Comoving Distance & Lookback Time Calculator

The comoving distance d_C between two objects is the spatial separation measured in coordinates that expand with the universe, so that the distance between two objects at rest in the Hubble flow is constant over time. It equals the proper distance today (at the current cosmic time). For a flat ΛCDM universe, d_C = (c/H₀) × ∫₀^z dz'/E(z') where E(z) = √(Ω_m(1+z)³ + Ω_Λ). It is the most natural measure of distance for cosmological calculations.

Lookback time is the time elapsed since light left the source and arrived at Earth. It equals the age of the universe minus the age of the universe when the light was emitted. For Planck 2018 cosmology and a galaxy at z = 1, the lookback time is about 7.7 Gyr — meaning you are observing the galaxy as it appeared 7.7 billion years ago. Lookback time is computed as t_L = (1/H₀) × ∫₀^z dz'/[(1+z')E(z')].

The proper distance is the physical separation between two objects at a given moment in cosmic time. For a galaxy at rest in the Hubble flow, the proper distance today equals the comoving distance, but the proper distance at the time of emission was d_proper(t_emit) = d_C / (1+z), which is the angular diameter distance. In other words: comoving distance is the present-day separation, angular diameter distance is the separation at the time the light was emitted, and luminosity distance is a larger quantity that preserves the inverse-square law for flux.

The luminosity distance d_L relates the observed flux F of an object to its intrinsic luminosity L via L = 4πd_L²F. For a flat universe d_L = (1+z) × d_C. The extra (1+z) factor comes from two effects: the photon energies are redshifted by 1/(1+z) and the photon arrival rate is also reduced by 1/(1+z). The luminosity distance is used to calibrate Type Ia supernovae and underpins the discovery of dark energy.

The angular diameter distance d_A is defined so that a rod of physical size D at redshift z subtends an angle θ = D/d_A (in radians) on the sky. It equals d_A = d_C / (1+z) for a flat universe. The angular diameter distance has a counterintuitive property: it reaches a maximum at z ≈ 1.5 in Planck 2018 cosmology and then decreases at higher redshifts. This means very high-z galaxies subtend a larger angle per unit physical size than those at z ~ 1, making them appear bigger.

The Hubble tension is the 5σ disagreement between the value of H₀ measured by Planck (67.4 km/s/Mpc) from the CMB power spectrum and the value measured by the SH0ES collaboration (73.04 km/s/Mpc) from Type Ia supernovae calibrated with Cepheid distances. A higher H₀ means the universe is expanding faster, making it younger and assigning smaller distances to high-z objects. The tension is unresolved as of 2025 and may indicate new physics beyond the standard ΛCDM model.

The CMB photons last scattered at z ≈ 1100 when the universe was 380,000 years old. In Planck 2018 cosmology, the comoving distance to the last scattering surface is approximately 14,163 Mpc = 46,200 Mly. This is also known as the radius of the observable universe. The diameter of the observable universe is therefore about 93 Gly. Objects beyond 14,163 Mpc comoving are in principle unobservable, as their light has not had time to reach us.

E(z) = H(z)/H₀ is the dimensionless Hubble parameter. The Hubble parameter H(z) gives the expansion rate of the universe at redshift z. In flat ΛCDM, E(z) = √(Ω_m(1+z)³ + Ω_Λ). The comoving distance integral 1/E(z) is large where E(z) is small (the expansion is slow, so light travels far per unit redshift), and small where E(z) is large (rapid expansion at high z). This is why most of the comoving distance to the CMB is accumulated at low-to-moderate redshifts.

In the Friedmann equation, the total energy density determines the spatial curvature of the universe. Ω_m is the fractional density in matter (baryonic + dark matter), Ω_Λ is the fractional density in dark energy (cosmological constant). If Ω_m + Ω_Λ = 1, the universe is spatially flat (zero curvature). CMB observations from Planck confirm flatness to within 0.5%. A flat universe means parallel geodesics stay parallel, and the sum of angles in a large triangle is exactly 180 degrees.

The light-travel distance (sometimes called the lookback distance) is c × t_L, the speed of light times the lookback time. For z = 1 in Planck 2018 cosmology: t_L ≈ 7.73 Gyr, so light-travel distance ≈ 7.73 Gly. But the comoving distance to z = 1 is about 3,310 Mpc ≈ 10,800 Mly. The difference arises because space itself was expanding while the light was travelling, so the source is now much farther away than the distance implied by how long the light travelled.

A megaparsec (Mpc) is one million parsecs. One parsec equals 3.0857 × 10¹³ km = 3.2616 light-years. One megaparsec = 3.2616 million light-years = 3.0857 × 10²² metres. Cosmological distances are routinely measured in Mpc: the Milky Way and Andromeda are about 0.77 Mpc apart, the Virgo Cluster is about 16 Mpc away, and the observable universe extends to about 14,163 Mpc comoving.

This calculator supports z up to 2,000. For z > 1100 you enter the radiation-dominated era where the Ω_r(1+z)⁴ term becomes significant and cannot be ignored. The E(z) formula used here omits the radiation density Ω_r (≈ 9.4 × 10⁻⁵), which introduces < 0.01% error for z < 100 but grows to a few percent near z = 1100. For the CMB at z = 1100, the error from neglecting Ω_r is about 1% in comoving distance and negligible for lookback time (which is dominated by low-z cosmic time).