Supernova Hubble Diagram

What is modelled

Light from a distant supernova travels through an expanding universe, so its apparent brightness depends on the history of the expansion, which depends on what the universe is made of. The Friedmann equation gives the expansion rate E(z) for matter, curvature and dark energy with an equation of state w. Integrating 1/E over redshift gives the comoving distance, and the luminosity distance follows. The page draws simulated supernovae around a flat universe with 30 per cent matter and compares them with your model.

Equations

• E(z) = √[Ωm(1+z)3 + Ωk(1+z)2 + ΩDE(1+z)3(1+w)].
• Distance dL = (1+z)(c/H0) ∫ dz / E (with a sinh or sin factor for curvature), modulus μ = 5 log10 dL + 25.
• Deceleration q = [Ωm(1+z)3 + (1+3w)ΩDE(1+z)3(1+w)] / (2E2).
• Age t0 = (1/H0) ∫ da / (a E).

Checks run in the page

• Matter-only distance against its closed form.
• De Sitter distance against its exact result.
• Age of a flat universe with a cosmological constant against the analytic formula.
• Low-redshift expansion with the deceleration parameter.

Honest simplifications

The supernovae are simulated, not real data, and their brightness offset is fitted away. Radiation, neutrinos, a changing w, the Hubble tension and the many other effects of a real analysis are left out, so the numbers show how the method works, not the true parameters of the universe.

• Press Matter only and see the points sit far from the curve.
• Press Open, no dark energy and compare the fit.
• Move w and watch the fit curve for the best value.