Trajectories with suppressed tensor-to-scalar ratio in Aligned Natural Inflation

TL;DR

Aligned Natural Inflation uses a two-axion potential with strong alignment to produce a light inflaton direction whose effective single-field description depends on four parameters (Λ, f_φ, r_f, r_Λ). The paper identifies two classes of inflationary trajectories: valleys connected to minima, yielding Natural-Inflation-like predictions with r in the 10^−4–10^−2 range, and saddle-point plateau trajectories disconnected from minima that give substantially smaller r, with analytic relations clarifying the ns–r behavior. By analyzing the 2-field valley/crest geometry and validating the 1-field reduction, the work shows that sub-Planckian axion scales can still generate viable inflation with potentially observable gauge-field effects due to enhanced couplings in the aligned setup. These results offer a mechanism to reconcile small tensor modes with Planck data and motivate further study of multi-field axion inflation and its UV completions.

Abstract

In Aligned Natural Inflation, an alignment between different potential terms produces an inflaton excursion greater than the axion scales in the potential. We show that, starting from a general potential of two axions with two aligned potential terms, the effective theory for the resulting light direction is characterized by four parameters: an effective potential scale, an effective axion constant, and two extra parameters (related to ratios of the axion scales and the potential scales in the $2-$field theory). For all choices of these extra parameters, the model can support inflation along valleys (in the $2-$field space) that end in minima of the potential. This leads to a phenomenology similar to that of single field Natural Inflation. For a significant range of the extra two parameters, the model possesses also higher altitude inflationary trajectories passing through saddle points of the $2-$field potential, and disconnected from any minimum. These plateaus end when the heavier direction becomes unstable, and therefore all of inflation takes place close to the saddle point, where - due to the higher altitude - the potential is flatter (smaller $ε$ parameter). As a consequence, a tensor-to-scalar ratio $r = {\rm O } \left( 10^{-4} - 10^{-2} \right)$ can be easily achieved in the allowed $n_s$ region, well within the latest $1 σ$ CMB contours.

Figures (10)

• Figure 1: Contour plot of the potential as a function of the rescaled fields ${\tilde{\phi}}$ (horizontal direction) and ${\tilde{\psi}}$ (vertical direction). Due to the different rescaling for the $2-$fields, the potential exhibits comparable curvature in both directions. However, for $\vert \alpha \vert \ll 1$, the field $\psi$ is significantly heavier than $\phi$, and the evolution proceeds along trajectories where $\frac{\partial V}{\partial {\tilde{\psi}}} = 0$
• Figure 2: Contour plot of the potential centered in the domain at the right of the origin. The $x-$axis is ${\tilde{\phi}} = \frac{\phi}{f_\phi}$, while the $y-$axis is ${\tilde{\psi}} = \frac{\psi}{f_\psi}$, see eq. (\ref{['tilde-fields']}). The green (magenta) curves are stable valleys (unstable crests) within the domain that pass through the origin, the maximum, and / or the saddle points ${\cal S}_{A} ,\, {\cal S}_B$. The plots are for $r_f=1.5$ and for different values of $r_\Lambda$ in the characteristic regions: $r_\Lambda \leq \frac{1}{r_f^4}$ (top-left); $\frac{1}{r_f^4} < r_\Lambda < \frac{1}{r_f^2}$ (top-right); $r_\Lambda = \frac{1}{r_f^2}$ (bottom-left); $r_\Lambda > \frac{1}{r_f^2}$ (bottom-right).
• Figure 3: Contour plot of (\ref{['V2']}), for parameters in the $\frac{1}{r_f^4} < r_\Lambda < \frac{1}{r_f^2}$ region, together with valleys (green) and crests (magenta). The two red curves are two distinct inflationary trajectories in this model. They are obtained from a numerical evolution of the exact model (\ref{['V-start']}). Both evolutions shown contain $60$ e-folds of inflation plus a brief transient moment after inflation in which the system reaches the minimum.
• Figure 4: Potential along the two valleys shown in Figure \ref{['fig:traj+evol']}.
• Figure 5: Predictions of Aligned Natural Inflation (with inflation along a trajectory connected to a minimum) in the $\left\{ n_s - r \right\}$ plane, confronted with the $1\sigma$ and $2 \sigma$ Planck contour lines (specifically, we choose to plot the more conservative red contour lines of Figure 12 of Ade:2015lrj). We fixed $r_f = 1.5$ and varied $r_\Lambda$ as follows: the dashed lines, from bottom to top, are for $r_\Lambda = 0.01, 0.07, 0.1, 0.19$; the solid lines, from top to bottom, are for $r_\Lambda = 0.3, 0.4144, 0.5, 1, 3$. The lowest theoretical curve, drawn as a dotted line, is for Natural Inflation. All the theoretical curves are done for $N=60$ e-folds of inflation.