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We generalize the ultraviolet sector of gravitation via a Born-Infeld actionusing lessons from massive gravity. The theory contains all of the elementarysymmetric polynomials and is treated in the Palatini formalism. We show how theconnection can be solved algebraically to be the Levi-Civita connection of aneffective metric. The non-linearity of the algebraic equations yields severalbranches, one of which always reduces to General Relativity at low curvatures.We explore in detail a {\it minimal} version of the theory, for which we studysolutions in the presence of a perfect fluid with special attention to thecosmological evolution. In vacuum we recover Ricci-flat solutions, but also anadditional physical solution corresponding to an Einstein space. The existenceof two physical branches remains for non-vacuum solutions and, in addition, thebranch that connects to the Einstein space in vacuum is not very sensitive tothe specific value of the energy density. For the branch that connects to theGeneral Relativity limit we generically find three behaviours for the Hubblefunction depending on the equation of state of the fluid, namely: either thereis a maximum value for the energy density that connects continuously withvacuum, or the energy density can be arbitrarily large but the Hubble functionsaturates and remains constant at high energy densities, or the energy densityis unbounded and the Hubble function grows faster than in General Relativity.The second case is particularly interesting because it could offer aninteresting inflationary epoch even in the presence of a dust component.Finally, we discuss the possibility of avoiding certain types of singularitieswithin the minimal model.

Infrared lessons for ultraviolet gravity: the case of massive gravity and Born-Infeld