Anisotropic Inflation from Charged Scalar Fields

TL;DR

This work investigates anisotropic inflation driven by a charged scalar field coupled to a background $U(1)$ gauge field with a time-dependent gauge kinetic function $f(\phi)$. The authors analyze three models—symmetry breaking hilltop inflation, charged hybrid inflation, and chaotic inflation—showing that an attractor regime emerges in which the anisotropy parameters become comparable to slow-roll parameters, with the gauge field energy fractions $R_1$ and $R_2$ following distinct evolutions. A key finding is that $R_1$ tracks the slow-roll parameter $\epsilon$ and the anisotropy $\delta$ approaches $\epsilon$ (or $\delta \simeq 2R_1/3$ in certain regimes), while $R_2$ grows near the end, triggering a final short stage with rapid gauge-field oscillations that ends inflation. In charged hybrid inflation, the gauge-field coupling can significantly modify the waterfall transition and has potential implications for tachyonic preheating and non-Gaussianities, motivating further study of cosmological perturbations in these backgrounds.

Abstract

We consider models of inflation with U(1) gauge fields and charged scalar fields including symmetry breaking potential, chaotic inflation and hybrid inflation. We show that there exist attractor solutions where the anisotropies produced during inflation becomes comparable to the slow-roll parameters. In the models where the inflaton field is a charged scalar field the gauge field becomes highly oscillatory at the end of inflation ending inflation quickly. Furthermore, in charged hybrid inflation the onset of waterfall phase transition at the end of inflation is affected significantly by the evolution of the background gauge field. Rapid oscillations of the gauge field and its coupling to inflaton can have interesting effects on preheating and non-Gaussianities.

Figures (6)

• Figure 1: Here we have plotted $\eta$ defined in Eq. (\ref{['epsilon-eta']}), with $\lambda= 2.5 \times 10^{-13}$, $M= 3\times 10^{-6} M_P$, $p=50$, $\rho_{in} = M_p/5$ and ${\bf{e}}=1$. The first phase change, at e-folding $\alpha\simeq 10$, happens when $R_1$ becomes comparable to $\epsilon$. The second phase change happens very close to the end of inflation (in about one e-folding towards the end of inflation) when $R_2$ also becomes comparable to $\epsilon$.
• Figure 2: Here we plot our analytical solution for $\rho(\alpha)$, Eq. (\ref{['hat-rho1']}), shown by the red dashed curve, and compare it to the full numerical solution denoted by the solid black curve. The agreement between them is very good. The left figure corresponds to ${\bf{e}}=1$ whereas for the right figure ${\bf{e}}=10^{-4}$. As argued, the time of first phase change, which here is at $\alpha_1 \simeq 10$, is independent of the value of ${\bf{e}}$ and is well approximated by our analytical formula Eq. (\ref{['alpha1']}). All other parameters are as in Fig.\ref{['eta-fig']}.
• Figure 3: Here we plot the evolution of the gauge field. The left graph represents $\ln A$ where the red dashed-dotted curve is our analytical solution Eq. (\ref{['A3-Bes']}) whereas the solid black curve is the full numerical solution. The agreement between our analytical solution Eq. (\ref{['A3-Bes']}) valid for the second and third phase and the full numerical result is good. Also the change of the slope of $\ln A$ form the first phase to the second phase is in good agreement with our other analytical result, Eq. (\ref{['A-prime2']}), valid for the first two phases. The right graph represents $A$ during the last few e-foldings. The oscillatory behavior suggested by Eq. (\ref{['A3-Bes']}) is clearly seen. The start of the third phase, corresponding to the first peak is well approximated by our analytical estimation Eq. (\ref{['alpha2']}). All parameters here are as in Fig.\ref{['eta-fig']}.
• Figure 4: In left figure we plot $\ln (R_1/\epsilon)$ ( upper solid green curve) and $\ln( R_2/\epsilon)$ (lower dashed-dotted red curve). The phase change takes place at $\alpha_1 \simeq 10$ followed by the attractor regime denoted by the almost horizontal line where $R_1 \propto \epsilon$. As explained in the text, $R_2$ is very small compared to $R_1$ until the very end of inflation when they become comparable and inflation ends shortly after that. Right: $\ln \delta$ is presented. The attractor behavior during the second inflationary stage is clear. All parameters here are as in Fig.\ref{['eta-fig']}.
• Figure 5: Here we plot the evolution of $\rho(\alpha)$ in chaotic inflation. The dashed-dotted red curve is the analytical solution whereas the solid black curve is the full numerical result. As can be seen, the agreement between them is very good. The left figure corresponds to ${\bf{e}}=0.1$ whereas for the right figure ${\bf{e}}=0$. As explained below, the position of the first kink is independent of the value of ${\bf{e}}$ and is well approximated by Eq. (\ref{['alpha1-chaotic']}). However, the total number of e-foldings depends logarithmically on ${\bf{e}}$. Other parameters are $m=10^{-6} M_P$, $\rho_{in} = 11.2 M_P$ and $c=2.5$.