Brane gases in the early universe: thermodynamics and cosmology

TL;DR

This work extends brane-gas cosmology into the early-universe regime within M-theory, combining a brane gas of M2-branes and a supergravity bath on a ten-torus to derive the thermodynamics, including a M-theory Hagedorn temperature $T_H=(2T_2/c_X)^{1/3}$, and to formulate Einstein–Boltzmann dynamics that track brane production and annihilation. The authors derive a Boltzmann equation for brane wrapping numbers and show a freeze-out behavior: brane annihilation becomes inefficient as the universe expands, leaving a possible relic winding density that could hinder decompactification. Numerical simulations reveal that the number of unwrapped dimensions at late times is highly sensitive to initial conditions (volume and anisotropy), rather than being uniquely predicted by the model. Imposing holographic bounds on initial conditions further constrains the viable parameter space, and in the holographically allowed region branes typically annihilate before freeze-out, leading to all dimensions becoming unwrapped and expanding isotropically. Together, these results suggest that, under these assumptions, the brane-gas mechanism does not naturally select a small number of large dimensions, though refinements—such as anisotropic holographic analyses or the string theory limit—could alter this conclusion.

Abstract

We consider the thermodynamic and cosmological properties of brane gases in the early universe. Working in the low energy limit of M-theory we assume the universe is a homogeneous but anisotropic 10-torus containing wrapped 2-branes and a supergravity gas. We describe the thermodynamics of this system and estimate a Hagedorn temperature associated with excitations on the branes. We investigate the cross-section for production of branes from the thermal bath and derive Boltzmann equations governing the number of wrapped branes. A brane gas may lead to decompactification of three spatial dimensions. To investigate this possibility we adopt initial conditions in which we fix the volume of the torus but otherwise assume all states are equally likely. We solve the Einstein-Boltzmann equations numerically, to determine the number of dimensions with no wrapped branes at late times; these unwrapped dimensions are expected to decompactify. Finally we consider holographic bounds on the initial volume, and find that for allowed initial volumes all branes typically annihilate before freeze-out can occur.

Figures (6)

• Figure 1: Probability distribution for the number of unwrapped dimensions at freeze out, for four different choices of the initial volume. The number of unwrapped dimensions is indicated on the horizontal axis. Each histogram is a Monte Carlo based on $10^3$ different sets of initial conditions. Note that it's impossible to have nine unwrapped dimensions, since the wrapping matrix is symmetric.
• Figure 2: Mean number of unwrapped dimensions at freeze-out (y-axis) versus log of the initial volume (x-axis).
• Figure 3: Probability distribution for the number of unwrapped dimensions at freeze out for two different choices of the initial velocity. The initial conditions are all $\dot{\lambda}_i = 0.55$ (left plot) and all $\dot{\lambda}_i = 1$ (right plot). In both plots the initial volume is fixed to $\log{V} = 20$. The plots are Monte Carlos based on $10^3$ different sets of initial conditions. There is relatively weak dependence on the initial velocity, as long as $\dot{\lambda}$ is large enough to start in the Hagedorn phase.
• Figure 4: Mean number of unwrapped dimensions at freeze-out (y-axis) versus initial velocity (x-axis).
• Figure 5: This contour plot shows the mean number of unwrapped dimensions as a function of both the log of the initial volume (y-axis) and the inverse initial velocity $1/\dot{\lambda}_i \equiv 1/H$ (x-axis). For each run the initial $\lambda_i$ are chosen randomly, but the initial $\dot{\lambda}_i$ are all identical. There is little dependence on $1/H$, provided we are in the Hagedorn phase to begin with ($\dot{\lambda} > 0.502$). The darkest shading corresponds to a mean number of unwrapped dimensions less than unity, while the lightest shading corresponds to a mean of 10 (fully unwrapped).