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Primordial power spectrum features and consequences

The present Cosmic Microwave Background (CMB) temperature and polarization anisotropy data is consistent with not only a power law scalar primordial power spectrum (PPS) with a small running but also with the scalar PPS having very sharp features. This has motivated inflationary models with such sharp features. Recently, even the possibility of having nulls in the power spectrum (at certain scales) has been considered. The existence of these nulls has been shown in linear perturbation theory. What shall be the effect of higher order corrections on such nulls? Inspired by this question, we attempt to calculate quantum radiative corrections to the Fourier transform of the two-point function in a toy field theory and address the issue of how these corrections to the power spectrum behave in models in which the tree-level power spectrum has a sharp dip (but not a null). In particular, we consider the possibility of the relative enhancement of radiative corrections in a model in which the tree-level spectrum goes through a dip in power at a certain scale. The mode functions of the field (whose power spectrum is to be evaluated) are chosen such that they undergo the kind of dynamics that leads to a sharp dip in the tree level power spectrum. Next we consider the situation in which this field has quartic self interactions and find one loop correction in a suitably chosen renormalization scheme. We thus attempt to answer the following key question in the context of this toy model (which is as important in the realistic case): in the chosen renormalization scheme, can quantum radiative corrections get enhanced relative to tree-level power spectrum at scales at which sharp dips appear in the tree-level spectrum?