Production of gravitational waves by inflationary transitions in aligned natural inflation

TL;DR

This work shows that aligned natural inflation with a U(1) gauge coupling generically supports two inflationary stages connected by a rapid transition. The authors analytically estimate the duration of the second phase via a light-direction valley and validate it with a numerical treatment that includes backreaction from gauge-field quanta. They demonstrate that the transition induces a substantial production of gravitational waves with a strongly scale-dependent spectrum peaking at the transition scale, and they find notable parity violation in the GW signal. This GW signature offers a tangible, testable imprint of inflationary transitions in multi-field axion models, with detectability depending on the transition duration and model parameters; however, a complete assessment requires incorporating scalar perturbations beyond the homogeneous-axion approximation.

Abstract

The original axion natural inflation model predicts a tensor-to-scalar ratio exceeding experimental limits. Conversely, in aligned axion inflation, inflation can proceed along trajectories emerging from near a saddle point of the two-field potential and ending through an instability in the orthogonal direction. Such solutions satisfy present observational limits and will be tested by future CMB experiments. Previous studies have suggested the possibility of two distinct inflationary stages separated by a transition characterized by rapid oscillations of the fields. In this work, we demonstrate that the existence of these two stages is a generic feature of the model. We explore a possible phenomenological signature of the transition when a U(1) gauge field is coupled to the axions, namely, the production of gravitational waves (GWs) sourced by gauge quanta generated during the transition. This mechanism produces a feature similar to those seen in spectator axion models or axion inflation with appropriate potentials, i.e. a strongly scale-dependent power spectrum. The scale at which the GW spectrum is produced is determined by the duration of the second inflationary phase. Consequently, the spectrum may peak at different frequencies, potentially detectable by future GW experiments.

Figures (8)

• Figure 1: Contour plots of the number of e-folds obtained in the second phase of inflation as a function of the parameters of the model and for two different values of alignment $\gamma$. The black dashed line is given by $\tilde{n}_2=\tilde{n}_{2,{\rm{max}}}(r_1)$ while the black star on the right panel marks the parameters used to compute the power spectrum in the left panel of fig. (\ref{['fig:power_spectrum']}).
• Figure 2: Field trajectory (left panel) and equation of state as a function of the number of e-folds (right panel) for $n_1 = \frac{70}{M_p^2} ,\, n_2 = \frac{500}{M_p^2} ,\, \gamma = 0.0067 ,\, r_\Lambda = 0.2$. In this figure the fields $\hat{\theta}$, $\hat{\rho}$ are shifted with respect to the original fields so that the saddle point $S_B$ is at the origin in these coordinates. $N=0$ marks the end of the first phase of inflation which is followed by oscillations about a new inflationary valley connected to the minimum ${\cal O}$, and by a second inflationary stage of duration $N_{\rm extra} \simeq 18$ along this valley, and, finally, by oscillations about the minimum. In this figure, the backreaction of the gauge field is neglected.
• Figure 3: Plot of the background behavior of $\xi(N-N_t)$, where $N$ is the number of e-folds and $N_t$ is the value of $N$ at the end of the first inflationary phase. We notice the oscillations caused by the transition between the two inflationary stages lead to the production of both polarizations. This run has been realized with the same set of parameters of fig. (\ref{['fig:traj']}) without including the backreaction of the gauge field on the axions.
• Figure 4: Comparison between $\xi$ in presence (red) and in absence (black) of the gauge field for $n_1 = \frac{70}{M_p^2} ,\, n_2 = \frac{500}{M_p^2} ,\, \gamma = 0.0067 ,\, r_\Lambda = 0.2,\; c_F=1.2,\; c_G=0$. As one can see, the backreaction slows the axions and dumps the oscillations.
• Figure 5: Comparison between the spectral energy density for the gauge field as a function of $(N-N_t)$ for different modes for left (solid) and right (dashed) polarizations. Except for the smallest wavelength modes showed in the figure, which does not contribute much to the GW power spectrum, the energy density for right-handed polarization is at least one order of magnitude smaller than the left-handed one asymptotically. We remind that $k_0$ is the value of momentum that becomes super-horizon at the end of the first phase of inflation.