Connection between Temperature and Redshift

6.4.1

Problem 6.4.1

Find the dependence of radiation temperature on the redshift.

solution

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

6.4.2

Problem 6.4.2

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

solution

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

6.4.3

Problem 6.4.3

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 $$ t_{rec} = t_0\left( \frac{a}{a_0} \right)^{3/2} = t_0\left(\frac{T_0}{T_{rec}} \right)^{3/2}\approx 300\,000\,years. $$