  One method to overcome the notorious problem of time in the quantisation of
gravity is the identification of a physically preferred time parameter, a
promising candidate being so-called `York time'. The dynamical equations for
matter and spatial geometry in York time may be obtained via Hamiltonian
reduction, that is, by solving the Hamiltonian constraint for the physical,
non-vanishing Hamiltonian density identified as the variable conjugate to the
chosen time parameter. Yet in general this equation cannot be solved
algebraically. Here we show how in a cosmological scenario, where one may treat
geometric and matter inhomogeneities as small perturbations, one is able to
obtain the physical Hamiltonian density by solving the constraint equation
perturbatively. By construction the Hamiltonian density is quadratic in the
perturbation variables, which makes it easily quantisable, although subtleties
arise due to the non-canonical form of the Poisson brackets and the
time-dependent coefficients. The latter are determined by the evolution of the
background variables.
