1. Hans Kristian Eriksen February 16th, 2010
AST5220 Recombination
Hans Kristian Eriksen
February 16th, 2010

2. Overview Some important points:
Photons scatter on free electrons
In the early universe, the temperature was very hot
Too hot for neutral hydrogen to exist; electrons and protons were ripped apart ultraviolet photons
Consequently, all electrons in the early universe were free
And therefore the universe was opaque
It was impossible to see more than ~1 meter ahead, because light was scattered from a different direction
Just like walking in a fog
But today the universe is transparent
Something must have happened at some point
Question: How did the universe become transparent?

3. Cosmic timeline

4. Zoom-in on recomination epoch

5. Why is this epoch important to us?
Recall our main goal:
We want to predict the CMB power spectrum given cosmological parameters
The CMB photons we observe today are those that decoupled during recombination
The fluctuations we see are mainly those that existed at the time of recombination
We therefore need to know
when recombination happened
how rapidly it happened

6. The optical depth I ( x ) = e ¡ ¿
To describe recombination quantitatively we introduce the concept of optical depth, τ
Imagine you are looking at a light source through a medium that absorbes light
The full intensity of the light source is called I0
The intensity you observe at position x is then where τ is called the optical depth
The further away you are, the higher τ is
The critical position is where τ ~ 1
If τ >> 1, you don’t see anything
If τ << 1, the medium doesn’t do anything anyway I ( x ) = e

7. The cosmological optical depth
In cosmology, the main source of ”absorption” is free electrons through Thompson scattering
The optical depth of Thompson scattering is given by
where
η is conformal time
ne = electron density
T = Thompson cross section (”probability of single scattering”?)
Our main job: Compute τ(η) ( ) = Z n e T a d

8. How do we compute τ? ¿ ( ´ ) = Z n ¾ a d ¾ = 8 ¼ 3 µ ® ~ m c ¶ 6 : 5 ¢n e T a d
The Thompson cross section is given by particle physics:
a is the scale factor, and a one-to-one function of the conformal time
The difficult one is the electron density, ne... T = 8 3 ~ m e c 2 6 : 5 1

9. Finding the electron density (1)
Assume that there are no helium or heavier elements in the universe, only electrons and protons
That is, there are four constituents in the universe:
Protons, p
Electrons, e
Neutral hydrogen, H
Photons, γ
These interact with each other through a two-particle process
We need to find ne, np, nH and nγ p + e $ H

10. Finding the electron density (2)
And from the last lecture, we know how to do this!
Recall first the Boltzmann equation for a two-particle process in an expanding universe:
After defining
this could (for thermal equlibrium) be written as a 3 d ( n e ) t = Z p 1 2 ~ E 4 + j M [ f ] n i = e c 2 k T Z d 3 p ( ) E n ( ) i = Z d 3 p 2 e E k T a 3 d ( n 1 ) t = 2 h v i 4

11. Finding the electron density (3)
Recall also that if the reaction rate is much larger than the Hubble expansion rate, then this implies
For our proton-electron-hydrogen-photon plasma, this therefore reads
This is the Saha equation on a symbolic form, and now we have to write this out in full. To the blackboard...  n 3 4 ( ) = 1 2 n e p ( ) = H

12. The Saha equation X 1 ¡ ¼ n µ m T ~ ¶ So, the Saha equation reads
This is a good approximation as long as the system is in strong thermodynamic equilibrium
But when the temperature and density fall, the system eventually goes out of thermodynamic equilibrium X 2 e 1 n b m T ~ 3 =

13. The general equation a d ( n ) t = h ¾ v i
When that happens, one has to go back to the full Boltzmann equation
If the only involved particles and states were photons, electrons and hydrogen, this would (as usual) look like this:
However, in real life this quickly becomes messy because of the atomic physics involved:
First, e + p → Hs1 + γ doesn’t count
the released photon will immediately ionize another nearby hydrogen atom, unless the distance between atoms is very large
The most important contribution is e + p → Hs2 + γ
This leads to the Peebles’ equation
For high precision, helium, litium and excited states must be included
This is what Recfast does, and it is messy a 3 d ( n e ) t = p h v i H

14. The Peebles’ equation a d ( n ) t = X a d ( n ) t = h ¾ v i
Let ne = nb Xe. Then the left-hand side of the Boltzmann equation becomes
The right-hand side becomes a 3 d ( n e ) t = p h v i H a 3 d ( n e ) t = X b n ( ) e p h v i H = b 1 X m T 2 ~ 3 k

15. The Peebles’ equation d X t = h ¾ v i ( 1 ¡ ) µ m T 2 ¼ ~ ¶ n
Cancelling terms, the equation then reads
With the correct constants, this leads to the Peebles’ equation
You will solve this numerically in the second milestone of the project
The full set of constants are given in the project description d X e t = h v i ( 1 ) m T 2 ~ 3 k n b

16. Numerical solution
Figure 1 of Callin (2005)

17. Optical depth and the visibility function
We now have everything we need to compute the optical depth since we finally know ne = Xe nb
However, we will also need the so-called ”visibility function”
This is simply the probability for a given observed photon to have scattered at conformal time η ( ) = Z n e T a d g ( ) = d e

18. Optical depth and the visibility function
Solid line =
Visibility function
z ~ 1100

19. Summary
We need to know the recombination history of the universe in order to predict the CMB spectrum
Specifically, we need the optical depth, τ, and the visibility function, g
To find these, we must solve the Boltzmann equation for the electron number density in the early universe
At early times, use the Saha approximation
At late times, use the Peebles’ equation
In order to do this job properly, one should really take into account heavier elements (helium, litium etc.) and many excitation states of each atom
Not trivial, and fortunately somebody else has spent a lot of time on this, and written Recfast
We will be happy with the simpler approximations above in this course
Recombination (and reionization) is messy... 