  A comprehensive, five-dimensional calculation of Higgs-boson production in
gluon fusion is performed for both the minimal and the custodially protected
Randall-Sundrum (RS) model, with Standard Model fields propagating in the bulk
and the scalar sector confined on or near the IR brane. For the first time, an
exact expression for the gg->h amplitude in terms of the five-dimensional
fermion propagator is derived, which includes the full dependence on the
Higgs-boson mass. Various results in the literature are reconciled and shown to
correspond to different incarnations of the RS model, in which the Higgs field
is either localized on the IR brane or is described in terms of a narrow bulk
state. The results in the two scenarios differ in a qualitative way: the gg->h
amplitude is suppressed in models where the scalar sector is localized on the
IR brane, while it tends to be enhanced in bulk Higgs models. In both cases,
effects of higher-dimensional operators contributing to the gg->h amplitude at
tree level are shown to be numerically suppressed under reasonable assumptions.
There is no smooth cross-over between the two scenarios, since the effective
field-theory description breaks down in the transition region. A detailed
phenomenological analysis of Higgs production in various RS scenarios is
presented, and for each scenario the regions of parameter space already
excluded by LHC data are derived.
