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and a positive cosmological constant, the universe necessarily expands

forever? The argument may be,

(a_t/a)^2 = (8*pi*G/3)*rho + lambda/3 (Friedmann equation)

where a_t refers to the first derivative of a with respect to t. Now

the right hand side is strictly positive (as rho is positive and

proportional to a^3 for dust), so a_t is always positive.

If this is true, why is it said that in a universe with a positive

cosmological constant, the fate of the universe has no direct relation

with the curvature (k) but depends on the exact proportion of

[(matter+radiation) density] / vacuum energy density?