Transcript Document

STATISTICAL ISOTROPY
of
CMB ANISOTROPY
Amir Hajian & Tarun Souradeep
I.U.C.A.A, Pune
http://www.iucaa.ernet.in/~tarun/pascos03.ppt
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
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
Cl
0
 tot
m
 CDM

n
Fig.. Bond 2002
The Angular power spectrum of
CMB anisotropy is considered a
powerful tool for constraining
cosmological parameters.
Statistics of CMB
T (nˆ )
smooth random function on a sphere (sky map).
General random CMB anisotropy: described by a
Probability Distribution Functional
 Ti 
– Covariance
P[T (nˆ )]
– Mean:
(2-point correlation)
 


Cij  C(ni , n j )  T (ni )T (n j ) 
– ...
– N-point correlation
 Ti Tj ...TN 
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
Cij
Statistics of CMB
CMB anisotropy completely specified by the
angular power spectrum
Only if
Cl
Statistically isotropic Gaussian random CMB anisotropy
1
 
 
C (n1 , n2 )  C (n1  n2 ) 
4
 
 (2l  1)Cl Pl (n1  n2 )
l
Statistics of CMB
•
:smooth random function on a (pix sphere.
• General random CMB
anisotropy:
described
a
Statistical
Isotropy
meansbythe
P[T (nˆ )]
probability distribution functional
assigning
a number
two point
correlation
function
(probability) to every CMB anisotropy sky map T (nˆ )
–
–
–
–
depends only on the angular
Mean:
 
 two
separation
the
Covariance(2-point correlation)
Cij  C(nbetween
,
n
)


T
(
n
i
j
i ), T (n j ) 
directions in the sky.
…..
N-point correlation
• Gaussian random CMB anisotropy:
Completely specified by the covariance:
• Statistically isotropic Gaussian random CMB
1
 
 
 
anisotropy:
C (n , n )  C (n  n ) 
(2l  1)C P (n  n )

4
1
2
1
2
l l
1
2
l
Completely specified by angular power spectrum Cl
Statistics of CMB
Statistical Isotropy implies the
:smooth random function on a P[T (nˆ)]
two point correlation function
) sphere.
T (nˆ )
depends
only
on
the
angular
• General random CMB anisotropy: described by a probability
distribution
i.e.,
functional
assigning
a number (probability)
to every
CMB
between

 two
 anisotropy
separation
the
C

C
(
n
,
n
)


T
(
n
),

T
(
n
ij
i
j
i
j) 
sky map
Correlation
is
– Mean:
directions in the sky.
•
–
•
Covariance(2-point correlation)
invariant
– …..
– N-point correlation
under random CMB anisotropy:
Gaussian
rotations Completely specified by the covariance
Statistically isotropic Gaussian random CMB anisotropy:
1
 
 
C (n1 , n2 )  C (n1  n2 ) 
4
 
 (2l  1)C P (n  n )
l l
1
2
l
Completely specified by angular power spectrum Cl
Radical breakdown of SI
disjoint iso-contours
multiple imaging
Mild breakdown of SI
Distorted iso-contours
Statistically isotropic (SI)
Circular iso-contours
Beautiful Correlation patterns
could underlie the CMB tapestry
Figs. J. Levin
Akin to
Leopard’s spots
Can we Measure Correlation Patterns?
Sure, maybe quite possible for Leopard spots
BUT for CMB anisotropy
the COSMIC CATCH is
Measuring the correlation
Statistical isotropy
C ( )
can be well estimated
by averaging over the temperature
product between all pixel pairs
separated by an angle  .
~


 
C ( )   T (n1 )T (n2 ) (n1  n2  cos  )
nˆ1
nˆ 2
Measuring the correlation
Violation of statistical isotropy
Estimate of the correlation function from
a sky map given by a single temperature
product
~  


C (n1 , n2 )  T (n1 )T (n2 )
is poorly determined.
Cl
is inadequate for model
comparison
A Measure of Statistical Anisotropy
  (2  1)

2
 d  d
n1
n2
[
1
8
2
ˆ
ˆ
 d ()C (n1 , n2 )]

2
A Measure of Statistical Anisotropy
  (2  1)

2
 d  d
n1
n2
[
1
8
2
ˆ
ˆ
 d ()C (n1 , n2 )]

2
 () 



D
 mm ()
m 
Characteristic
function
Wigner
rotation
matrix
A Measure of Statistical Anisotropy
  (2  1)

2
d  4 sin

 d  d
n1
2 
2
n2
[
1
8
2
ˆ
ˆ
 d ()C (n1 , n2 )]

2
d sin dd
is the three
dimensional rotation
through an angle  (0     )
about the axis n(,  )
0     , 0    2
 () 



D
 mm ()
m 
Characteristic
function
Wigner
rotation
matrix
A Measure of Statistical Anisotropy
  (2  1)

2
 d  d
n1
n2
[
1
8
2
ˆ
ˆ
 d ()C (n1 , n2 )]

2
A weighted average of the correlation function
over all rotations
Except for
  0
when
 ()  1
0
Why is

L
a measure of
statistical anisotropy.

L
 0, L  0 
statistical anisotropy.
 
Statistical Isotropy

   0
1
 d
0
Correlation is invariant
under rotations
C(nˆ1, nˆ2 )  C(nˆ1, nˆ2 )

  (2  1)
2
 d  d
n1
2
n2
C (nˆ1, nˆ2 )[
8
 d

2

2
()]
()    0
What exactly are

L
In Harmonic Space
• Correlation is a two point function on a sphere
Suggests a bipolar spherical harmonics expansion
 
C(n1, n2 ) 


 A {Yl1 (n1) Yl2 (n2 )}LM
LM
l1l2
l1l2 LM


{Yl1 (n1 )  Yl2 (n2 )}LM 


LM
C
Y
(
n
)
Y
(
n
 l1l2m1m2 l1m1 1 l2m2 2 )
m1m2
: Bipolar spherical harmonics.
In Harmonic Space
• Correlation is a two point function on a sphere
 


LM
C(n1, n2 )   Al1l2 {Yl1 (n1 )  Yl2 (n2 )}LM
l1l2 LM


{Yl1 (n1 )  Yl2 (n2 )}LM 


LM
C
Y
(
n
)
Y
(
n
 l1l2m1m2 l1m1 1 l2m2 2 )
m1m2
: Bipolar spherical harmonics.
• Inverse-transform
LM
l1l2
A
 

 *
  d n1  d n2 C (n1 , n2 ){Yl1 (n1 )  Yl2 (n2 )} LM
 
L
| A
| 0
LM 2
l1l2
M ,l1 ,l2
Statistical isotropy :
 
 
C(n1 , n2 )  C(n1  n2 )

 Pl1 (nˆ1  nˆ2 )


l1
(

1
)
2
l

1

{
Y
(
n
)

Y
(
n

1
l1
l1
1
l2
2 )}00  l1l2
l1l2 LM
LM
l1l2
A
 l1 l1l2  L0 M 0
    L0
L
0
L  0
What if we find
Statistical anisotropy
in CMB maps
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
Ultra Large scale structure of the universe
How Big is the Observable Universe ?
Relative to the local curvature & topological scales
Simple Torus
(Euclidean)
Homogenous & isotropic
but
Consider all Spaces of
Multiply
connected
universe
?
Constant Curvature
Compact hyperbolic
space
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
Spatial Correlations in …
• Simply connected space
(STATISTICALLY ISOTROPIC)
•A Toroidal Space
Iso-correlation
contours are no
more circular.
Back
THREE POSSIBILITIES
( Size of the compact space relative to horizon scale)
Small
Medium
Large
Multiply Imaged
Distorted
Isotropic
Equal Sided Torus
   fi ( )

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
i 1
,2i
 l is non-zero for even l.
2
4
6
8 10 12 14

BUT
2
is zero 
RSLS  L
Torus shows a strong characteristic  l pattern.
Unequal Sided Torus
   f i ( )

All i
• Non-zero
1.20
1.00
0.80

2
• Again non-zero
even l.
0.60
0.40
0.20
0.00
2
4
6
8
,2i


for
Squeezed Torus
 

f
k
( )
All k
,2k
  /3
1.00
 l is non-zero for even l.
0.80
0.60

0.40
0.20
2
And
is NOT zero 
0.00
2
4
6
8
Next
• Torus has three
preferred axes which
cause the statistical
anisotropy of Toroidal
Spaces.
Back

l
pattern related to preferred directions?

l
Pattern related to preferred directions?
Analytical Approach
• Leading order contributions to can
be calculated analytically for torus

C (q1, q2 ) 
 P(k ) exp[i
n
2R
L

n  (qˆ1  qˆ2 )]
n
(Bond, Pogosyan, Souradeep, 2000)

n  (n1 , n2 , n3 )
Well-known
periodic box
problem
Analytical Approach
1
0.8
0.6
0.4
0.2
0
• Equal sided torus
0
1
2
3
4
n1  1, n2  n3  0
n1  n2  1, n3  0
C(q1 , q2 )   (1  
  f ( )

2
 q
i
2

 g ( )
i
0 2

n1  n2  n3  1
is zero !
4
 q
i
i
4
4
)
Analytical Approach
1
0.8
0.6
0.4
0.2
0
• Un-equal sided torus
0
1
2
3
4
n1  1, n2  n3  0
C(q1 , q2 )  C 0[1 
2

(

q
)
 i i ]
2
i
  f ( ,  ,  )

0
 g ( ,  ,  )
2
Analytical Approach
1
0.8
0.6
0.4
0.2
0
• Squeezed torus
0
1
2
3
4
n1  1, n2  n3  0
C (q1 , q2 )   (1  
n1  n2  1, n3  0
2
 qi    i qi    i qi q j )
2
i
  f ( ,  ,  )

n1  n2  n3  1
2
i
0
2
2
i, j
 g ( ,  ,  )
2
A RECIPE for Estimating
1.
2.
3.
4.
Take two pixels, qˆ1 and qˆ2on the sky and the product T (qˆ1 )T (qˆ2 )
Rotate both pixels by an angle  around an axis nˆ to get pixels qˆ '1
and qˆ '1 . Compute the temperature product T (qˆ '1 )T (qˆ '2 )
Construct a function by summing over the temperature products
obtained by varying the rotation axis all over the sky,
Construct using
a series of terms

5.

L
d sin 2

2
L
Construct  by summing the square of
fl (qˆ1, qˆ2 )over
all pixel pairs,
Cosmic Bias
• Analytically calculate multi-D integrals over




 T (n1)T (n2 )T (n3 )T (n4 ) 
– Gaussian statistics => express as products of covariance.
Tedious exercise carried out for SI correlation


 Cl  Cl '  
l 0
• Numerically
l '0


LM
lml ' m ' product of 4 CG
M , M ' , m , m ' , m", m '"
– Make many realizations of CMB anisotropy.
For each of them measure
l
~
~
– For sufficiently large number of realizations the average value of  l will differ
from the ensemble average by the cosmic bias.
Cosmic Variance
• Analytically calculate multi-D integrals over
– Gaussian statistics => 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 
l
~
– For sufficiently large number of realizations the average value of  l will
tend to the ensemble average and the variance is a good estimate of the
ensemble variance.
Compact Hyperbolic Models
Compact hyperbolic (CH: m004) space at 0  0.3
when VLS / VM  153.
10
The number titled Tot is

 1

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.
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.
Finding these patterns leads
to geometric methods.
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.
Infrared cutoff
hC
k min 
2
The isometric
constant
Surface S divides the
space into two
subspaces
A( S )
hC  inf
min(V (M1 ),V ( M1 ))
Cheeger’s inequality
Torus:
k min
2

L