Early-time cosmic dynamics in $f(R)$ and $f(|\hat\Omega|)$ extensions of   Born-Infeld gravity
  We consider two types of modifications of Born-Infeld gravity in the Palatini formulation and explore their dynamics in the early universe. One of these families considers $f(R)$ corrections to the Born-Infeld Lagrangian, which can be seen as modifications of the dynamics produced by the quantum effects of matter, while the other consists on different powers of the elementary building block of the Born-Infeld Lagrangian, which we denote by $|\hat\Omega|$. We find that the two types of nonsingular solutions that arise in the original Born-Infeld theory are also present in these extensions, being bouncing solutions a stable and robust branch. Singular solutions with a period of approximate de Sitter inflation are found even in universes dominated by radiation. 
Andrey N. Makarenko, Sergei D. Odintsov, Gonzalo J. Olmo and Diego   Rubiera-Garcia
gr-qc astro-ph.CO hep-th