Connection between Temperature and Redshift

Problem 1

Find the dependence of radiation temperature on the redshift.

solution

$$ T \propto a^{ - 1};\quad T = T_0(1 + z). $$

Problem 2

Find the dependence of free non-relativistic gas temperature on the redshift.

solution

$$ T \propto a^{ - 2};\quad T = T_0(1 + z)^2. $$

Problem 3

Estimate the time moment when the recombination started, i.e. when ionized plasma transited to the gas of neutral atoms, and determine the corresponding redshift value. The recombination temperature equals to $T_{rec} \approx 0.3\,eV.$

solution

$$ z_{rec} = \frac{T_{rec}}{T_0} - 1. $$ As $1\mbox{\it eV} \approx 11600\,K$, then $$ z_{rec} \approx \frac{0.3}{2.73}11600 \approx 1300. $$ For a rough estimate assume that the transition $T_{rec} \to T_0$ occurs mainly in the matter-dominated epoch to obtain $$ t_{rec} = t_0\left( \frac{a}{a_0} \right)^{3/2} = t_0\left(\frac{T_0}{T_{rec}} \right)^{3/2}\approx 300\,000\,years. $$