It is a well known fact that quantum fields on Minkowski spacetime are correlated for each pair of spacetime regions. In Robertson - Walker spacetimes there are spacelike separated regions with disjoint past horizons but the absence of correlations in that case was never proved. We derive in this paper formulae for correlations of quantum fields on Robertson - Walker spacetimes. Such correlations could have reasonably influenced the formation of structure in the early universe. We use methods of algebraic and constructive quantum-field theory.