Master's Thesis in Theoretical Physics - Universiteit Utrecht

Chapter 4Nonminimal inflationIn this chapter we will consider a certain class of inflationary models, which I will callnonminimal inflationary models. The idea is that we have an additional term in the inflatonaction Eq. (2.33). This term is 1 2 ξRφ2 and couples the scalar field to the Ricci scalar througha coupling constant ξ. The action for the nonminimally coupled inflaton field is∫S M = d 4 x −g(− 1 2 gµν (∂ µ φ)(∂ ν φ) − V (φ) − 1 )2 ξRφ2 . (4.1)The sign of ξ is chosen such that for ξ = 1 6the action is invariant under a conformal transformation.We will learn more about the conformal transformation in section 4.2. Whenξ = 0, the coupling is said to be minimal. One way to interpret the 1 2 ξRφ2 term when ξ ≠ 0is to see it as an additional mass term for the scalar field φ, with mass m 2 = ξR that couldbe negative. The second interpretation is that the gravitational constant G N is effectivelychanged and now becomes time dependent.The nonminimal inflationary model is part of a wider class of models where a field is coupledto gravity, known as Brans-Dicke theories [13]. La and Steinhardt [14] used the Brans-Dicke theory and the interpretation that the gravitational constant changes in time to solvethe graceful exit problem in the old inflationary scenario. In their scenario of extendedinflation, the time-dependent gravitational constant effectively slows down inflation fromexponential to powerlaw inflation. The bubbles formed can overtake the expansion of theuniverse and the phase transition will be completed.Futamase and Maeda [15] have investigated the nonminimal inflation scenario for differentvalues of ξ in chaotic models with potentials m 2 φ 2 and λφ 4 . They find that for successfulinflation the nonminimal coupling ξ ≤ 10 −3 . The intuitive reason is that for largepositive ξ the effective gravitational constant can become negative in a chaotic inflationaryscenario, making the theory unstable. Fakir and Unruh[16] calculated the amplitudeof density perturbations in the nonminimal inflation scenario with a potential λφ 4 . Whenξ = 0, the amplitude of density perturbations δρ/ρ ∝ λ and the self-coupling of the fieldλ must be tuned to an extremely small value of < 10 −12 to agree with observations (seesection 3.6). However, when the coupling is non-minimal, Fakir and Unruh find that theamplitude of density perturbations is proportional to √ λ/ξ 2 . To calculate this Fakir andUnruh first write the action in the Einstein frame where the inflaton field is minimally coupledto gravity, which is related to the Jordan frame with nonminimal coupling through aconformal √ transformation. The proportionality of the amplitude of density perturbations toλ/ξ 2 means that λ can be much bigger as long as the nonminimal coupling ξ is sufficientlylarge. Note that the investigation for successful inflation by Futamase and Maeda does notexclude a large negative value of ξ! Salopek, Bond and Bardeen [17] have done an extensive19