Particle Production of Vector Fields: Scale Invariance is Attractive

TL;DR

The paper investigates a massive Abelian vector field whose gauge kinetic function and mass are modulated by the inflaton during inflation, showing that scale-invariant spectra for all vector components can appear as dynamical attractors. Using a dynamical-systems approach with expansion-normalised variables, it identifies SSR and vector scaling solutions (VSS) as the principal late-time attractors, and derives conditions under which scale invariance and mild anisotropy coexist with sustained inflation. A key result is that the scalings $f\propto a^{-4}$ and $m\propto a$ emerge as attractors (with a specific link $f'/f=-4 m'/m$) and yield backreaction that flattens the effective potential, enabling slow-roll even in supergravity contexts. The work covers massless, light, and heavy vector regimes, finds multiple VSS branches (including VSS$_1$, VSS$_2$, VSS$_3$ for the light case, and VSS for the heavy case), and demonstrates these attractors in concrete models (constant-parameter, chaotic, and supergravity-inspired potentials). The results imply potential observational signatures from statistical anisotropy in the curvature perturbation and offer a mechanism to mitigate the $\eta$-problem in inflationary model-building.

Abstract

In a model of an Abelian vector boson with a Maxwell kinetic term and non-negative mass-squared it is demonstrated that, under fairly general conditions during inflation, a scale-invariant spectrum of perturbations for the components of a vector field, massive or not, whose kinetic function (and mass) is modulated by the inflaton field is an attractor solution. If the field is massless, or if it remains light until the end of inflation, this attractor solution also generates anisotropic stress, which can render inflation weakly anisotropic. The above two characteristics of the attractor solution can source (independently or combined together) significant statistical anisotropy in the curvature perturbation, which may well be observable in the near future.

Figures (13)

• Figure 1: This numerical plot demonstrates the validity of our approximation in neglecting terms proportional to $\mu$ when considering a light vector field. The plot shows a numerical solution for the evolution of the kinetic function scaling $\frac{1}{f}\frac{\,\text{d} f}{\,\text{d} \alpha}=2x\Gamma$, given by Eq. (\ref{['scalingSol1']}), with respect to the number of elapsed e-folds $N$, for the particular case where the kinetic function, mass and scalar potential are exponential. The numerical solution (in blue-thick line) to the full system of equations, Eqs. (\ref{['fullsystem']})-(\ref{['fullsystem-m']}), is indistinguishable from the numerical solution (in red-thin line) where the terms proportional to $\mu$ have been removed until the vector field becomes heavy ($N\gtrsim 20$ in the plot). The plot is also an example of how the vector scaling solution $f_{\textrm{att}}\propto e^{-4\alpha}$ is attained in the light and heavy vector field case. Note that the attractors take only a handful of e-folds to be reached, corresponding to the oscillating behaviour for $N\sim5$ (light field) and $N\sim20$ (heavy field). More examples are shown in Sec. \ref{['examples']}.
• Figure 2: This numerical plot demonstrates the flow between the three vector scaling solutions found in the light vector field case, for the particular example where the kinetic function, mass and scalar potential are exponential. The plot shows how different initial conditions can give rise to solutions approaching different vector scaling critical points, depending on the relative magnitudes of $z$ and $s$. The model parameters are chosen so that the $\mathcal{VSS}_{\it3}$ is the late-time attractor. If a solution does not directly fall on this critical point, the small but positive eigenvalues $m_4$ for the $\mathcal{VSS}_{\it{1,2}}$ in Eqs. (\ref{['VSS1-eigen']}) and (\ref{['VSS2-eigen']}), and the small but negative eigenvalue $m_4$ for the $\mathcal{VSS}_{\it3}$ in Eq. (\ref{['VSS3-eigen']}), drive very weak flow of solutions along the arc (shown in red) towards the $\mathcal{VSS}_{\it3}$. However, throughout the flows, the solution $x\simeq-2/\Gamma$ is conserved and therefore so is the scaling solution attractor $f_{\textrm{att}}\propto a^{-4}$. The figure also depicts the $\mathcal{SSR}$ critical point which is an unstable saddle point for the parameter space considered.
• Figure 3: This numerical plot shows the evolution of the dimensionless variables $z$ and $s$ as the vector field becomes heavy. The plot demonstrates the validity of our approximation $\left\langle z^{2}\right\rangle =\left\langle s^{2}\right\rangle$, used when considering a heavy vector field.
• Figure 4: This numerical plot shows an example of the evolution of a solution for a massive vector field. As the backreaction is initially subdominant the $\mathcal{SSR}$ critical point is reached, where the anisotropy $\Sigma$ vanishes. Solutions may remain there for a long period of time until the backreaction becomes important. Solutions will then move to the $\mathcal{VSS}$ where $\Sigma\neq0$ and the attractor solution $x_{\textrm{att}}=-2/\Gamma$ is obtained. The vector field may then become heavy. As it does so the anisotropy once again vanishes but the solution $x_{\textrm{att}}=-2/\Gamma$ is conserved. This is clearly demonstrated by the plots in Fig. \ref{['fig-PhaseFlow-x-y-SSR-VSS-light-heavy']}.
• Figure 5: These plots show a projection of Fig. \ref{['fig-VSS-light-heavy-3d']} in the $x-y$ plane. They show in increasing detail how the attractor solution $x_{\textrm{att}}=-2/\Gamma$ is conserved as the solution jumps from the light to the heavy vector scaling solution.