Energy-Momentum Squared Gravity
  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
space-time, we find that there is a maximum energy density $\rho_{\text{max}}$,
and correspondingly a minimum length $a_{\text{min}}$, at 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. Also, we argue that although in
the context of the standard cosmological model the cosmological constant
$\Lambda$ 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 for
resolving the singularity.
