Ranku
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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 said:Is the curvature index κ necessarily zero in a flat universe with cosmological constant?
To extend the discussion, the curvature index ##\kappa## can also cast in terms of its energy densityGeorge Jones said: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.)
No.Ranku said: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?