What is the Bispectrum

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COSMOLOGY
Identifying Dust in the MBR using the Bispectrum
Vishal Kasliwal, Emory F. Bunn, Molly McCann, Rom
Chan Physics Department, University of Richmond,
VA 23173, USA
The Microwave Background Radiation (MBR) is
microwave radiation detectable over the entire
sky. The radiation is red-shifted light dating
back to shortly after the creation of the
Universe in the big bang. MBR data is a set of
temperature fluctuation values around the MBR
mean of 2.73K. Inflationary models of the early
Universe predict that the statistical variations
in this data follow a Gaussian distribution.
Unfortunately, Galactic dust generates a highly
non-Gaussian signal making it essential to come
up with a means of identifying dust-contaminated
portions of the sky in order to test whether the
theoretical results hold on the real data set.
The focus of our summer research was to find
statistical tests to quantify the amount of dust
contamination in small sections of the sky.
WWMAP Data
Aim The aim of the project is to evaluate the
value of the Bispectrum as a statistical test for
quantifying the amount of dust present in a small
section of the sky. The main difficulty in
quantifying the amount of dust in MBR lies in the
fact that while it is possible to explicitly
state what makes the statistical properties of a
map Gaussian, we do not know precisely what the
statistical properties of dust are, and hence it
is not possible to create a filter for dust. It
is hoped that the presence of dust will make
itself obvious in the bispectrum.
Procedure Computer code was written using IDL to
generate a number of random Gaussian maps, and
to extract an equal number of Dust maps from an
online archive of Dust maps. The bispectrum was
calculated for each Gaussian map and Dust map and
the results averaged over all the maps. The maps
generated were 10-degrees on a side, and were
modified to replicate the finite angular
resolution and noise characteristics of the WMAP
probe.
GGaussian Maps
DDust Maps
TThe WMAP probe
Technicalities Any instrument used to measure
MBR (e.g. the WMAP satellite) has finite angular
resolution and generates some noise. This has to
be factored into our calculation of the
bispectrum. Thus all our maps are first
smoothened so that the scale of the smallest
features on our maps matches the angular
resolution of the WMAP probe. Gaussian noise is
then added to the maps to reproduce the
electronic noise introduced by the probe. Another
technicality is that when performing a Fourier
transform, it is assumed that the data set
represents a period of some periodic function.
Thus the ends of the data set must match. This is
done by multiplying the data set by a function
that drops to zero at the ends, and the process
is known as anti-aliasing. A nice feature of the
bispectrum is that if the vectors forming the
triangle do not sum to zero, then the
contribution to the Bispectrum of all such
triangles is zero. This aids significantly when
computing the bispectrum.
What is the Bispectrum? The bispectrum is the
three point correlation function of the Fourier
transform of a map. We calculate the bispectrum
for small patches of the sky that may be treated
as flat. The Fourier transform of the map gives
us an idea of the scale of features present on
the map. Dust typically shows up in the form of
highly correlated nebulous patches much like in
pictures of nebulae taken in visible light.
Gaussian features show up as un-correlated
random patches. The bispectrum takes the Fourier
transform of these maps and picks all sets of
three points in vector space, and averages the
product of the Fourier transform values at these
three points for all sets of three points that
share some characteristic, for example, they may
all form identical triangles. Thus the bispectrum
becomes a function of the geometry of all
possible triangles. Since Gaussian maps are by
nature un-correlated, we expect that the
bispectrum of a Gaussian map would average out to
zero for a given triangle geometry, while since
dust maps are highly correlated , we would expect
the Bispectrum value to be non-zero for any given
triangle geometry.
The Power Spectrum A Two Point Analogue of the
Bispectrum The power spectrum is identical to
the bispectrum except that unlike the Bispectrum
which is a three-point function in Fourier space,
the power spectrum is a two-point function in
Fourier space. The power spectrum is used to
state what a Gaussian map is in very precise
terms. A Gaussian map is a map for which all the
information contained in the map is given by the
power spectrum. When we test a map for
non-Gaussianity, we are trying to determine
whether it contains information beyond the power
spectrum.
The Power Spectrum
P(k)
GGaussian Bispectrum
DDust Bispectrum
k
(only linear triangles shown)
CCurrent Work WWe are in the process of selecting
a good statistical test to pry information from
the bispectrum. Additionally, we are gathering
bispectrum data from larger sets of larger
maps. Since the bispectrum is faster to compute
than the three-point function is, we hope to find
a relation between the two functions, enabling us
to obtain the three-point function from the
bispectrum. Lastly we are looking at the accuracy
of a stochastic computation of the bispectrum,
which would greatly reduce the amount of time
required to compute the bispectrum.
The Results The bispectrum is a large function in
that it is a function of three variables
specifying the shape of a triangle in Fourier
space. As such, it contains a wealth of
information, and we are still in the process of
trying to find a good statistic to distinguish
between the bispectra of Gaussian maps and dust
maps. One feature that we have regularly observed
is that, as expected, the average of the
bispectrum of a Gaussian map is near zero while
the average of the bispectrum of a dust map is
distinctly non-zero. The problem is compounded by
the fact that computing the bispectrum is a very
computationally intensive task that takes a long
time to perform. Thus it is hard to obtain
sufficient amounts of data.