1. 30/8/2005COSMO-05, Bonn 20051 CMB and the Topology of the Universe Dmitri Pogosyan, UAlberta With: Tarun Souradeep Dick Bond and: Carlo Contaldi Adam Hincks Simon White, Garching, last week: Open questions in Cosmology: Is our Universe is finite or infinite? Is its space curved ? ….

2. 30/8/2005COSMO-05, Bonn 20052 Preferred parameter region,  tot =1.02 ± 0.02 Ω tot =1.01 R LSS /R curv =0.3

3. 30/8/2005COSMO-05, Bonn 20053 How Big is the Observable Universe ? Relative to the local curvature & topological scales

4. 30/8/2005COSMO-05, Bonn 20054 Dirichlet domain and dimensions of the compact space R LS < R <

5. 30/8/2005COSMO-05, Bonn 20055 Dirichlet domain and dimensions of the compact space R LS > R >

6. 30/8/2005COSMO-05, Bonn 20056 Absence of large scale power NASA/WMAP science team

7. 30/8/2005COSMO-05, Bonn 20057 Correlation function of isotropic CMB Oscillations in the angular spectrum come from this break

8. 30/8/2005COSMO-05, Bonn 20058 Perturbations in Compact space Spectrum has the lowest eigenvalue. Spectrum is discrete, hence statistics in general is anisotropic, especially at large scales. Statistical properties can be inhomogeneous. However, perturbations are Gaussian, and CMB temperature fluctuations are fully described by pixel-pixel correlation matrix C T (p,p’) We implemented general method of images to compute C T (p,p’) for arbitrary compact topology (Bond,Pogosyan, Souradeep, 1998,2002)

9. 30/8/2005COSMO-05, Bonn 20059 Pixel-pixel correlation with compact topology (using method of images, BPS, Phys Rev D. 2000)

10. 30/8/2005COSMO-05, Bonn 200510 Example of strong correlation on last scattering surface These two points on LSS are identical And so are these

11. 30/8/2005COSMO-05, Bonn 200511 Correlated Circles (after Cornish, Spergel, Starkman et al, 1998,2004) (Figure: Bond, Pogosyan & Souradeep 1998, 2002)

12. 30/8/2005COSMO-05, Bonn 200512 Temperature along the correlated circles (pure LSS signal)

13. 30/8/2005COSMO-05, Bonn 200513 Temperature along the correlated circles (ISW modification)

14. 30/8/2005COSMO-05, Bonn 200514

15. 30/8/2005COSMO-05, Bonn 200515 Positive curvature multiconnected universe ? : Perhaps, a Poincare dodecahedron “Soccer ball cosmos” ?

16. 30/8/2005COSMO-05, Bonn 200516 Isotropic Cpp’ D/R LSS =10

17. 30/8/2005COSMO-05, Bonn 200517 Surface term to DT/T D/R LSS =0.3

18. 30/8/2005COSMO-05, Bonn 200518 Complete, surface and integrated large-angle ΔT/T D/R LSS =0.3

19. 30/8/2005COSMO-05, Bonn 200519 Variation of the pixel variance D/R LSS =0.3

20. 30/8/2005COSMO-05, Bonn 200520 Constraining the models from maps Complete topological information is retained when comparison with data is done on map level Low res (Nside=16) maps contain most information, although special techniques as circle searching may benefit from finer pixalization. The cost – additional small scale effects which mask topological correlations. Main signal comes from effects, localized in space, e.q on LSS. But even integrated along the line of sight contributions retain signature of compact topology. Orientation of the space (and, possibly, position of observer) are additional parameters to consider. What is the prior for them ?

21. 30/8/2005COSMO-05, Bonn 200521 Likelehood comparison of compact closed versus flat Universe with   

22. 30/8/2005COSMO-05, Bonn 200522 NASA/WMAP science team

23. 30/8/2005COSMO-05, Bonn 200523 Single index n: (l,m) -> n Diagonal

24. 30/8/2005COSMO-05, Bonn 200524 Inadequacy of isotropized C l ’s: a lm cross correlation Very small spaceJust a bit smaller than LSS 2 3 4 5 6 7 8 9 10

25. 30/8/2005COSMO-05, Bonn 200525 Compression to isotropic Cls is lossy Inhanced cosmic variance of Cl’s

26. 30/8/2005COSMO-05, Bonn 200526 Bipolar Power spectrum (BiPS) : A Generic Measure of Statistical Anisotropy A weighted average of the correlation function over all rotations Except for when Bipolar multipole index (This slide is provided as a free advertisement for T. Souradeep talk, Wednesday)

27. 30/8/2005COSMO-05, Bonn 200527 Conclusions A fundamental question – whether our Universe is finite or infinite –is still open. The region near Otot=1 is rich with possibilities, with negatively or positively curved or flat spaces giving rise to distinct topological choices. Modern CMB data shows that small Universes with V < VLS are failing to describe the temperature maps. Reason – complex correlation are not really observed (in line with circle finding results). This is despite the fact that it is not too difficult to fit the low l suppression of isotropized angular power spectrum. Integrated along the line of sight contribution to temperature anisotropy masks and modifies topology signature. It must be taken into account for any accurate quantitative restrictions on compact spaces. Full likelihood analysis assumes knowledge of the models. Model independent search for statistical anisotropy calls for specialized techniques – see talk on BiPS by Tarun Souradeep on Wednesday.