How to find scale factor at recombination?

[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...

 

[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.

 

[eherrtelle59]

Using the redshift equation occurred 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 ?

 

[BillSaltLake]

a_then/a_now = 2.725/3000.

 

[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?

 

[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

 

[George Jones]

1) matter dominated era such that denisty goes a^-4

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

 

[George Jones]

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

 

[eherrtelle59]

@George

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

 

[George Jones]

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

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.

 

[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!)

 

[marcus]

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

...
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
...