Why is our physical Universe a de Sitter space? Wikipedia says,

When $n=4$ (3 space dimensions plus time), it is (the de Sitter space) a cosmological model for the physical universe; see de Sitter universe.

It appears to me that the statement means that our Universe described by the FRW metric is really a de Sitter Universe. I'm not sure that I correctly interpret the Wikipedia statement because it isn't obvious$^1$ to me how the FRW spacetime is related to the de Sitter spacetime. Moreover,the de Sitter spacetime arises as the maximally symmetric vacuum solution (i.e., for $T_{\mu\nu}=0$) of Einstein's field equations with a positive cosmological constant $\Lambda>0$ while for our Universe is $T_{\mu\nu}\neq 0$. Maybe this comment refers only to a very early universe dominated by $\Lambda$ that looked like de Sitter. I'm not quite sure what it means to convey.

$^1$ A de Sitter space is a submanifold of the Minkowski space described by the hyperboloid of one sheet $-x_0^2+\textbf{x}^2=\alpha^2$ where $\alpha$ is some nonzero constant having the dimension of length.
 A: 
it isn't obvious to me how the FRW spacetime is related to the de Sitter spacetime.

De Sitter space is the FRW solution in which there is no baryonic or dark matter, only dark energy.

Maybe this comment refers only to a very early universe dominated by Λ that looked like de Sitter.

You have this the wrong way around. In a cosmology with nonzero $\Lambda$, i.e., dark energy, dark energy always dominates at late times. This is because the contribution of dark energy to the stress energy stays the same as expansion continues, whereas contributions from other matter fields fall off like some negative power of the scale factor $a$. The early universe was radiation-dominated, because radiation has an exponent of $-4$, which is the biggest.
Our universe is currently quite well approximated by de Sitter space.
A: If the only contribution to density is dark energy, solving the Friedmann equations obtains $a\propto\exp Ht$ with $H:=\sqrt{\Lambda/3}$. This scale factor corresponds to one of several coordinate representations of a de Sitter space.
