Formal Topics in Inflation

Inflation can explain why our Universe is so flat, as well as homogeneous and isotropic on large scales, and it can give rise to the inhomogeneities observed in the CMB.
For inflation to accomplish all of this, only a limited number of assumptions about the potential of the inflaton field are necessary. In order to have a complete picture about the dynamics and the initial state of the Universe, we need to study correlation functions of the CMB inhomogeneities. The basic inflationary models predict that all of the higher-order correlation functions should be very much suppressed. Because of the limitations of the modern observational tools, only the power-spectrum of the scalar fluctuations has been detected so far. The first two topics of this thesis (chapter 2 and 3) concentrate on certain properties of two- and three-point correlation functions of the scalar fluctuations that arise in the basic single-field, slow-roll inflationary models. Both of these works are set up to include a general initial state that the system might have had, and not just the
traditional Bunch-Davies vacuum. Being able to treat general states is important, because even though current observations are consistent with the fluctuations being
initially in the Bunch-Davies state, it is possible that, having more precision, future observations might indicate the presence of small deviations from this state. The third topic (chapter 4) investigated here is more formal in the sense that we treat not the real curvature perturbations of the inflationary theories, but a toy model
of the massless scalar field with quartic interactions in the pure de Sitter background. We construct an effective theory of the long-wavelength part of the field which allows
us to study the late-time behavior of this system. We show that at leading order this effective theory matches with the stochastic description. In future work we are going to use this formalism to treat fluctuations in real inflationary models.