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## Homework Statement

The metric for trajectories along radial lines passing through observers on

Earth in comoving coordinates is

[tex]ds^2 = −c^2dt^2 + a(t)^2dr^2[/tex].

At time [tex]t_0[/tex], now, a galaxy comoving with the expansion that is not currently

observable here on Earth is receding from us at a recession velocity of [tex]v_p[/tex].

(a) In terms of [tex]v_p[/tex], the Hubble constant [tex]H_0[/tex] and the present value of the scale factor [/tex]a(t_0)[/tex], what are the current proper distance [tex]dp[/tex] and comoving distance r to the object?

(b) Consider a Universe where the time rate of change of the scale factor is

a constant. Show that the value of the scale factor at time t is given by

[tex]a(t) = a(t_0)(1 + H_0(t − t_0))[/tex].

(c) The source emits a photon directed towards us. Show that the time [tex]\tau[/tex] for

the photon to travel here from the source is given by

[tex]\tau = \frac{1}{H_0}(e^{\frac{v_p}{c}} - 1).[/tex]

## The Attempt at a Solution

I've got parts a and b done. For a,

[tex]d_p = \frac{v_p}{H_0}[/tex] and [tex]r = \frac{a(t_0)v_p}{H_0}[/tex]

and I can show whats asked of me in part b by integrating. However I am completely stumped by part c of the question.

I've started it by saying that for a photon, [tex]ds = 0[/tex] so that it can be said that

[tex]c\frac{dt}{a(t)} = dr[/tex], but then I got lost.

I tried substituting in for [tex]a(t)[/tex] and got three separate integrals that can be done, two of them being constants (and so just giving me first order powers of t) and the other giving me [tex]ln(t)[/tex], but after substituting the limits in (I assume these would be 0 and [tex]\tau[/tex] I don't know what to do with the ln0 that I get. And I have the linear powers of t to worry about.

If anyone could walk me through the steps I would appreciate it.