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A bstract It was recently shown that the homogeneous and isotropic cosmology of a massless scalar field coupled to general relativity exhibits a new hidden conformal invariance under Mobius transformation of the proper time, additionally to the invariance under time-reparamterization. The resulting Noether charges form a    $$ \mathfrak{sl}\left(2,\mathbb{R}\right) $$  sl 2 ℝ Lie algebra, which encapsulates the whole kinematics and dynamics of the geometry. This allows to map FLRW cosmology onto conformal mechanics and formulate quantum cosmology in CFT 1  terms. Here, we show that this conformal structure is embedded in a larger    $$ \mathfrak{so} $$  so (3, 2) algebra of observables, which allows to present all the Dirac observables for the whole gravity plus matter sectors in a unified picture. Not only this allows one to quantize the system and its whole algebra of observables as a single irreducible representation of    $$ \mathfrak{so} $$  so (3, 2), but this also gives access to a scalar field operator    $$ \hat{\phi} $$  ϕ ̂ opening the door to the inclusion of non-trivial potentials for the scalar field. As such, this extended conformal structure might allow to perform a group quantization of inflationary cosmological backgrounds.

Conformal structure of FLRW cosmology: spinorial representation and the $$ \mathfrak{so} $$ (2, 3) algebra of observables