Spectral Hierarchy of the Cosmic Web

Abstract

We introduce a spectral hierarchy of cosmic-web classifications obtained by applying simple scale-weighting kernels to the density field before performing a standard eigenvalue-based web classification. This unifies and extends several widely used web definitions within a single framework: the familiar potential/tidal web (large-scale, nonlocal), a curvature-based web (more local, peak- and ridge-sensitive), and additional higher-derivative levels that progressively emphasize smaller-scale structure. Because the classification is built from second derivatives of the filtered field, successive hierarchy levels align naturally with operator families that appear in renormalised bias and effective descriptions of large-scale structure, providing an explicit bridge between cosmic-web environments and long- and short-range nonlocal bias ingredients. We quantify the information content of the hierarchy with a compact statistic: we map each cell to one of four ordered web types (void, sheet, filament, knot), construct a corresponding ``web contrast'' field, and measure its cross-correlation with halos from the AbacusSummit simulation suite on a coarse mesh with $ΔL\simeq 5.5\,h^{-1}\mathrm{Mpc}$. We find that the hierarchy retains significant tracer-relevant information from very large scales down to the mesh Nyquist limit, with the more local (curvature/higher-derivative) levels dominating toward nonlinear scales. This makes the spectral hierarchy a practical, interpretable conditioning basis for fast mock-galaxy production and field-level modelling, and a flexible tool for studying environment-dependent clustering and assembly bias.

Figures (7)

• Figure 1: Scale-dependent growth with late-time UV activation.Top panel: Linear growth factor $D(a)$ obtained by solving Eq. \ref{['eq:growth_uv_final']} for a flat $\Lambda$CDM background. The solid black curve shows the standard dust solution (scale independent). Dashed coloured curves include scale-dependent UV response terms proportional to $(k/k_{\rm ref})^2$ and $(k/k_{\rm ref})^4$, activated at late times through the smooth switching function $S(a)$ defined in Eq. \ref{['eq:activation']}. Bottom panel: Ratio $D_{\rm modified}/D_{\rm ref}$ highlighting deviations. Deviations remain negligible at early times ($a\ll a_\star$), confirming that the dust growing mode is recovered prior to the onset of nonlinear clustering. At late times, high-$k$ modes experience controlled deviations, illustrating how such corrections become relevant toward the UV limit of coarse-resolution solvers.
• Figure 2: Top left: FastPM at $z=1$ assigned to a mesh. The side length is $100\,h^{-1}{\rm Mpc}$ and the cell size is ${\rm d}L\simeq 0.25\,h^{-1}{\rm Mpc}$. Other panels show the cosmic web classification contrast corresponding to the kernel hierarchy: ( top right) $-k^{-4}$, ( middle left) $-k^{-2}$, ( middle right) $-k^0$, ( bottom left) $-k^{2}$, ( bottom right) $-k^{4}$.
• Figure 3: Top left: Abacus "000" distinct halos at $z=1.1$ assigned to a mesh. The side length is $2000\,h^{-1}{\rm Mpc}$ and the cell size is ${\rm d}L\simeq 5.55\,h^{-1}{\rm Mpc}$. Other panels show the cosmic web classification contrast corresponding to the kernel hierarchy: ( top right) $-k^{-4}$, ( middle left) $-k^{-2}$, ( middle right) $-k^0$, ( bottom left) $-k^{2}$, ( bottom right) $-k^{4}$.
• Figure 4: Cross-correlation coefficient $C^{(i)}(k)$ for the different hierarchy levels $i=-4,-2,0,2,4$. The coefficient is defined as $C^{(i)}(k)=\left\langle \hat{\delta}^*_{\rm tr}(\mathbf{k})\hat{\delta}_{\rm web}^{(i)}(\mathbf{k})\right\rangle/\sqrt{P_{\rm tr}(k)P_{\rm web}^{(i)}(k)}$, where $\delta_{\rm tr}$ denotes the tracer field and $\delta_{\rm web}^{(i)}$ the compressed web-density-contrast field at hierarchy level $i$. Top panel: high-resolution FastPM matter field at $z=1$ with mesh spacing ${\rm d}L=0.25\,h^{-1}\,\mathrm{Mpc}$. Bottom panel:AbacusSummit halo field at $z=1.1$, analysed on a coarse mesh with ${\rm d}L=5.55\,h^{-1}\,\mathrm{Mpc}$; the corresponding web fields are computed from an ALPT matter field generated from the down-sampled initial conditions of the same simulation. The threshold is set to $\lambda_{\rm th}^{(i)}=0$ for all hierarchy levels except $i=4$, where a small stabilizing threshold $\lambda_{\rm th}^{(4)}=0.001$ is adopted. The figure shows that the classical tidal-web level $i=-2$ dominates on large scales, whereas the higher hierarchy levels $i=0,2,4$ retain significantly more information toward nonlinear scales, both in the high-resolution matter field and in the coarse-mesh halo analysis.
• Figure 5: Environmental footprint of the halo population across the spectral hierarchy. The figure shows the probability distribution functions (PDFs) of halo occupation within the 256 cosmic web regions obtained from four hierarchy levels ($i=-2, 0, 2, 4$), each subdivided into four morphological classes (voids, sheets, filaments, knots). For clarity, the PDF values are binned in intervals of width $0.2$ in the range $[0,1]$. Each bin corresponds to a distinct combination of web classifications across levels, forming a discrete multi-scale environmental signature of the tracer population. The non-uniform occupation across regions demonstrates that halos respond differently to long-range tidal structure, curvature, and higher-derivative local environment. This multi-level PDF can therefore be interpreted as the environmental footprint of the tracer population and provides a compact basis for bias characterization and environmental modelling.