Cosmic Web Environments

• Cosmic web environments are the large-scale network of voids, walls, filaments, and nodes formed via anisotropic gravitational collapse of primordial fluctuations.
• They are classified using the Hessian of the gravitational potential, where eigenvalue criteria robustly distinguish distinct cosmic structures across multiple scales.
• These environments drive galaxy and halo evolution by regulating mass flows, star formation, and assembly bias, as revealed by simulations and advanced segmentation algorithms.

The large-scale distribution of matter in the Universe organizes itself into a network of interconnected structures known as the cosmic web. This web consists of four main cosmological environments—voids, walls (sheets), filaments, and nodes (clusters)—that arise from the anisotropic gravitational collapse of primordial density fluctuations. The cosmic web is quantitatively defined through mathematical criteria based on the eigenvalues of various tensors such as the tidal tensor (the Hessian of the gravitational potential) or the deformation tensor derived from observational or simulation data. Cosmic web environments play a decisive role in the assembly, dynamics, and baryonic evolution of galaxies and dark matter haloes, and the statistical characterization and cosmological applications of these environments have become a central focus of contemporary large-scale structure studies.

1. Mathematical Formalism and Classification Schemes

The cosmic web environments are formally classified via the properties of the Hessian of an input field—most commonly, the normalized gravitational potential (tidal tensor) or the density field—smoothed over a range of spatial scales. The core scheme involves:

• Computation of the Hessian (or tidal tensor):

$$T_{ij}(\mathbf{x}) = \frac{\partial^2\phi(\mathbf{x})}{\partial x_i \partial x_j}$$

where $\phi$ is the gravitational potential, related to the density via Poisson’s equation $\nabla^2\phi = \delta$.

• Extraction of the three eigenvalues $\lambda_1 \leq \lambda_2 \leq \lambda_3$ at each location.
• Assigning environments: | Environment | Eigenvalue Criteria | |--------------|------------------------------| | Void | 0 positive eigenvalues | | Wall/Sheet | 1 positive eigenvalue | | Filament | 2 positive eigenvalues | | Node/Cluster | 3 positive eigenvalues |

In practice, a threshold $\lambda_\mathrm{th}$ is often introduced to further robustify the classification, and the identification is performed at multiple smoothing scales (scale-space analysis), combining the signature at each point into the final environment label using a maximization over scales (1209.2043, Cautun et al., 2014, Ayçoberry et al., 2023, Tojeiro et al., 27 Mar 2025).

Methods such as NEXUS/NEXUS+ (1209.2043), DisPerSE (topological persistence via Morse theory), and variants of the T-web and V-web (velocity field–based analogues) provide algorithmic implementations of these mathematical criteria. NEXUS+ notably improves filament and wall detection via log-Gaussian filtering of highly dynamic-range fields like density.

2. Environmental Evolution, Mass, and Volume Budgets

Application of these algorithms to cosmological simulations has revealed the following characteristic partitioning (Cautun et al., 2014):

• Voids: Occupy ~78–80% of the volume but only ~15% of the mass. Densities are extremely low ($1+\delta\sim0.1-0.2$), nearly devoid of massive halos.
• Walls (Sheets): Account for ~7–10% of the volume and ~11–15% of the mass, serving as boundaries between voids.
• Filaments: Contain ~50% of the cosmic mass in only ~6–8% of the volume, forming the primary channels for matter flows.
• Nodes (Clusters): Host ~10–11% of the mass in negligible volume, marking the sites of complete three-dimensional collapse and the most massive halo concentrations.

Over cosmic time, the web evolves from a state rich in tenuous, small-scale sheets and filaments (high redshift) to a mature network dominated by thick filaments and clusters at low redshift. Voids grow in volume as mass is drained toward higher-density components, and mean densities in filaments and clusters increase as non-linear evolution proceeds (Cautun et al., 2014).

3. Physical Processes and Dynamical Flows

The cosmic web serves as the scaffold for anisotropic mass flows and environmental pre-processing:

• Mass transport operates as a unidirectional sequence: matter flows from voids → walls → filaments → nodes.
• Velocity field properties:
  • In voids: flows are isotropic and expanding (cold, nearly super-Hubble), characterized by low velocity dispersions in both radial and tangential directions (Hellwing, 2014, Pomarede et al., 2017).
  • In walls and filaments: motions are increasingly anisotropic and dynamically hot, as matter converges toward filaments and, ultimately, nodes (clusters).
  • Nodes host highly virialized, turbulent regions with the largest velocity dispersions.

These kinematic features confirm that cosmic web environments are not merely geometric or topological entities—they possess distinct and coherent dynamical identities. Differences in mean infall velocities, pairwise velocity dispersions, and flow patterns across environments have become critical tests of web classification robustness and probe the mechanisms underlying gravitational clustering (Hellwing, 2014).

4. Impact on Galaxy and Halo Properties

Galaxy and dark matter halo characteristics exhibit marked, environment-dependent trends that are independent of global density alone:

• Halo assembly and bias:
  • Halo bias is not solely a function of halo mass; it is strongly modulated by environment. Halos in clusters (nodes) show suppressed bias for intermediate masses ($10^{11}-10^{13.5} M_\odot$), while those in voids can have biases enhanced by a factor $\sim 10$ compared to the average (Yang et al., 2017).
  • Halo assembly bias—the dependence of clustering on formation epoch at fixed mass—is prominent for small halos in clusters/filaments, but suppressed or absent in voids and sheets.
• Internal properties:
  • A universal threshold in halo mass ($M_\mathrm{th} \sim 6 \times 10^{10}h^{-1} M_\odot$) demarcates where environment starts to strongly modulate halo concentration and assembly history (Hellwing et al., 2020).
  • Below $M_\mathrm{th}$, filament halos exhibit up to 14% higher concentrations than void halos; wall halos track the mean. Above this mass, distinctions diminish.
  • Halo spin and triaxiality show weaker but non-negligible environmental dependence, more pronounced for massive halos, with void halos having reduced spin and higher triaxiality.
• Baryonic galaxy properties:
  • For low-to-intermediate mass galaxies ($M_* \lesssim 10^{10.8} M_\odot$), galaxies in voids have higher specific SFRs (sSFR) and dust content than those in filaments or nodes; this difference disappears for massive galaxies due to efficient environment-insensitive quenching (e.g., via AGN feedback from SMBH growth) (Parente et al., 2023).
  • Galaxy color distributions (rest-frame (u–r), D4000) and sSFR exhibit environment-driven gradients: redder, older, more metal-rich, and more α-enhanced central galaxies are found near filaments and nodes; sheets preferentially host blue, star-forming systems (Pandey et al., 2020, Winkel et al., 2021, Nandi et al., 29 Aug 2024).
  • The fraction of “active” (star-forming) galaxies decreases monotonically from voids to walls to filaments to knots (Xu et al., 2020).
  • Scaling relations such as mass–metallicity, mass–size, and the Tully–Fisher relation retain signatures of the large-scale environment, with detailed chemo-dynamical simulations confirming environment-induced secondary dependencies (Tissera et al., 10 Jan 2025).

Observationally, these trends are detected using both geometric (Hessian-eigenvalue) and topological (e.g., DisPerSE) definitions of environment, and are robust in both SDSS/GAMA galaxy samples and state-of-the-art hydrodynamical simulations (EAGLE, IllustrisTNG).

5. Applications: Cosmological Information and Statistical Methods

The statistical segmentation of the cosmic web has unlocked new constraining power for cosmological inference:

• Environmental Power Spectra:
  • Partitioning the matter distribution into voids, walls, filaments, and nodes, and computing the power spectrum for each segment, provides complementary cosmological information to that of the overall matter power spectrum (Bonnaire et al., 2021, Bonnaire et al., 2022).
  • Combining environment-dependent spectra breaks degeneracies between parameters such as the summed neutrino mass ($M_\nu$) and $\sigma_8$, or matter density ($\Omega_m$) and $\sigma_8$, improving constraints by factors up to 15 and increasing signal-to-noise by up to an order of magnitude.
  • These improvements persist up to $k_\mathrm{max} = 0.5\,h\,\mathrm{Mpc}^{-1}$ and are robust to the specific parameterization of environment-segmentation, indicating the underlying physical distinction of these environments.
• Mass reconstruction and noise reduction:
  • By optimally weighting galaxy or halo catalogs using both mass and environment, stochasticity in the mapping between tracers and the underlying matter field is significantly reduced, further enhancing the signal-to-noise of clustering and baryon acoustic oscillation (BAO) feature recovery (Fang et al., 2023).
  • Environmental information provides a complementary, and in some cases even superior, lever arm for bias correction and mass reconstruction compared to traditional mass-only approaches.
• Analytical predictions for environment abundances:
  • T-web–based frameworks leveraging the joint statistics of eigenvalues (with Gaussian and non-Gaussian corrections via Gram–Charlier expansions) yield accurate predictions for the fractional volume occupied by voids, walls, filaments, and nodes as a function of threshold, smoothing scale, and redshift (Ayçoberry et al., 2023).
  • Scaling the threshold with the variance of the density field ($\sigma(z)$) captures the redshift evolution of environment probabilities, providing a forward model for connecting theory and simulations.

6. Observational Diagnostics and Galaxy Evolution

Spectroscopic and imaging surveys (e.g., SDSS, GAMA, COSMOS/CHILES) in conjunction with cosmic web reconstruction methods (DisPerSE, filament proximity measures) have enabled precise observational tests:

• HI gas fraction and (differential) sSFRs of blue galaxies decline from filament cores to voids in CHILES, a behavior opposite to that predicted by some hydrodynamic simulations (TNG), emphasizing the sensitivity of feedback and gas processing prescriptions to environmental context (Luber et al., 4 Apr 2025).
• The persistence of bimodal color and age distributions in all environments, but with environment-dependent shifts in mean values and active/passive fractions, reveals that the geometry and topology of the cosmic web modulate—but do not fully determine—galaxy transformation pathways (Pandey et al., 2020, Nandi et al., 29 Aug 2024).
• Principal Component Analysis (PCA) and information-theoretic studies show that the interrelation of key galaxy properties (color, sSFR, D4000, morphology, metallicity) varies measurably across sheets, filaments, and clusters, as evidenced by statistically significant differences in principal component distributions and normalized mutual information correlations (Nandi et al., 29 Aug 2024, Nandi et al., 2023).

7. Stochasticity, Assembly Bias, and Limiting Cases

While cosmic web environments impart measurable trends in the statistical properties of galaxies and halos, certain aspects of galaxy structure show weaker sensitivity:

• For Milky Way–mass galaxies, the spatial and velocity anisotropy profiles of their stellar halos—quantified through entropy-based measures and the velocity anisotropy parameter—are not statistically distinguishable across environmental types (sheet, filament, cluster) at fixed halo mass (Mondal et al., 21 May 2025). The large scatter in profiles points to the dominance of stochastic, small-scale merger histories over large-scale geometric environment in setting halo structure, at least in this mass regime.

A plausible implication is that the influence of environment may rise with galaxy/halo mass, where anisotropic accretion and large-scale flows are more pronounced.

---

In summary, cosmic web environments provide a theoretically robust and observationally tractable framework for associating large-scale structure with the evolution of galaxies and halos. Their identification via scale-space Hessian or tidal tensor eigenanalysis underpins both cosmological parameter estimation and physical models of environmental regulation across cosmic time. The continuing interplay between advanced segmentation algorithms, high-precision simulations, and wide-field surveys is sharpening our understanding of how structure on the largest measurable scales shapes the fate of galaxies, stars, and dark matter in the Universe.