Fast-Roll Inflation

We show that in the simplest theories of spontaneous symmetry breaking onecan have a stage of a fast-roll inflation. In this regime the standardslow-roll condition |m^2| << H^2 is violated. Nevertheless, this stage can berather long if |m| is sufficiently small. Fast-roll inflation can be useful forgenerating proper initial conditions for the subsequent stage of slow-rollinflation in the very early universe. It may also be responsible for thepresent stage of accelerated expansion of the universe. We also make twoobservations of a more general nature. First of all, the universe after a longstage of inflation (either slow-roll or fast-roll) cannot reach anti-de Sitterregime even if the cosmological constant is negative. Secondly, the theorieswith the potentials with a "stable" minimum at V(\phi)<0 in the cosmologicalbackground exhibit the same instability as the theories with potentialsunbounded from below. This instability leads to the development of singularitywith the properties practically independent of V(\phi). However, thedevelopment of the instability in some cases may be so slow that the theorieswith the potentials unbounded from below can describe the present stage ofcosmic acceleration even if this acceleration occurs due to the fast-rollinflation.Comment: 21 pages, 4 figures, JHEP3, a discussion of initial conditions for the fast-roll inflation is extende