A Cosmological Mechanism for Stabilizing Moduli

Abstract

In this paper, we show how the generic coupling of moduli to the kinetic energy of ordinary matter fields results in a cosmological mechanism that influences the evolution and stability of moduli. As an example, we reconsider the problem of stabilizing the dilaton in a non-perturbative potential induced by gaugino condensates. A well-known difficulty is that the potential is so steep that the dilaton field tends to overrun the correct minimum and to evolve to an observationally unacceptable vacuum. We show that the dilaton coupling to the thermal energy of matter fields produces a natural mechanism for gently relaxing the dilaton field into the correct minimum of the potential without fine-tuning of initial conditions. The same mechanism is potentially relevant for stabilizing other moduli fields.

Figures (3)

• Figure 1: A schematic of the racetrack potential for the dilaton $\Phi= {\rm exp}( \lambda \phi)$, generated by gaugino condensates ($\lambda$ is a constant). This is represented by the solid curve. The desired minimum at $\Phi=\Phi_{min}$ is separated by a small barrier, peaked at $\Phi=\Phi_p$. Beyond $\Phi=\Phi_p$ (around $\Phi= 2.05$ in this example), there is an unacceptable anti-de Sitter vacuum. (The energy scale has been blown up by more than $60$ orders of magnitude to make the barrier visible.) The dashed line represents $V_{eff}$, the effective potential for $\Phi$ stemming from the dilaton coupling $f(\Phi)= g(\Phi) = 1/\Phi$ at temperature $T=T_i$. As $T$ decreases from $T_1$ to $T_2$ to zero, this contribution adiabatically decreases. The dotted line represents the total finite temperature potential for $\Phi$, $V_{T_i}$, which has a minimum at $\Phi=\Phi_{T_i}$.
• Figure 2: The evolution of the various energy densities for the case of dilaton coupling $f = g =1/\Phi$. $T_{RH}$ is the reheat temperature after inflation. The initial value of $\Phi$ was chosen to be $\Phi=10 \gg \Phi_{min}$. The figure shows how the zero point ($\rho_{zp}$), oscillation ($\rho_{osc}$), and perturbation ($\delta\rho$) energy densities evolve. In particular, note that, although the system begins with $\rho_{osc} \sim \rho_{zp}$, the oscillations are heavily damped after a few e-folds, leading to $\rho_{osc} \ll \rho_{zp}$. Furthermore, note that $\delta\rho$ (the contribution of inhomogeneity in all fields to the energy density) decays at the same rate as $\rho_{zp}$, so inhomogeneity in the universe does not come to dominate.
• Figure 3: A schematic illustration of initial phase space volume. The relative likelihood of an initial $\Phi$ is represented by the vertical distance between the curves bounding the shaded region. Naively, as shown in (a), all combinations of initial $1 \le \Phi \le \infty$ and $0 \le A \le 2 \pi m_{pl} /\alpha$ might appear equally probable, and the allowed volume of the shaded region is infinite. However, based on the arguments of Horne and Moore, the effective volume of moduli space is defined by the Kähler metric and is finite, as illustrated in (b). The initial conditions used in Fig. 2 are marked by "X."