Inflection Point Inflation and Time Dependent Potentials in String Theory

TL;DR

This work tackles inflection point inflation (IPI) and its overshoot problem by showing that string theory naturally supports IPI through a modular inflation framework with the radion as inflaton. It constructs a feasible, canonically normalized potential from multiple stringy contributions, notably a triple-exponential form $V(\phi) = a_1 e^{j_1 \alpha \phi} + a_2 e^{j_2 \alpha \phi} + a_3 e^{j_3 \alpha \phi}$ with $\alpha = 1/\sqrt{24}$, yielding an inflection point under specific ratios (e.g., $j_1=12$, $j_2=10$, $j_3=8$ and certain $a_i$), and matches COBE normalization via $L_{inflection} \approx 6.7 a_3^{1/8} \sqrt{N}$. The main novelty is a dynamical, stringy resolution to overshoot: time-dependent potentials from heavy $(0+p)$-branes, with $V_{0+p} = n_{0+p} T_{0+p} L_0^3 L^{p-3}$ and $n_{0+p}(t) \propto a^{-3}(t)$, slow the inflaton sufficiently to generate large $N$, even with generic initial conditions. Numerical examples show that the required initial brane densities are small and decrease with larger $L_{inflection}$, making the mechanism robust in the supergravity regime. Overall, the paper demonstrates that stringy degrees of freedom not only realize IPI but also naturally resolve its principal drawback, the overshoot problem, with broad applicability to string cosmology.

Abstract

We consider models of inflection point inflation. The main drawback of such models is that they suffer from the overshoot problem. Namely the initial condition should be fine tuned to be near the inflection point for the universe to inflate. We show that stringy realizations of inflection point inflation are common and offer a natural resolution to the overshoot problem.

Figures (2)

• Figure 1: A numerical illustration of the overshoot problem: In both cases we take $L_{inflection}=3$ and display $\phi$ as a function of time. In the picture to the left we take $\phi_{initial}=\frac{12}{13} \phi_{inflection}$ and obtain a decent amount of inflation when $\phi$ crosses $\phi_{inflection}\approx 5.38$. In the picture to the right we take $\phi_{initial}=\frac{6}{7} \phi_{inflection}$. Despite the fact that this is fairly close to $\phi_{inflection}$ inflation is not generated.
• Figure 2: A heuristic demonstration of how a time dependent potential, that scales like $1/a^3(t)$, can resolve the overshoot problem of IPI.