Correlation between momentum modes and winding modes in Brandenberger-Vafa's string cosmological model

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Brandenberger and Vafa have proposed the string cosmological model based on T-duality. In this model, they took the toroidal target space and introduced the new position x conjugate to the winding mode, in addition to the position x conjugate to the momentum mode. In this way they can describe a universe larger than the string scale with the coordinate x and one smaller with the coordinate x. Resultingly, they never encounter the singularity seen in the standard Big Bang scenario. The most interesting phenomenon in this model is the transition from x-space to x-space when the size of universe is nearly the string scale. Here, we define the dispersion of the momentum number m times the winding number w as the `correlation' of momentum modes and winding modes. Then using the statistical mechanics of strings on a torus, we calculate the correlation in low and high temperature limits, and we consider the possibility that we can observe this effect today, but we will see that this is unlikely.