Ghost instabilities of cosmological models with vector fields nonminimally coupled to the curvature

TL;DR

The paper demonstrates that cosmological models with nonminimally coupled vector fields to curvature generically harbor ghosts in the longitudinal polarization, due to the sign of the effective mass term $M^2 = -\frac{R}{6} + m^2$; it shows that these ghosts induce both UV ill-definition and linear instabilities when kinetic-term eigenvalues of the perturbation action cross zero. By constructing the quadratic action and analyzing the kinetic matrix $K$ in both cases of vanishing and nonvanishing vector vevs, the authors reveal ghosts across Turner-Widrow-like magnetic-field generation, vector inflation, and vector curvaton scenarios, with linearized solutions diverging at the crossovers where $\det K=0$. They provide explicit results in simplified setups (vector with no vev, vector with a small vev plus a cosmological constant, and vector-flavored inflation models) showing that the singular behavior coincides with horizon-crossing events and mass vanishing points, signaling a fundamental instability of the perturbative spectrum. The work concludes that these theories cannot be consistently UV-completed without modifying the vector sector (e.g., restoring gauge invariance or using alternate couplings $I(\phi) F^{\mu\nu}F_{\mu\nu}$), and that any robust phenomenology from such models must be re-evaluated in light of the ghost and singularity issues.

Abstract

We prove that many cosmological models characterized by vectors nonminimally coupled to the curvature (such as the Turner-Widrow mechanism for the production of magnetic fields during inflation, and models of vector inflation or vector curvaton) contain ghosts. The ghosts are associated with the longitudinal vector polarization present in these models, and are found from studying the sign of the eigenvalues of the kinetic matrix for the physical perturbations. Ghosts introduce two main problems: (1) they make the theories ill-defined at the quantum level in the high energy/sub horizon regime (and create serious problems for finding a well behaved UV completion); (2) they create an instability already at the linearized level. This happens because the eigenvalue corresponding to the ghost crosses zero during the cosmological evolution. At this point the linearized equations for the perturbations become singular (we show that this happens for all the models mentioned above). We explicitly solve the equations in the simplest cases of a vector without vev in a FRW geometry, and of a vector with vev plus a cosmological constant, and we show that indeed the solutions of the linearized equations diverge when these equations become singular.

Figures (9)

• Figure 1: Behavior of the linearized solutions representing the longitudinal mode. The system remains finite at the point $t_{\omega i}$, when the frequency vanishes, but it diverges at $t_M$, when the mass term $M^2$ of the vector vanishes. See the main text for the parameters chosen.
• Figure 2: Results from a numerical simulation with $m^2 = 0.1 \, H_0 \, B_{\rm in} = 0.1 ,\, p_{L,{\rm in}} = 400 H_{\rm in} ,\, p_{T,{\rm in}} = 900 H_{\rm in} ,\,$ (corresponding to $H/p \simeq 10^{-3}$; the modes are initially in the adiabatic vacuum; only the final part of the evolution is shown in the two figures). Since $H/p$ grows during inflation, we use this quantity as "time variable" in the Figure. Left panel: gauge invariant relativistic gravitational potential $\hat{\Phi}$. We show the real part in units of ${\hat{\Psi}}_{\rm in}$. We also show the eigenvalue $\lambda_1$ of the kinetic matrix (multiplied by $3 \times 10^5$, so that it is visible in the figure). We see that ${\hat{\Phi}}$ diverges when $\lambda_1 = 0 \,$. Right panel: real parts of the modes ${\hat{\alpha}}_0$ (in units of $100 \, H_0 \, {\hat{\Psi}}_{\rm in}$) and ${\hat{\alpha}}$ (in units of ${\hat{\Psi}}_{\rm in}$) . Also these modes (as all the modes of the system) diverge when $\lambda_1 = 0 \,$.
• Figure 3: Evolution of the eigenvalues of the kinetic matrix. The parameters and initial conditions are as in Figure \ref{['fig:plots']}. Due to the wide range spanned by the eigenvalues, in the $y$ axis, we have used a linear scale inside the interval $\left[ - 0.01 ,\, 0.01 \right]$, and a logarithmic scale outside.
• Figure 4: Determinant of the kinetic matrix, for the same choice of parameters and initial conditions as in the previous Figure. Compared with the previous figure, we show a close up of early times, around the point where the determinant vanishes. The black dashed curve shows the exact determinant, while the red curve shows the approximate expression given in eq. (\ref{['detapprx']}).
• Figure 5: Results from a numerical simulation starting at $t=0$ from $\phi = 16$ (providing about $60$ e-folds of inflationary expansion), $B_1 = 0.1$ (providing a $\sim B_1^2 \simeq 1 \%$ anisotropy). More precisely, we have considered a massive inflaton potential, with the inflaton mass equal to $m \,$. Left panel: inflaton (in units of $M_p$), hubble parameters (in units of $m$), and dimensionless rescaled vector$B_1$. The anisotropic rate $h$ and the vector are rescaled so that they are visible in the figure. Right panel: determinant of the kinetic matrix of the perturbations, for modes with initial momenta $p_{L,{\rm in}} = 100 H_{\rm in} ,\, p_{T,{\rm in}} = 200 H_{\rm in} ,\,$. The red curve shows the exact determinant, while the green points show the approximate expression given in eq. (\ref{['detapprx-2']}). The determinant vanishes at the time $m \, t \simeq 0.16$.