## How to find scale factor at recombination?

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....

 PhysOrg.com science news on PhysOrg.com >> Front-row seats to climate change>> Attacking MRSA with metals from antibacterial clays>> New formula invented for microscope viewing, substitutes for federally controlled drug
 Recognitions: Gold Member Science Advisor 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.
 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 ?

Recognitions:
Gold Member

## How to find scale factor at recombination?

athen/anow = 2.725/3000.

 @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?
 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

Mentor
 Quote by eherrtelle59 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.

 Mentor Or do you mean just the energy density of radiation, which goes as a^-4 in all eras.
 @George Sorry George, I meant at a^-3 for matter dominated, as recombination occurs after the radiation-dominated era.
 Mentor 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.
 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!)

Recognitions:
Gold Member
Here's a simple explanation offered earlier for George's point 2)
 Quote by marcus ... 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 ...

 Tags cmb, recombination, scale factor