A scaling relation of the evolving tidal fields in a ΛCDM cosmology

We report the finding of a scaling relation among the cosmic-web anisotropy parameter A, the linear density rms fluctuation σ(r) and the linear growth factor D(z). Using the tidal field derived from the Millennium Simulation on 5123 grids at z = 0,2,5 and 127, we calculate the largest eigenvalues λ of the local tidal tensor at each grid resolution and measure its distance-averaged two-point correlation function, ξλ, as a function of the cosines of polar angles cos θ in the local principal axis frame. We show that ξλ is quite anisotropic, increasing toward the directions of minimal matter compression, and that the anisotropy of ξλ increases as the redshift z decreases and as the upper distance cutoff rc decreases. Fitting the numerical results to an analytic fitting model ξλ(cos θ)∝(1+Acos nθ)−1, it is found that the best fit value of A, dubbed the cosmic-web anisotropy parameter, varies systematically with σ(rc) and D(z), allowing us to determine the simple empiral scaling relation A(rc,z) = 0.8 D0.76(z) σ(rc)