• $id In summary, The k term in the Friedmann equation represents the curvature term k = 0, -1 & +1 and is restricted to these values because coordinates in FRW models can always be rescaled so that k only takes these values. It represents the curvature of the spatial hypersurfaces in each fixed value of cosmological time, not the curvature of spacetime. Today, the focus is on measuring Omega_k instead of k and the freedom to rescale the coordinates is already taken up by the convenient choices made to apply the FRW equations. $id

Hello everyone,

Its been 4 years now since I posted, Last time i was asking for school homework and now I'm about to finish my physics master degree with 1 exam left in cosmology.

As I'm sure you are aware, the k term in the Friedmann equation represents the curvature term k = 0, -1 & +1. I have a few queries about this.

Why is this term restricted to these values? Is a continuous k not valid under the FRW model?
Secondly, more generally, does this k represent the "shape of the universe" and is different from the curvature in GR?

sid

$id said: Why is this term restricted to these values? Is a continuous k not valid under the FRW model? Yes, but coordinates in FRW models that allows this can always be rescaled so that k only takes the values -1, 0, 1.$id said:
Secondly, more generally, does this k represent the "shape of the universe" and is different from the curvature in GR?

k represents the curvature of the spatial hypersurfaces that result for each fixed value of cosmological time, not the curvature of spacetime.

George Jones said:
Yes, but coordinates in FRW models that allows this can always be rescaled so that k only takes the values -1, 0, 1.
Yup. However, today we often don't worry about measuring k, but instead Omega_k, which most certainly does take on a continuous value. Also, I may be mistaken, as it's been a while since I looked at this, but I think that the freedom to rescale the coordinates is already taken up by the convenient choices we make in order to apply the FRW equations to measuring the expansion (t = 0 at the false singularity, t = 1 today).