Curvature Index in Flat Universe with Cosmological Constant?

[Ranku]

Is the curvature index κ necessarily zero in a flat universe with cosmological constant?

 

[George Jones]

Is the curvature index κ necessarily zero in a flat universe with cosmological constant?

Yes. If a Friedmann-Robertson-Walker-Lemaitre universes has density (relative to critical density) $1 = \Omega = \Omega_r + \Omega_r +\Omega_\Lambda$, then it is flat and $\kappa = 0$. ("Flat" refers to spatial curvature (of 3-dimensional hypersurfaces), not to spacetime curvature.)

 

[Ranku]

Yes. If a Friedmann-Robertson-Walker-Lemaitre universes has density (relative to critical density) $1 = \Omega = \Omega_r + \Omega_r +\Omega_\Lambda$, then it is flat and $\kappa = 0$. ("Flat" refers to spatial curvature (of 3-dimensional hypersurfaces), not to spacetime curvature.)

To extend the discussion, the curvature index $\kappa$ can also cast in terms of its energy density
$\rho$_$\kappa$$=- \frac{3k}{8πGa^2}$. Can we identify the 'source' of $\rho$_$\kappa$? Is it the matter density in the universe?

 

[George Jones]

To extend the discussion, the curvature index $\kappa$ can also cast in terms of its energy density
$\rho$_$\kappa$$=- \frac{3k}{8πGa^2}$. Can we identify the 'source' of $\rho$_$\kappa$? Is it the matter density in the universe?

No.

The total mass/energy density of the universe $\rho_{total} = \rho_m + \rho_r + \rho_\Lambda$, where $\rho_m$ is the density of matter, $\rho_r$ is the density of radiation, $\rho_\Lambda$ is the density of the vacuum.

Critical density $\rho_{crit}$ is defined by
$$\rho_{crit} := \frac{3H^2}{8\pi G}.$$
If $\rho_{total} < \rho_{crit}$ then the universe is open with negative spatial curvature, if $\rho_{total} = \rho_{crit}$ then the universe is open and flat (zero spatial curvature), and If $\rho_{total} > \rho_{crit}$ then the universe is closed with positive spatial curvature. Note that I have not said anything about whether the universe expands forever or recollapses.

Now define $\rho_\kappa$ by $\rho_\kappa := \rho_{crit} -\rho_{total}$. This is just a definition, not a physical density, Sean Carroll writes "don't forget this just notational sleight of hand."