  We show that the simplest FLRW cosmological system consisting in the
homogeneous and isotropic massless Einstein-Scalar system enjoys a hidden
conformal symmetry under the 1D conformal group $SL(2,\mathbb{R})$ acting as
Mobius transformations in proper time. This invariance is made explicit through
the mapping of FLRW cosmology onto conformal mechanics. On the one hand, we
identify the corresponding conformal Noether charges, as combinations of the
Hamiltonian scalar constraint, the extrinsic curvature and the 3D volume, which
form a closed $sl(2,\mathbb{R})$ Lie algebra. On the other hand, this approach
allows to write FLRW cosmology in terms of a AdS${}_{2}$ phase space and a
Schwarzian action. Preserving this conformal structure at the quantum level
fixes the ordering ambiguities in the Wheeler-de Witt quantization and allows
to formulate FLRW quantum cosmology as a CFT${}_{1}$. We show that the CFT
two-points correlator is realized as the overlap of the evolution in proper
time of cosmological coherent wave-packets. In particular, the two-points
function is built from a vacuum state which, although not conformally
invariant, coincides with the cosmological vacuum annihilated by the scalar
constraint. These results suggest new perspectives in classical and quantum
cosmology, among which the possibility to apply the conformal bootstrap program
to quantize cosmological backgrounds.
