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Data Analysis Functionalities

With the aid of visualization, AstroMD allows the user to do several types of analyses of observational or simulated data. The scientist can quickly select the region of interest, using AstroMD sub-region selector, and the analysis modules can be applied to one. Moreover the analysis is done on click of mouse: the results are displayed in a few seconds.
The main following implemented functionalities are specifically directed at cosmological simulation data outputs. Some of them are statistical tools that one can deploy in order to discriminate among different cosmological models.

Particles Mass Density and Gravitational Field Calculations By using an eight-point Cloud-in-Cell smoothing algorithm, AstroMD computes the mass density field $\rho({\bf x})$ associated to the particle distribution by distributing the mass of each particle over the computational mesh. The computation can be done with the maximum accuracy, that is by considering all the particles over a uniform high resolution mesh. The user can also use a sample of the whole set of particles thus reducing the CPU time consumption and the memory request. The smoothing of the masses can be generally performed by using a coarse grid that can be refined where high resolution is necessary.

Power Spectrum and Correlation Function - The quantity $\rho({\bf k})$ is used to calculate the Power Spectrum $P(k)$ of the matter distribution, which is defined as the average value of the square norm of $\rho({\bf k})$:

\begin{displaymath}
P(k) = \langle \mid \rho({\bf k} \mid^{2} \rangle.
\end{displaymath} (2)

The Power Spectrum expresses the weight of each Fourier component of the mass distribution between $k_{min}$ and $k_{max}$ which represent the inverse of the size of the computational mesh and the Nyquist frequency, respectively. The spatial distributions shown in figures 1.31 and 1.32 (top) are about the visualization of the output data from the FLY code, for a box having size of $50 h^{-1}$ Mpc and with 262144 bodies, randomly extracted from a file with $256^{3}$ particles. It was considered the $\Lambda CDM$ model with $\Omega _{\Lambda} = 0.7$ at different redshifts.
Figure 1.31: Examples of the computation of the Power Spectral Density (PDS) with AstroMD, applied to data from FLY at a redshift of about zero, for different cell resolution values. The spectral index is approximately $2$ up to the turn-over, so Power Spectrum converges at large scale (small $k$). After the turn-over PSD decades as a power law with an exponent of approximately $-4$.
\begin{figure}\centerline{\psfig{file=becciani/AstroMD_clusterfield4_PS_1.ps,hei...
...{\psfig{file=becciani/AstroMD_clusterfield4_PS_3.ps,height=4.2cm}}\end{figure}
Figure 1.32: Graphical outputs of the Power Spectrum of data from FLY at a redshift of approximately $10$.
\begin{figure}\centerline{\psfig{file=becciani/AstroMD_clusterfield1_PS_1.ps,hei...
...}\psfig{file=becciani/AstroMD_clusterfield1_PS_3.ps,height=3.2cm}}\end{figure}

Several estimators have been used in the literature to measure, in particular, the two-point Correlation Function, that is defined in terms of probability $\delta P$ of finding a point in a randomly-chosen volume $\delta V_{1}$ and a point in another volume $\delta V_{2}$ separated by a distance $r$.
The two-point Correlation Function of AstroMD is based on the three-dimensional counterpart of the Peebles & Hauser estimator (1974):
\begin{displaymath}
\xi_{PH} = \frac{DD(r)}{RR(r)} (\frac{N_{rd}}{N}) - 1,
\end{displaymath} (3)

where $N_{rd}$ is the number of points of an auxiliary random sample, $DD(r)$ is the number of all pairs of points with separation inside the interval $[r-dr/2, r+dr/2]$, and $RR(r)$ is the number of pairs between the data and the random sample with separation in the same interval.

Minkowski Functionals The Minkowski Functionals (MFs) provide a novel tool to characterize the Large Scale Structure of the Universe. They describe the Geometry, the Curvature and the Topology of a point-set.
In a three-dimensional Euclidean space, these functionals have a direct geometric interpretation as listed in Table 1. The MFs algorithm inside AstroMD associates a ball of radius $r$ to each point of the point distribution. The size, the shape and the connectivity of the spatial pattern, formed by the union-set of these balls, change with the radius, which can be employed as a diagnostic parameter.

Table 1.1: Geometric interpretation of the Functionals. The first Functional equals the volume $V$ of the body, the second one is the surface area $A$. The third Functional corresponds to the integral mean curvature $H$ of the body's surface and provides information about the shape.
geometric quantity $\mu$ $M_{\mu}$ $V_{\mu}$  
V volume $0$ $V$ $V$  
A surface $1$ $A/8$ $A/6$  
H mean curvature $2$ $H/2 \pi^{2}$ $H/3\pi$  

Friend-of-Friend Algorithm Dynamical studies of groups of galaxies are an important method for estimating galaxy masses. The most common group-finding algorithm is known as Friend-of-Friend (FoF). This technique was first used by Huchra & Geller. A particle belongs to an FoF group if it lies within some linking length $\epsilon$ of any other particle in the group.
After all such groups are found, those with less than a specified minimum number of group members $num_{Members}$ are rejected.
AstroMD visualizes the positions of all grouped particles ,each of them being marked with their group identifier (see figure 1.33), and the positions of all centers of mass that are associated with the number of particles in the group; the radius, velocity (if present in input) and total mass are also visualized.

Figure 1.33: The FoF group finding algorithm inside AstroMD.
\begin{figure}\centerline{\psfig{file=becciani/AstroMD_FoF_1.ps,height=6.2cm}}\end{figure}

next up previous contents index
Next: VO-TECH Project Up: AstroMD a visualization tool: Previous: AstroMD a visualization tool:   Contents   Index
Innocenza Busa' 2005-11-14