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Comoving volume calculator

Q: How does the volume grow with redshift?

It keeps growing, but more slowly than the Euclidean D3 because the comoving distance flattens off with redshift. In the classic model each extra unit of redshift adds more volume than the one before up to z ≈ 2.5, and less beyond.

The volume of space, measured today, within redshift z over the whole sky.

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 volume V_C out to redshift z is the volume enclosed by the sphere at that redshift, measured in comoving coordinates. Number densities of galaxies, quasars or supernovae are quoted per comoving volume, so V_C is what turns counts into rates and densities.

For a survey covering only part of the sky, multiply by the fraction of the sky observed: area in square degrees ÷ 41,253.

Formula

Flat: V_C = (4π/3) D_M³
Curved: V_C = (4π D_H³ / 2Ω_k) [ (D_M/D_H) √(1 + Ω_k D_M²/D_H²) − sinh⁻¹(√|Ω_k| D_M/D_H) / √|Ω_k| ] (sin⁻¹ when closed)

Worked example

In the classic model (Ω_m = 0.3, Ω_Λ = 0.7, H₀ = 70 km s⁻¹ Mpc⁻¹), the whole-sky comoving volume out to z = 1 is 151.057 Gpc³, and out to z = 2 it is 582.16 Gpc³.

A survey of 1,000 square degrees out to z = 1 therefore covers 3.662 Gpc³.

Common questions

Is this the whole sky or per steradian?

The whole sky (4π steradians), as in classic iCosmos. Divide by 4π for the volume per steradian.

How does the volume grow with redshift?

It keeps growing, but more slowly than the Euclidean D³ because the comoving distance flattens off with redshift. In the classic model each extra unit of redshift adds more volume than the one before up to z ≈ 2.5, and less beyond.

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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.