  We show that if the Lagrangean for a scalar field coupled to General
Relativity only contains derivatives, then it is possible to completely
deparametrise the theory. This means that 1.Physical observables, i.e.
functions which Poisson commute with the spatial diffeomorphism and Hamiltonian
constraints of General Relativity, can be easily constructed. 2. The physical
time evolution of those observables is generated by a natural physical
Hamiltonian which is (constrained to be) positive. The mechanism by which this
works is due to Brown and Kucha\v{r}. In order that the physical Hamiltonian is
close to the Hamiltonian of the standard model and the one used in cosmology,
the required Lagrangean must be that of a Dirac -- Born -- Infeld type. Such
matter has been independently introduced previously by cosmologists in the
context of k -- essence due to Armendariz-Picon, Mukhanov and Steinhardt in
order to solve the cosmological coincidence (dark energy) problem. We arrive at
it by totally unrelated physical considerations originating from quantum
gravity. Our manifestly gauge invariant approach leads to important
modifictaions of the interpretation and the the analytical appearance of the
standard FRW equations of classical cosmology in the late universe. In
particular, our concrete model implies that the universe should recollapse at
late times on purely classical grounds.
