Inflection Point Inflation in Supergravity

TL;DR

The paper addresses inflection-point inflation within a minimal SUGRA framework using a single chiral superfield and a canonical Kähler potential. It combines analytic treatment of the inflection-point conditions in the full SUGRA potential with numerical analyses to map the observable predictions across small- and large-field regimes, and extends the model by including a Polonyi SUSY-breaking sector. The results show an extremely small tensor-to-scalar ratio $r$, a running of the spectral index $\alpha$ of order $10^{-3}$, and upper bounds on the inflationary scale $H_{\mathrm{inf}}$ and inflaton mass $m_{\phi}$, with $H_{\mathrm{inf}}$ in the range $10^{10}$–$10^{11}$ GeV and $m_{\phi}$ up to $10^{11}$–$10^{12}$ GeV depending on the inflection-point position $\phi_0$; the Polonyi sector imposes a nontrivial bound linking the SUSY-breaking scale $\mu$ to $H_{\mathrm{inf}}}$, e.g. $\mu<\frac{10.5}{\phi_0}H_{\mathrm{inf}}$ (or $\mu \lesssim 2.3\times 10^{4}(H_{\mathrm{inf}}/{\rm GeV})^{2/3}$ GeV when the Polonyi field sits near the origin), and implies a minimal $H_{\mathrm{inf}}$ for TeV-scale SUSY. Overall, the work demonstrates how inflationary observables and SUSY-breaking physics intertwine in minimal SUGRA constructions, offering testable predictions for future CMB probes and implications for reheating.

Abstract

In this paper, we study the inflection point inflation generated by a polynomial superpotential and a canonical Kähler potential under the supergravity framework, where only one chiral superfield is needed. We find that the special form of the scalar potential limits the inflationary Hubble parameter to values $\lesssim 10^{10}\, \textrm{GeV}$ and the inflaton mass to $\lesssim 10^{11} \, \textrm{GeV}$. We obtain analytic results for small field cases and present numerical results for large field ones. We find the tensor-to-scalar ratio $r<10^{-8}$ is always suppressed in these models, while the running of spectral index $α\approx \mathcal{O}(-10^{-3})$ may be testable in next-generation CMB experiments. We also discuss the possible effects of SUSY breaking Polonyi term presented in the superpotential where we find a general upper bound for the SUSY breaking scale for a given value of the Hubble parameter.

Figures (5)

• Figure 1: Rescaled inflation potential for different choices of the location of the inflection point $\phi_0$. Here $V_0 = V(\phi_0)$ is the value of the potential at the inflection point. Blue, orange, green, and red curves corresponding to $\phi_0=0.1,\,1,\,3$ and $5$, respectively.
• Figure 2: The dependence of the Hubble parameter $H_{\inf}$ during inflation, the inflaton mass $m_\phi$, tensor-to-scalar ratio $r$, and the running of spectral index $\alpha$ on the position $\phi_0$ of the inflection point. Different lines represent different choices of the number of e-folds: $N_\textrm{cmb}=65$ (blue) and $N_\textrm{cmb}=45$ (orange). We fixed $n_s = 0.9659$ and $P_\zeta = 2.1 \times 10^{-9}$.
• Figure 3: Position of the Polonyi field $Z$ during inflation. Different colors represent different choices of the relative SUSY breaking scale $\tilde{\mu}$. When $\tilde{\mu}\gg \phi_0^2$, the Polonyi field stays at $\sqrt{3}-1$, whereas for $\tilde{\mu}\ll \phi_0^2$ the Polonyi field stays close to the origin.
• Figure 4: The dependence of the Hubble parameter during inflation $H_\textrm{inf}$ (top left), the SUSY breaking scale $\mu$ (top right), the tensor-to-scalar ratio $r$ (bottom left), and the running of spectral index $\alpha$ (bottom right) on $\phi_0$. Different lines represent different choices for the number of e-folds: $N_\textrm{cmb}=65$ (blue) and $N_\textrm{cmb}=45$ (orange). We fixed $\tilde{\mu}=0.01$, $n_s = 0.9659$ and $P_\zeta = 2.1 \times 10^{-9}$ in this graph.
• Figure 5: The scale of SUSY breaking $\mu$ vs. the inflationary Hubble scale $H_{\inf}$ on a log-log scale. Different colors represent different choices of relative scale $\tilde{\mu}$. The straight line is the Polonyi field dominated case, where the SUSY breaking scale only depends on the inflection point positions. The right, flat region is an inflaton field dominated region, where the SUSY breaking scale depends linearly on the relative scale $\tilde{\mu}$.