Definitions

Olbers' paradox, described by the German astronomer Heinrich Wilhelm Olbers in 1823 (but not published until 1826 by Bode) and earlier by Johannes Kepler in 1610 and Halley and Cheseaux in the 18th century, is the argument that the darkness of the night sky conflicts with the supposition of an infinite and eternal static universe. It is one of the pieces of evidence for a non-static universe such as the current Big Bang model. This "paradox" is sometimes also known as the "dark night sky paradox" (see physical paradox).

## Assumptions

If the universe is assumed to contain an infinite number of uniformly distributed luminous stars, then:

1. The collective brightness received from a set of stars at a given distance is independent of that distance;
2. Every line of sight should terminate eventually on the surface of a star;
3. Every point in the sky should be as bright as the surface of a star.

Looking at trees within a big flat wood in the direction of the horizon shows the effect: The mass of dark trees will hide the horizon (imagine the trees now as lights).

The further away one looks, the older the image viewed by the observer. For stars to appear uniformly distributed in space, the light from the stars must have been emitted from places where the stellar density of the region at the time of emission was the same as the current local stellar density. A simple interpretation of Olbers' paradox assumes that there were no dramatic changes in the homogeneous distribution of stars in that time. This implies that if the universe is infinitely old and infinitely large, the flux received by stars would be infinite.

Kepler saw this as an argument for a finite observable universe, or at least for a finite number of stars. In general relativity theory, it is still possible for the paradox to hold in a finite universe: though the sky would not be infinitely bright, every point in the sky would still be like the surface of a star.

A more precise way to look at this is to place Earth in the centre of a "sphere". If the universe were homogeneous and infinite, then at a distance r away from the earth, the shell of the sphere would have a certain flux (viewed from Earth) due to the individual flux of the stars on the shell (brightness) and also the number of stars in the shell (cumulative flux). When an observer from Earth looks to a farther distance to another shell, r+x, the number of stars increases by the square of the distance, while the flux decreases by the inverse squared. Comparing the total brightness of the first shell to the second shell, one notices that both shells have equal flux, since the flux of each individual star decreases due to distance but is equally made up for by the number of stars. This means that no matter how far away an observer on Earth views the sky, the brightness of each consecutive shell would not diminish; rather, they would be equal. If the universe were infinite (age and volume) and had a regular distribution of stars, then there will be an infinite number of such shells and infinite amount of time for the light to reach Earth (infinite flux) as long as the earth remains, effectively meaning that there would never be night on Earth.

## The mainstream explanation

In order to explain Olbers' paradox, one would need to account for the relatively low brightness of the night sky in relation to the circle of our sun.

### Finite speed of light

The greater the distance of a star from an observer on Earth, the longer it takes the star's light to reach the observer. Thus, the farther we look into space, the farther we see into the past. This fact is a key ingredient in the mainstream explanation of Olbers' paradox, although it cannot alone explain the paradox, since the speed of light has no direct connection to the energy density and broadness of light received at any given point.

### Finite age of the universe; the origin of all light is a finite distance away

Edgar Allan Poe was the first to solve Olbers' paradox when he observed in his essay Eureka: A Prose Poem (1848):
"Were the succession of stars endless, then the background of the sky would present us a uniform luminosity, like that displayed by the Galaxy –since there could be absolutely no point, in all that background, at which would not exist a star. The only mode, therefore, in which, under such a state of affairs, we could comprehend the voids which our telescopes find in innumerable directions, would be by supposing the distance of the invisible background so immense that no ray from it has yet been able to reach us at all."

The universe, according to the mainstream theory of the universe, called the Big Bang Theory, is only finitely old; stars have existed only for part of that time. So, as Poe suggested, the earth receives no starlight from beyond a certain distance.

According to the Big Bang Theory, the sky was much brighter in the past, especially in the first few seconds of the universe. All points of the local sky at that era were therefore brighter than the circle of the sun, despite the finite and even more limited range that light could travel in that prehistoric era; this implies that most light rays will terminate not in a star but in the relic of the Big Bang.

### Expanding space

While the finite distance for the origin of any received light does not by itself solve the paradox, the Big Bang Theory also involves the expansion of the "fabric" of space itself (not just the distance of objects in that space) that can cause the energy of emitted light to be reduced via redshift. More specifically, the extreme levels of radiation from the Big Bang have been redshifted to microwave wavelengths as a result of the cosmic expansion, and thus form the cosmic microwave background radiation. This explains the relatively low light densities present in most of our sky despite the assumed bright nature of the Big Bang. The redshift also affects light from distant stars and quasars, but the diminution is only an order of magnitude or so, since the most distant galaxies and quasars have redshifts of only around 5. Thus, the mainstream explanation of Olbers' paradox requires a universe that is both finitely old and expanding.

## Alternative explanations

The redshift and expanding space hypothesised in the Big Bang model would by itself explain the darkness of the night sky, even if the universe were infinitely old. The steady state cosmological model assumes that the universe is indeed infinitely old and uniform in time as well as space. It is also expanding exponentially, producing a redshift. There is no Big Bang in this model, but there are stars and quasars at arbitrarily great distances. The light from these distant stars and quasars will be redshifted accordingly, so that the total light flux from the sky remains finite and dominated by the nearest light sources. However, the steady state model cannot explain the detailed behavior of distant starlight and the microwave background, since it requires a continuous transformation of the former into the latter at decreasing frequencies; this transformation is not observed.

### Finite age of stars

Stars have a finite age and a finite power, thereby implying that each star has a finite impact on a sky's light field density. But the finity of the influence from any given star does not imply that our sky will be darker than the circle of the sun in most areas of our sky. Only those stars whose worldlines intersect the light cone of a point would contribute to the luminosity there in any event, so the age of any given star is largely irrelevant. Despite being neither a sufficient nor a necessary explanation of the darkness of the sky, the finite age of stars is considered by some to be a reason for the dark sky, and accordingly, is seen as a solution to Olbers' paradox.

### Absorption

An alternative explanation, which is sometimes suggested by non-scientists, is that the universe is not transparent, and the light from distant stars is blocked by intermediate dark stars or absorbed by dust or gas, so that there is a bound on the distance from which light can reach the observer.

However, this reasoning alone would not resolve the paradox given the following argument: According to the second law of thermodynamics, there can be no material hotter than its surroundings that does not give off radiation and at the same time be uniformly distributed through space. Energy must be conserved, per the first law of thermodynamics. Therefore, the intermediate matter would heat up and soon reradiate the energy (possibly at different wavelengths). This would again result in intense uniform radiation as bright as the collective of stars themselves, which is not observed.

### Fractal star distribution

A different resolution, which does not rely on the Big Bang theory, was offered by Carl Charlier in 1908 and later rediscovered by Benoît Mandelbrot in 1974. They both postulated that if the stars in the universe were fractally distributed in a hierarchical cosmology (e.g., like a Cantor dust)—the average density of any region diminishes as the region considered increases—it would not be necessary to rely on the Big Bang theory to explain Olbers' paradox. This model would not rule out a Big Bang but would allow for a dark sky even if the Big Bang had not occurred. This is merely a demonstration of the consequences of fractal theory as a sufficient, but not necessary, resolution of the paradox.

Mathematically, the light received from stars as a function of distance from stars in a hypothetical fractal cosmos can be described via the following function of integration:

$text\left\{light\right\}=int_\left\{r_0\right\}^infty L\left(r\right) N\left(r\right),dr$

Where:

$r_0$ = the minimum distance from which light is received ≠ 0

$r$ = the variable of distance

$L\left(r\right)$ = average luminosity per star at $r$

$N\left(r\right)$ = number of stars at $r$

The function of luminosity from a given distance $L\left(r\right)N\left(r\right)$ determines whether the light received is finite or infinite. For any luminosity from a given distance $L\left(r\right)N\left(r\right)$ proportional to $r^a$, $text\left\{light\right\}$ is infinite for $a ge -1$ but finite for $a<-1$. So if $L\left(r\right)$ is proportional to $r^\left\{-2\right\}$, then for $text\left\{light\right\}$ to be finite, $N\left(r\right)$ must be proportional to $r^b$, where $b<1$. For $b=1$, the numbers of stars at a given radius is proportional to that radius. When integrated over the radius, this implies that for $b=1$, the total number of stars is proportional to $r^2$.

Mainstream cosmologists reject this type of fractal cosmology on the grounds that studies of large-scale structure in combination with the timeline of the universe have not produced conclusive evidence for it.