Master's Thesis in Theoretical Physics - Universiteit Utrecht

Φt252015105N8060402000 200 400 600 800 1000t(a) Evolution of φ, ξ = 000 5 10 15 20 25Φ(b) Number of e-folds NFigure 4.2: Numerical solution of the field φ and number of e-folds N for ξ = 0. The self-coupling λ = 10 −3 .The initial value of the field is φ in = 25, in units of M P . The initial velocity of the field is taken to be zero.Important differences with the nonminimal case are the observation that the field φ starts to oscillate almostimmediately and that the initial value of the field must be much larger in order to meet the condition N ≥ 70which is necessary for successful inflation.withA =12 ˙φ 2 + V (φ)3(M 2 p − ξφ 2 )(4.27)6ξφ ˙φB = −3(M 2 p − ξφ 2 ) . (4.28)Note that A is always positive and B > 0 during inflation since ξ < 0 and φ, ˙φ > 0. Thus wecan write for HH = − B 2 + √ ( B2) 2+ A, (4.29)where we have chosen only the positive solution for H. Now we can make certain approximations.Suppose that A ≫ (B/2) 2 . This roughly happens when λφ 2 ≫ (ξ ˙φ) 2 , i.e. when thefields are large and its derivatives small (slow-roll regime). Then we can approximate H by√Thus for large ξ,H = − B 2 + A1 + B24A ≃ A when A ≫( B2) 2. (4.30)√ √V (φ)H ≃−3ξφ 2 = − λφ212ξ , (4.31)which is of course the same result as we got previously for H in the inflationary regime. Thesituation changes however when the fields are very close to the minimum of the potential.Then λφ 2 ≪ (ξ ˙φ) 2 , which means (B/2) 2 ≫ A. We can then approximate H byH = − B ∣ √∣∣∣2 + B2∣ 1 + 4AB 2≃ − B ∣ ∣∣∣2 + B2A2∣ (1 +B 2 )= − B 2 + |B|2 + A( ) B 2when A ≪ .|B|2