  The first main result of this paper is to prove that the convergence of
Lott's delocalized eta invariant holds for all differential operators with a
sufficiently large spectral gap at zero. Furthermore, to each delocalized
cyclic cocycle, we define a higher analogue of Lott's delocalized eta invariant
and prove its convergence when the delocalized cyclic cocycle has at most
exponential growth.
  Our second main result is to obtain an explicit formula of the delocalized
Connes-Chern character of all $C^\ast$-algebraic secondary invariants for word
hyperbolic groups. Equivalently, we give an explicit formula for the pairing
between $C^\ast$-algebraic secondary invariants and delocalized cyclic cocycles
of the group algebra. When the $C^\ast$-algebraic secondary invariant is a
$K$-theoretic higher rho invariant of an invertible differential operator, we
show this pairing is precisely the higher analogue of Lott's delocalized eta
invariant alluded to above. Our work uses Puschnigg's smooth dense subalgebra
for word hyperbolic groups in an essential way. We emphasize that our
construction of the delocalized Connes-Chern character is at $C^\ast$-algebra
$K$-theory level. This is of essential importance for applications to geometry
and topology. As a consequence, we compute the paring between delocalized
cyclic cocycles and $C^\ast$-algebraic Atiyah-Patodi-Singer index classes for
manifolds with boundary, when the fundamental group of the given manifold is
hyperbolic.
