1. Cosmic 21-cm Fluctuations from Dark-Age Gas Kris Sigurdson Institute for Advanced Study Cosmo 2006 September 25, 2006 Kris Sigurdson Institute for Advanced Study Cosmo 2006 September 25, 2006

2. What are the properties of neutral hydrogen atoms 20 to 100 million years after the big bang? How do we calculate their observational signatures? Kris Sigurdson Institute for Advanced Study Cosmo 2006 September 25, 2006 Kris Sigurdson Institute for Advanced Study Cosmo 2006 September 25, 2006 C. Hirata and KS (astro-ph/0605071)

3. Cosmic 21-cm Fluctuations: Why? The Epoch of Reionization ( e.g. Furlanetto et. al 2004 ). (Covered by Steve, Peng, and Miguel.) Measure the Primordial Power Spectrum at high redshift! 3D instead of a 2D CMB. ( e.g. Loeb and Zaldarriaga 2004 ) Another probe of Inflation; exotic particle physics effects on the Matter Power Spectrum. ( e.g. KS and Cooray 2005; Profumo, KS, Ullio and Kamionkowski 2004 ) If measured they will leave us with an embarrassment of riches the likes of which the world has never seen! The Epoch of Reionization ( e.g. Furlanetto et. al 2004 ). (Covered by Steve, Peng, and Miguel.) Measure the Primordial Power Spectrum at high redshift! 3D instead of a 2D CMB. ( e.g. Loeb and Zaldarriaga 2004 ) Another probe of Inflation; exotic particle physics effects on the Matter Power Spectrum. ( e.g. KS and Cooray 2005; Profumo, KS, Ullio and Kamionkowski 2004 ) If measured they will leave us with an embarrassment of riches the likes of which the world has never seen!

4. What I am not talking about. 21-cm fluctuations from the epoch of reionization (EOR). (Steve, Peng, and Miguel will cover or already covered that!) 21-cm fluctuations from the epoch of reionization (EOR). (Steve, Peng, and Miguel will cover or already covered that!)

5. What I am talking about. 21-cm fluctuations before reionization physics becomes important. Bewtween recombination and reionization. Smooth, slightly lumpy Universe. Main Players: Neutral Gas and the CMB Roughly Speaking 20 < z < 100 21-cm fluctuations before reionization physics becomes important. Bewtween recombination and reionization. Smooth, slightly lumpy Universe. Main Players: Neutral Gas and the CMB Roughly Speaking 20 < z < 100

6. 21-cm Hyperfine Transition

7. Calculate: Atomic Distribution Function Determines the 21-cm line profile. The integrated line profile determines the total 21-cm emissivity. The 21-cm emissivity (and fluctuations in the emissivity) are needed when calculating the power spectrum of 21-cm fluctuations. Determines the 21-cm line profile. The integrated line profile determines the total 21-cm emissivity. The 21-cm emissivity (and fluctuations in the emissivity) are needed when calculating the power spectrum of 21-cm fluctuations.

8. The Plan First: Calculate the spin-resolved distribution function of atomic hydrogen. Then: Calculate the 21-cm Line Profile, the 21-cm Emissivity, and the 21-cm Power Spectrum. First: Calculate the spin-resolved distribution function of atomic hydrogen. Then: Calculate the 21-cm Line Profile, the 21-cm Emissivity, and the 21-cm Power Spectrum.

9. The Atomic H Distribution Function Statatistical Mechanics Basics: Maxwell-Boltzmann Number Density H atom distribution function

10. The Spin Temperature * Radiative interactions with the CMB vs. Atomic Collisions: * Before Ly-  photons and the Wouthuysen-Field Effect turns on Collision Threshold Thermal Spin-Change Cross Section Einstein A Coefficient (Dalgarno 1961; Allison & Dalgarno 1969)

11. Atomic Spin-Change Collisions Schrödinger Phase Shifts Spin-Change Cross Section (Dalgarno 1961; Allison and Dalgarno 1969)

12. Spin-Change Cross Section

13. Thermal Cross Section

14. Spin-Temperature Evolution Absorption Against the CMB (Loeb & Zaldarriaga, PRL 2004)

15. What’s Wrong? Some Clues: Thermal Spin-Change Cross Section (Velocity Independent) (A Velocity Independent Function of T)

16. Thermal Cross Section (A Velocity Independent Function of T)

17. Spin-Change Cross Section (A Velocity dependent Function of E)

18. What’s wrong? Distribution does not factor! Collision time comparable to the radiative time Spin degrees of freedom are correlated with the kinetic degrees of freedom! Distribution does not factor! Collision time comparable to the radiative time Spin degrees of freedom are correlated with the kinetic degrees of freedom!

19. Quantum Astrophysics Solve the Boltzmann equation: Dominant Terms No Ly  Early Mostly Neutral

20. Quantum Astrophysics Steady State Solution: Radiative Term Blackbody Formula

21. Quantum Astrophysics Collision Term: Product of Cross Section and Relative Velocity Scattering out of v Scattering in to v Probability of F

22. Quantum Astrophysics Equations are nonlinear and nontrivial to solve However as: May solve in a perturbation series in about the thermal equilibrium solution: Equations are nonlinear and nontrivial to solve However as: May solve in a perturbation series in about the thermal equilibrium solution: Perturbation Spins thermalized at T k

23. Quantum Astrophysics Expand in orthogonal modes: Smooth Hermite

24. The Solution The steady state solution is where The steady state solution is where The Answer!!!!

25. T s (v) The spin-resolved distribution functions are: For comparison define: The spin-resolved distribution functions are: For comparison define: Velocity-Dependent Spin Temperature

26. Ts(v)

27. The Observable: The Brightness Temperature A function of redshift, density, and velocity (and direction on the sky)

28. The Observable: The Brightness Temperature Linear Power Spectrum Direction cosine between wavevector and line of sight Fourier Space

29. The Observable: The Brightness Temperature

30. Power Spectra (Naoz and Barkana, astro-ph/0503196)

31. Power Spectra Change

33. 21-cm Line Profile

34. Line Profile Width

35. Fourier Transform of Profile

36. The End The spin and velocity degrees of atomic hydrogen in primordial gas are correlated and the spin-resolved distribution function of atomic hydrogen is nonthermal. The 21-cm line profile is not Gaussian. Total emissivity altered. Redshift and projection dependent effect of up to 5% on the large scale power spectrum, and an order unity effect on the small scale power spectrum of 21-cm fluctuations. Details: (See C. Hirata and KS, astro-ph/0605071) The spin and velocity degrees of atomic hydrogen in primordial gas are correlated and the spin-resolved distribution function of atomic hydrogen is nonthermal. The 21-cm line profile is not Gaussian. Total emissivity altered. Redshift and projection dependent effect of up to 5% on the large scale power spectrum, and an order unity effect on the small scale power spectrum of 21-cm fluctuations. Details: (See C. Hirata and KS, astro-ph/0605071)

37. The End

38. Ts(v)

39. The Observable: The Brightness Temperature

40. The Plan First: Calculate the distribution function of atomic hydrogen. Then: Calculate the 21-cm Line Profile, the 21-cm Emissivity, and the 21-cm Power Spectrum. First: Calculate the distribution function of atomic hydrogen. Then: Calculate the 21-cm Line Profile, the 21-cm Emissivity, and the 21-cm Power Spectrum.

41. 21-cm Emissivity Photon Phase Space Density Gaussian

42. 21-cm Line Profile

43. Solve the Equation Matrix Structure: Radiative H-H H-He

44. Rotate Basis The key to the solution: SumDifferenceHelium

45. A Simplification In the new basis: Note that both and have no source term and do not depend on In the new basis: Note that both and have no source term and do not depend on It can be shown

46. A Simplification We thus have: with the solution: We thus have: with the solution: Kinematic Distributions of H and He Relax to Thermal Equilibrium 

47. Quantum Astrophysics Most Generally: Simplifies If: A)Spin and velocity relaxation times are fast compared to the expansion, rotation, shearing, diffusion or free- streaming times. Steady State. Homogenous. B)Isotropic radiation field with smooth frequency dependence (such as the CMB). Radiative Rates Independent of Direction. C)Collisional transitions dominated by simple spin exchange mechanisms. No Atomic Polarization

48. Quantum Astrophysics Quantum Numbers Density Matrix How do we characterize neutral H atoms in the electronic ground state?

49. Quantum Astrophysics Spin-Resolved Distribution Function

50. Quantum Astrophysics Radiative HH Collision Matrix

51. H-H Atomic Collision Term The end result of all this formalism:

52. He-H Atomic Collision Term Should account for Helium as: Introduce: Helium collision term: Should account for Helium as: Introduce: Helium collision term: No F changing collisions as He is spin singlet

53. Solve the Equation Boltzmann Equation in Matrix form Relaxation Matrix Source Vector

54. 21-cm Line Profile

55. Fourier Transform of Profile 21-cm brightness temperature fluctuations must be convolved with the 21-cm line profile in the radial direction or in terms of power spectra multiplied by 21-cm brightness temperature fluctuations must be convolved with the 21-cm line profile in the radial direction or in terms of power spectra multiplied by Fourier Transform

56. 21-cm Line Profile Photon Phase Space Density Not Gaussian