The features of the Cosmic Web unveiled by the flip-flop field
  Understanding of the observed structure in the universe can be reached only
in the theoretical framework of dark matter. N-body simulations are
indispensable for the analysis of the formation and evolution of the dark
matter web. Two primary fields - density and velocity fields - are used in most
of studies. However dark matter provides two additional fields which are unique
for collisionless media only. These are the multi- stream field in Eulerian
space and flip-flop field in Lagrangian space. The flip-flop field represents
the number of sign reversals of an elementary volume of each collisionless
fluid element. This field can be estimated by counting the sign reversals of
the Jacobian at each particle at every time step of the simulation. The
Jacobian is evaluated by numerical differentiation of the Lagrangian
submanifold, i.e., the three-dimensional dark matter sheet in the
six-dimensional space formed by three Lagrangian and three Eulerian
coordinates. We present the results of the statistical study of the evolution
of the flip-flop field from z = 50 to the present time z = 0. A number of
statistical characteristics show that the pattern of the flip-flop field
remains remarkably stable from z = 30 to the present time. As a result the
flip-flop field evaluated at z = 0 stores a wealth of information about the
dynamical history of the dark matter web. In particular one of the most
intriguing properties of the flip-flop is a unique capability to preserve the
information about the merging history of dark matter haloes.
