Loop corrections in the separate universe picture

Abstract

Abstract In inflationary models that produce a spike of power on short scales, back-reaction of small-scale substructure onto large-scale modes is enhanced. Loop corrections that quantify this back-reaction have been evaluated by a number of authors. We argue that the separate universe framework provides a highly convenient tool for such computations. Each loop of interest is characterized by large hierarchies in wavenumber and horizon exit time. The separate universe framework highlights important factorizations involving these hierarchies. We interpret each loop correction in terms of a simple, classical, back-reaction model, and clarify the meaning of the different volume scalings that have been reported in the literature. We argue that significant back-reaction requires both short-scale nonlinearities and long-short couplings that modulate the short-scale power spectrum. In the absence of long-short couplings, only incoherent “shot noise”-like effects are present, which are volume-suppressed. Dropping the shot noise, back-reaction from a particular scale is controlled by a product of f NL-like parameters: an equilateral configuration measuring the nonlinearity of the short-scale modes, and a squeezed configuration measuring the long-short coupling. These may carry important scale dependence controlling the behaviour of the loop in the decoupling limit where the hierarchy of scales becomes large. In single-field models the long-short coupling may be controlled by this hierarchy, in which case the net back-reaction would be safely suppressed. We illustrate our framework using explicit computations in a 3-phase ultra-slow-roll scenario. Our analysis differs from earlier treatments of this model, which did not consistently include the effect of small-scale modes. Finally, we discuss different choices for the smoothing scale used in the separate universe framework and argue the effect can be absorbed into a renormalization of local operators. This complicates interpretation of the loop, because the analytic part of each loop integral is degenerate with unknown, ultraviolet-sensitive contributions.