Inflationary dynamics in modified gravity models

Abstract

Higher-order theories of gravity are a branch of modified gravity wherein the geometrodynamics of the four-dimensional Riemannian manifold is determined by field equations involving derivatives of the metric tensor of order higher than two. This paper considers a general action built with the Einstein-Hilbert term plus additional curvature-based invariants, viz. the Starobinsky $R^2$-type term, a term scaling with $R^3$, and a correction of the type $R\square R$. The focus is on the background inflationary regime accommodated by these three models. For that, the higher-order field equations are built and specified for the FLRW line element. The dynanical analysis in the phase space is carried in each case. This analysis shows that the Starobinsky-plus-$R^3$ model keeps the good features exhibited by the pure Starobinsky inflationary model, although the set of initial conditions for the inflaton field $\chi$ leading to a graceful exit scenario is more contrived; the coupling constant ${\alpha }_0$ of the $R^3$ invariant is also constrained by the dynamical analysis. The Starobinsky-plus-$R\square R$ model turns out being a double-field inflation model; it consistently enables an almost-exponential primordial acceleration followed by a radiation dominated universe if its coupling ${\beta }_0$ takes values in the interval $0\le {\beta }_0\le 3/4$. The models introducing higher-order correction to Starobinsky inflation are interesting due to the possibility of a running spectral index $n_s$, something that is allowed by current CMB observations.

11 pages, 4 figures. Chapter in: "Tribute to Ruben Aldrovandi", F. Caruso, J.G. Pereira, A. Santoro (eds.) (Editora Livraria da Física, São Paulo, 2024), pages 135-151