About Starobinsky inflation

It is believed that soon after the Planck era, space-time should have a semiclassical nature. According to this, the escape from the general relativity theory is unavoidable. Two geometric counterterms are needed to regularize the divergences which come from the expected value. These counterterms are responsible for a higher derivative metric gravitation. Starobinsky’s idea was that these higher derivatives could mimic a cosmological constant. In this work numerical solutions are considered for general Bianchi I anisotropic space-times in this higher derivative theory. The approach is “experimental” in the sense that there is no attempt for an analytical investigation of the results. It is shown that for zero cosmological constant Λ=0, there are sets of initial conditions which form basins of attraction that asymptote Minkowski space. The complement of this set of initial conditions form basins which are attracted to some singular solutions. It is also shown, for a cosmological constant Λ>0, that there are basins of attraction to a specific de Sitter solution. This result is consistent with Starobinsky’s initial idea. The complement of this set also forms basins that are attracted to some type of singular solution. Because the singularity is characterized by curvature scalars, it must be stressed that the basin structure obtained is a topological invariant, i.e., coordinate independent.