Inflation and Topological Phase Transition Driven by Exotic Smoothness

We will discuss a model which describes the cause of inflation by a topological transition. The guiding principle is the choice of an exotic smoothness structure for the space-time. Here we consider a space-time with topology S3×ℝ. In case of an exotic S3×ℝ, there is a change in the spatial topology from a 3-sphere to a homology 3-sphere which can carry a hyperbolic structure. From the physical point of view, we will discuss the path integral for the Einstein-Hilbert action with respect to a decomposition of the space-time. The inclusion of the boundary terms produces fermionic contributions to the partition function. The expectation value of an area (with respect to some surface) shows an exponential increase; that is, we obtain inflationary behavior. We will calculate the amount of this increase to be a topological invariant. Then we will describe this transition by an effective model, the Starobinski or R2 model which is consistent with the current measurement of the Planck satellite. The spectral index and other observables are also calculated