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Radiation conservation givesConsequentlyEvaluating at $t_0$, where $a=1$, shows thatPutThen $\dot{b}=2\sqrt{A+B{b}^2}$. Choosing the big bang as $t=0$ and integrating givesThereforewhere

At early times, $\sinh (\beta t)\sim \beta t$, so $a\propto t^{1/2}$, as in a radiation-dominated universe. At late times,the expected de Sitter expansion with $H\to \sqrt{\Lambda c^2/3}$.

Acceleration begins when the radiation deceleration and cosmological-constant acceleration balance:Thus $a=\alpha$, which in the exact solution means $\sinh (\beta t_{\Lambda })=1$. HenceThis is the radiation-to-cosmological-constant transition.