Tarun Souradeep
description
Transcript of Tarun Souradeep
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
CMB space missions
1991-94
2001-2010
2009-2011
CMBPol/COrE2020+
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
Transparent universe
Opaque universe
14 GPc
Here & Now
(14 Gyr)
0.5 Myr
Cosmic “Super–IMAX” theater
),(),(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
Current Angular power spectrum
Image Credit: NASA / WMAP Science Team
3rd
peak
6thpeak
4th peak 5th peak
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
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)
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
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 .
Can we measure correlation patterns?
the COSMIC CATCH is
Beautiful Correlation patterns could underlie the CMB tapestry
Figs. J. Levin
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
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)
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
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
• 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
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
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
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
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
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
)]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
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 )
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
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
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
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
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
'
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
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
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
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)
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:
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
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 !!!
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)
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)