A question about Friedmann 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

 

[George Jones]

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.

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.

 

[Chalnoth]

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).