Thermal Equilibrium of String Gas in Hagedorn Universe

TL;DR

This paper investigates how introducing dilaton potentials in a Hagedorn universe can achieve thermal equilibrium, potentially resolving a key obstacle in the Brandenberger-Vafa mechanism for explaining our universe's dimensionality.

Contribution

It explores minimal modifications with dilaton potentials to realize thermal equilibrium in the Hagedorn universe, addressing a critical flaw in previous models.

Findings

Thermal equilibrium can be achieved with exponential and double-well dilaton potentials.

Solutions include universe radius stabilization and growth, with dilaton stabilization at potential minima.

Initial phase thermal equilibrium is possible in these modified setups.

Abstract

The thermal equilibrium of string gas is necessary to activate the Brandenberger-Vafa mechanism, which makes our observed 4-dimensional universe enlarge. Nevertheless, the thermal equilibrium is not realized in the original setup, a problem that remains as a critical defect. We study thermal equilibrium in the Hagedorn universe, and explore possibilities for avoiding the issue aforementioned flaw. We employ a minimal modification of the original setup, introducing a dilaton potential. Two types of potential are investigated: exponential and double-well potentials. For the first type, the basic evolutions of universe and dilaton are such that both the radius of the universe and the dilaton asymptotically grow in over a short time, or that the radius converges to a constant value while the dilaton rolls down toward the weak coupling limit. For the second type, in addition to the above solutions, there is another solution in which the dilaton is stabilized at a minimum of potential and the radius grows in proportion to $t$. Thermal equilibrium is realized for both cases during the initial phase. These simple setups provide possible resolutions of the difficulty.