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Expanding universe: the maths

A smooth, uniform universe (the Friedmann equations) with matter, radiation and dark energy as a cosmological constant. The expansion is integrated step by step (fourth-order Runge–Kutta). Computed live.

Equations

H² = H₀² [Ω_r/a⁴ + Ω_m/a³ + Ω_k/a² + Ω_Λ]

How fast the universe expands (H) at each size a (a = 1 today). Ω are today's shares of radiation, matter, curvature and dark energy; Ω_k = 1 − Ω_m − Ω_Λ − Ω_r.

ä = H₀² [−Ω_r/a³ − Ω_m/(2a²) + Ω_Λ a]

The acceleration equation the lab integrates: matter and radiation brake, dark energy pushes.

age = time from a = 0 to a = 1, in units of 1/H₀ = 977.8/H₀ billion years

H₀ in km/s per megaparsec.

Constants

Ωm, ΩΛ	0.315, 0.685	Planck Collaboration (2020)
Ωr	9.1 × 10⁻⁵	photons + neutrinos
H₀	67.4 (or 73) km/s/Mpc	Planck (2020); SH0ES (Riess et al.)

Worked examples ✓ checked on every change

Age of our universe (Planck values)	13.79 billion years	Planck Collaboration (2020): 13.787 ± 0.020
The same universe with H₀ = 73	12.73 billion years	age scales as 1/H₀
Matter 3, dark energy 0.1: its fate	recollapses	closed, matter-dominated: gravity wins

The app's test suite puts the lab in each setup, reads the lab's own result, and fails if it strays from these values.

What's simplified

• Perfectly smooth on large scales — galaxies' own motions ignored.
• Dark energy is a constant; other forms would change the future, not the past much.

Where it breaks

Before about 10⁻³⁰ seconds (inflation, if it happened, and quantum gravity). And if the universe's lumpiness matters more than assumed — see Timescape on the Atlas.

Sources: A. Friedmann (1922); G. Lemaître (1927); Planck Collaboration, A&A 641, A6 (2020); A. Riess et al. (SH0ES).