Exact solutions of quintessence models of dark energy

[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.

 

[bapowell]

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?

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

 

[Diferansiyel]

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

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 $\phi(t)$

 

[bapowell]

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 $\phi(t)$

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.

 

[Diferansiyel]

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.

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.