f(R) Gravity with k-essence scaling relation and Cosmic acceleration

Abstract: A modified gravity theory with $f(R)=R^2$ coupled to a dark energy lagrangian $L=-V(\phi)F(X)$ , $X=\nabla_{\mu}\phi\nabla^{\mu}\phi$, gives plausible cosmological scenarios when the modified Friedman equations are solved subject to the scaling relation $X (\frac{dF}{dX})^{2}=Ca(t)^{-6}$. This relation is already known to be valid, for constant potential $V(\phi)$, when $L$ is coupled to Einstein gravity. $\phi$ is the k-essence scalar field and $a(t)$ is the scale factor. The various scenarios are: (1) Radiation dominated Ricci flat universe with deceleration parameter $Q=1$. The solution for $\phi$ is an inflaton field for small times. (2) $Q$ is always negative and we have accelerated expansion of the universe right from the beginning of time and $\phi$ is an inflaton for small times. (3)The deceleration parameter $Q= -5$, i.e. we have an accelerated expansion of the universe. $\phi$ is an inflaton for small times.(4)A generalisation to $f(R)= R^n$ shows that whenever $n > 1.780$ or $n < - 0.280$ , $Q$ will be negative and we will have accelerated expansion of the universe. At small times $\phi$ is again an inflaton.