A spectral sequence for string cohomology

Let X be a 1-connected space with free-loop space ΛX. We introduce two spectral sequences converging towards H*(ΛX;Z/p) and H*((ΛX)hT;Z/p). The E2-terms are certain non-Abelian-derived functors applied to H*(X;Z/p). When H*(X;Z/p) is a polynomial algebra, the spectral sequences collapse for more or less trivial reasons. If X is a sphere it is a surprising fact that the spectral sequences collapse for p=2.