Correction of a misunderstanding about Olbers' paradox

Abstract

It is sometimes said that, if the universe were not expanding, we all would burn up, (Olbers' paradox). In order to check on this, I have used a solution of Friedmann's equation that seems reasonable, and that predicts a contraction of the universe after it has reached a maximum size. We may then compare the radiation energy density U _n now at age t _n of the universe, with this density (=U ^* ) at a time t ^* during contraction when R(t) takes a value R ^* equal to its present value R _n .We simplify the calculation by choosing 2R _n as maximum of R(t). Then, t _n=(1/2 π−1)R _n/C=6.98 × 10⁹ year for R _n = 1.16× 10²⁸ cm.(The present density would then be ρ_n = 2.4× 10⁻²⁹ gram/cm³).We assume the number of galaxies per unit volume = N(t) = η/R(t) ³ with constant η,and we assume a constant average radiative power L per galaxy. Now at t _n we choose N _n L ≈ 10⁻³¹ erg/cm³sec,but our conclusions would be the same for much larger values of this.

We split U into three parts: U ₁ is the primordial energy density left over from t ≈ 0.We put U ₁(t _n ) ≈ 6× 10⁻¹³ erg/cm³ corresponding to T ≈ 3°K.U ₂ is the density of the energy flux near the earth emitted by all stars in our own Milky Way. Most of it is the density U _S =L _⊙/4ηr ² c = 4.5 × 10⁻⁵ erg/cm³ in the flux from the sun. Finally, U ₃ is the energy in the radiation emitted by all other galaxies since t ≈ 0, and was going to burn us according to Olbers. Since U ₁ is a function of R(t), we put U ₁(t^*) =U ₁ (t _n ).We shall also assume U ₂ to be unchanged, as it was not the effect of a change of U _s that we were investigating. In calculating U ₃ we shall overestimate it by neglecting the absorption by closer galaxies of some of the light emitted by farther ones.