1. Non-Gaussianity at low and high multipoles from WMAP data
Oleg Verkhodanov Special astrophysical observatory
Nizhnij Arkhyz, Russia
A.G.Doroshkevich, P.D.Naselsky, I.D.Novikov, L.-Y. Chiang

2. Analysis of CMB data Registration: time odered data
Pixelization: data accumuation in pixels
Component separation
Analysis of signal statistics
Analysis of power spectrum

4. Multipole expansion
L = 2 (quadrupole)
L =3 (octupole)
L = 5
L = 4

5. To the problem modern history of CMB Gaussianty
All the data are Gaussian:
WMAP team (Komatsu et al., 2003), Tegmark et al. (2003),
and Spanish team with wavelets, WMAP3 (Spergel, et al., 2006)
Detection of non-Gaussianity
Chiang, Naselsky, Verkhodanov, Way (2003) – phase analysis
Confirmation:
Dineen, Coles (2003) – phase analysis
Spanish (Vielva, Martinez-Gonzales et al., 2003) – wavelet analysis
Erikson et al, 2003 – Mikowski functionals and so on...

6. Foregrounds + systematic
NON-GAUSSIANITY
Primordial
Foregrounds + systematic
NEW
PHYSICS
S= G + NG

7. Why Gaussianity of signal is important ?
Test for the standard (now) simple inflational LCDM model.
Non-Gaussianity:
Complex inflation
Topological:
close Universe (non-trivial topology),
topological defects (strings),
fractals
Systematic (effect of observations and analysis)
Contribution of foregrounds

8. Non-Gaussianity and topological defects
LCDM
Strings

9. = ? + Why Gaussianity of signal is important ? What else ?
Power spectrum:
= ? +

10. Phase analysis and non-Gaussianity

11. Phase analysis

12. It is phase k that keep Max’s face, not amplitude |k| !!
transformed Planck
Planck satellite and transformed Planck have the same power spectrum (same |k| ), they have different “faces” due to different phases:
It is phase k that keep Max’s face, not amplitude |k| !!
Planck satellite
Max Planck
|k| exp(ik)
FT-1[ ]
|k|
exp(ik)
|k| exp(ik)

13. Gaussian Random Fields
Strict definition of Gaussianity:
They possess Fourier modes whose real and imaginary parts are independently distributed and both Gaussian,
Weak definition of Gaussianity : random phase hypothesis
the Fourier phases are uniformly random between 0 and (by Central Limit Theorem)

14. Non-Gaussianity: phase diagram

15. Non-Gaussian map, l=
Gaussian map, l=

16. Circular statistics of phases (Fisher 1993)

17. Phase cross-correlation between WMAP CMB signal and foregrounds. P
Phase cross-correlation between WMAP CMB signal and foregrounds. P.Naselsky, A.Doroshkevich and O.Verkhodanov, 2003,ApJ.Lett,599,53 P.Naselsky, A.Doroshkevich , O.Verkhodanov, 2004,MNRAS,347,795

18. Multipole expansion
L = 2 (quadrupole)
L =3 (octupole)
L = 5
L = 4

19. Quadrupole and component separation

20. Asymmetry of the WMAP signal

21. Quadrupole phase diagram
(Naselsky, Verkhodanov,2006)
Red ILC (III)
Black Foreground
Solid (l,m)=(2,1)
Dashed (l,m)=(2,2)
Red: ILC(III)
Black Solid: Dipole Cosmological
Black Dash: Dipole non-Cosmo

22. What is ILC map ?
Obtained from 5 channels by minimization of correlation with foregrounds

23. Distribution of correlation coefficients
(Naselsky, Verkhodanov,2007, not published )
10000 simulations of L=2 maps (by Eriksen et al., 2003)
Simulated input maps
ILC maps

24. The mode=0 (C ) defines correlations20
ILC C20 initial
ILC C20 corrected

25. Some tricks when find same phases
Even (l+m) harmonics
V channel
Odd (l+m) harmonics
ILC map

26. V channel,
odd (l+m)
ILC

27. V channel,
odd (l+m)
V channel,
odd (l+m)
ILC
ILC

28. Non-Gaussianity of the WMAP
It's time to think and to work hard !

29. Binning

31. |aℓm| |aℓmCMB| |aℓm| The importance of phases: morphology and phases
exp(iℓm)
|aℓmCMB|
exp(iℓmCMB)

|aℓm|
exp(iℓmCMB)