Comparison of dynamical and spectator models of a (pseudo)conformal universe

We consider two versions of the scenario for generating scalar cosmological perturbations based on the conformal symmetry: a spectator version with a scalar field conformally coupled to gravity and with a negligible energy density; a dynamical version with a scalar field minimally coupled to gravity and dominating the cosmological evolution. Using the Newtonian gauge, we show, first, that no UV strong-coupling scale is generated below $M_{\rm Pl}$, because of mixing with metric perturbations in the dynamical scenario, and, second, that both the dynamical and the spectator models yield identical results in the leading nonlinear order. These results, which include potentially observable effects like statistical anisotropy and non-Gaussianity, hold for the entire class of conformal models. As an example, in the dynamical scenario with the comoving gauge, we reproduce our result on the statistical anisotropy, previously obtained in the framework of the spectator approach.