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 $(n-1)$-Poisson operad $Pois_{n-1}^c$, which represents the cohomology of the operad of little $n$-discs $D_n$. We assume $n\geq 3$ in all cases. We explicitly prove that a Hopf cooperad in cochain graded dg-modules $A$ is weakly-equivalent (quasi-isomorphic) to $Pois_{n-1}^c$ as a Hopf cooperad as soon as we have an isomorphism at the cohomology level $H^*(A)\simeq Pois_{n-1}^c$ when $4\nmid n$. We just need the extra assumption that $A$ 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 $P$ which satisfies the relation $H^*(P,\mathbb{Q})\simeq Pois_{n-1}^c$ 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 $L G_{\bullet}(Pois_{n-1}^c)$ which we determine from the $(n-1)$-Poisson cooperad $Pois_{n-1}^c$. We also prove that the morphisms $\iota: D_m\rightarrow D_n$, 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 $n\geq 3$.
