# Time for a photons arrival

## 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 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, $$ds = 0$$ 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.