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This paper is to address using what a fluctuation of a metric tensor leads to, in Pre Planckian physics, namely . If so then, we pick the conditions for an equality, with a small , to come up with initial temperature, particle count and entropy affected by initial degrees of freedom in early Universe cosmology. This leads to an initial graviton production due to a minimum magnetic field, as established in our analysis. Which we relate to the inflaton as it initially would be configured and evaluated.

This article starts with updating what was done in [

〈 ( δ g u v ) 2 ( T ^ u v ) 2 〉 ≥ ℏ 2 V Volume 2 → u v → t t 〈 ( δ g t t ) 2 ( T ^ t t ) 2 〉 ≥ ℏ 2 V Volume 2 & δ g r r ~ δ g θ θ ~ δ g ϕ ϕ ~ 0 + (1)

In [

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

We assume δ g t t is a small perturbation and look at δ t Δ E = ℏ δ g t t with

Δ t time ( initial ) = ℏ / ( δ g t t E initial ) = 2 ℏ δ g t t ⋅ g ∗ s ( initial ) ⋅ T initial (3)

This would put a requirement upon a very large initial temperature T initial and so then, if

S ( initial ) ~ n ( particle-count ) ≈ g ∗ s ( initial ) ⋅ V volume ⋅ ( 2 π 2 45 ) ⋅ ( T initial ) 3 [

S ( initial ) ~ n ( particle-count ) ≈ V volume g ∗ s 2 ( initial ) ⋅ ( 2 π 2 45 ) ⋅ ( ℏ Δ t initial ⋅ δ g t t ) 3 (4)

And if we can write as given in [

V volume ( initial ) ~ V ( 4 ) = δ t ⋅ Δ A surface-area ⋅ ( r ≤ l Planck ) (5)

The volume in the Pre Planckian regime would be extremely small, i.e. if we are using the convention that Equation (4) holds, then it argues for a very large g s ∗ beyond the value of 102, as given in [

10 20 ≤ S ( initial ) ~ n ( particle-count ) | r ≤ l P ≤ 10 37 (6)

This is also assuming a δ t initial ≈ Δ t initial ∝ Plank-time , i.e. at or smaller than the usual Planck time interval.

The author is aware of the String theory minimum length and minimum time which is different from the usual Planck lengths, but are avoiding these, mainly due to a change in the assumed entropy formulae to read as the square root of the above results, namely [

10 10 ≤ S ( initial ) | String-Theory ~ n ( particle-count ) | r ≤ l P ≤ 10 16 (7)

The above is still non-zero, but it cannot be exactly posited as in the Pre Planckian regime of Space-time, since the minimum length may be larger than Planck Length, i.e. as of the sort given in [

If from Giovannini [

δ g t t ~ a 2 ( t ) ⋅ ϕ ≪ 1 (8)

Refining the inputs from Equation (8) means more study as to the possibility of a non-zero minimum scale factor [

α 0 = 4 π G 3 μ 0 c B 0 λ ⌢ ( defined ) = Λ c 2 / 3 a min = a 0 ⋅ [ α 0 2 λ ⌢ ( defined ) ( α 0 2 + 32 λ ⌢ ( defined ) ⋅ μ 0 ω ⋅ B 0 2 − α 0 ) ] 1 / 4 (9)

where the following is possibly linkable to minimum frequencies linked to E and M fields [

B > 1 2 ⋅ 10 μ 0 ⋅ ω (10)

This can be contrasted with looking at what happens if [

a ≈ a min t γ ⇔ ϕ ≈ γ 4 π G ⋅ ln { 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ t } ⇔ V ≈ V 0 ⋅ exp { − 16 π G γ ⋅ ϕ ( t ) } (11)

So as talked about with [

ρ Λ ≈ G ( E / c 2 ) 2 l l − 3 = G E 6 c 8 ℏ 4 (12)

And with the following substitution of

E → Pre-Planckian → Planckian Δ E ~ ℏ Δ t ⋅ ( δ g t t ≈ a min 2 ϕ initial ) (13)

Then to first order we would be looking at Equation (12) re written as leading to

ρ Λ ~ G c 8 ℏ 4 ⋅ ( ℏ Δ t ⋅ ( δ g t t ≈ a min 2 ϕ initial ) ) 6 (14)

And if Equation (15) holds,

Λ initial ⋅ H initial − 2 ≈ o ( 1 ) (15)

we would have by [

Λ initial ≈ H initial 2 ~ γ 2 / t 2 Λ initial ⋅ L p 2 ≈ 10 − 123 (16)

So

10 − 123 ~ γ 2 L P 2 ⋅ ( Δ E ⋅ δ g t t ) 2 / ℏ 2 (17)

Equation (17) would be key to the entire business, i.e. using this, we would have if

Δ E ∼ ℏ ω graviton (18)

Then

10 − 123 ~ γ 2 L P 2 ⋅ ( ℏ ω graviton ⋅ δ g t t ) 2 / ℏ 2 ~ γ 2 L P 2 ⋅ ( ω graviton ⋅ δ g t t ) 2 (19)

Then if we go to Equation (10) we have a threshold magnetic field for the production of gravitons which looks like if we apply the minimum scale factor condition [

B min ≥ 1 2 ⋅ 10 μ 0 ⋅ ω ≈ γ L P δ g t t 10 123 / 4 2 ⋅ 10 μ 0 ≈ γ L P ϕ initial a min 10 123 / 4 2 ⋅ 10 μ 0 & a min ~ ( 10 − 123 / 4 ) ⇒ B min ≥ γ L P ϕ initial 2 ⋅ 10 μ 0 (20)

i.e. we get graviton production if the last line of Equation (20) is satisfied, which means that the initial value of the inflaton, in this case is crucially important.

With that initial inflaton value determined in part by Equation (11).

The last line of Equation (20) helps establish a minimum magnetic field for the production of relic gravitons, with a magnetic field established through Equation (10) and subsequently modified by Equation (20).

This adds substance to what was brought up by Beckwith in [

a min ~ ( 10 − 123 / 4 ) ~ ( Δ E / E P ) 3 / 2 & ϕ initial 2 ~ o ( Δ E ⋅ γ ⋅ L P / ℏ 2 ) (21)

But Equation (21) and Equation (20) interplay and also give more substances to the use of Equation (19) with our guess of Equation (18) for the determination of the initial graviton frequency, which has to be at least of the order of 10^45 Hertz due to the fantastically small initial bubble of space time considered.

In doing so, we need to consider initial conditions so considered that Equation (20) and Equation (21) should be consistent with the inflaton and “gravity’s breath” document by Corda [

We conclude also with the hope that interpolating between the results of Equation (19), Equation (20) and Equation (21) will also in time confer answers as to the initial evaluative conditions for gravity as given in [

Work partially supported by National Nature Science Foundation of China grant No. 11375279.

Beckwith, A.W. (2017) Gedanken Experiment for B Field Contributions to Initial Conditions for Relic Graviton Production Based upon an Initial Inflaton Value. Journal of High Energy Physics, Gravitation and Cosmology, 3, 651-656. https://doi.org/10.4236/jhepgc.2017.34049