- #1

- 1,254

- 141

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

##\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##.

**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##.

Last edited: