How is the co-moving co-ordinate defined mathematically in cosmology?

Question

The relationship between proper distance and co-moving distance in cosmology is given by:

$$R(t)=a(t)r$$

Where

$R(t)=$ the proper distance which corresponds to where a distant object would be at a specific moment in time. This can change over time as the universe expands.

$a(t)=$ the cosmic scale factor which describes how the size of the universe is changing.Defined as the ratio of proper distance between two objects at time $t$ and a reference time $t_0$. i.e. $a(t)=\tfrac{d(t)}{d(0)}$

Now here is where my confusion lies:

The comoving co-ordinate r , I have read factors out the the expansion of the universe , giving a distance which does not change with time. But what is the equation that actually describes it ?

I think the following is rather circular reasoning but it's all I could come up with so far ( although I do not think it is correct ):

$a(t)=\tfrac{R(t)}{R(0)} \Rightarrow r=R(0) $

Answer 1

Theres no derivation of the comoving coordinate, but theres a definition where you can define it is as, $$r=a(t_0)R$$ or in words the comoving distance is the proper distance where scale factor to be taken as $1$ ($a(t_0)=1$).