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Homework Statement
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).
Homework Equations
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## 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})}##
The Attempt at a Solution
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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##
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