Plotting Density Parameters as a Function of Redshift

[slothwayne]

I'm trying to plot the density parameters against redshift in Python, so I suppose this is kind of a cross over of programming and physics. I've been given the following two equations in order to do so

$$r(z) = \lambda_H \int_{0}^{z} \frac{dz'}{E(z')}$$

$$E(z) = \frac{H(z)}{H_0} = \sqrt {\Omega_\Lambda + \Omega_M (1+z)^3}$$

where I have neglected the curvature term as I am assuming a flat universe.
I have a relatively basic understanding of Python, so I'm generally struggling on how to get any actual numbers out of Python using these two, the plotting itself shouldn't be a problem once I have the correct code.
How do I use these equations to find the density parameters as a function of the redshift, or maybe I could use the radial comoving distance r(z)? I'm not too sure.

I figured it would be better to post here than the programming section as most people with a higher understanding of Physics also have some coding experience.

Thanks in advance.

 

[Asim Riaz]

You can use Python to solve these equations. First, you need to define the functions for $r(z)$ and $E(z)$. Then you can calculate $r(z)$ for a given redshift ($z$) and a given value of $\lambda_H$:<code>def r(z, lambdah): integrand = lambda z: 1/E(z) return lambdah * integrate.quad(integrand, 0, z)[0]def E(z): return np.sqrt(Omegal + Omegam * (1+z)**3)# Set your values for Omega_Lambda and Omega_MOmegal = 0.7Omegam = 0.3# Calculate r(z) for a given redshift (z) and value of lambda_H (lambdah)z = 0.5lambdah = 70rz = r(z, lambdah)</code>Once you have calculated $r(z)$, you can then use this to calculate the density parameters as a function of redshift. For example, for a flat universe, the matter density parameter is given by:$$\Omega_M(z) = \frac{8 \pi G \rho_M(z)}{3 H^2(z)} = \frac{H_0^2 (1+z)^3}{H^2(z)}$$You can then implement this in Python:<code>def Omega_M(z): return (H0**2 * (1+z)**3) / (E(z)**2)# Calculate Omega_M for the given redshift (z)Omegam_z = Omega_M(z)</code>You can then plot the density parameters as a function of redshift using matplotlib.