Cosmology of light towers and swampland constraints

We study the dynamical evolution of FLRW cosmologies in the presence of a tower of scalar light states and a runaway exponential potential. Some of the attractor solutions have problematic behaviours from the EFT point of view, which we use to argue for restrictions on the possible exponential scalings of the potential and tower characteristic mass as we move towards asymptotic regions in moduli space. These serve as further evidence that the tower mass should not decay faster than the potential or the KK scale associated to the homogeneous decompactification of a single compact dimension. We provide support from different top-down compactifications and connect with previous arguments found in the literature.