# Future of a universe with scalar field

Assume that $\phi=\phi_0\gg M_{Pl}$, what can you say about the future of a universe in a model with $V(\phi)=\frac{1}{2}m^2\phi^2+\epsilon \phi$, where $|\epsilon|\ll m^2M_{Pl}^2$, is a small parameter and $m$ is such that $V(\phi_0)\approx \rho_c$.

My method is: I used Klein Gordon Equation $$\ddot{\phi}+3H\dot{\phi}+V'=0,$$

where $V'=m^2\phi+\epsilon$. After this I can solve for $\phi$ by using the slow roll conditions. But, what am I supposed to do afterwards? What does the future of universe mean? Am I supposed to show whether the universe will expand or contract?

• If $\Phi$ and $m$ have mass dimension $1$, $\epsilon$ must have mass dimension $3$ , not $4$ Commented Jul 15, 2014 at 9:07
• IHMO, you have to use the Einstein equations, see for instance equations $(7), (8)$, and the discussion in chapters $3 \to 8$ in this paper Commented Jul 15, 2014 at 9:26
• But $\epsilon$ has mass dimension of 3, I guess, because $M_{pl}^2$ has mass dimension of 1. Commented Jul 16, 2014 at 0:53
• No, $M_{pl}$ has mass dimension $1$, so $M_{pl}^2$ has mass dimensions $2$. Otherwise, give the precise mass dimension of $\Phi,m, M_{pl}..$ Commented Jul 16, 2014 at 8:11