  Whereas preheating after chaotic and hybrid inflation models has been
abundantly studied in the literature, preheating in small field inflation
models, where the curvature of the inflaton potential is negative during
inflation, remains less explored. In these models, a tachyonic instability at
the end of inflation leads to a succession of exponentially large increases and
\emph{decreases} of the inflaton fluctuations as the inflaton condensate
oscillates around the minimum of its potential. The net effect is a competition
between low-momentum modes which grow and decrease significantly, and modes
with higher momenta which grow less but also decrease less. We develop an
analytical description of this process, which is analogous to the quantum
mechanical problem of tunneling through a volcano-shaped potential. Depending
on the parameters, preheating may be so efficient that it completes in less
than one oscillation of the inflaton condensate. Preheating after small field
inflation may also be followed by a long matter-dominated stage before the
universe thermalizes, depending on the energy scale of inflation and the
details of the inflaton interactions. Finally, another feature of these models
is that the spectrum of the inflaton fluctuations at the end of preheating may
be peaked around the Hubble scale. In fact, because preheating starts when the
second slow-roll parameter $|\eta|$ becomes of order unity while the first
slow-roll parameter $\epsilon$ is still much smaller than one, the universe is
still inflating during preheating and the modes amplified by the initial
tachyonic instability leave the Hubble radius. This may lead to an abundant
production of primordial black holes and gravitational waves with frequencies
today which are naturally small enough to fall into the range accessible by
high-sensitivity interferometric experiments.
