- #1
CaptainMarvel
- 13
- 0
Using the FRW:
[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.
[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.
Last edited: