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?