Over the last decades, the emergence of conformal symmetry in gravitational systems has provided a powerful tool to investigate new aspects in classical, semi-classical and quantum general relativity. The best example is the conformal structure of tests fields in the near horizon region of black holes, which can be used to re-derived many features of black holes from a CFT perspective, such as their spectroscopic properties, the dynamics of super-radiant modes on Kerr, part of the quasi-normal modes spectrum, some quantities related to the emission of gravitational waves as well as their state counting. Similar structure have also been discussed in cosmological space-times, such as in FLRW and de Sitter backgrounds.

 

In this talk, I shall present a new conformal symmetry of the simplest gravitational system, namely the homogeneous and isotropic Einstein-Scalar system. Quite surprisingly, the action of this system exhibits a hidden SL(2,R) symmetry under Mobius transformation of the proper time. This new structure turns out to have far reaching consequences for classical and quantum cosmology. At the classical level, this structure allows i) to map this simple gravitational system onto conformal mechanics, ii) to endow the cosmological phase space with an AdS2 metric, iii) to use the Schwarzian as a generating solution method for inflationary backgrounds. At the quantum level, it opens the possibility to import the conformal bootstrap philosophy within quantum cosmology, using similar technics developed for conformal mechanics. In particular, one can show that the overlap between two coherent wave packets behaves as the CFT two point functions.

 

Related articles: arXiv:1909.13390, arXiv:2001.11807