How to normalize the curvature in the Robertson-Walker metric

[Ratzinger]

This metric is often written in an alternative form, here called the curvature normalized way. Unfortunately I can't follow how they rewrite it, could someone hint me how it is done or tell me a text where it is shown in more detail?

thanks

 

[Ich]

The derivation is straightforward once you know that $$sin^{-1}$$ means arcsin there.