Cosmological Production of Superheavy Magnetic Monopoles

There has been much interest recently in grand unified models of elementary particle interactions, ' in which a simple gauge group breaks down at a @cry large mass scale to a group containing the SU(3)SSU(2)U(1) of the strong, weak, and electromagnetic interactions. Because the unbroken symmetry group contains a U(1) factor, general arguments' imply that such models contain topologically stable solitons which carry U(1) magnetic charges and have masses of the order of the scale of the symmetry breakdown. In this note, I will argue that an unacceptably large number of such superheavy magnetic monopoles (M) and antimonopoles (M) might have been produced in the early universe, indicating a possible discrepancy between standard big-bang cosmology and grand unified models. The masses of the superheavy particles in grand unified models are characterized by a zerotemperature scalar-field' expectation value, ~ p, which is expected to be of the order of 10"GeV.' The M mass is approximately given by rn = ho~, where h is the M U(1) magnetic charge. " The smallest allowed magnetic charge is h =2w/q, where q is the minimal U(1) charge of a particle which transforms as a singlet under the unbroken subgroup. ' If symmetry breakdown occurs at many different mass scales, then the M mass is determined by the largest scale at which a U(1) factor appears in the unbroken subgroup, but its classical size at zero temperature is of the order of (v .„) ', where v „ is the smallest mass scale at which the U(1) factor is altered. If the temperature T is greater than a critical temperature T„which is of the order of v»' the full gauge symmetry is restored, and no M's are present. Suppose that very early in the history of the universe, T exceeds T', .' When the universe cools below T„M's can be produced. Because M's, unlike the other superheavy particles in these models, are absolutely stable, their density per comoving volume can be reduced only by annihilation of M-M pairs. We will see that the expansion of the universe halts this annihilation process. If M's are produced copiously when 7.' = T„ then many M's remain when the temperature is much lower —enough to dominate the mass density of the universe by many orders of magnitude. In a recent, paper, Zel'dovich and Khlopov' have considered the cosmological production of M's with a mass of order 104 GeV. However, these authors make the implausible assumption that collisions produce a thermal density of M's when T~ T', . I will argue that, if the phase transition at T =T, is second order (or weakly first order), the production of an appreciable density of M's is a consequence of the large fluctuations near the critical point. If the phase transition is strongly first order, the situation is less clear, and the density of M's may be tolerably small. We need to estimate both the M density produced initially and the rate at which the density per comoving volume decreases. Since the latter question can be answered more precisely, I consider it first. ' Below a temperature T; at which the M production rate is negligible compared with the expansion rate of the universe, the M density is governed by the rate of M-M anDBDlat10D. If we may ignore M-M correlations, we have