iCosmos › Curvature of the universe calculator

Curvature of the universe calculator

The shape of space follows from what the universe contains: Ω_k = 1 − Ω_m − Ω_Λ − Ω_r.

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

If the total density equals the critical density, space is flat (Euclidean). Below it, space is open, negatively curved like a saddle; above it, closed, positively curved like the surface of a sphere, finite but without an edge. In iCosmos Ω_k is never an input: it follows from the densities you choose, as the Friedmann equation requires.

Curvature changes how distances grow with redshift, the angular sizes of distant objects and the volume of space. Current measurements (Planck with baryon acoustic oscillations) find the universe flat to within a few parts in a thousand.

Formula

• Ω_k = 1 − Ω_m − Ω_Λ − Ω_r
• R₀ = c / (H₀ √|Ω_k|) (curvature radius)
• D_M = R₀ sinh(D_C/R₀) open, D_C flat, R₀ sin(D_C/R₀) closed

Worked example

In the classic model (Ω_m = 0.3, Ω_Λ = 0.7, H₀ = 70 km s⁻¹ Mpc⁻¹) Ω_k = 0, so space is flat and D_M(z = 1) = 3,303.83 Mpc.

Raising Ω_Λ to 0.9 makes Ω_k = −0.2: a closed universe with curvature radius 9.577 Gpc, in which D_M(z = 1) = 3,448.47 Mpc.

Common questions

Is the universe flat?

As far as we can measure, yes: Planck 2018 combined with baryon acoustic oscillations gives Ω_k = 0.0007 ± 0.0019. Whether it is exactly flat is a model-selection question, studied with Bayesian methods in the research behind iCosmos (Vardanyan, Trotta & Silk 2009, 2011).

Why is Ω_k not an input?

Because curvature is not an independent ingredient: in general relativity it is fixed by the total energy density. Changing Ω_m or Ω_Λ changes the shape of space automatically.

Related calculators

• Angular diameter distance calculator
• Comoving volume calculator
• Hubble parameter H(z) calculator
• The full iCosmos cosmology calculator

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.