The analytic surgery sequence is a long exact sequence of K-theory groups which combines topological information (the K-homology of manifolds), index theoretic information (the K-theory of group C*-algebras), and secondary index information (the analytic structure group of Higson-Roe). We will see how to give a definition of terms based entirely on algebras of pseudodifferential operators and their K-theory. We use this to systematically develop maps to an exact sequence of non-commutative de Rham homology/cyclic homology. Via pairings with cyclic cohomology classes, this gives rise to new numeric secondary index invariants (higher rho numbers) with explicit formulas and calculation tools due to the compatibility in the whole sequence. We use this for geometric applications. In particular, we derive new information about the moduli space of Riemannian metrics of positive scalar curvature, where we give new lower bounds on the number of its components.