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

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

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)

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.

This is the CMB theory!

n e = electron density (depends on recombination)

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

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

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

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

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

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

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

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).

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)

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.

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

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 =  x e = Plot by E. Switzer

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).

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:

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

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.

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).

3 1 S (1 pole) 3 1 D (1 pole)

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.

Why haven’t we solved hydrogen yet? It’s harder than helium! Larger optical depths: few x 10 8 vs. few x 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.

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?