research/cosmic-web/docs/; the model card is
MODEL-CARD.md.
The recipe, frozen
Input: a linear initial density field (equivalently its Zel’dovich displacement field) on a periodic grid. Output: approximate comoving particle positions at the present epoch, at a small fraction of an N-body run’s cost (~18 cheap steps versus 90 force-solving steps). The rule (E5d; MODEL-CARD):
- Evolve particles on straight Zel’dovich rays in growth-factor steps \(\Delta D = 0.05\); no Poisson solves beyond cheap density deposits.
- At each step, deposit the model’s own particles to a density grid.
- The first time a particle’s local density exceeds \(\rho_c = 5\), remove \(\beta = 60\%\) of its velocity components perpendicular to the local filament axis \(e_3\) — the minor eigenvector of the tidal tensor computed from the model’s own density via the Poisson equation, smoothed at 2 h⁻¹Mpc and refreshed every third step. Damping is applied in the simulation box frame: the implicit assumption is that a structure’s bulk transverse motion is small against the infall being arrested (adequate here; a momentum-conserving, relative-velocity version would be the Galilean-invariant refinement). The along-axis component is never touched.
Three fixed numbers (\(\beta\), \(\rho_c\), the smoothing scale). Two structural findings (E5c): partial damping (\(\beta < 1\)) is required for self-estimated frames to pay — at full damping the model’s own density feedback over-triggers crossings and cancels the frame gain — and pancake-ordered sequential damping (arrest along \(e_1\) first, as B2’s collapse ordering would suggest) underperforms both-perpendicular damping. That last result cuts against the tidy ordering story and deserves the honest gloss: with self-estimated frames and one density trigger, the estimator cannot reliably tell “which axis has collapsed so far”, and damping both transverse components hedges that frame noise better than trusting it.
The evidence chain, experiment by experiment
- E4 — the measurement. The residual between true particle-mesh (PM) N-body transport and each particle’s own Zel’dovich (ZA) prediction is large near the web (RMS ≈ 4.4 voxels along \(e_3\) alone within 2 voxels of spines) and points across the filament axis at every distance: the needed correction is transverse arrest (full table in Appendix B2).
- E5 — the first model. ZA rays + full transverse damping with proxy frames: best of three models overall (4.91 versus ZA 5.00 and isotropic sticking 5.34), clearest at the web, with a small 2–4-voxel deficit.
- E5b — the oracle. Same model with tidal frames from the true final field: beats ZA in every distance bin (4.47 overall, −20% at the web). The E5 deficit was frame-estimation error; 0.46 voxels of headroom priced.
- E5c/E5d — declared optimisation. Selection on seeds {2, 3}, validation on held-out {4, 5, 6}. Winner: self-density frames, β = 0.6, 2 h⁻¹Mpc smoothing — 4.52 ± 0.18, within 0.05 of the oracle bound.
- E6/E7 — transfer. Frozen knobs across voxel size, clustering amplitude and cosmology: the advantage persists everywhere and grows with clustering.
- E8 — field level. Extends the usable wavenumber range of a ZA-based mock on both phases and amplitudes at intermediate and small scales (r(k) trails plain ZA at k ≲ 0.14); isotropic sticking destroys phases.
- E9 — baselines. On identical initial conditions: statistical tie with MUSCLE, 2LPT degrades badly. The contribution is the mechanism, not a better engine.
The model ladder
Median per-particle transport error against PM truth, in voxels (= h⁻¹Mpc), held-out seeds — ZA reference 4.98 ± 0.27 on seeds {4, 5, 6} (E5–E5d, E9):
| stage | configuration | overall error (vox) | vs ZA |
|---|---|---|---|
| isotropic sticking (adhesion proxy) | full damping, all directions | 5.34 ± 0.15 | +7% |
| ZA baseline | straight rays, no damping | 4.98–5.00 | — |
| E5: first transverse model | β = 1, ZA-proxy frames | 4.91 ± 0.14 | −2% |
| E5c: refined | β = 0.75, self-density frames | 4.64 ± 0.18 | −7% |
| E5d: frozen recipe | β = 0.6, 2 h⁻¹Mpc frames | 4.52 ± 0.18 | −9% |
| E5b: oracle bound | true final-field frames | 4.47 ± 0.12 | −10% |
| MUSCLE (E9) | multiscale spherical collapse | 4.49 ± 0.12 | tie with frozen recipe |
| 2LPT (E9) | second-order perturbation theory | 8.07 ± 0.39 | +62% |
The frozen recipe closes 90% of the recoverable gap to the oracle. The largest gains sit where the correction was measured: at 0–2 voxels from spines the model reaches 5.78 ± 0.16 against ZA’s 6.93 ± 0.31 (−17%), with the oracle showing −20% available (E5b, E9). MUSCLE is slightly ahead at the web (5.57 ± 0.07); the reading of the tie is the mechanism result — a model with only the measured directional ingredient reproduces MUSCLE-class transport, so shell-crossing prescriptions work because they implement transverse arrest.
research/cosmic-web/.
Transfer with frozen knobs
No re-calibration anywhere; every error below is quoted in physical h⁻¹Mpc (the coarse run is converted from its native 2 h⁻¹Mpc voxels — compare relative advantages across rows, as E6 itself cautions):
| condition | seeds | ZA all | model all | Δ all | ZA web | model web | Δ web |
|---|---|---|---|---|---|---|---|
| base (EdS, σ₈ = 0.8, 1 h⁻¹Mpc vox) | 3 | 4.98 | 4.52 | −9% | 6.93 | 5.78 | −17% |
| coarse (2 h⁻¹Mpc voxels, ≈ physical) | 3 | 4.18 | 4.08 | −2% | 6.74 | 6.16 | −9% |
| σ₈ = 0.6 | 3 | 2.98 | 2.89 | −3% | 4.73 | 4.26 | −10% |
| σ₈ = 1.0 | 3 | 7.20 | 6.23 | −13% | 9.36 | 7.33 | −22% |
| flat ΛCDM, Ωm = 0.31 | 3 | 4.93 | 4.45 | −10% | 6.90 | 5.73 | −17% |
| 0.5 h⁻¹Mpc voxels (EdS) | 1 (+2 in E7b) | 4.95 | 4.45 | −10% | 6.76 | 5.72 | −15% |
Here σ₈ is the clustering amplitude of the initial conditions, “EdS” is Einstein–de Sitter expansion, and “web” means within 2 h⁻¹Mpc of the spine network. The advantage grows monotonically with clustering (−3% → −9% → −13% overall) — the behaviour of a physical shell-crossing correction, since higher σ₈ means more crossings — and is nearly identical between EdS and ΛCDM, as expected for a growth-factor-parametrised geometric term; physical-unit errors at 0.5 and 1 h⁻¹Mpc voxels match (4.45 vs 4.52), so the error scale is set by the physics, not the grid — the coarse 2 h⁻¹Mpc run’s smaller native-voxel numbers are a unit artifact, converted above.
Field-level fidelity
Density fields against PM truth, 3 seeds (E8, E9). The cross-correlation r(k) measures phase/structure fidelity at wavenumber k, in h Mpc⁻¹ (1 = perfect); the transfer function T(k) measures amplitude fidelity (1 = unbiased).
| k | r: ZA | r: 2LPT | r: MUSCLE | r: damp | T: ZA | T: 2LPT | T: MUSCLE | T: damp |
|---|---|---|---|---|---|---|---|---|
| 0.14 | 0.975 | 0.933 | 0.956 | 0.962 | 0.693 | 0.607 | 0.669 | 0.739 |
| 0.34 | 0.710 | 0.691 | 0.771 | 0.716 | 0.229 | 0.202 | 0.433 | 0.410 |
| 0.53 | 0.288 | 0.430 | 0.527 | 0.388 | 0.098 | 0.095 | 0.305 | 0.262 |
| 0.82 | 0.057 | 0.144 | 0.228 | 0.095 | 0.052 | 0.048 | 0.210 | 0.143 |
The damping model beats ZA on both statistics in the nonlinear regime; isotropic sticking (not shown) collapses to r = 0.089 at k = 0.53 (E8). Against MUSCLE the strengths are complementary — MUSCLE wins small-scale phases, the damping model mid-scale amplitudes — so a field-level hybrid is the natural future work; the naive trigger-level hybrid was tested and refuted (4.67, worse than both parents; E11).
Is the truth true? Two-code validation
All errors above are measured against the program’s own PM integrator, so the truth itself was audited. Time convergence: doubling to 180 steps shifts the median final position by 0.008 voxels and leaves the comparison invariant (E10). External validation: evolved from the CAMELS CV_0 initial conditions, the PM integrator reproduces the official final particle positions to a median matched-ID offset of 0.41 h⁻¹Mpc against both Arepo and MP-Gadget — and those two production codes agree with each other to 0.029 h⁻¹Mpc median (E12). The hierarchy is what matters: code consensus 0.03 ≪ our PM offset 0.41 ≪ measured effects 4–5 h⁻¹Mpc.
Limitations
From the model card (MODEL-CARD), quantified:
- One-shot damping. Only the first crossing is treated; multi-stream interiors are out of scope by design. Residual web error (5.5–6 voxels even at the oracle bound) is post-crossing physics; do not interpret positions within ~2 voxels of density peaks as resolved halo structure.
- Verified scope. Frozen knobs verified for voxels 0.5–2 h⁻¹Mpc, σ₈ ∈ [0.6, 1.0], EdS and flat ΛCDM (Ωm = 0.31). Beyond that, and at other box sizes or tracer densities, re-calibrate β and ρ_c against a small PM truth, optimising the competitor first (Appendix B3); the β optimum is a flat minimum possibly slightly below 0.6.
- Web-region correction, not global. Gains far from the web are −2%.
- Large-scale amplitude. ~7% T(k) deficit vs plain ZA in the largest bin measured (k = 0.06 h Mpc⁻¹: 0.849 vs 0.920; the deficit vs truth is larger); rescale before use in mocks.
Reproduce
Every table above regenerates from research/cosmic-web/
(Python virtualenv per requirements.txt):
cd research/cosmic-web
make residual # E4: the measured correction (Appendix B2 table)
make model-ladder # E5, E5b, E5c, E5d: the ladder and the oracle
make baselines # E9: 2LPT and MUSCLE on identical ICs
make transfer # E6, E7 (E7b lives inside the E8 report): frozen-knob transfer matrix
make field-level # E8: r(k) and T(k)
make truth-validation # E10 convergence; E12 external checks
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.
- S. N. Gurbatov, A. I. Saichev & S. F. Shandarin (1989). "The large-scale structure of the universe in the frame of the model equation of non-linear diffusion." MNRAS 236, 385–402.
- M. C. Neyrinck (2016). "Truthing the stretch: non-perturbative cosmological realizations with multiscale spherical collapse (MUSCLE)." MNRAS 455, 1204.
- F. R. Bouchet, S. Colombi, E. Hivon & R. Juszkiewicz (1995). "Perturbative Lagrangian approach to gravitational instability." A&A 296, 575.
- F. Villaescusa-Navarro et al. (2021). "The CAMELS project." ApJ 915, 71.
- Model card (
MODEL-CARD.md), experiment reports E4–E12 and the paper draft, inresearch/cosmic-web/docs/of the repository.