_{1}

^{*}

In 2003, Guth posed the following question in a KITP seminar in UCSB. Namely “Even if there exist 101000 vacuum states produced by String theory, does inflation produce overwhelmingly one preferred type of vacuum states over the other possible types of vacuum states”? This document tries to answer how a preferred vacuum state could be produced, and by what sort of process. We construct a di quark condensate leading to a cosmological constant in line with known physical observations. We use a phase transition bridge from a tilted washboard potential to the chaotic inflationary model pioneered by Guth which is congruent with the slow roll criteria. This permits criteria for initiation of graviton production from a domain wall formed after a transition to a chaotic inflationary potential. It also permits investigation of if or not axion wall contributions to inflation are necessary. If we reject an explicit axion mass drop off to infinitesimal values at high temperatures, we may use the Bogomolnyi inequality to rescale and reset initial conditions for the chaotic inflationary potential. Then the Randall-Sundrum brane world effective potential delineates the end of the dominant role of di quarks, and the beginning of inflation. And perhaps answers Freeman Dysons contention that Graviton production is unlikely given present astrophysical constraints upon detector systems. We end this with a description in the last appendix entry, Appendix VI, as to why, given the emphasis upon di quarks, as to the usefulness of using times before Planck time interval as to modeling our physical system and its importance as to emergent field structures used for cosmological modeling.

It is well known through conventional calculations via QCD that there is a huge disconnection between what is calculated for a cosmological constant [

As a general point, when we say S.H.O. I am referring to the acronym of simple harmonic oscillator. This paper is dedicated to physics way beyond the S.H.O. (simple harmonic oscillator) approximation and is dedicated to the premise that nonlinear contributions, which are not fully treated in the conventional literature are of decisive importance and this paper is a first attempt to include in effects which are often neglected for the same of convenience.

Note also that an acronym WKB is referred to as a semi-classical tunneling approximation in this paper. Semi-classical approximations abound all over physics, and what we are saying is that the construction we are emphasizing allows us to incorporate more quantum mechanical procedures in our treatment of cosmology. The reason for refinements of our protocol about tunneling, as in the initial phases of cosmology lies in our treatment of what is known about zero point fluctuations, which will be brought up next. We in looking at quantum tunneling, as in the WKB idea, and it is a way to form vacuum states, initially, in cosmology and also the concept of a vacuum energy density value, which will be the reason for Equation (1) in the paper below. More about the WKB approximation will be in the 2^{nd} to last appendix entry of this document.

The vacuum state idea is brought up, in our document because a quantum state is said to be a vacuum state if the expectation value of the Hamiltonian in a given theory is a local minimum (the Hamiltonian of course being part of the data that defines the theory). We do not necessarily have a Hamiltonian for our modeling of cosmology, although both loop quantum gravity and Wheeler De Witt theory do refer to cosmological states, although incompletely. Needless to state, if we can define a Hamiltonian for cosmological conditions, not impossible to do, then we can go to the next idea, which is that of a vacuum energy density, which is one of the biggest problems in physics today. To a large degree, this document is an attempt to explain what are the consequences of extremely large theoretically calculated vacuum energy density values, and how to ameliorate the serious problems such calculations bring to the study of cosmology.

Zero point fluctuations of quantum fields contribute to an enormous vacuum energy density value

What we are doing is to give a physical model as to what could be an alternative mechanism physically to the vacuum fluctuation model used in QCD calculations to give a bound to the cosmological constant which avoids having a value roughly 10^{120} times too large [^{ }

Recent field theory calculations have lowered the overshoot factor to being of the order of 10^{43} times larger than what is inferred experimentally [

The model which in the end I turned to is similar in part to what was attempted by Weinberg et al. in the early 1980s [

Dr. Abbot as of the mid 1980s attempted to lower a calculated cosmological constant value via application of a tilted Washboard potential for a scalar field [

QCD is the theory of quantum chromodynamics, which is well understood. What we are doing is to try to make QCD as commensurate to quantum gravity, and quantum gravity via the idea of use of di quarks, which show up in this paper.

If as I suspect traditional QCD calculations have missed any essential details, it is in a di quark condensate as a template [^{ }

We link an initial di quark configuration to instantons, spalererons, and the early universe. Since spalerons are not well understood, this will necessitate a discussion of spalerons and a thermal bath of temperature T [

We argue all of this will set up conditions for describing in detail an effective cosmological constant contribution to brane world production of gravitons in the early universe. This will be done, assuming that the final input of a cosmological constant is significantly less than what the initial cosmological tilted well potential would give, effectively leading to a large burst of gravitons in an early universe.

Doing this though will necessitate an additional fifth dimension. Usually in mathematics, this is done via the circle group, denoted by T (or by

The circle group forms a subgroup of C^{×}, the multiplicative group of all nonzero complex numbers. Since C^{×} is Abelian, it follows that T is as well.

The notation T for the circle group stems from the fact that T^{n} (the direct product of T with itself n times) is geometrically an n-torus. The circle group is then a 1-torus. As electromagnetism can essentially be formulated as a gauge theory on a fiber bundle, the circle bundle, with gauge group U (1). Once this geometrical interpretation is understood, it is relatively straightforward to replace U (1) by a general Lie group. Such generalizations are often called Yang-Mills theories. If a distinction is drawn, then it is that Yang?Mills theories occur on a flat space-time, whereas Kaluza-Klein treats the more general case of curved space-time. The base space of Kaluza- Klein theory need not be four-dimensional space-time; it can be any (pseudo-) Riemannian manifold, or even a supersymmetric manifold or orbifold. All this will show up in our discussion of how to relate a Randall-Sundrum effective potential to a four dimensional potential system which incorporate axion walls which collapse as we reach an interval of Planck’s time

This paper will finally conclude with a discussion of the relative merits of a space borne system as discussed by the DETF, like the JDEM project [

In any case, our work is taking a very physical interpretation of what was proved in “The embedding of the spacetime in five dimensions: an extension of Campbell-Magaard theorem” where the statement is made by F. Dahia et al. [

Beginning with Guth’s eternal inflation paradigm [

Let us now consider how eternal inflation is set up, and how variations in step size as concluded in this paradigm may affect a first order transition alluded to in this specific methodology first introduced by Guth in the 1980s [

1) Quantum fluctuations are important on small scales, if and only if one is working with a static space time (i.e. no expanding universe)

2) For inflating space times, quantum fluctuations are “expanded” to be congruent in magnitude with classical sizes (classical fluctuations)

3) Simple random walk picture: In each time interval of

4) Quantum fluctuations are equally likely to move field

Those who read the presentation should note the conclusion which is something which raises serious questions: i.e.

5) In equations, the probability of an upward fluctuation exceeds

But

6) Guth closes his presentation with a statement to the effect that

“Even if there are 10^{1000} vacuum states produced by String theory, then perhaps inflation produces overwhelmingly one preferred type of vacuum states over the other possible types of vacuum states”

The question then arises, How could this preferred type of vacuum state be produced, and by what sort of process?

Prior attempts to model the early stages of the universe by near ideal gas analogies for so called “super cooling” lead to thermodynamically oriented arguments which we will reproduce here [

To whit, from starting with the assumption that we are restricting ourselves to a non dark matter-dark energy regime of matter to be measured experimentally [

This is assuming that we are working with a degree of freedom for Fermionic and Bosonic matter contributions of the form [

This lead to, assuming

This would lead to

As well as a temperature dependent volume behaving as [

The radius of this “volume” is directly proportional to

Note that the expressions so put up are highly dependent upon the degree of freedom parameter

This leads to the 2^{nd} consideration of this document, which is how to put in explicit order calculations as to the order of the electroweak phase transition, which Trodden claims is crucial Trodden presents the following arguments [

For continuous transitions, the associated departure from equilibrium is insufficient to lead to relevant baryon number production. For a first order transition, quantum tunneling occurs about

At a particular temperature below

The final, and important point Trodden makes is what happens in the immediate aftermath of baryogenesis, there is an alteration of a Higgs field, with the Higgs VEV changing from

At this juncture, we wish to address how to incorporate a more accurate reading of phase evolution and the minimum requirements of phase evolution behavior in an evolving potential system which permits baryogenesis, and also incorporates dark energy production as well. This then answers the question raised earlier^{25}, namely does “inflation produces overwhelmingly one preferred type of vacuum states over the other possible types of vacuum states” so one can have not only baryogenesis, but that we also have inflation as well.

To do this, we start off with an embedding in higher dimensions for answering in what context we may have baryogenesis, and when this baryogenesis ceases to be a dominant factor. In addition, we will also, as a side note, answer the question of graviton production in a brane structure arising in pre Planckian physics cosmology. This in particular is not too different from what Wald and others have argued [

This investigation is attempting to show that the fifth dimension postulated by Randall-Sundrum theory helps give us an action integral which leads to a minimum physical potential we can use to good effect in determining initial conditions for the onset of inflation. The 5^{th} dimension of the Randall-Sundrum brane world is of the genre [

This lead to an additional embedding structure for typical GR fields, assuming as one may write up a scalar potential “field” with

This scalar field makes its way to an action integral structure which will be discussed later on, which Sundrum used to forming an effective potential. Our claim in this analysis can also be used as a way of either embedding a Bogomolyni inequality, perhaps up to five dimensions [

The potentials

We should keep in mind that

This in the context of the fluctuations having an upper bound of ^{ }

Here,^{ }

The consequences of the fifth dimension mentioned in Equation (10) above show up in a simple warped compactification involving two branes, i.e. a Planck world brane, and an IR brane [^{ }

This integral, will lead to the following equation to solve

Here, what is called ^{ }via (for n > 0)

This uses [_{5})

This is for a compactification scale, for

We then obtain after a non trivial vacuum averaging [

This is leading to an initial formulation of [^{ }

Now, if one is looking at an addition of a 2^{nd} scalar term of opposite sign, but of equal magnitude [^{ }

This is for when we set up an effective Randall-Sundrum potential looking like [^{ }

This above system has a meta stable vacuum for a given special value of

We are forced to consider two possible routes to the collapse of a complex potential system to the chaotic inflationary model promoted by Guth [

The first such model involves a simple reduction of the axion wall potential [

The simplest way to deal with Equation (13) is to set, when Kolb [

i.e. to declare that the axion “mass” vanishes, and to let this drop off in value give a simple truncated version of chaotic inflationary potentials along the lines given by a transition from Equation (3a) to Equation (3b) We should note that ^{120} larger than what it is observed to be today [

has a bearing on this situation. Not to mention the problems inherent in several proposed fixes to the cosmological constant problem [

Now if we want an equivalent explanation, which may involve baryogenesis, we need to look at the component behavior of each of the terms in Equation (23) without assuming

We then have to present a varying in magnitude value for the “scalar”

There has been credible work with instantons in higher dimensions, starting with Hawking’s 1999 article [

Clarifying what can be done with an instanton style quantum nucleation in multiple dimensions [

where

This leads, if done correctly to the quadratic sort of potential contribution as given by [^{st} to the 2^{nd} potential system,

This is for his chaotic inflation model using his potential; I call the 2^{nd} potential

Let us now view a toy problem involving use of a S-S’ pair which we may write as [^{ }

This is for a di quark pair along the lines given when looking at the first potential system, which is a take off upon Zhitinisky’s color super conductor model [

We should note what a common misconception as to the cosmological constant is. In dimensional terms we often see it referred to as a “natural” cosmological constant value in terms of Planck Energy values. This is similar to the problems one observes in a Quantum Field theoretic vacuum summation of zero point energy bosonic fields up to Planck energy values [^{ }

V.G. Gurzadyan, and She-Sheng Lue wrote a world scientific paper [^{ }giving a derivation to the effect that one can calculate a realistic value for the cosmological constant based upon a wave number based upon a vacuum fluctuation model which gives a Fourier style de composition of vacuum fluctuation wave modes such that if we assume no angular momentum “twisting” and a flat FRW metric

(30b)

Needless to say though, that any energy density so accumulated would be far, far less than what was assumed in the typical bosonic field calculation, above, especially since if

This equation would get dramatically smaller for increasing age of the universe to present conditions, with the initial values of it to be similar in “form” to the enormous values of initial energy density outlined above for an initial nucleating universe, especially if we model the initial energy as proportional to the square of Equation (30c) above. This, however, has a serious defect in that it does not give a genesis, or origins reference as to how the cosmological constant could evolve from initial big bang conditions. Mainly due to it being extremely difficult to form

Part of the misconception which I think is endemic in this field with respect to forming a cosmological constant which is consistent with known astrophysics observations lies in the difficulty of forming of an effective Field theoretic Hamiltonian for calculation of vacuum energy, i.e. for quasi particles making sense of

G.E. Volovick writes a candidate for an acceptable Hamiltonian in this above equation as having a chemical potential addition [

This assumes that one can actually define a number operator for quasi particles, i.e.

Again, for early universe conditions, how does one form

As of the mid 1980s, Abbot [^{8} value based upon a vacuum energy expression given below(with

As Abbot admitted though, this model, while giving certain qualitatively attractive features involved an unacceptably long period of final tunneling time based upon [

with

This idea assumes as Abbot postulated a cascading series of minimum values of ^{ }

Equation (4) lead to a cascading series of local minimum values, where Abbot scaled the local minimum values via setting his scalar field as

This assumes

Typical values for the constants above were

This lead to, for final values of tunneling time of the order of

We looked at Ariel Zhitnitsky’s formulation of how to form a condensate of a stable instanton style configuration of cold dark matter as a starting point for how an axion field can initiate forming a so called QCD ball [

Note that in classical GR, potential diagrams have proved useful in analyzing orbits of particles and photons in Schwartzschild geometry. Here, we use the di quark model so formed here as to add in a physically realistic upper bound to forming a cosmological constant input into analyzing when gravitons could be released from an early universe brane. This is similar in part to the use of potential diagrams in investigating Lorentz violations in

The important thing to keep in mind, as brought up in a review of an earlier proposal made on this topic is that the di-quark potential satisfies the slow roll condition for inflation. Does one want to have inflation at 100 MeV? This is not impossible but difficult to achieve and it is necessary for a presenter of any such di quark model how to get the right amplitude of fluctuations [

We also claim that the process of forming instantons as spoken of in the formation of di quark pairs in itself is useful for the later formation of cold dark matter in the form of QCD (Quantum Chromodynamics) balls, as is elucidated upon later in this document. In addition is the datum that the main part of the potential for forming the single tunneling washboard potential used in the formation of di quark pairs is extremely similar to a cosmic axion potential which is known to have a temperature dependence. This axion potential, as is stated in one of the appendix entries disappears as temperature increases, leading to the more typical Guth chaotic inflationary potential [

As referred to in Mukhanov’s book on foundations of cosmology, spalerons are a way to introduce motion of a “quasi particle” in a Euclidian metric via use of Wick rotations ^{ }

This assumes

Next

This assumes

When we have the energy of the system close to Equation (35bi), we are in the realm of a first order approximation of escape probability of constituents of a scalar field

This assumes that the energy (mass) of a sphaleron is defined via

The exact particulars of forming an appropriate instaton

We looked at Zhitnitsky’s formulation of how to form a baryon condensate to initiate forming a so called QCD ball [

And, Zhitnitsky [^{ }

He furthermore states that stability, albeit not absolute stability is still guaranteed with

This is done in such a manner as to use

It is important to note that Abbots potential [

Then I claim that the tunneling problem Abbot spoke of can be scaled out as to obtain tunneling times not incongruent with respect to a reasonable age of the universe.

We should note that ^{120} larger than what it is observed to be today. However, if axions are involved in the formation of instanton physics for early universe nucleation, then Equation (40) tells us that as can be expected for very high initial temperatures that axions are without mass but exist as an energy construct, which is not so surprising

Why in this model did I work with a 2^{nd} potential system as well? The reason is because that the first potential system in its emergent fashion is congruent only with the first slow roll condition assumed as necessary for inflation, i.e. if we look at

We find that

However, if we work with

An easy, straightforward manner to calculate tunneling time in the case of a false vacuum is to use a WKB type bounce calculation for forming an energy based tunneling [

We need now to do this for a potential system given in part by Equation (5) above, and to do it consistently. Assuming that ^{ }

Here,

If one defines the minimum of the potential as being due to the 1^{st} tilted washboard potential ^{ }

If^{st} tilted washboard potential given in Equation (13a), this construction leads to a non zero, but not enormous tunneling time for instantons in the bubble of space time used for an early universe configuration, this will lead to [

This explicitly assumes that one is using the tilted washboard potential of Equation (13a)

What suffices to initiate chaotic inflation? Note that a potential

Involves a change of scale factor of the form [^{ }

What is involved is a reduction of the degrees of freedom of the initial physical system which permitted the di quark condensate to form in the first place. This would cause the physical system to shift from the 1^{st} to the final Guth chaotic potential [

This would be in tandem with [

In forming this problem, we offer an initial review of the basic physics we are assuming leads up to this problem. The physics shown in Equation (13a) to Equation (13b) above, corresponds to a phase transformation from

Our discussion in part is a brief introduction to the. brane structure built by Leach, and Lesame in their article about escape of gravitons from a brane [^{st} and 2^{nd} potentials of Equation (5) above, and our analysis of the eventual bound to the cosmological constant is commensurate with an early universe producing most of gravitons we would expect in our present universe. Our candidate for graviton production would be when we are producing cold dark matter in the form of QCD balls as written up by Zhitinisky in his model of QCD color super conductors as a cold dark matter candidate [

To begin we present using the paper written by Leach et al. on conditions from a FRW brane [^{nd} type given in Equation (13b).

We are assuming explicitly that we can use a Friedmann-Lemaître-Robertson-Walker brane world at or near the end of inflation, and that it is embedded in a structure similar to a 5-dimensional anti-de Sitter Schwartzshield bulk [

We shall briefly summarize their findings and put in our work on di quark contributions to an upper bound to the cosmological constant as an enabler of graviton production from this brane structure. A further unstated assumption is that the gravitons were likely released in a brief instant after di quark nucleation. This will be discussed in lieu of the presumed experimental difficulties as to graviton detection via conventional detectors, with recommendations as to how to find gravitons experimentally if they are as I suspect primarily released in early universe inflationary expansion.

Leach et al. obtain a condition for a gravitational signal to be going through a FRW brane via use of an effective potential of a graviton, which is defined via a potential impact parameter [^{ }

Also there exists an “impact parameter”

This leads to, practically, a condition of “accessibility” via [^{ }

Note, that in this definition, R so defined is with respect to “bulk dimensions” and from considerations of a five dimensional anti de Sitter Schwartzshield metric

Here, k = 0 for flat space, k = −1 for hyperbolic three space, and k = 1 for a three sphere, while we also set an anti-de Sitter radius of curvature [^{ }

This assumes a negative bulk cosmological constant

Then we have a maximum effective potential of gravitons defined via

This leads to a bound with respect to release of a graviton from an anti De Sitter brane as defined by Leach et al. as

How do we link this to our problem with respect to di quark contributions to a cosmological constant? Here I make several claims.

Claim 1.

It is possible to re define

Here, I am defining ^{st} and the 2^{nd} potentials of Equation (5) above. This new value for

Claim 2.

Claim 3.

Equation (5) has a 1^{st} potential which tends to be for a di quark nucleation procedure which just before a defined Planck’s time ^{120} time greater. i.e. that there was in fact for the region in between the 1^{st} and the 2^{nd} potential systems of. Equation (5)

With furthermore

So then that there would be a great release of gravitons at or about time

Claim 4.

Few gravitons would be produced significantly after time

Before we reach our conclusion, we should note that negative bulk cosmological constant used in brane world with the sort of metric referred to above assumes that there exist positive and negative tensions ^{ }

Still though this assumes as in Randall-Sundrum brane-worlds that the negative tension brane world has the standard model fields in the negative tension brane [

Now for the question the paper is raising, Can we realistically state the following for initial conditions of a nucleating universe? If so, then what are the consequences?

The right hand side of Equation (20) can be stated as having

We can insist that this

So, this leads to the following question. Does a reduction of axion wall mass for the first potential system given in Equation (13a) being transformed to Equation(13b) above give us consistent physics, due to temperature dependence in axion “mass”, or should we instead look at what can be done with S-S’ (soliton-anti soliton) instanton physics and the Bogolmyi inequality [

Finally, does this process of baryogenesis, if it occurs lend then to the regime where there is a bridge between classical applications of the Wheeler De Witt equation to the quantum bounce condition raised by Ashtekar [

Ashtekar’s quantum bounce [

In the common versions of Wheeler De Witt theory a potential system using a scale radius

Here we have that if

As well as

Now, Henriques [

Using a momentum operator as give by

This is assuming a real scalar field

As well as an energy term

This is for a “cosmic” Schrodinger equation as given by

This has

And

Here

Now Ashtekar in his longer arXIV article [

With the advent of this re definition of momentum we are seeing what Ashtekar works with as a sympletic structure with a revision of the differential equation assumed in Wheeler-De Witt theory to a form characterized by [^{ }

Also, and more importantly this

This is for a crucial “momentum” value

and

Which leads to, for an initial point in “trajectory space” given by the following relation

So that if we consider eignfunctions of the De Witt (difference) operator, as contributing toward

With each

This equation above has a “symmetry” as seen in

Our model is in part a response to an assertion by Dr. Steinbeck about the feasibility of information exchange in a cyclic universe model. As stated in Appendix III, [^{120} magnitude overshoot of current QCD vacuum state values which are reported all over the literature without using his cyclic universe model. QCD balls [

Furthermore, we also argue that the semi classical analysis of the initial potential system as given by Equation (6) above and its subsequent collapse is de facto evidence for a phase transition to conditions allowing for cold dark energy to be created at the beginning of inflationary cosmology. And that this production of dark energy would be at the same time as a surge in early universe graviton production this is done with Zhitinisky’s QCD ball condensate construction [

Any further work in analyzing this problem will require reconciling the modification of Abbot’s hypothesis stated above with what is manifestly wrong in the ideas presented in standard quantum chromodynamics references [^{ }which may permit time dependent effects in the beginning of cosmic nucleation of inflationary universe initial states.

Let us now briefly review what we can say now about standard graviton detection schemes. As mentioned earlier, Rothman states that the Dyson seriously doubts we will be able to detect gravitons via present detector technology [

Here,

This has

ton source, to a detector. Furthermore,

My entire point is that Dysons conjecture will be true for most candidates for graviton production, but that if we can define early universe graviton production more succinctly via a refinement of the above presented arguments that in due time, most likely by space detection systems looking for dark energy, that we may in fact be able to identify gravitons from an early universe environment. In addition, how the tunneling occurs should be investigated further via a refinement of the bounce solutions and vacuum tunneling procedures used by J. C. Hackworth and E.J. Weinberg in October 2004 [

We can point out via a heuristic argument a linkage between the di quark scalar field and fluctuations of a scalar field toward the end of the inflationary era which we do believe will support a proposal for refining this inquiry as to getting appropriate data sets from planned JDEM space borne instrument platforms. As pointed out by Zhitinisky [

Let us state for the beginning of the inflationary era, where the di quark pairs were already synthesized an amplitude for admissible fluctuations to be written as reaching a peak value when

This assumes that ^{st} potential of Equation (13a) already utilized to obtain the di quark nucleation value of a scalar potential. Here, ^{st} and 2^{nd} potentials of Equation (13a), we end up with the production of prototype QCD ball style instantons during this time which would eventually be blended into the quadratic 2^{nd} potential of Equation (13b). So being the case we would get a metric fluctuation value^{ }

This assumes we define

This is assuming k so defined is given by Equation (69) above, while

This last equation represents the value of the Hubble parameter at the end of the inflationary epoch, with

This leads to fluctuations for a quadratic potential as assumed by Guth

As Mukhanov [

could be detected in the

beginning of the manuscript, a base line requirement as to the development of realistic inflationary models, tying into the amplitude of the initial inflationary model is that for inflation to be on going, due to a di quark nucleation at the beginning, with a mass m defined in part by Equation (32)

All this needs to be vetted and confirmed by space borne observations.

A final consideration to be investigated, for the sake of how we define scale factors lies in an issue that Ellis raised recently in “On Horizons and the Cosmic Landscape” [

or

If as I assert that we can not realistically define ^{st} potential in Equation (5), we have to consider that there is a total wipe out of information at the start of nucleation of a new universe, and that we cannot appeal to the multi universe hypothesis in order to explain via an anthropic principle selection criteria for how our physical constants came into being as they are, with a surviving universe, as we see it today. The last two appendix entries, Appendix IV, and Appendix V, deal with, first (Appendix IV) what is the WKB approximation, with the understanding that we are improving on the WKB approximation, and Appendix V, which elaborates upon how the eventual detection of gravitational waves will allow us to distinguish among different gravitational theories, with a suggestion as to how to improve upon the ideas brought up in section IV. Furthermore we also are considering elaborations as to what is brought up in Appendix VI, which outlines the importance of Pre-Planckian space-time contributions to the physics which we model in Equation (12) and right afterwards, which is instrumental in terms of giving conditions in modeling cosmology, so as to allow for experimental proofs of the di-quark hypothesis, as presented in our document. The Appendix IV and Appendix VI entries should be further developed and inter related in future documents as to turn our suppositions into falsifiable experimental science. Doing so will allow examining Corda’s suppositions [

While checking Dr. Corda’s supposition [

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

In addition: the author wishes to thank Dr. Steinhardt for his elucidation of the importance of the Abbot model in the UCLA 2006 dark matter convention. Previously, the author had no awareness of this work, or of its deep significance as to parameterization of a realistic value to the cosmological constant. In addition, several researchers at the Kalvi institute Eastern gravity conference in March 2006, noticeably Dr. Thorsten Battefeld stated that much of what I presented was a re hash of Guth and Weinberg’s inflationary models, which forced me to enunciate graviton production and detection as crucially important. Finally, reviewers in South Africa pointed out the necessity of linking inflationary magnitude to a given specific upper bound which I claim in part is partly answered in the conclusion leading up to Equation (34) above.

In addition, Dr. Xi Yang is thanked for motivating me to continue physics, in spite of every hardship put my way.

Andrew WalcottBeckwith, (2016) Does a Randall-Sundrum Brane World Effective Potential Influence Axion Walls Helping to Form a Cosmological Constant Affecting Inflation?. Journal of High Energy Physics, Gravitation and Cosmology,02,125-153. doi: 10.4236/jhepgc.2016.21013

How we show phase evolves in both Equation (13a) and Equation (13b) above

Appendix I AFirst, the physics of how phase evolves due to potential Equation (13b) leading to chaotic inflation.

We shall make a plausibility argument linking the two potentials mentioned above in Equation 13a and Equation 13b. This will be using a slow roll approximation which is de rigor in inflationary cosmology, namely using

To start off, we will be using a nearly constant Hubble value of H in an early universe nucleation scheme. The reason for doing this will be considering

This will be done in spite of the well known relationship of

Furthermore, as a very rough approximation, we will be setting

Let us now review how a di quark phase

Then, what I write as Equation (1) may be viewed as

From now on, we will write

If so we can write Equation (1d) above as

We re write this Equation (1f) as

This leads to a general equation solution of

This leads to an equation for

Next, let us view the particular solution as given by

So, then we can look at the general situation created by the chaotic potential given by Equation (1b), as given by

This assumes we may write

This in the very small time limit is similar to what Guth wrote up in the 1980s.

Appendix I BWhen we look at Equation (6a) of the main text

Next, let us review the far more complicated situation arising with the tilted washboard potential given in Equation (13a) we will write up as, with

This has a derivative value of the form

Which we write as

Let us first look at the first to third order contributions to this potential, and then get a slow roll equation for evolution of

This has a general solution equation we may write up as

If we pick an exponential value for the general phase, and solve for

Implies, if we set the time unit as

This would lead for very small time intervals

This is almost always negative, since it would require the following for it to be positive, namely

If so then for the re do of the washboard potential mentioned, we have an exponentially growing

A particular solution for this phase for the washboard potential would lead to the cubic equation of the form

In general this leads to the di quark phase

This allows us to form a bridge between Equation (1b) and Equation (2). This is incidentally in line with the following graphs given in Dr. Kolb’s book, namely

Kolb’s book [

Here, he has the mass of the Axion potential as given by

As given by Mirjana Dalarsson and Nils Dalarsson [

With a corresponding time length of

This however is due to a wave functional argument for the universe which is time independent and depends only on the space time geometry. We do not have a way to include in an evolutionary history of states of matter contributing to these order of magnitude estimates other than a probalistic argument about space time geometry and the matter field content.

The di quark hypothesis, is intended to be a start to giving more definition to the Wheeler-De Whitt equation, for a wave function of the universe

This is where z is a dimensionless scale variable and

And

The enormous initial energy densities given in forming the bounce radius effectively preclude information exchange in prior universe cycles with our own, at least from the stand point of entropy of information at such high energy values assumed in the formation of our present universe.

As given by Griffiths [

This is the solution to what is actually a Hamilton-Jacobi equation. The solution given above, with no requirement that E be the energy operator of the Schrodinger equation is inherently semi classical. And an approximation used in tunneling problems.

We take the following discussion as quoted directly from [

Quote: from [

Quoting from [

“Thus, if advanced projects on the detection of GWs will improve their sensitivity allowing to perform a GWs astronomy (this is due because signals from GWs are quite weak) [

We argue that a third polarization in Gravitational waves from the early universe may be detected, if there is proof positive that in the pre Planckian regime that the Corda conjecture [

“The case of massless Scalar-Tensor Gravity has been discussed in [

This ends our recap of the section given in [

What we are arguing for is that the choice of the vacuum energy as given by Equation (27) may give conclusive proof as to satisfy the Corda conjecture and his supposition as to the existence of an additional polarization [^{rd} polarization. Which will be a way to determine the final disposition of GR as THE theory of Cosmology, or open up the possibility of alternate theories. It is an issue which we think will require extreme diligence. While ending our query as to the possible existence of a third polarization we should mention what would be the supreme benefit of our upcoming analysis of Equation (27), namely how to avoid the conflating of dust, with gravitational waves, i.e. the tragic Bicep 2 mistake [

End of quote from reference [

Which indicate if we have a scalar-tensor theory of gravity, or something else as is discussed in [

What we are referencing in Equation (12) in the main text is a situation for which there is a pre-Planckian space time for emergent phenomena. The Equation (12) is part of using a publication, [^{−}^{62} grams after 10^{−62} grams. We pick this procedure as part of our also wanting to know how and why there would be a high degree of flat space, as opposed to curved space, right in the aftermath of Planck time, and an emergent space-time construction would be instrumental in terms of understanding the background as given in publication [