Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 2 / 10A (Cosmology)

Set $s=a(t)/a(t_{\mathrm{dec}})$. Free propagation in the homogeneous expanding universe redshifts frequencies by $\nu ={\nu }_{\mathrm{dec}}/s$ and dilutes physical number density by $s^{-3}$. Number conservation in corresponding frequency intervals givesInserting the initial Planck spectrum, the factor $s^{-2}(s\nu {)}^2$ becomes ${\nu }^2$, while its exponent is $hs\nu /(k{T}_{\mathrm{dec}})$. HenceThe shape is preserved even without collisions maintaining thermal equilibrium; it follows from redshift of an initially Planckian distribution.

For the total photon number density put $x=h\nu /(kT)$. For the energy density include the energy $h\nu$ of each photon. Thenwhere $\alpha =(8\pi /c^3)(k/h{)}^3{\int }_0^{\infty }{x}^2/(e^x-1)dx$ and $\xi =(8\pi h/c^3)(k/h{)}^4{\int }_0^{\infty }{x}^3/(e^x-1)dx$. Both integrals are finite positive constants. Thus $n\propto a^{-3}$ and $ϵ\propto a^{-4}$, with the extra power supplied by photon-energy redshift.