During hybrid inflation, the slowly-rolling inflaton field has a significant
coupling to the trigger field which is responsible for most of the potential.
Barring a fine-tuned accidental cancellation, this coupling induces a minimal
one-loop contribution to the inflaton potential. The requirement that this
contribution be not too large constrains a wide class of hybrid inflation
models. Assuming that the inflaton perturbation generates structure in the
Universe, the inflaton field and/or the trigger field after inflation have to
be bigger than 10^9\GeV. This and other results make hybrid inflation at or
below the TeV scale problematical. (There is no problem with hybrid inflation
at the high energy scales normally considered.) `New' and thermal inflation
seem to be viable alternatives for inflation at or below the TeV scale,
including the case that quantum gravity is at the TeV scale. In any case,
supersymmetry is needed required during inflation, in order to protect a scalar
mass.Comment: 15 pages, one ref added in V
