On the Nonlinear Evolution of Cosmic Web: Lagrangian Dynamics Revisited
  We investigate the nonlinear evolution of cosmic morphologies of the
large-scale structure by examining the Lagrangian dynamics of various tensors
of a cosmic fluid element, including the velocity gradient tensor, the Hessian
matrix of the gravitational potential as well as the deformation tensor.
Instead of the eigenvalue representation, the first two tensors, which
associate with the "kinematic" and "dynamical" cosmic web classification
algorithm respectively, are studied in a more convenient parameter space. These
parameters are defined as the rotational invariant coefficients of the
characteristic equation of the tensor. In the nonlinear local model (NLM) where
the magnetic part of Weyl tensor vanishes, these invariants are fully capable
of characterizing the dynamics. Unlike the Zeldovich approximation (ZA), where
various morphologies do not change before approaching a one-dimensional
singularity, the sheets in NLM are unstable for both overdense and underdense
perturbations. While it has long been known that the coupling between tidal
tensor and velocity shear would cause a filamentary final configuration of a
collapsing region, we show that the underdense perturbation are more subtle, as
the balance between the shear rate (tidal force) and the divergence (density)
could lead to different morphologies. Interestingly, this instability also sets
the basis for understanding some distinctions of the cosmic web identified
dynamically and kinematically. We show that the sheets with negative density
perturbation in the potential based algorithm would turn to filaments faster
than in the kinematic method, which could explain the distorted dynamical
filamentary structure observed in the simulation.
