  Using a novel, string theory-inspired formalism based on a Hamiltonian
constraint, we obtain a conformal mechanical system for the spatially flat
four-dimensional Robertson-Walker Universe. Depending on parameter choices,
this system describes either a relativistic particle in the Robertson-Walker
background, or metric fluctuations of the Robertson-Walker geometry. Moreover
we derive a tree-level M-theory matrix model in this time-dependent background.
Imposing the Hamiltonian constraint forces the spacetime geometry to be fuzzy
near the big bang, while the classical Robertson-Walker geometry emerges as the
Universe expands. From our approach we also derive the temperature of the
Universe interpolating between the radiation and matter dominated eras.
