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Cosmological perturbation theory with York time

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.