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Cosmological Polytopes and the Wavefunction of the Universe
(NBIA (Niels Bohr International Academy), Copenhagen)
Aula Teorici (Dipartimento di Fisica e Astronomia)
Dipartimento di Fisica e Astronomia
Via Irnerio, 46
In this talk we present a novel approach to understanding cosmological observables, such as the wavefunction of the universe and the late-time correlators. The general questions we ask are: Which features should these observables have to come from a consistent unitary evolution in cosmological space-time? If someone hands them to us, how do we check that they are correct? Concretely, we focus on the wavefunction of the universe for a class toy models of scalars with time-dependent coupling constants, including conformally coupled scalars (with non-conformal interactions) in FRW cosmologies as a special case. We discuss novel representations for such an observable. Strikingly, we found a connection between the physics of cosmological time evolution and the wavefunction of the universe, and the mathematics of positive geometries. The contribution of each Feynman diagram to the wavefunction of the universe is associated with a certain universal rational integrand. We show that this integrand can be identified with the canonical form of a “cosmological polytope”. These polytopes have an intrinsic definition making no reference to physics, and the connection to “time”, along with familiar properties of the wavefunction, arises from this definition. In particular the singularity structure of the wavefunction for this toy model of scalars is common to all theories, and is geometrized by the cosmological polytope. A natural triangulation of the polytope is associated with the time-integral representation of the wavefunction; another natural triangulation of the dual polytope reproduces “old-fashioned perturbation theory”. Other triangulations are associated with efficient recursive computations of the wavefunction, while recently discovered new representations for the canonical forms of general polytopes give new representations of the wavefunction with no extant physical interpretation. We discuss in some examples how symmetries of the cosmological polytope descend to symmetries of the wavefunction, (such as conformal invariance of deSitter wavefunctions). Finally, we will briefly comment how the flat-space S-matrix is contained in this picture as a facet of the cosmological polytope.