  We establish that $E_n$-operads satisfy a rational intrinsic formality
theorem for $n\geq 3$. We gain our results in the category of Hopf cooperads in
cochain graded dg-modules which defines a model for the rational homotopy of
operads in spaces. We consider, in this context, the dual cooperad of the
$n$-Poisson operad $\mathsf{Pois}_n^c$, which represents the cohomology of the
operad of little $n$-discs $\mathsf{D}_n$. We assume $n\geq 3$. We explicitly
prove that a Hopf cooperad in cochain graded dg-modules $\mathsf{K}$ is
weakly-equivalent (quasi-isomorphic) to $\mathsf{Pois}_n^c$ as a Hopf cooperad
as soon as we have an isomorphism at the cohomology level
$H^*(\mathsf{K})\simeq\mathsf{Pois}_n^c$ when $4\nmid n$. We just need the
extra assumption that $\mathsf{K}$ is equipped with an involutive isomorphism
mimicking the action of a hyperplane reflection on the little $n$-discs operad
in order to extend this formality statement in the case $4\mid n$. We deduce
from these results that any operad in simplicial sets $\mathsf{P}$ which
satisfies the relation $H^*(\mathsf{P},\mathbb{Q})\simeq\mathsf{Pois}_n^c$ in
rational cohomology (and an analogue of our extra involution requirement in the
case $4\mid n$) is rationally weakly equivalent to an operad in simplicial sets
$LG_{\bullet}(\mathsf{Pois}_n^c)$ which we determine from the $n$-Poisson
cooperad $\mathsf{Pois}_n^c$. We also prove that the morphisms $\iota:
\mathsf{D}_m\rightarrow\mathsf{D}_n$, which link the little discs operads
together, are rationally formal as soon as $n-m\geq 2$. These results enable us
to retrieve the (real) formality theorems of Kontsevich by a new approach, and
to sort out the question of the existence of formality quasi-isomorphisms
defined over the rationals (and not only over the reals) in the case of the
little discs operads of dimension $n\geq 3$.
