1
$\begingroup$

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.

$\endgroup$
3
  • $\begingroup$ 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 $V$. $\endgroup$
    – OON
    Aug 28, 2016 at 0:57
  • $\begingroup$ @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 $\phi(t)$ $\endgroup$ Aug 28, 2016 at 13:20
  • $\begingroup$ 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. $\endgroup$
    – DanielSank
    Aug 28, 2016 at 23:05

1 Answer 1

2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.