The correlation functions C(phi) and C(phi)2, associated with the order-parameter field phi over arrow pointing right(r,t) and its square, respectively, are discussed using heuristic arguments and an approximate analytical approach. Topological defects (walls, strings, monopoles) in the field, seeded by a quench from the high- to the low-temperature phase, lead to singular short-distance behavior in the scaling functions, and power-law tails in the corresponding structure factors. For superfluid helium, the structure factor S(phi)2(k,t) is measurable in principle using small-angle scattering (whereas S(phi) is inaccessible). It is predicted to exhibit a power-law tail, approximately [a4/L(t)2](lnka)2/k, where L(t) is the characteristic scale at time t after the quench and a is the core size of a vortex line. Correlation functions for the defect density are also discussed.
