Quantifying
Statistical Isotropy
Violation
in the CMB
Indo-UK Seminar
IUCAA, Pune
(Aug. 10, 2011)
Tarun Souradeep
I.U.C.A.A, Pune, India
CMB space missions
1991-94
2001-2010
2009-2011
CMBPol/COrE
2020+
CMB Anisotropy & Polarization
CMB temperature
Tcmb = 2.725 K
-200  K <  T < 200  K
 Trms ~ 70 K
TpE ~ 5  K
TpB ~ 10-100 nK
Temperature anisotropy T + two polarization
modes E&B
Four CMB spectra : ClTT,
ClEE,ClBB,ClTE
Parity violation/sys. issues: ClTB,ClEB
Cosmic “Super–IMAX” theater
0.5 Myr
Here
& Now
(14 Gyr)
Transparent universe
Opaque universe
Statistics of CMB
CMB Anisotropy Sky map => Spherical Harmonic decomposition

l
T ( ,  )    almYlm ( ,  )
l 2 m l
Gaussian CMB anisotropy completely specified by the
angular power spectrum
Statistical
isotropy
IF
alm al*'m'  Cl  ll ' mm'
=> Correlation function C(n,n’) is rotationally invariant
Current Angular power spectrum
3rd
peak
4th peak
5th peak
6thpeak
Image Credit: NASA / WMAP Science Team
Good old Cosmology, … New trend !
Total energy
density
Baryonic matter
density
Dark energy
density
‘Standard’ cosmological model:
Flat, CDM (with
nearly
Power Law primordial power spectrum)
Talk: Dhiraj Hazra
NASA/WMAP science team
PPS with features
Non-Parametric: Peak Location
(Amir Aghamousa, Mihir Arjunwadkar, TS arXiv:1107.0516)
Talk: AmiR Aghamousa
Day1
Beyond Cl :
Detecting patterns in CMB
Universe on Ultra-Large scales:
• Global topology
• Global anisotropy/rotation
• Breakdown of global syms, Magnetic field,…
Deflection fields
Observational artifacts:
• Foreground residuals
• Inhomogeneous noise, coverage
• Non-circular beams (eg., Hanson et al. 2010)
Statistics of CMB
C (nˆ1 , nˆ2 )  C (nˆ1  nˆ2 )
Possibilities:
•
•
•
•
Statistically Isotropic, Gaussian models
Statistically Isotropic, non-Gaussian models
Statistically An-isotropic, Gaussian models
Statistically An-isotropic, non-Gaussian models
Ferreira & Magueijo 1997,
Bunn & Scott 2000,
Bond, Pogosyan & TS 1998, 2000
f (nˆ )  C (nˆ , zˆ )
Radical breakdown of SI
disjoint iso-contours
multiple imaging
Mild breakdown of SI
Distorted iso-contours
Statistically isotropic (SI)
Circular iso-contours
E.g.. Compact hyperbolic
Universe .
(Bond, Pogosyan & Souradeep 1998, 2002)
Beautiful Correlation patterns
could underlie the CMB tapestry
Can we measure correlation patterns?
Figs. J. Levin
the COSMIC CATCH is
Measuring the SI 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
C (nˆ1  nˆ2 ) 
nˆ 2
1
8
2
 d
C (nˆ1 , nˆ2 )
Measuring the non-SI correlation
In the absence 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!!
(unless it is a KNOWN pattern)
•Matched circles statistics (Cornish, Starkman, Spergel ‘98)
•Anticorrelated ISW circle centers (Bond, Pogosyan,TS ‘98,’02)
• Planar reflective symmetries (de OliveiraCosta, Smoot Starobinsky ’96)
Bipolar Power spectrum (BiPS) :
A Generic Measure of Statistical Anisotropy
1
Recall : C (nˆ1  nˆ2 )  2  d C (nˆ1 , nˆ2 )
Bipolar multipole index
8

   dn
 dn
1
2
 1


 8 2  d  () C (nˆ1 , nˆ2 )
A weighted average of the
correlation function over all
rotations
 () 



D
 mm ()
m 
Characteristic
function
Wigner
rotation
matrix
2

     0
Statistical Isotropy
0
Correlation is invariant
under rotations
C(nˆ1, nˆ2 )  C(nˆ1, nˆ2 )

  (2  1)
2
 d  d
n1
2
n2
C (nˆ1, nˆ2 )[
1
8
2
 d

()]
d


(

)


0


2
Bipolar Power spectrum (BiPS) :
A Generic Measure of Statistical Anisotropy
• Correlation is a two point function on a sphere
 
C (n1 , n2 ) 


 A {Yl1 (n1 )  Yl2 (n2 )}LM
LM
l1l2
l1l2 LM
• Inverse-transform
A
Bipolar spherical
harmonics.


{Yl1 (n1 )  Yl2 (n2 )}LM


LM
  Cl1l2m1m2Yl1m1 (n1 )Yl2m2 (n2 )
2l  1
C (n1  n2 )  
Cl Pl (n1  n2 )
4
LM
l1l2
BiPoSH
m1m2 Clebsch-Gordan
 

 *
  dn1  dn2 C (n1 , n2 ){Yl1 (n1 )  Yl2 (n2 )}LM
  al1m1 al2m2
m1m2
LM
Cl m l m
1 12 2
Linear combination of
off-diagonal elements
Recall: Coupling of angular momentum states
l1m1l2m2 | M
l1    l2  l1  , m1  m2  M  0
M
*
BiPoSH
Al1l2   al1m1 al2 M m1
coefficients :
m1
M
Cl m l
1 1 2 M m1
• Complete,Independent linear combinations of off-diagonal correlations.
• Encompasses other specific measures of off-diagonal terms, such as
- Durrer et al. ’98 :
M
M
D

a
a

A
C
l
lm l  2 m
ll '
l 2 m l m
- Prunet et al. ’04 :

M
M
ll '
Dl(i )  almal 1 mi   A ClM1 mi l m
M
BiPS:
rotationally invariant
 

| A
| 0
M 2
l1l2
M ,l1 ,l2
Understanding BiPoSH
coefficients
SI violation:
alm al*' m '  Cl  ll ' mm '
4M
ll '
A
AllLM

'
alm al ' m '
C lmlLM' m '
mm '
Measure cross correlation in alm
2M
ll '
A
Spherical
harmonics
alm
Bipolar spherical
harmonics
M
ll '
A
Spherical Harmonic BiPoSH coefficents
coefficents
Cl
Angular power
spectrum


BiPS
Bipolar Power spectrum (BiPS) :
A Generic Measure of Statistical Anisotropy
Spherical
harmonics
alm
Bipolar spherical
harmonics
M
ll '
A
Spherical Harmonic BipoSH Transforms
Transforms
Cl
Angular power
spectrum
Statistical Isotropy
i.e., NO Patterns


BiPS
 

   0
0
BIPOLAR maps of WMAP
Hajian & Souradeep (PRD 2007)
ILC-3
Reduced BipoSH

Bipolar representation
M
•
AM 
All 'Measure of statistical isotropy
ILC-1
ll ' • Spectroscopy of Cosmic topology
Visualizing non-SI
• Anisotropic power spectrum
Bipolar map
correlations
• Deflection fields (WL,…)
• Diagnostic of systematic effects/observational
artifacts in the map
Diff.
ˆ
 ( nˆ )  •A Differentiate
Y
(
n
)
Cosmic vs. Galactic B-mode
M M
M
polarization

•SI part corresponds to the
“monopole” of the map.
Preferred Directions in the universe
Correlation function on a Torus (periodic box)
C (q1 , q2 ) 
 P(k ) exp[ i
n
2R
L

n  (qˆ1  qˆ2 )]
n
SI violation at low wave-number, kR<1
C (q1 , q2 )  C 0 [1    i (qi ) 2 ]
2
    0 ( i )

  2 ( i )
i
0

2
More generally,
P(k )  P0 (k )[1  g LM (k )YLM (kˆ)]
LM
ll '
A
C
L0
0 '0
(Hajian & Souradeep 2003)
(Pullen & Kamionkowski 2008)
dk
*
P
(
k
)
g
(
k
)

(
k
,

)

, o ) Moumita Aich
o
' ( kTalk:
 k 0 LM
BipoSH for Torus
Testing Statistical Isotropy of WMAP-3yr
Retain orientation
information:
Reduced BipoSH
Hajian & Souradeep, (PRD 2007 )
Outliers (> 95%)
A M   AllM'
ll '
Filters
(  1)  M  1
BIPOLAR measurements by WMAP-7 team
(  ) LM
l1l2
L0
l 0l '0
A
C
(Bennet et al. 2010)
Non-zero Bipolar coeffs.!!!
Sys. effect : beam distortion ?
(Hanson et al. 2010)
TALK by Santanu Das
Image Credit: NASA / WMAP Science Team
Revisting Statistics of BipoSH
(Nidhi Joshi, Aditya Rotti, TS, 2011)
• ~Analytic PDF for BipoSH
• Confirmed with MC simulations
• Faster BipoSH code (20x)
• Compute up to high multipoles
Talk: Nidhi Joshi
Day 2
Even & odd Bipolar coefficients
 


C (n , n )   A {Y (n )  Y (n )}
1
LM
l1l2
2
l1
1
l2
2
LM
l1l2 LM
 C (n1 , n2 ) 
(  ) LM [1 ( 1) L  l  l ' ]
ll '
2
A
{Yl1 (n1 )  Yl2 ( n2 )}LM
ll ' LM
(  ) LM [1( 1) L  l  l ' 1 ]
ll '
2
 A
{Yl1 ( n1 )  Yl2 ( n2 )}LM
ll ' LM
even L+l’+l
contribute, it is
becoming
popular to use
If only
Al(1l2) LM 
Al(1l2) LM
ClL00l '0
2 L 1
(2l1 1)(2 l2 1)
• Anisotropic P(k)
• Linear order templates
Even & odd parity BipoSH
Al(2l1) LM  Al(1l2 ) LM
symmetric
Al(2l1) LM   Al(1l2) LM
antisymm.
[ Al(1l2 ) LM ]*  ( 1) M Al(1l2) L , M
Even parity
[ Al(1l2) LM ]*  ( 1) M 1 Al(1l2) L , M Odd parity
SI violation : Deflection field
n̂
nˆ '
T ( nˆ ')  T ( nˆ  )  T ( nˆ )    T ( nˆ )
oo
   ( nˆ )    ( nˆ )
 i ( nˆ )   ij  j ( nˆ )
Gradient
Curl
WL:scalar
WL: tensor/GW
SI violation : Deflection field
Brooks, Kamionkowski & Souradeep
S
T ( nˆ )  alm  alm
  alm ,
 (nˆ ), ( nˆ )  LM ,  LM
1
LM
 alm   alS' m ' LM EllL'  i LM OlLl '  GllL'Clml
'm '
2 LM l ' m '
GllL'  [...] ClL01l ' 1

L0
(  [....]C l 0 l '0 for l  l '  L : even )
1
L
l l '  L
even: E ll ' 
1  ( 1)
2


1
L
l l '  L
& odd: Oll ' 
1  ( 1)
2
SI violation: alm al ' m '  Cl ll ' mm '

Deflection field: Even & Odd parity BipoSH
Brooks, Kamionkowski & Souradeep 2011
All( ' ) LM  LM
Al(2l1) LM  i LM
 Cl GlL' l
Cl 'GllL' 



l (l  1) 
 l '(l ' 1)
WL: scalar
L
 Cl GlL' l
Cl 'Gll '  WL: tensor



l (l  1) 
 l '(l ' 1)
BipoSH Measures of deflection field
Estimators
LM 
Variance
 (  ) LM
2 LM
Q
A
/

 ll ' ll '
ll '
ll '
 (Q
) /
 2
ll '
 
var LM
2 LM
ll '

 2
2 LM 
   (Qll ' ) /  ll ' 
 ll '

1
ll '
 LM 
 (  ) LM
2 LM
Q
A
/

 ll ' ll '
ll '

 2
2 LM 
var   LM     (Qll ' ) /  ll ' 
 ll '

ll '
 2
2 LM
(
Q
)
/

 ll ' ll '
ll '
(  ) LM
l1l2
A

Al(1l2) LM
L1
l 0 l '1
C
... 
Al(1l2) LM
L
ll '
G
1
CMB BipoSHs & Bispectra
(Kamionkowski & Souradeep, PRD 2011)
For deflection field
AllLM
'
S
alm  alm
  alm
LM
LM  alms als' m ' Clml
'm'
mm '
LM  aLM
 AllLM
'

LM
aLM almal ' m ' Clmlm
'
mm '
BipoSH related to Bispectrum
BLll '

LM
aLM alm al ' m ' Clml
' m ' (...)
Mmm '
(  ) LM
A
 ll '
M
Consider only: l  l ' L  even
Odd parity Bispectra ?
( )
BLll
'
(  ) LM
A
 ll '
M
Flat sky intuition:
l2
l2
l3
l3
l1
B
()
l1  l2
l1l2
l1  l2  l3
l1
has opposite sign in the
two mirror configurations.
Odd parity Bispectra
For local NG model

odd l1  l2 
B(l1, l2 ) = 2  f nl  f nl
 (Cl1Cl2 
l1l2 

In general
f nl  
f
odd
nl

2
f nl
2
f nl
 

l3
l1l2
6G
l1 l2 l3
2
f nl

(Cl1 Cl2 
l1 l2 l3
(Cl1 Cl2 
perms.
perms.
2
)
Ell13l2
)
Cl1 Cl2  Cl3 Cl2  Cl1 Cl3
6 G (Cl1 Cl2  perms.) 
 
l1 l2 l3 Cl1 Cl2  Cl3 Cl2  Cl1 Cl3
l3
l1l2
)
Cl1 Cl2  Cl3 Cl2  Cl1Cl3
l3
l1l2
6G
perms.
Flat sky approx
Oll13l2
Summary
•
•
•
Current observations now allow a meaningful search for deviations from the `standard’
flat, CDM cosmology.
Anomalies in WMAP suggest possible breakdown down of statistical isotropy.
Thank you !!!
Bipolar harmonics provide a mathematically complete, well defined,
representation of SI violation.
– Possible to include SI violation in CMB arising both from direction dependent Primordial
Power Spectrum , as well as, SI violation in the CMB photon distribution function.
– BipoSH provide a well structured representation of the systematic breakdown of rotational
symmetry.
– Bipolar observables have been measured in the WMAP data.
– Systematic effects are important and must be quantified similar to signal
•
BipoSH coefficients can be separated into even and odd parity parts.
–
For a general deflection field, gradient & curl parts are represented by even & odd parity
BipoSH, respectively. Eg., Weak lensing by scalar & tensor (or 2nd order scalar) perturbations.
– Estimators for grad/curl deflections field harmonics in terms of even/odd BipoSH
• BipoSH for correlated deflection field relate to Bispectra
– Pointed to, hitherto unexplored, odd-parity bispectrum.
– Minor modification to existing estimation methods for even-parity bispectra
– Odd parity bispectrum may arise in exotic parity violations, but, also an interesting
null test for usual bispectrum analysis.
Statistical Isotropy: CMB Photon distribution
(Moumita Aich & TS, PRD 2010)
 (k , rec )
Statistical
isotropy
 ... j (k  )

l
'l

General:
NonStatistical
isotropy

  (k , 0 )
Free stream
LM
3 4
(k , rec )
C
2
  ' (k , rec )
 
Free stream
 ... j (k  ) C
l
l0
0 '0
L0
0 10
C
L0
0 10
3 4
  LM
(
k
,

)
re
c
3 4
LM
1 2



( k , 0 )
4
L
3
1
l
2



Statistical Isotropy: CMB Photon distribution
(Moumita Aich & TS, PRD 2010)
Non-Statistical isotropic terms also freestream to large multipole values
L
 4

 1s ls
 Large s
   ... C
2  s
3
3
4
( 1 l )
( 2 l ) L ( 1 
l  1  l , 2  l , L
LM
 LM
(
k
,

)


...
j
(
k


)

  l
0
1 2
1 l ,
l
2 l
2)
(k , rec )