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

In summary, we have a scalar field ##\phi## and the Klein-Gordon equations and Friedmann equations for the field. We need to solve these equations for two cases: 1) Slow-roll approximation and 2) Oscillation part. Under the assumption that ##\dot{\phi}^2/2 \ll V_n(\phi)##, the equations simplify to ##3H \dot{\phi} + \frac{dV(\phi)}{d\phi} = 0## and ##H^2 = \frac{8 \pi G}{3} V(\phi)##. However, there is always one extra parameter in the solution and it is unclear if a general solution can be defined or if
  • #1
Arman777
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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 ?
 
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  • #2
Maybe that helps:
-divide (5) by (6)
-make the approximation V'=V/phi
That gives phi as a function of H and t.
 

Related to EDE - Solving the Klein - Gordon Equation for a scalar field

1. What is the Klein-Gordon equation?

The Klein-Gordon equation is a relativistic wave equation that describes the behavior of a scalar field, which is a field that has only one value at each point in space and time. It was first proposed by physicists Oskar Klein and Walter Gordon in 1926.

2. What is the significance of solving the Klein-Gordon equation for a scalar field?

Solving the Klein-Gordon equation for a scalar field allows us to understand the behavior of quantum particles, such as mesons, which are described by scalar fields. It also has applications in fields such as quantum field theory and particle physics.

3. How is the Klein-Gordon equation solved for a scalar field?

The Klein-Gordon equation can be solved using various mathematical techniques, such as separation of variables, Fourier transforms, and Green's function methods. The specific method used depends on the boundary conditions and physical properties of the system being studied.

4. What are some real-world applications of the Klein-Gordon equation?

The Klein-Gordon equation has been applied in various fields, including quantum mechanics, quantum field theory, and condensed matter physics. It has also been used to study the behavior of particles in accelerators, as well as in the development of quantum computing and quantum cryptography.

5. Are there any limitations to the Klein-Gordon equation?

One limitation of the Klein-Gordon equation is that it only describes scalar fields, which means it cannot be used to study particles with spin. It also does not account for interactions between particles, and therefore, it is often combined with other equations, such as the Dirac equation, to provide a more complete description of quantum systems.

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