A topological model for inflation

Abstract

In this paper we will discuss a new model for inflation based on topological ideas. For that purpose we will consider the change of the topology of the spatial component seen as compact 3-manifold. We analyzed the topology change by using Morse theory and handle body decomposition of manifolds. For the general case of a topology change of a $n-$manifold, we are forced to introduce a scalar field with quadratic potential or double well potential. Unfortunately these cases are ruled out by the CMB results of the Planck misssion. In case of 3-manifolds there is another possibility which uses deep results in differential topology of 4-manifolds. With the help of these results we will show that in case of a fixed homology of the 3-manifolds one will obtain a scalar field potential which is conformally equivalent to the Starobinsky model. The free parameter of the Starobinsky model can be expressed by the topological invariants of the 3-manifold. Furthermore we are able to express the number of e-folds as well as the energy and length scale by the Chern-Simons invariant of the final 3-manifold. We will apply these result to a specific model which was used by us to discuss the appearance of the cosmological constant with an experimentally confirmed value.