Coordinate Distance of Object at Redshift z: Solving w/ k=0, n<1

[Logarythmic]

Homework Statement

For a universe with [tex]k=0[/tex] and in which [tex](a/a_0) = (t/t_0)^n[/tex] where [tex]n<1[/tex], show that the coordinate distance of an object seen at redshift z is

[tex]r=\frac{ct_0}{(1-n)a_0}[1-(1+z)^{1-1/n}][/tex].2. The attempt at a solution
I have used

[tex]r=f(r)=\int_{t}^{t_0} \frac{cdt}{a(t)}=\frac{ct_0}{(1-n)a_0}\left(t_{0}^{1-n}-t^{1-n}\right)[/tex]

but then what? I know that [tex]1+z=\frac{a_0}{a}[/tex] but I can't get it right.

 

[HallsofIvy]

You're missing a power of "n".
[tex]f(r)=\int_{t}^{t_0} \frac{cdt}{a(t)}=\frac{1}{a_0}\int_{t}^{t_0}\fract_0^n t^{-n}dt= \frac{t_0^n}{a_0(1-n)}\left(t_0^{1-n}- t^{1-n}\right)[/tex]