STATISTICAL ISOTROPY of CMB ANISOTROPY

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STATISTICAL ISOTROPYofCMB ANISOTROPY
Amir Hajian Tarun Souradeep
I.U.C.A.A, Pune
http//www.iucaa.ernet.in/tarun/pascos03.ppt
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Cosmic Microwave Background a probe beyond
the cosmic horizon
Pristine relic of a hot, dense smooth early
universe - Hot Big Bang model
Pre-recombination Tightly coupled to, and in
thermal equilibrium with, ionized
matter. Post-recombination Freely propagating
through (weakly perturbed) homogeneous
isotropic cosmos.
CMB anisotropy and Large scale Structure formed
from tiny primordial fluctuations through
gravitational instability
Simple linear physics allows for accurate
predictions Consequently a powerful cosmological
probe
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Fig. M. White 1997
The Angular power spectrum of the CMB anisotropy
depends sensitively on the present matter
current of the universe and the n spectrum of
primordial perturbations
The Angular power spectrum of CMB anisotropy is
considered a powerful tool for constraining
cosmological parameters.
Fig.. Bond 2002
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Statistics of CMB

• smooth random function on a sphere (sky map).
• General random CMB anisotropy described by a
• Probability Distribution Functional
• Mean
• Covariance
• (2-point correlation)
• ...
• N-point correlation

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Statistics of CMB
The primordial perturbations are believed to be
Gaussian random field. The most popular idea
about their origin is quantum fluctuations during
inflation.
Gaussian Random CMB anisotropy
Completely specified by the covariance matrix
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Statistics of CMB
CMB anisotropy completely specified by
the angular power spectrum
Only if

• Statistically isotropic Gaussian random CMB
  anisotropy

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Statistics of CMB

• smooth random function on a (pix sphere.
• General random CMB anisotropy described by a
  probability distribution functional assigning a
  number (probability) to every CMB anisotropy sky
  map
• Mean
• Covariance(2-point correlation)
• ..
• N-point correlation
• Gaussian random CMB anisotropy
• Completely specified by the covariance
  
• Statistically isotropic Gaussian random CMB
  anisotropy

Statistical Isotropy means the two point
correlation function depends only on the angular
separation between the two directions in the sky.
Completely specified by angular power spectrum
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Statistics of CMB
Statistical Isotropy implies the two point
correlation function depends only on the angular
separation between the two directions in the sky.

• smooth random function on a
• ) sphere.
• General random CMB anisotropy described by a
  probability distribution functional assigning a
  number (probability) to every CMB anisotropy sky
  map
• Mean
• Covariance(2-point correlation)
• ..
• N-point correlation
• Gaussian random CMB anisotropy
• Completely specified by the covariance
• Statistically isotropic Gaussian random CMB
  anisotropy

i.e., Correlation is invariant under rotations
Completely specified by angular power spectrum
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Iso-contours of correlation around a point
Radical breakdown of SI disjoint iso-contours
multiple imaging
Mild breakdown of SI Distorted iso-contours
Statistically isotropic (SI) Circular iso-contours
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Beautiful Correlation patterns could underlie
the CMB tapestry
Figs. J. Levin
Akin to Leopards spots
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Can we Measure Correlation Patterns?
Sure, maybe quite possible for Leopard spots
BUT for CMB anisotropy the COSMIC CATCH is
there is only one CMB sky.
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Measuring the correlation

• Statistical isotropy
• can be well estimated
• by averaging over the temperature
• product between all pixel pairs
• separated by an angle .

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Measuring the correlation

• Violation of statistical isotropy
• Estimate of the correlation function from
• a sky map given by a single temperature
• product
• is poorly determined.
• is inadequate for model
  comparison

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A Measure of Statistical Anisotropy


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A Measure of Statistical Anisotropy


Wigner rotation matrix
Characteristic function
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A Measure of Statistical Anisotropy


is the three dimensional rotation through an
angle about the axis
Wigner rotation matrix
Characteristic function
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A Measure of Statistical Anisotropy


A weighted average of the correlation function
over all rotations
Except for
when
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Why is a measure of statistical
anisotropy.
statistical anisotropy.
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Statistical Isotropy


Correlation is invariant under rotations
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What exactly are
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In Harmonic Space

• Correlation is a two point function on a sphere
• Suggests a bipolar spherical harmonics expansion
• 
  
• Bipolar spherical harmonics.

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In Harmonic Space

• Correlation is a two point function on a sphere
• 
  Bipolar spherical harmonics.
• Inverse-transform

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• Statistical isotropy

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What if we find Statistical anisotropy in CMB
maps
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Sources of Statistical Anisotropy

• Ultra large scale structure and cosmic topology
  GR is a local theory and does not dictate the
  global topology of space-time. Space can be
  multiply connected, e.g. Torus universe with
  Euclidean geometry.

SIGNAL

• Observational artifacts
• Anisotropic noise
• Non-circular beam
• Incomplete/unequal sky coverage
• Residuals from foreground removal

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Ultra Large scale structure of the universe
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How Big is the Observable Universe ?
Relative to the local curvature topological
scales
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Simple Torus (Euclidean)
Consider all Spaces of Constant Curvature
Homogenous isotropic but Multiply connected
universe ?
Compact hyperbolic space
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A Toroidal Universe
The Euclidean 2-torus is a flat square whose
opposite sides are connected.
Light from the yellow galaxy can reach them along
several different paths. So they can see more one
image of it.
Pictures Weeks et. al. 1999
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Spatial Correlations in

• Simply connected space
• (STATISTICALLY ISOTROPIC)

• A Toroidal Space

Iso-correlation contours are no more circular.
Back
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THREE POSSIBILITIES ( Size of the
compact space relative to horizon scale)
Large
Medium
Small
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Multiply Imaged
Distorted
Isotropic
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Equal Sided Torus
is non-zero for even l.
BUT
is zero ?
Torus shows a strong characteristic pattern.
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Unequal Sided Torus

• Non-zero
• Again non-zero for even l.

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Squeezed Torus
is non-zero for even l.
And
is NOT zero ?
Next
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• Torus has three preferred axes which cause the
  statistical anisotropy of Toroidal Spaces.

Back
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• pattern related to preferred directions?

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Pattern related to preferred directions?
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Analytical Approach

• Leading order contributions to can be
  calculated analytically for torus

(Bond, Pogosyan, Souradeep, 2000)
Well-known periodic box problem
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Analytical Approach

• Equal sided torus

is zero !
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Analytical Approach

• Un-equal sided torus

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Analytical Approach

• Squeezed torus

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A RECIPE for Estimating

1. Take two pixels, and on the sky and the
   product
2. Rotate both pixels by an angle around an axis
   to get pixels and . Compute the
   temperature product
3. Construct a function by summing over the
   temperature products obtained by varying the
   rotation axis all over the sky,
4. Construct using
   a series of terms
5. Construct by summing the square of
   over all pixel pairs,

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Cosmic Bias

• Analytically calculate multi-D integrals over
• Gaussian statistics gt express as products of
  covariance.
• Tedious exercise carried out for SI correlation
• Numerically
• Make many realizations of CMB anisotropy.
• For each of them measure
• For sufficiently large number of realizations the
  average value of will differ from the
  ensemble average by the cosmic bias.

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Cosmic Variance

• Analytically calculate multi-D integrals over
• Gaussian statistics gt express as products of
  covariance.
• 105 terms. 56 connected terms..
• But we have the terms !
• Tedious exercise similar the bias but more
  complicated.
• Numerically
• Make many realizations of CMB anisotropy.
• For each of them measure
• For sufficiently large number of realizations the
  average value of will tend to the ensemble
  average and the variance is a good estimate of
  the ensemble variance.

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Compact Hyperbolic Models
Compact hyperbolic (CH m004) space at
when The number titled Tot is
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Summary

• A generic measure for detecting and quantifying
  Statistical isotropy violations
• Can search the most generic signature of
  cosmic topology and Ultra large scale structure
• The measures can be neatly related to existence
  of preferred directions in correlation
• But measure is insensitive to the overall
  orientation
• SI breakdown (orientation of preferred axes).
  Hence
• limits on SI are not orientation specific.

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Future Plans

• Identify compute Statistical anisotropy
  signatures in other scenarios with SI violating
  correlations
• Address and remove observational artifacts.
• Apply to high-sensitivity full-sky data from the
  MAP satellite in early 2003.
• Search for signatures of cosmic topology
• and Ultra large scale structure.

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Finding these patterns leads to geometric methods.
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Three notable alterations to the predicted
fluctuations when the manifold is compact

• The eigenvalue spectrum is discrete not
  continuous.
• A cutoff in the power of fluctuations on
  wavelengths larger than the size of the space
• Two point correlation function depends on
  orientation and is not simply a function of the
  angular separation.

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Infrared cutoff
The isometric constant
Surface S divides the space into two subspaces
Cheegers inequality
Torus