Cosmological Calculations (astropy.cosmology)

Cosmological Calculations (astropy.cosmology)

The astropy.cosmology subpackage provides tools for cosmological calculations based on various cosmological models.

Using Built-in Cosmologies

Preloaded cosmologies based on WMAP and Planck observations:

from astropy.cosmology import Planck18, Planck15, Planck13
from astropy.cosmology import WMAP9, WMAP7, WMAP5
from astropy import units as u

# Use Planck 2018 cosmology
cosmo = Planck18

# Calculate distance to z=4
d = cosmo.luminosity_distance(4)
print(f"Luminosity distance at z=4: {d}")

# Age of universe at z=0
age = cosmo.age(0)
print(f"Current age of universe: {age.to(u.Gyr)}")

Creating Custom Cosmologies

FlatLambdaCDM (Most Common)

Flat universe with cosmological constant:

from astropy.cosmology import FlatLambdaCDM

# Define cosmology
cosmo = FlatLambdaCDM(
    H0=70 * u.km / u.s / u.Mpc,  # Hubble constant at z=0
    Om0=0.3,                      # Matter density parameter at z=0
    Tcmb0=2.725 * u.K             # CMB temperature (optional)
)

LambdaCDM (Non-Flat)

Non-flat universe with cosmological constant:

from astropy.cosmology import LambdaCDM

cosmo = LambdaCDM(
    H0=70 * u.km / u.s / u.Mpc,
    Om0=0.3,
    Ode0=0.7  # Dark energy density parameter
)

wCDM and w0wzCDM

Dark energy with equation of state parameter:

from astropy.cosmology import FlatwCDM, w0wzCDM

# Constant w
cosmo_w = FlatwCDM(H0=70 * u.km/u.s/u.Mpc, Om0=0.3, w0=-0.9)

# Evolving w(z) = w0 + wz * z
cosmo_wz = w0wzCDM(H0=70 * u.km/u.s/u.Mpc, Om0=0.3, Ode0=0.7,
                   w0=-1.0, wz=0.1)

Distance Calculations

Comoving Distance

Line-of-sight comoving distance:

d_c = cosmo.comoving_distance(z)

Luminosity Distance

Distance for calculating luminosity from observed flux:

d_L = cosmo.luminosity_distance(z)

# Calculate absolute magnitude from apparent magnitude
M = m - 5*np.log10(d_L.to(u.pc).value) + 5

Angular Diameter Distance

Distance for calculating physical size from angular size:

d_A = cosmo.angular_diameter_distance(z)

# Calculate physical size from angular size
theta = 10 * u.arcsec  # Angular size
physical_size = d_A * theta.to(u.radian).value

Comoving Transverse Distance

Transverse comoving distance (equals comoving distance in flat universe):

d_M = cosmo.comoving_transverse_distance(z)

Distance Modulus

dm = cosmo.distmod(z)
# Relates apparent and absolute magnitudes: m - M = dm

Scale Calculations

kpc per Arcminute

Physical scale at a given redshift:

scale = cosmo.kpc_proper_per_arcmin(z)
# e.g., "50 kpc per arcminute at z=1"

Comoving Volume

Volume element for survey volume calculations:

vol = cosmo.comoving_volume(z)  # Total volume to redshift z
vol_element = cosmo.differential_comoving_volume(z)  # dV/dz

Time Calculations

Age of Universe

Age at a given redshift:

age = cosmo.age(z)
age_now = cosmo.age(0)  # Current age
age_at_z1 = cosmo.age(1)  # Age at z=1

Lookback Time

Time since photons were emitted:

t_lookback = cosmo.lookback_time(z)
# Time between z and z=0

Hubble Parameter

Hubble parameter as function of redshift:

H_z = cosmo.H(z)  # H(z) in km/s/Mpc
E_z = cosmo.efunc(z)  # E(z) = H(z)/H0

Density Parameters

Evolution of density parameters with redshift:

Om_z = cosmo.Om(z)        # Matter density at z
Ode_z = cosmo.Ode(z)      # Dark energy density at z
Ok_z = cosmo.Ok(z)        # Curvature density at z
Ogamma_z = cosmo.Ogamma(z)  # Photon density at z      # Neutrino density at z

Critical and Characteristic Densities

rho_c = cosmo.critical_density(z)  # Critical density at z
rho_m = cosmo.critical_density(z) * cosmo.Om(z)  # Matter density

Inverse Calculations

Find redshift corresponding to a specific value:

from astropy.cosmology import z_at_value

# Find z at specific lookback time
z = z_at_value(cosmo.lookback_time, 10*u.Gyr)

# Find z at specific luminosity distance
z = z_at_value(cosmo.luminosity_distance, 1000*u.Mpc)

# Find z at specific age
z = z_at_value(cosmo.age, 1*u.Gyr)

Array Operations

All methods accept array inputs:

import numpy as np

z_array = np.linspace(0, 5, 100)
d_L_array = cosmo.luminosity_distance(z_array)
H_array = cosmo.H(z_array)
age_array = cosmo.age(z_array)

Neutrino Effects

Include massive neutrinos:

from astropy.cosmology import FlatLambdaCDM

# With massive neutrinos
cosmo = FlatLambdaCDM(
    H0=70 * u.km/u.s/u.Mpc,
    Om0=0.3,
    Tcmb0=2.725 * u.K,
    Neff=3.04,  # Effective number of neutrino species
    m_nu=[0., 0., 0.06] * u.eV  # Neutrino masses
)

Note: Massive neutrinos reduce performance by 3-4x but provide more accurate results.

Cloning and Modifying Cosmologies

Cosmology objects are immutable. Create modified copies:

# Clone with different H0
cosmo_new = cosmo.clone(H0=72 * u.km/u.s/u.Mpc)

# Clone with modified name
cosmo_named = cosmo.clone(name="My Custom Cosmology")

Common Use Cases

Calculating Absolute Magnitude

# From apparent magnitude and redshift
z = 1.5
m_app = 24.5  # Apparent magnitude
d_L = cosmo.luminosity_distance(z)
M_abs = m_app - cosmo.distmod(z).value

Survey Volume Calculations

# Volume between two redshifts
z_min, z_max = 0.5, 1.5
volume = cosmo.comoving_volume(z_max) - cosmo.comoving_volume(z_min)

# Convert to Gpc^3
volume_gpc3 = volume.to(u.Gpc**3)

Physical Size from Angular Size

theta = 1 * u.arcsec  # Angular size
z = 2.0
d_A = cosmo.angular_diameter_distance(z)
size_kpc = (d_A * theta.to(u.radian)).to(u.kpc)

Time Since Big Bang

# Age at specific redshift
z_formation = 6
age_at_formation = cosmo.age(z_formation)
time_since_formation = cosmo.age(0) - age_at_formation

Comparison of Cosmologies

# Compare different models
from astropy.cosmology import Planck18, WMAP9

z = 1.0
print(f"Planck18 d_L: {Planck18.luminosity_distance(z)}")
print(f"WMAP9 d_L: {WMAP9.luminosity_distance(z)}")

Performance Considerations

• Calculations are fast for most purposes
• Massive neutrinos reduce speed significantly
• Array operations are vectorized and efficient
• Results valid for z < 5000-6000 (depends on model)