Title: Fuzzy Bounces

Abstract: We observe that the energy and the enthalpy densities can be smeared by two fudge factors that are constrained by the contracted Bianchi identities. Depending on the analytic properties of the smearing functions the underlying cosmological solutions belong to two physically different classes, namely the bounces of the scale factor and the curvature bounces. While the curvature bounces are naturally compatible with a stage of accelerated expansion, the bounces of the scale factor demand an early phase of accelerated contraction even if a short inflationary stage may arise prior to the decelerated regime. Despite the regularity of the underlying solutions, gradient instabilities and singularities do occasionally appear in the evolution of curvature inhomogeneities. After deducing the specific criteria behind these occurrences, the background-independent conclusions are corroborated by a series of concrete examples associated with different forms of the smearing functions. The evolution of the curvature inhomogeneities restricts the ranges of the solutions that turn out to be unsuitable even for a limited description of the pre-inflationary initial data. The same observation holds in the case of the gauge-invariant evolution of the matter density contrast. It is however not excluded that a class of scenarios (mainly associated with the curvature bounces) could indeed avoid the potential instabilities. All in all the present analysis explore a general approach whose results are relevant in all the contexts where bouncing solutions are invoked either as complementary or as alternative to the conventional inflationary scenarios.