How to find scale factor at recombination?

1. Apr 30, 2012

eherrtelle59

If we know that the temperature of photons was apprx. 3000 K at recombination and the temperature of the CMB is apprx. 2.725 K today, how can we extrapolate the value of the scale factor at recombination?

I know that recombination happens at a matter-dominated era, such that the density goes a^-3 ****(not a^-4) and that at this era a(t) goes t^2/3

But this doesn't tell me how a(t) is related to temperature....

Last edited: Apr 30, 2012
2. Apr 30, 2012

marcus

1+z is usually given as about 1090, at recombination.
So that would make a(recomb.) = 1/1090

The temperature of the ancient light goes as 1/a

You can think of it as the number of photons per unit volume going as 1/a3
and the energy of an individual photon going as 1/a (as its wavelength lengthens).
So the energy density of the ancient light goes as 1/a4
And you know the fourth power law of temperature.

Last edited: Apr 30, 2012
3. Apr 30, 2012

eherrtelle59

Using the redshift equation occured to me, but according to the problem I should be using T=3000 and T=2.725

So, you're saying T goes 1/a. This is for relativistic fluids at "ancient times"?

I'm not quite sure I understand your explanation for the derivation of this....

In any sense, if I use T=3000, then a is around 3*10^-3 ?

4. Apr 30, 2012

BillSaltLake

athen/anow = 2.725/3000.

5. Apr 30, 2012

eherrtelle59

@BillSaltLake

I'm afraid there's something I'm missing conceptually. Why can we conclude that we can put temperature values into the redshift equation as you have done?

6. Apr 30, 2012

eherrtelle59

I should also point out I'm confused as to how this relates to the matter-dominated era.

To be clear, my estimate for the scale factor at recombination should take two things into account
1) matter dominated era such that denisty goes a^-4 and
2) T of CMB is about 2.725

7. Apr 30, 2012

George Jones

Staff Emeritus
In the matter-dominated era, density goes as a^-3; in the radiation-dominated era, density goes as a^-4.

8. Apr 30, 2012

George Jones

Staff Emeritus
Or do you mean just the energy density of radiation, which goes as a^-4 in all eras.

9. Apr 30, 2012

eherrtelle59

@George

Sorry George, I meant at a^-3 for matter dominated, as recombination occurs after the radiation-dominated era.

10. Apr 30, 2012

George Jones

Staff Emeritus
1)The energy density of a blackbody photon gas is proportional to T^4
http://en.wikipedia.org/wiki/Photon_gas; [Broken]

2) the energy density of radiation is proportional to a^-4, even in the matter-dominated era;

3) therefore, T is proportional to 1/a.

Last edited by a moderator: May 5, 2017
11. Apr 30, 2012

eherrtelle59

Aha!

2) was the conceptual issue. In any era, the energy density of relativistic matter goes a^-4.

Thanks George! (and the others who answered!)

12. May 1, 2012

marcus

Here's a simple explanation offered earlier for George's point 2)