CMB Anisotropy & Polarization in Multiply Connected Universes

1. 8th B A H M A N 1384 * * * CMB Anisotropy & Polarization in Multiply Connected Universes By: Ehsan Kourkchi IUCAA & Sharif Univ. of Tech. Supervisors: T. Souradeep & S. Rahvar Saturday Jan. 28, 2006

2. Outline • What is the CMB? • The Statistics of CMB • The Different Possible Topologies of the Flat Universe • The Simplest Toroidal Compact Universe • Calculation of Correlation Function Using Naive Sachs-Wolf effect • CMB Map Generating • Considering the Other Physical Sources in Correlation Function • Map Analyzing

4. What is CMB? WMAP: First year results announced on Feb. 11, 2003 !

5. WMAP map of CMB anisotropy NASA/WMAP science team

9. Isotropy and Homogeneity

11. Statistics of CMB CMB can be treated as a Gaussian Random Field. . Mean . Correlation . N point Correlation <…> is ensemble average, i.e. an average over all possible realizations The Whole information could be found in two-point correlation function

12. TOPOLOGY

14. A Toroidal Universe Pictures: Weeks et. al. 1999 Slide by: Amir Hajian

15. Different Flat Topologies Table: Riazuelo et al. arXiv:astro-ph/0311314 v1 13 Nov 2003

16. Table: Riazuelo et al. arXiv:astro-ph/0311314 v1 13 Nov 2003

17. Different Flat Topologies Table: Riazuelo et al. arXiv:astro-ph/0311314 v1 13 Nov 2003

18. Table: Riazuelo et al. arXiv:astro-ph/0311314 v1 13 Nov 2003

19. Different Flat Topologies Table: Riazuelo et al. arXiv:astro-ph/0311314 v1 13 Nov 2003

21. Different Flat Topologies Table: Riazuelo et al. arXiv:astro-ph/0311314 v1 13 Nov 2003

22. Slab Space Slab Space With Flip

23. 2d Torus Imagine a cube which each parallel pair of its faces has been identified Then Confine the Last Scattering Surface into a 3d Torus

24. * Calculation of Correlation Function On large angular scales where topological effect becomes important, Sachs-Wolf effect is dominant and the relation between temperature of Last Scattering Surface and gravitational potential is: Conformal time Correlation using only Sachs-Wolf effect

25. Considering homogeneity dictate that: * Harrison-Zeldovich Spectrum Calculation of Correlation Function . . . Fourier Transform *

26. Calculation of Correlation Function . . . Correlation in a compact toroidal universe

27. Correlation Function in a Compact Toroidal Universe R x x’ Using FFT method one can easily find the two point correlation function for each pair very fast

28. Using FFT method and generating map realization 1) First we need to generate correlation matrix for each two point. For the last scattering surface we use HEALPix pixelization. 2) Decompose the covariance matrix into two matrices. 3) Multiply the decomposed matrix into a random matrix to have a map realization. Random matrix, < >

29. Correlation maps … The correlations between the pole of last scattering surface and the other points of the sphere. R/L = 1.5 The correlations between the pole of last scattering surface and the other points of the sphere. R/L = 1 R L

30. Correlation function between two points on a surface R=L/2

31. Correlation function between two points on a circle vs. angle separation R=L Correlation

34. Correlation Function in a Compact Torus Universe Using all physical sources. To considering all physical effect (not only naive Sachs-Wolf effect, we have such relation: What to do ! ? If we have statistical isotropy, the angular parts could be taken out and calculated easily to reduce the relation to: But, to investigate the topological effects we can no longer do the previous method. The integral over 3 dimensional k space is also taking the huge time (e.g. its order of magnitude is something like the Universe age ) S is the source function which contains all information since the CMB photons emitted to this point we observe them  Regarding to this condition the above integral should be taken over 1 dimensional k space and the process is fast enough.

37. Correlation Function in a Compact Torus Universe Using all physical sources. Separation of the Integral

38. More calculations …. Adding topological constraints, only some special Ks contribute in the summation,

41. It is under progress …

42. Statistical analysis of different generated maps … Symmetrical Maps (smaps): For each point of map it can be defined another temperature which is the square root of mean square of difference of each point temperature and its image regarding to the plain which it normal vector is the axis of symmetry connecting the main point and the center of the sphere. Symmetrical Maps (smaps): The point which has lower temperature shows the axis around which the map is most symmetrical. Doing some statistical analysis might enable us to get some particular limits on most probable volume of the compact space. St = < (T1-T2)2 >1/2 T1 T2 Oliveira-Costa & Smoot 1995 Oliveira-Costa 2003

43. Statistical analysis of different generated maps … (Naive Sachs-Wolf effect) Smap generated using a map with R/L=1 Some cool points show that there are some proffered axis in our universe. Absolutely, having a torus topology make the Universe some symmetrical axis.

44. Smap Smap generated using a map with R/L=1.5

45. Smap Analysis … <St>min = S0 S0 Map Number

46. Smap <St>min = S0 Under Progress …. generating different map realization containing all physical sources using appropriate calculated correlation matrices, we will be able to predict the properties of real toroidal compact spaces … Probability of finding a map which its Smin is less than S0 Different figures for different R/L ratios. S0

47. Hope to be done  • Calculating correlation matrix using faster methods • Generating map realization using all physical sources using appropriate correlation matrices • Analyzing our generated maps for different compact spaces • Investigating of non-statistical isotropic maps using different methods (Bips, S-map, …) and put some constraint on the type and size of the possible compact fundamental domains • Using the source functions of polarization in compact spaces and do everything again 

48. Thank You ...