How to Obtain a Mass of a Graviton, and Does This Methodology Lead to Voids?

Using the Klauder enhanced quantization as a way to specify the cosmological constant as a baseline for the mass of a graviton, we eventually come up and then we will go to the relationship of a Planck Length to a De Broglie length in order to link how we construct a massive graviton mass, with cosmological constant and to interface that with entropy in the early universe. We then close with a reference to the possible quantum origins of e folding and inflation. This objective once achieved is connected with a possible mechanism for the creation of voids, in the later universe, using a construction of shock fronts from J. P Onstriker, 1991 and followed up afterwards with Mukhanov’s physical foundations to Cosmology book section as to indicate how variable input into self reproduction of the Universe structures may lead to void formation in the present era. A connection with Wesson’s 5 dimensional cosmology is brought up in terms of a generalized uncertainty principle which may lead to variations of varying energy input into self reproducing cosmological structures which could enable non uniform structure formation and hence voids. One of the stunning results is that the figure of number of gravitons, about 10^58 , early on, is commensurate with a need for negative pressure, (middle of manuscript) which is a stunning result, partly based on Volovik and weakly interacting Bose gas model for pressure, which is completely unexpected. Note that in quantum physics, the idea statistically is that at large quantum numbers, we have an approach to classical physics results. We will do the same as to our cosmological work. This means that the quantum number n −  , in our last set of equations, which as we indicate has the surprise condition that for Pre – Planckian space-time that a very large value for initial Pre Planckian dimensions dim d which is the dimensional input into the Pre Planckian state, prior to emergence into Planckian cosmology conditions. We conclude by stating the following question. Can extra dimensions come from a Multiverse feed in to PrePlanckian space-time? See Theorem at the end of this publication. Our answer is in the affirmative, and it with Black hole physics.


Start with the General Relativity First Integral
We use the Padmanabhan 1 st integral [1], of the form, with the third entry of Equation (1) having a Ricci scalar defined via [2] and usually the curvature ℵ is set as extremely small, with the general relativity [3] [4] ( ) Also, the variation 2 min tt g a δ φ ≈ by [5] [6] will have an inflaton, φ given by [7]. Leading to the inflaton which is combined into other procedures for a solution to the cosmological constant problem. Here min a is a minimum value of the scale factor and is not zero, but close to it.

Next for the Idea from Klauder
We are going to go to page 78 by Klauder [4] of what he calls on page 78 a restricted Quantum action principle which he writes as 2 S and we write a 1-1 equivalence as in [1], which is also seen in [3] ( ) ( ) ( ) ( ) ( ) Our assumption is that Λ is a constant, hence we assume then the following approximation, from [3] which is the precursor of activity as given in [4] [5] [6] [7] we have ( ) Our innovation is to then set 0 0 q q p t φ = ± ≈ and assume small time step values. Then as in [7] ( ) These are terms within the bubble of space-time given in [1] using the same inflaton potential. The scale factor is presumed here to obey the value of the scale factor in [8].

Why This Is Linked to Gravity/Massive Gravitons, and Possibly Early Universe Entropy
Klauder's program [4] is to embed via Equation (3) as a quantum mechanical well for a Pre-Planckian-system for inflaton physics as given by Equation (3).
And Equation (3a) and Equation (3b) as given in Klauder's treatment of the action integral as of page 87 of [4] where Klauder talks of the weak correspondence principle, where an enhanced classical Hamiltonian, is given 1-1 correspondence with quantum effects, in a non-vanishing fashion. If so, by Novello [9] and Equation (3) and Equation (3a) and Equation (3b), we have then for early universe conditions, that we will be leading up to using an algorithm for massive gravitons, as in [7], and [9]. If so then use Equation (4) and then get the mass of a graviton via g m c The long and short of it is, to tie this value of the cosmological constant, and the production of gravitons due to early universe conditions, to a relationship between De Broglie wavelength, Planck length, and if the velocity v gets to a partial value close to the speed of light, that, we have, say by using [10] as given by Diosi, in Dice (2018) for quantum systems, if we have instead of a velocity much smaller than the speed of light, a situation where the particle moves very quickly (a fraction of the speed of light) that instead of the slow massive particle post- If so then, we will be looking at using Ng version of entropy via use of infinite quantum Statistics [11], we have for a clearly specified value of mass of the gra- What we wish to explore will be if Equation (9) above is consistent with ( ) 58 Entropy 10 10 58 Doing so may involve use of the Corda article, as given in [11].

Now for Foundational Treatment as to if We May Have an Influence of the 5 th Dimension in Our Problem
Wesson, [12] has a procedure as far as a five-dimensional uncertainty principle which is written as, if n L l  .
Where L is for 4 th dimensions, and l is a five-dimensional representation, so we have Then we have an uncertainty principle in 5 dimensions as by Wessson [12] for which we can do if we look at the zeroth contribution Using an expansion of the form from CRC tables [13]. n n n n n n n n = ⋅ − − ⋅ − + ⋅ − + (13) Up to cubic roots, we obtain one real root and 2 conjugate complex roots of, if we use minimum uncertainty of 1 E t ∆ ∆ = =  and set 1 c = , we have then one real root, and two conjugate complex roots, so that 1 1.54715 n  (as real root for a cubic equation for n) ( We will then look at the consequences of the real root, first, in terms of variation of minimum time step before going to other cases, but for the record, we have then the weird case of, for real root n in Equation (14) that to other cases, but for the record, we have then the weird case of, for real root n in Equation Equation (16) is real valued only if 0 E ∆ < .

Under What Conditions Is E ∆ ≤ 0 How Would Negative Energy Tie into Negative Pressure Which Is Normally Expected in the Onset of Inflation?
First of all, look at conditions for rapid acceleration of the Universe, i.e. to have this according to the GR theory we have by [14] if ( ) a t is a scale factor, then the Friedman equations read as ( ) Then if we go to Gravitons again, and j = 1 we can state Equation (17) ( ) ( ) Now, look at a concept of pressure. Here. If the first expression is tabulated about Planck time (or just before).
We can then make the identification that we have negative pressure, we then have if we have both pressure and energy negative then we can make the following pairing of terms, i.e. first for the negative terms in Equation (18) and also looking at Equation ( Usually, the α is small so then the momentum term is such that the pres-A. W. Beckwith Journal of High Energy Physics, Gravitation and Cosmology sure is negative. As seen in Equation (19), and we furthermore elaborate upon this in the next section, via what is brought up by [14] from Volovok. But before doing this: We will after this is described go to the positive terms in Equation (18). The density in our problem we assert is due to the same positive terms in Equation We will then be looking at how we can then equate out a negative energy and a negative pressure for this Pre Planckian to Planckian physics transition.

Explicit Calculation for a Negative Pressure in This Pre-Planckian to Planckian Physics Transition
We will transition to Reference [14] Or an upper bound of say for graviton mass of 10 −62 grams, we have that we have negative pressure in our system for the number of gravitons being less than 10 58 , in a volume about 27 times, the cube of Planck length. This is stunning because in Equation (7) we have an entropy number 10 57 to 10 58 , which is amazing because it suggests that the entropy generation we pick is tied in explicitly for the generation of negative pressure which is essential for inflation.

Now for How We Could Consider Having E ∆ Drop as Negative Energy, in Our Problem of Pre-Planckian Physics Right before the Onset of Inflation. With A Flip over to Ultra High Temperature-Energy Conditions.
From [17], we have the following relationship, i.e. see referenced [17] have in its Equation (8) the following value i.e. if d = Dimensions, P = Pressure, V = Volume, then the basic energy expression is given as The discussion as to implementation of Equation (25) has that if the conditions in section 6 above are obtained for negative pressure, that in the Pre Planckian state we have at a chance, a quadratic dispersion relationship. In addition, Reference [17] claims that this is a result of a derivation from the Virial theorem as given in [18], so then that we may look at This is in turn directly related to the Schrodinger-Ehrenfest theorem we can write as This is in a way of referring to [17] and [18] a way to ascertain the correctness of using Equation (25) in the Pre-Planckian to Planckian transition in spacetime. Having said that. We will then state that what we believe is that V as volume, as given in Equation (25) would be roughly about 27 times the cube of Planck length, as a starting point, for investigation and that we would then have a transition up to the Planck length. Prior to nucleation of space-time.

A. W. Beckwith Journal of High Energy Physics, Gravitation and Cosmology
Our hypothesis is that breaching the barrier to full emergence would entail a simultaneous flip from negative (bound energy states) to Positive energy, whereas we would be using a variant of positive energy given as a restatement of (27) i.e. a release of bound state to unrestrained positive energy would be commenced from the Pre Planckian to Planckian transition.
i.e. eventually, if there is a barrier, of space-time at the surface of a sphere of about. 27 times the cube of Plank length, in "volume" that when the barrier was breached, there would be a switch from negative energy, to positive energy, but that the pressure would still be negative, hence "inside" the initial near singularity sphere we would have a negative value of Equation (27) signifying a BOUND state. Once the barrier collapsed, Equation (27) would switch to positive, but that in lieu of inflation that the pressure of our system would still follow Equation (21) and Equation (22).
All this may be tied into an issue of semi classical reasoning as given below.
We include this in to motivate readers to consider how a semi classical set of approximations may lead to bridging the gap between General Relativity and Quantum mechanics. We argue that the challenge in our present problem is to reduplicate the same methodology, but to also find a suitable potential system, instead of just a hierarchy of kinetic energy expressions.

Lesson Learned, i.e. a Way to Ascertain if Quantum Gravity Has a Chance to Be Applied Quantum Geometrodynamics and Semi Classical Approximations, as Reference [19] and Evolutionary Equations, for Quantum States, and Its Relationships to Quantum Issues Arising in [20]
We wish now to refer to another result which we view as largely in tandem with our quest as to come up with precursors to quantum gravity, i.e. from Kieffer.
Due to how huge this literature is, we will be by necessity restricting ourselves to pages 172 to 177 of [19] as that encompasses Hamiltonian style formalism and also has some connections to the Hamilton Jacobi equation.
We will make this limitation so our methods are not too far removed from the Solvay conference, 1927, i.e. the Hamilton-Jacobi equation makes an appearance, as well as a full stationary Schrodinger equation.
In this discussion, the wave functions are often quantized, or nearly so, albeit usually added gravitational background is semi classical.
To begin our inquiry as to Geometrodynamics, which has some fidelity to the Solvay 1927 conference, we look at the following expansion of the Klein Gordon Equation, without an external potential. i.e.
A. W. Beckwith Journal of High Energy Physics, Gravitation and Cosmology Which has a series expansion wavefunction solution we can write as The First, second and third terms in Equation (28a) are as follows and lead to the subsequent Equations, in terms of series expansion powers Then we acquire the term, at no power of c, so that ( ) Leading to a free Schrodinger equation of the form Then, the 2 nd term of the Klein Gordon Equation in terms of powers of ( ) Which leads to a Schrodinger like term, with an additional radiation correc- Here we have the following Radiative corrective term added ( ) As a Klein Gordon result, this leads directly to the idea of quantum mechanics, as embedded within a larger theory.
i.e. this methodology as brought up by Kieffer, in page 177 of [10] in its own way is fully in sync with some of the investigations of the embedding of quantum mechanics within a larger structure, as has been mentioned in a far more abstract manner by t'Hooft, in [21], although to make further connections, it would be advisable to have a potential term put in, as well as to have more said about relativistic corrections.
As mentioned by [21], Lammerzahl, C. in [22] has extended this sort of reasoning to quantum optics in a gravitational field. The virtue of this is that one is NOT using the functional Schrodinger equation, as seen in page 149 of the Wheeler De Witt equations, given in [19]. i.e. the above derivation, within the context of the orders of c, given above, has explicit time dependence put in its evolution equations, and avoids some of the issues of the Wheeler De Witt program. i.e. read page 149 and beyond in [19] as to some of the perils and promises as to this approach.
In addition, the recovery of the Schrodinger equation and the other recovery of the Schrodinger equation with a radiative corrective term added within the context of the Klein Gordon equation is fully in sync with some of the Solvay 1927 deliberations. As given in [20]. And also directly linkable to [21].

A. W. Beckwith Journal of High Energy Physics, Gravitation and Cosmology
What we wish to do is to reduplicate the same sort of power expansion picking off of terms given in Equation (28)  However, before tying an evolution Equation, from Equation (28) suitably modified to use parts of Equation (3) and Equation (4) [23] ( ) Whereas we can write if ( )  [24] where we have for a scalar field driving the expansion of the universe, with a scalar field being bigger than the square root of the mass of the universe for domain production as given in [24], page 353.

What if We Wish to Consider Mukhanov Self Reproduction of the Universe Criteria?
First of all, we will give pertinent background before we go to the Mukhanov criteria.
And then we can look at the consequences for self reproduction of the universe, given on page 353 of [24] and its Figure 1 below which is as seen in page 353, of [24] which is with a perpetuating continual expansion of the universe, given a mass, m, for which the scalar field of Equation (34) obeys The results of Equation (35) are accessible in Figure 1 below.
If we use Clifford Will, as in [15] for velocity of a massive graviton and make Which can be if (after we set 1 =  , and Then we have the inequality for self reproduction of the universe as Also keep in mind the numerical density N, as given above, can be linked to a "particle count" due to Entropy.  , which makes entropy density initially proportional to 58 10 g g N * ≈ ≈ will lead to a weird situation later on where one has an infinite (or nearly infinite) number of contributions from parallel or contributing universes from the meta multiverse to start the big bang, in our present universe. In addition, we can look at a shock wave prior to the transition to the inflationary regime that we have the following situation As given on page 44 of [28] Shock wave shock wave velocity Here the mass is defined by  (41) is specifying change in energy, and this can happen before nucleation of the universe, if there is a negative temperature. Which is striking. See also change in scale factor, with change of the temperature of the universe as given in page 401 of [29].
In doing so, we would then in this case see if we use the real root of n L l  , given in Equation (14) above Then a shock front, right at the starting gate of expansion would look like for the first root of n L l  Equation (14) Here the volume, in this case would be 27 times the cube of Planck length, and the mass of a graviton is approximately 10 62 grams.

Self Reproduction of the Universe May Entail Varying Values of Equation (43) if We Look at Three Roots of n Given in Equation (14), Which Influences a Minimum Time Step
We state that using the conjugate complex roots of n L l  given in Equation (14) This is obviously semi classical, and we will ask readers to consider that what may be used to add more rigor to our analysis would be the process of Bosonification, as seen in [30] Equation (44) is for the real root of Equation (14). Very likely the two complex roots of Equation (14) would yield different numerator values for the shock wave front formula, and the mixing of all three versions of shock waves, would be itself enough to induce chaos, or at least some of the phenomenology seen in [31]. And if we are lucky in our formulation we may be able to get a potential added to the deliberations of Equation (28), in terms of hierarchy of embedding space time in terms of a power law development. To do that though would require identifying though a suitable potential added, and we need to find that commensurate potential.

More as to a Cosmological Link to the (Weak) Correspondence Principle
In physics, we have that the correspondence principle is commonly held to be that at large quantum numbers we have an approach to classical results. A request was given to me to quantify that, in terms of mathematics, and the closest which I can come to that is to do the following. I.e. first look at this [32].

Quote
Even if one restricts oneself to Bohr's writings, however, there is still a disagreement among Bohr scholars regarding precisely which of the several relations between classical and quantum mechanics that Bohr discovered should be designated as the correspondence principle. There are three primary candidate-definitions in the literature. First, there is the frequency interpretation, according to which the correspondence principle is a statistical asymptotic agreement between one component in the Fourier decomposition of the classical frequency and the quantum frequency in the limit of large quantum numbers.
Second, there is the intensity interpretation according to which it is a statistical agreement in the limit of large quantum numbers between the quantum intensity, understood in terms of the probability of a quantum transition, and the classical intensity, understood as the square of the amplitude of one component of the classical motion. Finally, there is the selection rule interpretation, according to which the correspondence principle is the statement that each allowed quantum transition between stationary states corresponds to one harmonic component of the classical motion.

End of quote
In our situation, we most certainly would prefer the first definition, i.e. to look at.

Quote
First, there is the frequency interpretation, according to which the correspondence principle is a statistical asymptotic agreement between one component in the Fourier decomposition of the classical frequency and the quantum frequency in the limit of large quantum numbers. In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers.

End of quote
What we are doing is to assume a large quantum number will be generated just about the transition from the interior to the exterior of a min .
As it is, I expect that the transition of steps given in Equation (45) and Equation (45a) will lead to the following, i.e. as we have the transition from small to large values of a potential given in Equation (3)    Vacuum energy 1 1 (48) i.e. and this gets into one of the issues brought up by Christian Corda who asked about it. i.e. is there a way to reconcile the value of a cosmological constant as given by Wesson, in 5-dimensional cosmology with that of what is in official data sets. Before going to this issue, we should consider [35] Vacuum energy 8 G As we have a value of minimum time step t ∆ from Equation (36) The terms in the brackets refer to a 3-dimensional space, with four-dimensional time component, whereas 2 dl is for the 5 th dimension. In this context, the cosmological constant, is then according to [36], assuming that L is for fourdimensional Space-time A word of explanation is due here. What I assumed in the calculation of t ∆ , in terms of time step is to look at a projection and interaction of the fourth and 5 th dimensions to come up with the MININUM time step, and then from there to insert it into Equation (48).
Our working assumption is as follows, i.e. that what we have, as of Equation (48) should be virtually identical in magnitude to Equation (51) but it should be understood that L in Equation (51) is really the present day value of the assumed "radius of the universe". i.e. we are assuming then from Pre Planckian conditions to our present day that the Cosmological constant does not change. In any case, the approximate value of the Cosmological constant in Equation (51) should be understood to be by observations, approximately as follows, i.e. the true dimension of ( )  Planckian results [38], which is the backbone of the delta t term used in Equation (48)

More on a Linkage to Pre-Planckian to Planckian Physics
One of the striking results in [38] is their treatment of entropy, as given in their Equation (40), which is brought up to take into consideration the possibility of tunneling. i.e. the variation in entropy, S ∆ , is given as My first conclusion is that if there is a tie into the formula 27 of my manuscript that in fact what was done in [38] may be a way to tie in energy, E, with entropy, and make the analogy to Tunneling from the interior to the exterior of a boundary between pre Planckian to Planckian space time more exact.
I would be inclined to take the absolute magnitude of this above entropy expression and to assume the following, i.e. in the aftermath of tunneling right at the nexus of a boundary we would see approximately have for entropy generation, using the absolute magnitude of [38] as well as delta S ~ n (particle counting) by infinite quantum statistics as given by Ng. [39]. An advantage of Equation (52) if confirmed would be a way to examine the Weak correspondence principle more exactly. We shall comment upon this in our conclusion. Here we take the absolute value of Equation (52) and we will use that in our conclusion.
We should before proceeding also note that we would also be utilizing having Equation (41)

Conclusion, Part B: Can Extra Dimensions Come from a Multiverse Feed into Pre-Planckian Space-Time? See Theorem
To do this what we do is to state the multiverse done in [43] and [44] and cite the number, N so brought up with changes in g * , which is, the degree of freedom so assumed.
This idea is extremely speculative, but it embodies using this version of an idea which is in a recent conference proceedings in Spain used these two refer- i.e. the DNA of the idea was to refer to a Multiverse version of what is known as the Penrose Cyclic Conformal cosmology conjecture, i.e. [45] use this construction.
We are extending Penrose's suggestion of cyclic universes, black hole evaporation, and the embedding structure our universe is contained within, This multiverse embeds BHs and may resolve what appears to be an impossible dichotomy.
The following is largely taken from [43] and [44] and has serious relevance to the final part of the conclusion. That there are no fewer than N universes undergoing Penrose "infinite expansion" (Penrose) [45] contained in a mega universe structure. Furthermore, each of the N universes has black hole evaporation, with the Hawking radiation from decaying black holes. If each of the N un- n value, will be using  ( ) entropy~f S n [28]. How to tie in this energy expression, will be to look at the Journal of High Energy Physics, Gravitation and Cosmology What is done in Claim 1 and Claim 2 is to come up with a protocol as to how a multi-dimensional representation of black hole physics enables continual mixing of spacetime [47] largely as a way to avoid the Anthropic principle, as to a preferred set of initial conditions.

What this Ergodic condition of mixing of different contributions in the Pre
Planckian space-time would do is to add, via using up to N (almost infinite, say) multiverse contributions to a CCC version of space time is to add a statistical averaging of an initial start from a Pre Planckian to Planckian transition.
Prior to working with the theorem, we wish to bring up the following, i.e. that we would write.
The number, N of different multiverse contributions to a pre Planckian space-time would then lead to the following theorem. Here, we will define, in terms of what is given in Kolb and Turner [27], see that usually [27] has a value of, in the very early un- We furthermore state that this procedure, as similar to a black hole (not identical) and, has much overlap with Dr. George Chapline's et al. [48].
If this theorem is upheld as far as being proven, a road to quantum, gravity exists. This idea will be significantly developed in future publications.
Chapline, et al., state as follows on page 1 of their article [48].

Quote
The black hole event horizon is a continuous quantum phase transition of the vacuum of space-time roughly analogous to the quantum liquid-vapor critical point of an interacting bose fluid.

End of quote
We are doing much the same sort of thing, in the Pre Planckian to Planckian transition, and we will add far more detail relevant to experimental confirmation in a future article follow up which conceivably could be tested via experimental.