Cosmological stabilization of moduli with steep potentials

TL;DR

Problem: the cosmological overshoot problem for stabilizing moduli with steep $V(\phi)$ in string theory; Method: propose that a radiation-like source with $\rho_{rad}\propto a^{-4}$ eventually dominates the energy budget and provides cosmic friction to dissipate $KE=\tfrac{1}{2}\dot\phi^2$, enabling stabilization; Analytic four-epoch framework plus numerical verification using a KKLT-inspired potential $V(\sigma)$ with $\sigma= e^{\sqrt{2/3}\phi}$; Findings: finite windows of initial conditions lead to bound stabilization, and radiation can widen the viable region; Significance: broadens viable moduli stabilization in the outer region and supports potential inflationary scenarios with flat barriers.

Abstract

A scenario which overcomes the well-known cosmological overshoot problem associated with stabilizing moduli with steep potentials in string theory is proposed. Our proposal relies on the fact that moduli potentials are very steep and that generically their kinetic energy quickly becomes dominant. However, moduli kinetic energy red-shifts faster than other sources when the universe expands. So, if any additional sources are present, even in very small amounts, they will inevitably become dominant. We show that in this case cosmic friction allows the dissipation of the large amount of moduli kinetic energy that is required for the field to be able to find an extremely shallow minimum. We present the idea using analytic methods and verify with some numerical examples.

Figures (5)

• Figure 1: A typical moduli potential. The region of the shallow minimum had to be magnified by 26 orders of magnitude so that it can be seen. The vertical axis is in units of $M_p^4$, and the horizontal axis is in units of $M_p$.
• Figure 2: A bound case. Shown in the top panel are fractional densities of potential energy (blue), kinetic energy (red) and radiation energy (green) as a function of time. KE becomes dominant and then the radiation. Shown in the bottom panel is the evolution of the field as a function of time ending in the shallow minimum of the potential.
• Figure 3: A bound case. Shown in the top panel are the various energy densities (color coded as in Fig. \ref{['F2']}) as a function of the scalar field position. Shown in the bottom panel is the potential and the starting position of the field. Note that the scale is logarithmic and that the difference in potential energy between the starting point and the minimum is about 30 orders of magnitude!
• Figure 4: An unbound case: various energy densities (color coded as in Fig. \ref{['F2']}) as a function of time. The only difference from the bound case is where the field ends up at the end of the radiation dominated epoch. In this case it lands to the right of the shallow minimum and continues to run on the potential in a "tracking" solution.
• Figure 5: Window of allowed initial conditions for bound solutions. Shown in the top panel are the various energy densities as a function of scalar field position (color coded as in Fig. \ref{['F2']}) for two bound solutions with two different initial conditions, depicted by solid and dashed curves. In the lower panel we show the potential as a function of the scalar field. The green dots on the potential are the end points of the two regions of initial conditions that lead to bound solutions. One of the regions is around the minimum, and the other is way up on the potential.