Abstract
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 have attempted to calculate quantum radiative corrections to the Fourier transform of the 2-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 have considered 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 have considered the situation in which this field has quartic self interactions, and found one loop correction in a suitably chosen renormalization scheme. Thus, we have attempted 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 be enhanced relative to tree-level power spectrum at scales, at which sharp dips appear in the tree-level spectrum?
Highlights
It is a well known fact that the correlations of Cosmic Microwave Background (CMB) anisotropies depend on the values of various background cosmological parameters (e.g. Ωb, Ωc etc) and on the assumed form of the primordial power spectrum (PPS)
At this stage, there is no way to favour models that give smooth PPS with those that do not. At this stage, even observationally, it is not possible to favour one of these. In many such theoretical scenarios, the scalar PPS has cuspy dips that sometimes correspond to a null in the PPS i.e. precisely zero scalar power at some wave number
An exact null in scalar PPS can have interesting consequences, such as on processed non-linear matter power spectrum. This leads to an interesting possibility: since the power spectrum calculation is done perturbatively, and since the leading order answer is too small at some scale, could the higher order corrections be important at this scale? If there is no power at some scale, can higher order corrections become so important that they become dominant at this scale? We attempt to answer such questions in this paper in the context of a toy field theory
Summary
It is a well known fact that the correlations of CMB anisotropies depend on the values of various (late time) background cosmological parameters (e.g. Ωb, Ωc etc) and on the assumed form of the PPS. This leads to an interesting possibility: since the power spectrum calculation is done perturbatively, and since the leading order answer is too small (or zero) at some scale, could the higher order corrections be important at this scale?
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