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I Comoving distance and redshift relationship derivation

  1. Jun 21, 2017 #1
    Hello PhysicsForum,

    There is something I don't get at the end of this course notes PDF file. In the last section, titled "Comoving distance and redshift", which I have copied below, we have a short derivation of the comoving distance and redshift relation.

    Almost all is well, the only thing that troubles me is : why is there no minus sign after da has been replaced by -a2dz ?

    notes3_dvi.png

    I have searched the web and found almost identical derivations in other courses or publications, but I never read the explanation for why the minus sign drops. I have found what seems to be the source material for most of those derivations : this paper from 93 (see section 6.3, "The General Redshift-Distance Relation" on 3rd page). It is referenced quite often by others when this comoving distance and redshift relationship shows up.

    Maybe I am just missing some mathematical trick ? This is not super important of course, but it bugs me.
     
  2. jcsd
  3. Jun 21, 2017 #2

    Bandersnatch

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    Could it be because the integration limits have been swapped? (consider what it means when the limits are ##a_e -> a_0##, where e stands for emission, and ##0 -> z##.)
     
  4. Jun 21, 2017 #3

    kimbyd

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    Yes. The change of variables leads to two minus signs which cancel one another: ##da = -a^2 dz##, and reversing the limits of integration.
     
  5. Jun 21, 2017 #4
    Thank you !

    Thay's what I thought, but then it means that the ae limit corresponds to z when you do the change of variable, and a0 to 0 redshift ? Sounds weird, shouldn't it be the other way around ? Since a0 = a(t0) is the expansion when we receive the redshifted signal.
     
  6. Jun 21, 2017 #5

    kimbyd

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    This is probably easiest to see if you look at the equation for the scale factor in terms of the redshift:
    [tex]a = {1 \over 1+z}[/tex]

    Here note that for ##z=0##, ##a=1##. That's the current scale factor and redshift. A far-away object, at, say, a redshift of ##z=2## is at a scale factor of ##a=1/3##. The integral above over ##da## would integrate from ##1/3## to 1, while the integral over ##dz## integrates from 0 to 2.
     
  7. Jun 21, 2017 #6

    Bandersnatch

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    Think of how far you need to look. Higher z is seen farther than lower z, while lower a is seen farther than high a.
     
  8. Jun 22, 2017 #7
    Makes sense now. Thank you !
     
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