Connection between Temperature and Redshift

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Problem 1

Find the dependence of radiation temperature on the redshift.

Problem 2

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

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 $T_{rec} \approx 0.3\,eV.$