Olbers paradox

Why, Olbers asks, the night sky isn’t as bright as during the day?

The paradox §

One of the most famous cosmological paradox from 1826 by the omonim german astronomer.

The paradox is simple, the hypotesis are:

1. space is euclidean, thus infinite;
2. space is uniformley filled with stars (today we woulds say galaxies) at a sufficiently large scale;
3. there aren’t important movements of stars (galaxies), the universe is immutable;
4. the universe has been around forever.

These hypotesis, in essence, are what follows naturally from the cosmologial principle.

From these premises, let’s imagine drawing concentrical spheres from our perspective, each sphere having a greater radius, increased by a fixed step. From hypotesis $1$ and $2$, each sphere has in average a number of stars (galaxies) proportional to its volume, which is proportional to its radius squared:

$$\#\text{ stars in sphere}\propto r^2$$

In turn, the magnitude of light we receive from each star (galaxy) is proportional to the inverse of the square distance:

$$\text{ light received}\propto \frac{1}{r^2}$$

From this follows that each shell provides the same amount of light, and since by assumption $1$ space is infinite, there are an infinite number of shells, so we receive an infinite amount of light. Of course that isn’t the case. That’s why it’s a paradox.

The explanation §

For a long time, it was believed that the extent of the cosmos was limited and that a dark background could be seen among the stars. This assumption naturally presumes being at the center of the universe, and it was rendered obsolete by the philosophical collapse of geocentrism.

In the seventeenth century, the presence of interstellar clouds of dust in empty space that obscure distant stars was hypothesized. This solution does not hold up to thermodynamic analysis, as the absorbed radiation would heat the matter until it re-emits the same amount of light (blackbody radiation). Olbers himself had leaned towards this erroneous solution.

Another possibility is that the speed of light is limited and the universe has existed for a limited time. In reality, the speed of light had been approximately known since the seventeenth century, based on the measurements of Ole Rømer, but this solution, strangely, was never highly regarded.

A purely statistical possibility is that the visible universe has a fractal distribution, with a fractal dimension less than 2. In this way, the limit as r → ∞ would still tend towards a finite number.

Modern Solution In 1929, the American astronomer Edwin Hubble discovered that the present universe is expanding (the temporal evolution of a homogeneous and isotropic universe like ours had already been predicted by Aleksandr Aleksandrovič Friedman). The visible radiation emitted by stars, traveling through an expanding spacetime, becomes infrared due to the redshift.

In addition to stars, the sky is illuminated by cosmic background radiation. Redshift is responsible for shifting it, this time from visible light to microwaves.

Only a limited number of stars manage to send us visible light, hence the sky appears dark to us.

Thanks to redshift, even in an infinite universe with infinite age, we wouldn’t have the paradox. The possibility of a static universe is definitively discarded.