Hydrodynamics on (mini)superspace, or a non-linear extension of quantum cosmology

We outline the content and theoretical support for the proposal of "hydrodynamics on (mini)superspace" (or a non-linear extension of quantum cosmology) as an effective framework for quantum gravity in a cosmological context. The basis for the proposal is a general correspondence between hydrodynamics and cosmology, and a picture of the universe as a quantum gravity condensate. The support comes from several directions. One is from mathematical physics: the hydrodynamics of quantum fluids can be mapped to relativistic cosmological dynamics, and both share the same conformal symmetries, which can be unraveled via geometric methods in superspace. The second is that the proposed framework is realized in quantum gravity formalisms like group field theory, in which an emergent cosmological dynamics can be extracted from the hydrodynamics of fundamental quantum simplices in a condensate phase. The same proposal can be motivated from the idea of 3rd quantization of gravity and from recent work on quantum cosmology, as an effective way of incorporating topology changing processes or cosmological inhomogeneities, respectively. A key conceptual ingredient is the relational understanding of space and time, which makes superspace the natural arena for gravitational dynamics, as opposed to the "spacetime" manifold, together with the general idea of emergent spacetime. The proposal and the results supporting it suggest an exciting dialogue between quantum gravity, the theory of quantum fluids and cosmology, as well as a new direction for analogue gravity simulations in the lab.