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How to find scale factor at recombination?

  1. Apr 30, 2012 #1
    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. jcsd
  3. Apr 30, 2012 #2

    marcus

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    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
  4. Apr 30, 2012 #3
    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 ?
     
  5. Apr 30, 2012 #4

    BillSaltLake

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    athen/anow = 2.725/3000.
     
  6. Apr 30, 2012 #5
    @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?
     
  7. Apr 30, 2012 #6
    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
     
  8. Apr 30, 2012 #7

    George Jones

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    In the matter-dominated era, density goes as a^-3; in the radiation-dominated era, density goes as a^-4.
     
  9. Apr 30, 2012 #8

    George Jones

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    Or do you mean just the energy density of radiation, which goes as a^-4 in all eras.
     
  10. Apr 30, 2012 #9
    @George

    Sorry George, I meant at a^-3 for matter dominated, as recombination occurs after the radiation-dominated era.
     
  11. Apr 30, 2012 #10

    George Jones

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    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
  12. Apr 30, 2012 #11
    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!)
     
  13. May 1, 2012 #12

    marcus

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