Exact solutions of quintessence models of dark energy I got the basic ideas quintessence (minimally coupled) and derived the KG equation for scalar field:
$$ \ddot{\phi} + 3 H \dot{\phi} + \frac{\partial V(\phi)}{\partial \phi}  = 0 $$
where $$H=\frac{\dot{a}}{a}$$ and $\phi$ is the scalar field.
There are various models depending on the choice of potential of the field. How do we construct cosmological model from these potentials? Obviously, there is a differential equation that must be solved but does it even have an analytical solution?
I am open your resource suggestions.
 A: The equation for the inflaton comes from the Lagrangian density
$$
{\cal L}~=~\frac{1}{2v}\partial_\mu\phi\partial^\mu\phi^*~-~V(|\phi|),
$$
where $v~=~\frac{4\pi}{3}a^3$ is the volume measure. The differential equaiton for $\phi$  is 
$$
\partial^\mu\partial_\mu\phi~+~3\frac{\partial_\mu a}{a}\partial^\mu\phi~+~\frac{\partial V(|\phi|}{\partial\phi}~=~0
$$
For quintessence was may be concerned with the $\nabla\phi$ terms in the partial derivatives. We assume however that these are zero and the differential equation of importance is
$$
\ddot\phi~+~3H\dot\phi~+~\frac{\partial V(|\phi|)}{\partial\phi^*}~=~0,
$$
for $H~=~\frac{\partial_\mu a}{a}$.
The inflaton potential is typically of the form $V(\phi)~=~\frac{\alpha}{2}|\phi|^2$. This is a typical linear differential equation where a proposed solution $\phi~=~\phi_0e^{i\sigma t}$ easily gives
$$
\sigma~=~i\frac{3H}{2}~\pm~\frac{\sqrt{4\alpha~-~9H^2}}{2}
$$
The condition for the solution to be all imaginary is for $H^2~>~4\alpha/9$. The solution is then over damped and is what I label in the diagram below hypercritical. For $H^2~<~4\alpha/9$ there is a real term to the $\sigma$ and the solution oscillates with time. 
I also considered the potential to be such that $\partial V/\partial \phi~=~sin(\phi)$ that results in a differential equation related to the damped Sine-Gordon equation. The frequency of the field is decreased, but the generic behavior is much the same.
