Download - 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


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


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 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


General:Non-


(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)