For nearby galaxies the redshift and the distance are simply proportional: Hubble's law, $v = H_0d$. Running the expansion backward gives a first estimate of the age of the universe. Here is how to use the law well, and where it breaks.
Why it works
The expansion rate is $H = \dot a/a$; today it is $H_0$. Two galaxies a distance $d$ apart move apart at $v = \dot a\,(d/a) = Hd$. For small $z$, the recession speed is $v \approx cz$, so $d \approx cz/H_0$. If galaxies had always moved apart at today's speeds, they would all have been together a time $d/v = 1/H_0$ ago: the Hubble time. With $H_0$ in km/s/Mpc, $$\frac{1}{H_0} = \frac{3.09 \times 10^{19}\text{ km}}{H_0\text{ km/s}} \approx \frac{978}{H_0}\text{ Gyr}.$$
Key idea
For $z \ll 1$: $d \approx cz/H_0$. The Hubble time $1/H_0$ is about 14 Gyr for $H_0 = 70$ km/s/Mpc. Measurements of $H_0$ range from about 67 (from the cosmic microwave background) to about 73 km/s/Mpc (from Cepheids and supernovae); why they differ is an open question.
Worked examples
Worked example: A nearby galaxy
A galaxy has $z = 0.010$. With $H_0 = 70$ km/s/Mpc, how far is it?
- $v \approx cz = 3.00 \times 10^{5} \times 0.010 = 3000$ km/s.
- $d = v/H_0 = 3000/70$.
Answer: About 43 Mpc.
Worked example: The Hubble time
Find $1/H_0$ for $H_0 = 70$ km/s/Mpc.
- $1/H_0 = 3.09 \times 10^{19}\text{ km}/(70\text{ km/s}) = 4.4 \times 10^{17}$ s.
- Divide by $3.16 \times 10^{16}$ s per Gyr.
Answer: About 14.0 Gyr, close to the age of 13.8 Gyr from detailed models: a coincidence of our era, as the next topics show.
Worked example: Peculiar velocities
A galaxy recedes at 1,000 km/s, but 300 km/s of that is its own motion away from us within its group. How far is it, with $H_0 = 70$?
- Expansion part: $1000 - 300 = 700$ km/s.
- $d = 700/70 = 10$ Mpc.
Answer: 10 Mpc; ignoring the peculiar velocity would give 14 Mpc, 40% too far. Far galaxies, with large $cz$, are hardly affected.
When Edwin Hubble first measured the law in 1929, he found about 500 km/s/Mpc, seven times today's value. His velocities were fine; his distances were too small, because the brightness of Cepheids was not yet calibrated correctly. It is a vivid example of how an error low on the distance ladder carries up into $H_0$, and of why every galaxy, not just ours, sees the same law: the expansion has no center.
Common mistake
Using $d = cz/H_0$ for large redshifts. It is a first-order approximation for $z \ll 1$. At $z = 1$, for example, it gives 4,300 Mpc, but the distance to such a galaxy depends on how the expansion rate changed while the light traveled, and "distance" itself must be defined carefully (now, then, or by brightness). Check: use the linear law only for $z$ below about 0.1.