What is the cosmic scale factor?

The scale factor a(t) is a dimensionless number that describes the relative size of the universe at time t compared to today. By convention a = 1 now. At redshift z, a = 1/(1+z). When a galaxy emitted its light at a = 0.5, every distance in the universe was half of its current value.

Why does a = 1/(1+z)?

Cosmological redshift is caused by the stretching of light as space expands. A photon emitted when the universe had scale factor a is observed at z = (1/a) − 1, so a = 1/(1+z). This is exact within general relativity and does not assume a particular cosmological model.

What cosmological parameters does this use?

The defaults are Planck 2018 best-fit flat ΛCDM values: H₀ = 67.74 km/s/Mpc, Ω_M = 0.3089, Ω_Λ = 0.6911, flat geometry (Ω_k = 0). You can override all three in the advanced panel.

How accurate is the lookback time?

The integration uses 1000-step Simpson's rule over the ΛCDM Friedmann integral. For z 1100 (near recombination) radiation energy density becomes non-negligible; the tool omits Ω_R and will slightly underestimate lookback time there.

What is H(z) and why does it matter?

H(z) = H₀ · √(Ω_M(1+z)³ + Ω_Λ) is the Hubble parameter at redshift z. It gives the instantaneous expansion rate of the universe at that epoch. At high z the matter term dominates and H(z) ≫ H₀; in the distant future the cosmological constant dominates and H approaches a constant.

Cosmic Scale Factor Calculator

Name: Cosmic Scale Factor Calculator
Availability: InStock
Author: Nham Vu

Convert between cosmological redshift, scale factor a(z), and lookback time using standard ΛCDM parameters.

Input

Redshift (z ≥ 0)

Scale Factor (0 < a ≤ 1)

Advanced Parameters

H₀ (km/s/Mpc)

Ω_M

Ω_Λ

Enter a redshift or scale factor to see the results.

Redshift z

—

Scale Factor a(z)

—

Cosmological Quantities

Lookback Time —

Age of Universe at z —

Current Age of Universe (t₀) —

Hubble Parameter H(z) —

Expansion Rate Ratio H(z)/H₀ —

Universe Size vs. Today —

Universe Size at This Epoch

Big Bang

Today

Common Cosmic Epochs

Epoch	z	a(z)	Lookback (Gyr)

Summary

Convert between cosmological redshift, scale factor a(z), and lookback time using standard ΛCDM parameters.

How it works

Enter a redshift z (≥ 0) or a scale factor a (0 < a ≤ 1) — the two fields are linked.
The scale factor is computed directly: a = 1 / (1 + z).
The Hubble parameter at that epoch is E(z) = √(Ω_M (1+z)³ + Ω_Λ), giving H(z) = H₀ · E(z).
Lookback time integrates 1 / [(1+z′) H(z′)] from z′ = 0 to z using 1000-step Gaussian quadrature.
The age of the universe at that redshift is t(z) = t₀ − t_lookback.
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

Last updated: 2026-06-11 · Reviewed by Nham Vu