Redshift preservation of a thermal photon spectrum
= Redshift preservation of a thermal photon spectrum
After photon decoupling in an expanding universe, every frequency scales as $\nu\propto a^{-1}$. Therefore a Planck occupation factor retains its thermal form when its temperature is assigned the same scaling,
$$
T(t)=\frac{a(t_{\rm dec})}{a(t)}T_{\rm dec}.
$$