Born-Infeld Gravity with a Massless Graviton in Four Dimensions

We construct Born-Infeld (BI) type gravity theories which describe tree-level unitary (non-ghost and non-tachyonic) massless spin-2 modes around their maximally symmetric vacua in four dimensions. Building unitary BI actions around flat vacuum is straightforward; but, this is a complicated task around (anti)-de Sitter backgrounds. In this work, we solve the issue and give details of constructing perturbatively viable determinantal BI theories. It is interesting that the Gauss-Bonnet combination, which is a total derivative in four dimensions, plays an important role in the construction of viable BI theories.