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 ($λ_1, λ_2, λ_3$) 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 ${\cal A} = {\rm Tr}(T^3)$, which are algebraically independent of the density contrast delta and capture anisotropic deformation invisible to the density alone. The information dimension of each morphological component is related to its Hausdorff (fractal) dimension through the multifractal formalism: clusters ($D_H = 1.2$), filaments ($D_H = 1.8$), walls ($D_H = 2.5$), and voids ($D_H = 3$) define a spectrum of generalized Rényi dimensions $D_q$, whose $q = 1$ limit recovers the Shannon information dimension. The resulting entropy budget identifies filaments as the dominant information carriers of the matter distribution, while the tidal eigenvalue entropy is maximized in wall-like configurations near the saddle points of the gravitational potential. We also compute the redshift evolution of the multifractal entropy and derive its relation to the linear growth rate $f(z)$, providing an independent constraint complementary to redshift-space-distortion measurements of $fσ_8$.