Cosmological density perturbations in a conformal scalar field theory

We consider a scenario in which primordial scalar perturbations are generated when a complex conformal scalar field rolls down its negative quartic potential. Initially, these are perturbations of the phase of this field, which are then converted into adiabatic perturbations of the density. The existence of perturbations in the radial field direction, which have a red power spectrum, is a potentially dangerous feature of this scenario. But we show that in the linear order in the small parameter, the self-coupling, the infrared effects are completely nullified by an appropriate field redefinition. We evaluate the statistical anisotropy inherent in the model because of the presence of the long-wave perturbations of the radial field component. In the linear order in the self-coupling, the infrared effects do not affect the statistical anisotropy. They are manifested only at the quadratic order in the self-coupling, weakly (logarithmically) enhancing the corresponding contribution to the statistical anisotropy. The resulting statistical anisotropy is a combination of a large term, which decreases as the momentum increases, and a momentum-independent nonamplified term.