Does the homogeneity and isotropy of space imply that the expansion of the universe is uniform?

Question

I have asked this question. Now I wonder what could happen if I take a step further. If space is assumed to be BOTH homogeneous AND isotropic, can I prove that the expansion of the universe is uniform? I tried to add this into my previous question, but people suggest I should open a new thread.

Again, by uniformity, I mean that if I pick three galaxies to form a triangle, then the ratio of the side lengths will never change over time.

This is NOT a duplicate of the previous question, because it is actually a completely different issue.

Answer 1

I am not sure how to prove it mathematically but the answer is yes. Now, we are assuming that the universe is homogeneous and isotropic so when the universe expands this homogeneity and isotropy should remain the same (it cannot be distorted).

While we describe such expansion, scale factor cannot have dependence on the position. Actually thats the only important thing.

Since scale factor is only time dependent. The ratios of the side lengths of this triangle will be the same all the time.

If the scale factor were also position dependent (such as $a(r,t)$, then when the distance was increasing between two galaxies the ratios of the side length wouldnt be the same.

For instance

$r_{12}(t) = a(t)r_{12}(t_0)$

$r_{13}(t) = a(t)r_{12}(t_0)$

so

$\frac{r_{12}(t)}{r_{13}(t)} = \frac{r_{12}(t_0)}{r_{12}(t_0)}$

but when $a(r,t)$ we cannot simply cancel them out.