1. 1 Analytical Spectra of RGWs and CMB Yang Zhang Astrophysics Center University of Science and Technology of China (USTC)

2. 2 Topics 1.Relic Gravitational Waves （ RGWs ） 2.C l XX of CMB generated by RGWS and re-ionized 3.C l XX generated by scalar perturbations in synchronous gauge

3. 3 1. RGWs Robertson-Walker metric ds 2 =a 2 (t) [ -dt 2 + (δ ij +h ij ) ] perturbations h ij ＝ hδ ij /3+h ij || (scalar) +h ij ┴ (vector) +h ij T (TT: RGWs) after generated by inflation, existing all the time, existing everywhere, broad distribution over (10 － 18 － 10 10 ) Hz,

4. 4 medium frequencies cavity: ν ＝ 10 4 Hz MAGO, EXPLORER laser interferometer: ground, ν ＝ 10 2 － 10 3 Hz LIGO, VIRGO, etc space, ν ＝ 10 -3 － 10 0 Hz LISA ， ASTROD, etc high frequencies Gaussian laser beam ν ＝ 10 9 － 10 10 Hz waveguide ν ＝ 10 8 Hz (Cruise & Ingley) low frequencies CMB ν ＝ (10 -18 － 10 -14 ) Hz WMAP, Planck, CMBPol, etc pulsar timing ν ＝ 10 -9 Hz PPTA, etc

5. 5 Analytic calculations equation ： solution ：

6. 6 Initial condition: A, β, α T : behavior during expansion ： Wavelengths > horizon ， |h ij | = constant ； Wavelengths < horizon ， |h ij | = h/a(t); → lower |h ij | at short wavelength λ

7. 7 Other modification processes : ν free-streaming ； uud→p, QCD phase transition ； e + e - →2γ, annihilation ； accelerating expansion (dark energy Ω Λ ) ；

8. 8 h(ν) and Ω g (ν) depend on inflation index β ： Class. Quant. Grav. 23, 3783 （ 2006 ）

9. 9 h(ν) and Ω g (ν) depend on running index α T ： Phys.Rev.D80 084022 （ 2009 ）

10. 10 neutrino free-streaming; Phys.Rev.D75, 104009 (2007 ）

11. 11 Phys.Rev.D77, 104016 (2008) QCD phase transition, e - e + annihilation,

12. 12 Dark energy reduces h(v) by a factor: Ω m /Ω Λ Class. Quant. Grav. 22, 3405 （ 2005 ）

13. 13 LIGO, Adv LIGO, LISA, DECIGO Phys.Rev.D80 (2009) 084022

14. 14 PPTA

15. 15 BBN constraints:

16. 16 MAGO, EXPLORER: Still short by ~7 orders of magnitudes in sensitivity; PRD80, (2009) 084022 Gaussian beam: Still short by ~5 orders in sensitivity; PRD 78, 024041 (2008);

17. 17 RGWs might be directly detected via CMB Scalar : C TT, C EE, C TE RGWs: C TT, C EE, C TE, C BB WMAP5

18. 18 2 、 Analytic C l TT, C l TE, C l EE, C l BB generated by RGWs, and re-ionized

19. 19 anisotropies polarization Boltzmann eq for CMB photons ： Equivalent to ： with

20. 20 The formal integrations ： where the visibility function for the decoupling process fitted by two half Gaussian functions: Carrying out time integration, one has

21. 21 Approximate, analytical solution ： where with c ~ 0.6, b ~ 0.8

22. 22 Analytical CMB spectra ：

23. 23 analytical, and numerical(CAMB) Phys.Rev.D74 (2006) 083006; Phys.Rev. D78 （ 2008 ） 123005

24. 24 improvements : 1.effective range : l < 300  l < 600, covering the first 3 peaks; 2. errors only ~ 3%; 3. C TT l and C TE l are also obtained ;

25. 25 ν- free-streaming has small modifications:

26. 26 ν- free-streaming ： 1. amplitude reduced by 20 ～ 35 ％ for l > 100; 2. C l XX are shifted slightly to larger l with Δl ∝ l. Δl =(1~5) ； (for the first two peaks)

27. 27 Zero multipole method: to examine the value l ～ 50,where C TE l crosses 0. Our results: Δl by NFS is the same order of magnitude as those caused by inflation index β inf and Ω b. More works are needed before a conclusion can be made.

28. 28 inflation index β ：

29. 29 WMAP5 constraint on C BB l : Phys.Rev.D78, 123005 (2008)

30. 30 Reionized case ------- possibly by first generation of luminous stellar objects ------- likely occurred z = (6~ 20), uncertain yet; WMAP5 : （ sudden re-ionization ） z = 11 (95%CL). ------- a major process secondary only to the decoupling V(t) consists of two parts : around z~1100 and around z~11

31. 31

32. 32 three models of reionization: Sudden ： η-linear ： Z-linear ：

33. 33

34. 34

35. 35

36. 36 Approximate, analytical solution ： with the coefficients

37. 37 a 1 -- the probability of a polarized photon last scattered during decoupling, a 2 -- the probability of a polarized photon last scattered during reionization, both depending upon the optical depth κr ： PRD79, 083002 (2009)

38. 38 Where h k (η) and dot h k (η) at decoupling and reionization

39. 39 Re-ionized CMB spectra

40. 40 Phys.Rev.D79, 083002 (2009)

41. 41 profile of C XX l determined by RGWs.

42. 42 Re-ionization bump location is sensitive to time η r. height is sensitive to duration Δη r.

43. 43 κ r - A degeneracy Phys.Rev.D79, 083002 (2009)

44. 44 κ r - β degeneracy broken from the 2nd peak

45. 45 Re-ionization also shifts l 0 around l=50 Therefore, Re-ionization has to be well studied before one can determine major cosmological parameters from CMB observational data.

46. 46 Contribution of baryon isocurvature is minor

47. 47 3 、 Analytic C l TT, C l TE, C l EE generated by scalar perturbations (in synchronous gauge)

48. 48 scalar perturbations of metric (synchronous gauge) ：

49. 49 Boltzmann eq.: formal sol. ：

50. 50 several technique treatments ： 1 。 Time integration 2 。 Removing gauge modes 3 。 Joining at R=M 4 。 Initial condition

51. 51 We get ： where

52. 52 Analytic result: (valid for l>400)

53. 53 Contributions by each term

54. 54 C l XX depend on inflation scalar index n S

55. 55 C l XX depend on baryon Ω b

56. 56 conclusion: Still need to fully understand CMB, The “precision cosmology” is yet to be reached. Thank you!