Abstract:
Cosmic Microwave Background (CMB) is a relic from the early Universe. It was
generated due to the physical processes in the early Universe during an epoch
known as the recombination or decoupling epoch. The CMB has highly uniform
temperature over the entire sky but with small variations in different directions. Due to the Thomson scattering between photons and electrons and also
because of the quadrupole anisotropies induced in the cosmic plasma during
the decoupling epoch, the CMB is linearly polarized. The CMB radiation in each
line of sight is associated with temperature (T) and polarization. The polarization can be decomposed into Stokes parameters Q/U, or E mode (E)
and B mode (B) fields. Here, Q/U fields transform as spin ±2 objects under
rotation transformation while the E/B fields remain invariant. The fluctuations observed in CMB is due to the quantum fluctuations generated during the
inflationary phase, which is a period of exponential expansion moments after
the Big Bang in the early Universe. The CMB fluctuations will have statistical
properties similar to this primordial fluctuations only for the linear evolution of
fluctuations. Statistical observable can be used to capture the morphological
properties of the CMB fluctuations. Then these morphological properties can
be studied in relation to the parameters describing the physical mechanisms of
the inflationary phase. In this research work, we use the geometrical and topological observables to study the CMB polarization fields and we also introduce
a novel statistical observable for the analysis of CMB fields.
The models about the inflationary phase predict that the Probability Distribution
Function (PDF) of primordial fluctuations are close to the Gaussian distribution
but with small deviation. The information about the exact form of deviation
in the PDF of primordial fluctuation will be encapsulated in the CMB fields. We
investigate the local type non-Gaussian features in the CMB polarization fields,
which is parametrized by fNL. We derive the analytic expression for the PDF
of any general local type non-Gaussian field such as the T and E fields of CMB,
and also for the local type non-Gaussian polarization intensity (IP ) constructed
from local type non-Gaussian Q and U fields. We use the analytic expression
and simulations of local type non-Gaussian CMB fields, namely T, E and IP ,
and study the deviation in their PDF relative to the Gaussian PDF. We found
from the analytic expression that the non-Gaussian deviation in the PDF of the
T and E fields are proportional to (fNLσ) while that of IP field is proportional
to (fNLσ)
2
. The numerical calculations show that the non-Gaussian deviation
in the PDF of E field is similar to that of the T field. While the non-Gaussian
deviation corresponding to the IP field has smaller amplitude and large error
bars in comparison to that of T field. This analysis was repeated using the
geometrical and topological observables, namely Scalar Minkowski Functionals
(SMFs) and Betti numbers of fields. These observables capture different morphological features of a given field. The results obtained using these observables
are similar to those from the PDF of the fields. Hence from the theoretical point
of view, these results imply that the E field can provide an independent constraint on fNL similar to the T field. Further, the results also show that when
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the IP field is used independently for such analysis, it cannot provide any statistically significant information. In the realistic scenario, the observational
data contains instrumental systematics which will lead to the reduction in the
statistical significance of the above results.
The CMB polarization is usually analyzed using the E/B fields as they are
scalar fields. We investigate the theoretical aspects of using the Q/U fields
as a complementary analysis of CMB polarization. We show that the variance
of Q/U and its gradient fields are invariant under rotation transformation,
and hence the invariance of the SMFs of a Gaussian Q/U fields. However, this
statement breaks down for incomplete sky. Then we studied the non-Gaussian
deviation in Q/U fields constructed from the simulations of local type nonGaussian E field. These simulations use the same x−y coordinates along each
line of sight. We found that its amplitude is about an order of magnitude
lower than that of T field and has different shape. This finding will be useful
for distinguishing different non-Gaussian signals in the observational data from
future experiments. Further, we studied the effect of the presence of primordial
tensor perturbation, which is parametrized by r, on the SMFs of Q/U and IP
fields, and the number density of singularities in IP field. Here, a singularity is
a point on the CMB field where IP = 0. We found that the amplitude of SMFs of
these fields are sensitive to the presence of primordial tensor perturbation and
it decreases with r. We also show that the number density of singularities in IP
field decreases with r. This finding will be useful for the searches of primordial
tensor perturbation in the future experiments. The instrumental systematics
in the observational data will decrease the statistical significance of the above
results.
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We introduce Tensor Minkowski Functionals (TMFs), which are tensor generalization of Minkowski Functionals, as a new statistical observable for the
analysis of CMB data. Since these are tensor quantities, they are capable of
capturing more morphological properties in a given field than their scalar counterparts. We have developed a code, referred as TMFCode, to compute the TMFs
for any general field on an Euclidean plane. In order to apply the TMFs, specifically W
1,1
2 which is a tensor of rank 2, to CMB fields which lies on a 2-d spherical
surface, we map each point on the sphere with a point on a plane using stereographic projection. The code computes W
1,1
2
, and then the net anisotropy (β)
and net orientation (α) of the structures are estimated. We investigated the numerical error in this computation due to pixelization. We found the error in β
increases with the increasing curvature of the boundaries of the structure. The
error in α is negligible when the structures are completely unoriented with each
other and it increases as the structures become more and more aligned with
each other. We present the numerical calculation of the systematic variation of
α and β with the threshold value for the simulated Gaussian and isotropic CMB
T and E fields. We found that the value of β shows a characteristic variation
with the threshold value while α is flat. We show that according to the standard model, β = 0.62 for T and β = 0.63 for E, where the values are corrected
for pixelization. The value of α is one for both the fields, which is as expected
for an isotropic field.
We applied W
1,1
2
for the analysis of PLANCK data as an illustration of its
application. The instrumental systematics and the gravitational lensing due to
large scale structure affects the morphological features of the CMB fields. We
study the effect of these factors on the value of α and β using the simulations
of CMB frequency bands, namely 44GHz and 70GHz provided in PLANCK data,
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which contains the respective instrumental characteristics. We found that the
percentage difference in α and β due to these factors are less than 2% and it
significantly increases the size of their error bars. We use the CMB simulations
corresponding to the frequency band 44GHz as the basis for testing the consistency of different PLANCK data sets with theoretical expectations. We estimated
the deviation in α and β for the foreground cleaned CMB maps namely SMICA,
COMMANDER, SEVEM and NILC corresponding to full mission, half mission 1, half
mission 2, half ring 1 and half ring 2 provided in the PLANCK data. These calculations showed that β is consistent with the standard model within 2 − σ for
all data sets, except the T map of NILC half mission 2 which has slightly higher
deviation. We found the values of α for T map of different data sets to be in
excellent agreement with the standard model within 1.2 − σ. The deviation in
α of E map of all data sets are higher than 3 −σ except the SMICA full mission
data. Further, α for E map corresponding to the half mission 1 of all data
sets showed consistently higher deviation of 5 − σ. These results imply that
the structures in the E map has an extent of alignment with each other. This
alignment could be cosmological or due to instrumental systematics. Since we
are comparing the PLANCK maps which are obtained by co-adding all frequency
bands with that of the simulations with the instrumental characteristics of a
specific frequency band, namely 44GHz, the instrumental systematics is more
probable reason for the alignment measured in E map.