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Radiative Transfer Effects in Hydrogen Recombination: 2γ Transitions and Lyα Diffusion

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). Outline. Motivation, CMB Standard picture He recombination

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Radiative Transfer Effects in Hydrogen Recombination: 2γ Transitions and Lyα Diffusion

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  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 • Motivation, CMB • Standard picture • He recombination • Two-photon decays in H • Lyman-α diffusion

  3. 1. Motivation, CMB 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)

  4. 1. Motivation, CMB CMB and inflation • Primordial scalar power spectrum: ns, 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)

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

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

  7. 2. Standard picture Recombination history He2+ + e- He+ no effect He+ + e- He z: damping tail degenerate with ns H+ + e- H z: acoustic peak positions degenerate with DA z: polarization amplitude z … as computed by RECFAST1.3 (Seager, Sasselov, Scott 2000) The “standard” recombination code until Feb. 2008.

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

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

  10. 3. He recombination Helium level diagram TRIPLETS (S=1) SINGLETS (S=0) 1s3p 31P1 1s3d 31D2 1s3d 33D1,2,3 1s3p 33P0,1,2 1s3s 31S0 1s3s 33S1 1s2p 21P1 1s2p 23P0,1,2 1s2s 21S0 1, 584Å 1.8x109/s 1s2s 23S1 1, 591Å 170/s from 23P1 only 2 52/s 1, 626Å 1.3x10-4/s 1s2 11S0

  11. 3. He recombination Issues in He recombination • Mostly similar to H recombination except: • Two line escape processesHe(21P1)  He(11S0) + 584ÅHe(23P1)  He(11S0) + 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(21P1)  He(11S0) + 584ÅH(1s) + 584Å  H+ + e-(Hu et al 1995)

  12. 3. He recombination He III recombination history Accel. fromH opacity Accel. from23P1 decay Seager et al 2000 Switzer & Hirata 2007 thermal equilibrium

  13. 3. He recombination Current He recombination histories

  14. 4. Two-photon decays in H 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.

  15. 4. Two-photon decays in H 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.

  16. 4. Two-photon decays in H Calculation p. 2 • Two diagrams for 2 decay:M= + • Total decay rate: (=k for on-shell photon) • Problem: infinite because Mcontains a pole ifn1=2 … n-1 (n1p intermediate state is on-shell).

  17. 4. Two-photon decays in H

  18. 4. Two-photon decays in H ∞ • 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. Λ3d = 6.5×107 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.

  19. 4. Two-photon decays in H 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).

  20. 4. Two-photon decays in H 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:

  21. 4. Two-photon decays in H 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, thevirtual level can be treatedjust like a real level: • 2γ decay • 2γ excitation • Feedback • But it’s just a mathematicaldevice! nl ν’ virtual ν 1s

  22. 4. Two-photon decays in H Physical effects 1 • Definitions: a 2 decay is “sub-Ly” if both photons have E<E(Ly). a 2 decay is “super-Ly” if one photon has 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.

  23. 4. Two-photon decays in H

  24. 4. Two-photon decays in H 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) + <1216Å + >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.

  25. 4. Two-photon decays in H

  26. 4. Two-photon decays in H

  27. 4. Two-photon decays in H Phase space density Ly Ly Ly Full resultWithout 2 decaysBlackbody

  28. 4. Two-photon decays in H

  29. 4. Two-photon decays in H Correction:0.19 ACBAR 0.27 WMAP5 7 Planck L

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

  31. 5. Lyman-α diffusion 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, fν~e-hν/T. • hfν/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.

  32. 5. Lyman-α diffusion 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: • fν @ blue boundary; atomic level populations. • Solve iteratively with atomic level + 2γ code.

  33. 5. Lyman-α diffusion

  34. 5. Lyman-α diffusion

  35. 5. Lyman-α diffusion L

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