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!

This is the CMB theory!

eTna

ne = electron density(depends on

recombination)

Recombination history

z

H

ee n

nx

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

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

z: polarization amplitude

He+ + e- Hez: damping taildegenerate with ns

He2+ + 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

2s 2p

3s 3p 3d

H+ + e-

2 Lyman-resonanceescape

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* (6ALyPesc)

• photoionization( )

* Pesc~1/~8H/3nHIALyLy3.1s

2s 2p

3s 3p 3d

H+ + e-

2 Lyman-resonanceescape

radiative recombination+ photoionization

ii

kTEEi

ieg /)( 2

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

2s 2p

3s 3p 3d

H+ + e-

2 Lyman-resonanceescape

radiative recombination+ photoionization

2,

/)( 262

62

t V

rec#

nnlnle

pee

ii

kTEEiescLy

escLy nnegPA

PAi

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

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

Standard theory of He+ He recombination

HeIkTHeIe

HHeIIee

ii

kTEEi

kTEEescpss

kTEEescpssHeI xe

h

kTmnxx

egePA

ePA

dt

dxssissps

ssps

/)(2/3

2/)(/)(

211

/)(

211 24

3

3

2121212

21212

• 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.1s2

1s2s 1s2p

1s3s 1s3p 1s3d

He+ + e-

2 1s2-1s2presonanceescape

radiative recombination+ photoionization

Is this all the physics?

1. Resonance escape from higher-order lines: H Ly, Ly, etc. and He 1s2-1snp (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-LyH(2s).

Is this all the physics?

5. Resonance escape from semiforbiddenHe 1s2(S=0)-1snp(S=1) transition (Dubrovich & Grachev 2005)

6. Effect of absorption of He resonance and continuum photons by hydrogen (increases Pesc) (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

xe=0.006

xe=0.001

Plot by E. Switzer

Stimulated 2-photon decays and absorption of redshifted Lyman- photons

Stimulated 2 decayIncluding re-absorption of redshifted resonance photons

He I

H I

xe = 0.0008

xe = 0.00003

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).

e

sSps

H)eV2.21(H

)eV2.21()1(He)0,21(He 2

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:

2000 @ THz 2~4

2

crdlineline

z

fτ

)decreasingally (exponenti ionHIH

HIHI cxn

H

Helium recombination history(including effects 1-6)

OLD

NEW

SAHAEQUILIBRIUM

line < HIline > 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 rr1+r2).

• Photon energies E+E’=Enl,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.

'

11''1

)1)(1()12(27

'8)1(

1,1,'

2

'

3362

EEEEnlrpnpnrs

l

EE

dE

snld

snlsnln

EE

M

MNN

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).

)(1,'10

91,

'

22lnlnrnllnrnl

n

Hes 1045

Hs 895

1

1(nonres)

1(nonres)

1n

nAA sndsns

31S (1 pole)

31D (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 Tn3, 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 108 vs. few x 107.• 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.

THz 70~ THz; 160

)Lyfor (max. THz 60~4

LyLy

2crdline

line

h

kT

fτ

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?