  We study cosmological perturbation theory with scalar field and pressureless
dust in the Hamiltonian formulation, with the dust field chosen as a
matter-time gauge. The corresponding canonical action describes the dynamics of
the scalar field and metric degrees of freedom with a non-vanishing physical
Hamiltonian and spatial diffeomorphism constraint. We construct a momentum
space Hamiltonian that describes linear perturbations, and show that the
constraints to this order form a first class system. We then write the
Hamiltonian as a function of certain gauge invariant canonical variables and
show that it takes the form of an oscillator with time dependent mass and
frequency coupled to an ultralocal field. We compare our analysis with other
Hamiltonian approaches to cosmological perturbation theory that do not use dust
time.
