  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, both
from the point of view of the string frame (in which it is easier to understand
the generation of fluctuations) and the Einstein frame (in which the evolution
equations are well known). It is shown that fixing the dilaton at some early
stage is important in order to obtain a scale-invariant spectrum of
cosmological fluctuations in string gas cosmology.
