Testing THE STATISTICAL ISOTROPY OF CMB maps

1. Open Question in Cosmology : Testing THE STATISTICAL ISOTROPYOF CMB maps Tarun Souradeep MPA, Garching (Aug 22 , 2005) Amir Hajian, Tuhin Ghosh I.U.C.A.A, Pune

2. WMAP: Angular correlation function Intriguing: Lack of power at large angular scales Can imply more than just the suppression of power in the low multipoles ! NASA/WMAP science team

3. Asymmetries in the CMB anisotropy N-S asymmetry Eriksen, et al. 2004, Hansen et al. 2004 (in local power) Larson & Wandelt 2004, Park 2004 (genus stat.) Special directions Tegmark 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?) . . WMAP first year data Bianchi template Difference map Broadly, statistical properties are notinvariant under rotations I.e., Breakdown of Statistical isotropy ? Low N-S asymmetry High N-S asymmetry Fig: H. K. Eriksen, et al. 2003

4. Statistics of CMB 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 • Gaussian Random CMB anisotropy • Completely specified by the covariance matrix

5. Statistics of CMB Possibilities: • Statistically Isotropic, Gaussian models • Statistically Isotropic, non-Gaussian models • Statistically An-isotropic, Gaussian models • Statistically An-isotropic, non-Gaussian models

6. Statistics of CMB CMB Anisotropy Sky map => Spherical Harmonic decomposition CMB anisotropy completely specified by the angular power spectrum Statistical isotropy

7. Single index n: (l,m) -> n Diagonal

8. Mild breakdown (Bond, Pogosyan & Souradeep 1998, 2002)

9. Radical breakdown (Bond, Pogosyan & Souradeep 1998, 2002)

10. Bipolar Power spectrum (BiPS) :A Generic Measure of Statistical Anisotropy BiPoSH • Correlation is a two point function on a sphere • Inverse-transform Bipolar spherical harmonics. Clebsch-Gordan Linear combination of off-diagonal elements

11. Recall: Coupling of angular momentum states 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 : BiPS:rotationally invariant

12. BiPS: real space construct Bipolar multipole index A weighted average of the correlation function over all rotations Wigner rotation matrix Characteristic function

13. Radical breakdown (Bond, Pogosyan & Souradeep 1998, 2002)

14. Understanding BiPoSH coefficients

15. Spherical harmonics Bipolar spherical harmonics

16. Spherical harmonics Bipolar spherical harmonics

17. Measure of Statistical Isotropy SH transform of the map bias • 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.

18. “True” Cl Cosmic Bias • Analytically calculate multi-D integrals over • Gaussian statistics => express as products of covariance. For SI correlation (A. Hajian and Souradeep, ApJ Lett. 2003)

19. Cosmic Variance • Analytically calculate multi-D integrals over • Gaussian statistics => express as products of covariance. Tedious exercise: 105 terms, 96 connected terms. “True” underlying theory (A. Hajian and Souradeep, ApJ Lett. 2003)

20. Bias corrected BiPS measurement (A. Hajian and Souradeep, ApJ Lett. 2003) Bias Cosmic Variance Analytic estimate for bias and cosmic variance match numerical measurements on simulated statistically isotropic maps !

21. Testing Statistical Isotropy of WMAP (for WMAP best fit model) (Hajian, TS, Cornish, ApJLett in press ) Foreground cleaned map (Tegmark et al. 2003) ILC NASA/WMAP science team Circles search (Cornish, Starkman, Spergel, Komatsu 2004)

22. Angular power spectra of the maps (compared to the WMAP best fit model) • `Tegmark’ Foreground cleaned map • `Spergel’ Circles search map • ILC: WMAP internal combination map • WMAP best fit curve • Average of 1000 realizations

23. Cosmic variance >> Noise • Tegmark’s map is ‘foreground’ free 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)

24. Testing Statistical Isotropy of WMAP Low pass Gaussian filter at l= 40

25. Statistically isotropic! Low pass Gaussian filter at l= 40 (assuming WMAP best fit model)

26. 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)

27. Probability of a Map being SI Bayesian probability Low pass Gaussian filter at l= 40

28. Probability of a Map being SI Bayesian probability Band pass filter between multipoles 20-30 BiPS imply WMAP is Statistically Isotropic !!

29. What does the null BiPS meaurement of CMB mapsimply

30. Sources 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

31. Simple Torus (Euclidean) Eg., Zeldovich & Starobinsky 1972 Cosmic topology Multiply connected universe ? MC spherical space (“soccer ball”) Compact hyperbolic space Eg., Gott ’70, Cornish et al. 1996, Linde ‘04

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

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

34. Measured BiPS for a “soccer ball” universe (Hajian, Pogosyan, TS, Contaldi, Bond : in progress.) 2.5 2.0 1000 simulated full sky maps with WMAP noise 1.5 1.0 0.5

35. WMAP first year data Rotating Universe Template Subtraction of above two maps Is there a hidden pattern? Jaffe et al. 2005

36. Bianchi CMB Map & Power spectrum Template of a rotating universe looking along the axis of rotation.

37.  BiPS of Bianchi plus random map

38. BiPS of Bianchi plus random map  =1e-3

39. BiPS of Bianchi plus random map  =4e-4

40. Target specific l-space with different windows • Maps can be filtered by isotropic window to retain power on certain angular scales, Low pass Gaussian filters Band pass filters

41. Null BiPS Probability for Bianchi VIIh ( W=0.5) Band pass filter Gaussian filter

42. Null BiPS Probability for Bianchi VIIh ( W=0.5) Band pass filter Cl filter

43. Null BiPS Probability for Bianchi VIIh ( W=0.99) Gaussian filter Band pass filter (s/H)

44. 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 ?

45. Thank you !!!

46. Testing maps for observational artifacts: • Anisotropic noise • Non-circular beam • Incomplete/unequal sky coverage • Residuals from foreground removal

47. Simplest case: Anisotropic Noise

48. Simplest case: Anisotropic Noise

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

50. Wiener Filtered Map