_{1}

This paper is to address using what a fluctuation of a metric tensor leads to, in pre Planckian physics. If so then, we pick the conditions for an equality, with a small
δg_{n}, to come up with restraints which are in line with modifications of the Friedman equation in a quantum bounce, with removal of the Penrose theorem initial singularity. In line with super negative pressure being applied, so as to understand what we can present as far as
H = 0 (quantum bounce) in terms of density of the Universe. And also considering what to expect when
P = wΔρ ~ (-1+ε
^{+})Δ
ρ, i.e. we have a negative energy density in Pre Planckian space-time. This leads to a causal discontinuity between Pre Planckian to Planckian space-time due to the sign of the inflaton changing from minus to positive, for reasons brought up in this manuscript,
i.e. looking at Equations (9)-(11) of this document, with explanations as to what is going on physically.

We will here, in Equations (9)-(11) of the following document, outline the point of the document, i.e. a change in the inflaton field, from a “negative” to a “positive” field contribution, leading to a counter intuitive result, namely that there would be a causal barrier when the inflaton field would vanish in the denominator of the derived energy density expression, about at the boundary between Pre Planckian to Planckian space-time physics. The rest of the paper will be to explain the reasons for this startling model and its possible implications.

We use Freeze et al. Phantom bounce [

H 2 = 8 π 3 M Planck 2 ⋅ ( ρ − ρ 2 2 | σ | ) . (1)

Which when this is set equal to zero, at the time of a quantum bounce for a non singular universe, with

3 ⋅ ( 1 + p ρ ) ⋅ ρ 2 | σ | − ρ 2 | σ | − ( 1 + 3 p ρ ) ⋅ ρ = 3 ⋅ ( 1 + p ρ ) ⋅ ρ . (2)

This Equation (2) will have a modification of the density along the lines of ρ → Δ ρ .

We also will be examining the influence of [

Δ ρ Δ t ~ ( visc ) × ( H int 2 ) × a 4 . (3)

With here as given by [

Δ ρ ~ ( visc ) × ( H int 2 ) × a 4 × 2 ℏ δ g t t k B T initial ~ ( visc ) × ( H int 2 ) × a init 2 × 2 ℏ ϕ inf k B T initial . (4)

Our task will be to be looking at what this becomes with Equation (4) put into Equation (2) when ρ → Δ ρ .

The term for pressure we will be using is, then from [

P = w Δ ρ ~ ( − 1 + ε + ) Δ ρ . (5)

Then, we will be looking at Equation (2) written as

3 ⋅ ( 1 + ( − 1 + ε + ) ) ⋅ Δ ρ 2 | σ | − Δ ρ 2 | σ | − ( 1 + 3 ( − 1 + ε + ) ) ⋅ Δ ρ = 3 ⋅ ( 1 + ( − 1 + ε + ) ) ⋅ Δ ρ . (6)

Leading to

Δ ρ ⋅ ( 1 − 1 3 ⋅ ε + ) = | σ | ⋅ ( 1 + ( 2 − 3 ⋅ ε + ) ( 3 ⋅ ε + ) ) . (7)

Or then, if we use [

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

We get in the regime of Pre Planckian physics, the situation that we would have

Δ ρ ≈ − 2 | σ | ≈ ( visc ) × ( H int 2 ) × a i n i t 2 × 2 ℏ ϕ inf k B T initial ≈ ( visc ) × ( H int 2 ) × a i n i t 2 × 2 ℏ ϕ inf k B T initial ≈ − ( visc ) × ( H int 2 ) × a init 2 × 2 ℏ | γ 4 π G ⋅ ln { 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ t min } | k B T initial . (9)

In the regime of boundary between Pre Plankian to Planckian physics, we would have, instead

Δ ρ ≈ − 2 | σ | ≈ ( visc ) × ( H int 2 ) × a init 2 × 2 ℏ ( ϕ inf + δ + ) ⋅ k B ⋅ ( T Pre-Planck → Planck ) ≈ ( visc ) × ( H int 2 ) × a init 2 × 2 ℏ | γ 4 π G ⋅ ln { 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ ( ( t min + ε + ) ≤ t Plank ) } | k B ⋅ ( T Pre-Planck → Planck ) . (10)

What will be examined, in this document will be what we will be considering i.e. when the bracket in the LN expression approaches zero, namely

8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ ( ( t min ) < t Plank ) < 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ ( ( t min + ε + ) ≤ t Plank ) < 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ ( t Plank ) & 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ ( ( t min + ε + ) ≤ t Plank ) ≈ 1 . (11)

The terms Visc, a init 2 and H int 2 will be considered to be invariant in the area of the surface of the spherical (?) regime for where we have our analysis as to what this causal discontinuity implies, and why. This will be an addition to [

In reference [

ϕ ˙ 2 ≫ V SUSY . (12)

In so many words, we believe that the dynamics of Equation (11) as it applies to Equation (9) and Equation (10) fit this bill and also add, perforce a way as to confirm the existence of such behavior.

With the initial Hubble parameter, in this situation a constant value in the Pre Planckian regime of space-time, instead of the usual

H Hubble = a ˙ / a . (13)

Also, visc in Equation (1) is for a viscous “fluid” approximation in a non-singular regime of space-time namely, that we have initially due to [

Δ t initial ~ ℏ δ g t t E initial ~ 2 ℏ δ g t t k B T initial . (14)

At about this time interval, and beyond, we are examining a init 2 as given by [

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 . (15)

To get to the bottom of what this is implying as far as causal structure and how we modify it, we will be examining what Dowker brought up in [

The key point is that we wish to determine what is a minimum amount of information bits/attendant entropy values needed for transmission of ℏ initial [ t initial ≤ t Planck ] . If we specify a mass of about 10^{−}^{60} grams per graviton, then to get at least one photon, and if we use photons as a way of “encapsulating” ℏ initial [ t initial ≤ t Planck ] , then to first order, we need about 10^{12} gravitons/entropy units (each graviton, in the beginning being designated as one “carrier container” of information for one unit of ℏ initial [ t initial ≤ t Planck ] ). If as an example, as calculated by Beckwith [^{21} gravitons introduced during the onset of inflaton , this means a minimum copy of about one billion ℏ initial [ t initial ≤ t Planck ] information packets being introduced from a prior universe, to our present universe, i.e. more than enough to insure introducing enough copies of ℏ initial [ t initial ≤ t Planck ] to insure continuity of physical processes.

The dynamics of ϕ ˙ 2 ≫ V SUSY actually gives us a clue as to how this is possible, i.e. to use, due to the brevity of time interval, the equivalent of quantum teleportation between both sides of the causal barrier, to insure continuity of physical processes, along the lines of [

In other words, only enough information between both sides of the causal barrier would be swapped as to insure the continuity of physical processes, and this would be commensurate with an inquiry as to issues we will bring up next.

In order to have a positive inflaton, we would need to satisfy [

ϕ > 0 iff 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ δ t > 1 . (16)

This Equation (16) has to be taken in light of preserving also, ϕ ˙ 2 ≫ V SUSY , as given in Equation (12).

This also is the same condition for which we would have to have visc, i.e. the viscosity of the initial spherical starting point for expansion, nonzero as well as reviewing the issues as of [

Whereas how we do it may allow for the Corda references [

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

Beckwith, A. (2018) Gedankenexperiment, Assuming Nonsingular Quantum Bounce Friedman Equations Leading to a Causal Discontinuity between Pre Planckian to Planckian Physics Space-Time Regime. Journal of High Energy Physics, Gravitation and Cosmology, 4, 14-19. https://doi.org/10.4236/jhepgc.2018.41003