Imperfect Dark Matter

We consider cosmology of the recently introduced mimetic matter with higher derivatives (HD). Without HD this system describes irrotational dust - Dark Matter (DM) as we see it on cosmologically large scales. DM particles correspond to the shift-charges Noether charges of the shifts in the field space. Higher derivative corrections usually describe a deviation from the thermodynamical equilibrium in the relativistic hydrodynamics. Thus we show that mimetic matter with HD corresponds to an imperfect DM which: i) renormalises the Newton's constant in the Friedmann equations, ii) has zero pressure when there is no extra matter in the universe, iii) survives the inflationary expansion which puts the system on a dynamical attractor with a vanishing shift-charge, iv) perfectly tracks any external matter on this attractor, v) can become the main (and possibly the only) source of DM, provided the shift-symmetry in the HD terms is broken during some small time interval in the radiation domination epoque. In the second part of the paper we present a hydrodynamical description of general anisotropic and inhomogeneous configurations of the system. This imperfect mimetic fluid has an energy flow in the field's rest frame. We find that in the Eckart and in the Landau-Lifshitz frames the mimetic fluid possesses nonvanishing vorticity appearing already at the first order in the HD. Thus, the structure formation and gravitational collapse should proceed in a rather different fashion from the simple irrotational DM models.