Born-Infeld type modification of the gravity

We study a non-linear modification to General Relativity in which the standard Einstein-Hilbert action is replaced by a Born-Infeld type action. Also study us stability issues to judge about viability of this modification. We establish the conditions that this modification must satisfy for to avoid the problems associated with the Dolgov-Kawasaki instability, tachyon instability and negative effective gravitational coupling. The particle content of gravitational spectrum of the linearized Born-Infeld theory shows the existence of massless gravitons plus new degree of freedom of 0-spin associated with $R^2$ term. For a toy model we proved that for appropriate values of the background curvature, this theory is free of ghost, Dolgov-Kawasaki instability and tachyon instability also it has a positive effective gravitational coupling. We find the scalar-tensor theory equivalent at this $f(R)$-theory in the Einstein frame, and we study some properties of the scalar potential.