The Delaunay tessellation field estimator (DTFE), (or Delone tessellation field estimator (DTFE)) is a mathematical tool for reconstructing a volume-covering and continuous density or intensity field from a discrete point set. The DTFE has various
astrophysical applications, such as the analysis of
numerical simulations of
cosmic structure formation, the mapping of the
large-scale structure of the universe
The observable universe is a ball-shaped region of the universe comprising all matter that can be observed from Earth or its space-based telescopes and exploratory probes at the present time, because the electromagnetic radiation from these obj ...
and improving computer simulation programs of cosmic structure formation. It has been developed by Willem Schaap and Rien van de Weijgaert. The main advantage of the DTFE is that it automatically adapts to (strong) variations in density and geometry. It is therefore very well suited for studies of the large scale galaxy distribution.
Method
The DTFE consists of three main steps:
Step 1
The starting point is a given discrete point
distribution. In the upper left-hand frame of the figure, a point distribution is plotted in which at the center of the frame an object is located whose density diminishes radially outwards. In the
first step of the DTFE, the
Delaunay tessellation of the point
distribution is constructed. This is a volume-covering division
of space into triangles (tetrahedra in three dimensions), whose
vertices are formed by the point distribution (see figure, upper right-hand frame). The Delaunay tessellation is defined such
that inside the interior of the circumcircle of each Delaunay triangle
no other points from the defining point distribution are present.
Step 2
The Delaunay tessellation forms the heart of
the DTFE. In the figure it is clearly visible that the tessellation
automatically adapts to both the local density and geometry of the
point distribution: where the density is high, the triangles are small
and vice versa. The size of the triangles is therefore a measure of
the local density of the point distribution.
This property of the Delaunay tessellation is exploited in step 2 of
the DTFE, in which the local density is estimated at the locations of
the sampling points. For this purpose the density is defined at the
location of each sampling point as the inverse of the area of its
surrounding Delaunay triangles (times a normalization constant, see
figure, lower right-hand frame).
Step 3
In step 3 these density estimates are interpolated to any other point,
by assuming that inside each Delaunay triangle the density field varies
linearly (see figure, lower left-hand frame).
Applications
An atlas of the nearby universe
One of the main applications of the DTFE is the rendering of our cosmic neighborhood.
Below the DTFE reconstruction of the
2dF Galaxy Redshift Survey
In astronomy, the 2dF Galaxy Redshift Survey (Two-degree-Field Galaxy Redshift Survey), 2dF or 2dFGRS is a redshift survey conducted by the Australian Astronomical Observatory (AAO) with the 3.9m Anglo-Australian Telescope between 1997 and 11 A ...
is shown, revealing
an impressive view on the cosmic structures in the nearby universe. Several
supercluster
A supercluster is a large group of smaller galaxy clusters or galaxy groups; they are among the largest known structures in the universe. The Milky Way is part of the Local Group galaxy group (which contains more than 54 galaxies), which in turn ...
s stand out, such as the
Sloan Great Wall
The Sloan Great Wall (SGW) is a cosmic structure formed by a giant wall of galaxies (a galaxy filament). Its discovery was announced from Princeton University on October 20, 2003, by J. Richard Gott III, Mario Jurić, and their colleagues, b ...
, one of the largest structures in the universe.
Image:2dfgrs.png, The 2dF Galaxy Redshift Survey
Image:2dfdtfe.gif, DTFE reconstruction of the inner parts of the 2dF Galaxy Redshift Survey
Numerical simulations of structure formation
Most algorithms for simulating cosmic structure formation are
particle hydrodynamics codes. At the core of these codes is the
smoothed particle hydrodynamics
Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid flows. It was developed by Gingold and Monaghan and Lucy in 1977, initially for astrophysica ...
(SPH) density estimation procedure.
Replacing it by the DTFE density estimate will yield a major
improvement for simulations incorporating feedback processes,
which play a major role in
galaxy
A galaxy is a system of stars, stellar remnants, interstellar gas, dust, dark matter, bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar System. ...
and
.
Cosmic velocity field
The DTFE has been designed for reconstructing density or intensity
fields from a discrete set of irregularly distributed points sampling
this field. However, it can also be used to
reconstruct other continuous fields which have been sampled at the
locations of these points, for example the cosmic velocity field. The use of the DTFE for this purpose has
the same advantages as it has for reconstructing density fields. The
fields are reconstructed locally without the application of an artificial or user-dependent
smoothing procedure, resulting in an optimal
resolution
Resolution(s) may refer to:
Common meanings
* Resolution (debate), the statement which is debated in policy debate
* Resolution (law), a written motion adopted by a deliberative body
* New Year's resolution, a commitment that an individual mak ...
and the suppression of
shot noise
Shot noise or Poisson noise is a type of noise which can be modeled by a Poisson process.
In electronics shot noise originates from the discrete nature of electric charge. Shot noise also occurs in photon counting in optical devices, where shot ...
effects. The
estimated quantities are volume-covering and allow for a direct
comparison with theoretical predictions.
Evolution and dynamics of the cosmic web
The DTFE has been specifically designed for describing the
complex properties of the cosmic web. It can therefore be used to study the evolution of voids and superclusters in the large scale matter galaxy distribution.
Image:voidevolution.jpeg, Evolution of a void
Image:filamentevolution.jpeg, Evolution of a supercluster
External links
DTFE: the Delaunay Tessellation Field Estimator Willem Schaap, 2007, PhD Thesis, Rijksuniversiteit Groningen, The Netherlands
*
Probing cosmic velocity flows in the local universe Emilio Romano-Diaz, 2004, PhD Thesis, Rijksuniversiteit Groningen, The Netherlands
The cosmic web: geometric analysis Rien van de Weygaert and Willem Schaap, 2004
{{Portal bar, Physics, Mathematics, Astronomy, Stars, Spaceflight, Outer space, Solar System
Large-scale structure of the cosmos
Cosmological simulation
Astronomical surveys
Geometric algorithms