A new covariant generalization of Einstein’s general relativity is developed which allows the existence of a term proportional to $T_{\alpha \beta }{T}^{\alpha \beta }$ in the action functional of the theory ($T_{\alpha \beta }$ 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 ${\rho }_{\max }$, 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 $\mathrm{\Lambda }$ does not play any important role in the early times and becomes important only after the matter-dominated era, in this theory the “repulsive” nature of the cosmological constant plays a crucial role at early times in resolving the singularity.

• Received 9 February 2016

DOI:https://doi.org/10.1103/PhysRevD.94.044002