About Starobinsky inflation

TL;DR

This paper investigates numerical solutions in a higher derivative gravity theory inspired by Starobinsky inflation, revealing basins of attraction to Minkowski or de Sitter spaces and singular solutions, emphasizing the topological invariance of these structures.

Contribution

It provides a numerical analysis of anisotropic Bianchi I space-times in higher derivative gravity, demonstrating basin structures leading to different asymptotic states.

Findings

Basins of attraction to Minkowski space for zero cosmological constant

Basins of attraction to de Sitter space for positive cosmological constant

Basin structures are topologically invariant and coordinate independent

Abstract

It is believed that soon after the Planck era, space time should have a semi-classical nature. According to this, the escape from General Relativity theory is unavoidable. Two geometric counter-terms are needed to regularize the divergences which come from the expected value. These counter-terms are responsible for a higher derivative metric gravitation. Starobinsky idea was that these higher derivatives could mimic a cosmological constant. In this work it is considered numerical solutions for general Bianchi I anisotropic space-times in this higher derivative theory. The approach is ``experimental'' in the sense that there is no attempt to an analytical investigation of the results. It is shown that for zero cosmological constant $\Lambda=0$, 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 $\Lambda> 0$ 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.