# Coordinate distance

1. Jan 23, 2007

### Logarythmic

1. The problem statement, all variables and given/known data
For a universe with $$k=0$$ and in which $$(a/a_0) = (t/t_0)^n$$ where $$n<1$$, show that the coordinate distance of an object seen at redshift z is

$$r=\frac{ct_0}{(1-n)a_0}[1-(1+z)^{1-1/n}]$$.

2. The attempt at a solution
I have used

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

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

Last edited: Jan 23, 2007
2. Jan 23, 2007

### HallsofIvy

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