Late-time dynamics of brane gas cosmology

TL;DR

The paper analyzes the late-time dynamics of brane gas cosmologies by explicitly modeling the decay of winding brane modes into non-winding loops and coupling this to the dilaton-driven background. A key finding is that decay into static loops (equation-of-state parameter $eta=0$) prevents the three large spatial dimensions from growing, whereas allowing the brane gas to be non-static (nonzero effective velocity) with $0<eta eq 0$ enables expansion and can resolve the brane problem. A phase-space analysis identifies attractor lines $l=-eta f$ and shows that maintaining a small string coupling requires $eta o 1/3$ for efficient expansion, while loitering phases naturally arise and help restore causal contact. The work also derives scaling relations for the winding network and demonstrates that, for $eta>0$, the horizon scales with cosmic time, whereas $eta=0$ leads to extended loitering; incorporating a non-static brane gas broadens the parameter space in which a realistic, expanding universe emerges. Overall, the results highlight the crucial role of the loop equation of state and dilaton dynamics in determining whether brane gas cosmologies can reproduce our observed 3+1-dimensional expanding universe, and point to non-static branes as a robust mechanism to overcome the brane problem.

Abstract

Brane gas cosmology is a scenario inspired by string theory which proposes a simple resolution to the initial singularity problem and gives a dynamical explanation for the number of spatial dimensions of our universe. In this work we have studied analytically and numerically the late-time behaviour of these type of cosmologies taking a proper care of the annihilation of winding modes. This has help us to clarify and extend several aspects of their dynamics. We have found that the decay of winding states into non-winding states behaving like a gas of ordinary non-relativistic particles precludes the existence of a late expansion phase of the universe and obstructs the growth of three large spatial dimensions as we observe today. We propose a generic solution to this problem by considering the dynamics of a gas of non-static branes. We have also obtained a simple criterion on the initial conditions to ensure the small string coupling approximation along the whole dynamical evolution, and consequently, the consistency of an effective low-energy description. Finally, we have reexamined the general conditions for a loitering period in the evolution of the universe which could serve as a mechanism to resolve the {\sl brane problem} - a problem equivalent to the {\sl domain wall problem} in standard cosmology - and discussed the scaling properties of a self-interacting network of winding modes taking into account the effects of the dilaton dynamics.

Figures (5)

• Figure 1: Phase space for $(f,l)$. In the dark grey area ($f^2-3l^2<0$) the total energy of the matter sources is negative and therefore it is excluded from the dynamical analysis. The light grey dotted wedge, defined by the lines $l=-f/3$ and $l=-f/\sqrt{3}$, is a region where the smallness of the string coupling cannot be guaranteed. We have plotted the numerical solutions of the equations of motion for several values of the physical parameters $c$, the efficiency of the winding mode decay, and $\gamma$, the parameter characterising the equation of state of the loops created. The dashed lines correspond to solutions with $\gamma=0$ whereas the dark continuous lines to solutions with $\gamma=1/3$. In both cases $c$ takes values $(0.1,1.0,10)$ from bottom to top. For comparison we have also included the solution corresponding to no winding mode decay $c=0$ (light continuous line).
• Figure 2: Hubble parameter $l=\dot\lambda$ as a function of cosmic time. The plotted curves represent solutions of the equations of motion with parameters $c$ and $\gamma$ chosen in the same manner as in Fig. \ref{['fig:phase_space']}.
• Figure 3: Scale factor as a function of cosmic time. The plotted curves represent solutions of the equations of motion with parameters $c$ and $\gamma$ chosen in the same manner as in Fig. \ref{['fig:phase_space']}.
• Figure 4: Energy profiles for $\gamma = 0$ (top) and $\gamma = 1/3$ (bottom). In both plots the dark lines correspond to $c=1.0$, the light dashed lines to $c=0.1$, and the light continuous line to $c=0$, that is, the case without winding mode decay (Recall that in this particular case $E_l=0$ and then the total energy is equal to $E_w$). The curves that asymptotically approach the zero axis represent the energy of winding modes and the curves starting at zero the energy of the loops produced. The curves tracked at late times by the loop energy is the total energy of the system $E_l+E_w$.
• Figure 5: Characteristic length of the winding mode network as a function of cosmic time. This graph shows the scaling behaviour of the decay of winding modes. The plotted curves represent solutions of the equations of motion with parameters $c$ and $\gamma$ chosen in the same manner as in Fig. \ref{['fig:phase_space']}.