Information Content of the Cosmic Web

Abstract

We present an information-theoretic analysis of the Cosmic Web that goes beyond the scalar density contrast and exploits the full structure of the tidal deformation tensor. The three eigenvalues (${\lambda }_1,{\lambda }_2,{\lambda }_3$) of the tidal Hessian furnish a natural morphological classifier: clusters, filaments, walls, and voids correspond to (+,+,+), (+,+,-), (+,-,-), and (-,-,-) sign patterns, and their joint probability distribution function (PDF), known analytically in the linear regime from Doroshkevich (1970), defines a continuous Shannon entropy that quantifies the information encoded in the geometry of large-scale structure. Additional information resides in the shear invariants $\mathcal{Q}=\mathrm{Tr}(T^2)$ and $\mathcal{A}=\mathrm{Tr}(T^3)$, independent of the density. We derive the {\it Log-Doroshkevich distribution}, and show the ${\mathbb{Z}}_2$ symmetry survives the nonlinear map, broken only in matter fractions. Gravitational collapse breaks it through the density skewness: an Edgeworth treatment gives the cluster--void asymmetry ${\mathcal{R}}_{\mathrm{cv}}-1=0.609{\epsilon }_3$ and entropy difference $\Delta H_{\mathrm{CV}}=0.107{\epsilon }_3$ bits, with ${\epsilon }_3=S_3^{\mathrm{tot}}(R){\sigma }_0(R)D(z)$. The total skewness combines a gravitational term $S_3^{\mathrm{grav}}=34/7+n_{\mathrm{eff}}(R)$ and a primordial one $S_3^{\mathrm{PNG}}=S_3^{(1)}(R)f_{\mathrm{NL}}$ with $S_3^{(1)}(8h^{-1}\mathrm{Mpc})\simeq 3\times 10^{-4}$, smaller by four to five orders of magnitude, so the one-point asymmetry is gravity-dominated and only weakly sensitive to $f_{\mathrm{NL}}$. The multifractal entropy rate obeys ${\dot{H}}_{\mathrm{mf}}(z)=-3f(z)/(1+z)$, a growth probe complementary to $f{\sigma }_8$.

arXiv comment15 pages, 4 figures, to appear in Physics of the Dark Universe