Using the Friedmann equation (9.1.1)

\[
\frac{d a}{d t}=\sqrt{\frac{8 \pi G u}{3 c^{2}}} a
\]

and the connection between scale factor \(a\) and blackbody temperature \(T\), \(T a=T_{0} a_{0}\), along with (9.3.4b) we get

\[
\frac{d T}{d t}=-\sqrt{\frac{8 \pi G u}{3 c^{2}}} T=-\sqrt{\frac{8 \pi^{3} G g k^{4}}{45 \hbar^{3} c^{5}}} T^{3},
\]

where \(g=43 / 8\) is the effective number of relativistic species from equation (9.3.6b). The solution of the differential equation is

\[
T(t)=T_{0} \sqrt{\frac{t_{0}}{T}}
\]

where

\[
t_{0}=\frac{1}{2} \sqrt{\frac{45 \hbar^{3} c^{5}}{8 \pi^{3} G g\left(k T_{0} \right)^{4}}} \simeq 0.99 \mathrm{~s}
\]

for the case of \(T_{0}=10^{10} \mathrm{~K}\).