Density parameter and curvature index

[Ranku]

For Ω=1, κ=0. Does the value of κ simply follow from the value of Ω, or can its value have an independent existence? So if Ω>1, does κ have to be 1?

 

[mathman]

Context would be useful. I don't want to guess what you mean.

 

[Bandersnatch]

So if Ω>1, does κ have to be 1?

Yes, by definition. Since Ω=1 is the critical density - the density at which the universe is flat - any other value of Ω necessitates that the k parameter is not 0 and has the same sign as Ω-1.

 

[mathman]

Define k for me.

 

[Ibix]

Define k for me.

See the discussion in the General Metric section of https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric, for example.

 

[mathman]

The definition of k seems to have more than one option. For example the (-1,0,1) option simplifies your question.