Dynamical Systems and Superstring Phases in the Early Universe

TL;DR

The paper investigates how a volume modulus rolling on an exponential potential interacts with a population of fundamental string loops whose tension decreases with the modulus, during the post-inflation epoch. Using a dynamical-systems framework, it derives fixed points for zero and nonzero radiation, focusing on LVS with $\lambda=3$ and a canonical case with $\lambda=2$, and shows that a String Loop Tracker with $\Omega_{loops}=3/4$ (for $\beta=1/2$) acts as a global attractor in LVS, while radiation can induce transient tracker phases. The work reveals that a loop-dominated epoch is a natural late-time attractor in LVS, stabilized by the time-dependent tension, and remains compatible with eventual modulus stabilization and a Hot Big Bang. It also outlines potential gravitational-wave signatures and future extensions to spatial inhomogeneities and loop interactions. These findings illuminate a novel post-inflationary cosmological phase where string loops play a leading energetic role.

Abstract

We study the string theory dynamics of the volume scalar rolling down an exponential potential during the period between inflation and reheating, in a background of cosmic superstring loops. In the context of the LVS potential, we demonstrate the existence of a novel string loop attractor tracker solution, in which 75% of the energy density of the universe is in the form of a gas of fundamental cosmic superstring loops (a configuration preferred over the standard radiation tracker). On this tracker, it is the continual reduction in the string tension as the volume scalar evolves that makes the loops stable against decay. For more general non-LVS potentials, mixed radiation-loop trackers can also occur.

Figures (6)

• Figure 1: Evolution in phase space towards the mixed tracker labeled as point D, for the case of fundamental strings in a potential of the form $V\sim m_s^4$ , i.e. $\beta=1/2$ and $\lambda=2$. Points $A_\pm$ correspond to kination and point B to the string loop tracker
• Figure 2: Evolution in phase space towards the attractor corresponding to the string loop tracker, labeled as point B for the case of fundamental strings in LVS ($\beta=1/2$ and $\lambda=3$). Points $A_\pm$ correspond to kination.
• Figure 3: Evolution of a initially kinating background, with a small initial $\Omega_{\rm loops}^i=10^{-2}$, towards the string loop tracker for $\lambda =3$ and $\beta =1/2$, plotted in terms of the number of e-foldings $N$.
• Figure 4: Evolution in phase space towards a loop-radiation tracker (red line, labeled F) for fundamental strings in a potential of the form $V\sim m_s^4$, i.e. $\beta=1/2$ and $\lambda =2$. This tracker corresponds to a line rather than a point because there is a flat direction in phase space.
• Figure 5: Evolution in phase space towards the loop tracker attractor (B) through a transient radiation tracker (E) for fundamental strings in LVS (with $\beta=1/2$ and $\lambda =3$).