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Fast-Roll Inflation

Andrei Linde

TL;DR

This work expands the inflationary paradigm by exploring fast-roll inflation, a regime where $|m^2|$ is of order $H^2$ and slow-roll conditions are violated yet inflation can persist if $|m|$ is small. By analyzing a tachyonic potential $V(\phi)=V_0-\tfrac{m^2}{2}\phi^2$ in a flat FRW background, it derives the growth rate through $F(m^2/H^2)$ and shows how the total expansion factor $e^{Ht_*}$ scales with initial and threshold field values, allowing substantial expansion even in the fast-roll regime. The paper then discusses broad implications, including impossibility of AdS domination after inflation, instability of potentials with $V(\phi)\le0$, and the potential for fast-roll stages to seed proper initial conditions for slow-roll, generate adiabatic and isocurvature perturbations, or address moduli and gravitino problems, with caution about perturbation amplitudes. Overall, it argues that fast-roll inflation can supplement or, in some scenarios, precede slow-roll inflation and may contribute to present-day acceleration, inviting further exploration of its cosmological consequences and constraints.

Abstract

We show that in the simplest theories of spontaneous symmetry breaking one can have a stage of a fast-roll inflation. In this regime the standard slow-roll condition |m^2| << H^2 is violated. Nevertheless, this stage can be rather long if |m| is sufficiently small. Fast-roll inflation can be useful for generating proper initial conditions for the subsequent stage of slow-roll inflation in the very early universe. It may also be responsible for the present stage of accelerated expansion of the universe. We also make two observations of a more general nature. First of all, the universe after a long stage of inflation (either slow-roll or fast-roll) cannot reach anti-de Sitter regime even if the cosmological constant is negative. Secondly, the theories with the potentials with a "stable" minimum at V(φ)<0 in the cosmological background exhibit the same instability as the theories with potentials unbounded from below. This instability leads to the development of singularity with the properties practically independent of V(φ). However, the development of the instability in some cases may be so slow that the theories with the potentials unbounded from below can describe the present stage of cosmic acceleration even if this acceleration occurs due to the fast-roll inflation.

Fast-Roll Inflation

TL;DR

This work expands the inflationary paradigm by exploring fast-roll inflation, a regime where is of order and slow-roll conditions are violated yet inflation can persist if is small. By analyzing a tachyonic potential in a flat FRW background, it derives the growth rate through and shows how the total expansion factor scales with initial and threshold field values, allowing substantial expansion even in the fast-roll regime. The paper then discusses broad implications, including impossibility of AdS domination after inflation, instability of potentials with , and the potential for fast-roll stages to seed proper initial conditions for slow-roll, generate adiabatic and isocurvature perturbations, or address moduli and gravitino problems, with caution about perturbation amplitudes. Overall, it argues that fast-roll inflation can supplement or, in some scenarios, precede slow-roll inflation and may contribute to present-day acceleration, inviting further exploration of its cosmological consequences and constraints.

Abstract

We show that in the simplest theories of spontaneous symmetry breaking one can have a stage of a fast-roll inflation. In this regime the standard slow-roll condition |m^2| << H^2 is violated. Nevertheless, this stage can be rather long if |m| is sufficiently small. Fast-roll inflation can be useful for generating proper initial conditions for the subsequent stage of slow-roll inflation in the very early universe. It may also be responsible for the present stage of accelerated expansion of the universe. We also make two observations of a more general nature. First of all, the universe after a long stage of inflation (either slow-roll or fast-roll) cannot reach anti-de Sitter regime even if the cosmological constant is negative. Secondly, the theories with the potentials with a "stable" minimum at V(φ)<0 in the cosmological background exhibit the same instability as the theories with potentials unbounded from below. This instability leads to the development of singularity with the properties practically independent of V(φ). However, the development of the instability in some cases may be so slow that the theories with the potentials unbounded from below can describe the present stage of cosmic acceleration even if this acceleration occurs due to the fast-roll inflation.

Paper Structure

This paper contains 9 sections, 36 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Dynamics of spontaneous symmetry breaking and fast-roll inflation
  3. Possible implications of the fast-roll inflation
  4. Fast-roll inflation and non-eternal quintessence
  5. Long stage of inflation cannot lead to the universe dominated by $\Lambda < 0$.
  6. The potentials with minima at $V(\phi)<0$ lead to a singular behavior of the field $\phi$, just like the potentials unbounded from below.
  7. Can we use the fast-roll inflation to solve the moduli and gravitino problems?
  8. Adiabatic and isocurvature perturbations from the fast-roll inflation
  9. Conclusions

Figures (2)

  • Figure 1: Evolution of the scalar field and the scale factor in the model with $V(\phi) = V_0 - {m^2\phi^2\over 2}$.
  • Figure 2: Evolution of the scalar field and the scale factor in the model with $V(\phi) = {\lambda\over 4} (\phi^2-v^2)^2 + \Lambda$, with $\Lambda <0$.