Inflation is the theory that governs the earliest moments of the universe, the first fraction of a second after the big bang. Inflation purports that, just after the big bang there was a scalar field (dubbed the inflaton field) that caused the universe to undergo exponential expansion. Since its conception, the infl ationary theory has been very widely studied, however not a lot has been done with non-minimally coupled in ation. Inflation is non-minimally coupled (rather than minimally coupled, which is what is generally studied) when the inflaton field is coupled to gravity, meaning that the motion of the scalar field and the local curvature of space-time are dependent on each other. In this work I use phase portraits, and a specific type of phase portrait called a Poincaré map, to study the behavior of a non-minimally coupled inflaton. In essence, this thesis works to determine whether or not certain models of inflation that lead to in ation with a minimally coupled scalar field, would produce inflation when that field is non-minimally coupled to gravity, and vice versa. We determine this using several different plotting techniques in Mathematica that clarify if and where the universe instantiations (trajectories) on the phase portraits and Poincaré maps indicate inflation, if at all. These techniques prove especially useful in the non-minimally coupled case where the equations of motion are complicated. In this study, we focus on two models, the widely studied m2ø2 model and a warped brane inflation model that was shown to lead to inflation when the inflaton is non-minimally coupled.  We found that for the m2ø2 model there is inflation in the minimally coupled case, but not when the inflaton is non-minimally coupled to gravity. With the brane inflation potential, I found that there is inflation in both the minimally and non-minimally coupled cases.
Bergman, Amanda Stevie, "Study of non-minimally coupled inflation of the early universe using phase portraits and Poincaré maps" (2009). Theses, Dissertations, and Projects. 1437.