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

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

1. $$a(t)$$ 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 ?

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.

• Thanks, could you tell what is the most widespread convention taken by cosmologists ( a(t) dimensionless and r length or the contrary) ? Jun 7, 2023 at 4:33
• Dimensionless $a(t)$ is more widely adopted. See wiki page here: en.wikipedia.org/wiki/… Jun 7, 2023 at 15:15