  Gromov proved a quadratic decay inequality of scalar curvature for a class of
complete manifolds. In this paper, we prove that for any uniformly contractible
manifold with finite asymptotic dimension, its scalar curvature decays to zero
at a rate depending only on the contractibility radius of the manifold and the
diameter control of the asymptotic dimension. We construct examples of
uniformly contractible manifolds with finite asymptotic dimension whose scalar
curvature functions decay arbitrarily slowly. This shows that our result is the
best possible. We prove our result by studying the index pairing between Dirac
operators and compactly supported vector bundles with Lipschitz control. A key
technical ingredient for the proof of our main result is a Lipschitz control
for the topological $K$-theory of finite dimensional simplicial complexes.
