# 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.

• If you can also derive Friedmann eqs I can't get what's your question is? What solutions are and whether they are analytical is determined of course by potential $V$.
– OON
Aug 28, 2016 at 0:57
• @OON It is possible to use the energy density of the scalar field ($\rho_\phi$) in Friedmann equations, the problem is that we don't know the exact form of the scalar field $\phi(t)$, therefore integration can't be accomplished. There must be some other constrains on $\phi(t)$ Aug 28, 2016 at 13:20
• Quick note on writing questions: putting a question mark at the end of a declarative statement doesn't make it a question. For example, "I do not understand XYZ?" doesn't really make sense. It's better to write "Why XYZ?" I've edited the post to fix this. Aug 28, 2016 at 23:05

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.