Recently, it has been proposed to relate properties of primordial scalar perturbations to the conformal invariance of some very early Universe models. A concrete realization of this idea is given in the context of the conformal rolling scenario. The main ingredient of the latter is the complex scalar field conformally coupled to gravity and rolling down the negative quartic potential. During the conformal rolling, phase perturbations acquire flat power spectrum, which can be converted into adiabatic perturbations at some later epoch by one of another mechanism. There are two sub-scenarios of the model depending on the behavior of the cosmologically interesting modes by the end of the rolling: one with superhorizon modes and the other with subhorizon ones. In the latter case, phase perturbations proceed to evolve at the intermediate stage between conformal rolling and conventional epoch. This evolution results into the small negative tilt, statistical anisotropy of all even multipoles starting from quadrupole of general structure and non-Gaussianity of the peculiar form. The signatures of the former sub-scenario are the quadrupole statistical anisotropy of both general and quadrupole types and the non-Gaussianity with a (fairly mild) singularity in the folded limit. We review analogous predictions in the context of the inflation and conclude that the conformal rolling scenario can be descriminated from the inflation in future experiments. We also discuss the novel cosmological scenario of the early Universe, where the potential role of conformal symmetries is understood from a much broader prospective. In this picture, proposed by Hinterbichler and Khoury, conformal rolling scenario is just a particular case in a myriad of possible models. We review the main novelties and assumptions of the general setup, and show that predictions made in the framework of the conformal rolling scenario hold for a broad class of models, e.g. the Galilean Genesis.

Starting from the particular prediction, the statistical anisotropy, we constrain the unique parameter h2 of the conformal rolling scenario. For this purpose, we employ the quadratic maximum likelihood method, and apply it to the search of the statistical anisotropy in the seven-year WMAP data. We confirm the large quadrupole anisotropy detected in V and W bands, which has been argued to originate from systematic eects rather than from cosmology. We construct an estimator for the parameter h2. In the case of the sub-scenario with the intermediate stage we set an upper limit h2 < 0:045 at the 95% condence level. The constraint on h2 is much weaker in the case of another sub-scenario, where the intermediate stage is absent. We also comment on the statistical anisotropy recently detected in the CMB low multipoles. This anomaly is unlikely to be explained by the primordial physics. On the other hand, the uncounted foregrounds appear to be a natural source of the low multipoles correlations. We show that the strong quadrupole-octupole anomaly is, in principle, resolved by the account of the Kuiper belt. Simultaneously, the latter can provide the resolution to the parity asymmetry.