1. Radiative Transfer Effects in Hydrogen Recombination: 2γ Transitions and Lyα Diffusion Christopher Hirata Orsay – July 2009 With special thanks to: E. Switzer (Chicago) D. Grin, J. Forbes, Y. Ali-Haïmoud (Caltech)

2. Outline 1.Motivation, CMB 2.Standard picture 3.He recombination 4.Two-photon decays in H 5.Lyman-α diffusion

3. Cosmic microwave background The CMB has revolutionized cosmology: - Tight parameter constraints (in combination with other data sets) - Stringent test of standard assumptions: Gaussianity, adiabatic initial conditions - Physically robust: understood from first principles. (Linear perturbation theory.) WMAP Science Team (2008) 1. Motivation, CMB

4. CMB and inflation Primordial scalar power spectrum: n s,  s  measured by broad-band shape of CMB power spectrum  the damping tail will play a key role in the future (Planck, ACT, SPT, …)  probe of inflationary slow-roll parameters: (for single field inflation; but measurements key for all models) 1. Motivation, CMB

5. This is the CMB theory! 1. Motivation, CMB

6. This is the CMB theory! n e = electron density (depends on recombination) 1. Motivation, CMB

7. Recombination history z … as computed by RECFAST1.3 (Seager, Sasselov, Scott 2000) The “standard” recombination code until Feb. 2008. H + + e -  H z: acoustic peak positions degenerate with D A  z: polarization amplitude He + + e -  He z: damping tail degenerate with n s He 2+ + e -  He + no effect 2. Standard picture

8. Standard theory of H recombination (Peebles 1968, Zel’dovich et al 1968)  = 2-photon decay rate from 2s P esc = escape probability from Lyman-  line A Ly  = Lyman-  decay rate  e = recombination rate to excited states g i = degeneracy of level i  i = photoionization rate from level i R = Rydberg 1s 2s2p 3s3p3d H + + e - 22 Lyman-  resonance escape radiative recombination + photoionization 2. Standard picture

9. Standard theory of H recombination (Peebles 1968, Zel’dovich et al 1968)  = 2-photon decay rate from 2s P esc = escape probability from Lyman-  line = probability that Lyman-  photon will not re-excite another H atom. Higher  or P esc  faster recombination. If  or P esc is large we have approximate Saha recombination. 1s 2s2p 3s3p3d H + + e - 22 Lyman-  resonance escape radiative recombination + photoionization 2. Standard picture

10. Helium level diagram 3. He recombination 1s 2 1 1 S 0 1s2s 2 1 S 0 1s2p 2 1 P 1 1s2s 2 3 S 1 1s2p 2 3 P 0,1,2 1s3s 3 1 S 0 1s3p 3 1 P 1 1s3d 3 1 D 2 1s3s 3 3 S 1 1s3p 3 3 P 0,1,2 1s3d 3 3 D 1,2,3 SINGLETS (S=0) TRIPLETS (S=1) 2  52/s 1 , 584Å 1.8x10 9 /s 1 , 591Å 170/s from 2 3 P 1 only 1 , 626Å 1.3x10 -4 /s

11. Issues in He recombination Mostly similar to H recombination except: Two line escape processes He(2 1 P 1 )  He(1 1 S 0 ) +  584Å He(2 3 P 1 )  He(1 1 S 0 ) +  591Å These are of comparable importance (Dubrovich & Grachev 2005). Feedback: redshifted radiation from blue line absorbed in redder line. Enhancement of escape probability by H opacity: He(2 1 P 1 )  He(1 1 S 0 ) +  584Å H(1s) +  584Å  H + + e - ( Hu et al 1995 ) 3. He recombination

12. He II  I recombination history 3. He recombination thermal equilibrium Seager et al 2000 Switzer & Hirata 2007 Accel. from 2 3 P 1 decay Accel. from H opacity

13. 3. He recombination Current He recombination histories

14. Two-photon decays H(2s)  H(1s) +  +  (8.2 s -1 ) included in all codes. But what about 2  decays from other states? Selection rules: ns,nd only. Negligible under ordinary circumstances: H(3s,3d)  H(2p) +  6563Å, depopulates n  3 levels. In cosmology: so 3s,3d 2  decays might compete with 2s (Dubrovich & Grachev 2005). Obvious solution: compute 2  decay coefficients  3s,3d, add to multilevel atom code. 4. Two-photon decays in H

15. Calculation p. 1 Easy! This is tree-level QED. Feynman rules (in atomic basis set): electron propagator photon propagator vertex (electric dipole) Ignore positrons and electron spin – okay in nonrelativistic limit. 4. Two-photon decays in H

16. Calculation p. 2 Two diagrams for 2  decay: M= + Total decay rate: (  =k for on-shell photon) Problem: infinite because M contains a pole if n 1 =2 … n-1 (n 1 p intermediate state is on-shell). 4. Two-photon decays in H

18. ∞ The resolution to this problem in the cosmological context has provided some controversy. (See Dubrovich & Grachev 2005; Wong & Scott 2007; Hirata & Switzer 2007; Chluba & Sunyaev 2007; Hirata 2008). Lots more this afternoon! Pole displacement: rate still large, e.g. Λ 3d = 6.5×10 7 s -1. This includes sequential 1  decays, 3d  2p  1s. Re-absorption of 2  radiation. No large rates or double-counting in optically thick limit. 4. Two-photon decays in H

19. Radiative transfer calculation A radiative transfer calculation is the only way to solve the problem. Must consistently include:  Stimulated 2  emission (Chluba & Sunyaev 2006)  Absorption of spectral distortion (Kholupenko & Ivanchik 2006)  Decays from n  3 levels.  Raman scattering – similar physics to 2  decay, except one photon in initial state: H(2s) +   H(1s) +   Two-photon recombination/photoionization. 2008 code did not have Lyman-  diffusion (now included – thanks to J. Forbes). 4. Two-photon decays in H

20. Radiative transfer calculation The Boltzmann equation: f = photon phase space density  = number of decays / H nucleus / Hz / second Ill-conditioned at Lyman lines: coefficients   (or large).  But solution is convergent: 4. Two-photon decays in H

21. Numerical approach Key is to discretize the 2γ continuum. Each frequency bin (ν,Δν) is treated as a virtual level. Included in multi- level atom code. In the steady-state limit, the virtual level can be treated just like a real level: –2γ decay –2γ excitation –Feedback But it’s just a mathematical device! 4. Two-photon decays in H 1s nl virtual ν’ ν

22. Physical effects 1 Definitions:  a 2  decay is “sub-Ly  ” if both photons have E E(Ly  ). Sub-Ly  decays:  Accelerate recombination by providing additional path to the ground state.  Delay recombination by absorbing thermal + redshifted Ly  photons.  The acceleration always wins, i.e. reaction: H(nl)  H(1s) +  +  proceeds forward. 4. Two-photon decays in H

24. Physical effects 2 Super-Ly  decays are trickier!  Also provide additional path to ground state.  But for every super-Ly  decay there will later be a Ly  excitation, e.g.: H(3d)  H(1s) +  6563Å H(1s) +  1216Å  H(2p) The net number of decays to the ground state is zero. But there is an effect:  Early, z>1260: accelerated recombination.  Later, z<1260: delayed recombination. Same situation for Raman scattering. 4. Two-photon decays in H

27. Phase space density Full result Without 2  decays Blackbody Ly  Ly  Ly 

28. 4. Two-photon decays in H

29. L Correction: 0.19  ACBAR 0.27  WMAP5 7  Planck

30. Doppler shifts & Lyman-α diffusion So far we’ve neglected thermal motions of atoms. Main effect is in Lyman-α where resonant scattering leads to diffusion in frequency space due to Doppler shift of H atoms. Diffusion coefficient is Rapid diffusion near line center, very slow in wings. 5. Lyman-α diffusion

31. Doppler shifts & Lyman-α diffusion Can construct Fokker-Planck equation from two physical conditions (e.g. Rybicki 2006):  Exactly conserve photons in scattering  Respect the second law of thermodynamics: must preserve blackbody with μ-distortion, f ν ~e -hν/T. hf ν /T term can be physically interpreted as due to recoil (e.g. Krolik 1990; Grachev & Dubrovich 2008). Effect is to push photons to red side of Lyman-α, speeding up recombination. With J. Forbes, diffusion now patched on to 2γ radiative transfer code. 5. Lyman-α diffusion

32. Numerical details Solve Fokker-Planck equation (FPE) in frequency region near Lyα. –Discretize FPE without Hubble term in ν direction to convert to system of stiff ODEs. (Default 2000 bins.) –No photons allowed to diffuse out of bounds. –Hubble redshift implemented by stepping all photons 1 bin to the red at each time step. Required for solution: –f ν @ blue boundary; atomic level populations. –Solve iteratively with atomic level + 2γ code. 5. Lyman-α diffusion

35. L