1. Challenging
the
Isotropic Cosmos
CTACC Colloquium Tarun Souradeep
AIMS, Cape Town I.U.C.A.A, Pune, India
(Apr. 13, 2012)

2. CMB space missions
1991-94
2001-2010
2009-2011
CMBPol/COrE
2020+

3. Cosmic Microwave Background
Pristine relic of a
hot, dense & smooth
early universe -
Hot Big Bang model
Post-recombination :Freely
propagating through (weakly
perturbed) homogeneous &
isotropic cosmos.
Pre-recombination : Tightly
coupled to, and in thermal
equilibrium with, ionized
matter.
(text background: W.

4. Cosmic “Super–IMAX” theater
0.5 Myr
Here
& Now
(14 Gyr)
Transparent universe
Opaque universe

5. CMB Anisotropy & Polarization
CMB temperature
Tcmb = 2.725 K
-200 μ K < Δ T < 200 μ K
Δ Trms ~ 70μ K
ΔTpE ~ 5 μ K
ΔTpB ~ 10-100 nK
Temperature anisotropy T + two polarization
modes E&B Four CMB spectra : ClTT,
ClEE,ClBB,ClTE
Parity violation/sys. issues: ClTB,ClEB

6. Statistics of CMB
CMB Anisotropy Sky map => Spherical Harmonic decomposition
∞ l
Δ T (θ , φ ) = ∑ ∑a Y (θ , φ )
lm lm
l =2 m=− l
alm a *
l 'm' = Cl δ ll 'δ mm '
Gaussian Random field => Completely specified by
angular power spectrum l(l+1)Cl :
Power in fluctuations on angular scales of ~ π/l

7. Fig. M. White 1997
The Angular power spectrum of
CMB anisotropy depends
C
sensitively on Cosmological l
parameters
Hence, a powerful tool for
constraining cosmological
parameters.
Multi-parameter Joint likelihood (MCMC)

8. Dissected CMB Angular power spectrum
•Low multipole : • Moderate multipole : • High multipole :
Sachs-Wolfe plateau Acoustic “Doppler” peaks Damping tail
CMB physics is very
well understood !!!
(fig credit: W. Hu)

9. Cosmic Acoustics: Ping the ‘Cosmic drum’
150 Mpc
More technically,
(Fig: Einsentein ) the Green function

10. WMAP: Angular power spectrum
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
Good match to
WMAP team

11. Peaks of the angular power spectrum
(74.1±0.3, 219.8±0.8)
(74.7 ±0.5, 220.1 ±0.8
Ω0K = 0
Ω0 B = 0.04
(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)
(Saha, Jain, Souradeep Apj Lett 2006)

12. Peak heights and ratios Cosmological Parameters
ωb ≡ Ωb h 2 = 0.0224 ± 0.0009, ω m ≡ Ω m h 2
ΔH 2 Δω b Δω m
= 0.88Δns − 0.67 + 0.04
H2 ωb ωm
ΔH 3 Δω b Δω m
= 1.28Δns − 0.39 + 0.46
H3 ωb ωm
ΔH 2
TE Δω b Δω m
= −0.66Δns + 0.095 + 0.45
H2
TE ωb ωm

13. WMAP 5 & 7: Angular power spectrum
3rd
peak
Fig.: Tuhin Ghosh

14. Current Angular power spectrum
3rd
peak
4th peak
5th peak
6thpeak
Image Credit: NASA / WMAP Science Team

15. Ω00m + Ω Λ + Ω0 K + Ω0 r = 1
Ω m + ΩΛ +
Image Credit: NASA / WMAP Science Team
Fig.: Moumita Aich

16. Good old Cosmology, … New trend !
Total energy
density
Dark energy
Baryonic matter density
density
‘Standard’ cosmological model:
Flat, ΛCDM (with nearly
Power Law primordial power spectrum)
NASA/WMAP science team

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

18. Implied ‘cosmological parameter’ estimation
(Amir Aghamousa, Mihir Arjunwadkar, TS, in progress, 2012)

19. Non-Parametric fit to CMB spectrum
(Amir Aghamousa, Mihir Arjunwadkar, TS in progress)

20. Statistics of CMB
CMB Anisotropy Sky map => Spherical Harmonic decomposition
∞ l
Δ T (θ , φ ) = ∑ ∑a Y (θ , φ )
lm lm
l =2 m=− l
Gaussian CMB anisotropy completely specified by the
angular power spectrum IF
Statistical alm a *
= Cl δ ll 'δ mm '
isotropy l 'm'
=>Correlation function C(n,n’)=<ΔT(n) ΔT(n’)>
is Rotationally Invariant

21. 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)

22. ‘Anomalies’ in the WMAP CMB maps
North-South asymmetry
Eriksen, et al. 2004,2006; Hansen et al. 2004 (in local power)
Larson & Wandelt 2004 … , Park 2004 (genus stat.)
Cosmic topology
. (Poincare Dodecahedron)
.
Special directions (“Axis of Evil”)
Tegmark et al. 2004 (l=2,3 aligned), 2006
Copi et al. 2004 (multipole vectors), … ,2006
Land & Magueijo 2004 (cubic anomalies), …
Prunet et al., 2004 (mode coupling)
Bernui et al. 2005 (separation histogram)
Wiaux et al. 2006 Anisotropic,
rotating cosmos
Underlying patterns (Bianchi VIIh)
T.Jaffe et al. 2005,2006
.
.
Statistical properties are not invariant under rotation of the sky
Breakdown of Statistical Isotropy !

23. Statistics of CMB
C(n1 , n2 ) ≡ C(n1 • n2 )
ˆ ˆ ˆ ˆ
Possibilities:
• Statistically Isotropic, Gaussian models
• Statistically Isotropic, non-Gaussian models
• Statistically An-isotropic, Gaussian models
• Statistically An-isotropic, non-Gaussian models
Ferreira & Magueijo 1997,
Bunn & Scott 2000,
Bond, Pogosyan & TS 1998, 2000

24. f ( n ) ≡ C ( n, z )
ˆ ˆ ˆ
Radical breakdown of SI
disjoint iso-contours
multiple imaging
Mild breakdown of SI
Distorted iso-contours
Statistically isotropic (SI)
Circular iso-contours
E.g.. Compact hyperbolic
Universe . (Bond, Pogosyan & Souradeep 1998, 2002)

25. Beautiful Correlation patterns
could underlie the CMB tapestry
Can we measure correlation patterns?
Figs. J. Levin
the COSMIC CATCH is

26. Measuring the SI correlation
Statistical isotropy
C (θ ) can be well estimated
by averaging over the temperature
product between all pixel pairs
separated by an angle θ .
~
C(θ ) = ∑∑ΔT (n1)ΔT (n2 )δ (n1 ⋅ n2 − cosθ )
ˆ
n1 ˆ
n2
1
C (n1 • n2 ) =
ˆ ˆ
8π 2 ∫ dℜ C (ℜn1 , ℜn2 )
ˆ ˆ

27. 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 ~
C ( n1 , n 2 ) = Δ T ( n1 ) Δ T ( n 2 )
is poorly determined!!
(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)

28. Bipolar Power spectrum (BiPS) :
A Generic Measure of Statistical Anisotropy
1
Recall : C (n1 • n2 ) = 2 ∫ dℜ C (ℜn1 , ℜn2 )
ˆ ˆ ˆ ˆ
Bipolar multipole index 8π
2
⎡ 1 ⎤
κ = ∫ dΩ n ∫ d Ω n
1 2 ⎢ 8π 2 ∫ dℜ χ (ℜ) C (ℜn1 , ℜn2 )⎥
⎣
ˆ ˆ
⎦
A weighted average of the
correlation function over all χ (ℜ ) = ∑D
m=−
mm (ℜ )
rotations Wigner
Characteristic
function rotation
matrix

29. ⇒κ =κ δ
0
Statistical Isotropy 0
Correlation is invariant
under rotations
C (ℜn1 , ℜn2 ) = C (n1 , n2 )
ˆ ˆ ˆ ˆ
1
κ = (2 + 1) ∫ dΩ ∫ dΩ ∫ dℜ χ
2 2 2
n1 n2
ˆ ˆ
C (n1 , n2 )[ (ℜ)]
8π
2
∫ dℜχ (ℜ) = δ 0

30. Bipolar Power spectrum (BiPS) :
A Generic Measure of Statistical Anisotropy
• Correlation is a two point function on a sphere
BiPoSH
C ( n1 , n2 ) = ∑A
l1l2 LM
LM
l1l2 {Yl1 ( n1 ) ⊗ Yl2 ( n2 )}LM Bipolar spherical
harmonics.
C (n1 • n2 ) = ∑
2l + 1
Cl Pl (n1 • n2 ) {Yl1 (n1 ) ⊗ Yl2 (n2 )}LM
= ∑ Cl1l2m1m2Yl1m1 (n1 )Yl2m2 (n2 )
4π LM
• Inverse-transform m1m2 Clebsch-Gordan
Al1l2 = ∫ dΩn1 ∫ dΩn2C(n1, n2 ){Yl1 (n1) ⊗Yl2 (n2 )}*
LM
LM
= ∑ al1m1al2m2 Cl1m1l2m2 LM Linear combination of
off-diagonal elements
m1m2

31. Recall: Coupling of angular momentum states
l1m1l2 m2 | M l1 − ≤ l2 ≤ l1 + , m1 + m2 + M = 0
BiPoSH Al1l2 = ∑ al1m1 al2 M +m1
M * M
Cl1m1l2 M +m1
coefficients : m1
• 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 : D l ≡ a lm a l + 2 m = ∑
All M C l +M m l m
'
M
2
Dl( i ) ≡ alm al +1 m+i = ∑ A Cl +M m+i l m
M
ll ' 1
M
BiPS:
rotationally invariant
κ ≡ ∑| A
M ,l1 ,l2
M 2
l1l2 | ≥0

32. Understanding BiPoSH
SI violation: coefficients
alm al*' m ' ≠ Cl δ ll 'δ mm '
4M
A ll '
2M
All ' = ∑
LM alm al ' m ' LM
C lml ' m ' A ll '
mm '
Measure cross correlation in alm

33. Spherical Bipolar spherical
harmonics harmonics
M
alm All '
Spherical Harmonic BiPoSH coefficents
coefficents
Cl κ
Angular power BiPS
spectrum
Bipolar Power spectrum (BiPS) :
A Generic Measure of Statistical Anisotropy

34. Spherical Bipolar spherical
harmonics harmonics
M
alm All '
Spherical Harmonic BipoSH Transforms
Transforms
Cl κ
Angular power BiPS
spectrum
Statistical Isotropy
⇒ κ = κ δ
0
i.e., NO Patterns 0

35. BIPOLAR maps of WMAP
Hajian & Souradeep (PRD 2007) ILC-3
Reduced BipoSH
Bipolar representation
AM = ∑ • M
All 'Measure of statistical isotropy
ll ' • Spectroscopy of Cosmic topology
ILC-1
Visualizing non-SI
• Anisotropic power spectrum
Bipolar map correlations
• Deflection fields (WL,…)
• Diagnostic of systematic effects/observational
∑
artifacts in the map
θ ( n ) = •A Differentiate Cosmic vs. Galactic B-mode Diff.
ˆ M Y M (n)
ˆ
M polarization
•SI part corresponds to the
“monopole” of the map.

36. Is the Universe Compact ?
Simple Torus
(Euclidean)
Multiply connected Spherical space
(Poincare dodecahedron)
Compact hyperbolic space Post WMAP Nature article
(Luminet et al 2003)

37. BiPS signature of a “soccer ball” universe
(Hajian, Pogosyan, TS, Contaldi, Bond : in progress.)
ΩK =
Ideal, noise free maps
predictions
κ

38. BiPS signature of a “soccer ball” universe
(Hajian, Pogosyan, TS, Contaldi, Bond :
in progress.)
Ωtot = 1.013
Ideal, noise free maps
κ predictions

39. BiPS signature of Flat Torus spaces
BiPS Spectroscopy of
κ Cosmic topology !?!
Hajian & Souradeep (astro-
ph/0301590)

40. 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

41. BIPOLAR measurements by WMAP-7 team
( + ) LM
(Bennet et al. 2010)
A
l1l2
L0
Non-zero Bipolar coeffs.!!!
C l 0 l '0 9‐σ Detections !! Sys. effect : beam distortion ?
(Souradeep & Ratra 2000, Mitra etal 2004, 2009
Hanson et al. 2010, Joshi, Mitra, TS 2012)
Image Credit: NASA / WMAP Science Team

42. Statistical Isotropy: CMB Photon distribution
(Moumita Aich & TS, PRD 2010)
Δ ( x , p,τ )
ˆ ⎯⎯→FT
Δ ( k , p, τ ) ≡ Δ ( k , k , p, τ )
ˆ ˆ ˆ
Δ(k , k , p,τ ) ≡ Δ(k , k • p,τ )
ˆ ˆ ˆ ˆ
Δ(k , k , p,τ ) SH ↓ Expansion
ˆ ˆ
Bipolar ≡ Δ ( k , k • p,τ )
ˆ ˆ
↓ Expansion
LegendreΔ LM (k ,τ )
'
Δ ( k ,τ )
dk
= ∫ P(k ) ∑ Δ 1 ' (k2 τ o ) ⎡Δ 1 ' (k ,τ o ) ⎤
*
LM L'M '
a ma 'm' dk , ⎣ ⎦
C ∫ LML(' M ) 1Δ (k ,τ )
k= P k ' m1
k
× C LM1 m C L1m1 '' m '
1m
'M

43. Statistical Isotropy: CMB Photon distribution
(Moumita Aich & TS, PRD 2010)
Δ (k ,τ rec ) ⎯⎯⎯⎯ Δ (k ,τ 0 )
→
Free stream
Statistical
∑ [...] j (k Δτ )
2
isotropy
= l
⎡C
⎣
l0
0 '0
⎤ Δ ' (k ,τ rec )
⎦
'l
Δ LM4 (k ,τ rec )
3
⎯⎯⎯⎯ Δ L1M2 (k ,τ 0 )
Free stream
→
General: ⎧ L ⎫
Non- = ∑ [...] j (k Δτ ) C
l
L0
0 10 C L0
0 10 ⎨
4 3
⎬
Statistical 3 4 ⎩ 1 l 2 ⎭
isotropy
× Δ LM4 (k ,τ rec )
3

44. Even & odd parity BipoSH
A l(2+1 ) L M = A l(1 l+2 ) L M
l s y m m e tric
A l(2−1 ) L M = − A l(1 l−2 ) L M
l a n tis y m m .
[ A l(1 l+2 ) L M ]* = ( − 1) M A l(1 l+2 ) L , − M E v e n p a rity
[ A l(1 l−2 ) L M ]* = ( − 1) M + 1 A l(1 l−2 ) L , − M O d d p a rit y

45. Weak Lensing

46. SI violation : Deflection field
ˆ
n
ˆ
n'
T (n ') = T (n + Θ) = T ( n ) + Θ • ∇T ( n )
ˆ ˆ ˆ ˆ
oo
Θ = ∇φ ( n ) + ∇ × Ω ( n )
ˆ ˆ
= ∇iφ ( n ) + ε ij ∇ j Ω( n )
ˆ ˆ
Gradient Curl
WL:scalar WL: tensor/GW

47. Deflection field: Even & Odd parity BipoSH
Book, Kamionkowski & Souradeep, PRD 2012
⎡ Cl GlL' l Cl 'Gll ' ⎤
L
All+ ) LM = φLM
(
⎢ + ⎥ WL: scalar
⎣ l '(l '+ 1) l (l + 1) ⎦
'
⎡ Cl GlL' l Cl 'Gll ' ⎤ WL: tensor
L
Al(2−1) LM = iΩ LM ⎢ − ⎥
⎣ l '(l '+ 1) l (l + 1) ⎦
l

48. BipoSH Measures of deflection field
Estimators Variance
∑Q A + ( + ) LM
/σ 2 LM
⎡ 2 LM ⎤
−1
( )
ll ' ll ' ll '
φLM = ll '
var φLM = ⎢ ∑ (Qll ' ) / σ ll ' ⎥
+ 2
∑ (Q ) ll '
+ 2
ll ' /σ 2 LM
ll ' ⎣ ll ' ⎦
∑ Qll ' All− ) LM / σ ll ' LM
− (
'
2
−1
⎡ 2 LM ⎤
Ω LM = var ( Ω LM ) = ⎢ ∑ (Qll ' ) / σ ll ' ⎥
ll ' − 2
∑ (Qll ' ) 2 / σ ll ' LM
−
ll '
2
⎣ ll ' ⎦
( ± ) LM ( ± ) LM
A A
A ( ± ) LM
l1l2 →
l1l2
L1 [...] = l1l2
L
C l 0 l '1 G ll '

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

50. 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.

51. • 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

52. Status of Non-Gaussianity
Mild 2.5σ deviation hinted in the WMAP 3 data !
Yadav & Wandelt (2008); Smith, Senatore and
Zaldarriaga 2009)
WMAP5&7 consistent with zero
Slide adapted from Amit Yadav

53. CMB BipoSHs & Bispectra
(Kamionkowski & Souradeep, PRD 2011)
For deflection field alm = alm + δ alm
S
All ' ∼ φLM ∑ alm als' m ' Clml ' m '
LM s LM
mm '
φ LM → a LM ⇒ All ' ∼ ∑ aLM alm al ' m ' Clmlm '
LM LM
mm '
BipoSH related to Bispectrum
BLll ' ∼ ∑
Mmm '
LM
aLM alm al ' m ' Clml ' m ' (...)
∼ ∑A
M
( + ) LM
ll ' Consider only: l + l '+ L = even

54. Odd parity Bispectra ?
(−
BLll )' ∼ ∑ '
M
All− ) LM
(
Flat sky intuition:
l2 l2
l3 l3
l1 < l2 < l3
l1 l1
(−) l1 × l2 has opposite sign in the
B ∼ two mirror configurations.
l1l2

55. Odd parity Bispectra
For local NG model
Flat sky approx
⎡ odd l1 × l2 ⎤
B(l1 , l2 ) = 2 ⎢ f nl + f nl ⎥ (Cl1 Cl2 + perms. )
⎣ l1l2 ⎦
In general
(Cl1 Cl2 + )
∑
perms.
f nl = σ 2
6G l3
Ell13l2
Cl1 Cl2 + Cl3 Cl2 + Cl1 Cl3
f nl l1l2
l1 <l2 <l3
(Cl1 Cl2 + )
∑
perms.
f odd
= σ 2
6G l3
Oll13l2
Cl1 Cl2 + Cl3 Cl2 + Cl1 Cl3
nl f nl l1l2
l1 <l2 <l3
2
⎡6 G (Cl1 Cl2 + perms.) ⎤
l3
σ −2
= ∑ ⎣ ⎦
l1l2
l1 <l2 <l3 Cl1 Cl2 + Cl3 Cl2 + Cl1 Cl3
f nl

56. Planck Surveyor Satellite
European Space Agency: Launched May 14, 2009 HFI completed Jan 2012
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 ClTT
• Polarization ClEE as good WMAP ClTT
• Unlikely, but may be lucky with ClBB
• Planck HFI core team members @IUCAA working
on SI measurements using BipoSH
(Sanjit Mitra, Rajib Saha, TS)

57. 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.

58. Ultra Large scale structure of the universe
Thank you !!!