Natura Non Facit Saltum: An Analytical Model of Smooth Slow-Roll to Ultra-Slow-Roll Transition

Abstract

In this letter, we propose a single-field inflation model that realizes a slow-roll-to-ultra-slow-roll transition while keeping the second slow-roll parameter smoothly varying throughout. The model is built through a minimal modification by introducing a simple time dependence in the effective mass term of the Mukhanov-Sasaki equation. We obtain fully analytical solutions for both the background evolution and the curvature perturbations, which makes the parameter dependence of the curvature power spectrum easy to track. To the best of our knowledge, this is the first analytical model that describes a smooth transition of this kind. We also compare its signatures with those of the corresponding sharp-transition counterpart.

Figures (5)

• Figure 1: The evolution of $\epsilon_2(N)$ with different choices of the parameter $\alpha$, while we fix the combination $\left(\mu^2-9/4\right)/q^2=0.09$. $\nu^2(N)-9/4$ with $\alpha=22.63$ and $\mu=2.0294$ is shown as the red dashed line.
• Figure 2: Top: The evolution of the leftmost slow-roll parameter $\epsilon_1(N)$. $\textit{Bottom}$: The inflaton potential $V(\phi)$ of our model. The first dashed line (the origin of the horizontal axis) marks the time $\tau_\star$ and the other one indicates when $\epsilon_2$ reaches value $-6$. By setting $\epsilon_2=-6$ and $\mathrm{d}\epsilon_2 /\mathrm{d} N=0$ in Eq. \ref{['evol_epsilon2_N']}, the $e$-fold number corresponding to $\epsilon_2=-6$ is given as $N=-\ln[\alpha/(\alpha-\mu^2+9/4)-1]$. The parameters used in these plots are $\alpha=22.63$ and $\mu=2.0294$ and $\epsilon_{\text{SR1}}=10^{-6}$.
• Figure 3: Comparison of the analytically derived power spectrum (red line) with the approximation formulae in different regions (yellow for the IR region, green and blue for the growth region, purple for the UV region), as well as with the numerical results (gray dots). The dip and the peak location are shown by the black dashed lines. The parameters are set to $\tau_{\star}=-1$, $\alpha=22.63$ and $\mu=2.0294$.
• Figure 4: The comparison of $\epsilon_2(N)$ between our smooth model (green line) and the transient model (purple line). For the smooth one we take $\alpha=22.63$ and $\mu=2.0294$, while the transient one is specified by Eq. \ref{['eq:sharpmodel']}. The shaded regions correspond to the areas where the models enter the non-attractor phase.
• Figure 5: The comparison of power spectra between our smooth model (green line) with $\tau_{\star}=-1$, $\alpha=22.63$ and $\mu=2.0294$, and the transient model (purple line) given by Eq. \ref{['eq:sharpmodel']}. Horizontal dashed lines denotes the dip and peak positions, while vertical dashed lines show the amplitude of peaks.