_{1}

This paper is to address using what a fluctuation of a metric tensor leads to, in pre Planckian physics, namely with a small δg
_{n}, affected by a small nonsingular region of space-time. The resulting density will be of the form
with the first term, on the right viscosity, of space-time, the 2
^{nd} on the right the square of an initial expansion rate, and due to the nonsingular nature of initial space time the fourth power of a scale factor, with a ~ a
_{init} ~ 10
^{-55} . We apply these density alteration tools to a criterion of self-replication of the universe, as written up by Mukhanov, as to how classical and quantum inflaton variations lead to understanding if the initial inflaton field of the Universe will be “growing”, or shrinking with. The first contribution to the initial alteration of the inflaton is classical, equivalent to minus the inverse of the inflaton field, and the second quantum mechanically based alteration of the inflaton is a “mass” term times an initial inflaton field. If this change in inflaton is positive, it means that domains of space time are increasing, and this is dependent upon the effective mass term we calculate in this manuscript. Finally after we do this we state how this relates to a formulation of the initial change in the Cosmological “constant” as given by ΔΛ
_{initial} ~ M
_{total-space-time-mass} as well as heavy gravity issues.

We use Mukhanov’s “self-reproduction of the universe” criteria [

Δ ρ Δ t ~ ( visc ) × ( H int 2 ) × a 4 . (1)

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 . (2)

Also, visc in Equation (1) is for a viscous “fluid” approximation in a non-singular regime of space-time , where we are using [

Δ t initial ~ ℏ δ g t t E initial ~ 2 ℏ δ g t t k B T initial . (3)

If so, then Equation (1) and Equation (3) will give, to first approximation

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)

Here, we will be using an inflaton given by [

a ≈ a min t γ ⇔ ϕ ≈ γ 4 π G ⋅ ln { 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ t } . (5)

Which is also in tandem with a Potential term given by

V ≈ V 0 ⋅ exp { − 16 π G γ ⋅ ϕ ( t ) } . (6)

These will be used in the rest of this paper, for our derivations, which will be in tandem with an emergent Cosmological Constant parameter which is in tandem with an alteration of the initial Penrose singularity theorem as brought up in [

Start off with a definition of

Δ Λ ~ ( α 2 ~ 1 ( 137 ) 2 ) × M initial 4 . (7)

With volume defined in four space by

V volume ( initial ) ~ V ( 4 ) = δ t ⋅ Δ A surface-area ⋅ ( r ≤ l Planck ) . (8)

And with the “three volume” defined above, with the time factored out. So then we will be looking at initial mass given by

M initial ~ Δ A surface-area ⋅ ( r ≤ l Planck ) × ( visc ) × ( H int 2 ) × a init 2 × 2 ℏ ϕ inf k B T initial . (9)

And initial scale factor given as [

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 . (10)

Here, the minimum scale factor has a factor of Λ which we interpret as today’s value of the cosmological constant. B is the early cosmological B field, the Frequency of the order of 10 ^ 40 Hz, and a min ~ a initial ~ 10 − 55 .

We will be combining the above into a commentary on Equation (7) to Equation (10) next.

We also can restate the above behavior with an initial mass density we can give as

Δ ρ ~ [ V 3 ( volume ) ] − 1 × N initial-count × m graviton . (11)

And, after Using Ng. Infinite quantum statistics [

M initial ~ Δ A surface-area ⋅ ( r ≤ l Planck ) × ( visc ) × ( H int 2 ) × a init 2 × 2 ℏ ϕ inf k B T initial ~ N gravitons ⋅ m gravitons ⇔ N gravitons ~ Δ A surface-area m gravitons ⋅ ( r ≤ l Planck ) × ( visc ) × ( H int 2 ) × a init 2 × 2 ℏ ϕ inf k B T initial . (12)

We found that the above would yield an N~10^{4} or so, which is not zero, but is only nonzero if we have the visc term not equal to zero in the initial bubble of space-time, and also that we observe having

a ≈ a min t γ ⇔ ϕ ≈ γ 4 π G ⋅ ln { 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ t } & ϕ > 0 iff 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ δ t > 1 . (13)

This puts a major restriction upon admissible V 0 and δ t terms, for our problem.

In addition we postulate that the existende of massive gravitons is syominous with the classical-quantum mechanics linkage as given in [

v group ∝ E / 2 ( E − V ) < c . (14)

Presumably in [

Note that as given in [

All this should also be tied into an investigation of how the viscosity of Equation (4) would also tie into the results above, with the interplay of Equation (3) and Equation (4) maybe giving by default some information as to condition for which the quantization condition linkage in the classical regime (represented by the Classical De Alembert equation) and the quantum Schrodinger equation have analogies in our model.

And now back to [

Δ ϕ total = Δ ϕ classical + Δ ϕ quantum ~ − ϕ inf − 1 + M total ϕ inf > 0 ⇔ ϕ inf > ( 1 / M total ) 1 / 2 . (15)

We need to satisfy Equation (11) and Equation (12) in order to make sense out of Equation (15).

Furthermore, though, the number N, of Equation (12) will be nonzero and well behaved with a nonzero real value for positive N and entropy only if we have

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

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 [

We argue that the formulation of Equation (15) and Equation (16) would be, with the inclusion of the mass of the graviton, especially as in ϕ inf > ( 1 / M total ) 1 / 2 as a precondition for steady growth of the inflaton, in cosmological expansion as out lined in [

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

Beckwith, A.W. (2018) Gedankenexperiment for Contributions to Cosmological Constant from Kinematic Viscosity Assuming Self Reproduction of the Universe with Non-Zero Initial Entropy. Journal of High Energy Physics, Gravitation and Cosmology, 4, 8-13. https://doi.org/10.4236/jhepgc.2018.41002