Tarun Souradeep

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Tarun Souradeep I.U.C.A.A, Pune, India Indo-UK Seminar IUCAA, Pune (Aug. 10, 2011) Quantifying Quantifying Statistical Isotropy Statistical Isotropy Violation Violation in the CMB in the CMB

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Quantifying Statistical Isotropy Violation in the CMB. Tarun Souradeep. Indo-UK Seminar IUCAA, Pune (Aug. 10, 2011). I.U.C.A.A, Pune, India. CMB space missions. 1991-94. 2001-2010. 2009-2011. CMBPol/COrE 2020+. CMB Anisotropy & Polarization. CMB temperature T cmb = 2.725 K. - PowerPoint PPT Presentation

Transcript of Tarun Souradeep

Page 1: 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

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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)