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I am busy with an effort to show how the energy density parameters evolve over time in an update of the LightCone7 calculator. See the posts on the thread Steps on the way to Lightcone cosmological calculator. As part of this effort, I ran into some difficulties with deciding how to find and present the evolution of the various Omega's over time for non-flat geometries.
##\Omega## without subscript is usually defined as the ratio: present total energy density to the present critical density (##\rho_{crit} = 3Ho^2/(8 \pi G)##), with the latter the density required to make space flat, given the present expansion rate Ho.
##\Omega = \Omega_\Lambda + \Omega_m + \Omega_r = 1## for a spatially flat universe.
The evolution of Omega over time (or redshifts) is a 'weighted' sum of the components of present matter density ##\Omega_m##, radiation density ##\Omega_r## and the cosmological constant's energy equivalent, ##\Omega_\Lambda##, i.e.
##\Omega(z) = \Omega_\Lambda + \Omega_m (z+1)^3 + \Omega_r (z+1)^4##
and the individual components are given by the terms in the function.
However, it seems not to be so simple when ##\Omega <> 1##. I have tried an approach through the actual energy densities, i.e.
## \rho_{z=0} = \Omega \rho_{crit} ##
## \rho_\Lambda = ## constant
## \rho_{m} = \rho_{z=0} - \rho_\Lambda ##
## \rho_{r} = \rho_{m}/(z_{eq}+1) ## (z_eq indicates radiation-matter density equality)
These present densities can be translated into density evolution as functions of z:
## \rho(z) = \rho_\Lambda + \rho_{m}(z+1)^3 +\rho_{r}(z+1)^4##
My problem is to decide by which ##\rho_{crit}## should I divide the above individual components to get the corresponding Omega's for the ##\Omega <> 1## case. In the latest test version of LightCone7, I have used the ##\rho_{crit}(z)## values for the flat (##\Omega = 1##) case. But is this correct? Should I have not have used the "non-flat critical density" based on the "non-flat H(z)", that now has a different profile?
Does anyone know of a straightforward method that has been published?
PS. I've corrected the last (##\rho(z)##) equation. There is no (z) required for the ##\rho_\Lambda## term, because it is ##(1+z)^0##.
##\Omega## without subscript is usually defined as the ratio: present total energy density to the present critical density (##\rho_{crit} = 3Ho^2/(8 \pi G)##), with the latter the density required to make space flat, given the present expansion rate Ho.
##\Omega = \Omega_\Lambda + \Omega_m + \Omega_r = 1## for a spatially flat universe.
The evolution of Omega over time (or redshifts) is a 'weighted' sum of the components of present matter density ##\Omega_m##, radiation density ##\Omega_r## and the cosmological constant's energy equivalent, ##\Omega_\Lambda##, i.e.
##\Omega(z) = \Omega_\Lambda + \Omega_m (z+1)^3 + \Omega_r (z+1)^4##
and the individual components are given by the terms in the function.
However, it seems not to be so simple when ##\Omega <> 1##. I have tried an approach through the actual energy densities, i.e.
## \rho_{z=0} = \Omega \rho_{crit} ##
## \rho_\Lambda = ## constant
## \rho_{m} = \rho_{z=0} - \rho_\Lambda ##
## \rho_{r} = \rho_{m}/(z_{eq}+1) ## (z_eq indicates radiation-matter density equality)
These present densities can be translated into density evolution as functions of z:
## \rho(z) = \rho_\Lambda + \rho_{m}(z+1)^3 +\rho_{r}(z+1)^4##
My problem is to decide by which ##\rho_{crit}## should I divide the above individual components to get the corresponding Omega's for the ##\Omega <> 1## case. In the latest test version of LightCone7, I have used the ##\rho_{crit}(z)## values for the flat (##\Omega = 1##) case. But is this correct? Should I have not have used the "non-flat critical density" based on the "non-flat H(z)", that now has a different profile?
Does anyone know of a straightforward method that has been published?
PS. I've corrected the last (##\rho(z)##) equation. There is no (z) required for the ##\rho_\Lambda## term, because it is ##(1+z)^0##.
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