research/cosmic-web/docs/ by name.
The tidal tensor and its eigenframe
Let \(\Phi(\mathbf{x})\) be the peculiar gravitational potential — the part of the potential sourced by density fluctuations \(\delta\) about the cosmic mean, via the Poisson equation \(\nabla^2 \Phi \propto \delta\). The tidal tensor is its Hessian, the matrix of second derivatives:
\[T_{ij}(\mathbf{x}) \;=\; \frac{\partial^{2} \Phi}{\partial x_i\, \partial x_j}.\]It records how the gravitational pull differs across a small region — the stretching and squeezing that a cloud of test particles feels. Being symmetric, it has an orthonormal eigenframe: eigenvectors \(e_1, e_2, e_3\) with eigenvalues ordered \(\lambda_1 \ge \lambda_2 \ge \lambda_3\). A positive eigenvalue means matter is being compressed along that eigenvector; the ordering says the squeeze is strongest along \(e_1\) and weakest along \(e_3\) — so the filament axis is \(e_3\), the bottom eigenvector. One convention trap, flagged once for the whole series: the results post’s ridge detector uses the Hessian of the density, where — under the same \(\lambda_1 \ge \lambda_2 \ge \lambda_3\) ordering — the filament axis is the top eigenvector. Same symbol, opposite end of the spectrum; potential-Hessian \(e_3\) here, density-Hessian \(e_1\) there.
Collapse happens one axis at a time
The Zel’dovich approximation (Appendix B1) makes the consequence exact. Its map \(\mathbf{x} = \mathbf{q} - D\,\nabla_q \Phi_0(\mathbf{q})\) has Jacobian
\[\frac{\partial x_i}{\partial q_j} \;=\; \delta_{ij} \;-\; D\,\frac{\partial^{2} \Phi_0}{\partial q_i\, \partial q_j},\]whose determinant — the inverse of the local density — first vanishes when \(D\,\lambda_1 = 1\). Collapse therefore happens first along \(e_1\), the strongest-squeeze axis, flattening the cloud into a sheet (a Zel’dovich “pancake”); then along \(e_2\), draining the sheet into a filament; and last along \(e_3\), pooling the filament into a node. A filament is the structure that exists after two collapses and before the third: it is extended along \(e_3\), the axis gravity squeezes last. Bond, Kofman & Pogosyan (1996) showed the resulting filamentary pattern is already encoded in the initial tidal field around proto-clusters — hence “cosmic web”.
The T-web and V-web classifications
The eigenframe gives a pointwise map of the web. The T-web (Hahn et al. 2007; Forero-Romero et al. 2009) counts how many eigenvalues of \(T_{ij}\) exceed a threshold \(\lambda_{\mathrm{th}}\):
| eigenvalues above threshold | environment | axis carried |
|---|---|---|
| 0 | void | — |
| 1 | sheet | normal along e₁ |
| 2 | filament | spine along e₃ |
| 3 | node | — |
The V-web (Hoffman et al. 2012) applies the same counting to the velocity shear tensor
\[\Sigma_{ij} \;=\; -\frac{1}{2H_0} \left( \frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i} \right),\]with \(H_0\) the Hubble constant, which resolves finer structure because the velocity field is smoother than the density but responds to the same tides. Libeskind et al. (2018) compared twelve web finders on one simulation and found substantial disagreement at filament boundaries and junctions — the observation that motivated the series’ benchmark in the first place.
Why the measurement had to be made in this frame
The series’ physical hypothesis (H4 and its refinement H4’) claimed the lifted geometry supplies a correction to standard transport that is aligned with filaments: cheap motion along \(e_3\). The tidal frame is the only frame in which that claim is falsifiable — an isotropic statistic would average the signature away. Experiment E4 therefore measured, for every tracked particle, the residual between truth and the particle’s own Zel’dovich prediction from the same initial conditions,
\[\mathbf{R} \;=\; \mathbf{x}_{\mathrm{true}}(a{=}1) \;-\; \mathbf{x}_{\mathrm{ZA}}(a{=}1),\]and its velocity analogue \(\mathbf{R}_v\), projected both onto the local eigenframe, and binned by distance to the filament spine network. The direction statistic is the squared projection of the unit residual on the filament axis, with isotropic null 1/3: values above 1/3 mean the correction points along filaments (the H4’ prediction), below it across them.
The E4 measurement in full
Six particle-mesh (PM) N-body simulations (128³, Einstein–de Sitter), 50,000 tracked particles each; mean ± across-seed standard deviation; distances in voxels of 1 h⁻¹Mpc (E4):
| d to spine (vox) | n per seed | ⟨(R̂·e₃)²⟩ | ⟨(R̂ᵥ·e₃)²⟩ | RMS R∥ (vox) | RMS R⊥ per axis (vox) | R∥/R⊥ |
|---|---|---|---|---|---|---|
| 0–2 | 20,658 | 0.298 ± 0.002 | 0.312 ± 0.003 | 4.36 | 4.92 | 0.89 |
| 2–4 | 7,269 | 0.297 ± 0.004 | 0.300 ± 0.004 | 3.77 | 4.19 | 0.90 |
| 4–8 | 6,632 | 0.264 ± 0.004 | 0.280 ± 0.003 | 2.40 | 3.07 | 0.78 |
| 8–16 | 10,320 | 0.218 ± 0.002 | 0.218 ± 0.003 | 1.86 | 2.91 | 0.64 |
| 16–64 | 5,120 | 0.208 ± 0.003 | 0.203 ± 0.003 | 1.77 | 2.83 | 0.63 |
Column key: ⟨(R̂·e₃)²⟩ and ⟨(R̂ᵥ·e₃)²⟩ are the mean squared projections of the unit position and velocity residuals on the filament axis (isotropic null 1/3); RMS R∥ is the root-mean-square residual component along e₃; RMS R⊥ per axis is the same for each perpendicular axis; R∥/R⊥ is their ratio (1 = isotropic correction, above 1 = along-filament, below 1 = across-filament). One interpretive caveat: in the outermost bin (16–64 voxels) “the nearest spine” is tens of h⁻¹Mpc away, so its \(e_3\) is no longer a physically meaningful local frame — the residual anisotropy there reflects sheet-and-void kinematics, not filament-relative infall.
Three facts follow. The correction is large where structure forms — RMS ≈ 4.4 voxels along \(e_3\) alone within 2 voxels of spines. It is organised by the tidal frame — every entry deviates strongly from isotropy. And it points across the filament axis at every distance: the direction statistic sits at 0.21–0.30, below the 1/3 null everywhere, and R∥/R⊥ = 0.63–0.89 with no bin reaching 1. Physically: ballistic transport overshoots through forming walls and filaments, and the correction real gravity applies is transverse arrest at the web — the adhesion model’s viscosity made anisotropic by the tidal frame — not enhanced transport along it. H4’ is refuted as stated, and the measured form of the true correction became the model of Appendix B4. The result is consistent with the model-independent precursor (E2: bulk deviations-from-chord at 0.21–0.23 versus 1/3) and with the theory note’s shock picture (T1).
The tidal frame as an anisotropy judge
The eigenframe also grades filament finders. Define the alignment statistic (M3): for spine samples with unit tangent \(\hat{t}_i\) at points \(\mathbf{x}_i\),
\[A \;=\; \bigl\langle\, \lvert \hat{t}_i \cdot e_3(\mathbf{x}_i) \rvert \,\bigr\rangle, \qquad \mathbb{E}[A] = \tfrac{1}{2} \ \text{under an isotropic null}.\]Neither detector is shown the tidal field; A measures how well each one’s spines recover the frame from density data alone. On simulations, the orientation-lift spines align at 0.734–0.759 versus the Hessian detector’s 0.665–0.674 (E1). On the real Universe the separation replicates: across 18 independent tiles of BOSS CMASS galaxies stacked in the E3 campaign, the lift reaches 0.677 ± 0.016 versus the Hessian’s 0.617 ± 0.016 (null 0.5) — a clean, tile-replicated win (E3). This is the lifted geometry’s one surviving advantage, and the theory explains why it is a descriptor rather than a mechanism: the orientation manifold captures the tangent structure of the already-formed shock set (T1), even though the transport that builds it flows across, not along, the filaments.
Back to the series
Back to the series: The Geometry of the Cosmic Web: A Research Program · Two Ways to See a Cosmic Filament · From Cosmic Filaments to Curved Spacetime.
References
- Ya. B. Zel'dovich (1970). "Gravitational instability: an approximate theory for large density perturbations." Astron. Astrophys. 5, 84–89.
- J. R. Bond, L. Kofman & D. Pogosyan (1996). "How filaments of galaxies are woven into the cosmic web." Nature 380, 603–606.
- O. Hahn, C. Porciani, C. M. Carollo & A. Dekel (2007). "Properties of dark matter haloes in clusters, filaments, sheets and voids." MNRAS 375, 489–499.
- J. E. Forero-Romero et al. (2009). "A dynamical classification of the cosmic web." MNRAS 396, 1815–1824.
- Y. Hoffman et al. (2012). "A kinematic classification of the cosmic web." MNRAS 425, 2049–2057.
- N. I. Libeskind et al. (2018). "Tracing the cosmic web." MNRAS 473, 1195–1217.
- Experiment reports E1, E2, E3, E4 and the theory note T1, in
research/cosmic-web/docs/of the repository.