More on the spectrum of perturbations in string gas cosmology

String gas cosmology is rewritten in the Einstein frame. In an effective theory in which a gas of closed strings is coupled to a dilaton gravity background without any potential for the dilaton, the Hagedorn phase which is quasi-static in the string frame corresponds to an expanding, non-accelerating phase from the point of view of the Einstein frame. The Einstein frame curvature singularity which appears in this toy model is related to the blowing up of the dilaton in the string frame. However, for large values of the dilaton, the toy model clearly is inapplicable. Thus, there must be a new string phase which is likely to be static with frozen dilaton. With such a phase, the horizon problem can be successfully addressed in string gas cosmology. The generation of cosmological perturbations in the Hagedorn phase seeded by a gas of long strings in thermal equilibrium is reconsidered, from both the point of view of the string frame (in which it is easier to understand the generation of fluctuations) and that of the Einstein frame (in which the evolution equations are well known). It is shown that fixing the dilaton at some early stage is important for obtaining a scale-invariant spectrum of cosmological fluctuations in string gas cosmology.