Hubble's Law and the Age of the Universe

Hubble's law — the speed at which a galaxy recedes is directly proportional to its distance from us:

v = H₀ d

where v is the speed of recession in m/s, d the distance in metres, and H₀ the Hubble constant, measured in s⁻¹.

A galaxy twice as far away recedes twice as fast. This is exactly what an expanding space produces, and it is the quantitative form of the red shift observations.

Rearranging:

H₀ = v / d and d = v / H₀

Measuring the two quantities. The speed of recession is found from the red shift of the galaxy's absorption lines. The distance is much harder, and is found using objects of known brightness — the uncertainty in H₀ comes mostly from the difficulty of measuring astronomical distances, not from the speeds.

Estimating the age of the universe. If galaxies have been separating at a steady rate since the beginning, the time since they were all together is distance divided by speed:

age ≈ 1 / H₀

This falls straight out of v = H₀d: rearranging gives d / v = 1 / H₀, and distance divided by speed is a time.

Why it is only an estimate. The calculation assumes the expansion rate has been constant throughout the history of the universe, which it has not — gravity slowed the early expansion, and the rate appears to have increased more recently. So 1/H₀ gives the right order of magnitude rather than a precise figure.

Units matter enormously here. With H₀ in s⁻¹, the answer comes out in seconds, and dividing by 3.15 × 10⁷ converts it to years. An answer of around 4 × 10¹⁷ s, or roughly 14 billion years, is the sensible range — anything wildly different means a conversion has gone astray.

Example: with H₀ = 2.2 × 10⁻¹⁸ s⁻¹, the age is 1/H₀ = 4.5 × 10¹⁷ s. Dividing by 3.15 × 10⁷ s per year gives about 1.4 × 10¹⁰ years — roughly 14 billion, which matches the accepted figure of 13.8 billion.