Master's Thesis in Theoretical Physics - Universiteit Utrecht

drives inflation is called the inflaton.The first inflationary models suffered from some problems, such as the graceful exit andfine-tuning problems. The most accepted scenario is the chaotic inflationary model, wherea scalar field initially has a very large value and then slowly rolls down the potential wellto its minimum. We can define the slow-roll conditions that must generically be satisfiedin order for successful inflation to take place. These slow-roll conditions are determined bythe potential of the scalar field. So far there have not been any direct measurements thatconfirm whether or not inflation took place in the early universe, and we therefore also donot know the form of the scalar potential. This is also a hard nut to crack, since inflationconcerns the expansion of space and the only thing we can observe are photons and particles,which have in principle not much to do with inflation. However, cosmological inflationdoes make some predictions about our universe. One of the most important is that inflationpredicts a nearly scale invariant spectrum of density perturbations. This has actually beenmeasured from the CMBR and many cosmologists consider this to be the confirmation ofthe inflationary hypothesis. The deviations from a scale invariant spectrum can actuallybe expressed in terms of the slow-roll parameters, and since these parameters are derivedfrom the inflationary potential, measurements of the spectrum constrain the form of theinflaton potential. Inflation also predicts primordial gravitational waves that leave theirimprint on the CMBR, and cosmologists hope to see this very soon with the new Plancksatellite that was launched in May 2009. The question remains what the correct inflationarymodel really is. Chaotic inflation is possible in most scalar field models, and this is onthe one hand nice because the most general models are allowed, but on the other hand itleaves room for many exotic models.In this thesis we considered a specific type of inflationary models, the nonminimal inflationarymodels. In these models the scalar inflaton field is coupled to gravity, which in theaction formalism corresponds to the Ricci scalar R in the Einstein-Hilbert action. The termnonminimal coupling term ξRφ 2 effectively changes the gravitational constant G N suchthat it can become much smaller. On the other hand it acts as an additional mass termin the potential. We considered specifically nonminimal inflationary models with strongnegative coupling, ξ ≪ −1. In section 4.1 we considered the simplest model of a real scalarfield φ with a quartic potential with self-coupling λ that is nonminimally coupled to R. Wederived the dynamical equations for H and for the field φ and solved these analytically inthe slow-roll approximation and in the strong negative coupling limit. We saw that the fieldrolls down slowly to its minimum and that the large negative ξ reduces the initial valueof φ necessary for successful inflation, i.e. to solve the cosmological puzzles. The Hubbleparameter H has the special property that during inflation it is proportional to the fieldφ. Numerically we have solved the complete field equations and found that our analyticalresults are verified to great accuracy.Besides providing a successful inflationary model, the nonminimally coupled inflaton fieldhas the additional benefit that it lowers the constraint on the quartic self-coupling λ. Asmentioned, the measurements of the spectrum of density perturbations put constraints onthe slow-roll parameters, and since these are derived from the inflaton potential they thereforeconstrain the form of the potential. For minimally coupled models, the spectrum of densityperturbation is proportional to λ and the value of the self-coupling must therefore beλ ≃ 10 −12 . However, if we include a nonminimal coupling term, we find that the spectrumis proportional to √ λ/ξ 2 . In order to derive this, we first needed to perform a conformaltransformation of the metric that removes the nonminimal coupling term. We introducedthe general conformal transformation in section 4.2 and performed the specific conformaltransformation that removes the ξRφ 2 term in section ??. We found that we could write the