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

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 ):

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

• Related: "The difference between comoving and proper distances in defining the observable universe" (physics.stackexchange.com/q/400358/206691) – Chiral Anomaly Jan 9 at 4:17
• Yes, you see the thing is , in my college we do cosmology first and general relativity second so I have only seen the Robertson-Walker metric at a cursory glance and I don't yet have the skills necessary to define anything from it. – bhapi Jan 9 at 5:10
• Here's another recent post that might help: physics.stackexchange.com/a/453213/206691 – Chiral Anomaly Jan 10 at 2:28

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 $$1$$ ($$a(t_0)=1$$).