A new covariant generalization of Einstein’s general relativity is developed which allows the existence of a term proportional to T α β T α β in the action functional of the theory (T α β is the energy-momentum tensor). Consequently, the relevant field equations are different from general relativity only in the presence of matter sources. In the case of a charged black hole, we find exact solutions for the field equations. Applying this theory to a homogeneous and isotropic spacetime, we find that there is a maximum energy density ρ max, and correspondingly a minimum length a min, at the early Universe. This means that there is a bounce at early times, and this theory avoids the existence of an early-time singularity. Moreover, we show that this theory possesses a true sequence of cosmological eras. We also argue that, although in the context of the standard cosmological model the cosmological constant Λ does not play …