Correlation between Momentum Modes and Winding Modes in Brandenberger-Vafa's String Cosmological Model

Brandenberger and Vafa have proposed the string cosmological model based onT-duality. In this model, they took the toroidal target space and introducedthe new position \tilde{x} conjugate to the winding mode, in addition to theposition x conjugate to the momentum mode. In this way they can describe auniverse larger than the string scale with the coordinate x and one smallerwith the coordinate \tilde{x}. Resultingly, they never encounter thesingularity seen in the standard Big Bang scenario. The most interestingphenomenon in this model is the transition from \tilde{x}-space to x-space whenthe size of universe is nearly the string scale. Here, we define the dispersionof the momentum number m times the winding number w as the `correlation' ofmomentum modes and winding modes. Then using the statistical mechanics ofstrings on a torus, we calculate the correlation in low and high temperaturelimits, and we consider the possibility that we can observe this effect today,but we will see that this is unlikely.Comment: 21 pages, LaTeX, five figures, minor errors correcte