_{1}

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Beginning with Pebble’s restatement of the Roberson-Walker line element, we obtain a way, afterwards, to calculate the relationship between an initial value of the “cosmological constant” and the value of fluctuations in the time component of the metric tensor g (tot). We assume, in doing so that the value of the cosmological “constant” does not change from its initial formation. We close with speculations as to how this ties into other issues in the conclusion.

We start off with using the Roberson Walker metric, i.e. using [

V ( 3-sphere-volume ) = 2 π 2 ( a ( t ) ⋅ R ) 3 (1)

Here, a ( t ) is a scale factor, with the scale factor = 1 in the present era, and being as low as 10 - 55 in Planck time regimes. If so then, if we speculate upon a density drop off, given phenomenologically by

ρ ( Space-time-energy-density ) ~ ρ = ρ ( initial ) ⋅ exp ( − α ˜ ˜ t ) (2)

And, then use of the evolution Equation (3), on the LHS, the time derivative of density, as given by

ρ ˙ = − 3 H ⋅ ( ρ + ( P / c 2 ) ) (3)

As well as looking at the generalized Chapyron Gas model for DM and DE [

P = − A / ρ ω ; 0 ≤ ω ≤ 1 (4)

Then the density function for space time, as referenced in Equation (2) has an initial value of the form, if the volume is proportional to the cube of Planck length, l P 3 , then we will write, to first approximation, where A = 1/3 by the radiation regime of space-time. Here we are assuming an invariant Λ for the cosmological constant, with its value in early time the same as today, i.e. no Quintessence

ρ ~ 3 α ˜ ˜ ⋅ ( 1 ± A ) ⋅ Λ + H . O . T (5)

The rest of this document will use a derivation by the author modified HUP [

( Δ l ) i j = δ g i j g i j ⋅ l 2 ( Δ p ) i j = Δ T i j ⋅ δ t ⋅ Δ A (6)

If we use the following, from the Roberson-Walker metric [

g t t = 1 g r r = − a 2 ( t ) 1 − k ⋅ r 2 g θ θ = − a 2 ( t ) ⋅ r 2 g ϕ ϕ = − a 2 ( t ) ⋅ sin 2 θ ⋅ d ϕ 2 (7)

Following Unruh [

a 2 ( t ) ~ 10 − 110 , r ≡ l P ~ 10 − 35 meters (8)

Then, the surviving version of Equation (6) and Equation (7) is, then, if Δ T t t ~ Δ ρ [

V ( 4 ) = δ t ⋅ Δ A ⋅ r δ g t t ⋅ Δ T t t ⋅ δ t ⋅ Δ A ⋅ r 2 ≥ ℏ 2 ⇔ δ g t t ⋅ Δ T t t ≥ ℏ V ( 4 ) (9)

Equation (9) is such that we can extract, up to a point the HUP principle for uncertainty in time and energy, with one very large caveat added, namely if we use the fluid approximation of space-time [

T i i = d i a g ( ρ , − p , − p , − p ) (10)

Then by [

Δ T t t ~ Δ ρ ~ Δ E V ( 3 ) (11)

Then, by [

δ t Δ E ≥ ℏ δ g t t ≠ ℏ 2 Unless δ g t t ~ O ( 1 ) (12)

The summary of what we obtain here, is if

ρ ~ 3 α ˜ ˜ ⋅ ( 1 ± A ) ⋅ Λ + H . O . T ~ Δ E l p 3 & A = 1 / 3 ( radiation ) ⇔ Δ g t t ~ ℏ α ˜ ˜ ( t min ~ Planck-time ) ⋅ l p 3 ⋅ ( 1 ± A ) ⋅ Λ Today's-value (13)

For our purposes, this corresponds to having α ˜ ˜ fairly large but not infinite, but also the decisive factor in the reduction of energy density as given in Equation (2), i.e. that even in the Pre Planckian regime, that we position the energy density for a dramatic drop in value. We do this preparation for a reduction in the energy density so that the value of Δ g t t is very small and consistent with [

δ t Δ E ≥ ℏ δ g t t | Pre-Octonionic → changeinphase,givenbyp phase δ 0 δ t Δ E ≥ ℏ | Octonionic with δ t ≥ ℏ δ g t t Δ E FIXED (14)

This matter of Octonionic and Pre Octonionic is being pursued separately by the author, but the notice of a phase shift, is in work which is consistent with work which Dr. Li and Dr. Yang did, in [

Keep in mind one basic fact. If we restrict ourselves solely to Octonionic geometry, we are embedded deeply in only what the Standard Model of physics allows. We should though understand what is implied by the physics of the Octonionic structure and so the rest of this first discussion is devoted to it.

In [

Quote:

(A linkage to the) mathematics of the division algebras and the Standard Model of quarks and leptons with U (1) × SU (2) × SU (3) gauge fields.

End of quote:

Once again, if we have only U (1) × SU (2) × SU (3) gauge fields, we have only the standard model, and that if we wish to have a minimum time step, we need to go beyond the standard model.

The division algebras are linked to Octonionic structure in a way which is touched upon by Crowell [

It is worth reviewing if this construction meets the experimental gravitational tests mentioned in [^{nd} Abbot paper, [

In addition, all these can be used to also vet if [

This work is supported in part by National Nature Science Foundation of China grant No. 11375279.

Beckwith, A.W. (2017) Upper Bound in Change in Initial Value for “Time Component” of Pre Planckian Metric Tensor, for a Cosmological “Constant” Universe. Journal of High Energy Physics, Gravitation and Cosmology, 3, 657-662. https://doi.org/10.4236/jhepgc.2017.34050