_{1}

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First we review what was done by Klauber, in his quantum field theory calculation of the Vacuum energy density, and in doing so, use, instead of Planck Mass, which has 1019 GeV, which leads to an answer 10122 times too large, a cut-off value of instead, a number, N, of gravitons, times graviton mass (assumed to be about 10°43 GeV) to get a number, N, count of about 1031 if the vacuum energy is to avoid an overshoot of 10122, and instead have a vacuum energy 10°47 GeV4. Afterwards, we use the results of Mueller and Lousto, to compare the number N, of 1031, assumed to be entropy using Ng’s infinite quantum statistics, to the ratio of the square of (the Hubble (observational) radius over a calculated grid size which we call a), here, a ~ a minimum time step we call delta t, times, the speed of light. Now in doing so, we use a root finder procedure to obtain where we use an inflaton value due to use of a scale factor if we furthermore use as the variation of the time component of the metric tensor in Pre-Planckian Space-time up to the Planckian space-time initial values.

We will reference what was doing by Klauber [^{−}^{43} GeV, in order to come up with a count value of 10^{31} for gravitons. Furthermore, if we then next go to Muller and Lousto’s entangled entropy [^{31} with the Muller and Lousto’s entropy result, in order to calculate the initial event horizon radius, which we find has a value of about 10^{−}^{20} meters. Small enough, according to the comparative calculations, and this depends upon the presumed grid size having a value of a ~ c times Dt. Here we will reference [^{44} seconds.

The summary result is that we get a set of conditions for a cosmological version of the event horizon, which is equivalent to stating that the early universe, has some qualitative similarities to a black hole, initially.

In doing this, we are assuming that the entropy as calculated by [

Finally, after establishing that, we reference a modified version of Hawkings power spectrum for black holes given in Calmert, Car and Winstanley, [

In applying [

So, then we will start our inquiry, and to do so, we have the formation of Dt so as to create the grid value, a, which is in the Muller and Lousto definition of entropy, but notice that we will start the idea of entropy, via an arrow of time argument, [

We initiate our work, citing [^{31}, and if this is a measure of entropy, as by [

So let us now begin to look at the step size for time step, as to get the grid size for [

Δ t ⋅ | ( 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) − ( 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) 2 2 + ( 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) 3 3 − ⋯ | ≈ ( γ π G ) − 1 48 π ℏ a min 2 ⋅ Λ (1)

From here, we then cited, in [

S Λ | Arrow-of-time = π ⋅ ( R c | initial ~ c ⋅ Δ t l Planck ) 2 ≠ 0 (2)

This leads to the following, namely in [

( R c | initial ~ c ⋅ Δ t l Planck ) ~ ϑ ( 1 ) (3)

The rest of this article will be contingent upon making the following assumptions FTR.

In short, our view is that the formation of a minimum time step, if it satisfies Equation (1) is a necessary and sufficient condition for the formation of an arrow of time, at the start of cosmological evolution we have a necessary and sufficient condition for the initiation of an arrow of time. With causal structure, along the lines of Dowker, as in [

Δ E Δ t Volume ~ [ ℏ / Volume ⋅ ( δ g t t ~ a min 2 ⋅ ϕ initial ) ] | Pre-Planckian → ( Pre-Planckian ) → ( Planckian ) Δ E Δ t ~ ℏ | Planckian (4)

i.e. the regime of where we have the initiation of causal structure, if allowed would be contingent upon the behavior of [

g t t ~ δ g t t ≈ a min 2 ϕ initial ≪ 1 → Pre-Planck → Planck δ g t t ≈ a min 2 ϕ Planck ~ 1 ⇔ ( R c | initial ~ c ⋅ Δ t l Planck ) ~ ϑ ( 1 ) | Planck (5)

These conditions are enough to have our creating of Dt, which is part of how we will come up, with the grid, for the Muller and Lousto calculation of entanglement entropy [

a ~ c ⋅ Δ t (6)

The equivalence of Equations ((5) and (6)) should be obvious, and we will be using Equation (6) to use [

In short, after a very long set of calculations, their model of entropy comes up with [

S ( Lousto ) ~ 0.31 ⋅ ( r H / a ) 2 ~ 0.31 ⋅ ( r H / c ⋅ Δ t ) 2 ~ 0.31 ⋅ ( r H / l P ) 2 (7)

We argue that this entropy, is equivalent to the count, N, of gravitons, in accordance to Ng’s infinite quantum statistics, which will then lead to us adapting the next sections results, as to obtain the number of gravitons, initially.

In [

ρ ( vacuum-energy-density ) = 1 2π ∫ 0 Δ k 3 d k = Δ 4 8 π → Δ = planck-mass 2.80 × 10 74 GeV 4 (8)

Our supposition is to make the following change in the above calculation, namely

Δ → early-universe N ⋅ m g ⇒ ρ ( vacuum-energy-density ) = 1 2π ∫ 0 Δ k 3 d k = ( N ⋅ m g ) 4 8 π → Δ = planck-mass 10 − 47 GeV 4 ⇔ N = 10 31 & m g = 10 − 43 GeV (9)

If, the above N, in Equation (9) is the same as entropy, then we can state the following, namely

N = 10 31 ~ 0.31 ⋅ ( r H / l P ) 2 ⇒ r H ~ 10 15 × l P ~ 10 − 20 meters (10)

Here for the satisfying of Equation (10) is contingent upon R c | initial ~ c ⋅ Δ t as of being about Planck Length. Then, we have to deal with inflation. If there are roughly 65 e folds, according to Freeze [

R initial ⋅ 10 27 = R final (11)

If so, using by [^{20} Hertz, or about 10^{11} GHz, which is then divided by 10^{27}, which would then indicate a frequency of about 10^{−}^{7} Hertz today, which sounds horrendous. But is it clear that the frequency would be of the order of 10^{20} Hertz, initially?

Conceivably, depending upon the production scheme, our estimate of r H ~ 10 15 × l P ~ 10 − 20 meters depends upon a very specific treatment of an obtained R c | initial ~ c ⋅ Δ t being of the same value as Planck length, and also of r H ~ 10 15 × l P ~ 10 − 20 meters being of the order of 10^{−}^{20}. Also, it depends upon the magnitude of the presumed inflationary expansion i.e. a value of instead of 65 e folds for expansion may instead be 50 e folds of expansion, which would correspondingly dramatically shrink the expansion of the wave function form for gravitational waves.

The only way such questions can be answered, is by experimental inquiry. The above are rough estimates only.

Also the introduction of more analysis from Cosmological Non Linear dynamics, as given in [

In addition, fidelity, or facts in dispute of the Penrose reference [

The experimental gravity considerations are covered in [

Reference [

We hope our document and its spin offs are in congruence with these last 3 references, and will work on the assumption they are synergistic with respect to our inquiry as stated here.

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

Beckwith, A.W. (2018) How Does a Setting of the Vacuum Energy Density, as Given Today, Lead to an Initial Hubble Radius for the Early Universe, i.e. How Does the Early Universe Partly Mimic a Black Hole? Journal of High Energy Physics, Gravitation and Cosmology, 4, 354-360. https://doi.org/10.4236/jhepgc.2018.42022