  We determine the rational homology of the space of long knots in R^d for
$d\geq4$. Our main result is that the Vassiliev spectral sequence computing
this rational homology collapses at the E^1 page. As a corollary we get that
the homology of long knots (modulo immersions) is the Hochschild homology of
the Poisson algebras operad with a bracket of degree d-1, which can be obtained
as the homology of an explicit graph complex and is in theory completely
computable.
  Our proof is a combination of a relative version of Kontsevich's formality of
the little d-disks operad and of Sinha's cosimplicial model for the space of
long knots arising from Goodwillie-Weiss embedding calculus. As another
ingredient in our proof, we introduce a generalization of a construction that
associates a cosimplicial object to a multiplicative operad. Along the way we
also establish some results about the Bousfield-Kan spectral sequences of a
truncated cosimplicial space.
