🌠Cosmology Calculator - Redshift to Distance, Age of the Universe, Lookback Time

How to use Cosmology Calculator

1. 1Pick a preset or enter H₀, Ωm and ΩΛ, and choose whether the universe is flat and includes radiation
2. 2In 'Redshift calculator' enter z (or the age then) and read the age, lookback time and distances
3. 3Open 'Charts' to compare a(t) for open, closed and flat models and see distances, age and H(z) versus redshift
4. 4Use 'Hubble's law' for v = H₀d conversions and 'CMB temperature' for the radiation temperature at any epoch

FAQ

What is the difference between luminosity, angular diameter and comoving distance?

In an expanding universe "distance" has several definitions. Comoving distance is measured along the line of sight today. Luminosity distance D_L makes F = L/(4πD_L²) hold and equals (1+z) times the transverse comoving distance; it turns brightness into distance. Angular diameter distance D_A makes size = angle × D_A hold and equals the transverse comoving distance divided by (1+z). D_A peaks near z ≈ 1.6, so more distant galaxies can look larger. Light-travel distance is simply the light travel time times c.

Why is the luminosity distance at z = 1 about 6.8 Gpc?

Luminosity distance is not where the object is now. It is the equivalent distance implied by its dimming, which includes a (1+z)² factor from redshifted photon energies and stretched arrival times. A z = 1 galaxy is now about 3.4 Gpc away in comoving terms (about 11 billion light years), and its light travelled for about 7.95 billion years. The observable universe has a comoving radius of about 14 Gpc (46 billion light years), larger than 13.8 billion light years because space kept expanding while the light was in transit.

How is it computed, and how accurate is it?

Everything follows from E(z) = √(Ωr(1+z)⁴ + Ωm(1+z)³ + Ωk(1+z)² + ΩΛ). Age and distances are integrals of 1/E, done with adaptive Simpson and the substitution a = u² that removes the singularity at the Big Bang, giving relative errors far below 10⁻⁶. Non-flat models get the sinh or sin curvature correction. Planck 2018 parameters with radiation give 13.79 Gyr and D_L = 6.80 Gpc at z = 1, in line with standard calculators, and Ωm = 1 reproduces the exact 2/(3H₀).

When does it say there is no Big Bang?

If ΩΛ is too large compared with Ωm, running the expansion backwards makes H² negative at some finite redshift: the universe bounced from an earlier contraction and never reached a = 0. Such bouncing models contradict observations, so the tool stops and explains. Closed models such as Ωm = 2, ΩΛ = 0 do the opposite: they stop expanding and recollapse, and the a(t) chart marks the Big Crunch.

When does Hubble's law v = H₀d stop working?

v = H₀d works well in the nearby universe (z ≲ 0.1), where v ≈ cz too. Farther out, redshift no longer maps linearly to velocity and you must say which distance you mean, so use the ΛCDM integrals instead. Using today's comoving distance, H₀·D_C exceeds the speed of light beyond z ≈ 1.5. That does not break relativity, because it is space itself expanding, not galaxies moving through space.