Monday, December 01, 2014

Olbers' paradox is the argument that the darkness of the night sky conflicts with the assumption of an infinite and eternal static universe.

Let n be the average number density of galaxies in the universe. Let L be the average stellar luminosity. The flux f(r) received on earth from a galaxy at a distant r is,

(1) f(r) = L/(4πr2)

Consider now a thin spherical shell of galaxies of thickness dr. The intensity of radiation from that shell is,

(2) dJ(r) = flux x number of galaxies in the thin shell.

= L/(4πr2) x n x r2dr

= (nL/4π)dr

We can see that the intensity only depends on the thickness of the shell, not its distance.

The total intensity is found by integrating over shells of all radii.

(3) J = (nL/4π) 0dr = ∞

Accordingly, the night sky should be bombarded by an infinite number of photons.

Explanation

(A) BBT

The primary argument of the Olbers' paradox from the Big Bang Theory is the universe has a finite age (by extrapolating backward in time), and the galaxies beyond a finite distance, called the horizon distance, are invisible to us simply because they are moving faster than lightspeed and therefore their light can’t reach us.

(B) The static model

The paradox is only an apparent contradiction. In this case, if humans had eyes that could see 2mm wavelength (the Cosmic Microwave Background), then one would see the night sky being illuminated from every direction. What's missing in the above calculation is that the wavelength of light travelling immense distance is shifted more and more towards the red. In terms of the wave model, the next peak would take an infinite of time to reach us. This is the surface of infinite redshift. What we see in the CMB are the photons released from a distance slightly less than the surface of infinite redshift. Any photons released from galaxies beyond that distance will not reach us.