Seeking reference for math related to the age of Recombination

[Buzz Bloom]

TL;DR Summary

Wikipedia's article about Recombination says it occurred at about time T =370,000 years after the Big Bang. I have tried (and failed) to search for the math that calculates Recombination as happening at this time T.

The Wikipedia references is

https://en.wikipedia.org/wiki/Recombination_(cosmology).​

It says:

Recombination occurred about 370,000 years after the Big Bang (at a redshift of z = 1100),​

and

the cosmic background radiation is infrared [and some red] black-body radiation emitted when the universe was at a temperature of some 3000 K, redshifted by a factor of 1100 from the visible spectrum to the microwave spectrum).​

I get that scale factor a(t) =1/(z+1) corresponds to the fact that the (about) 3000 K production of photons at time T is perceived now as 2.7260±0.0013 K photons. (See reference

https://en.wikipedia.org/wiki/Cosmic_microwave_background​

paragraph 4 under the heading"Features".)​

What seems to be missing is how the value of T is calculated. I get that the value of H(a) can be calculated for the time T. If H(a) is known, then it would then be possible to calculate the distance at time T between (1) the source of the CMB produced at time T (and observed at time now) at (2) the place in the universe at T which is now Earth .

I hope some reader will be able to post a source for the math producing the value T.

 

[George Jones]

I hope some reader will be able to post a source for the math producing the value T.

A simplified version of the calculation is given in section 2.3.3 of David Tong's lecture notes on cosmology.

http://www.damtp.cam.ac.uk/user/tong/cosmo.html

This simplified calculation is still fairly involved, but the real calculation is more complicated; see Mea Culpa at the end of this section.

 

[Buzz Bloom]

ADDED: I FOUND MY MISTAKE! I fixed it below.

I very much want to thank @George Jones for his post to this thread. I have been trying for a week to grasp the process leading to the conclusion regarding the age at the universe at the time of Recombination, but I keep coming up with a wrong answer. I am hopeful a reader will be able to find my error and explain it to me.

It may be useful to start with references.
R1:
https://originoftheuniverse.fandom.com/wiki/Cosmic_microwave_background_radiation

R2:
https://www.damtp.cam.ac.uk/user/tong/cosmo/two.pdf

R3:
https://en.wikipedia.org/wiki/Friedmann_equations#Detailed_derivation

R1 says:
the current temperature of the CMB is
T_now = 2.725K.

R2 says on page 101 what is below. If you want to find this page, I suggest you search for the text saying “101”.
The temperature at the time of Recombination is
T_rec ~= 3600 K.
The time at recombination is given as
t_rec = 300,000 years.

The value for z_rec is given as 1300. My guess is that this is because of the lack of precision to the value of the temperature T_rec. I choose to use a bit more precision.
z_rec = T_rec/T_now = 1308

I calculate a_rec as follows:
a_rec = 1/(1+z_rec) = 1/1309 = 0.00076394 .

It will be also needed below to calculate the value:
a_rec^(3/2) = 0.000021115 .

R2 also calculates the time t_rec based on simplification of the R3 Friedmann equation, based on
Omega_m/a^3 >>(Omega_r/a^4 + Omega_k/a^2 + Omega_Lambda).
However, it is not clear in the text exactly how this is done. I assume that R2 uses the R3 equation which is then simplified based on the above, assuming that Omega_m =0.3 remains in the process.
H(a) = (da/dt)/a = H_0 SQRT(Omega_m/a^3) = H_0 SQRT(0.3/a^3)
Therefore:
dt = (da/a) / (H_0 SQRT(a^3/0.3)) = [a^(1/2) x (1/0.3)^(1/2) x (1/H_0)] da
= [(1/0.3)^(1/2) x (1/H_0)] x a^(1/2) da
= 1.8257 x 14.4 Gyr x a^(1/2) da

Integrating I get:
t_rec = (2/3) x 1.8257 x 14.4 Gyr x a_rec^(3/2)
= 17.53 Gyr x a_rec^(3/2)
= 0.000021115 x 17.53 Gyr
= 21,115 x 17.53 yr
= 370,000 yr

I will see what a numerical integration using the full combinationof Omega_m and Omega_r.

 

[Buzz Bloom]

I decided to take a different approach to this topic in another thread. It will take me some time to figure out how to organize it. I have found some improved reliable values for the necessary parameters. The following are integration results using these "new" parameter values and using both integration methods., one with just Omega_m and the other also with Omega_r. These results obviously have more numerical digits than are appropriate for the actual precision of these values.

With Omega_m only:
t_rec = 358,590 years.

With both Omegas:
t_rec = 273,341 years.

The 300,000 years presented in

https://www.damtp.cam.ac.uk/user/tong/cosmo/two.pdf pg 101​

is obviously a very very rough approximate value.

By the way, if you want to find page 101, I suggest searching for the text "101".