If galaxies are flying apart now, as Hubble's law shows, then in the past they were closer together. Run the expansion backward and everything comes together at one time in the past: the start of the expansion. How long ago?
Why it works
A galaxy at distance $d$ moving at $v = H_0 d$ would have taken $t = \frac{d}{v} = \frac{d}{H_0 d} = \frac{1}{H_0}$ to get there, if the speed never changed. Notice that $d$ cancels: every galaxy gives the same time. This Hubble time, $1/H_0$, estimates the age of the universe. The units need care: $H_0 = 70$ (km/s)/Mpc is a speed divided by a distance. Writing $1$ Mpc as $3.086 \times 10^{19}$ km makes the kilometers cancel: $H_0 = \frac{70}{3.086 \times 10^{19}}$ per second $= 2.27 \times 10^{-18}$ s⁻¹. Then $\frac{1}{H_0} = 4.41 \times 10^{17}$ s $= 14.0$ billion years. A handy shortcut: $\frac{1}{H_0} \approx \frac{978}{H_0}$ billion years, with $H_0$ in km/s/Mpc.
Key idea
Age of the universe $\approx \dfrac{1}{H_0}$. With $H_0 = 70$ km/s/Mpc, $1/H_0 \approx 14$ billion years; detailed models give $13.8$ billion years.
Worked examples
Worked example: Units that cancel
Find $1/H_0$ for $H_0 = 70$ km/s/Mpc.
- $H_0 = \frac{70\ \text{km/s}}{3.086 \times 10^{19}\ \text{km}} = 2.27 \times 10^{-18}$ per second.
- $\frac{1}{H_0} = 4.41 \times 10^{17}$ s.
- $\frac{4.41 \times 10^{17}}{3.156 \times 10^{7}} = 1.40 \times 10^{10}$ years.
Answer: About $14.0$ billion years.
Worked example: A different H₀
If $H_0$ were $73$ km/s/Mpc, what would $1/H_0$ be?
- $\frac{978}{73} = 13.4$ billion years: faster expansion means less time to reach today's distances.
Answer: About $13.4$ billion years.
Common mistake
The most common slip is with the units of $H_0$. Dividing 1 by 70 gives 0.014, which is not an age in any unit. You must first convert megaparsecs to kilometers so the distance units cancel, leaving 1/seconds. Check the size: the answer should be about 14 billion years, a little older than the oldest stars we know (about 12 to 13 billion years).
$1/H_0$ assumes the expansion rate never changed. In fact gravity slowed it early on and dark energy has sped it up lately; the effects nearly cancel, so the measured age, $13.8$ billion years (from the Planck mission), is close to $1/H_0$.