Not much to add beyond the title. The Robertson-Walker metric solution to the field equations has the form

$$g\_{\mu\nu}dx^\mu dx^\nu=-dt^2+a^2(t)\biggl(\frac{dr^2}{1-Kr^2}+r^2(d\theta^2+sin^2\theta \phi^2)\biggr)$$

in which the scaling factor $a(t)$ is squared. I cannot see any reason given for this so far, is this for dimensional reasons or is there a more important reason?