MODULAR COSMOLOGY

We begin a study of the cosmology of moduli in string theory. The quan tum field theory requirement of ''naturality'' is shown to be incompat ible with slow roll inflationary cosmology unless very stringent const raints are satisfied. In most cases, these constraints imply the exist ence of fields with the properties of string moduli: their natural ran ge of variation must be the Planck scale. The scale which characterize s their potential energy (the inflation scale) must be two to three or ders of magnitude smaller than the Planck mass in order to explain the observed magnitude of the fluctuations in the cosmic microwave backgr ound. Even if these constraints are satisfied, generic initial conditi ons near the Planck energy density do not lead to inflation unless the theory contains topological defects. In this case inflation can arise naturally at the cores of the defects. We show that string theory has two generic types of domain walls which could be the seeds for inflat ion, and argue that modular physics provides a very robust model of in flation. Two scenarios are presented to explain the discrepancy betwee n the inflation scale and the scale of supersymmetry breaking. One of them is favored because it leads to a natural understanding of why the dilaton does not run out to the weak coupling region in the postinfla tionary period.