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 ${\cal Q} = {\rm Tr}(T^2)$ and ${\cal A} = {\rm Tr}(T^3)$, 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 $\mathcal{R}_{\rm cv}-1=0.609\,\varepsilon_3$ and entropy difference $\Delta H_{\rm CV}=0.107\,\varepsilon_3$ bits, with $\varepsilon_3=S_3^{\rm tot}(R)\,\sigma_0(R)\,D(z)$. The total skewness combines a gravitational term $S_3^{\rm grav}=34/7+n_{\rm eff}(R)$ and a primordial one $S_3^{\rm PNG}=S_3^{(1)}(R)\,f_{\rm NL}$ with $S_3^{(1)}(8\,h^{-1}{\rm Mpc})\simeq3\times10^{-4}$, smaller by four to five orders of magnitude, so the one-point asymmetry is gravity-dominated and only weakly sensitive to $f_{\rm NL}$. The multifractal entropy rate obeys $\dot H_{\rm mf}(z)=-3f(z)/(1+z)$, a growth probe complementary to $f\sigma_8$.