Why is the sky dark at night?
In 1826, Heinrich Wilhelm Olbers’ paper “Über die Durchsichtigkei des Weltraumes“, or On the Transparency of Space, attempted to answer this question based on the understanding of the Universe* at the time. He showed that, in an infinite, static universe, the radiation density at any point was exactly equal to that of the surface of a star, so an observer should see nothing but blinding light. The paradox is, of course, that the night sky is dark. This is known as Olbers’ paradox.
*In following with convention, uppercase Universe refers to the actual Universe, while lowercase universe refers to any theoretical or model universe.
All-Encompassing Flux
Let’s present a simple classical argument before delving into the radiative aspects, and we’ll soon see where the issues lie. Consider an infinitely large universe, with the stars distributed uniformly such that the number of stars per unit volume is given by some number density \( n \). Let’s consider a spherical shell of radius \( r \) and infinitesimally small thickness \( dr \). The number of stars in this shell is approximately equal to \( n \) times the volume element, or \[ 4\pi n r^2dr \quad dr \ll 1\] Now, if each star shines with some average luminosity given by \( L \), then the overall intensity of the light received at the origin is \[ I \approx \frac{4\pi n r^2dr}{4\pi r^2} = nLdr \] Notice that the \( r^2 \) cancels! Thus, no matter how far away the shell is, the light received at the origin will be the same. Since the universe is infinitely large, there should be infinitely many shells, hence an infinite amount of light! Alas, the night sky is dark.
There are other (mathematical) approaches that essentially lead to the same conclusion. We can perform a Taylor expansion to show that, in the limit to infinity, the overall solid angle subtended by all stars approaches exactly 4\( \pi \) (i.e. the entire sky should be covered). Another way of stating the paradox is that, no matter what direction you look, your line-of-sight should eventually hit a star. If there are an infinite number of stars, it stands to reason they’ll cover your entire field of view.
Seems like an awful amount of assumptions going on here
Indeed. However it’s not as simple as merely saying “clearly the universe is not infinite”, as it is perfectly feasible for an infinite universe to have dark skies. In order for Olbers’ paradox to make complete sense:
The universe must be homogeneous, static and infinite.
When Olbers wrote his 1826 paper, the prevailing view was that the Universe was infinite and static, a view largely unchanged (and unchallenged) from the time of Newton. Even Einstein was so tempted by the idea of a static universe that he introduced a cosmological constant into his field equations of general relativity, a mistake he would later refer to as his “greatest blunder”. It was not until the work of Georges Lemaître and Edwin Hubble, who showed that the Universe was expanding, that the idea of a static universe was mostly dismissed.
Stronger principles and steadier states
The cosmological principle states that, on a large enough scale, the Universe is homogeneous (uniform at all locations) and isotropic (uniform in all directions), and so the Universe should look the same (to an observer) from any location. You’d be forgiven for thinking this seems to echo the assumptions of Olbers’ paradox. An extension to this principle, called the perfect cosmological principle, also applies the homogeneity and isotropy requirements to time as well. This perfect principle underpins the steady state theory, a cosmological theory that emerged in the late 1940s with a series of influential papers, including one authored by Sir Fred Hoyle. The steady-state theory states that although the universe is expanding, it is infinitely old and maintains a steady overall density via a replenishing stream of new matter. It was not until the accidental discovery of the cosmic microwave background radiation (CMB) in 1964 (published in 1965) that the steady state theory began to decline in significance. In particular, the CMB implies a finite age of the universe. It had been postulated as early as the 1930s that the universe came into being from some initial singularity, which Lemaître referred to as a “primeval atom”, later popularised as a “cosmic egg”. Sir Fred Hoyle, a proponent of the steady state theory, coined the term “Big Bang” during a 1949 BBC radio interview (popular legend says that he deliberately used this term pejoratively, but this is now regarded to be a myth).
Dust to dust
One does not even need to conjure a finitely-old universe in order to resolve Olbers’ paradox. Stars live and die, and there is only a finite amount of matter to go around. Thus stars can only radiate for a finite amount of time. To see why, let’s discuss energy. Olbers’ original argument was based on radiative density rather than luminosity. Olbers argued that the radiative density of any position in an infinite, static universe was equal to the (average) density at the surface of the stars. The problem here is that this theory predated Einstein’s \( E = mc^2 \). In particular, suppose the stars radiate an average energy of \( E_s \), have an average mass of \( M_s \), and have been shining for some time \( t \). Then clearly this condition must hold: \[ \frac{E_st}{M_s} \leq c^2 \] To suggest otherwise would violate the conservation of energy. This necessarily restricts the overall value of \( t \). I should note that this is a valid argument only if there is a finite amount of matter in the universe; eventually there won’t be enough hydrogen and helium to form stars. The steady state theory gets around this by introducing a continuous stream of matter with which to provide more fuel for stars.
Beyond the horizon
The most common, and widely accepted, reason for resolving Olbers’ paradox is due to the expansion of the Universe (under the Big Bang model). Distant stars in distant galaxies are redshifted; the further away, the more extreme the redshift due to the expansion of space. There is eventually a distance beyond which stars are redshifted into the infrared spectrum, and are no longer visible to the (human) eye. Further beyond that lies the ultimate boundary – the limit of the observable universe – a cosmic event horizon beyond which objects can never be detected as the light will never get to reach us (even if we were to wait infinitely long). Even now, the CMB acts as an effective particle horizon; we cannot currently observe objects beyond it. Olbers’ paradox is a statement of faith in the concept of infinity. But the Universe’s ultimate speed limit – the speed of light – confines us.
We are living in a rather fortuitous age where plenty of galaxies are observable for us to infer the overall structure and nature of our Universe. As galaxies recede they get fainter and fainter. Many billions of years from now, distant galaxies will have receded to the point where they can no longer be detected (i.e. indistinguishable from the background). What would a future, advanced civilisation see, but an endless expanse of darkness? Would their “Universe” be a single galaxy adrift in an ocean of night? At least when we go outside and gaze up the night sky, we can confidently answer the question “Why is the sky dark at night?” Future civilisations may not be so lucky.