Born-Infeld-like $f(R)$ gravity

Several features of an $f(R)$ theory in which there is a maximum value for the curvature are analyzed. The theory admits all the vacuum solutions of General Relativity, and also the radiation evolution for the scale factor of the standard cosmological model. Working in the Jordan frame, a complete analysis of the phase space is performed, and shown to be in agreement with examples obtained by numerical integration. In particular, nonsingular cosmological solutions were obtained, which after the bounce enter a phase of de Sitter expansion and subsequently relax to a GR-like radiation-dominated evolution.

Figure 1

The effective potential $U(\varphi )$ displays a local minimum at $\varphi =1$, corresponding to GR, and a local maximum at $\varphi =3.315$.

Figure 2

Evolution for $H$ and $\varphi$ around Minkowski spacetime. The detailed frame shows the oscillation of the Hubble parameter between $t=-80$ to $t=-20$.

Figure 3

Evolution of a universe that emerges from a past singularity, enters a de Sitter phase and reaches Minkowski spacetime.

Figure 4

In this case the Hubble parameter oscillates around the Minkowski state, and goes through a contracting de Sitter phase which culminates in a big crunch singularity.

Figure 5

The model in the figure begins with growing oscillations around the Minkowski state, followed by a contracting de Sitter phase, a bounce to a de Sitter expansion, to finally reach again the Minkowski geometry.

Figure 6

In this case the evolution, having started from de Sitter spacetime, displays a bounce, and reaches Minkowski in the end.

Figure 7

The figure shows a de Sitter to de Sitter evolution through a bounce.