Common choice in FLRW between dimensionless of scale factor (and coordinates of r lenght dimension) or the contrary

Question

I have an old cosmology book in which the scale factor is expressed in the Roberston-Walker metric as:

$\mathrm{d} s^2=c^2 \mathrm{~d} t^2-R(t)^2 \mathrm{~d} l^2$ with: $\mathrm{d} l^2=\dfrac{\mathrm{d} r^2}{1-k r^2}+r^2\left(\mathrm{~d} \theta^2+\ sin ^2 \theta \mathrm{d} \phi^2\right)$

Now, I mainly see the notation $a(t)$ to designate the scale factor, which is 1 when $t=t_0$.

To try to make the connection with the first notation, I deduce that $a(t)=R(t) / R_0$. But if I take that, I take automatically the comoving distance (and not just the comoving coordinates) that appears:

$$ \mathrm{d} s^2=c^2 \mathrm{~d} t^2-\frac{R(t)^2}{R_0^2} \frac{\mathrm{d}\left(R_0^2\, r\right)^2}{1-k r^2}+(R_0\, r)^2\left(\mathrm{~d} \theta^2+\sin ^2 \theta \mathrm{d } \phi^2\right) $$ When I say comoving distance, I mean physical distance. to t=to, i.e. $: \mathrm{d} l=R 0\times r$.

But the problem is that the coordinate r contained in the denominator $1-k r^2$ always remains a coordinate and not a comoving distance ( i.e. $\mathrm{d} l=R_0 \times r$ ), which means that I end up with a hybrid expression having comoving coordinates and a coordinate in the denominator of $1-k r^2$.

So Finally, which is the most common choice into the cosmologist community ? :

1. $a(t)=R(t)/R_0$ dimensionless and $R(t)$ and $R_0$ are distances and $r$ is dimensionless ?

OR

2. $a(t)$ is length dimension and $r$ is coordinate ?

EDIT 1: @MadMax. Is there a link between space curvature $\dfrac{k}{R_0^2}$ present into FLRW metric and the term $\Omega_k=-\dfrac{k\,c^2}{H_0^2\,R_0^2}$ present in Friedmann equation ?

Answer 1

I don't see a problem. The denominator $$1-k r^2$$ can be rewritten as $$1-\frac{k}{R_0^2} (R_0r)^2$$ then $\frac{k}{R_0^2}$ is the curvature constant (of length dimension -2) of the space.