Gravity grows a small ripple

Cosmic Web and Shell Crossing

Why does nearly uniform matter form a cosmic web? Gravity draws more matter into slightly denser regions. Compression proceeds at different rates along different directions, forming sheets and filaments around emptier volumes. Collisionless matter can pass through the same place with different velocities, so its motion keeps a folded history in phase space.

Change an initial ripple and watch matter fold into sheets and filaments. Compare a displacement approximation with three-dimensional mesh gravity.

Expansion scale factor $1.09$

RMS density contrast $6.36$

Peak mesh density / mean $318$

Power in one long-wave mode $2790\,\mathrm{Mpc}^3$

Grow a cosmic web

Initial ripple amplitude

2.10

A

Expansion scale factor

1.09

a

Overlay the displacement reference

Drag the gold handle to change the initial ripple. Drag the background to turn the box.

The folded default is visible; computing its full evolution.

Coordinates, resolution and export

$L=20\,\mathrm{Mpc},\quad N_p=N_g^3=16^3$

The box is comoving. Physical distances also grow with the expansion scale. Each point represents equal mass; colour shows the smoothed mesh density.

$\boldsymbol r=aL\boldsymbol y,\qquad \boldsymbol v_{\rm pec}=\frac{H_\star L}{a}\boldsymbol u$

The model has collisionless matter and matter-only expansion. Gas, dark energy and galaxy light are omitted. Symmetric presets compare directions; a dense node is a coarse structure, before any halo definition.

The displacement reference keeps the initial growing map after crossing. The solid coloured particles follow updated mesh gravity. Playback selects native saved frames.

$M=1.09\times10^{15}\,\mathrm{M}_\odot$

Model assumptions

This teaching model evolves collisionless matter in a periodic, twenty-megaparsec comoving box in an Einstein–de Sitter universe. Equal-mass particles start on a regular lattice with a few specified waves. Mesh gravity uses sixteen cells along each axis, CIC assignment, a fixed Fourier cutoff and a four-point potential gradient. It resolves large-scale folding; dense nodes are not identified halos. Gas, dark energy, baryonic feedback and galaxy light are absent. The displacement reference continues its initial growth after crossing, whereas mesh gravity follows the changing density. The model saves one hundred and twenty-nine native states without time interpolation. Colours show mesh density. The default snapshot and five presets have independent numerical checks; their initial waves are synthetic and are not an IllustrisTNG sample.

Mesh densityDisplacement referenceOne material line

A section through the middle of the mesh

The same density scale colours the box and this section. The highest colours saturate.

How a long wave grows

$P_{100}=L^3\left|\frac1{N_p}\sum_j e^{-2\pi i y_{j,x}}\right|^2$

Both curves use the same initial waves. The dotted curve extrapolates the displacement map through crossing.

Start with the essentials

Focus question: What changes when particles first pass through one another?
One-sentence intuition: Several velocities can coexist at one position; density sums their positive mass.

Core mathematical model

A displacement that grows with expansion

\boldsymbol y(\boldsymbol q,a)=\boldsymbol q+a\,\boldsymbol s(\boldsymbol q),\qquad \boldsymbol r=aL\boldsymbol y

The first relation is the Zel’dovich reference in this matter-only universe. The labels identify the same matter. The second relation converts the displayed comoving box into physical position.

The first fold

J=\det\!\left(I+a\,\frac{\partial\boldsymbol s}{\partial\boldsymbol q}\right),\qquad \frac{\rho}{\bar\rho}=\frac1J\quad\text{before crossing}

Before crossing, a smaller volume holds the same mass. At the first fold this one-stream formula becomes singular. After crossing, each stream contributes positive mass; a negative determinant marks reversed orientation.

Gravity follows the changing density

\nabla_y^2U=\frac32\delta,\qquad\frac{d\boldsymbol y}{da}=a^{-3/2}\boldsymbol u,\qquad\frac{d\boldsymbol u}{da}=a^{-1/2}(-\nabla_yU)

Mesh gravity updates the density and force at every integration step. The dimensionless canonical momentum includes the expansion factor. The finite mesh smooths small structures.

Common difficulties

Crossing does not stop the streams

Typical misconceptionA dense sheet must make all particles stick together.

Better mental modelThe model follows collisionless matter. Streams pass through and continue responding to gravity; gas shocks would require another model.

A dense node and a halo

Typical misconceptionEvery bright knot already measures a halo mass.

Better mental modelColour shows coarse mesh density. Identifying a halo needs a separate definition and adequate force and mass resolution.

Run the experiment

01
Open the sheet
Choose A sheet folds, then move the time slider across the collapse.
What to observe: The selected material line folds. Its position–velocity plot separates streams that overlap in position.
02
Change the original ripple
Drag the gold handle horizontally. Compare the new displacement reference with the completed mesh calculation.
What to observe: A larger initial ripple reaches strong compression earlier. The camera can turn independently.
03
Compare the two evolutions
Choose An oblique web and enable the displacement reference. Inspect the power curve below the box.
What to observe: Both start from the same waves. Mesh gravity responds to the later density, so their positions and long-wave power separate.

Sources and numerical scope

The displacement map follows Zel’dovich’s 1970 approximation. The cosmological variables, CIC mesh method and time factors follow the cited GADGET-2 methods; this small model uses a fixed low-pass mesh force.

Independent checks cover matter-only linear growth, actual half-cell particle response, positive mass deposition, a periodic sheet reference across crossing and a separate adaptive integrator. The finite mesh remains a large-scale teaching model.

Cosmic Web and Shell Crossing

Matter can fold without sticking together

Core mathematical model

A displacement that grows with expansion

The first fold

Gravity follows the changing density

Common difficulties

Crossing does not stop the streams

A dense node and a halo

Run the experiment

Open the sheet

Change the original ripple

Compare the two evolutions

Sources and numerical scope

Related experiments and articles

Expansion, crossing and independently tested gravity