Symmetries of cosmological perturbations: The residual low multipole ambiguity

Abstract: In cosmology, long-wavelength modes are related to large-gauge transformations (LGT), i.e. changes of coordinates that modify the physical geometry of the cosmological patch. These LGTs stand as bona-fide symmetries of cosmological perturbation theory with various applications, from consistency relations constraining cosmological correlators to non-linear conservation laws in the separate-universe approach. In this work, we revisit LGTs and derive two new results. First, we show that the global symmetries already identified in the literature can be extended to local infinite-dimensional symmetries. The associated generators depend on arbitrary functions of time, and generate low-multipole modes that modify the mean curvature energy and the angular momentum of the patch, demonstrating their physical nature. We propose to interpret these low-multipole soft modes as a new cosmological-frame ambiguity that needs to be fixed prior to evaluating cosmological observables. Second, we demonstrate that the adiabatic cosmological perturbations generated by LGTs deform but preserve all the explicit and hidden Killing symmetries of the background geometry. As such, long-wavelength modes stand as a concrete example of algebraically-special cosmological perturbations of Petrov-type O, and inherit the conformal group as isometries and a set of four deformed Killing-Yano tensors and their associated Killing tensors. This opens the possibility to study their effect on cosmological observables in a fully analytic manner.