Radiation-to-cosmological-constant transition
= Radiation-to-cosmological-constant transition
In a flat radiation-plus-$\Lambda$ universe,
$$
a(t)=\left(\frac{\Omega_{R0}}{1-\Omega_{R0}}\right)^{1/4}
\left[\sinh\!\left(2H_0\sqrt{1-\Omega_{R0}}\,t\right)\right]^{1/2}.
$$
The expansion changes from deceleration to acceleration when the radiation and cosmological-constant terms are equal, at
$$
t_\Lambda=\frac{\operatorname{arsinh}1}
{2H_0\sqrt{1-\Omega_{R0}}}.
$$