gotta question.
Consider, if you will, a non-homogenously expanding sphere driven by an expansion regime. This sphere has a time-evolving maximum radius $R_M(t)$, driven by position and time-dependent scale factor $a((x_A,x_B),t)$ - i.e. it expands different rates across the length of this line.
Considering this discussion about the origin to the circumference, the relationship between the two is:
$$R(t)=1[m] \cdot a((0,R(t)),t)$$
The original radius of this sphere, at start-time $t=t_{00}$ (please forgive the poor labelling), is $R_{M00}$.
Consider the radius-line of this sphere seen along the $x$ axis and see that I have selected an arbitrary point $A$ that has position $R_{A00}$ at time $t_{00}$.
At time $t_{E}$ (end), the length of this line is $R_{ME}$, and the length from origin to $A$ is $R_{AE}$.
My question is: given the inhomogeneous nature of expansion, is the amount both lengths increase by a time and radius-independent ratio, $B$? ie. is this true:
$$\frac{R_{A00}}{R_{AE}} = B\frac{R_{M00}}{R_{ME}}$$
If not, what would be the relationship between the increase in the two?
Thanks for your insights, H