Post Inflationary Behaviour of String Moduli
  We analyze the behaviour of moduli fields in string effective models between the end of inflation and reheating. The effective moduli potential during this era is derived for a class of simple models. We argue that this potential significantly stabilizes the modulus at its high energy minimum, if some restrictions on modular weights are met. Two mechanisms to further stabilize the moduli to their low energy minima are discussed explicitly: coinciding minima at a point of enhanced symmetry, and the smooth transition from high to low energy minimum by an effective mass term C^2 H^2. For both cases we present explicit examples, and C^2 is found to be O(10) at most. In addition, we show that during a smooth transition the reduction of the modulus amplitude strongly depends on the shape of the low energy potential. 
Andre Lukas, Alexander Niemeyer, Masahiro Yamaguchi