10.2.1.1. Hubble's Law

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Seeing the Universe Expand

Hubble's law is the simple rule that, on very large scales, galaxies are moving away from us with a speed that is proportional to their distance from us. The farther away a galaxy is, the faster it recedes.

This idea is one of the foundations of modern cosmology because it shows that the universe is expanding. It does not mean that Earth is at the center. Instead, space itself is stretching, so distant galaxies become more separated from one another everywhere in the universe.

The Basic Law

The mathematical form of Hubble's law is

$$
v = H_0 d
$$

where $v$ is the recession speed of a galaxy, $d$ is its distance from us, and $H_0$ is the Hubble constant.

The Hubble constant tells us how fast the universe is expanding today. Its unit is usually written as

$$
\text{km s}^{-1}\text{Mpc}^{-1}
$$

This means kilometers per second for every megaparsec of distance.

For example, if

$$
H_0 = 70 \, \text{km s}^{-1}\text{Mpc}^{-1}
$$

then a galaxy at a distance of $1 \, \text{Mpc}$ recedes at about $70 \, \text{km/s}$, and a galaxy at $100 \, \text{Mpc}$ recedes at about $7000 \, \text{km/s}$.

Important relation:
$$
v = H_0 d
$$
For an expanding universe on large scales, recession speed is proportional to distance.

What the Law Means Physically

Hubble's law tells us that the expansion is uniform on large scales. Every observer in a galaxy would see distant galaxies receding in the same general way. This is a key sign that expansion is a property of space itself, not an explosion from one special point.

A useful picture is a rising loaf of raisin bread. As the dough expands, every raisin sees all the other raisins moving farther away. The more distant raisins separate faster because more dough lies between them.

Interpreting the Hubble Constant

Although it is called a constant, $H_0$ refers to the present value of the expansion rate. In more advanced cosmology, the expansion rate can change with time, but Hubble's law in its basic form uses the current value.

A rough estimate of the age scale of the expanding universe comes from the inverse of the Hubble constant:

$$
t \sim \frac{1}{H_0}
$$

This gives a characteristic timescale for expansion. It is not exactly the age of the universe in all models, but it gives the right order of magnitude.

$H_0$ is the present expansion rate, not a permanent unchanging rate over all cosmic history.

Typical Values

Different measurement methods give values of $H_0$ that are close, but not exactly identical. For beginners, it is enough to know that the commonly quoted value is around $70 \, \text{km s}^{-1}\text{Mpc}^{-1}$.

Distance $d$	Recession speed $v$ for $H_0 = 70 \, \text{km s}^{-1}\text{Mpc}^{-1}$
$1 \, \text{Mpc}$	$70 \, \text{km/s}$
$10 \, \text{Mpc}$	$700 \, \text{km/s}$
$100 \, \text{Mpc}$	$7000 \, \text{km/s}$
$1000 \, \text{Mpc}$	$70000 \, \text{km/s}$

A Simple Graph

If we plot recession speed versus distance, Hubble's law predicts a straight line through the origin. The slope of that line is the Hubble constant.

Rendering scientific sketch…

Hubble's law as a straight-line relation

Recession Speed and Ordinary Motion

The speed in Hubble's law is called recession speed. It is related to the expansion of space between galaxies. This is different from local motions caused by gravity, such as stars orbiting in a galaxy or galaxies moving inside a cluster.

For nearby galaxies, these local motions can disturb the simple linear relation. For very distant galaxies, the expansion effect becomes dominant and Hubble's law works much better.

Worked Example

Suppose a galaxy is $50 \, \text{Mpc}$ away, and take

$$
H_0 = 70 \, \text{km s}^{-1}\text{Mpc}^{-1}
$$

Then

$$
v = H_0 d = 70 \times 50 = 3500 \, \text{km/s}
$$

So the galaxy recedes at about $3500 \, \text{km/s}$.

Why Hubble's Law Matters

Hubble's law was the first clear observational evidence that the universe is not static. It showed that the large-scale universe changes with time. This discovery strongly supported the idea that the universe began in a denser and hotter state, which later became the basis of the Big Bang model.

Hubble's law does not mean that galaxies are flying away from a central point in preexisting empty space. It means that the distances between galaxies increase because space expands.

Limits of the Simple Law

The simple formula $v = H_0 d$ works best for large-scale cosmic expansion and for not too extreme interpretations. At very great distances, cosmology uses more precise models based on general relativity. Still, Hubble's law remains the essential first description of the expanding universe for beginners.

Summary

Hubble's law states that the recession speed of a galaxy is proportional to its distance: