- #1

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- 193

\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 ?