helium recombination christopher hirata (ias) in collaboration with eric switzer (princeton)...
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Helium Recombination
Christopher Hirata (IAS)
in collaboration with Eric Switzer (Princeton)
astro-ph/0609XXX
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Recombination Physics
1. Role of recombination in the CMB
2. Standard recombination history
3. New physics
4. Preliminary results for helium(hydrogen coming later)
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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)
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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.
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This is the CMB theory!
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This is the CMB theory!
eTna
ne = electron density(depends on
recombination)
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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
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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
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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
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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
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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
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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
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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
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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).
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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)
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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.
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Effect of Feedback
He I
H I
xe=0.006
xe=0.001
Plot by E. Switzer
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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
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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
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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
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Helium recombination history(including effects 1-6)
OLD
NEW
SAHAEQUILIBRIUM
line < HIline > HI
Plot by E. Switzer
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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
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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
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31S (1 pole)
31D (1 pole)
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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.
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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
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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?