Calculate cosmological distances with CCL¶

In this example, we will calculate various cosmological distances for an example cosmology.

In [1]:

import numpy as np
import pylab as plt
import pyccl as ccl
%matplotlib inline

Set up a Cosmology object¶

Cosmology objects contain the parameters and metadata needed as inputs to most functions. Each Cosmology object has a set of cosmological parameters attached to it. In this example, we will only use the parameters of a vanilla LCDM model, but simple extensions (like curvature, neutrino mass, and w0/wa) are also supported.

Cosmology objects also contain precomputed data (e.g. splines) to help speed-up certain calculations. As such, Cosmology objects are supposed to be immutable; you should create a new Cosmology object when you want to change the values of any cosmological parameters.

In [2]:

cosmo = ccl.Cosmology(Omega_c=0.27, Omega_b=0.045, h=0.67, A_s=2.1e-9, n_s=0.96)

In [3]:

print(cosmo)

<pyccl.cosmology.Cosmology>
	A_s     = 2.1e-09
	Neff    = 3.044
	Omega_b = 0.045
	Omega_c = 0.27
	h       = 0.67
	n_s     = 0.96
	extra_parameters =
		emu = {'neutrinos': 'strict'}
	HASH_ACCURACY_PARAMS = 0xa0af9c5b75d6de86

Parameter values can be accessed from the Cosmology object contains, like so:

In [4]:

print(cosmo['Omega_c'])

0.27

In [5]:

print(cosmo['sigma8'])

nan

Notice $\sigma_8$ is nan here because it has not yet been computed. Instead, the normalization of the power spectrum was defined via $A_s$ above.

Cosmological Distances¶

With a cosmology in hand, we can begin performing some calculations. We can start with the most basic measure, the comoving radial distance.

In [6]:

z = 0.5 
ccl.comoving_radial_distance(cosmo, 1/(1+z)) # Mpc

Out[6]:

1962.9390685778556

Note that all distance function calls require scale factors, not redshifts. This function can take a numpy array of values as well.

In [7]:

zs = np.arange(0, 1, 0.1)
ccl.comoving_radial_distance(cosmo, 1/(1+zs))

Out[7]:

array([   0.        ,  436.6985067 ,  851.3944111 , 1243.78925379,
       1614.09910482, 1962.93906858, 2291.20721282, 2599.98198852,
       2890.43959663, 3163.79072723])

CCL also supports calculation of the comoving angular distance. In flat spacetime (like the cosmology we have here) it is the same as the radial distance.

In [8]:

ccl.comoving_angular_distance(cosmo, 1/(1+z))

Out[8]:

1962.9390685778556

If we create a cosmology with curvature, we'll get a different result.

In [9]:

curved_cosmo = ccl.Cosmology(Omega_k = 0.1, Omega_c=0.17, Omega_b=0.045, h=0.67, A_s=2.1e-9, n_s=0.96)

chi_rad  = ccl.comoving_radial_distance(curved_cosmo, 1/(1+z))
chi_curved = ccl.comoving_angular_distance(curved_cosmo, 1/(1+z))

print ('Radial Dist. = %.2f Mpc \t Angular Dist. = %.2f Mpc'%(chi_rad, chi_curved))

Radial Dist. = 1992.53 Mpc 	 Angular Dist. = 1999.12 Mpc

CCL explictly supports the calculation of the luminosity distance and the distance modulus too:

In [10]:

chi_lum = ccl.luminosity_distance(cosmo, 1/(1+z))
DM = ccl.distance_modulus(cosmo, 1/(1+z))
print('Luminosity Dist = %.2f Mpc \t Distance Modulus = %.2f ' % (chi_lum, DM))

Luminosity Dist = 2944.41 Mpc 	 Distance Modulus = 42.34

Finally, CCL supports an inverse operation, which calculates the scale factor for a given comoving distance:

In [11]:

ccl.scale_factor_of_chi(cosmo, 1962.96)

Out[11]:

0.6666639215879805