Do Friedmann-Lemaitre Equations Determine the Shape and Fate of the Universe?

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Mikeal
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The solutions to the Friedmann-Lemaitre equations have curvature values of K = -1 (open universe), 0 (flat universe), +1 (closed universe).

The corresponding critical density values at the current time are: ρc(k = -1), ρc(k = 0) and ρc(k = +1)

If (k = 0) and the current density is less than ρc(k = 0), does this mean that the universe will expand at an ever-increasing rate. If so, does it mean the universe is in-fact open, rather than flat?

Conversely, if (k = -1) and the current density is greater than ρc(k = -1), does this mean that the universe will reach a maximum and then collapse. If so, does it mean the universe is in-fact flat or closed, rather than open?
 
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Mikeal said:
The solutions to the Friedmann-Lemaitre equations have curvature values of K = -1 (open universe), 0 (flat universe), +1 (closed universe).

The corresponding critical density values at the current time are: ρc(k = -1), ρc(k = 0) and ρc(k = +1)

If (k = 0) and the current density is less than ρc(k = 0), does this mean that the universe will expand at an ever-increasing rate. If so, does it mean the universe is in-fact open, rather than flat?

Conversely, if (k = -1) and the current density is greater than ρc(k = -1), does this mean that the universe will reach a maximum and then collapse. If so, does it mean the universe is in-fact flat or closed, rather than open?

I thought it was the other way, positive k collapses. I get signs mixed up.

Can we be called "flat earthers" if we use k = 0?