# A Exact solutions of quintessence models of dark energy

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1. Aug 27, 2016

### Diferansiyel

Hi everyone,

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, however I do not understand how to construct cosmological model from these potentials? Obviously, there is an differential equation that must be solved but does it even have an analytical solution?

P.S: I am open your resource suggestions.

Last edited: Aug 27, 2016
2. Aug 27, 2016

### bapowell

Have you looked at the Friedmann Equations with the scalar field as the energy source?

3. Aug 28, 2016

### Diferansiyel

Dear bapowell,

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 $\phi(t)$

4. Aug 28, 2016

### bapowell

For a given $V(\phi)$, you must specify $\phi(t_0)$ and $\dot{\phi}(t_0)$. Then you can solve the Klein-Gordon and Friedmann equations together to obtain your cosmology.

5. Aug 28, 2016

### Diferansiyel

So there are "special" solutions depending on the behaviour of scalar field like slow-roll approximation in the inflationary cosmology. In fact, tracker quintessence models propose some solutions to cosmological constant problems. Maybe I should examine these fields.