Topological inflation from the Starobinsky model in supergravity

We consider the ghost-free higher order corrections to the Starobinsky model in the old minimal supergravity. In general, higher order corrections cannot be forbidden by symmetries, which likely violate the flatness of the scalaron potential and makes inflation difficult to explain the present Universe. We find a severe constraint on the dimensionless coupling of the $R^4$ correction as $-5.5 \times 10^{-8}<s<9.1 \times 10^{-8}$ from the recent results of the Planck observation. If we start from the chaotic initial condition, the constraint becomes much severer. However, in the case where the coupling of the $R^4$ correction is positive, the scalaron potential has a local maximum with two local minimum at the origin and infinity, which admits topological inflation. In this case, inflation can take place naturally if the coupling satisfies the observational constraints.