Relaxing the cosmological moduli problem.

Typically the moduli fields acquire mass mφ = ±C H in the early universe, which shifts the position of the minimum of their effective potential and leads to an excessively large energy density of the oscillating moduli fields at the later stages of the evolution of the universe. This constitutes the cosmological moduli problem, or Polonyi field problem. We show that the cosmological moduli problem can be solved or at least significantly relaxed in the theories in which C 1, as well as in some models with C 1. String moduli couple to standard model fields only through Planck scale suppressed interactions. Their effective potential is exactly flat in perturbation theory perturbatively in the supersymmetric limit, but it may become curved due to nonperturbative effects or because of supersymmetry breaking. If these fields originally were far from the minimum of their effective potential, the energy of their oscillations decrease in an expanding universe in the same way as the energy density of nonrelativistic matter, ρm ∼ a(t). Meanwhile energy density of relativistic plasma decreases as a. Therefore the relative contribution of moduli to the energy density of the universe may soon become quite significant. They are expected to decay after the stage of nucleosynthesis, violating the standard nucleosynthesis predictions, unless the initial amplitude of the moduli oscillations φ0 is sufficiently small. The constraint on φ0 depends on details of the theory. The most stringent constraint appears because of the photodissociation and photoproduction of light elements by the decay products, φ0 <∼ 10 Mp, see [1, 2] and references therein. However, if one makes an assumption that moduli decay only to the particles in the hidden sector, the constraint becomes less stringent, φ0 <∼ 10 Mp. For greater values of φ0 the energy density of the oscillating field dominates the energy density of the universe at the epoch of nucleosynthesis, which leads to a significant overproduction of He (i.e. to the absence of hydrogen). Meanwhile one would expect the initial amplitude of oscillations φ0 to be of the same order as Mp. (Here we use stringy normalization for the Planck mass, Mp = 1 √ 8πG ∼ 2× 10 GeV.) This is the essence of the cosmological moduli problem, which is the string version [3, 2] of the Polonyi problem [4]. There were many suggestions how to solve this problem. For example, it was suggested that the moduli fields slowly slide down to the minimum of their effective potential during inflation, and do not oscillate there anymore [5]. This regime would be possible even for very light moduli if inflation is long enough. Moreover, according to [6], moduli fields typically acquire massmφ ∼ H during inflation. Thus, their effective mass during inflation was not that small, and they could roll down to their minimum even if inflation was not very long [5, 7]. However, as was argued by Goncharov, Linde and Vysotsky [8], this does not solve the problem since typically the minimum of the effective potential during inflation does not coincide with the minimum of the effective potential at the present time with an accuracy 10Mp − 10Mp. Recently this problem was investigated by Dine, Randall and Thomas [9], who have argued that the positions of the two minima may in fact coincide if one invokes some additional symmetries. If the mass of the moduli fields is very large (which may happen in certain models, see e.g. [10]), then they decay very early and do not pose any problems. A more general solution would be to have an additional stage of inflation which would dilute the energy of the oscillating moduli fields [2]. The most elegant realization of this idea is the “thermal inflation” scenario suggested by Lyth and Stewart [11]. It appears that in many models where scalar potentials have flat directions a secondary stage of inflation may indeed take place when the temperature becomes sufficiently small. This stage is short, but it may be long enough to resolve the cosmological moduli problem. A similar (or maybe even somewhat longer) stage of “nonthermal” inflation may occur due to nonthermal phase transitions after reheating [12]. Thus, it may happen that after all the cosmological moduli problem may be not too severe. However, since “thermal” (or “nonthermal”) inflation is very short, it solves the cosmological mod-