The homogeneous expanding universe

Look around a room and it is clearly not the same in all directions. Even on the scale of our galactic neighbourhood it isn't: the direction of the Virgo cluster looks very different from the opposite direction. But average over a few hundred million light-years and the universe becomes isotropic: the same in every direction.

The strongest evidence is the cosmic microwave background (CMB), the oldest light we can see. Until about 400,000 years after the Big Bang the universe was filled with a hot plasma. Photons could not travel far in it: they kept scattering off free electrons. Then the gas cooled enough to become neutral and transparent, and the photons have travelled in straight lines ever since. Looking at the CMB, we see a snapshot of the universe at that moment. (The event is called recombination, oddly, since electrons and nuclei were combining for the first time.)

The CMB temperature is the same in every direction to about one part in a thousand. That small difference has the pattern expected from our own motion through the CMB. Remove that, and what remains varies by only about one part in 100,000.

Homogeneity means the same at every place. It is harder to test, because it needs distances, and distances in astronomy are hard to measure. As far as we can tell, the universe is homogeneous on the same large scales.

The two ideas are logically different, and examples of one without the other make that clear:

• Homogeneous but not isotropic: a universe in which every galaxy spins about the same axis. Every place is alike, but one direction is special.
• Isotropic but not homogeneous: a universe of concentric shells centred on us. It looks the same in every direction — but only from the centre.

The second example puts us at a special place. The Copernican principle says we should not assume that. If the universe is isotropic about every point, it must also be homogeneous.

The third large-scale fact is that the universe is expanding. Distant galaxies recede from us with a velocity proportional to their distance:

Here $v$ is the recession velocity, $r$ the distance and $H$ the Hubble constant (constant in space, but, as we will see, not in time). The law holds on average; individual galaxies also have their own random motions.

Georges Lemaître derived this relation for a uniformly expanding universe in 1927 and estimated $H$ from existing data. Edwin Hubble, combining his own distance measurements with velocities measured largely by Vesto Slipher, showed in 1929 that the data supported it.

Units: the parsec

Astronomers quote $H$ in kilometres per second per megaparsec. A parsec is the distance at which the Earth–Sun distance (1 AU) appears under an angle of one arcsecond:

One parsec is about 3.26 light-years, and a megaparsec (Mpc) is a million parsecs. Since kilometres and megaparsecs are both lengths, $H$ really has units of 1/time. Guth gives a conversion worth remembering:

This is an identity between units, not a measured value of $H$.

A long history of measurement

Early values were far too large: Lemaître found about 575–625 km/s/Mpc and Hubble about 500. For decades, astronomers argued over values between 50 and 100, because the distance scale was hard to calibrate. Precise values arrived around 2001. In his 2013 lecture Guth quotes the Hubble Space Telescope Key Project (72 ± 8), WMAP (70.2 ± 1.4) and the then-new Planck result (67.3 ± 1.2), already noting a mild disagreement between them.

Since then the disagreement has grown into what astronomers call the Hubble tension. Measurements based on the CMB give about 67 km/s/Mpc, while the local distance ladder of Cepheids and supernovae gives about 73; some other local methods land near 70. Whether this points to new physics or to hidden measurement errors is an open question.

The value matters because the age of the universe scales as $1/H$. Hubble's value, seven times too large, implied an age of only about 2 billion years — younger than the Earth. That conflict helped the rival steady-state theory for years.

If everything recedes from us, are we at the centre? No. Velocities depend on the frame of reference, and switching to another galaxy's frame (a Galilean change of velocity is enough here) gives the same law around that galaxy.

Try it with the generated diagram: tap any galaxy to stand on it, and stretch the universe with the slider.

A homogeneous expansion is like a sequence of photographic enlargements: every distance between galaxies grows by the same factor. The galaxies themselves do not grow. They formed from slightly denser regions, the same ripples seen at the $10^{-5}$ level in the CMB, whose gravity halted their expansion until they collapsed into bound systems.

Guth's analogy is a map of a state that keeps growing. The map never has to be redrawn: the towns stay at the same map positions, and only the scale in the legend changes, from "1 inch = 7 miles" to 8, then 9.

Cosmologists do the same. Give each galaxy fixed comoving coordinates. The real, physical distance between two galaxies is the coordinate distance times the scale factor $a(t)$:

Guth measures coordinates in arbitrary units he calls "notches", so $a$ has units of metres per notch. Most textbooks instead make $a$ a pure number equal to 1 today. Either way, the notches (or the choice of normalisation) must drop out of every physical result, which makes a useful check.

Now take the time derivative. A galaxy at rest in comoving coordinates keeps the same $\ell_c$, so

Uniform expansion automatically produces Hubble's law. The Hubble "constant" is the same everywhere at a given moment, but it changes with time. The single function $a(t)$ describes the expansion of the entire universe. Finding it is the goal of the next lectures.

Light always moves at speed $c$ relative to the local observers it passes. An observer at rest in comoving coordinates is locally an inertial observer, so in comoving coordinates a light ray moves at

As the universe stretches, light covers fewer notches per second. This equation is the starting point for the cosmological redshift and for working out how far light can have travelled since the Big Bang.

One last puzzle: in relativity, clocks moving relative to each other disagree about simultaneity, and comoving galaxies are moving apart. Homogeneity saves the day. Because every comoving observer sees the same history, they can agree on a common time by a physical signal, for example "noon is when $H$ reaches a certain value" or when the CMB has cooled to a certain temperature. This shared clock is called cosmic time, and it is the $t$ in $a(t)$.

• On scales above a few hundred million light-years the universe is isotropic (the CMB to 1 part in $10^5$) and homogeneous.
• Galaxies recede according to Hubble's law $v = Hr$, which looks the same from every galaxy, so it implies no centre.
• Physical distances are $\ell_p = a(t)\,\ell_c$; uniform expansion gives $H = \dot a/a$, and $1/H$ sets the age scale ($\approx 14$ billion years for $H = 70$ km/s/Mpc).
• In comoving coordinates light moves at $dx/dt = c/a$, and homogeneity defines a universal cosmic time.