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Comoving distance calculator

Q: What is the Hubble distance?

c/H0, the natural unit of cosmological distance: 4282.7 Mpc (about 14 billion light-years) for H0 = 70 km s−1 Mpc−1.

The distance to an object today, in coordinates that expand with the universe.

The interactive calculator loads here: change the redshift and the cosmological parameters (Ω_m, Ω_Λ, Ω_r, w₀, w_a, H₀) and the result and plot update instantly.

What it is

The comoving distance D_C is the distance between us and a distant galaxy measured today, along the line of sight, in coordinates that stretch with the expansion. Two galaxies that simply move with the expansion keep the same comoving distance forever.

It is the basic distance from which all the others follow: D_M (transverse), D_A and D_L. Because space has expanded while the light travelled, D_C is larger than the light-travel distance c × t_L.

Formula

D_C = (c / H₀) ∫₀^z dz′ / E(z′)
E(z) = H(z) / H₀
c / H₀ = Hubble distance (4282.7 Mpc for H₀ = 70)

Worked example

In the classic model (Ω_m = 0.3, Ω_Λ = 0.7, H₀ = 70 km s⁻¹ Mpc⁻¹), D_C(z = 1) = 3,303.83 Mpc, which is 10.78 billion light-years. The light took 7.715 billion years to reach us, so the galaxy is now farther away than the distance light could have covered in that time.

At z = 2, D_C = 5,179.86 Mpc (16.89 billion light-years).

Common questions

How can a galaxy be more than 13.8 billion light-years away?

Space has been expanding while the light was on its way, so the galaxy has moved farther away since the light left it. The comoving distance to the most distant observable regions is around 46 billion light-years in standard cosmology.

What is the difference between comoving and proper distance?

They are equal today. The proper distance at an earlier time t is a(t) × D_C, where a is the scale factor: at redshift z the galaxy was D_C/(1 + z) away from us.

What is the Hubble distance?

c/H₀, the natural unit of cosmological distance: 4282.7 Mpc (about 14 billion light-years) for H₀ = 70 km s⁻¹ Mpc⁻¹.

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About iCosmos

iCosmos was created by Mihran Vardanyan. Mihran Vardanyan is a cosmologist who earned his DPhil in cosmology at the University of Oxford, supervised by Joseph Silk and Roberto Trotta. He developed iCosmos during his research years at Oxford Astrophysics, where his work applied Bayesian model selection to the curvature and size of the Universe.

How to cite: Vardanyan, Mihran. iCosmos V4: Cosmology Calculator. https://icosmos.co.uk/. Include your access date, model parameters and exported data when reporting calculations.