Cosmic Scale Factor Calculator

Summary

The cosmic scale factor a(z) describes how the size of the universe at a given epoch compares to its present size. At redshift z, a = 1/(1+z), so a = 1 today, a = 0.5 when the universe was half its current size, and a → 0 near the Big Bang. This calculator converts between redshift, scale factor, and lookback time, and displays the corresponding age of the universe and the expansion rate H(z) using the standard Planck 2018 ΛCDM parameters (H₀ = 67.74 km/s/Mpc, Ω_M = 0.3089, Ω_Λ = 0.6911).

How it works

1. Enter a redshift z (≥ 0) or a scale factor a (0 < a ≤ 1) — the two fields are linked.
2. The scale factor is computed directly: a = 1 / (1 + z).
3. The Hubble parameter at that epoch is E(z) = √(Ω_M (1+z)³ + Ω_Λ), giving H(z) = H₀ · E(z).
4. Lookback time uses 1,000-step Simpson integration over scale factor, avoiding loss of accuracy from uniformly stepping across very large redshifts.
5. The age of the universe at that redshift is t(z) = t₀ − t_lookback.
6. Adjust H₀, Ω_M, and Ω_Λ to explore non-standard cosmologies.

Use cases

• Determine when a galaxy at a known redshift emitted the light you observe.
• Check how large the observable universe was at the epoch of reionization (z ≈ 6–10).
• Verify the age of the universe at matter–radiation equality (z ≈ 3400).
• Explore alternative cosmologies by adjusting Ω_M and Ω_Λ.
• Convert scale factor values from N-body simulation snapshots to physical redshifts.
• Teach or learn the relationship between redshift and cosmic expansion history.

Frequently Asked Questions