I'm trying to determine the redshift observed for a light signal between 2 observers (let's call them observers one and two) in a flat matter-dominated universe. Observer one sends a signal ($t_1$) at light speed to observer 2 and she records it as a z of 5 ($t_2$). Observer two then sends a signal directly back. At what redshift does observer one see this incoming signal ($t_3$)?
We know the scale factor for a matter-dominated universe is as follows: $$ \frac{a}{a_0}=\frac{{t^{2/3}}}{t_0} $$
and $a_0$ and $t_0$ are $1$ for the present time
Additionally, the comoving coordinate will stay the same for both time intervals by definintion and can be described as follows: $$ r=c\int\frac{dt}{a(t)} $$
Using the relation between the scale factor and z, we can find the following relationship: $$ a(t_2) = 6\cdot a(t_1) $$
Describe the comoving coordinates for the time intervals and set them equal to each other: $$ \int_{t_1}^{t_2}\frac{dt}{{t}^{\frac{2}{3}}} = \int_{t_2}^{t_3}\frac{dt}{{t}^{\frac{2}{3}}} $$
After integration, substitute the relation found above in terms of t to get the ratio between $t_1$ and $t_3$: $$ t_2 = 6^{\frac{3}{2}}\cdot t_1 $$ ...to get: $$ 2(14.7\cdot t_1)^{1/3}-t_1^{1/3} = t_3^{1/3} $$
Lastly, now that we have a relationship between the initial signal emitted ($t_1$) and the final signal received ($t_3$), we can determine $z$: $$ \frac{a(t_3)}{a(t_1)}=1+z=\frac{t_3^{\frac{2}{3}}}{\left [ \frac{t_3}{59} \right ]} \therefore z\approx 14 $$
Above is the work I did but I'm not confident my answer is correct although I don't see anything glaring wrong with what I did. I expect the redshift to be greater which is consistent with the redshift calculated, but I don't have much greater intuition that this. Perhaps there is a more elegant way to solve it. I'd appreciate if someone could let me know if the solution seems sound.
Thanks