  The general properties of a perfect relativistic fluid resulting from the
quantum gravitational anomaly are investigated. It is found that, in the limit
of a weak gravitational field, this fluid possesses a polytropic equation of
state characterized by two universal constants: the polytropic constant and the
natural polytropic index. Based on the astrophysical data, the estimates for
the polytropic constant are given. It is shown that this fluid can describe a
considerable part of the cold dark matter. The quantum theory of such a fluid
is constructed in the framework of the background field method. The Ward
identities associated with the entropy and vorticity conservation laws are
derived. The leading gradient corrections to the pressure of the perfect fluid
are found and the restrictions on their form are obtained. These restrictions
guarantee, in particular, the absence of ghosts in the model. The second order
nonlinear corrections to the equations of motion of a perfect relativistic
fluid are analyzed and the explicit expressions for the transverse and
longitudinal perturbations induced by a sufficiently strong sound wave are
obtained. A dynamical solution to the problem of time in quantum gravity is
proposed.
