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I'm currently putting together a basic summary of the Lambda-CDM model and I have a slight issue with the fact that the equation to calculate lambda (which includes factors to convert physical units into geometric units) is incorporated into the omega_lambda calculation (which incorporates the critical density equation that appears to be simply expressed in physical units). Is this the norm? I'd appreciate it if someone could confirm that the following is correct (I've added footnotes in bold)-
The equation for lambda is taken from Einstein's modified field equation for general relativity.
Einstein's field equation G = 8 pi G T/C^4
G is the Einstein tensor of curvature (space time).
T is the energy tensor of matter (matter energy).
8 pi is the concentration factor.
C and G are introduced to convert the quantity T (which is expressed in physical units) to geometric units (G/C^4 is used to convert units of energy into geometric units while G/C^2 is used to convert units of mass, when density is used instead of energy, the C^4 can be replaced with C^2). {evidence of converting to geometric units.}
Lambda = 8 pi G x vacuum energy/C^4 = 8 pi G x vacuum density/C^2
= 8 x 3.14159 x 6.6742x10^-11 x 0.67 x 10^-26/(3x10^8)^2
= 1.252 x 10^-52 m^-2 {geometric units?}
Based on the vacuum density being 0.67 x 10^-26 kg/m^3, the figure above is 1.252 x 10^-52 m^-2 which is expressed in geometric units. This divided by G/C^2 (7.43 x 10^-28) converts it to mass SI units of kg/m^3, providing a figure of 1.685 x 10^-25 kg/m^3 (or 1.685 x 10^-28 g/cm^3) in physical units for the cosmological constant.
Critical density = 3H^2/8 pi G
= 3 x (2.26x10^-18)^2/8 x 3.14159 x 6.6742x10^-11
= 0.918 x 10^-26 kg/m^3
{no evidence of converting to geometric units!}
Omega = actual density/critical density
-Normalized matter density (incorporating the critical density equation. Omega = density of universe/critical density, if we put the above critical density equation into this simple formula, we get the equation below).
Baryonic matter (b)
Omega b = 8 pi G x density of b/3H^2
= 8 x 3.14159 x 6.6742x10^-11x 0.4x10^-27/3 x (2.26x10^-18)^2
Omega b = 0.044 for baryonic matter (matter composed of protons, neutrons and electrons)
Dark matter (dm)
Omega dm = 8 pi G x density of dm/3H^2
= 8 x 3.14159 x 6.6742x10^-11 x 0.202x10^-26/3 x (2.26x10^-18)^2
Omega dm = 0.222 for dark matter
Omega b + omega dm = 0.044 + 0.222
-Normalized vacuum energy density (incorporating lambda equation. By altering the lambda equation shown above slightly, we get lambda C^2 = 8 pi G x vacuum density, therefore lambda C^2 replaces 8 pi G x vacuum density in the omega equation).
Dark energy (l)
Omega l = lambda C^2/3H^2
= 1.252x10^-52 x (3x10^8)^2/3 x (2.26x10^-18)^2
{equation based on combining lambda equation- geometric units and critical density equation- physical units?}
Omega l = 0.732 for dark energy
Hopefully this is clear. I would appreciate any feedback.
regards
Steve
The equation for lambda is taken from Einstein's modified field equation for general relativity.
Einstein's field equation G = 8 pi G T/C^4
G is the Einstein tensor of curvature (space time).
T is the energy tensor of matter (matter energy).
8 pi is the concentration factor.
C and G are introduced to convert the quantity T (which is expressed in physical units) to geometric units (G/C^4 is used to convert units of energy into geometric units while G/C^2 is used to convert units of mass, when density is used instead of energy, the C^4 can be replaced with C^2). {evidence of converting to geometric units.}
Lambda = 8 pi G x vacuum energy/C^4 = 8 pi G x vacuum density/C^2
= 8 x 3.14159 x 6.6742x10^-11 x 0.67 x 10^-26/(3x10^8)^2
= 1.252 x 10^-52 m^-2 {geometric units?}
Based on the vacuum density being 0.67 x 10^-26 kg/m^3, the figure above is 1.252 x 10^-52 m^-2 which is expressed in geometric units. This divided by G/C^2 (7.43 x 10^-28) converts it to mass SI units of kg/m^3, providing a figure of 1.685 x 10^-25 kg/m^3 (or 1.685 x 10^-28 g/cm^3) in physical units for the cosmological constant.
Critical density = 3H^2/8 pi G
= 3 x (2.26x10^-18)^2/8 x 3.14159 x 6.6742x10^-11
= 0.918 x 10^-26 kg/m^3
{no evidence of converting to geometric units!}
Omega = actual density/critical density
-Normalized matter density (incorporating the critical density equation. Omega = density of universe/critical density, if we put the above critical density equation into this simple formula, we get the equation below).
Baryonic matter (b)
Omega b = 8 pi G x density of b/3H^2
= 8 x 3.14159 x 6.6742x10^-11x 0.4x10^-27/3 x (2.26x10^-18)^2
Omega b = 0.044 for baryonic matter (matter composed of protons, neutrons and electrons)
Dark matter (dm)
Omega dm = 8 pi G x density of dm/3H^2
= 8 x 3.14159 x 6.6742x10^-11 x 0.202x10^-26/3 x (2.26x10^-18)^2
Omega dm = 0.222 for dark matter
Omega b + omega dm = 0.044 + 0.222
-Normalized vacuum energy density (incorporating lambda equation. By altering the lambda equation shown above slightly, we get lambda C^2 = 8 pi G x vacuum density, therefore lambda C^2 replaces 8 pi G x vacuum density in the omega equation).
Dark energy (l)
Omega l = lambda C^2/3H^2
= 1.252x10^-52 x (3x10^8)^2/3 x (2.26x10^-18)^2
{equation based on combining lambda equation- geometric units and critical density equation- physical units?}
Omega l = 0.732 for dark energy
Hopefully this is clear. I would appreciate any feedback.
regards
Steve