Undergrad EDE - Solving the Klein - Gordon Equation for a scalar field

[Arman777]

Let us suppose we have a scalar field $\phi$. The Klein-Gordon equations for the field can be written as

\begin{equation}
\ddot{\phi} + 3H \dot{\phi} + \frac{dV(\phi)}{d\phi} = 0
\end{equation}

The other two are the Friedmann equations written in terms of the $\phi$

\begin{equation}
H^2 = \frac{8 \pi G}{3} [\frac{1}{2}\dot{\phi}^2 + V(\phi)]
\end{equation}

\begin{equation}
\dot{H} = -4 \pi G \dot{\phi}^2
\end{equation}

Now I need to solve these equations for the two cases.
1) Slow-roll approximation
2) Oscillation part.

In (1) we assume that
\begin{equation}
\dot{\phi}^2/2 \ll V_n(\phi)
\end{equation}

thus

$$|\ddot{\phi}| \ll |\frac{dV(\phi)}{d\phi}|$$

Under these conditions, equation (1) and (2) becomes\begin{equation}
3H \dot{\phi} + \frac{dV(\phi)}{d\phi} = 0
\end{equation}\begin{equation}
H^2 = \frac{8 \pi G}{3} V(\phi)
\end{equation}

The problem is that I cannot solve this equation. There is always one extra parameter. Can we define some sort of a general solution for this type of equation or do I need to provide some potential ?

I am trying to work on the Early Dark Energy (EDE) model. In the EDE model, similar to the inflation, we have two phases initially the field must have some initial value $\phi_i$ where the potential is constant. And then the field makes damped oscillations due to some reasons about the Hubble tension.

Do I need slow-roll parameters in order to solve it ?

 

[mbond]

Maybe that helps:
-divide (5) by (6)
-make the approximation V'=V/phi
That gives phi as a function of H and t.