Ambiguous power spectrum from a quantum bounce

A quantum cosmological bouncing model may exhibit an ambiguity stemming from the nonclassical nature of the background evolution: Two classically equivalent theories can produce two qualitatively different potentials sourcing the perturbations. It reflects the general ambiguity in quantization of the gravitational field at linear order. We derive explicitly the quantum canonical transformation of linear perturbations involving the quantum background to show how it leads to inequivalent theories. We identify the relevant quantum parameter describing the difference and expand upon the ambiguity by calculating the expected power spectra produced for initial quantum vacuum fluctuations in the contracting phase of both potentials. We find that one spectral index corresponds to all values of this parameter but one, while the other thus represents a set of measure zero.