Cosmological Calculations (astropy.cosmology)

The astropy.cosmology subpackage provides tools for cosmological calculations based on various cosmological models. Requires SciPy for these workflows. Pass redshifts positionally in 8.0.1. The shorter formula fragments below assume cosmo, z, and any measured magnitudes are defined for the science case.

Import paths: Always import classes and functions directly from astropy.cosmology (as shown in all examples below). The old submodule import paths (astropy.cosmology.flrw, .core, .funcs, .connect, .parameter) were deprecated in v7.1 and removed in Astropy 8.0.

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)

w0wzCDM extrapolates w(z) linearly; upstream does not recommend it for z > 1. Its age integral samples to infinity and can fail numerically for parameters like wz=0.1; a finite low-redshift distance does not validate that extrapolation. Choose an appropriate physical model before interpreting ages.

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)
# Returns proper kpc / arcmin; keep the angular unit when applying the scale.

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 dOmega), Mpc3 / sr

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
Onu_z = cosmo.Onu(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)

Angular-diameter distance is not monotonic. Select a scientifically justified zmin/zmax branch and check the forward residual; a two-point Brent bracket does not constrain the solution to that interval.

z_low = z_at_value(Planck18.angular_diameter_distance, 1500*u.Mpc, zmax=1.5)
z_high = z_at_value(Planck18.angular_diameter_distance, 1500*u.Mpc, zmin=2.5, zmax=10)

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
)

Massive neutrinos can make evaluations slower. Choose their masses and Neff from the intended physical model; adding them does not establish model accuracy. The default Tcmb0=0 disables photons and neutrinos, even when m_nu is given.

Cloning and Modifying Cosmologies

Cosmology objects are immutable. Clone an already named model. Astropy 8.0.1 raises TypeError when cloning an unnamed model with changes, even if a new name is passed; give custom models a name when constructing them.

# Start from a named built-in model
cosmo = Planck18
# Clone with different H0
cosmo_new = cosmo.clone(name="Modified H0", 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  # Distance-only; no extinction or K correction

Survey Volume Calculations

# Full-sky comoving volume between two redshifts; multiply by survey solid angle / (4*pi*sr)
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_value(u.rad)).to(u.kpc)  # Small-angle approximation

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
• Numerical evaluation is not evidence of physical validity at a chosen redshift; record model parameters, radiation/neutrino assumptions and domain.
• Check distance duality d_L = (1+z)**2 * d_A for these standard FLRW models, and record the named cosmology or all custom parameters in output provenance.