Interaction Rates in String Gas Cosmology

TL;DR

The paper reconsiders Brandenberger–Vafa string-gas cosmology in a fully compact geometry, demonstrating that the interaction rate of strings in the pressureless, string-radius phase is far smaller than the expansion rate, thus invalidating the assumption of thermal equilibrium ($\Gamma \ll H$). It develops an effective time-dependent quantum-mechanical framework for winding annihilation, grounded in tree-level string amplitudes with finite widths, to handle energy nonconservation in an expanding background. Numerical results corroborate the analytic conclusion of rapid freeze-out, and the work highlights that achieving the BV mechanism likely requires significant modifications (e.g., dilaton potentials, brane gases, or stochastic dynamics). Overall, the study provides both non-equilibrium insights and a practical calculational approach for BV-style cosmologies, with implications for how many dimensions may become macroscopic.

Abstract

We study string interaction rates in the Brandenberger-Vafa scenario, the very early universe cosmology of a gas of strings. This cosmology starts with the assumption that all spatial dimensions are compact and initially have string scale radii; some dimensions grow due to some thermal or quantum fluctuation which acts as an initial expansion velocity. Based on simple arguments from the low energy equations of motion and string thermodynamics, we demonstrate that the interaction rates of strings are negligible, so the common assumption of thermal equilibrium cannot apply. We also present a new analysis of the cosmological evolution of strings on compact manifolds of large radius. Then we discuss modifications that should be considered to the usual Brandenberger-Vafa scenario. To confirm our simple arguments, we give a numerical calculation of the annihilation rate of winding strings. In calculating the rate, we also show that the quantum mechanics of strings in small spaces is important.

Figures (3)

• Figure 1: Cosmological evolution of the scale factor and dimensionally reduced dilaton for $\dot{\mu}_0=0.7$, $\psi_0=-4$, and $E_0=100$ for 3 (solid curve, diamonds) and 9 (dashed curve, stars) expanding dimensions.
• Figure 2: Evolution of the scale factor and dilaton in the large-radius Hagedorn phase for $d=3$ expanding dimensions. The initial conditions are given by the pressureless phase evolution of the initial conditions given in figure \ref{['f:era1']}.
• Figure 3: The interaction rate for strings in the pressureless phase for $d=3$ expanding dimensions. The initial conditions are those given in figure \ref{['f:era1']}. The lower, solid curve has $\mathcal{N}$ calculated for $w=-1$, and the dashed curve takes $\mathcal{N}=\ln E$.