Problem 5.6.1: Gamov, 1953
The paper "Extended Universe and creation of Galactics" by G.A. Gamov was published in 1953.
In that paper Gamov took two numbers - the age of the world and average density of matter in the Universe - and basing on them he determined the third number: the temperature of the relic radiation (cosmic microwave background). Try to repeat this scientific feat of Gamov.
Gamov considered the simplest variant of cosmological dynamics---the inertial regime of the expansion of the Universe: \( R \propto t,\ v = const. \) Density of homogeneous Universe filled by non-relativistic matter is \[\rho _m = \frac{M}{(4/3\pi )R^3 } = \rho _m (t_0 )\left( \frac{t_0}{t} \right)^3. \] Gamov used the following numerical values for the current age of the Universe and matter density respectively: \( t_0 \simeq 3\cdot^10^9\mbox{years}, \rho _m (t_0 ) \simeq 10^{-30} \mbox{g/cm}^3.\) Energy density in the radiation dominated Universe is \[ \rho _r = \frac{3}{32\pi Gt^2 }. \] Then Gamov used the condition $\rho _r (t^*) = \rho _m (t^*)$ to find the moment of time when radiation-dominated epoch changed to matter-dominated one: \( t^* \simeq 73\cdot10^6\ \mbox{years}\) Then he made use of the Stephan-Boltzmann law to determine the temperature at the matching point, i.e. at the moment $t^*$ , \( T(t^*) = 320K.\) Then he only heeded to use the condition $aT = const,$ in order to convert the temperature for the present time to obtain \[ T(t_0 ) = T\left( {t^*} \right)\frac{t^*}{t_0 } \simeq 7K. \] "Is it possible to take two asymptotes, join them and determine the CMB temperature? The answer is, firstly, it is impossible, and secondly, Gamov did it already long ago." (A.D. Chernin)