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Cosmological Redshift

  1. Feb 13, 2019 #1
    1. The problem statement, all variables and given/known data

    Consider a point in the intergalactic medium at some cosmic time ## t_{obs}##, the time of arrival of a photon of wavelength ##λ_{obs}## as seen by a hydrogen atom at that location. The source of this photon a comoving distance ##r## away emitted it at wavelength ##λ_{em}## at time ##t_{em}##. Assume the scale factor at present ##a_0= 1##.

    (a) Express ##r## as a function of ##t_{obs}##, ##t_{em}## and the scale factor ##a(t)##.

    (b) Solve for the dependence of ##a(t)## on ##t## for a universe in which the Hubble constant varies with time according to
    [tex]\ H(t)=\frac{2}{3t}. [/tex]

    (c) What is the ratio of ##λ_{em}## to ##λ_{obs}## in terms of ##t_{em}## and ##t_{obs}##, in a universe described in (b)?

    (d) The same photon will later reach a telescope on Earth today at ##λ_0=3645 Angstroms##. Suppose ##λ_{obs}=1215 Angstroms##, the H atom Lyman-alpha line transition. What is ##λ_{em}## if the source is located a comoving distance ##r=500 Mpc## (in present-day units) away from the H atom? Assume ##H_0=70 km s^{-1} Mpc^{-1}##.

    I mainly just want help with part (d).


    2. Relevant equations

    ## ds^2=-c^2dt^2+a(t)^2[dr^2+s_k(r)^2d\Omega^2## but ##ds=0## and ##d\Omega=0##
    ##H(t)=\frac{1}{a}*\frac{da}{dt}##
    ##\frac{\lambda_{em}}{a(t_{em})}=\frac{\lambda_{obs}}{a(t_{obs})}##


    3. The attempt at a solution

    For part (a) I intergrated and got ## r=c\int\limits_{t_{em}}^{t_{obs}} \frac{1}{a(t)} \ dt ##.
    For part (b) I used the second equation to get ##a(t)=t^{2/3}##
    For part (c) I used the third equation to get ##\frac{\lambda_{em}}{\lambda_{obs}}=\frac{t_{em}^{2/3}}{t_{obs}^{2/3}}##
    I'm really lost on part (d) though. I was told that I'm supposed to integrate my answer from (a), but I am not quite sure how to go about that.
    ##r=c\int\limits_{t_{em}}^{t_{obs}} \frac{1}{t^{2/3}} \ dt = 3c(t_{obs}^{1/3}-t_{em}^{1/3})##
    I'm not sure how to get ##\lambda's## from this, and I'm also not sure how to include ##t_0/\lambda_0##
     
    Last edited: Feb 13, 2019
  2. jcsd
  3. Feb 13, 2019 #2

    Orodruin

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    ##\lambda_{obs}## is not the wavelength observed on Earth at the present time. It is the wavelength observed by a comoving hydrogen atom at some intermediate time ##t_{obs}##.
     
  4. Feb 13, 2019 #3
    Thanks! I somehow over looked that it said hydrogen atom. For part (d) do you think it's still assuming the same universe as in part (b)?
     
  5. Feb 14, 2019 at 12:25 AM #4

    Orodruin

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    Yes.
     
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