Calculate the time of the surface of last scattering

[g.lemaitre]

Homework Statement

By extrapolating the Fricdmann equations back in time, they discovered that the energy density of this radiation field must have initially considerably exceeded that of matter (as defined by E = mc^2). Radiation density is the amount of energy in a given volume of space, and it can be expressed as the temperature of the black body emitting the same energy. The observed 75 25 per cent hydrogen/helium ratio* implied that the temperature of the radiation must have been 10^9 K. Electromagnetic radiation readily interacts with electrically charged protons and electrons, so while the fireball was dominated by the radiation field the mean distance traveled by a photon was minuscule. As the Universe expanded, the radiation field would have cooled*. After some 300,000 years, its temperature would have fallen to 10^3 K, cool enough for electrons to associate with protons to produce neutral hydrogen atoms.

Homework Equations

The Attempt at a Solution

I would like to do this calculation. I don't even know where to begin since there are quite a lot of Friedmann equations out there. Which Friedman equation do I use and how do I use it?

 

[g.lemaitre]

I should also mention that I'm trying to figure out how many years elapsed from the BB to the surface of last scattering. I also want to figure out the temperature at certain times.

 

[clamtrox]

This is the standard way: let's denote time today as [itex] t_0 [/itex] and time of last scattering as [itex] t_* [/itex], (and normalize so that [itex]a_0 = 1 [/itex])

[tex] t_0 - t_* = \int_{t_*}^{t_0} dt = \int_{a_*}^{a_0} da \frac{dt}{da} = \int_{a_*}^{1} \frac{da}{a H} [/tex]

Then plug in Hubble parameter from Friedmann equations and solve the integral. So for example, if you had a flat dust universe, then
[tex] H = H_0 a^{-3/2} [/tex]
and
[tex] t_0 - t_* = H_0^{-1} \int_{a_*}^{1} da \sqrt{a} = \frac{2}{3 H_0} (1 - a_*^{3/2}) [/tex]

 

[g.lemaitre]

I don't know how to normalize. I'm assuming you learn that in QM. Is it easy or should I just wait until I finish QM or is there a youtube video that will show me how to do it easily.

 

[clamtrox]

I don't know how to normalize. I'm assuming you learn that in QM. Is it easy or should I just wait until I finish QM or is there a youtube video that will show me how to do it easily.

You do it by saying that [itex] a_0 = 1 [/itex]. It is not mathematically very demanding :-) Basically, you are free to choose the value of scale factor on a single moment of time (this corresponds to a choice of units basically) and making the choice above is very reasonable, as it simplifies most formulas.

 

[g.lemaitre]

Is all I have to do is plug the numbers into here and solve?

[tex] = \frac{2}{3 H_0} (1 - a_*^{3/2}) [/tex][/QUOTE]

I don't think the value for [tex]a_*[/tex] is known. For [tex]H_0 [/tex] I'm just going to use 22.2 (km/s)Mly^-1 which I think is the same as 71 (km/s)Mpc^-1.

Try to use less jargon with me when you're answering my question. My knowledge of math is a bit shaky.

 

[clamtrox]

I don't think the value for [tex]a_*[/tex] is known.

What makes you think that? What do you know then?

 

[g.lemaitre]

Well, in every equation you've given me there is at least one unknown, so I can't get the answer

 

[clamtrox]

Well, in every equation you've given me there is at least one unknown, so I can't get the answer

Of course there is one unknown: this is the thing that determines when last scattering happened. In the original post you gave the temperature for it. I gave a formula which has the scale factor. They are related via [itex] \frac{T_1}{T_2} = \frac{a_2}{a_1} [/itex]