Perturbations in Chromo-Natural Inflation

TL;DR

Chromo-Natural Inflation posits slow-roll inflation driven by magnetic-drift from an axion–non-Abelian gauge-field CS coupling. The authors provide a comprehensive linear perturbation analysis, uncovering a decoupled scalar perturbation channel and a parity-violating, chiral tensor sector that sources gravitational waves with potentially large chirality. They show that, in the parameter regions where the scalar tilt matches Planck data, the chiral tensor amplitude is too large, while regions with modest tensor power yield an excessively red scalar spectrum, placing the model in tension with observations. The work also develops a reduced, magnetic-drift EFT in the heavy gauge-field limit with $c_s^2=1/3$ and discusses connections to Gauge-flation, suggesting variants (e.g., more gauge fields or multiple axions) that might salvage viable phenomenology while preserving distinctive observational signatures like chiral gravitational waves.

Abstract

Chromo-Natural Inflation is the first worked example of a model of inflation in which slow-roll inflation is achieved by "magnetic drift" as opposed to Hubble friction. In this work, we give an account of the perturbations at linear order in this model. Our analysis uncovers two novel phenomena. First, the amplitude of scalar curvature perturbations is not directly tied to the shape of the inflationary potential. This allows the theory to violate naïve formulations of the Lyth bound. Second, the tensor sector of the theory is significantly altered from the usual case: the non-Abelian gauge field perturbations have a tensor degree of freedom. One chirality of the this tensor can be exponentially enhanced by a temporary instability near horizon crossing; this chiral instability exists because of the classical gauge field background, which violates parity. These tensor fluctuations of the gauge field also couple to gravitational waves at linear order in perturbation theory and source a chiral spectrum of gravitational waves. This spectrum can be exponentially enhanced over the usual inflationary spectrum due to the instability in the gauge sector. These new features cause the theory in its present form to be in significant tension with current observational data. This is because the new scalar physics leads to a significant reddening of the spectral tilt in the same region of parameter space where the exponential enhancement of the gravitational wave amplitude is small enough to satisfy current constraints on the tensor-to-scalar index. Hence, the model either predicts a spectral tilt that is too red, or it overproduces gravitational waves, or both.

Figures (13)

• Figure 1: The evolution of the amplitudes of the axion, $|\hat{ X}|$ (black), and gauge scalar perturbations, $|\hat{z}|$ (red) and $|\delta \hat{\phi}|$ (blue), both allowing the background to evolve (solid lines) and with the background fixed at its horizon crossing values (dashed lines). Although there are small differences between the two cases, the dynamics, especially at late times, is almost entirely captured by taking the background to be fixed at its horizon crossing values for a given $k$ mode. The model plotted has parameters $g=0.001$, $\mu=0.01$, $\lambda=2000$, $f=0.2$, $m_\psi \simeq 2.6$ and ${\boldsymbol\Lambda}\simeq 350$. The transition times between different behaviors mentioned in the text are marked with vertical dashed lines and labelled. Note that in making this plot we have made use of the insights given in Sec. \ref{['sec:WKB']} to initialize the mode functions in such a way as to isolate the 'slow' magnetic drift mode of the system, which has slowly evolving amplitudes.
• Figure 2: Example of fast mode scaling: Numerical evaluation of the solution for the axion, with generic initial conditions, showing the Real part of ${\hat{X}}(x)/\sqrt{x}$ (black solid curve), ${\hat{X}}(x/10)/\sqrt{x/10}$ (blue dashed curve), ${\hat{X}}(x/100)/\sqrt{x/100}$ (red dot-dashed curve). The amplitudes and the frequencies match, apart from an overall drift due to the slow mode oscillation (the red curve shows the start of the growth in the axion just after horizon crossing). The parameters used here are $\{{\boldsymbol\Lambda}=350,m_{\psi}=2.6\}$.
• Figure 3: The gray shaded plot shows the numerical solution for the axion over the range $0.25<x<100$ with generic initial conditions; the plot is dense due to the rapid oscillation of the fast mode. A running average reveals that the slow magnetic drift mode has approximately constant frequency, and amplitude decreasing linearly in conformal time, down to $x\sim m_{\psi}$.
• Figure 4: Comparison of numerical integration of the full dynamics (solid lines) and reduced dynamics (dashed lines). We show the absolute values of the fields, with boundary conditions set by the WKB solution for the slow mode in the far past, and the relative amplitudes fixed by e.g. Eqn. $(\ref{['fudgefactor']})$; parameters are chosen so that ${\boldsymbol\Lambda}=350$, $m_{\psi}=2.6$. The value of ${\hat{X}}$ in the reduced system is determined from Eqn. $(\ref{['xsoln']})$. ${\hat{\delta\!\phi}}$ has been multiplied by 10, and ${\hat{z}}$ by 100, so that the curves don't lie on top of one another. The agreement persists until $x\gtrsim {\boldsymbol\Lambda}m_{\psi}$, where the terms dropped in the reduced system contribute significantly to the dynamics. The difference between the light and dark solid lines is that the dark solid lines represent solutions that omit, and the light solid lines solutions that include, the contribution to the dynamics coming from the axion's mass term, $m_\mathcal{X}^2 = V"(\mathcal{X}/f)$. In the reduced system, the gauge perturbations do not have a late-time growing solution. Their late time growth as $1/x$ is entirely due to their coupling to the axion when it has a mass term; this term is dropped in the reduced dynamics. As we show, the deviation between the two solutions commences around $x \simeq \sqrt{2} m_\mathcal{X}/ H{\boldsymbol\Lambda}$.
• Figure 5: In this figure, we show the late-time value of ${\boldsymbol\varphi}$ obtained from exact numerical integration of the reduced system as a function of $(m_{\psi}-\sqrt{2})$. We take initial conditions that match the unit amplitude WKB slow mode.