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})$.