The Big Bang — Question 8

Question 8

Entropy transfer heats photons relative to decoupled neutrinos

Assume instantaneous neutrino decoupling before electron-positron annihilation. The coupled electromagnetic entropy degrees of freedom drop from 11/2 to 2. Its comoving entropy obeys $g_sT_\gamma^3a^3=$ constant, while $T_\nu a=$ constant.

1. Derive the final neutrino-to-photon temperature ratio.

2. Find the neutrino temperature for Tgamma=2.725 K and the photon-temperature enhancement relative to pure a inverse cooling.

Question 8 - Solution

Entropy transfer heats photons relative to decoupled neutrinos

(a) Method and calculation

$$g_{s,i}(T_i a_i)^3=g_{s,f}(T_{\gamma,f}a_f)^3.$$$$\boxed{T_\nu/T_\gamma=(2/(11/2))^{1/3}=(4/11)^{1/3}=.713766}.$$

(b) Interpretation and further calculation

$$\boxed{T_\nu=.713766(2.725)=1.94501\ {\mathrm{K}}}.$$$$\boxed{(T_\gamma a)_f/(T_\gamma a)_i=(11/4)^{1/3}=1.40102}.$$

Physical check

Finite-duration decoupling gives small corrections. The exercise isolates entropy transfer and explains why all relic species need not share one temperature.