  Given a smooth fibration of closed manifolds and a family of generalised
Dirac operators along the fibers, we define an associated bivariant JLO
cocycle. We then prove that, for any $\ell \geq 0$, our bivariant JLO cocycle
is entire when we endow smoooth functions on the total manifold with the
$C^{\ell+1}$ topology and functions on the base manifold with the $C^\ell$
topology. As a by-product of our theorem, we deduce that the bivariant JLO
cocycle is entire for the Fr\'echet smooth topologies. We then prove that our
JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher
spectral flow.
