Cosmology · The cosmic web, voids and intergalactic matter

Filaments, sheets and connected structures

Use three prescribed geometric compressions to compare morphology and density, while keeping spatial connection separate from gravitational binding.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Distinguish galaxies, groups and clusters from larger web patterns.
Calculate shape and density changes in an explicitly kinematic affine patch.
Explain why a visible connection or dense node does not by itself establish a bound system.

Galaxies, groups and clusters sit within larger patterns

Galaxies can belong to groups and clusters, while their distribution also forms larger filaments, walls and supercluster patterns. The labels describe different levels of organization and do not all imply one common dynamical state.

A filament is elongated; a sheet or wall is extended mainly in two directions; nodes mark concentrations or intersections in a chosen description. Matter and selected galaxy tracers need not reveal these patterns in exactly the same way.

A spatial connection or shared supercluster label does not generally prove that the entire grouping is gravitationally bound. Mass distribution, relative velocities, surroundings and a dynamical definition are needed for a binding assessment.

A local geometric example can show how directional compression changes shape and density. It cannot by itself construct the entire connected cosmic web, identify real cluster boundaries or solve those systems’ motions.

Count how many directions become short

Begin with a cube of comoving side L and assigned positive mass M. Use a prescribed map x_i = (1−Dλ_i)q_i along its three principal axes, where q_i and x_i are comoving coordinates.

D and the three λ_i are dimensionless. Display 0 ≤ D ≤ 1. D is a deformation parameter, not an assigned cosmic time, redshift or calculated growth factor. No law relating it to the scale factor is supplied.

Choose three separate patches with λ lists (0,0,3/4), (0,3/4,3/4) and (3/4,3/4,3/4). They have one, two or three shrinking directions respectively.

At a common dimensionless scale factor a, physical mapped volume is a³L³J, where J is the product of the three factors 1−Dλ_i. Every factor stays positive in the displayed interval, so the maps remain one-to-one.

The same assigned mass occupies different volumes

At D = 1, the sheet-like patch has axis lengths (1,1,1/4)L, the filament-like patch (1,1/4,1/4)L and the node-like patch (1/4,1/4,1/4)L in comoving units.

Their volume fractions J are 1/4, 1/16 and 1/64. For a uniform inventory in each patch, ρ_m = M/(a³L³J). Relative to its own unmapped reference ρ_ref = M/(a³L³), the density ratio is 1/J.

The D = 1 density ratios are therefore 4, 16 and 64. At the intermediate D = 0.5 they are 1.6, 2.56 and 4.096. Different directional compressions produce different volume changes even with the same initial side and assigned mass.

The node-like name identifies morphology in this construction. A large density ratio does not add a velocity distribution, a relaxation mechanism or a proof that the patch is a bound halo.

Separate prescribed affine patches at D = 1 and the same a, L and assigned M; no solved gravitational evolution
Patch shape	Comoving axes / L	Compressed directions	J	ρ_m/ρ_ref	Mapped mass/M
Sheet-like	1, 1, 1/4	1	1/4	4	1
Filament-like	1, 1/4, 1/4	2	1/16	16	1
Node-like	1/4, 1/4, 1/4	3	1/64	64	1

Density and volume compensate in the inventory

Multiplying each physical density by its physical mapped volume returns M. A larger density ratio is not creation of mass; it comes from the smaller assigned volume under this mass-preserving construction.

The reference ρ_ref belongs to the corresponding unmapped cube at the same a. It is not automatically the measured cosmic mean density or a fixed physical density across different epochs.

The first formal zero of a shrinking factor would occur at D = 4/3, outside the displayed interval. Continuing past a zero Jacobian would invalidate the one-to-one inventory and would not supply a physical treatment of orbit crossing.

The affine maps prescribe geometry only. They are not claimed exact or calibrated nonlinear gravitational evolution, a solved pressure model or a numerical simulation of how these particular patches formed.

Begin with three separate cubes of comoving side L and assigned positive mass M. Prescribe diagonal maps x_i = (1−Dλ_i)q_i, with dimensionless deformation D from zero to one. D is not assigned cosmic time, redshift or a solved growth factor. Choose λ tuples (0, 0, 3/4), (0, 3/4, 3/4), and (3/4, 3/4, 3/4), named sheet-like, filament-like and node-like. The three wireframe boxes at D = one use the same fixed oblique projection: one unit of each comoving axis projects as vectors (100, 0), (0, −100) and (50, −35) pixels. Projection lengths are drawing coordinates, not physical angles or a measured cosmic map. All twelve edges of each box are computed from its axis factors. The sheet has comoving axes (1, 1, 1/4)L, filament (1, 1/4, 1/4)L, and node (1/4, 1/4, 1/4)L. Their dimensionless volume fractions J are 1/4, 1/16 and 1/64. At a common dimensionless scale factor a, physical volume is a³L³J and uniform physical mass density is M/(a³L³J), relative to each own unmapped reference ρ_ref = M/(a³L³). Thus density ratios at D = one are 4, 16, 64, and physical density times physical mapped volume always returns the assigned M. The lower graph uses linear D from zero to one and J from zero to one. Solid blue circles identify sheet J = 1−3D/4; dashed orange squares filament J = (1−3D/4)²; dotted green diamonds node J = (1−3D/4)³. Samples at D = 0, 0.25, 0.5, 0.75, 1 include exact coincident initial markers at J = one. At D = 0.5 the density ratios are 1.6, 2.56 and 4.096. No factor is smaller than 1/4 in this domain. The first formal zero would be D = 4/3, outside the drawing; no orbit-crossing continuation is supplied. This is mass-preserving prescribed geometry, not an exact nonlinear gravitational evolution or a sequence of stages in one history. The reference is not a measured cosmic mean, and shape or high density does not prove binding. A visible network connection, tracer pattern, galaxy-bias relation, dynamical membership and full cosmic-web classifier require additional information. No observed box, calibrated formation time, halo or full web simulation is shown. Original Ark teaching model.

A network description needs an explicit field and method

A full web analysis must specify how the density or tracer field is estimated, which scales are smoothed, and what rule identifies ridges, sheets, nodes or connections. Different operational choices can emphasize different features.

Survey selection and galaxy bias also affect which structures are visible in a tracer map. A thin visual bridge is not by itself a measured mass connection, and a density threshold is not a dynamical binding criterion.

Gravitational theories and simulations can model anisotropic growth and connected patterns with stated initial conditions and approximation limits. Comparing those predictions with observations additionally needs the survey and uncertainty model.

Keep local shape, large-scale connectivity and bound membership as separate questions. Doing so lets groups, clusters, filaments and superclusters contribute to a coherent picture without assigning them all the same physical boundary or fate.

Worked reasoning

Compute one compressed patch

At D = 1, why is the sheet-like patch four times its unmapped reference density?

Its comoving sides are L, L and L/4, giving J = 1/4.
At the same a, physical volume is a³L³/4.
Fixed assigned mass M gives density 4M/(a³L³) = 4ρ_ref.

Its assigned volume is four times smaller at fixed mass and common scale factor. This is a geometric density comparison, not a binding result.

Try the reasoning

Which supplied patch has two compressed directions?
Show the answer and reasoning
The filament-like patch with λ = (0,3/4,3/4).
One axis remains L while the other two shrink by the chosen factors.
Does a visible filament connection establish that an entire supercluster is bound?
Show the answer and reasoning
No.
A spatial pattern supplies neither the complete mass and velocity information nor a dynamical binding calculation.
Are the three λ lists successive stages of one calculated cosmic history?
Show the answer and reasoning
No.
They define separate local kinematic comparisons; no evolution law for D(a) is solved.

What to carry forward

Groups and clusters occupy a web whose larger visual connections do not automatically imply common binding.
Directional compression changes morphology, volume and density in a mass-preserving geometric illustration.
A full cosmic-web inference requires an explicit field, classifier, dynamics and observational selection model.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.