Download - Challenging the Isotropic Cosmos
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Tarun SouradeepI.U.C.A.A, Pune, India
CTACC ColloquiumAIMS, Cape Town
(Apr. 13, 2012)
Challenging the
Isotropic Cosmos
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CMB space missions
1991-94
2001-2010
2009-2011
CMBPol/COrE2020+
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Post-recombination :Freely propagating through (weakly perturbed) homogeneous & isotropic cosmos.
Pre-recombination : Tightly coupled to, and in thermal equilibrium with, ionized matter.
Pristine relic of a hot, dense & smooth early universe -Hot Big Bang model
(text background: W.
Cosmic Microwave Background
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Transparent universe
Opaque universe
Here & Now
(14 Gyr)
0.5 Myr
Cosmic “Super–IMAX” theater
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Temperature anisotropy T + two polarization modes E&B Four CMB spectra : Cl
TT, Cl
EE,ClBB,Cl
TE
CMB Anisotropy & PolarizationCMB temperature
Tcmb = 2.725 K
-200 μ K < Δ T < 200 μ KΔ Trms ~ 70μ K ΔTpE ~ 5 μ K
ΔTpB ~ 10-100 nK
Parity violation/sys. issues: ClTB,Cl
EB
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),(),(2
φθφθ ∑ ∑∞
= −=
=Δl
l
lmlmlmYaT
CMB Anisotropy Sky map => Spherical Harmonic decompositionStatistics of CMB
Gaussian Random field => Completely specified byangular power spectrum l(l+1)Cl :
Power in fluctuations on angular scales of ~ π/l
*' ' ' 'lm l m l ll mma a C δ δ=
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Hence, a powerful tool for constraining cosmological
parameters.
Fig. M. White 1997
The Angular power spectrum of CMB anisotropy depends
sensitively on Cosmological parameters
lC
Multi-parameter Joint likelihood (MCMC)
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•Low multipole : Sachs-Wolfe plateau
• Moderate multipole : Acoustic “Doppler” peaks
• High multipole : Damping tail
Dissected CMB Angular power spectrum
(fig credit: W. Hu)
CMB physics is verywell understood !!!
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Cosmic Acoustics: Ping the ‘Cosmic drum’
More technically,the Green function(Fig: Einsentein )
150 Mpc
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Independent, self contained analysis of WMAP multi-frequency maps
Saha, Jain, Souradeep(WMAP1: Apj Lett 2006)
WMAP3 2nd release : TS,Saha, Jain: Irvine proc.06
Eriksen et al. ApJ. 2006
WMAP: Angular power spectrum
Good match to WMAP team
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(48.3 ±1.2, 544 ±17)
(48.8 ±0.9, 546 ±10)
(41.7 ±1.0, 419.2 ±5.6)
(41.0 ± 0.5, 411.7 ±3.5)
(74.1±0.3, 219.8±0.8)
(74.7 ±0.5, 220.1 ±0.8
Peaks of the angular power spectrum
(Saha, Jain, Souradeep Apj Lett 2006)
0K 0Ω =0 0.04BΩ =
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Peak heights and ratios Cosmological Parameters
22 ,0009.00224.0 hh mmbb Ω≡±=Ω≡ ωω
m
m
b
bsn
HH
ωω
ωω Δ
+Δ
−Δ=Δ 04.067.088.0
2
2
m
m
b
bsn
HH
ωω
ωω Δ
+Δ
−Δ=Δ 46.039.028.1
3
3
m
m
b
bs
TE
TEn
H
H
ω
ω
ω
ω Δ+
Δ+Δ−=
Δ 45.0095.066.02
2
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WMAP 5 & 7: Angular power spectrum
3rd
peak
Fig.: Tuhin Ghosh
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Current Angular power spectrum
Image Credit: NASA / WMAP Science Team
3rd
peak
6thpeak
4th peak5th peak
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0 0 0 1m K rΛΩ +Ω +Ω + Ω =0 0 0 1m K rΛΩ +Ω + Ω + Ω =
Image Credit: NASA / WMAP Science Team Fig.: Moumita Aich
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NASA/WMAP science team
Total energy density
Baryonic matter density
‘Standard’ cosmological model:Flat, ΛCDM (with nearly
Power Law primordial power spectrum)
Dark energy density
Good old Cosmology, … New trend !
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Non-Parametric: Peak Location(Amir Aghamousa, Mihir Arjunwadkar, TS ApJ 2012)
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Implied ‘cosmological parameter’ estimation
(Amir Aghamousa, Mihir Arjunwadkar, TS, in progress, 2012)
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(Amir Aghamousa, Mihir Arjunwadkar, TS in progress)
Non-Parametric fit to CMB spectrum
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),(),(2
φθφθ ∑ ∑∞
= −=
=Δl
l
lmlmlmYaT
CMB Anisotropy Sky map => Spherical Harmonic decomposition
Statistics of CMB
Statistical isotropy
*' ' ' 'lm l m l ll mma a C δ δ=
Gaussian CMB anisotropy completely specified by theangular power spectrum IF
=>Correlation function C(n,n’)=<ΔT(n) ΔT(n’)>
is Rotationally Invariant
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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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Statistical properties are not invariant under rotation of the sky
Breakdown of Statistical Isotropy !
North-South asymmetry Eriksen, et al. 2004,2006; Hansen et al. 2004 (in local power)Larson & Wandelt 2004 … , Park 2004 (genus stat.)
.
.
Special directions (“Axis of Evil”)Tegmark et al. 2004 (l=2,3 aligned), 2006Copi et al. 2004 (multipole vectors), … ,2006Land & Magueijo 2004 (cubic anomalies), …Prunet et al., 2004 (mode coupling)Bernui et al. 2005 (separation histogram)Wiaux et al. 2006
Underlying patternsT.Jaffe et al. 2005,2006..
Anisotropic, rotating cosmos(Bianchi VIIh)
Cosmic topology(Poincare Dodecahedron)
‘Anomalies’ in the WMAP CMB maps
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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 isotropycan be well estimated
by averaging over the temperature product between all pixel pairs separated by an angle .
Measuring the SI correlation
)(θC
θ
)cos()()()(~21
ˆ ˆ21
1 2
θδθ −⋅ΔΔ=∑∑ nnnTnTCn n
)ˆ,ˆ(8
1)ˆˆ( 21221 nnCdnnC ℜℜℜ=• ∫π
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Measuring the non-SI correlationIn the absence of statistical isotropyEstimate of the correlation function from a sky map given by a single temperature product
is poorly determined!!
)()(),(~2121 nTnTnnC ΔΔ=
(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()(
81
⎥⎦⎤
⎢⎣⎡ ℜℜℜℜ= ∫∫ Ω∫ Ω nnCdndnd χπ
κ
A weighted average of the correlation function over all
rotations
)ˆ,ˆ(8
1)ˆˆ( :Recall 21221 nnCdnnC ℜℜℜ=• ∫πBipolar multipole index
Wigner rotation matrix
)()( ℜ=ℜ ∑−=m
mmDχ
Characteristic function
Bipolar Power spectrum (BiPS) :A Generic Measure of Statistical Anisotropy
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2221
22 )](8
1[)ˆ,ˆ()12(21
ℜℜΩΩ+= ∫∫∫ χπ
κ dnnCdd nn
0)( δχ =ℜℜ∫ d
Statistical Isotropy
Correlation is invariant under rotations
)ˆ,ˆ()ˆ,ˆ( 2121 nnCnnC =ℜℜ
00δκκ =⇒
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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
221121
2211
212121*
2121 )}()(){,(
∑∫∫
=
⊗ΩΩ=
BiPoSH
Linear combination of off-diagonal elements
Bipolar spherical harmonics.
Clebsch-Gordan
Bipolar Power spectrum (BiPS) :A Generic Measure of Statistical Anisotropy
1 2 1 22 1( ) ( )4 l llC n n C P n nπ+
• = •∑
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0||21
21,,
2 ≥≡ ∑llM
MllAκBiPS:
rotationally invariant
M
mmMlml
Mll mMlml
CaaA1211
1
121121
*+∑ +=BiPoSH
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 statesMmlml |2211 0, 21121 =+++≤≤− Mmmlll
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MllA 2
'
MllA 4
'
Understanding BiPoSH coefficients
*' ' ' '
SI violation
:
lm l m l ll mma a C δ δ≠
' '' ''
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 κ
Spherical harmonics
Bipolar Power spectrum (BiPS) :A Generic Measure of Statistical Anisotropy
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Bipolar spherical harmonics
Spherical Harmonic Transforms
BipoSH Transforms
Angular power spectrum
BiPS
lma MllA '
lC κ
Spherical harmonics
Statistical Isotropyi.e., NO Patterns 0
0δκκ =⇒
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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 nθ
=
=
∑
∑
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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Simple Torus (Euclidean)
Compact hyperbolic space
Multiply connected Spherical space (Poincare dodecahedron)
Is the Universe Compact ?
Post WMAP Nature article(Luminet et al 2003)
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BiPS signature of a “soccer ball” universe
κ
(Hajian, Pogosyan, TS, Contaldi, Bond : in progress.)
=ΩK
Ideal, noise free maps predictions
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BiPS signature of a “soccer ball” universe
κ
(Hajian, Pogosyan, TS, Contaldi, Bond : in progress.)
013.1=Ωtot
Ideal, noise free maps predictions
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BiPS signature of Flat Torus spaces
κ
Hajian & Souradeep (astro-ph/0301590)
BiPS Spectroscopy of
Cosmic topology !?!
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Spaces that have must have only Even multipole BiPS ?
Flat compact spaces
Single-action spherical compact spaces
r No hyperbolic compact spaces
HM, TS: Discussion with Jeff Weeks
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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 ?(Souradeep & Ratra 2000, Mitra etal 2004, 2009
Hanson et al. 2010, Joshi, Mitra, TS 2012)
1 2
( )
00 '0
LMl l
Ll l
AC
+
9‐σ Detections !!
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2
ˆ ˆˆ ˆ( , , , ) ( , , )Legendre Expansion
( , )
( ) ( , )
k k p k k p
k
dkC P k kk
τ τ
τ
τ
Δ ≡ Δ •
↓Δ
= Δ∫1 1
1 1
1 1 1 1
'
*' '' ' ' '
' '
' '' '
ˆ ˆˆ ˆ( , , , ) ( , , )Bipolar SH Expansion
( , )
( ) ( , ) ( , )
LM
LM L Mm m o o
LML M m
LM L Mm m m m
k k p k k p
k
dka a P k k kk
C C
τ τ
τ
τ τ
Δ ≡ Δ •
↓
Δ
⎡ ⎤= Δ Δ⎣ ⎦
×
∑∫
Statistical Isotropy: CMB Photon distribution
FT ˆˆ ˆ ˆ( , , ) ( , , ) ( , , , )x p k p k k pτ τ τΔ ⎯⎯→ Δ ≡ Δ(Moumita Aich & TS, PRD 2010)
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Statistical Isotropy: CMB Photon distribution
[ ]
Free stream0
200 '0
''
(( , )
( ,
, )
... ( )) ll
rec
ecl
r
k
j k C
k
k τ
τ
τ
τ ⎯⎯⎯⎯→ Δ
⎡ ⎤= ⎦
Δ
ΔΔ ⎣∑
[ ]
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
τ
τ
τ
τ
⎯⎯⎯⎯→ Δ
⎧ ⎫= Δ ⎨ ⎬
⎭
Δ
⎩× Δ
∑
Statistical isotropy
General:Non-
Statistical isotropy
(Moumita Aich & TS, PRD 2010)
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2 1 1 2
2 1 1 2
1 2 1 2
1 2 1 2
( ) ( )
( ) ( )
( ) * ( ) ,
( ) * 1 ( ) ,
sym m e tric
a n tisym
E ve n p a rity
O d d p a rit
m .
[ ] ( 1)
[ ] ( 1) y
L M L Ml l l l
L M L Ml l l l
L M M L Ml l l l
L M M L Ml l l l
A A
A A
A A
A A
+ +
− −
+ + −
− + − −
=
= −
= −
= −
Even & odd parity BipoSH
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Weak Lensing
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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
nnφφ ε
Θ =
= + Θ = +Θ•
+= +∇
Ω∇
∇
∇×∇ Ω
oo
n̂ˆ 'n
CurlGradient WL: tensor/GWWL:scalar
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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 GAl l l l
C G C GA il l l l
φ+
−
⎡ ⎤= +⎢ ⎥
+ +⎣ ⎦
⎡ ⎤= Ω −⎢ ⎥
+ +⎣ ⎦
Deflection field: Even & Odd parity BipoSHBook, Kamionkowski & Souradeep, PRD 2012
WL: scalar
WL: tensor
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( )
( )
12 2
' ''
12 2
' ''
var ( ) /
var ( ) /
LMLM ll ll
ll
LMLM ll ll
ll
Q
Q
φ σ
σ
−+
−−
⎡ ⎤= ⎢ ⎥⎣ ⎦
⎡ ⎤Ω = ⎢ ⎥⎣ ⎦
∑
∑
[ ]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
σφ
σ
σ
σ
+ +
+
− −
−
=
Ω =
∑∑
∑∑
VarianceEstimators
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WMAP-7 BIPOLAR ‘anomaly’ from weak lensing?(Aditya Rotti, Moumita Aich & TS arXiv:1111.3357)ϕ20∼ 0.02
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Implications :
• The quadrupole of the projected lensing potential is large and cannot be accomodated in the standard LCDM cosmology.
• The BipoSH detection could be suggesting a strong deviation from standard cosmology. Primordial non-Gaussianity / alternative theories of gravity could possibly explain the large value of the quadrupole.
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• To probe violations of isotropy, measuring the large scale distribution of dark matter surrounding us will be of utmost
importance.
• Making measurements of the LSS on the largest angular scales will be an extremely challenging task. However future
experiments like LSST, DES and EUCLID might make this possible.
Lewis and
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Status of Non-GaussianityMild 2.5σ deviation hinted in the WMAP 3 data !
WMAP5&7 consistent with zero Yadav & Wandelt (2008); Smith, Senatore and
Zaldarriaga 2009)
Slide adapted from Amit Yadav
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CMB BipoSHs & Bispectra
' ' ' ' ''
LM s s LMll LM lm l m lml m
mm
A a a Cφ ∑∼
' ' ' ''
LM LMll LM lm l m lmlm
mm
A a a a C⇒ ∑∼
For deflection field
LM LMaφ →
BipoSH related to Bispectrum
' ' ' ' ''
( )'
(...)
LMLll LM lm l m lml m
MmmLM
llM
B a a a C
A +
∑
∑
∼
∼
Slm lm lma a aδ= +
Consider only: ' evenl l L+ + =
(Kamionkowski & Souradeep, PRD 2011)
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Odd parity Bispectra ?
1l 1l
2l2l
3l3l1 2 3l l l< <
( ) 1 2
1 2
l lBl l
− ×∼ has opposite sign in thetwo mirror configurations.
( ) ( )' ' LM
Lll llM
B A− −∑∼
Flat sky intuition:
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Odd parity Bispectra
1 2
odd 1 2 perms.1 2 nl nl1 2
( , ) = 2 ( )l ll lB l l f f C Cl 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 CC C C C
E
O
σ
σ
σ
< <
< <
−
+=
+ +
+=
+ +
⎡ ⎤+⎣ ⎦=+
∑
∑
1 2 3 2 1 3l l l l l lC C< < +∑
For local NG modelFlat sky approx
In general
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Planck Satellite on display at Cannes, France (Feb. 1, 2007)
Capabilities:•3x angular resolution of WMAP•5 to 20 x sensitivity of WMAP
Promises:• Cosmic Variance limited primary Cl
TT
• Polarization ClEE as good WMAP Cl
TT
• Unlikely, but may be lucky with ClBB
• Planck HFI core team members @IUCAA working on SI measurements using BipoSH
(Sanjit Mitra, Rajib Saha, TS)
Planck Surveyor SatelliteEuropean Space Agency: Launched May 14, 2009 HFI completed Jan 2012
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Summary• Current observations now allow a meaningful search for deviations from the `standard’
flat, ΛCDM 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.
• 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.
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Ultra Large scale structure of the universe
Thank you !!!