How Space Itself Expands

Here is a claim worth resisting: the galaxies rushing away from us are not, for the most part, moving. They sit more or less still in their own patch of space, drifting only slightly the way any object drifts. What grows is the distance between us, and it grows because the space in the gap is being manufactured. New space, continuously, everywhere at once. The galaxy isn't travelling away from you down a road. The road itself is getting longer underneath it.

That sounds like wordplay until you notice it makes a different prediction. If galaxies were shrapnel from an explosion, their redshift would be an ordinary Doppler shift, the same thing that drops the pitch of a passing siren. But the cosmological redshift isn't that. The light leaves a distant galaxy with some wavelength, travels for a billion years, and arrives stretched, because the space it was crossing stretched while it was in flight. The photon got longer along with everything else. Same equations of motion, completely different cause, and the difference is the whole subject of this essay.

The scale factor: one number for the size of everything

Start with the trick that makes cosmology tractable. Instead of tracking the absolute position of every galaxy, lay down a coordinate grid that expands along with the universe and let the galaxies sit on it. These are comoving coordinates. A galaxy that isn't doing anything special keeps the same comoving coordinates forever, even as the real, physical distance to it balloons. The grid stretches; the labels stay fixed.

All of the stretching is then bundled into a single dimensionless function of time, the scale factor $a(t)$. By convention $a = 1$ today. At half the present size $a = 0.5$, and at the Big Bang $a = 0$. The physical distance between two galaxies, the proper distance you'd measure with an absurdly long ruler frozen at one instant, is just the fixed comoving distance multiplied by this one number:

$$d_{\text{proper}}(t) = a(t)\, d_{\text{comoving}}$$

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This is the move that tames the problem. The comoving distance is a property of the pair of galaxies and never changes. Every bit of time-dependence, the entire expanding-universe story, is squeezed into $a(t)$. Find $a(t)$ and you know how the size of the universe has run from the Big Bang to now and onward. Cosmology becomes the project of solving for one function.

Hubble's law falls out for free

Once distance is $a(t)$ times a constant, the recession velocity is forced. Differentiate equation (1) with respect to time. The comoving part is fixed, so only $a$ varies, and you get $\dot{d}_{\text{proper}} = \dot{a}\, d_{\text{comoving}}$. Divide and substitute back, and the comoving distance cancels into the proper distance:

$$v = \dot{d}_{\text{proper}} = \frac{\dot{a}}{a}\, d_{\text{proper}}$$

The factor out front is the Hubble parameter, $H = \dot{a}/a$, the fractional rate at which the scale factor is growing.

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

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Evaluate it now and you have the famous Hubble constant $H_0$, and the whole thing collapses to the linear relation Edwin Hubble announced in 1929: recession velocity proportional to distance, $v = H_0 D$. He measured it the hard way, distances and Doppler-like redshifts for 22 galaxies, and found the line. We just derived it in three lines of algebra from the assumption that space scales uniformly.

The linearity is the tell. A Doppler shift from random motion would have no reason to grow with distance; near and far galaxies would scatter. But if every proper distance is the same comoving distance times the same $a(t)$, then twice as far means twice the recession speed, automatically. The straight line through Hubble's data is the fingerprint of uniform expansion. Georges Lemaître had in fact derived the relation from general relativity two years earlier, in a Belgian journal almost nobody read, which is why it's now the Hubble–Lemaître law.

So far this is all kinematics: if space scales by $a(t)$, then galaxies obey Hubble's law. It says nothing about why $a(t)$ does what it does, or what it'll do next. For that we need the dynamics, and the dynamics come from gravity.

The Friedmann equation: gravity writes the script

In 1922 Alexander Friedmann took Einstein's field equations and asked what they say about a universe that's the same everywhere and in every direction. Einstein had assumed the universe was static and bolted on a term to keep it that way. Friedmann didn't assume; he solved, and the solution refused to sit still. A homogeneous, isotropic universe must expand or contract. It cannot hover. Out of that came the equation that governs $a(t)$:

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

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The left side is $H^2$, the square of the expansion rate. The right side is the bill: three contributions that decide how fast space grows. Read it as a budget.

The first term, $\frac{8\pi G}{3}\rho$, is ordinary gravitating stuff, matter and radiation. Density $\rho$ pulls; more of it means a faster-changing $a$, and crucially it brakes the expansion over time as gravity tugs back. The second term carries the spatial curvature $k$, which is $+1$, $0$, or $-1$ for a universe that's closed (like a 3-sphere), flat, or open (saddle-shaped). The third term is the cosmological constant $\Lambda$, Einstein's old fudge, resurrected. Unlike matter it doesn't dilute as space grows; it's a fixed energy of space itself, and it pushes outward without ever weakening.

That single equation is the engine. Hand it the present density, curvature, and $\Lambda$, and it tells you $\dot{a}$ at every value of $a$, which you integrate to get $a(t)$ over all of cosmic history, forward and backward. The past asks: run it backward and $a$ shrinks to zero in finite time. That zero is the Big Bang. Not an explosion somewhere in space, but the moment $a(t) \to 0$ where every proper distance vanishes at once, everywhere.

The critical density and the shape of space

The Friedmann equation hides a beautiful piece of accounting. Set the curvature term to zero and ask what density makes the universe exactly flat. Rearranging equation (3) gives a threshold:

$$\rho_c = \frac{3H_0^2}{8\pi G}$$

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Plug in the measured $H_0$ and this works out to roughly $10^{-26}$ kilograms per cubic metre, about five hydrogen atoms in a volume the size of a phone box. That's the whole universe averaged out: emptier than the best laboratory vacuum by an absurd margin. The ratio of the actual density to this critical value is written $\Omega$, and it decides the geometry. If $\Omega = 1$, the density is exactly critical and space is flat. More than critical ($\Omega > 1$) curves it closed; less ($\Omega < 1$) leaves it open.

The remarkable observational fact is that $\Omega$ comes out within a per cent of $1$. Space is flat, as far as anyone can measure, which is itself a puzzle the standard model handles with inflation but doesn't fully explain. The budget that adds up to that flat $\Omega = 1$ is strange, though.

Dark energy

68.3%

the cosmological constant Λ, pushing expansion

Dark matter

26.8%

gravitates, but emits no light

Ordinary matter

4.9%

everything we've ever seen or touched

Everything made of atoms, every star and planet and person, is under five per cent of the total. The rest is dark matter that gravitates without shining and dark energy that does the opposite of gravitate. These aren't optional extras you sprinkle on top. They are terms in equation (3): the density $\rho$ includes dark matter, and $\Lambda$ is the dark energy. The composition of the universe is literally the coefficients of the equation that sets its fate.

Why expansion is accelerating

For billions of years the matter term dominated. Density was high, gravity was winning, and the expansion decelerated. Picture a ball thrown upward, slowing as gravity pulls it back. Everyone expected the universe to do the same, the only question being whether it had enough density to eventually halt and recollapse, or would coast outward forever.

In 1998 two teams measuring distant supernovae found the opposite. The expansion isn't slowing. It bottomed out around six billion years ago and has been speeding up since. The ball, halfway up, started rising faster. The culprit is the $\Lambda$ term. As space expanded, matter thinned out and its term in equation (3) faded, while the cosmological constant held steady, because it's an energy of space itself and there's only ever more space. Once $\Lambda$ overtook matter, the brake became an accelerator.

Matter dilutes and weakens; the cosmological constant doesn't. Give expansion enough time and the term that never fades must win. The universe's long-run fate was decided the moment dark energy outpaced gravity.

Look again at equation (2) under a dominant $\Lambda$. If the right-hand side of the Friedmann equation is a constant, then $H = \dot{a}/a$ is constant, which means $a$ grows exponentially: $a(t) \propto e^{Ht}$. The Hubble parameter, after falling for most of history, is now levelling off toward a fixed value and the universe is sliding into exponential growth. Distant galaxies will redshift away until they fade from view entirely. The night sky of the far future is lonelier than ours.

What the equation actually buys you

Run the machine forward from the measured ingredients and one number falls out that's worth pausing on: the age of the universe. Integrate equation (3) backward from today's expansion rate and composition until $a = 0$, and the elapsed time is 13.787 billion years, pinned to about twenty million years by the Planck satellite's data. That figure isn't read off a fossil. It's computed, the time it takes to wind $a(t)$ down to zero given how fast it's growing now and what it's made of. The Big Bang has a birthday because the Friedmann equation has a definite integral.

There's a crack in the picture, and it's currently the liveliest argument in cosmology. Measure $H_0$ from the nearby universe, using supernovae and the distance ladder, and you get about 73 km/s/Mpc. Measure it from the early universe, by reading the cosmic microwave background and running the Friedmann equation forward, and you get about 67.7. Both methods are careful, both have shrunk their error bars, and the two numbers stubbornly refuse to meet. This is the Hubble tension, and nobody yet knows whether it's a measurement subtlety or a sign that equation (3) is missing a term. A clean discrepancy in a number this fundamental is exactly where new physics tends to hide.

Step back and the shape of the whole subject is one function and one equation. The scale factor $a(t)$ is the size of everything, comoving coordinates hold the galaxies still while it grows, and Hubble's law is the immediate consequence. The Friedmann equation takes the universe's inventory, ordinary matter, dark matter, dark energy, curvature, and turns it into the complete history of $a(t)$: a Big Bang at one end, deceleration under gravity, a handover to dark energy six billion years ago, and an accelerating, exponential future at the other. The galaxies were never the story. The space between them was. Some food for thought, next time one of them looks like it's running away.

Reading further

• Friedmann, On the Curvature of Space (Zeitschrift für Physik 10, 1922). The founding paper, where the dynamic-universe solution first emerges from Einstein's equations against the prevailing belief in a static cosmos.
• Hubble, A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae (PNAS 15, 1929). The observational confirmation: 22 galaxies, a straight line, and the empirical birth of $v = H_0 D$.
• Ryden, Introduction to Cosmology (2nd ed., Cambridge). The standard pedagogical treatment of the scale factor, the Friedmann equations, and the FLRW geometry, drawn carefully and from first principles.
• Lemaître's expanding universe and the long road to credit (Science). How Lemaître derived the expanding solution and the Hubble constant two years before Hubble, and why recognition took decades.