The Scale Unifier

The cosmic web — the filamentary large-scale structure of the universe — has been classified by at least half a dozen different methods. Some use the tidal tensor (second derivatives of the gravitational potential). Others use the velocity shear. Others use the Hessian of the smoothed density field. Each method defines voids, sheets, filaments, and knots differently, producing maps that agree qualitatively but disagree quantitatively. The field has classification schemes but no classification theory.

Kitaura and Sinigaglia show that these methods are slices of a single hierarchy. The unifying operation is scale-weighting: apply a kernel that weights the density field by scale before computing the eigenvalues of the Hessian. Different kernels recover different established classification schemes. The tidal tensor method corresponds to one kernel. The velocity-shear method corresponds to another. What appeared to be different definitions of cosmic structure are the same definition at different scales.

The hierarchy is ordered. Large-scale kernels produce nonlocal classifications (the tidal tensor sees gravitational influence from distant overdensities). Small-scale kernels produce local classifications (the density Hessian sees only the immediate curvature). Between the two extremes, the classification transitions smoothly. Filaments identified at large scales break into smaller filaments at small scales. Voids merge. The cosmic web is not a single structure but a nested set of structures, each visible at its characteristic scale.

The practical payoff: the hierarchy preserves environmental information across scales, from the linear regime through the nonlinear regime, making it useful for field-level cosmological inference and mock-galaxy generation. But the conceptual payoff is larger. When multiple classification schemes disagree, the instinct is that one is right and the others are approximations. Here, all are right — they are projections of the same object onto different scale axes. The disagreement was the theory, waiting to be recognized.