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[tex]

\left( \frac {\dot{a}} {a} \right)^2 = \frac {8 \pi G \rho} {3} - \frac {k c^2} {a^2}

[/tex]

We define critical density by setting k = 0 and rearranging to get:

[tex]

\rho_c = \frac {3 H^2} {8 \pi G}

[/tex]

Where:

[tex]

H = \left( \frac {\dot{a}} {a} \right)

[/tex]

My question is does [tex]\rho[/tex] include the density contribution for Cosmological Constant (dark energy) [tex] \Lambda [/tex] or is this derivation only for a Universe with no cosmological constant?

How does one then actually measure the density of Universe?

I know that the density has been measured to be slightly less than the critical density, but I thought we are meant to live in a flat Universe? Is this due to the cosmological constant and how is this reconciled with [tex]\rho[/tex] not being exactly [tex]\rho_c[/tex]?

Finally, I am right in saying that a Universe with [tex]\rho_c[/tex] will stop expanding after infinite time, one with [tex]\rho > \rho_c[/tex] will collapse back on itself and one with [tex]\rho < \rho_c[/tex] will expand forever?

Many thanks.

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# Critical Density

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