I Seeking reference for math related to the age of Recombination

AI Thread Summary
Recombination occurred approximately 370,000 years after the Big Bang, with a corresponding redshift of z = 1100, resulting in cosmic background radiation that is now observed at a temperature of about 2.726 K. The calculation of the temperature at the time of recombination (T_rec) is estimated to be around 3600 K, leading to a redshift value of z_rec = 1308. The discussion highlights the complexity of calculating the age of the universe at recombination, referencing various sources for the necessary equations and methodologies. Numerical integration methods were explored, yielding results for t_rec that vary based on the parameters used, with one estimate being 358,590 years when considering only matter density. The thread emphasizes the need for clarity and precision in these calculations, inviting further input from knowledgeable readers.
Buzz Bloom
Gold Member
Messages
2,517
Reaction score
465
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
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
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.
 
Space news on Phys.org
Buzz Bloom said:
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.
 
  • Like
Likes Buzz Bloom and PeterDonis
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.
 
Last edited:
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
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".
 
Last edited:
Abstract The Event Horizon Telescope (EHT) has significantly advanced our ability to study black holes, achieving unprecedented spatial resolution and revealing horizon-scale structures. Notably, these observations feature a distinctive dark shadow—primarily arising from faint jet emissions—surrounded by a bright photon ring. Anticipated upgrades of the EHT promise substantial improvements in dynamic range, enabling deeper exploration of low-background regions, particularly the inner shadow...
https://en.wikipedia.org/wiki/Recombination_(cosmology) Was a matter density right after the decoupling low enough to consider the vacuum as the actual vacuum, and not the medium through which the light propagates with the speed lower than ##({\epsilon_0\mu_0})^{-1/2}##? I'm asking this in context of the calculation of the observable universe radius, where the time integral of the inverse of the scale factor is multiplied by the constant speed of light ##c##.
Back
Top