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Instability of the ACW model, and problems with massive vectors during inflation

Burak Himmetoglu, Carlo R. Contaldi, Marco Peloso

TL;DR

The paper analyzes the stability of the Ackerman–Carroll–Wise (ACW) anisotropic inflation model, showing that the background with a fixed-norm vector is linearly unstable due to a ghost mode arising near horizon crossing. By performing a complete linearized perturbation analysis and decomposing perturbations into gauge-invariant 2d scalar and vector sectors, the authors demonstrate that Fourier modes diverge when a specific momentum–Hubble combination is crossed, with the divergence associated to a vanishing/negative kinetic term. The quadratic action confirms a ghost via a kinetic matrix whose determinant changes sign at the same horizon-crossing moment; a simplified vector-only calculation corroborates the result. The instability persists in related models with nonminimal curvature couplings, suggesting a generic ghost problem for longitudinal vector modes in these frameworks, and raises serious questions about UV completion and the viability of vector-driven anisotropic inflation for cosmology.

Abstract

We prove that the anisotropic inflationary background of the Ackerman-Carroll-Wise model, characterized by a fixed-norm vector field, is unstable. We found the instability by explicitly solving the linearized equations for the most general set of perturbations around this background, and by noticing that the solutions diverge close to horizon crossing. This happens because one perturbation becomes a ghost at that moment. A simplified computation, with only the perturbations of the vector field included, shows the same instability, clarifying the origin of the problem. We then discuss several other models, with a particular emphasis on the case of a nonminimal coupling to the curvature, in which vector fields are used either to support an anisotropic expansion, or to generate cosmological perturbations on an isotropic background. In many cases, the mass term of the vector needs to have the ``wrong'' sign; we show that, as a consequence, the longitudinal vector mode is a ghost (a field with negative kinetic term, and negative energy; not simply a tachyon). We comment on problems that arise at the quantum level. In particular, the presence of a ghost can be a serious difficulty for the UV completion that such models require in the sub-horizon regime.

Instability of the ACW model, and problems with massive vectors during inflation

TL;DR

The paper analyzes the stability of the Ackerman–Carroll–Wise (ACW) anisotropic inflation model, showing that the background with a fixed-norm vector is linearly unstable due to a ghost mode arising near horizon crossing. By performing a complete linearized perturbation analysis and decomposing perturbations into gauge-invariant 2d scalar and vector sectors, the authors demonstrate that Fourier modes diverge when a specific momentum–Hubble combination is crossed, with the divergence associated to a vanishing/negative kinetic term. The quadratic action confirms a ghost via a kinetic matrix whose determinant changes sign at the same horizon-crossing moment; a simplified vector-only calculation corroborates the result. The instability persists in related models with nonminimal curvature couplings, suggesting a generic ghost problem for longitudinal vector modes in these frameworks, and raises serious questions about UV completion and the viability of vector-driven anisotropic inflation for cosmology.

Abstract

We prove that the anisotropic inflationary background of the Ackerman-Carroll-Wise model, characterized by a fixed-norm vector field, is unstable. We found the instability by explicitly solving the linearized equations for the most general set of perturbations around this background, and by noticing that the solutions diverge close to horizon crossing. This happens because one perturbation becomes a ghost at that moment. A simplified computation, with only the perturbations of the vector field included, shows the same instability, clarifying the origin of the problem. We then discuss several other models, with a particular emphasis on the case of a nonminimal coupling to the curvature, in which vector fields are used either to support an anisotropic expansion, or to generate cosmological perturbations on an isotropic background. In many cases, the mass term of the vector needs to have the ``wrong'' sign; we show that, as a consequence, the longitudinal vector mode is a ghost (a field with negative kinetic term, and negative energy; not simply a tachyon). We comment on problems that arise at the quantum level. In particular, the presence of a ghost can be a serious difficulty for the UV completion that such models require in the sub-horizon regime.

Paper Structure

This paper contains 10 sections, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Instability from the linearized equations
  3. The ACW model
  4. Classification of the perturbations and gauge invariant combinations
  5. Solutions to the linearized equations for the perturbations
  6. Ghost from the quadratic equation
  7. Simplified computation
  8. Appearance of a ghost in other theories
  9. Discussion
  10. Explicit expressions for the computation of Subsection \ref{['subsect-linsol']}

Figures (2)

  • Figure 1: The right panel shows a zoom of the left panel in the region $-0.6 \leq H_a \, t \leq - 10^{-5 } \,$ (the time is shown in log units). Both modes show a 1/t divergence.
  • Figure 2: The right panel shows a zoom of the left panel in the region $-1 \leq H_a \, t \leq - 2.5 \cdot10^{-3 } \,$ (the time is shown in log units). Both modes blow up when the system (\ref{['appacw:maxwell-2dS']}), becomes singular, $\alpha \propto {\rm log } t$ and $\alpha_0 \propto 1/t \,$, in agreement with the analytic approximate solution (\ref{['appacw:apprx-soln-2dS']}). Notice also that $\alpha_0 \approx \dot{\alpha}$ close to the singularity, in agreement with the analytic computation.