  From four-dimensional $N=4$ matter-coupled gauged supergravity, we study
smooth time-dependent cosmological solutions interpolating between a
$dS_2\times \Sigma_2$ spacetime, with $\Sigma_2 = S^2$ and $H^2$, in the
infinite past and a $dS_4$ spacetime in the infinite future. The solutions were
obtained by solving the second-order equations of motion from all the ten
gauged theories known to admit $dS_4$ solutions, of which there are two types.
Type I $dS$ gauged theories can admit both $dS$ solutions as well as
supersymmetric $AdS$ solutions while type II $dS$ gauged theories only admit
$dS$ solutions. We also study the extent to which the first-order equations
that solve the aforementioned second-order field equations fail to admit the
$dS_4$ vacua and their associated cosmological solutions.
