S TATISTICAL I SOTROPY of CMB A NISOTROPY

1. STATISTICALISOTROPYofCMBANISOTROPY Amir Hajian & Tarun Souradeep I.U.C.A.A, Pune http://www.iucaa.ernet.in/~tarun/pascos03.ppt

2. Cosmic Microwave Background – a probe beyond the cosmic horizon Pristine relic of a hot, dense & smooth early universe - Hot Big Bang model Pre-recombination : Tightly coupled to, and in thermal equilibrium with, ionized matter. Post-recombination :Freely propagating through (weakly perturbed) homogeneous & isotropic cosmos. CMB anisotropy and Large scale Structure formed from tiny primordial fluctuations through gravitational instability Simple linear physics allows for accurate predictions Consequently a powerful cosmological probe

3. Fig. M. White 1997 The Angular power spectrum of the CMB anisotropy depends sensitively on the present matter current of the universe and the n spectrum of primordial perturbations The Angular power spectrum of CMB anisotropy is considered a powerful tool for constraining cosmological parameters. Fig.. Bond 2002

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

5. Statistics of CMB The primordial perturbations are believed to be Gaussian random field. The most popular idea about their origin is quantum fluctuations during inflation. • Gaussian Random CMB anisotropy • Completely specified by the covariance matrix

6. Statistics of CMB CMB anisotropy completely specified by the angular power spectrum Statistically isotropicGaussian randomCMB anisotropy Only if

7. Statistics of CMB Statistical Isotropy means the two point correlation function depends only on the angular separation between the two directions in the sky. • :smooth random function on a (pix sphere. • General random CMB anisotropy:described by a probability distribution functional assigning a number (probability) to every CMB anisotropy sky map • Mean: • Covariance(2-point correlation) • ….. • N-point correlation • Gaussian random CMB anisotropy: Completely specified by the covariance: • Statistically isotropicGaussian random CMB anisotropy: Completely specified by angular power spectrum

8. Statistics of CMB Statistical Isotropy implies the two point correlation function depends only on the angular separation between the two directions in the sky. • :smooth random function on a ) sphere. • General random CMB anisotropy:described by a probability distribution functional assigning a number (probability) to every CMB anisotropy sky map • Mean: • Covariance(2-point correlation) • ….. • N-point correlation • Gaussian random CMB anisotropy: Completely specified by the covariance Statistically isotropic Gaussian random CMB anisotropy: i.e., Correlation is invariant under rotations Completely specified by angular power spectrum

9. Iso-contours of correlation around a point Radical breakdown of SI disjoint iso-contours multiple imaging Mild breakdown of SI Distorted iso-contours Statistically isotropic (SI) Circular iso-contours

10. Beautiful Correlationpatterns could underlie the CMB tapestry Figs. J. Levin Akin to Leopard’s spots

11. Can we Measure Correlation Patterns? Sure, maybe quite possible for Leopard spots BUT for CMB anisotropy the COSMIC CATCH is there is only one CMB sky.

12. Measuring the correlation Statistical isotropy can be well estimated by averaging over the temperature product between all pixel pairs separated by an angle .

13. Measuring the correlation Violation of statistical isotropy Estimate of the correlation function from a sky map given by a single temperature product is poorly determined. is inadequate for model comparison

14. A Measure of Statistical Anisotropy

15. A Measure of Statistical Anisotropy Characteristic function Wigner rotation matrix

16. A Measure of Statistical Anisotropy is the three dimensional rotation through an angle about the axis Characteristic function Wigner rotation matrix

17. A Measure of Statistical Anisotropy A weighted average of the correlation function over all rotations Except for when

18. Why is a measure ofstatistical anisotropy. statistical anisotropy.

19. Statistical Isotropy Correlation is invariant under rotations

20. What exactly are

21. In Harmonic Space • Correlation is a two point function on a sphere Suggests a bipolar spherical harmonics expansion : Bipolar spherical harmonics.

22. In Harmonic Space • Correlation is a two point function on a sphere : Bipolar spherical harmonics. • Inverse-transform

23. Statistical isotropy :

24. What if we find Statistical anisotropy in CMB maps

25. Sources of Statistical Anisotropy • Ultra large scale structure and cosmic topology:GR is a local theory and does not dictate the global topology of space-time. Space can be multiply connected, e.g. Torus universe with Euclidean geometry. SIGNAL • Observational artifacts: • Anisotropic noise • Non-circular beam • Incomplete/unequal sky coverage • Residuals from foreground removal

26. Ultra Large scale structure of the universe

27. How Big is the Observable Universe ? Relative to the local curvature & topological scales

28. Simple Torus (Euclidean) Consider all Spaces of Constant Curvature Homogenous & isotropic but Multiply connected universe ? Compact hyperbolic space

29. A Toroidal Universe The Euclidean 2-torus is a flat square whose opposite sides are connected. Light from the yellow galaxy can reach them along several different paths. So they can see more one image of it. Pictures: Weeks et. al. 1999

30. Spatial Correlations in … • Simply connected space • (STATISTICALLY ISOTROPIC) • A Toroidal Space Iso-correlation contours are no more circular. Back

31. THREE POSSIBILITIES ( Size of the compact space relative to horizon scale) Medium Large Small

32. Multiply Imaged Distorted Isotropic

33. Equal Sided Torus is non-zero for even l. BUT is zero  Torus shows a strong characteristic pattern.

34. Unequal Sided Torus • Non-zero • Again non-zero for even l.

35. Squeezed Torus is non-zero for even l. And is NOT zero  Next

36. Torus has three preferred axes which cause the statistical anisotropy of Toroidal Spaces. Back

37. pattern related to preferred directions?

38. Pattern related to preferred directions?

39. Analytical Approach • Leading order contributions to can be calculated analytically for torus (Bond, Pogosyan, Souradeep, 2000) Well-known periodic box problem

40. Analytical Approach • Equal sided torus is zero !

41. Analytical Approach • Un-equal sided torus

42. Analytical Approach • Squeezed torus

43. A RECIPE for Estimating • Take two pixels, and on the sky and the product • Rotate both pixels by an angle around an axis to get pixels and . Compute the temperature product • Construct a function by summing over the temperature products obtained by varying the rotation axis all over the sky, • Construct using a series of terms • Construct by summing the square of over all pixel pairs,

44. Cosmic Bias • Analytically calculate multi-D integrals over • Gaussian statistics => express as products of covariance. Tedious exercise carried out for SI correlation • Numerically • Make many realizations of CMB anisotropy. For each of them measure • For sufficiently large number of realizations the average value of will differ from the ensemble average by the cosmic bias.

45. Cosmic Variance • Analytically calculate multi-D integrals over • Gaussian statistics => express as products of covariance. 105 terms. 56 connected terms.. But we have the terms ! Tedious exercise similar the bias but more complicated. • Numerically: • Make many realizations of CMB anisotropy. For each of them measure • For sufficiently large number of realizations the average value of will tend to the ensemble average and the variance is a good estimate of the ensemble variance.

46. Compact Hyperbolic Models Compact hyperbolic (CH: m004) space at when The number titled Tot is

48. Summary • A generic measure for detecting and quantifying Statistical isotropy violations • Can search the most generic signature of cosmic topology and Ultra large scale structure • The measures can be neatly related to existence of preferred directions in correlation • But measure is insensitive to the overall orientation SI breakdown (orientation of preferred axes). Hence limits on SI are not orientation specific.

49. Future Plans • Identify & compute Statistical anisotropy signatures in other scenarios with SI violating correlations • Address and remove observational artifacts. • Apply to high-sensitivity full-sky data from the MAP satellite in early 2003. • Search for signatures of cosmic topology and Ultra large scale structure.

50. Finding these patterns leads to geometric methods.