Cosmological Dynamics of the Thermal Scalar Near the Hagedorn Temperature
We study the cosmological dynamics of the thermal scalar the winding string mode that becomes massless at the Hagedorn transition by coupling it to the string-frame gravi dilaton effective action. This provides a field-theoretic framework for investigating winding mode dynamics near the Hagedorn temperature and their role in string cosmology. Below the Hagedorn temperature, the phase space contains static configurations in which the thermal scalar balances the shifted dilaton evolution. These configurations are boundary states rather than attractors, and when the thermal scalar mass depends on the scale factor, winding mode back reaction opposes expansion and can reverse it. Above the Hagedorn temperature, the tachyonic thermal scalar generates negative effective energy density while preserving the null energy condition, enabling branch changes of the Brustein Veneziano type. These transitions do not provide the graceful exit required to connect the Hagedorn phase to standard cosmological evolution. At the Hagedorn temperature itself, the quadratic effective theory breaks down and higher order interactions become essential. Because the thermal scalar originates as a Euclidean order parameter, our Lorentzian treatment should be viewed as an effective dynamical model of the Hagedorn transition. Within this framework, the thermal scalar clarifies the dynamical structure surrounding the Hagedorn transition and shows how the Hagedorn exit problem cannot be resolved within the quadratic effective theory alone.