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

• I
Gold Member
Let us suppose we have a scalar field ##\phi##. The Klein-Gordon equations for the field can be written as

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

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

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

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

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

In (1) we assume that

\dot{\phi}^2/2 \ll V_n(\phi)

thus

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

Under these conditions, equation (1) and (2) becomes

3H \dot{\phi} + \frac{dV(\phi)}{d\phi} = 0

H^2 = \frac{8 \pi G}{3} V(\phi)

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 ?