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Astronomy and Cosmology
Cosmology
EDE - Solving the Klein - Gordon Equation for a scalar field
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[QUOTE="Arman777, post: 6413163, member: 579807"] 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 ? [/QUOTE]
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EDE - Solving the Klein - Gordon Equation for a scalar field
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