Dynamical System Analysis - L-CDM
Published:
$\Lambda-\mathrm{CDM}$ is the simplest model that is widely consistent with the observations.
The following set of equations describe the system:
\[\begin{align} \begin{split} \Omega'_{\text{m}} &= \Omega_{\text{m}}(\Omega_{\text{r}} -3 \Omega_\Lambda )\\[1ex] \Omega'_{\text{r}} &= \Omega_{\text{r}}(\Omega_{\text{r}} -3 \Omega_\Lambda - 1 )\\[1ex] \Omega'_\Lambda &= \Omega_\Lambda (\Omega_{\text{r}} - 3\Omega_\Lambda + 3) \end{split} \end{align}\]Using the constraint equation $\Omega_{\text{m}} + \Omega_{\text{r}} + \Omega_{\Lambda} = 1$, the system can be reduced to 2 dimensions,
\(\begin{align} \begin{split} \Omega'_{\text{m}} &= \Omega_{\text{m}}(3\Omega_{\text{m}} + 4\Omega_{\text{r}} -3 )\\[1ex] \Omega'_{\text{r}} &= \Omega_{\text{r}}(3\Omega_{\text{m}} + 4\Omega_{\text{r}} -4 ) \end{split} \end{align}\) where we have eliminated the $\Omega_\Lambda$ term.
You can use the following code to generate phase space diagram..
import numpy as np
import matplotib.pyplot as plt
import sympy as sp
import scipy.odeint
def system(m,r,l,t):
.
.
.
.
add gist