Could the acceleration of expansion have a simple geometric reason ?

[stephaneww]

I have copy and translate this from a french forum :

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The Hubble parameter defined from the scale factor of the FLRW metric is [latex]H(t) = \frac{\dot{a}(t)}{a(t)}[/latex]
a parameter representing the expansion rate of the "physical" volume of a compact spatial domain whose coordinates of the points of the surface are constant in co-moving coordinates (i.e., whose material content is always the same) can be defined by
[latex]\theta(t) = \frac{\dot{V}(t)}{V(t)}[/latex]
We then simply obtain 
[latex]\theta(t) = 3 H(t)[/latex]

Indeed, if we reason on a ball (a volume [latex]V[/latex] whose surface is a sphere), with a radius [latex]R[/latex] in co-moving coordinates, its physical volume is then 
[latex]V(t) = \frac{4\pi}{3} a(t)^3 R^3[/latex]
and, by drifting in relation to time, 
[latex]\dot{V}(t) = \frac{4\pi}{3}[3 a(t)^2\dot{a}(t)] R^3[/latex]
Hence the fact that 
[latex]\theta(t) = \frac{\dot{V}(t)}{V(t)} = 3 \frac{\dot{\dot{a}(t)}}{a(t)} = 3 H(t)[/latex]

 

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When [latex]\Lambda[/latex] cosmological constant is expressed in [latex]s^{-2}[/latex] we have

 [latex]\Lambda/(3H(t)^2)=\Omega_{\Lambda(t)}[/latex], 

which corresponds, if I am not mistaken, to Friedman's equation with only the cosmological constant as its component, i.e. for a spherical universe expanding under the effect of the cosmological constant and Hubble constant. It seems interesting to me to see that we can simplify by [latex]H(t)[/latex] (expansion = constant Hubble effect) and that as a result, there remains [latex]3H(t)*\Omega_{\Lambda(t)}[/latex] as a factor that could be interpreted as the acceleration of expansion (but I am far from being sure that this is a valid interpretation) As a result, matter (ordinary and dark) is diluted under this double effect and would further increase the share of dark energy in the total energy of the universe in the LambdaCDM model.
The vaccum created by the expansion in the universe would be of constant density energy and the new space created by the expansion would be part of the quantum vaccum. 

[latex]\Lambda/H(t)=3*H(t)*\Omega_{\Lambda(t)}[/latex], 

[latex]3*H(t)*\Omega_{\Lambda(t)}[/latex], would be the automatic (geometric effect) expression on the cosmological constant, i.e.the acceleration of the expansion, due to the effect of the Hubble constant on a sphere region of space

 

Thanks in advance for yours comments

Edited March 28, 2019 by Strange
latex then pb with edition / corrections requested

[stephaneww]

1 hour ago, stephaneww said:

..would be the automatic (geometric effect) expression on the cosmological constant, i.e.the expansion of the universe, due to the effect of the Hubble constant on a sphere region of space

 

sorry, please read :

in title  :"... acceleration of the expansion",

 

"comobil" instead ",commotional",

 

 "The vaccum created by the expansion in the universe would be of constant density energy and the new space created by the expansion would be quantum vaccum." instead:

"The vaccum created by the expansion in the universe would be of constant density energy and the new space created by the expansion would be part of the quantum vaccum."

 

and

"...would be the automatic (geometric effect) expression on the cosmological constant, i.e. the acceleration of the expansion of the universe, due to the effect of the Hubble constant on a sphere region of space"

Edited March 25, 2019 by stephaneww

[Strange]

On 3/25/2019 at 8:37 PM, stephaneww said:

"comobil" instead ",commotional",

I have made the changes you mention. I changed this to "co-moving" which, I think is the  correct term. (Sorry I have nothing more constructive to add!)

[stephaneww]

Thank you for the changes Strange