A Scaling Relation of the Evolving Tidal Fields in a LCDM Cosmology
  We report the finding of a scaling relation among the cosmic-web anisotropy
parameter $A$, the linear density rms fluctuation sigma(r) and the linear
growth factor D(z). Using the tidal field derived from the Millennium
Simulation on 512^{3} grids at z=0, 2, 5 and 127, we calculate the largest
eigenvalues $\lambda$ of the local tidal tensor at each grid resolution and
measure its distance-averaged two-point correlation function, xi_{lambda}, as a
function of the cosines of polar angles cos(theta) in the local principal axis
frame. We show that xi_{lambda} is quite anisotropic, increasing toward the
directions of minimal matter compression, and that the anisotropy of
xi_{lambda} increases as the redshift, z, decreases and as the upper distance
cutoff r_{c} decreases. Fitting the numerical results to an analytic fitting
model \xi_{\lambda}(\cos\theta)\propto (1+A\cos^{n}\theta)^{-1}, it is found
that the best fit value of A, dubbed the cosmic-web anisotropy parameter,
varies systematically with sigma(r_{c}) and D(z), allowing us to determine the
simple empiral scaling relation A(r_{c},z)=0.8, D^{0.76}(z), sigma (r_{c})$.
