Inflation from a generalized exponential plateau: towards extra suppressed tensor-to-scalar ratios

TL;DR

This work tackles the challenge of achieving ultra-small $r$ in single-field inflation without non-minimal couplings. It introduces a minimally coupled scalar with a three-parameter potential featuring a generalized exponential plateau, connecting plateau flattening to a steeper quadratic minimum. Using slow-roll analysis and numerical results, it demonstrates $r$ values below $10^{-5}$ while yielding $n_s \approx 0.965$ and $A_s \approx 2\times10^{-9}$, with reheating governed by oscillations about the quadratic minimum. In the Jordan frame, the potential maps to higher-order corrections to Starobinsky $f(R)$ gravity, offering a gravity-motivated mechanism for tensor suppression and connecting scalar-field and modified gravity perspectives.

Abstract

We investigate a standard minimally-coupled scalar-field inflationary scenario, which is based on a new potential, with suitably generalized plateau features, that leads to extra small tensor-to-scalar ratios. In particular, we consider a specific three-parameter potential, which has a flatter plateau and a steeper well compared to the Starobinsky potential in the Einstein frame. We study the inflationary realization and we show that it guarantees a prolonged period of slow-roll inflation and a successful exit. Additionally, the steeper minimum leads to significantly suppressed tensor perturbations, and thus to an extra-small tensor-to-scalar ratio $r$, and we show that we are able to obtain $r$ values less than $10^{-5}$. Moreover, we calculate the reheating temperature showing that in order to be in agreement with observations one of the potential parameters should remain within specific bounds. Finally, performing an inverse conformal transformation to the Jordan frame we show that the considered potential corresponds to higher-order corrections to Starobinsky potential in the Einstein frame, and these corrections are the reason for the improved behavior of the tensor-to-scalar ratio.

Figures (4)

• Figure 1: The potential (\ref{['potential']}) with generalized exponential plateau proposed in this work, for $\alpha = \gamma = 1$, $\beta = 1.7\times 10^{-7}$, and $V_0=2 \times 10^{60}GeV^4$.
• Figure 2: The predictions of the inflationary scenario based on the potential (\ref{['potential']}) with generalized exponential plateau on the $n_{\mathrm{s}}-r$ plane. The left set of curves corresponds to e-folding number $N_*=50$, for $\gamma = 0.1$ (blue dashed curve), $\gamma = 1$ (black thick curve), and $\gamma = 2$ (red curve). The right set of curves corresponds to e-folding number $N_*=60$, for $\gamma = 0.1$ (green dashed curve), $\gamma = 1$ (black curve), and $\gamma = 2$ (magenta curve). In all curves $\beta = 1.7\times 10^{-7}$, $V_0$ is set around $V_0=2 \times 10^{60}GeV^4$ in order to obtain $A_s = 2 \times 10^{-9}$, and $\alpha$ runs from $0.1$ to $10$ .
• Figure 3: The predictions of Fig. \ref{['rns']} on top of the 1$\sigma$ and 2$\sigma$ Planck 2018 TT,TE,EE+lowE+lensing +BK15+BAO results Planck:2018jri (note that at the scale of the figure all curves degenerate at a point very close to the horizontal axis).
• Figure 4: The potential $V(\phi) = V_0 \left( 1-e^{\frac{-\alpha \phi^2}{\beta M_{Pl}^2 +\gamma \phi^2}}\right)$ with generalized exponential plateau, for $\alpha = \gamma = 1$, $\beta = 1.7$ and $V_0=2 \times 10^{66}GeV^4$ (blue curve), alongside Starobinsky potential $V_S(\phi)=V_{S0} \left(1 - e^{-\sqrt{\frac{2}{3}} \frac{\phi}{M_{\text{Pl}}}} \right)^2$ for $V_{S0} = 1.66\times 10^{66}GeV^4$ (yellow curve).