Challenging the Isotropic Cosmos

Tarun SouradeepI.U.C.A.A, Pune, India

CTACC ColloquiumAIMS, Cape Town

(Apr. 13, 2012)

Challenging the

Isotropic Cosmos

CMB space missions

1991-94

2001-2010

2009-2011

CMBPol/COrE2020+

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

Transparent universe

Opaque universe

Here & Now

(14 Gyr)

0.5 Myr

Cosmic “Super–IMAX” theater

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

),(),(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=

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)

•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 !!!

Cosmic Acoustics: Ping the ‘Cosmic drum’

More technically,the Green function(Fig: Einsentein )

150 Mpc

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

(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 =

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

WMAP 5 & 7: Angular power spectrum

3rd

peak

Fig.: Tuhin Ghosh

Current Angular power spectrum

Image Credit: NASA / WMAP Science Team

3rd

peak

6thpeak

4th peak5th peak

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

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 !

Non-Parametric: Peak Location(Amir Aghamousa, Mihir Arjunwadkar,  TS  ApJ 2012)

Implied ‘cosmological parameter’ estimation

(Amir Aghamousa, Mihir Arjunwadkar,  TS, in progress, 2012)

(Amir Aghamousa, Mihir Arjunwadkar,  TS  in progress)

Non-Parametric fit to CMB spectrum

),(),(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

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)

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

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 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

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)

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

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 =⇒

• 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+

• = •∑

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

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 =∑

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

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 =⇒

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

Simple Torus (Euclidean)

Compact hyperbolic space

Multiply connected Spherical space (Poincare dodecahedron)

Is the Universe Compact ?

Post WMAP Nature article(Luminet et al 2003)

BiPS signature of a “soccer ball” universe

κ

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

=ΩK

Ideal, noise free maps predictions

BiPS signature of a “soccer ball” universe

κ

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

013.1=Ωtot

Ideal, noise free maps predictions

BiPS signature of Flat Torus spaces

κ

Hajian & Souradeep (astro-ph/0301590)

BiPS Spectroscopy of

Cosmic topology !?!

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

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 !!

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)

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)

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

Weak Lensing

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

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

( )

( )

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

WMAP-7 BIPOLAR ‘anomaly’ from weak lensing?(Aditya Rotti, Moumita Aich & TS arXiv:1111.3357)ϕ20∼ 0.02

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.

• 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

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

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)

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:

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

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

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.

Ultra Large scale structure of the universe

Thank you !!!