What is the Wavelength of Emitted Photon in a Cosmological Redshift Scenario?

[Joella Kait]

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
$$\ H(t)=\frac{2}{3t}.$$

(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 $t_0/\lambda_0$

 

[Orodruin]

$\lambda_{obs}$ 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 $t_{obs}$.

 

[Joella Kait]

$\lambda_{obs}$ 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 $t_{obs}$.

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)?

 

[Orodruin]

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)?

Yes.