Finding the Time for a Photon's Arrival

[Brewer]

Homework Statement

The metric for trajectories along radial lines passing through observers on
Earth in comoving coordinates is
ds^2 = −c^2dt^2 + a(t)^2dr^2.
At time t_0, now, a galaxy comoving with the expansion that is not currently
observable here on Earth is receding from us at a recession velocity of v_p.

(a) In terms of v_p, the Hubble constant H_0 and the present value of the scale factor [/tex]a(t_0)[/tex], what are the current proper distance dp 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
a(t) = a(t_0)(1 + H_0(t − t_0)).

(c) The source emits a photon directed towards us. Show that the time \tau for
the photon to travel here from the source is given by
\tau = \frac{1}{H_0}(e^{\frac{v_p}{c}} - 1).

The Attempt at a Solution

I've got parts a and b done. For a,
d_p = \frac{v_p}{H_0} and r = \frac{a(t_0)v_p}{H_0}

and I can show what's 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, ds = 0 so that it can be said that
c\frac{dt}{a(t)} = dr, but then I got lost.

I tried substituting in for a(t) 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 ln(t), but after substituting the limits in (I assume these would be 0 and \tau 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.

 

[Brewer]

I totally forgot how to integrate and how to split up fractions. I think I have it now.