|THE EXPANSION OF THE UNIVERSE|
Olbers' Paradox Solution
Olbers started with a uniform distribution of stars in an infinite large space (number of stars per ly^3 - cubic lightyear - is constant) and the Sun is one of them. Consider a spherical shell around the Sun of 1 ly thickness and radius of R ly. The volume of the shell, and thus the number of stars in it, varies with R^2. When R is 3 times as big, then there are 9 times more stars in the shell. (Two dimension, length and width, vary while the 3rd dimension, its thickness, is fixed at 1 ly. Therefore the volume of the shell varies with R^2 and not with R^3). The brightness of a star in the shell decreases with R^2. When R is 3 times larger the luminosity of a star in the shell is 9 times smaller. So the increase due to the number of stars in a shell is just canceled by the decrease of luminosity per star. As a result on the average the luminosity of the shell as a whole doesn't change. A shell around the Sun of 1 ly thickness at 100 ly distance is as bright as a shell of 1 ly thickness at 1 billion ly around the Sun. Now realize the space of Olbers is build from an infinite of this kind of shells around the Sun, one neatly on top of the other. The sky at night should be infinite bright.
A Thought Experiment
Let's perform a thought experiment. Assumed is the cosmological principle to be true, stating the spatial distribution of matter in the universe is homogeneous and isotropic when viewed on a sufficiently large scale. We start to assume it is the light from stars that determine the brightness of the sky at night. We neglect e.g. the 2.7 K Background Radiation that did not origin in stars. Imagine you had a magic button that speeds up star evolution. With this button all molecular clouds in the universe immediately convert into stars and all stars (old ones and the newly formed) immediately convert entirely into radiation, according to E=mc. The situation cannot become worse. The resulting radiation bath or photon gas as it sometimes is called, when considered at scales large enough for the cosmological principle to become apparent, is uniform and thus in equilibrium with respect to the amount of photons per area. An area can only gain more radiation at the cost of nearby areas and that on the average will counteract statistics. Normally an area will radiate out to nearby areas the same amount as it receives from that areas. Mind the light of stars IS the radiation bath. The amount of radiation that enters a steradian in one second then is the brightness of the sky at night, that is, its upper limit. This is a finite amount and so this suffices as a solution of the Paradox of Olbers.
Mind this solution does not depend on the size of the universe, nor on its age. It does not depend on whether the universe is expanding or not; the solution works also for the Steady State cosmological model. That is, the considered areas must be large enough for the cosmological principle to become apparent. And the age must be sufficient long for light to have crossed that areas, allowing the photon bath in a single area to become uniform. But it is not important whether the size or age is finite or infinite. It only depends on the density of the universe.
Of course expansion of the universe dilutes the radiation bath and lowers its average brightness. In the course of time from the Big Bang up to now the universe passes all densities from, well let's start at 10^17 kg/m^3 (neutron star density) down to 10^-26 kg/m^3 (critical density of the universe, that it is now, nearly). It depends on the precise moment in this elapse of time where you perform this thought experiment, what precisely is the value of the upper limit of the brightness of the sky at night.
Nowadays most molecular clouds have not yet contracted to stars, and not all star matter will finally convert to radiation. So, what is the actual radiation bath right now? The density of the universe as it is now consists of 73% Dark Energy, 23 % Dark Matter (WIMPs, neutrinos), 4 % ordinary matter (protons, neutrons, electrons) and only 0.005 % consists of photons (starlight and Background Radiation).
NEXT PAGE Up CONTACT