Helium Recombination
Christopher Hirata (IAS)
in collaboration with Eric Switzer (Princeton)
astro-ph/0609XXX
Recombination Physics
1.
2.
3.
4.
Role of recombination in the CMB
Standard recombination history
New physics
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!
  a T ne
ne = electron density
(depends on
recombination)
Recombination history
ne
xe 
nH
He+ + e-  He
z: damping tail
degenerate with ns
He2+ + e-  He+
no effect
H+ + e-  H
z: acoustic peak positions
degenerate with DA
z: polarization amplitude
z
… as computed by RECFAST (Seager, Sasselov, Scott 2000)
The “standard” recombination code.
Standard theory of H recombination
(Peebles 1968, Zel’dovich et al 1968)
H+ + eradiative recombination
+ photoionization
3s
3p
2s
2p
3d
Lyman-
resonance
escape
2
1s
• 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.)
Standard theory of H recombination
(Peebles 1968, Zel’dovich et al 1968)
For H atom in excited level, 3
possible fates:
H+ + eradiative recombination
+ photoionization
3s
3p
2s
2p
3d
Lyman-
resonance
escape
2
1s
• 2 decay to ground state
(2)
• Lyman- resonance escape*
(6ALyPesc)
• photoionization
(  gi e ( E  E ) / kT  i )
i
2
i
* Pesc~1/~8H/3nHIALyLy3.
Standard theory of H recombination
(Peebles 1968, Zel’dovich et al 1968)
H+ + eradiative recombination
+ photoionization
3s
3p
2s
2p
3d
Lyman-
resonance
escape
2
1s
• Effective recombination rate
is recombination coefficient to
excited states times
branching fraction to ground
state:
2  6 ALy Pesc
# rec

 e ne n p
( Ei  E2 ) / kT
V t 2  6 ALy Pesc   g i e
i
e 

nl , n  2
i
nl
Standard theory of H recombination
(Peebles 1968, Zel’dovich et al 1968)
3/ 2

2  6 ALy Pesc
2

m
kT
dxHI

  R / kT 
e

 e  xe x p nH  
xHI 
 e
( Ei  E2 ) / kT
2
dt
2  6 ALy Pesc   g i e
 i 
 h


i
H+ + eradiative recombination
+ photoionization
3s
3p
2s
2p
3d
Lyman-
resonance
escape
2
1s
 = 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
Standard theory of H recombination
(Peebles 1968, Zel’dovich et al 1968)
3/ 2

2  6 ALy Pesc
2

m
kT
dxHI

  R / kT 
e

 e  xe x p nH  
xHI 
 e
( Ei  E2 ) / kT
2
dt
2  6 ALy Pesc   g i e
 i 
 h


i
H+ + eradiative recombination
+ photoionization
3s
3p
2s
2p
3d
Lyman-
resonance
escape
2
1s
 = 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.
Standard theory of He+  He
recombination
( E
E
3/ 2
  3 A1s 2 1s 2 p Pesc e 1s 2 p 1s 2 s


dxHeI
 2me kT 
 I ( He ) / kT

 e  xe xHeII nH  4
xHeI 
 e
 ( E1 s 2 p  E1 s 2 s ) / kT
2
 ( Ei  E1 s 2 s ) / kT
dt
h
  3 A1s 2 1s 2 p Pesc e
  gie
 i 



) / kT
i
He+ + eradiative recombination
+ photoionization
1s3s
1s3p
1s2s
1s2p
1s3d
1s2-1s2p
resonance
escape
2
1s2
• 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.
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-LyH(2s).
Is this all the physics?
5. Resonance escape from semiforbidden
He 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, 1sns and 1s-nd (Dubrovich & Grachev 2005)
Revisiting Recombination
• Project underway at Princeton/IAS to “resolve” 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
xe=0.006
xe=0.001
He I
HI
Plot by E. Switzer
Stimulated 2-photon decays and
absorption of redshifted Lyman- photons
He I
xe = 0.00003
HI
xe = 0.0008
Stimulated 2 decay
Including re-absorption of redshifted resonance photons
Plot by E. Switzer
HI effect on Helium recombination I
• Small amount of neutral hydrogen can speed up
helium recombination:
He(1s 2 p, S  0)  He(1s 2 )   (21.2eV)
H   (21.2eV)  H   e 
• 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).
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
lineτ f crd
 line 
~ 2 THz @ z  2000
2
4
Compare to frequency range over which H I is
optically thick:
 HI 
H HI
nH xHI ionc
(exponenti ally decreasing )
Helium recombination history
(including effects 1-6)
SAHA
EQUILIBRIUM
OLD
NEW
line > HI
line < 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)
d2 (nl  1s )
dE
M 
n'
•
•
•
•
8 6 E 3 E '3
2

(1  N E )(1  N E ' ) M
27 (2l  1)


1
1


1s r n' p n' p r nl 


 Enl ,1s  E Enl ,1s  E ' 
Sum includes continuum levels.
Same equation for He (replace rr1+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.
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).
Largest dipole matrix element from ns or nd state is to np:
2.
nl r n, l  1
3.
2
9
  nl r n' , l  1
10 n '
2
(n  l )
Therefore take only this term in sum over intermediate states
and get:
1

89
n
s
H
(nonres)
(nonres)
Ans1s  5 And 1s  
1
1045
n
s
He

Compare to two-photon rates from 2s: 8s-1 (H) and 51s-1 (He).
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
Tn3, 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:
lineτ f crd
 line 
~ 60 THz (max. for Ly )
2
4
kT
 Ly  Ly  160 THz;
~ 70 THz
h
• 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.
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?