Cosmic 21-cm Fluctuations from Dark-Age Gas

1. Cosmic 21-cm Fluctuations from Dark-Age Gas 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 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 richesthe 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!)

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

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.

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.

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

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

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

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 dependentFunction 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!

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: Perturbation Spins thermalized at Tk

23. Quantum Astrophysics Expand in orthogonal modes: Smooth Hermite

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

25. Ts(v) 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 Fourier Space Power Spectrum Direction cosine between wavevector and line of sight

29. The Observable:The Brightness Temperature

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

31. Power Spectra Change

32. 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)

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.

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: Difference Sum Helium

45. A Simplification 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:  Kinematic Distributions of H and He Relax to Thermal Equilibrium

47. Quantum Astrophysics Most Generally: Simplifies If: Spin and velocity relaxation times are fast compared to the expansion, rotation, shearing, diffusion or free-streaming times. Steady State. Homogenous. 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 How do we characterize neutral H atoms in the electronic ground state? Quantum Numbers Density Matrix

49. Quantum Astrophysics Spin-Resolved Distribution Function

50. Quantum Astrophysics Radiative A Big Mess HH Collision Matrix