Expansion of the Universe

Cosmic expansion, Hubble law, scale factor, and redshift. Theory, formulas, worked examples and interactive graphs.

Conceptual context

The expansion of the Universe is not best pictured as galaxies flying through empty space from a central explosion. It is the growth of the geometric scale that separates comoving points. When light from distant galaxies reaches us, spectral lines are shifted toward longer wavelengths; $z$ records that the light has been stretched during its journey. The central question is how $D$ and $v_rec$ are connected. In the local regime, the connection is nearly linear: a larger instantaneous physical separation corresponds to a larger rate of separation. The Coord graph gives this idea a visual spine: $D$ on the horizontal axis, $v_rec$ on the vertical axis.

🟢 Essential level

Cosmic expansion is easiest to read as a changing scale for a network of galaxies that keep fixed comoving labels.

The $D$ between two such galaxies grows because the $a$ grows with $t$. This creates a $v_rec$, not as an engine-driven local motion, but as the rate at which physical separation increases. In the nearby Universe, a more distant galaxy generally shows larger $z$ and larger $v_rec$. The $H0$ tells how much recession corresponds to each unit of $D$ today. If the $D$ is doubled, the local model predicts doubled $v_rec$. This reading applies to objects following the average cosmic flow, not to tightly bound systems such as stars inside a galaxy or planets inside a solar system. This is a scale effect.

Edwin Hubble published this relation in 1929 after measuring distances and redshifts for several nearby galaxies. With the modern value of $H0$, about 70 km/s per megaparsec, a galaxy at 100 Mpc should recede at roughly 7000 km/s, a figure that is readily checked with spectroscopy.

🔵 Formal level

Basic formulas

The local relation between $v_rec$ and $D$ is expressed by the present Hubble law:

Local Hubble law

Here $H0$ is the slope of the Coord graph: it converts $D$ into $v_rec$. This slope has the dimension of inverse time, although it is often written as km/s/Mpc. For small $z$, recession can be estimated with the kinematic approximation:

Low-redshift velocity approximation

Real student questions

Does Earth expand with the Universe? No. Bound systems such as planets, stars, and galaxies do not follow cosmological expansion the way large separations between distant clusters do.

Does a $v_rec$ greater than $c$ break relativity? No in general cosmology. The local limit set by $c$ applies to motion through space, not to the global increase of $D$ between widely separated regions.

Does $z$ always measure velocity? At low $z$, it can be approximated that way. In deep cosmology, it mainly measures change in $a$.

Final synthesis

Expansion relates $D$ and $v_rec$ through a local law whose slope is $H0$. Its deeper meaning appears by joining $a$, $z$, and $\chi$: galaxies do not flee from a center; the cosmic ruler changes.