Einstein and Jordan frame correspondence in quantum cosmology: expansion-collapse duality

Abstract

The conformal correspondence between FLRW universes in the Einstein and Jordan frames allows for an expansion-collapse duality – an always expanding Einstein frame universe can have a dual Jordan frame description that is contracting forever. The scenario eventually runs into an apparent paradox. When the contracting Jordan frame universe becomes sufficiently small, the classical description becomes inadequate and the universe is expected to develop quantum characteristics. However, at this time, the corresponding Einstein frame universe is expected to behave classically, due to the arbitrarily large size it has grown to. The conformal map here appears to be providing a duality between a quantum effect-dominated universe and a classical universe. We investigate the status of the conformal map at the quantum level in such a scenario, focusing on addressing this paradox. The Einstein and Jordan frame universes are quantized using the Wheeler-DeWitt prescription. We show that the classical conformal map holds at the quantum level when compared through expectation values of scale factors. The relative quantum fluctuation in the scale factor becomes conformally invariant, it increases in both the past and future directions according to the internal clock. Expectedly, the quantum fluctuations in the collapsing Jordan frame increase as it shrinks towards singularity. More surprisingly, the quantum fluctuations in the expanding Einstein frame increase as well, even as its classical scale factor becomes larger. Despite having drastically different cosmological evolutions, the rise in quantum characteristics in a collapsing frame implies the same in its expanding counterpart, thereby resolving the apparent paradox.