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Analytical Spectra of RGWs and CMB Yang Zhang Astrophysics Center University of Science and Technology of China (USTC). Topics. Relic Gravitational Waves （ RGWs ） C l XX of CMB generated by RGWS and re-ionized C l XX generated by scalar perturbations

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  1. Analytical Spectra ofRGWs and CMBYang Zhang Astrophysics CenterUniversity of Science and Technology of China(USTC)

  2. Topics • Relic Gravitational Waves（RGWs） • ClXX of CMB generated by RGWS and re-ionized • ClXX generated by scalar perturbations in synchronous gauge

  3. 1. RGWs Robertson-Walker metric ds2=a2(t) [ -dt2 + (δij+hij) ] perturbations hij ＝ hδij/3+hij|| (scalar) +hij┴ (vector) +hijT (TT: RGWs) after generated by inflation, existing all the time, existing everywhere, broad distribution over (10－18－1010 ) Hz,

  4. medium frequenciescavity: ν＝104 Hz MAGO, EXPLORER laser interferometer: ground,ν＝102－103 Hz LIGO, VIRGO, etcspace,ν＝ 10-3－100 Hz LISA，ASTROD, etchigh frequenciesGaussian laser beamν＝ 109－1010 Hz waveguide ν＝ 108 Hz (Cruise & Ingley)low frequenciesCMB ν＝(10-18－10-14) Hz WMAP,Planck, CMBPol, etcpulsar timingν＝ 10-9HzPPTA, etc

  5. Analytic calculations equation： solution：

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

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

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

  9. h(ν) and Ωg (ν) depend on running index αT： Phys.Rev.D80 084022 （2009）

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

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

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

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

  14. PPTA

  15. BBN constraints:

  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. RGWs might be directly detected via CMB Scalar : CTT, CEE, CTE RGWs: CTT, CEE, CTE, CBB WMAP5

  18. 2、Analytic ClTT, ClTE, ClEE, ClBBgenerated by RGWs , and re-ionized

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

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

  22. Analytical CMB spectra：

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

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

  25. ν- free-streaming has small modifications:

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

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

  28. inflation index β：

  29. WMAP5 constraint on CBBl : Phys.Rev.D78, 123005   (2008)

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

  32. Approximate, analytical solution ： with the coefficients

  33. a1 -- the probability of a polarized photon last scattered during decoupling, a2 -- the probability of a polarized photon last scattered during reionization, both depending upon the optical depth κr： PRD79, 083002 (2009)

  34. Where hk(η) and dot hk(η) at decoupling and reionization

  35. Re-ionized CMB spectra

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

  37. profile of CXXl determined by RGWs .

  38. Re-ionization bump • location is sensitive to time ηr. • height is sensitive to duration Δηr.

  39. κr - A degeneracy Phys.Rev.D79, 083002 (2009)

  40. κr - β degeneracy broken from the 2nd peak

  41. Re-ionization also shifts l0 around l=50 Therefore, Re-ionization has to be well studied before one can determine major cosmological parameters from CMB observational data.

  42. Contribution of baryon isocurvature is minor

  43. 3、Analytic ClTT, ClTE, ClEEgenerated by scalar perturbations (in synchronous gauge)

  44. scalar perturbations of metric(synchronous gauge)：

  45. Boltzmann eq.: formal sol. ：

  46. several technique treatments：1。Time integration2。Removing gauge modes 3。Joining at R=M4。Initial condition

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