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Ultra slow-turn inflation

Ana Achúcarro, Perseas Christodoulidis, Jinn-Ouk Gong, Oksana Iarygina

Abstract

In standard multi-field models, tachyonic isocurvature perturbations generally indicate the presence of an instability. We revisit the stability of some known counterexamples and show that, in a certain class of models that we call ultra slow-turn, an exponentially decreasing turn rate can shut off this potential instability. We argue that the stability of a given model can be correctly inferred by the total entropy perturbation, even if the effective mass squared of the isocurvature perturbation is negative. Several recent supergravity- or string-inspired models such as fibre inflation, SL(2,$\mathbb{Z}$) attractors and modular inflation fall into the ultra slow-turn class.

Ultra slow-turn inflation

Abstract

In standard multi-field models, tachyonic isocurvature perturbations generally indicate the presence of an instability. We revisit the stability of some known counterexamples and show that, in a certain class of models that we call ultra slow-turn, an exponentially decreasing turn rate can shut off this potential instability. We argue that the stability of a given model can be correctly inferred by the total entropy perturbation, even if the effective mass squared of the isocurvature perturbation is negative. Several recent supergravity- or string-inspired models such as fibre inflation, SL(2,) attractors and modular inflation fall into the ultra slow-turn class.
Paper Structure (9 sections, 40 equations, 4 figures)

This paper contains 9 sections, 40 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Stability in the linear theory
  3. Symmetric models
  4. When is $\mu_{\rm eff}^2<0$ an instability?
  5. Examples
  6. Long-lived instability
  7. Massless total entropy perturbation
  8. Summary and discussion
  9. Notation and useful expressions

Figures (4)

  • Figure 1: Demonstration of the ultra slow-turn regime: evolution of the turn-rate (left) and its logarithmic time derivative (right) as functions of the number of e-folds $N$ for SL(2,$\mathbb{Z}$) model \ref{['eq:VSL2Z']} and \ref{['eq:GSL2Z']}. Solid curves correspond to the full numerical solution, while dashed curves represent the analytical expressions given in \ref{['eq:omega_1']} and \ref{['eq:log_turn_rate']}, with excellent agreement between the two. The parameter values we use are $\alpha=1/3$, $\beta=12^3$, $\phi_0=3.3$, $\chi_0=0.3$, $\phi'_0=1$, $\chi'_0=1/g(\phi_0)$.
  • Figure 2: The two stability conditions (left) from \ref{['eq:stability_conditions']} and the effective isocurvature mass on superhorizon scales $\mu^2_{\rm eff}/H^2$ (right) for the SL(2,$\mathbb{Z}$) model for the same parameters as in Figure \ref{['Fig:OmegaEtaSL2Z']}. Even though $\mu^2_{\rm eff}/H^2<0$, both stability conditions remain positive, implying the stability of the perturbations.
  • Figure 3: The turn-rate $\Omega$ (left) and $\mu^2_{\rm eff}/H^2$ (right) in the modular inflation model as functions of the number of e-folds $N$, shown for zero initial angular velocity $\chi'_0 = 0$ (black dashed) and for $\chi'_0 = 5$ (solid blue and solid green). A nonzero initial angular velocity qualitatively alters the evolution of the turn rate. The effective mass $\mu^2_{\rm eff}/H^2$ shown in the right panel remains largely insensitive to this change due to the extremely small magnitude of the turn rate, except at early times. Consequently, the curves are nearly indistinguishable. We have set $\alpha=1/3$ and $\beta=1/5$, $\phi_0=4.35$, $\chi_0=0.3$ and $\phi'_0=0$.
  • Figure 4: The stability conditions \ref{['eq:stability_conditions']} for the modular inflation model, shown for zero initial angular velocity $\chi'_0 = 0$ (dashed black curves) and $\chi'_0 = 5$ (solid red). Both conditions fail at the end of inflation, $M_s^2/H^2<0$ and $(3-\epsilon+\eta-2\eta_{\Omega})<0$, indicating that the system is unstable.