A new variable in scalar cosmology with exponential potential

We present a new way describing the solution of the Einstein-scalar field theory with exponential potential $V\propto e^{\sqrt{6}\beta \phi/M_{Pl}}$ in spatially flat Friedmann-Robertson-Walker space-time. We introduced a new time variable, $L$, which may vary in $[-1,1]$. The new time represents the state of the universe clearly because the equation of state at a given time takes the simple form, $w= -1+ 2L^2$. The universe will inflate when $|L|<1/\sqrt{3}$. For $\beta\leq 1$, the universe ends with its evolution at $L=\beta$. This implies that the equation of state at the end of the universe is nothing but $w=-1+2\beta^2$. For $\beta \geq 1$, the universe ends at L=1, where the equation of state of the universe is one. On the other hand, the universe always begins with $w=1$ at $L=\pm 1$.