling of thescalar field by the scale factor. We will come back to and show this in section 4.2.Let us continue by inserting Eq. (4.10) into Eq. (4.9). We find that the field φ obeys theequation of motion−ξ(1 − 6ξ)φ¨φ + 3H ˙φ + ˙φ 2(M 2 p − ξ(1 − 6ξ)φ 2 )[]1=(M 2 p − ξ(1 − 6ξ)φ 2 −4ξφV (φ) − (M 2 P)− dV (φ)ξφ2 ) . (4.11)dφIn the next section we will solve this equation analytically by using the slow-roll approximationsand the assumption that ξ ≪ −1.4.1.2 Analytical solutions of the dynamical equationsTo be complete I will give again the three equations we obtained from the action (4.2). Firstthe constraint equations (4.7) for H 2 and the dynamical equation (4.8) for Ḣ,[H 2 1 1=3(M 2 p − ξφ 2 ) 2 ˙φ 2 + V (φ) + 6ξHφ ˙φ][1Ḣ =(M 2 p − ξφ 2 − 1 ]) 2 ˙φ 2 + ξ ˙φ2 + ξφ ¨φ − ξHφ ˙φ ,