Shalom, Netsivi!  I see this in a different way:

  or
, where  is the 
cosmic scale parameter, which is the same in all space points of the 
universe and may be obtained from the global Friedmann equations for 
. Only for a closed universe  you can it consider as "radius" 
of the universe. In a certain place you can define the infinitesimal distance 
between two points:  ; wherecan be a dimensionless 
quantity and then you take for the dimensions of  length or vice versa: 
 you give the dim. of length and r is dimensionless. Take into account 
that  has not to be "coordinated" in a certain area. It is a 
cosmic-GLOBAL function. 

The first Friedmann EQ.:

, here all functions depend solely on the 
cosmic time t.