Large blue spectral index from a conformal limit of a rotating complex scalar

One well-known method of generating a large blue spectral index for axionic isocurvature perturbations is through a flat direction not having a quartic potential term for the radial partner of the axion field. In this work, we show how one can obtain a large blue spectral index even with a quartic potential term associated with the Peccei-Quinn symmetry breaking radial partner. We use the fact that a large radial direction with a quartic term can naturally induce a conformal limit which generates an isocurvature spectral index of 3. We point out that this conformal representation is intrinsically different from both the ordinary equilibrium axion scenario or massless fields in Minkowski spacetime. Another way to view this limit is as a scenario where the angular momentum of the initial conditions slows down the radial field or as a superfluid limit. Quantization of the nonstatic system in which the derivative of the radial field and the derivative of the angular field do not commute is treated with great care to compute the vacuum state. The parametric region consistent with axion dark matter and isocurvature cosmology is discussed.