An \'Etude on the Regularization and Renormalization of Divergences in Primordial Observables

Many cosmological observables of interest derive from primordial vacuum fluctuations evolved to late times. These observables represent statistical draws from some underlying quantum or statistical field theoretic framework where infinities arise and require regularization. After subtracting divergences, renormalization conditions must be imposed by measurements or observations at some scale, mindful of scheme and background dependence. We review this process on backgrounds that transition from finite duration inflation to radiation domination, and show how in spite of the ubiquity of scaleless integrals, UV divergences can still be meaningfully extracted from quantities that nominally vanish when dimensionally regularized. In this way, one can contextualize calculations with hard cutoffs, distinguishing between UV and IR scales corresponding to the beginning and end of inflation from UV and IR scales corresponding the unknown completion of the theory and its observables. This distinction has significance as observable quantities cannot depend on the latter although they will certainly depend on the former. One can also explicitly show the scheme independence of the coefficients of UV divergent logarithms. Furthermore, certain IR divergences can be shown to be an artifact of the de Sitter limit and are cured for finite duration inflation. For gravitational wave observables, we stress the need to regularize stress tensors that do not presume a prior scale separation in their construction (as with the standard Isaacson form), deriving an improved stress tensor fit to purpose. We conclude by highlighting the inextricable connection between inferring $N_{\rm eff}$ bounds from vacuum tensor perturbations and the process of background renormalization.