Olbers' paradox

Step outside on a clear night and the sky is mostly black, with scattered points of light. That seems too obvious to question. But here is the trouble. If the universe goes on forever and is full of stars, then every direction you look should, sooner or later, land on the surface of a star. Near stars look big and bright, far ones look tiny and faint, but pack enough of them into every direction and there should be no gaps at all. The whole sky should blaze, as bright as the surface of the Sun, everywhere at once.

It does not. The night is dark. That contradiction is Olbers' paradox, and the answer is not that there are too few stars.

The answer is that the universe is not infinitely old, and it is growing. Light from the most distant places has not had time to reach us yet. And the light that does arrive from far enough away has been stretched and dimmed by the expansion of space until it slips out of sight. The dark sky is quietly telling you that the universe had a beginning.

The argument is simple, and that is what makes it sharp. Picture the sky split into nested shells around you, each a little farther out. A shell twice as far away holds four times as many stars, because it has four times the area, \(N \propto r^2\). But each of those stars looks four times fainter, by the inverse-square law, \(f \propto 1/r^2\). The two effects cancel exactly. Every shell, near or far, sends you the same total amount of light. Add up endless shells and the sky should be uniformly, blindingly bright.

People noticed this for centuries. Kepler used the dark night sky in 1610 to argue that the universe must be finite. Edmond Halley and Jean-Philippe de Chéseaux wrestled with it in the 1700s, and Heinrich Olbers restated it in 1823, which is how it got his name, even though he did not discover it.

The tempting escape is to say that dust soaks up the distant starlight. It does not work. In a universe that had been shining forever, the dust would absorb starlight until it heated up and glowed just as brightly as the stars it was meant to be hiding. You cannot hide light with matter forever. The energy has to go somewhere.

The real resolution arrived in pieces, and oddly, the writer Edgar Allan Poe sketched it first, in 1848. The universe is not eternal. It is about 13.8 billion years old, so we can only see out to the distance light has managed to cross in that time. Beyond that horizon there are surely more stars, but their light simply has not reached us. The shells do not go on forever in practice, so most lines of sight run out at the horizon without ever striking a star. On top of that, space is expanding, which stretches light from distant sources toward longer, redder wavelengths and drains its energy. The most distant light of all is redshifted clean out of the visible range.

And here is the twist. The sky really does end, in every direction, on a hot bright surface. It is the afterglow of the early universe, the cosmic microwave background. The instinct behind the paradox was almost right. That surface has just been stretched by expansion, from the glare of something thousands of degrees hot down to a faint microwave hum the eye cannot see. The night is dark only because the wall of light at the edge has been shifted to where we cannot look.

Why the sum saturates rather than diverges. The naive shell sum gives infinite brightness, but it ignores nearer stars blocking farther ones. Once you account for occultation, distant stellar surfaces hide behind closer ones, and the sky brightness saturates at the surface brightness of a typical star instead of running to infinity. That is the sharper form of the paradox: not an infinitely bright sky, but a sky uniformly as bright as the surface of a star. Still catastrophically far from black.

Finite age does most of the work. It is common to credit expansion and redshift as the main reason the sky is dark. They contribute, but a careful accounting, set out by Edward Harrison, shows that finite age is the dominant factor, and would keep the sky dark even in a static universe of the same age. The reason is energy. To fill the sky to stellar surface brightness, stars would have to have poured out far more light than they have had time to produce. Stars also burn out, so at any moment only a thin slice of cosmic history is shining. There has not been enough time, and there is not enough fuel. Run the numbers and the universe would need to be trillions of years old, with stars to match, before the night sky approached daylight.

Redshift is the supporting actor. Expansion then helps in two ways. It dilutes the radiation as space grows, and it shifts the spectrum of distant light toward the red and out of the visible band, cutting each far source's contribution by a factor that grows with redshift. In the full relativistic treatment these effects lower the predicted background further, but they are corrections on top of the finite-age result, not the heart of it.

What about clumping? One can ask whether the uneven, clustered distribution of stars and galaxies saves the day. A sufficiently hierarchical, fractal arrangement, with matter spread ever more thinly on larger scales, could in principle keep lines of sight from landing on stars. Mandelbrot explored exactly this. But observations show the universe is close to homogeneous on the largest scales, so real clustering does not resolve the paradox. The age of the universe does.

The sky's true surface. Push every line of sight far enough and it does terminate, not on a star but on the surface of last scattering, the moment about 380,000 years after the Big Bang when the universe turned transparent. That surface filled the whole sky at a temperature of roughly 3,000 kelvin, genuinely Olbers-bright. Expansion has since redshifted it by a factor of about 1,100, cooling it to 2.7 kelvin, the cosmic microwave background. So the paradox's instinct was sound. The sky is uniformly bright in every direction. We just cannot see the wavelength, and the reason we cannot is the same reason the night is dark: the universe is expanding, and it had a beginning.