Cosmological Models in the Generalized Einstein Action

We have studied the evolution of the Universe in the generalized Einstein action of the form $R+\beta R^2$, where $R$ is the scalar curvature and $\beta=\rm const.$. We have found exact cosmological solutions that predict the present cosmic acceleration. These models also allow an inflationary de-Sitter era occurring in the early Universe. The cosmological constant ($\Lambda$) is found to decay with the Hubble constant ($H$) as, $\Lambda\propto H^4$. In this scenario the cosmological constant varies quadratically with the energy density ($\rho$), i.e., $\Lambda\propto \rho^2$. Such a variation is found to describe a two-component cosmic fluid in the Universe. One of the component accelerated the Universe in the early era, and the other in the present era. The scale factor of the Universe varies as $a\sim t^{n}$, $n=1/2$ in the radiation era. The cosmological constant vanishes when $n=4/3$ and $n=1/2$. We have found that the inclusion of the term $R^2$ mimics a cosmic matter that could substitute the ordinary matter.