Testing THE STATISTICAL ISOTROPY OF CMB maps

Testing Testing

THE THE STATISTICAL STATISTICAL ISOTROPYISOTROPY

OF OF CMB mapsCMB maps

Testing Testing

THE THE STATISTICAL STATISTICAL ISOTROPYISOTROPY

OF OF CMB mapsCMB maps

Tarun Souradeep

I.U.C.A.A, Pune

Amir Hajian, Tuhin Ghosh

MPA, Garching

(Aug 22 , 2005)

Open Question in Cosmology :

NASA/WMAP science team

Intriguing: Lack of power at large angular scales )60( o

Can imply more than just the

suppression of power in the

low multipoles !

Asymmetries in the CMB anisotropy

Broadly, statistical properties are not invariant under rotations

I.e., Breakdown of Statistical isotropy ?

N-S asymmetry Eriksen, et al. 2004, Hansen et al. 2004 (in local power)Larson & Wandelt 2004, Park 2004 (genus stat.)

Special directionsTegmark et al. 2004 (l=2,3 aligned)Copi et al. 2004 (multipole vectors)Land & Magueijo 2004 (cubic anomalies)Prunet et al., 2004 (mode coupling)

Underlying `template’ pattern Jaffe et al. 2005 (Bianchi VIIh:mild cosmic rotation?)..

High N-S asymmetry

Low N-S asymmetry

Fig: H. K. Eriksen, et al. 2003

WMAP first year data

Bianchi template

Difference map

smooth random function on a sphere (sky map).

General random CMB anisotropy: described by a

Probability Distribution Functional – Mean:

– Covariance(2-point correlation)

– ...

– N-point correlation

Statistics of CMB

)]ˆ([ nTP 0 iT

Nji TTT ...

)()(),( jijiij nTnTnnCC

)ˆ(nT

Gaussian Random CMB anisotropy

Completely specified by the covariance matrix ijC

Possibilities:• Statistically Isotropic, Gaussian models

• Statistically Isotropic, non-Gaussian models

• Statistically An-isotropic, Gaussian models

• Statistically An-isotropic, non-Gaussian models

Statistics of CMB

),(),(2

l

l

lmlmlmYaT

CMB Anisotropy Sky map => Spherical Harmonic decomposition

Statistics of CMB

Statistical isotropy

''*

'' mmlllmllm Caa

CMB anisotropy completely specified by the

angular power spectrumangular power spectrum

''*

'' mmlllmllm Caa

Single index n: (l,m) -> n

Diagonal

''*

'' : violationSI mmlllmllm Caa

Mild breakdown

**''''

*''

lmlmmlml

mllm

aaaa

aa

(Bond, Pogosyan & Souradeep 1998, 2002)

''*


Radical breakdown

**''''

*''

lmlmmlml

mllm

aaaa

aa


• 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

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

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

BiPS: real space constructBiPS: real space construct

''*


Radical breakdown

**''''

*''

lmlmmlml

mllm

aaaa

aa


MllA2

'

MllA4

'

Understanding BiPoSH coefficients

*' ' ' '

SI violation

:

lm l m l ll mma a C

' '''

'

Measure cross correlation in

lml m

LMlm lm

mm

LM

lm

ll a a CA

a

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

Measure of Statistical IsotropyMeasure of Statistical Isotropy

00

isotropy Stat.

2

''

'''

' ''

BA

CaaA

Mll

Mll

Mmllm

mm

Mll mlml

• Averaging over l,l’& M beats down Cosmic variance .

• Fast: Advantage of fast SH transform. (1 min. /alpha 1.25 GHz proc.: Healpix 512, BiPS upto 20 )

• Orientation independent.

bias

SH transform of the map

Cosmic Bias

• Analytically calculate multi-D integrals over

– Gaussian statistics => express as products of covariance.

For SI correlation

)()()()( 4321 nTnTnTnT

))1(1()12(212

1

1

12

12

||

||lll

l

l

lll CCB

~B

(A. Hajian and Souradeep, ApJ Lett. 2003)

“True” Cl

Cosmic Variance

• Analytically calculate multi-D integrals over

– Gaussian statistics => express as products of covariance.

Tedious exercise: 105 terms, 96 connected terms.

''

', , ,

2

,

2

''

',

4

,,

2

1

2,

3

1

22

,

2

1

24

312211424231

1

131

2

242

22112

21

1

311141214131

1

1

2111

1

132

321

1

2

21

12

21

1

1

1

4

))1(2(412

)12(8

12

)12()1(15)12(4))12()1(4

12

)12(9()var(

Mmlml

Mmlml

Mmlml

MM

l

lmm

l

lmm

Mmlmll

lll

Mmlml

Mmlml

Mmlml

MM

l

lm

Mmlml

llll

llll

lll

llll

ll

l

CCCCCC

CCCCCCCCl

CCl

CCl

C

i

222 ~~)(

)()()()()()()()( 87654321 nTnTnTnTnTnTnTnT

(A. Hajian and Souradeep, ApJ Lett. 2003)“True” underlying theory

Bias corrected BiPS measurement

Analytic estimate for bias and cosmic variance match numerical measurements on simulated statistically isotropic maps !

~B

(A. Hajian and Souradeep, ApJ Lett. 2003)

222 ~~)(

Bias

Cosmic Variance

1

Testing Statistical Isotropy of WMAP

Circles search (Cornish, Starkman, Spergel, Komatsu 2004)

Foreground cleaned map

(Tegmark et al. 2003)

ILCNASA/WMAP science team

(for WMAP best fit model)

(Hajian, TS, Cornish, ApJLett in press )

Angular power spectra of the maps

• `Spergel’ Circles search map

• `Tegmark’ Foreground cleaned map

• ILC: WMAP internal combination map

(compared to the WMAP best fit model)

• WMAP best fit curve

• Average of 1000 realizations

Scanning the l-space with different windows•Maps can be filtered by isotropic window to retain power on certain angular scales, (eg., l~30 to 70)

lmllm aWa

•Cosmic variance >> Noise

•Tegmark’s map is ‘foreground’ free

Testing Statistical Isotropy of WMAPLow pass Gaussian filter at l= 40

(assuming WMAP best fit model)

Low pass Gaussian filter at l= 40

Statistically isotropic!

Probability Distribution of BiPS

Obtained from measurements of 1000 simulated SI CMB maps.

Can compute a Bayesian probability of map being SI for each BiPS multipole

(Given theory Cl)

Probability of a Map being SI

Low pass Gaussian filter at l= 40

Bayesian probability


Band pass filter between multipoles 20-30


BiPS imply

WMAP is Statistically

Isotropic !!

What does the null BiPS

meaurement of CMB maps imply

Sources of Statistical AnisotropySources of Statistical Anisotropy• Ultra large scale structure and cosmic topology.

• Anisotropic cosmology• Primordial magnetic fields (based on Durrer et al. 98, Chen et al. 04)

• Observational artifacts: – Anisotropic noise – Non-circular beam – Incomplete/unequal sky coverage – Residuals from foreground removal

Cosmic topologyCosmic topology

Multiply connected universe ?Multiply connected universe ?

Simple Torus (Euclidean)

Compact hyperbolic space

MC spherical space (“soccer ball”)

Eg., Zeldovich & Starobinsky 1972

Eg., Gott ’70, Cornish et al. 1996, Linde ‘04

BiPS signature of a “soccer ball” universe

(Hajian, Pogosyan, TS, Contaldi, Bond : in progress.)

K

Ideal, noise free maps predictions

BiPS signature of a “soccer ball” universe


013.1tot

Ideal, noise free maps predictions

Measured BiPS for a “soccer ball” universe


013.1tot

0.5

2.5

2.0

1.0

1.51000 simulated full

sky maps with WMAP noise

Is there a hidden pattern?

Jaffe et al. 2005

WMAP first year data

Rotating Universe Template

Subtraction of above two maps

Bianchi CMB Map & Power spectrum

Template of a rotating universe looking along the axis of rotation.

BiPS of Bianchi plus random map


=1e-3

=4e-4


Target specific l-space with different windows•Maps can be filtered by isotropic window to retain power on certain angular scales,

lmllm aWa

Low pass Gaussian filters

Band pass filters

Null BiPS Probability for Bianchi VIIh ( =0.5)

Gaussian filter

Band pass filter


Band pass filterCl filter

(/H)

Gaussian filter

Band passfilter


Summary • Propose BiPS as a generic measure for detecting

and quantifying Statistical isotropy violations. BiPS is insensitive to the overall orientation of SI breakdown (e.g., orientation of preferred axes). Hence constraints are not orientation specific. Computationally fast method

• Null results on some WMAP full sky maps. SI improves for a theory that predicts low power on low multipoles.

• Can constrain/detect cosmic topology.. BiPS constrains Dodecahedron universe strongly.

• Constrain anisotropic cosmological models BianchiVIIh claim (Jaffe et al. 2005), (/H)0 < 2.5*10-10 (99%CL)

• Diagnostic tool for observational artifacts in CMB maps• BipoSH & BiPS of CMB Polarization maps ?

Thank you !!!Thank you !!!

Testing maps for observational artifacts:

• Anisotropic noise

• Non-circular beam

• Incomplete/unequal sky coverage

• Residuals from foreground removal

Anisotropic Noise• Simplest case:

ijiijC 2

lmlmlm

N

mlmL

nYfnnC

f

)ˆ()ˆ,ˆ(

|| 2

Anisotropic Noise• Simplest case:

ijiijC 2

Effect of the Mask(AH, Souradeep astro-ph/0501001)

Wiener Filtered Map

Foregrounds ….


low pass filter to retain the lowest multipoles


Lowest multipoless l=2-4


(assuming WMAP bf

theory Cl )

Statistical Isotropy of WMAPProbability depend on the ‘true’ model

WMAP best fit theory spectrum over-predicts power on low

multipoles

Statistical Isotropy of WMAP

WMAP maps are SI if the model fits the power on low multipoles !!!!

Probability depend on the ‘true’ model

Statistical Isotropy of WMAPProbability depend on the ‘true’ model

WMAP best fit theory spectrum over-predicts power on low

multipoles

CSSK

TOH

(Shafeiloo & Souradeep )

Improved Error sensitive iterative Richardson-Lucy deconvolution method

)()( kGkPk

dkC ll

Primordial power spectrum from Early universe can deconvolved from CMB anisotropyspectrum

Recovered spectrum shows an infra-red cut-off on Horizon scale !!!

Is it cosmic topology ? Signature of pre-inflationary phase ? Trans-Planckian physics ? ….

Recovering the primordial power spectrum Recovering the primordial power spectrum

Horizon scale

Angular power spectrum from the recovered P(k)(Shafieloo & Souradeep 2003)


Lowest multipoles l=2-4


(assuming theory Cl for an ‘optimal’ primordial power

spectrum )

BiPS imply

WMAP is Statistically

Isotropic !!


Band pass filter between multipoles 20-30

Testing Statistical Isotropy of WMAPGalactic mask!

(assuming WMAP best fit model)

Ultra Large scale structure of the universe

BiPS of Primordial Magnetic Fields

B=0 B=30nG

BiPS of Primordial Magnetic Fields

Hajian, Chen, Souradeep, Kahniashvili, Ratra: in progress

I.U.C.A.A., Pune, India I.U.C.A.A., Pune, India

T= TR+5*10-3 TB

T= TR+3*10-3 TB

T= TR+2*10-3 TB

T= TR+1*10-3 TB

+

+ =

=

ClT=Cl

Random+2ClBianchi

2

• Where max 2

22

2

( )

l

l T CDMl l

C

C C

ClT=Cl

Random+2ClBianchi

2 P(2)

22

Testing THE STATISTICAL ISOTROPY OF CMB maps

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