Finding an expression for the rate of change of Comoving volume with respect to redshift

I am trying to find the expression for the rate of change of comoving volume with respect to redshift, that is $$\frac{\mathrm{d}V_c}{\mathrm{d}z}$$. In this paper (Hogg, David W.), the comoving volume differential is given as follows:

$$\mathrm{d}V_c=D_\mathrm{H}\frac{(1+z)^2D_\mathrm{A}^2}{E(z)}\mathrm{d}\Omega\mathrm{d}z$$

$$\implies\frac{\mathrm{d}V_c}{\mathrm{d}z}=D_\mathrm{H}\frac{(1+z)^2D_\mathrm{A}^2}{E(z)}\mathrm{d}\Omega$$

Where $$D_\mathrm{H}$$ is the Hubble distance, $$D_\mathrm{A}$$ is the angular diameter distance. These expressions can be expressed as follows:

$$D_\mathrm{H}=\frac{c}{H_0}$$

Where $$c$$ is the speed of light in vacuum, and $$H_0$$ is the Hubble's constant.

$$D_\mathrm{A}=\frac{D_\mathrm{M}}{1+z}$$

Where $$D_\mathrm{M}$$ is the transverse comoving distance given by:

$$D_\mathrm{M}=D_\mathrm{C},\textrm{ }\Omega_k=0$$

Where $$\Omega_k=0$$ implies spatial curvature 0, and $$D_\mathrm{C}$$ is the comoving distance given by:

$$D_\mathrm{C}=D_\mathrm{H}\int_0^z\frac{\,\mathrm{d}z'}{E(z')}$$

Now, this derivation can be substituted in $$\frac{\mathrm{d}V_c}{\mathrm{d}z}=4\pi D_\mathrm{H}\frac{(1+z)^2D_\mathrm{A}^2}{E(z)}$$, where $$4\pi$$ substitutes the solid angle element $$\mathrm{d}\Omega$$. Now, I want to plot this result as a function of redshift, for which I have written the following code in Python:

import numpy as np
import matplotlib.pyplot as plt
from astropy.cosmology import FlatLambdaCDM

cosmopar = FlatLambdaCDM(H0 = 67.8, Om0 = 0.3) #Defining the Cosmological model

speed_light = 3*10**5 #km/s
D_H = (speed_light/cosmopar.H(0).value)/10**3 #Gpc

Omega_m = 0.3
Omega_k = 0
Omega_Lambda = 1-Omega_m

def E(z_m): #Redshift z_m
return math.sqrt((1+z_m)^2*Omega_m+(1+z_m)^2*Omega_k+Omega_Lambda)

def Comov_vol(z_m): #in Gpc^3
return 4*math.pi*D_H*(((1+z_m)^2*(cosmopar.angular_diameter_distance(z_m).value/10**3)^2)/(E(z_m)))

#Plotting dVc_dz as a function of z:

ztrial = np.linspace(0,10,100) #Creating an array of redshift
comovol_arr = np.array([Comov_vol(v) for v in ztrial]) #Creating an array for corresponding comoving volume per redshift

plt.xlabel(r'Redshift')
plt.ylabel(r'$$\frac{\mathrm{d}V_c}{\mathrm{d}z}$$')

plt.plot(ztrial, comovol_arr)


This gives me the following plot:

Now, as one might notice, the magnitude of $$\frac{\mathrm{d}V_c}{\mathrm{d}z}$$ (in Gpc$$^3$$) quite large as compared to literature (Karathanasis, Christos, et. al.). I am referring to this paper for a project I am working on and the comoving volume per unit redshift vs redshift graph provided in this paper is as follows:

This graph is again given in Gpc$$^3$$, and the shape of both my plot and this literature plot is the same; however, the magnitude of $$\frac{\mathrm{d}V_c}{\mathrm{d}z}$$ seems to differ vastly. Can someone please help me understand where I am going wrong?

For context: I am new to both Astronomy and Coding. As an undergraduate Physics student, I am working on a project as a part of my summer internship and the paper which I refer to for the literature plot is the one required by my mentor, thank you.

• The plot in question comes in the middle of 5 pages of appendices all devoted to calculating Bayesian probabilities based on a set of expected distributions. Only a normalized distribution makes any sense here. Plus, the question itself goes into complete detail about how to calculate it in units of Gpc$^3$ per z. If you want me to move it to a comment, I can. But, it answers the question, and it would be a waste of time for others to work on it. Jul 29 at 23:01