The Expanding Universe

Section 3 of 6

Hubble's Law as an Expansion Pattern

Part 2: Hubble’s Law Is an Expansion Pattern

At low redshift, the observed pattern is summarized by Hubble’s law:

Unpack it: $v$ is the recession speed inferred from redshift (usually km/s), $d$ is distance (often megaparsecs for galaxies), and $H_0$ is the Hubble constant, the present-day expansion rate, about $70~\mathrm{km\,s^{-1}\,Mpc^{-1}}$ . On large enough scales, recession speed increases linearly with distance.

Hubble's law

The large-scale relation $v = H_0 d$ , connecting galaxy recession speed and distance at low redshift. It is the empirical signature of cosmic expansion.

Hubble constant

The present-day expansion rate $H_0$ in Hubble’s law, usually written in $\mathrm{km\,s^{-1}\,Mpc^{-1}}$ . Independent measurement chains currently disagree — early-universe methods give roughly $67$ , local distance-ladder methods roughly $73$ — a gap called the Hubble tension, an active research frontier.

This is not a rule for every object near us. The Moon is not expanding away from Earth by Hubble’s law. The Solar System is bound. The Milky Way is bound. The Local Group is bound. Hubble’s law describes the large-scale expansion pattern after local gravitational motions average out.

Infographic showing three methods for measuring the Hubble constant, including distance ladder measurements and early-universe or large-scale-structure approaches. — Figure 4What to notice: the Hubble constant can be inferred through multiple chains of evidence. Agreement or tension between methods is a test of the model, not just a bookkeeping detail.

The Hubble constant can be measured in more than one way. That is powerful because independent methods test the model. It is also where modern cosmology becomes alive: when different measurement chains disagree, astronomers have to decide whether the issue is unrecognized systematic error, incomplete modeling, or new physics. The expansion rate is not just a number; it is a constraint on cosmic history — including its age.

Worked Example 1A Recession Speed and the Age of the Universe

Problem

Take $H_0 = 70~\mathrm{km\,s^{-1}\,Mpc^{-1}}$ . (a) A galaxy lies at $d = 100~\mathrm{Mpc}$ ; find its recession speed. (b) Invert $H_0$ to estimate the age of the universe. Use $1~\mathrm{Mpc} = 3.086\times10^{19}~\mathrm{km}$ and $1~\mathrm{yr} = 3.16\times10^{7}~\mathrm{s}$ .

StepRecession speed from v = H₀ d

$v = H_0 d = (70~\mathrm{km\,s^{-1}\,Mpc^{-1}})(100~\mathrm{Mpc}) = 7{,}000~\mathrm{km\,s^{-1}}$ — about $2\%$ of the speed of light.

StepThe Hubble time, 1/H₀

Convert $H_0$ to inverse seconds: $H_0 = 70~\mathrm{km\,s^{-1}\,Mpc^{-1}} / (3.086\times10^{19}~\mathrm{km\,Mpc^{-1}}) = 2.27\times10^{-18}~\mathrm{s^{-1}}$ . Then $1/H_0 = 4.4\times10^{17}~\mathrm{s} \approx 14~\mathrm{Gyr}$ .

Dimensional check

$[H_0 d] = (\mathrm{km\,s^{-1}\,Mpc^{-1}})(\mathrm{Mpc}) = \mathrm{km\,s^{-1}}$ , a speed; and $[1/H_0] = 1/(\mathrm{s^{-1}}) = \mathrm{s}$ , a time. Both units land where they should.

Result

The Hubble time $1/H_0 \approx 14~\mathrm{Gyr}$ is a first estimate of the age — what you would get if the universe had always expanded at the same speed (neither slowing nor speeding up). The true age (about $13.8~\mathrm{Gyr}$ ) is close, but getting it exactly requires the full expansion history — how the rate sped up and slowed down — which is the next reading’s job. Keep the number; we will earn it properly in L4.

Now push the law where it surprises us.

Misconception — nothing can recede faster than light

Galaxies beyond a certain distance recede faster than light, and that does not break relativity.

Addresses: misconception-nothing-can-recede-faster-than-light

refutation-text· Tippett 2010

Extend $v = H_0 d$ outward. At the Hubble distance $d_H = c/H_0 \approx 4.3~\mathrm{Gpc}$ , the recession speed equals $c$ ; beyond it, $v > c$ . A galaxy at $6~\mathrm{Gpc}$ “recedes” at $\sim 1.4c$ . This is allowed because it is not motion through space — nothing is overtaking a light beam locally. It is the expansion of space itself carrying distant galaxies apart, and special relativity’s speed limit applies only to motion through space, not to the stretching of space.

There is a second, related caution: the simple linear forms $v = H_0 d$ and $v \approx cz$ are low-redshift approximations. A galaxy at $z = 7$ is not receding at $7c$ — at high redshift you do not convert redshift to a velocity at all. You read it as the scale-factor stretch, $1+z = a_0/a$ (next part), and turning it into a precise distance needs the cosmological model of L4.

Quick check

A student computes $v = H_0 d$ for a very distant galaxy and gets $v = 1.3c$ , then concludes the calculation must be wrong because nothing travels faster than light. What is the student missing?

Quick check

Why does Hubble’s law work better for large-scale galaxy samples than for nearby bound systems such as the Local Group?