  The cosmological field equations sourced by a self-interacting scalar field
are dynamically equivalent to a closed system of equations obtained by applying
the moment method to non-linear Schr\"odinger equations possessing an
underlying non-relativistic conformal $SL(2,\mathbb{R})$ symmetry. We consider
the one-dimensional, quintic Schr\"odinger equation relevant to strongly
repulsive, dilute Bose gases. The action of the diffeomorphism group on the
space of Schr\"odinger operators generates an harmonic trapping potential that
can be identified with the kinetic energy of the cosmological scalar field.
Inflationary cosmologies are represented by points on the orbit of the de
Sitter solution, which is the quotient manifold ${\rm
Diff}(\mathbb{R})/SL(2,\mathbb{R})$. Key roles are played by the Schwarzian
derivative of the diffeomorphism and the Ermakov-Pinney equation. The
underlying $SL(2,\mathbb{R})$ symmetry results in a first integral constraint
which ensures energy-momentum conservation. When the analysis is restricted to
the universal cover group of diffeomorphisms on the circle, the generation of a
rolling scalar field can be understood in terms of the Virasoro coadjoint
action. The corresponding symplectic two-form and Hamiltonian generator of the
coadjoint orbit are determined by the scalar field kinetic energy.
