  We study the behavior of the heat kernel of the Hodge Laplacian on a contact
manifold endowed with a family of Riemannian metrics that blow-up the
directions transverse to the contact distribution. We apply this to analyze the
behavior of global spectral invariants such as the eta-invariant and the
determinant of the Laplacian. In particular we prove that contact versions of
the relative eta-invariant and the relative analytic torsion are equal to their
Riemannian analogues and hence topological.
