Questions about Gravitational Instability of Axion Density Fluctuations in an Axion Dominated Universe

Abstract

It has recently been suggested that the axion with its spontaneous symmetry breaking mass scale of IA'10 12 GeV would dominate the energy density of the present universe and possibly explain the observed structure of the universe.') However most of previous analyses on this topic seem to rely on a too simplified argument on the development of density fluctuations in an axion dominated universe (e.g. treating axions as a ,usual bosonic fluid). Since in the scenario of our interest, the energy density of the universe is not dominated by an axion fluid but by a coherent energy of the axion field, we must take account of its coherent nature when considering the development of density fluctuations. Keeping this point in mind, we investigate perturbations of the coherent axion fi~ld in this letter. The result strongly indicates that the coherent nature of the axion field restrains the growth of density fluctuations. Although only the case of the axion is considered here explicitly, our result applies to other cosmologically possible coherent fields if they have a chance to dominate the energy density of the universe. When the Peccei-Quinn symmetry breaks down spontaneous1y at cosmic time t = to, the corresponding' U(1)PQ angle 1J (modulo 2ir) would attain some finite expectation value which would be coherent over at least the horizon size of the universe (Hubble radius) to. Although the expectation value of If at this epoch is not quite physical, it will eventually affect the physics at later times and hence should be regarded as a physical quantity. Therefore the axion field, which is defined by ? = fA If, should be considered as being materialized classically and its energy momentum tensor as contributing to the r.h.s. of the Einstein equations directly. This is in' contrast with the case when the field' is highly incoherent and its expectation value is essentially zero; in such a case we know that a fluid picture becomes more adequate. Now, since rp= is realized at least over the horizon scale at t = to, spatial fluctuations of rp on scales l~to are real classical (observable) fluctuations. Hence their Fourier amplitudes rp k (where k is the comoving wavenumber) are classically observable. Although the phases of rp k cannot be predicted a priori, this probabilistic nature of rp k is completely classical and nothing to do with quantum uncertainty. Note that the same is true in the usual case of cosmological density fluctuations where each Fourier component 8p k represents an observable density fluctuation though fluctuations may be quite incoherent as a whole. This is a characteristic feature in cosmology where scales of fluctuations are so large. The energy momentum tensor of the field is expressed as