Tarun Souradeep
description
Transcript of Tarun Souradeep
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Tarun SouradeepI.U.C.A.A, Pune, India
Indo-UK SeminarIUCAA, Pune
(Aug. 10, 2011)
QuantifyingQuantifying
Statistical Isotropy Statistical Isotropy Violation Violation in the CMBin the CMB
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CMB space missions
1991-94
2001-2010
2009-2011
CMBPol/COrE2020+
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Temperature anisotropy T + two polarization
modes E&B Four CMB spectra : ClTT,
ClEE,Cl
BB,ClTE
CMB Anisotropy & Polarization
CMB temperature Tcmb = 2.725 K
-200 K < T < 200 KTrms ~ 70 K
TpE ~ 5 KTpB ~ 10-100 nK
Parity violation/sys. issues: ClTB,Cl
EB
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Transparent universe
Opaque universe
14 GPc
Here & Now
(14 Gyr)
0.5 Myr
Cosmic “Super–IMAX” theater
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),(),(2
l
l
lmlmlmYaT
CMB Anisotropy Sky map => Spherical Harmonic decomposition
Statistics of CMB
Statistical isotropy
''*
'' mmlllmllm Caa
Gaussian CMB anisotropy completely specified by the
angular power spectrumangular power spectrum IF
=> Correlation function C(n,n’) is rotationally invariant
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Current Angular power spectrum
Image Credit: NASA / WMAP Science Team
3rd
peak
6thpeak
4th peak 5th peak
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NASA/WMAP science team
Total energy density
Baryonic matter density
‘Standard’ cosmological model:Flat, CDM (with nearly
Power Law primordial power spectrum)
Dark energy density
Good old Cosmology, … New trend !
Talk: Dhiraj Hazra
PPS with features
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Non-Parametric: Peak Location(Amir Aghamousa, Mihir Arjunwadkar, TS arXiv:1107.0516)
Talk: AmiR Aghamousa
Day1
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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)
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Possibilities:• Statistically Isotropic, Gaussian models
• Statistically Isotropic, non-Gaussian models
• Statistically An-isotropic, Gaussian models
• Statistically An-isotropic, non-Gaussian models
Statistics of CMB
1 2 1 2ˆ ˆ ˆ ˆ( , ) ( )C n n C n n
Ferreira & Magueijo 1997,Bunn & Scott 2000,
Bond, Pogosyan & TS 1998, 2000
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Statistically isotropic (SI)Circular iso-contours
Radical breakdown of SI disjoint iso-contours
multiple imaging
Mild breakdown of SIDistorted iso-contours
(Bond, Pogosyan & Souradeep 1998, 2002)
)ˆ,ˆ()ˆ( znCnf
E.g.. Compact hyperbolic Universe .
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Can we measure correlation patterns?
the COSMIC CATCH is
Beautiful Correlation patterns could underlie the CMB tapestry
Figs. J. Levin
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Statistical isotropy
can be well estimated by averaging over the temperature product between all pixel pairs separated by an angle .
Measuring the SI correlation
)(C
)cos()()()(~
21ˆ ˆ
21
1 2
nnnTnTCn n
)ˆ,ˆ(8
1)ˆˆ( 21221 nnCdnnC
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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
is poorly determined!!
)()(),(~
2121 nTnTnnC
(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)
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2
21221)ˆ,ˆ()(
8
1
nnCdndnd
A A weighted averageweighted average of the of the correlation function over all correlation function over all
rotationsrotations
)ˆ,ˆ(8
1)ˆˆ( :Recall 21221 nnCdnnC Bipolar multipole index
Wigner Wigner rotation rotation matrixmatrix
)()(
mmmD
Characteristic Characteristic functionfunction
Bipolar Power spectrum (BiPS) :Bipolar Power spectrum (BiPS) :A Generic Measure of Statistical AnisotropyA Generic Measure of Statistical Anisotropy
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2
221
22)](
8
1[)ˆ,ˆ()12(
21
dnnCdd nn
0)( d
Statistical IsotropyStatistical Isotropy
Correlation is invariant under rotations
)ˆ,ˆ()ˆ,ˆ( 2121 nnCnnC
0
0
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• Correlation is a two point function on a sphere
• Inverse-transform )()(
)}()({
21
21
2211
21
2121
21
nYnYCnYnY
mlmlmm
LMmmll
LMll
LMllLMll
LMll nYnYAnnC )}()({),( 2121 21
21
21
LM
mmmlml
LMllnnLMll
mlmlCaa
nYnYnnCddA
2211
21
2211
212121
*2121 )}()(){,(
BiPoSH
Linear combination of off-diagonal elements
Bipolar spherical harmonics.
Clebsch-Gordan
Bipolar Power spectrum (BiPS) :Bipolar Power spectrum (BiPS) :A Generic Measure of Statistical AnisotropyA Generic Measure of Statistical Anisotropy
1 2 1 2
2 1( ) ( )
4 l l
lC n n C P n n
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0||21
21,,
2 llM
MllABiPS:BiPS:
rotationally invariantrotationally invariant
M
mmMlml
Mll mMlml
CaaA
1211
1
121121
*
BiPoSH BiPoSH coefficients :coefficients :
• Complete,Independent linear combinations of off-diagonal correlations.• Encompasses other specific measures of off-diagonal terms, such as
- Durrer et al. ’98 :- Prunet et al. ’04 :
M
Mmliml
Mllimllm
il CAaaD
1'1
)(
M
Mmlml
Mllmllml CAaaD
2'2
Recall: Coupling of angular momentum states Mmlml |2211 0, 21121 Mmmlll
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MllA2
'
MllA4
'
Understanding BiPoSH coefficients
*' ' ' '
SI violation
:
lm l m l ll mma a C
' '' ''
Measure cross correlation in
' lml m
LMlm l m
mm
lm
a a C
a
LMllA
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Bipolar spherical harmonics
Spherical Harmonic coefficents
BiPoSH coefficents
Angular power spectrum
BiPS
lma MllA'
lC
Spherical harmonics
Bipolar Power spectrum (BiPS) :Bipolar Power spectrum (BiPS) :A Generic Measure of Statistical AnisotropyA Generic Measure of Statistical Anisotropy
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Bipolar spherical harmonics
Spherical Harmonic Transforms
BipoSH Transforms
Angular power spectrum
BiPS
lma MllA'
lC
Spherical harmonics
Statistical IsotropyStatistical Isotropyi.e., NO Patternsi.e., NO Patterns 0
0
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BIPOLAR maps of WMAPHajian & Souradeep (PRD 2007)
''
Reduced BipoSH
Bipolar map
ˆ ˆ( ) ( )
MM ll
l
M M
l
M
n
A A
A Y n
Visualizing non-SI correlations
•SI part corresponds to the “monopole” of the map.
Diff.
ILC-3
ILC-1
Bipolar representation• Measure of statistical isotropy
• Spectroscopy of Cosmic topology
• Anisotropic power spectrum
• Deflection fields (WL,…)
• Diagnostic of systematic effects/observational artifacts in the map
• Differentiate Cosmic vs. Galactic B-mode polarization
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)]ˆˆ(exp[)(),( 212
21 qqnikPqqCn
LR
n
Preferred Directions in the universeCorrelation function on a Torus (periodic box)
SI violation at low wave-number, kR<1
])(1[),( 22 021 i
ii qCqqC
20 )()( 20 ii
(Hajian & Souradeep 2003)
0ˆ( ) ( )[1 ( ) ( )]LM LMP k P k g k Y k
(Pullen & Kamionkowski 2008)
More generally,
0 *' 0 '0 0 '( ) ( ) ( , ) ( , )LM L
ll LM o o
dkA C P k g k k k
k Talk: Moumita Aich
BipoSH for Torus
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Testing Statistical Isotropy of WMAP-3yr
Filters ( 1) 1M
Retain orientation information:Reduced BipoSH
''
MM ll
ll
A A
Outliers (> 95%)
Hajian & Souradeep, (PRD 2007 )
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BIPOLAR measurements by WMAP-7 team
Image Credit: NASA / WMAP Science Team
(Bennet et al. 2010)
Non-zero Bipolar coeffs.!!!Sys. effect : beam distortion ?
(Hanson et al. 2010)TALK by Santanu Das
1 2
( )
00 '0
LMl l
Ll l
A
C
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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
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Even & odd Bipolar coefficients
LMllLMll
LMll nYnYAnnC )}()({),( 2121 21
21
21
1 2
1 2
[1 ( 1) ]1 2 1 22
'
[
'( )'
1 ( 1) ]' 1(22
'
)1'
( , ) { ( ) ( )}
{ ( ) ( )}L l lLM
l
l l LMll LM
l l LMll LM
L
l
l lLMllC n n Y n Y n
Y n YA
A
n
1 2
1 2 1 2
( )2 1( )
0 (2 1)(2 1)0 '0
LMl l LLM
l l L l ll l
AA
C
If only even L+l’+l contribute, it is becoming popular to use
• Anisotropic P(k)• Linear order templates
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2 1 1 2
2 1 1 2
1 2 1 2
1 2 1 2
( ) ( )
( ) ( )
( ) * ( ) ,
( ) * 1 ( ) ,
symmetric
antisym
Even parity
Odd parit
m.
[ ] ( 1)
[ ] ( 1) y
LM LMl l l l
LM LMl l l l
LM M L Ml l l l
LM M L Ml l l l
A A
A A
A A
A A
Even & odd parity BipoSH
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SI violation : Deflection field
ˆ( )
ˆ(
ˆ ˆ ˆ ˆ( ') ( ) ( ) ( )
ˆ( )
ˆ( )) ji ij
T n T n T
n
n n
n
T
nn
oo
n̂ˆ 'n
CurlGradient WL: tensor/GWWL:scalar
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'
0( [..
' ' ' '
..
''
1even: 1 ( 1)
] for ' :
& odd:
'
'
1' 0 ' 1 0
'
)'0
2
' '
ˆ ˆ ˆ( ) , ( ), ( ) ,
1
2
[...] l l L
LOl
Slm lm lm LM LM
S LMlm l m LM LM lml m
LM l m
L
LE
Lll
L Lll
ll
L LE O
C l l L evenl
ll ll
l
l
l l
G
G
T n a a a n n
a a i C
C
'
' '
11 ( 1
'
)
'
2
SI violation:
l l L
lm l m l ll mma a C
SI violation : Deflection fieldBrooks, Kamionkowski & Souradeep
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2 1
( ) ' ' ''
( ) ' ' '
'( ' 1) ( 1)
'( ' 1) ( 1)
L LLM l l l l ll
ll LM
L LLM l l l l ll
l l LM
C G C GA
l l l l
C G C GA i
l l l l
Deflection field: Even & Odd parity BipoSH
WL: scalar
WL: tensor
Brooks, Kamionkowski & Souradeep 2011
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1
2 2' '
'
1
2 2' '
'
var ( ) /
var ( ) /
LMLM ll ll
ll
LMLM ll ll
ll
Q
Q
1 2 1 2
1 2 10
( ) ( )
1
( )
' '
...L Ll l
LM LMl l l lL
ll
Ml l
A AA
C G
BipoSH Measures of deflection field
( ) 2' ' '
'2 2
' ''
( ) 2' ' '
'2 2
' ''
/
( ) /
/
( ) /
LM LMll ll ll
llLM LM
ll llll
LM LMll ll ll
llLM LM
ll llll
Q A
Q
Q A
Q
VarianceEstimators
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CMB BipoSHs & Bispectra
' ' ' ' ''
LM s s LMll LM lm l m lml m
mm
A a a C
' ' ' ''
LM LMll LM lm l m lmlm
mm
A a a a C
For deflection field
LM LMa
BipoSH related to Bispectrum
' ' ' ' ''
( )'
(...)
LMLll LM lm l m lml m
Mmm
LMll
M
B a a a C
A
Slm lm lma a a
Consider only: ' evenl l L
(Kamionkowski & Souradeep, PRD 2011)
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Odd parity Bispectra ?
1l
1l
2l
2l
3l
3l
1 2 3l l l
( ) 1 2
1 2
l lB
l l
has opposite sign in the
two mirror configurations.
( ) ( )' ' LM
Lll llM
B A
Flat sky intuition:
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Odd parity Bispectra
1 2
odd 1 2perms.1 2 nl nl
1 2
( , ) = 2 ( )l l
l lB l l f f C C
l l
1 23
nl 1 2
1 2 3 1 2 3 2 1 3
1 23
nl 1 2
1 2 3 1 2 3 2 1 3
3
1 2 1 2
nl
3
1 2
2 3
3
1 2
1
perms.2nl
perms.2nl
2per
d
s
d
m .2
o
( ) 6
( ) 6
6 ( )
l llf l l
l l l l l l l l l
l llf l l
l l l l l l l l l
ll l l l
f
ll
ll
l l l
l
l
C Cf G
C C C C C C
C Cf G
C C C C C C
G C C
C C C C
E
O
1 2 3 2 1 3l l l l l lC C
For local NG modelFlat sky approx
In general
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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.
• 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.
Thank you !!! Thank you !!!
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Statistical Isotropy: CMB Photon Statistical Isotropy: CMB Photon distributiondistribution
Free stream0
200 '0
''
(( , )
( ,
, )
... ( )) ll
rec
ecl
r
k
j k C
k
k
1 2
1
3 4
3 4
1
3 4
Free stream0
4 30 00 0 0 0
1 2
( ( , )
... ( )
, )
( , )
LLMrec
LMre
M
Ll
c
L
k
Lk C
k
k
j Cl
Statistical isotropy
General:Non-
Statistical isotropy
(Moumita Aich & TS, PRD 2010)
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Statistical Isotropy: CMB Photon Statistical Isotropy: CMB Photon distributiondistribution
3 1
4 2 1 2
1
2 1
2
1 2
4 3 ( )Large ( ) ( )
1 2
,,
,0
...
( , ) ... ( ( , )) LMl l r
lsl L
l l l LLMel
lc
LC
s l s s
k j k k
Non-Statistical isotropic terms also free-stream to large multipole values
(Moumita Aich & TS, PRD 2010)