1. Helium Recombination Christopher Hirata (IAS) in collaboration with Eric Switzer (Princeton) astro-ph/0609XXX

2. Recombination Physics 1.Role of recombination in the CMB 2.Standard recombination history 3.New physics 4.Preliminary results for helium (hydrogen coming later)

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 WMAP Science Team (2006)

4. Need for CMB Theory This trend will continue in the future with Planck, ACT/SPT, and E/B polarization experiments. But the theory will have to be solved to <<1% accuracy in order to make full use of these data. Theory is straightforward and tractable: linear GR perturbation theory + Boltzmann equation.

5. This is the CMB theory!

6. n e = electron density (depends on recombination)

7. Recombination history z … as computed by RECFAST (Seager, Sasselov, Scott 2000) The “standard” recombination code. 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

8. Standard theory of H recombination (Peebles 1968, Zel’dovich et al 1968) Effective “three level atom”: H ground state, H excited states, and continuum Direct recombination to ground state ineffective. Excited states originally assumed in equilibrium. (Seager et al followed each level individually and found a slightly faster recombination.) 1s 2s2p 3s3p3d H + + e - 22 Lyman-  resonance escape radiative recombination + photoionization

9. Standard theory of H recombination (Peebles 1968, Zel’dovich et al 1968) For H atom in excited level, 3 possible fates: 2  decay to ground state (  2  ) Lyman-  resonance escape* (  6A Ly  P esc ) photoionization (  ) * P esc ~1/  ~8  H/3n HI A Ly  Ly  3. 1s 2s2p 3s3p3d H + + e - 22 Lyman-  resonance escape radiative recombination + photoionization

10. Standard theory of H recombination (Peebles 1968, Zel’dovich et al 1968) Effective recombination rate is recombination coefficient to excited states times branching fraction to ground state: 1s 2s2p 3s3p3d H + + e - 22 Lyman-  resonance escape radiative recombination + photoionization

11. 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

12. 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

13. Standard theory of He +  He recombination Essentially the same equation as H. Only spin singlet He is relevant in standard theory (triplet not connected to ground state). Differences are degeneracy factors, rate coefficients, and 1s2s-1s2p nondegeneracy. Excited states are in equilibrium (even in full level code). This is exactly the equation integrated in RECFAST. 1s 2 1s2s1s2p 1s3s1s3p1s3d He + + e - 22 1s 2 -1s2p resonance escape radiative recombination + photoionization

14. Is this all the physics? 1. Resonance escape from higher-order lines: H Ly , Ly , etc. and He 1s 2 -1snp (Dubrovich & Grachev 2005) 2. Feedback: Ly  photons redshift, become Ly , and re-excite H atoms. 3. Stimulated two-photon transitions (Chluba & Sunyaev 2006) 4. Two-photon absorption of redshifted Ly  photons: H(1s)+  CMB +  red-Ly   H(2s).

15. Is this all the physics? 5. Resonance escape from semiforbidden He 1s 2 (S=0)-1snp(S=1) transition (Dubrovich & Grachev 2005) 6.Effect of absorption of He resonance and continuum photons by hydrogen (increases P esc ) (e.g. Hu et al 1995) 7.Higher-order two-photon transitions, 1s- ns and 1s-nd (Dubrovich & Grachev 2005)

16. Revisiting Recombination Project underway at Princeton/IAS to “re- solve” recombination including all these effects. Preliminary results are presented here for helium. Hydrogen will require more work due to higher optical depth in resonance lines.

17. Effect of Feedback He I H I  x e =0.006  x e =0.001 Plot by E. Switzer

18. Stimulated 2-photon decays and absorption of redshifted Lyman-  photons Stimulated 2  decay Including re-absorption of redshifted resonance photons He I H I  x e = 0.0008  x e = 0.00003 Plot by E. Switzer

19. HI effect on Helium recombination I Small amount of neutral hydrogen can speed up helium recombination: Issue debated during the 1990s (Hu et al 1995, Seager et al 2000) but not definitively settled. Must consider effect of H on photon escape probability. This is a line transfer problem and is not solved by any simple analytic argument. We use Monte Carlo simulation (9 days x 32 CPUs).

20. HI effect on Helium recombination II Must follow 4 effects: -- emission/absorption in He line (complete redistribution) -- coherent scattering in He line (partial redistribution) -- HI continuum emission/absorption -- Hubble redshifting Conceptually, as long as complete redistribution is efficient, He line is optically thick out to Compare to frequency range over which H I is optically thick:

21. Helium recombination history (including effects 1-6) OLD NEW SAHA EQUILIBRIUM  line <  HI  line >  HI Plot by E. Switzer

22. What about 2-photon decays? 2-photon decays from excited states n≥3 have been proposed to speed up recombination (Dubrovich & Grachev 2005) Rate: (in atomic units) Sum includes continuum levels. Same equation for He (replace r  r 1 +r 2 ). Photon energies E+E’=E nl,1s. (Raman scattering if E or E’<0.) The 2-photon decays are simply the coherent superposition of the damping wings of 1-photon processes.

23. 2-photon decays (cont.) How to find contribution to recombination? Argument by Dubrovich & Grachev rests on three points: 1.Photons emitted in a Lyman line (resonance) are likely to be immediately re-absorbed, hence no net production of H(1s). 2.Largest dipole matrix element from ns or nd state is to np: 3.Therefore take only this term in sum over intermediate states and get: Compare to two-photon rates from 2s: 8s -1 (H) and 51s -1 (He).

25. 3 1 S (1 pole) 3 1 D (1 pole)

26. What’s going on? Large negative contribution to 2-photon rate from interference of n’=n and n’≠n terms in summation. Cancellation becomes more exact as n . For large values of n and fixed upper photon energy E, rate scales as n -3, not n. (e.g. Florescu et al 1987) Semiclassical reason is that 2-photon decay occurs when electron is near nucleus. The period of the electron’s orbit is T  n 3, so probability of being near nucleus is  n -3. (Same argument in He.) Bottom line for recombination: n=2,3 dominate 2-photon rate; smaller contribution from successively higher n.

27. Why haven’t we solved hydrogen yet? It’s harder than helium! Larger optical depths: few x 10 8 vs. few x 10 7. Consequently damping wings of Lyman lines in H overlap: The Lyman series of hydrogen contains broad regions of the spectrum with optical depth of order unity. This can only be solved by a radiative transfer code.

28. Summary Recombination must be solved to high accuracy in order to realize full potential of CMB experiments. There are significant new effects in helium recombination, especially H opacity. Extension to H recombination is in progress. Is there a way to be sure we haven’t missed anything?