Ultra Slow-Roll Inflation and the non-Gaussianity Consistency Relation

TL;DR

Ultra slow-roll inflation defines a one-parameter family of models with the background dynamics $\ddot{\phi}=nH\dot{\phi}$ that can yield a scale-invariant power spectrum while producing a local bispectrum with $f_{\rm NL}=\frac{5}{4}(3-n_{\mathrm{S}})$. The analysis shows USR is dynamically unstable and generally cannot sustain $60$ e-folds without extreme initial-condition fine-tuning, while end-of-inflation normalization forces an unrealistically small mass scale $M$ in the potential. Despite the potential for order-one non-Gaussianity, the combination of instability, normalization, and quantum-diffusion concerns renders USR an unlikely counterexample to the Maldacena consistency relation. Thus, USR offers a peculiar but not physically generic scenario for testing non-Gaussianity and the standard consistency conditions. The work derives the power spectrum and non-Gaussianity across the one-parameter class and discusses observational implications and theoretical hurdles.

Abstract

Ultra slow-roll inflation has recently been used to challenge the non-Gaussianity consistency relation. We show that this inflationary scenario belongs to a one parameter class of models and we study its properties and observational predictions. We demonstrate that the power spectrum remains scale-invariant and that the bi-spectrum is of the local type with fnl=5(3-ns)/4 which, indeed, represents a modification of the consistency relation. However, we also show that the system is unstable and suffers from many physical problems among which is the difficulty to correctly WMAP normalize the model. We conclude that ultra slow-roll inflation remains a very peculiar case, the physical relevance of which is probably not sufficient to call into question the validity of the consistency relation.

Figures (6)

• Figure 1: Left panel: new family of ultra slow-roll potentials for different values of the parameter $n$. Right panel: classical ultra slow-roll and slow-roll trajectories for $n=-3.01$ (solid green line and dotted blue line) and $n=-2.99$ (dashed pink line and dotted dashed red line). The initial condition for the scalar field is chosen to be $\phi_{\rm ini}=0.1\, \phi_{\rm lim}$.
• Figure 2: Left panel: exact (numerical) evolution of the field (solid green line) compared to the ultra slow-roll solution (dashed blue line) and to the slow-roll solution (dotted red line). The parameter $n$ is taken to be $n=-2.99$ and the initial condition is the same as in Fig. \ref{['fig:potential']} (right panel), namely $\phi_{\rm ini}=0.1\, \phi_{\rm lim}$. The inset shows the global evolution of the system on a larger time scale. Right panel: same as left panel but for $n=-3.01$.
• Figure 3: Top left panel: numerical (exact) evolution of the first horizon-flow parameter $\epsilon_1$ (solid green line) compared to its ultra-slow-roll behavior (dashed blue line) for $n=-2.99$ and $\phi_{\rm ini}=0.1\, \phi_{\rm lim}$. Top right panel: same as top left panel but for the choice $n=-3.01$. Bottom left panel: Evolution of the quantity $\delta\equiv \ddot{\phi}/(H\dot{\phi})$ for $n=-2.99$ and $\phi_{\rm ini}=0.1\, \phi_{\rm lim}$. Bottom right panel: same as bottom left panel but with $n=-3.01$.
• Figure 4: Number of e-folds at which the ultra slow-roll solution is left as a function of the initial value of the field. The exact numerical result (solid black line) is in excellent agreement with the analytical estimate of Eq. (\ref{['eq:Ndev']}) (dashed red curve).
• Figure 5: Spectral index versus parameter $n$ for the new family of potentials.