12.5 — The Expanding Universe

In 1929 Edwin Hubble published a graph with 24 points on it. Distance on one axis, recession velocity on the other, and a straight line through them.

The scatter was substantial and the distance calibration was wrong by a factor of about seven. The conclusion was right: the universe is expanding.

And it should not have been a surprise. Friedmann had derived expanding solutions from Einstein's equations in 1922, and Lemaître had both derived them and predicted the velocity–distance relation in 1927, two years before Hubble measured it. Einstein told Lemaître his mathematics was correct and his physics abominable.

Hubble's law

\boxed{v = H_0d}

H_0 is the Hubble constant, currently about 70 km/s per megaparsec.

Read the units. For every megaparsec of distance, the recession speed increases by 70 km/s.

Its dimensions are inverse time:

H_0 = \frac{70\ \text{km/s}}{3.086\times10^{19}\ \text{km}} = 2.27\times10^{-18}\ \text{s}^{-1}

\frac{1}{H_0} = 4.41\times10^{17}\ \text{s} = 14.0\ \text{billion years}

The Hubble time, which is close to the actual age of 13.8 billion years — a coincidence that holds only because the deceleration from matter and the later acceleration from dark energy roughly cancel.

And the Hubble distance:

\frac{c}{H_0} = \frac{3\times10^{5}\ \text{km/s}}{70\ \text{km/s/Mpc}} = 4286\ \text{Mpc} = 14.0\ \text{billion light years}

Beyond this distance, the recession speed exceeds c. That is not a violation of relativity, for reasons below.

What is actually expanding

This is the point most often got wrong, and it matters for everything that follows.

Galaxies are not flying apart through space. Space itself is expanding, and the galaxies are carried along.

Two consequences that distinguish the pictures:

There is no centre. Every observer sees everything receding from them. The raisin bread analogy: as the dough rises, every raisin moves away from every other, and no raisin is the centre. Better still, a two-dimensional analogy: dots on the surface of an inflating balloon. The surface has no centre — the balloon's centre is not on the surface at all.

Recession can exceed c. Relativity forbids anything moving through space faster than light past a local observer. It says nothing about the expansion of space itself. Galaxies beyond 14 billion light years recede faster than c and no signal is being sent — quite the reverse, since they are becoming permanently unreachable.

And bound systems do not expand. The Earth, the solar system, the Milky Way and even the Local Group are held together by gravity, which overwhelms the expansion locally. Atoms do not grow. Rulers do not stretch.

\boxed{\text{Expansion happens between gravitationally unbound systems and nowhere else.}}

The scale factor

Define a(t) as the relative size of the universe, normalised so a = 1 today.

The proper distance between two galaxies:

d(t) = a(t)\,\chi

where \chi is the comoving distance — a fixed label that does not change as the universe expands.

Differentiate:

v = \dot{d} = \dot{a}\chi = \frac{\dot{a}}{a}\,d

\boxed{H(t) = \frac{\dot{a}}{a}}

Hubble's law falls out immediately, and H is revealed to be a function of time rather than a constant. The subscript in H_0 means "now".

And H has been decreasing for most of cosmic history, despite the expansion accelerating — which sounds contradictory and is not. H = \dot{a}/a can fall while \dot{a} rises, if a rises faster.

Redshift is not a Doppler shift

Light emitted with wavelength \lambda_e at scale factor a_e arrives with:

\boxed{\frac{\lambda_o}{\lambda_e} = \frac{a_o}{a_e} = \frac{1}{a_e}}

The wavelength stretches in exact proportion to the universe.

Define redshift:

z = \frac{\lambda_o-\lambda_e}{\lambda_e} \quad\Longrightarrow\quad 1+z = \frac{1}{a}

Worked example. A quasar at z = 6:

a = \frac{1}{7} = 0.143

The universe was one seventh its present size when that light left. Its wavelength has been stretched sevenfold — ultraviolet at 121.6 nm arrives at 851 nm, in the infrared.

Why it is not Doppler. A Doppler shift is caused by relative motion at the moment of emission and reception. Cosmological redshift accumulates continuously during the journey, and it depends on the total expansion between emission and reception, not on any velocity.

Two galaxies could be momentarily at rest relative to each other and still show a redshift, if space expanded while the light was in transit.

At low redshift the two pictures agree numerically, which is why v = cz works for nearby galaxies. At high redshift they diverge completely — z = 6 would give v = 6c under the naive formula, and the relativistic Doppler formula would give 0.96c, and neither is the right description.

Where does the energy go? A redshifted photon has less energy than when it was emitted, and it did not give it to anything.

The honest answer: energy conservation in general relativity requires a time-translation symmetry, and an expanding universe does not have one (this is Noether's theorem). There is no global conserved energy in an expanding universe, and the question does not have the answer it seems to demand.

Deriving the Friedmann equations

The full derivation uses the Einstein field equations (Chapter 6.8) with a homogeneous, isotropic metric. A Newtonian derivation gives the same answer and is worth doing, because it makes the physics visible.

Take a sphere of radius r in a uniform universe of density \rho. By the shell theorem (Chapter 1.9), only the mass inside matters:

M = \frac{4}{3}\pi r^3\rho

A test particle on the surface has energy:

E = \frac{1}{2}m\dot{r}^2-\frac{GMm}{r} = \frac{1}{2}m\dot{r}^2-\frac{4\pi G\rho r^2m}{3}

Write r = a\chi, so \dot{r} = \dot{a}\chi:

E = \frac{1}{2}m\dot{a}^2\chi^2-\frac{4\pi G\rho a^2\chi^2m}{3}

Divide by \frac{1}{2}ma^2\chi^2:

\frac{2E}{ma^2\chi^2} = \frac{\dot{a}^2}{a^2}-\frac{8\pi G\rho}{3}

Define kc^2 = -2E/m\chi^2 and rearrange:

\boxed{\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G\rho}{3}-\frac{kc^2}{a^2}}

The first Friedmann equation.

Read the terms. The left is H^2. The first term on the right is gravity from all the matter and energy. The second is the curvature term, and the Newtonian derivation reveals what it means: k is the total energy of the expansion.

• k > 0: negative total energy, bound, expansion eventually reverses. Closed, positively curved.
• k = 0: exactly zero energy, expansion asymptotically halts. Flat.
• k < 0: positive energy, expands forever. Open, negatively curved.

Exactly the escape velocity problem of Chapter 1.9, applied to the universe.

With a cosmological constant (Chapter 6.8):

\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G\rho}{3}-\frac{kc^2}{a^2}+\frac{\Lambda c^2}{3}

The second Friedmann equation comes from energy conservation and gives the acceleration:

\boxed{\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\left(\rho+\frac{3P}{c^2}\right)+\frac{\Lambda c^2}{3}}

Note the pressure term, which has no Newtonian analogue. In general relativity pressure gravitates (Chapter 6.8), so positive pressure decelerates the expansion.

And negative pressure accelerates it, which is exactly what dark energy does.

How the contents dilute

Different components thin out differently as the universe expands.

Matter. The number of particles is fixed and the volume grows as a^3:

\rho_m \propto a^{-3}

Radiation. Photon number density falls as a^{-3}, and each photon is redshifted, losing energy as a^{-1}:

\rho_r \propto a^{-4}

Dark energy. If it is a property of space itself, its density does not change:

\rho_\Lambda \propto a^{0} = \text{constant}

\boxed{\text{Radiation dilutes fastest, then matter, and dark energy not at all.}}

This single fact organises the whole history of the universe.

Era	Dominant	a(t)	When
Radiation	\rho_r	t^{1/2}	0 to 47,000 y
Matter	\rho_m	t^{2/3}	47,000 y to 9.8 Gy
Dark energy	\rho_\Lambda	e^{Ht}	9.8 Gy onwards

Radiation dominated first because it dilutes fastest, so going backwards it grows fastest.

Dark energy dominates last because it does not dilute at all, so eventually everything else falls below it.

And we live near the transition, at about 9.8 billion years, which is either a coincidence or a hint. The "why now?" problem is one of the reasons some physicists are uncomfortable with a pure cosmological constant.

The critical density

Set k = 0 in the first Friedmann equation:

H^2 = \frac{8\pi G\rho_c}{3} \quad\Longrightarrow\quad \boxed{\rho_c = \frac{3H^2}{8\pi G}}

Evaluate at H_0 = 70 km/s/Mpc = 2.27\times10^{-18} s⁻¹:

\rho_c = \frac{3(2.27\times10^{-18})^2}{8\pi(6.674\times10^{-11})} = \frac{3(5.153\times10^{-36})}{1.677\times10^{-9}} = \frac{1.546\times10^{-35}}{1.677\times10^{-9}}

\rho_c = 9.22\times10^{-27}\ \text{kg/m}^3

About five hydrogen atoms per cubic metre. The best laboratory vacuum is 10^{14} times denser.

Express everything as a fraction:

\Omega_i = \frac{\rho_i}{\rho_c}

Current values (Planck 2018):

Component	\Omega
Dark energy	0.685
Dark matter	0.265
Ordinary matter	0.049
Radiation	9\times10^{-5}
Curvature	0.001\pm0.002
Total	1.000

\boxed{\text{The universe is flat to within 0.4 \%.}}

And 95 % of it is something nobody has identified.

Distances in an expanding universe

"Distance" becomes ambiguous, and getting it wrong produces nonsense.

Comoving distance — the separation measured today, with the expansion factored out. Fixed for objects moving with the expansion.

Proper distance — the actual separation at a given moment. Changes with time.

Luminosity distance — inferred from apparent brightness:

d_L = \sqrt{\frac{L}{4\pi F}}

Angular diameter distance — inferred from apparent size:

d_A = \frac{\text{physical size}}{\text{angular size}}

And they are related by:

d_L = (1+z)^2d_A

Worked example: a galaxy at z = 1.

Comoving distance: about 3.3 Gpc. Luminosity distance: d_L = (1+z)\times3.3 = 6.6 Gpc. Angular diameter distance: d_A = 3.3/(1+z) = 1.65 Gpc.

A factor of four between two perfectly reasonable definitions of "how far away".

The strangest consequence

Beyond z \approx 1.6, objects start to look bigger as they get further away.

Because d_A = d_C/(1+z), and while d_C grows with redshift it eventually grows more slowly than (1+z).

The physical reason: a very distant object emitted its light when the universe was small, so it was close at emission and subtended a large angle. The light has been travelling ever since, but the angle was set at emission.

This is a genuine, measured effect and it means the sky has a minimum apparent size for any given physical size, at around z = 1.6.

Ages and horizons

The age of the universe comes from integrating the Friedmann equation:

t_0 = \int_0^1\frac{da}{aH(a)}

With the measured parameters: 13.797 \pm 0.023 billion years.

The observable universe. Light has travelled for 13.8 billion years, so the naive answer is 13.8 billion light years.

The correct answer is 46.5 billion light years, because the space the light crossed has been expanding while it travelled.

d_{\text{horizon}} = c\int_0^{t_0}\frac{dt}{a(t)} = 14.3\ \text{Gpc} = 46.5\ \text{billion light years}

So the observable universe is 93 billion light years across, and it contains an estimated 2\times10^{12} galaxies.

And there is a second horizon that matters more.

The event horizon. Because the expansion is accelerating, there is a distance beyond which light emitted today will never reach us.

d_{\text{event}} \approx 16\ \text{billion light years}

Galaxies beyond about z = 1.6 are already emitting light that will never arrive. We can see them as they were; we cannot see them as they will be.

\boxed{\text{The observable universe is shrinking in comoving terms. Things are leaving, permanently.}}

Chapter 12.8 follows this to its conclusion.

The Hubble tension

Chapter 11.9 raised it. Here is the current state.

Route 1: the distance ladder. Parallax → Cepheids → Type Ia supernovae, measuring the local expansion directly.

H_0 = 73.0 \pm 1.0\ \text{km/s/Mpc}

Route 2: the early universe. The CMB gives the sound horizon at recombination — a standard ruler — and combined with the standard cosmological model that fixes H_0.

H_0 = 67.4 \pm 0.5\ \text{km/s/Mpc}

The gap is about 5 sigma.

Independent checks have not resolved it:

Gravitational wave standard sirens (Chapter 6.10) give H_0 with no calibration ladder at all. Current precision is too poor to discriminate — about 70^{+12}_{-8} from GW170817 — and it will improve.

The tip of the red giant branch, an alternative standard candle avoiding Cepheids, gives about 70 — between the two, and the error bars overlap both.

Time-delay lensing of quasars gives around 73, supporting the ladder.

JWST re-measured the Cepheids with better resolution, testing whether crowding had biased the Hubble results. It confirmed them, removing the most-cited possible systematic.

Possible resolutions:

An unidentified systematic in one method. Increasingly hard to sustain given the cross-checks.

Early dark energy — a brief burst of dark energy before recombination, which would shrink the sound horizon and raise the inferred H_0. Fits reasonably; requires a new field with finely tuned timing.

Extra relativistic species in the early universe.

Something wrong with the standard model at a more basic level.

It is unresolved, and it is the sharpest quantitative disagreement in cosmology.

Where this shows up in your life

Every distance quoted for a galaxy depends on which definition is being used, which is why different sources give different numbers for the same object.

The CMB temperature of 2.725 K is a direct measurement of how much the universe has expanded since it was 3000 K.

Redshift surveys map the universe's structure and are the primary cosmological dataset.

And the expansion is why the night sky is dark. Olbers' paradox asks why, in an infinite eternal universe, every line of sight should not end on a star and the sky blaze uniformly. The answers are that the universe has a finite age, so only light from within the horizon has arrived, and that the expansion redshifts distant light out of the visible band. The dark sky is evidence for the Big Bang, and you can see it from your garden.

What the next chapter fixes

Running the expansion backwards makes the universe smaller, denser and hotter without limit. Chapter 12.6 follows that history second by second: the fundamental forces separating, the quark–gluon plasma, the three minutes in which the light elements were forged with abundances that can be checked, the moment the universe became transparent, and the radiation from that moment which is still arriving and which encodes the parameters of everything.