  We show that in hybrid inflation it is possible to generate large
second-order perturbations in the cosmic microwave background due to the
instability of the tachyonic field during preheating. We carefully calculate
this effect from the tachyon contribution to the gauge-invariant curvature
perturbation, clarifying some confusion in the literature concerning nonlocal
terms in the tachyon curvature perturbation; we show explicitly that such terms
are absent. We quantitatively compute the nongaussianity generated by the
tachyon field during the preheating phase and translate the experimental
constraints on the nonlinearity parameter f_{NL} into constraints on the
parameters of the model. We also show that nonscale-invariant second-order
perturbations from the tachyon field can become larger than the
inflaton-generated first-order perturbations, leading to stronger constraints
than those coming from nongaussianity. The width of the excluded region in
terms of the logarithm of the dimensionless coupling g, grows linearly with the
log of the ratio of the Planck mass to the tachyon VEV, \log(M_p/v); hence very
large regions are ruled out if the inflationary scale v is small. We apply
these results to string-theoretic brane-antibrane inflation, and find a
stringent upper bound on the string coupling, g_s < 10^{-4.5}.
