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

Christopher HirataOrsay – July 2009

With special thanks to:E. Switzer (Chicago)

D. Grin, J. Forbes, Y. Ali-Haïmoud (Caltech)

Outline

1. Motivation, CMB

2. Standard picture

3. He recombination

4. Two-photon decays in H

5. Lyman-α diffusion

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

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)

22 22416261 ssn

...ln2

1ln)1(const)(ln

2

**

3

k

k

k

knkPk ss

1. Motivation, CMB

This is the CMB theory!1. Motivation, CMB

This is the CMB theory!

eT na

ne = electron density(depends on

recombination)

1. Motivation, CMB

Recombination history

z

H

ee n

nx

… as computed by RECFAST1.3 (Seager, Sasselov, Scott 2000)The “standard” recombination code until Feb. 2008.

H+ + e- Hz: acoustic peak positionsdegenerate with DA

z: polarization amplitude

He+ + e- Hez: damping taildegenerate with ns

He2+ + e- He+

no effect

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 iR = Rydberg

1s

2s 2p

3s 3p 3d

H+ + e-

2 Lyman-resonanceescape

radiative recombination+ photoionization

HI

kTReHpee

ii

kTEEiescLy

escLyHI xeh

kTmnxx

egPA

PA

dt

dxi

/2/3

2/)(

2

62

622

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.

1s

2s 2p

3s 3p 3d

H+ + e-

2 Lyman-resonanceescape

radiative recombination+ photoionization

HI

kTReHpee

ii

kTEEiescLy

escLyHI xeh

kTmnxx

egPA

PA

dt

dxi

/2/3

2/)(

2

62

622

2. Standard picture

Helium level diagram

3. He recombination

1s2

11S0

1s2s21S0

1s2p21P1

1s2s23S1

1s2p23P0,1,2

1s3s31S0

1s3p31P1

1s3d31D2

1s3s33S1

1s3p33P0,1,2

1s3d33D1,2,3

SINGLETS (S=0) TRIPLETS (S=1)

252/s

1, 584Å1.8x109/s 1, 591Å

170/s from 23P1 only1, 626Å1.3x10-4/s

Issues in He recombination

• Mostly similar to H recombination except:• Two line escape processes

He(21P1) He(11S0) + 584Å

He(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) + 584Å

H(1s) + 584Å H+ + e-

(Hu et al 1995)

3. He recombination

He III recombination history3. He recombination

thermal equilibrium

Seager et al2000

Switzer & Hirata2007

Accel. from23P1 decay

Accel. fromH opacity

3. He recombination

Current He recombination histories

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 n3 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

01.05 /

2

3 3,2 CMBkTE

s

d en

n

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.

€

1

E − Enlm + iε

4. Two-photon decays in H

ikE 1

nlm

kJM

1,2/3'

'

1''')1( J

m

mMm

llnlrlnek

Calculation p. 2

• Two diagrams for 2 decay:

M= +

• Total decay rate: (=k for on-shell photon)

• Problem: infinite because M contains a pole ifn1=2 … n-1 (n1p intermediate state is on-shell).

100

4. Two-photon decays in H

nlm 111mn

M1 '1' M

100nlm 111mn

M1 '1' M

)1('

||)1(

0

221

snlE

dsnlE

nl

M

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, 3d2p1s.

• Re-absorption of 2 radiation.

No large rates or double-counting in optically thick limit.

4. Two-photon decays in H

€

H(3d) ↔ H(1s) + γ>Lyα + γ<Hα

H(2p) ↔ H(1s) + γ Lyα

H(3d) ↔ H(1s) + γ<Lyα + γ>Hα

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 n3 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

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:

rec2γRaman2γ2

3Hν

ν πν8νν

nlnl

nlnl

cnfHf

4. Two-photon decays in H

snlnl

nl xfflxffd

d1'νν'νν

2γ )12()1)(1(ν

s

p

nnlspnlpnl

nnlnlpnlpnl

sppnlnl

x

x

xfAl

xfA

fAA

d

d

1

2

3,1)2,(ν2,

3,)2,(ν2,

ν(Lyα)2Lyα

2

1,22,

3)12(

)1(

;)νν(4ν

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!

4. Two-photon decays in H

1s

nl

virtual

€

Anl,b =dΛnl

dνΔν

€

Ab,1sPesc =Π b

1− Π b

gnl

gb

fν ' Anl ,b

nl

∑

Π b =1− e−Δτ b

Δτ b

Δτ b =c 3nHx1s

8πHν 3

gnl

g1s

fν ' Anl,b

nl

∑

ν’

ν

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.

4. Two-photon decays in H

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.

4. Two-photon decays in H

4. Two-photon decays in H

4. Two-photon decays in H

4. Two-photon decays in H

Phase space density

Full resultWithout 2 decaysBlackbody

Ly

Ly

Ly

4. Two-photon decays in H

4. Two-photon decays in H

L

Correction:0.19 ACBAR0.27 WMAP57 Planck

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

€

H(1s) + γ Lyα → H(2p)virtual → H(1s) + γ Lyα

€

D(ν ) = Hν Lyα ⋅τ Lyα f scatφ(ν ) ⋅Tν Lyα

2

mH

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.

5. Lyman-α diffusion

€

˙ f ν diff=

1

ν 2

∂

∂νν 2D(ν )

∂fν

∂ν+

hfν

T

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

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.

5. Lyman-α diffusion

5. Lyman-α diffusion

5. Lyman-α diffusion

L

5. Lyman-α diffusion