# Groot onderzoek blog

## Inleiding in het Nederlands

Help! Ik weet niets over sterrenkunde / dit onderwerp ...

Het kan gebeuren dat je hier verzeild raakt en niets van het onderstaande begrijpt. Daarvoor heb ik hier inleiding in het Nederlands.

## In short

We are looking at a possible connection between the somewhat elderly technique of adhesion approximation and the more recently popular Morse theory. This concerns the theory of structure formation; to put it in a single question: "How does the current foam-like structure of the large scale distibution of galaxies relate to the initial density fluctuations in the Universe?". Adhesion is one of the simplest non-linear approximations there are of gravitational structure formation. In fact it is so simple it has an analytical solution. Morse theory is a topological way of identifying the structures in a density distribution. They share the property of looking at extrema in the potential field or density distribution.

## Klaar!

My thesis is ready :) get it here. Note: This is a version with rasterised figures for on-screen reading.

My advisor is Rien van de Weygaert

## Morphology #3

Morphology constraints from potential for clusters in red and filaments in blue, after smoothing the density on 5 different scales.

## Wall dynamics #2

It's raining galaxies in this wall!

## Different powerspectrum

Four runs with P(k) = kn. When n goes to zero, behaviour becomes more chaotic, and we need to integrate over more timesteps to prevent noise. These snapshots are all taken at z~1.

## Geometric method

From Lagrangian viewpoint it is possible to see whether a particle q is stuck at some time b or if it is still free. If its trajectory crosses that of another particle it is bound in some structure.

From Eulerian viewpoint it is possible to see where a particle that is at x now came from. if there is one solution, the particle is still free, otherwise it is bound. This way we can 'scan' the surface for particles that are just about to enter a structure.
To make the Lagrangian animation, we needed to enter some precision parameter to determine whether the parabola dipped the surface of the potential. Whereas the Eularian view looks if there is a sudden jump in the point connecting to the parabola. Also the Eulerian method doesn't need the derivative of the potential.

## 3D geometry

As you can see above, it is possible to connect a Lagrangian coordinate to an Eulerian one. This results in the function q(x). dq/dx is a measure of density, and it is what you see in the profile seen in the image above. Wherever there is a sudden jump in q(x) there is a structure. We need to improve the method to subpixel precision, and identify the nodes an filaments that are detected this way.

## Convex hull method

Three timesteps in a 1D situation. Notice the slinking areas.

## Fast Legendre Transform

Using the method described in Vergassola et al. (1994), we made a 2D simulation, and it seems to work (click to enlarge). The same is also working in 3D.

## dx/dq vs. dq/dx

The first image is the derivative of the map x(q), the second is the derivative of q(x) of the same area centered on a void.

## Structures identified

Using the lagrangian <-> eulerian maps it is possible to extract the voids, clusters and filaments. Left to right, top to bottom: deriv. of Eulerian map; deriv. of lagr. map; voids found in L-space projected back in E-space; boundary points of voids.

## New Movie!

There is a problem with the accuracy of the Legendre Transform that is illustrated next. This is a zoom-in on the L-space representation of a 2D simulation. The red pixels are stuck particles, the yellow ones are free, black are stuck boundary particles. The flood fill was done with 4-connectivity, but the problem is also present when using all 8 surrounding pixels to do the fill. We define a void as a single connected area on this map. The highligthed area shows points where voids are connected, then disconnected, to be connected again in the next frame. This "blinking" probably arises due to inaccuracy in the calculation of the Legendre transform.

## Smooth density extracted (in 2D)!

Density at ~ D = 0.1.
The image shows the square root of the density to enhance contrast in the voids. (at ~ D = 0.5)

Using a bi-cubic interpolation scheme, I was able to extract a smooth density. After finding the Legendre transform of the Lagrangian potential using FLT, the interpolation helped to find a sub-pixel, minimum value where the discrete minimum was used as starting point.

Here is another animated .gif of a 2D smooth animation showing both E and L space.

## Smooth density in 3D

It required a tri-cubic spline interpolation to minimize every voxel in this image, which is relatively heavy on computation time. But the result is very nice. This is a 1283 voxel cube spanning 256 Mpc. Next is the Lagrangian "space density" of a 3D simulation. This shows the voids in red surface contour and the hyper-facets of the 4D convex hull are shown with as a volume plot.

## A Watershed Pollock

This watershed image reminds me of some works by Jackson Pollock.

## Powerlaw cosmologies

Pure power-law cosmologies are numerically more unstable than LCDM. We may fix this by blurring the initial conditions a bit, losing some resolution. However the following result was still obtained without blurring.

From top to bottom (P(k) = k^n) n = 1, 0, -1; which is similar to having n = 0, -1, -2 in 3D. From left to right is a time series.

## Problems going to 3D

When going to 3 dimensions, new problems arise. The connectivity in Lagrangian space no longer suffices to detect voids, as can be seen in the above image. Only at the latest times can we detect most of the structures. From top to bottom: Lagrangian domain of adhesion mapping; boundaries of connected regions in Eulerian space; density (logarithmic colour map).

Last updated september 19th 2009, Contact me at: