_{1}

^{*}

The author uses a low temperature and low entropy pre inflation state to create a bridge between String theory and loop quantum gravity. We use this analysis in lieu of the CMB barrier as of z = 1000 since it is our way to come up with a working model of quintessence scalar fields, which permits relic generation of dark matter and dark energy. Not only referencing this bridge, we do it in such a way as to utilize the low entropy condition which the Brane world model of Randal and Sundrum creates, and to show how it is in common with what Caroll and Chen wrote up in 2005., i.e. when the universe was about 1000 times smaller and 100,000 times younger than today.

Our task is to first answer if there are “vacuum states” created out of “nothing”, and this is done because vacuum fluctuations create dark energy density. The dark energy density is connected with an axion wall, which is dissolved by inflationary expansion. It is given by Weinberg [^{32} Kelvin, i.e. the high temperature of 10^{32} Kelvin is the beginning of the onset of quantum gravity [

Still though, as written up by L. Crowell [

Also, relatively recently, quintessence models were for large time scales ruled out, and Padmanabhan [

Contemporary graviton theory states that there is a thermal upsurge which initiates the growth of graviton physics. This is shown in K. E. Kunzes’ well written (2002) article [^{12} Kelvin forming at or before a Planck interval of time

A way of getting to all of this is to work with a variant of the Holographic principle and an upper bound to entropy calculations. R. Busso and L. Randall (2001) [

Recently, Bo Feng et al. [

In addition this approach accounts for data suggesting that the four-dimensional version of the “cosmological constant” in fact varies with respect to external background temperature. If this temperature significantly varies during early universe baryogenesis, the end result is that there would be a huge release of spin-two gravitons in the early stages of cosmic nucleation of a new universe. It also answers whether “Even if there are 10^{100}^{0} vacuum states produced by String theory, then does inflation produces overwhelmingly one preferred type of vacuum states over the other possible types of vacuum states?” [

Also, we also account for the evolution of an equation of motion of a quintessence field, via equations given to us by M. Li, X. Wang, B. Feng, and X. Zhang [

Finally, we can state unequivocally that the typical dark energy SUGRA potential, i.e. for times

blends/overlaps well with the finalized Guth style inflationary quadratic potential so derived in our formulation of this problem after we have a thermal input from a quantum bounce which melts axion walls. This is the case only because the intervals of time are so short as to merely indicate that Equation (1) merely indicates the growth behavior of a quantum vacuum state. In addition, we should also mention that the super nova survey, called Essence, has experimentally measured the “dark energy”―the thing that is causing the acceleration of the universe―to better than within 10% accuracy. The feature of this dark energy that was measured is its “equation of state”. So far it looks that the strange acceleration of the expansion of the universe can be explained by Einstein’s “cosmological constant”. Note that in modern terms the cosmological constant is viewed as a quantum mechanical phenomenon called the “energy of the vacuum”, in other words, the energy of empty space. And our work leads to the standard cosmological constant value well within the 1000 year limit of observational accuracy after the big bang, due to thermal cooling off which would occur about the z = 1000 or so red shift limit. This being the case, we will construct the following two tables for outlining the principles involved.

We look at the cosmological constant in 5 dimensions, and after it is given an absolute value give it as in Equation (2) below, we are finding that the absolute value is inversely proportional to temperature, T, as given below.

This is with regards to

Note that the non absolute value version of the contents of Equation (2) is negative valued which is very important, as the negative sign given to Equation (2) contents is put in string theory, to exhibit a different character for the string theory’s version of the cosmological constant than what is given in the four dimensional case.

In comparison, the four dimensional cosmological constant is positive valued we write as

In comparison, if we have a temperature decrease of cosmological temperature to 10^{12} Kelvin, both Equation (2) and Equation (3) exhibit the following claimed linkage as given in Equation (4), noting that if

It is useful to note a spatial relationship between the four and five dimensional cosmological constant parameters i.e. that the parameterization of the absolute value of the four and five dimensional constant have the following behavior.. I.e. the length scale between the two representations is similar

So being the case, we get a simple case of where we can analyze vacuum energy density fluctuations in the region of space smaller than the radii given in Equation (5) above, via

This is with respect to working with dimensions of the order of Plancks length, i.e. the volume of space where the volume of space has a radii which is of the order of, for

where initially we have temperatures of the order of 1.4 times 10 to the 32 power Kelvin as a thresh hold for the existence of quantum effects. This would pre suppose answering the issue raised by Weinberg [

This is pre supposing that we have a working cosmology which actually gets to such temperatures at the instance of quantum nucleation of a new universe. And if there is no temperature dependence, in the 5th dimensional cosmological constant set as having magnitude Λ, we still can get a five dimensional line element [

We shall reference a simple Lypunov Exponet argument as to adjustment of the initial quantum flux on the brane world picture. This will next be followed up by a description of how to link the estimated requirement of heat influx needed to get the quantum spatial variation flux in line with inflation expansion parameters.

To begin this, we access the article “Quantum theory without Measurements” to ascertain the role of a Lyapunov exponent

and

Here, we define where a wave functional forms via the minimum time requirement as to the formation of a wave functional via a minimum time of the order of Planck’s time

If we have a specified minimum length as to how to define

Two reasons First of all, we have that our description of a link of the sort between a brane world effective potential and Guth style inflation has been partly replicated by Sago, Himenoto, and Sasaki in November 2001 [

Their model is in part governed by a restriction of their 5-dimensional metric to be of the form, with l = brane world curvature radius, and H their version [

i.e. if we take

Our difference with Equation (102) of Sago et al. [

which assumes one is still working with a modified Gaussian potential all the way through. This is assuming that there exists an effective five dimensional cosmological parameter which is still less than zero, with

It is simply a matter of having

And of making the following identification

of _{P} being a Planck mass. This identifies an imbedding structure we will elaborate upon later on. His will in its own way lead us to make sense of a phase transition we will write as a four dimensional embedded structure within the 5 dimensional Sundrum brane world structure and the four dimensional

The potentials

(22)

This is where for low temperatures

axion walls specified by Kolb’s book [

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

So that the axion “matter” will oscillate with a “frequency” proportional to

perature^{2}

and then relate it to Guth inflationary potential scalar fields [^{ }

Or

This axion treatment is similar, in a lower dimensional sense to the work which is presented in [

The dissolving of axion walls is necessary for dark matter-dark energy production and we need to incorporate this in a potential system in four dimensions, and relate it to a bigger five dimensional potential systems. We need to find a way to, using brane theory, to investigate how we can have non zero axion mass conditions to begin with. This will be done after we bring up a brief interlude of Quintessence evolution of the scalar field, which for long periods of time is unworkable, but which would be appropriate up to times in the order of magnitude of the Planck’s time coefficient. Note after this description of Quintessence, we will be looking at the mechanism of thermal input leading to the time dependence of the axion mass, as given in Equation (26) above.

This discussion is modeled on an paper on Quintessence and spontaneous Leptogenesis (baryogenesis) by M. Li, X. Wang, B. Feng, and Z. Zhang [

Let us now look at a different effective Lagrangian which has some similarities to equations of motion for Quintessence scalar fields, assuming that specifying a non zero value to

What will be significant will be the constant,

Here I am making the following assumption about the axion contribution scalar potential system

For low temperatures, we can assume that prior to inflation, as given by Carroll and Chen [

And that right at the point where we have a thermal input with back ground temperatures at or greater than

This entails having at high enough temperatures

Let us now review the four cases so mentioned and to use them to analyze new physics.

CASE I:

Look now at a low temperate slow roll case, which is also true when we get to time

CASE II:

If temperature is T very large and time of the order of Planck time. We ignore slow roll, and we use

We then use the following approximations

CASE III:

T very large and time almost Planck time

Case IV:

T not large, so the Axion mass is not negligible, and time is almost

Then we obtain

This will lead to as the temperature rises we get that the general solution obeys

The upshot is, that we have a real field if the axion mass disappears. First of all, though, we have to understand how the conditions presented by S. Carroll, and J. Chen came about via brane theory.

Our starting point here is first showing equivalence of entropy formulations in both the Brane world and the more typical four dimensional systems. A Randall-Sundrum Brane world will have the following as a line element and we will continue from here to discuss how it relates to holographic upper bounds to both anti De sitter metric entropy expressions and the physics of dark energy generating systems.

To begin with, let us first start with the following as a

Furthermore, the ^{ }

We can then speak of a four dimensional volume

And if a Brane world gravitational constant expression ^{ }

If we look at an area “boundary” ^{ }

We link this to the principle of the Jeans inequality for gravitational physics and a bound to entropy and early universe conditions, as given by S. Carroll and J. Chen (2005) [

Low entropy conditions for initial conditions, as stated above give a clue as to the likely hood of low temperatures as a starting point via R. Easther et al. (1998) [

Similar reasoning, albeit from the stand point of the Jeans inequality and instability criteria lead to Sean Carroll and J. Chen [

We shall next refer to how this relates to, considering a low entropy system as a start an expression Wheeler wrote for graviton production and its implications for early relic graviton production, and its connection to axion walls and how they subsequently vanish at or slightly past the Planck time

We use below the Wheeler black body approximation used in [

Equation (49) is useful for, in a black body cavity setting for generating a burst of gravitons in a given frequency range. Now, here Equation (49) predicts a surge in gravitons being produced if there is a sharp increase in temperature. If the Early universe prior to the big bang were low temperature in character, with a subsequent build up to Planck temperature, then Equation (49) models an extreme graviton burst.

Here is how we can build up a scenario for just that. Equation (49) suggests that at low temperatures we have large busts of gravitons.

Now, how do we get a way to get the ^{15} cm)^{−}^{1}. This is though if we connect massive gravitons with dark matter candidates, and not necessarily with relic gravitons. Having said this we can note that Massimo Giovannini [_{0}^{2}Ω_{GW}) is of the order of 10^{−6}, roughly eight orders of magnitude larger than in ordinary inflationary models. That roughly corresponds with what could be expected in our brane world model for relic graviton production.

We also are as stated earlier, stating that the energy input into the frequency range so delineated comes from a prior universe collapse, as modeled by Ashtekar, A., Pawlowski, T. and Singh, P (2006) [

The subsequent analysis is for a low temperature axion wall domain wall in early universe structure. This will change, with the axion domain wall being ‘melted’ with a surge to Planck Temperature. We will next then talk more explicitly about Graviton production, aside from using the Wheeler Black Body model, given in Equation (49) above.

We need to understand what is required for realistic space propulsion. To do this, we need to refer to a power spectrum value which can be associated with the emission of a graviton. Fortunately, the literature contains a working expression as to power generation for a graviton being produced for a rod spinning at a frequency per second^{−}^{60} - 10^{−62} grans, If so, then from Fontana [

First of all, we will integrate Equation (50), and also give a Planck length value to the rotating rod, and then we get Equation (51). One can see the results of integrating Equation (50), Note this expression for numerical production of gravitons is extremely sensitive to temperature, T, and se obtain

N1 = 1.794E − 6 for | Power = 0 |
---|---|

N2 = 1.133E − 4 for | Power = 0 |

N3 = 7.872E + 21 for | Power = 1.058E + 16 |

N4 = 3.612E + 16 for | Power @ very small value |

N5 = 4.205E − 3 for | Power = 0 |

Then, if

Assuming when one does this that the back ground in the initial inflation state causes a thermal heat up of the axion wall “material” due to a thermal input from a prior universe quantum bounce. Our next task will be to configure the conditions via brane world dynamics leading to graviton production. This necessitates using a brane world potential to accommodate the building of a structure accommodating a transition from relic graviton production to the onset of Guth style chaotic inflation.

The consequences of the fifth-dimension show up in a simple warped compactification involving two branes, i.e., a Planck world brane, and an IR brane. Let’s call the brane where gravity is localized the Planck brane This construction permits (assuming K is a constant picked to fit brane world requirements) [^{ }

Here, what is called

To use Kaluza-Klein theory, as in [^{1} that is the fiber of the U (1)-bundle of electromagnetism. This leads to construction of, by [

Start with

The above in our integral, as in this treatment makes use of the following quadratic approximation. I.e. if Q is a charge, then

Part of the integrand in Equation (54) is known as an action integral,

And

We should note that the quantity _{false min,} and a true vacuum minimum energy E_{true min}, with the difference in energy reflected in Equation (59). Note what was done by the Japanese theorists. We obtain the following simplification, namely

For convenience we have made the substitution as given in Equation (61).

The upshot is that the following simplification appears to hold as in Equation (62).

All these final steps need to be confirmed in the future by rigorous analysis.

We use our bound to the cosmological constant to obtain a conditional escape of gravitons from an early universe brane. To begin, we present conditions (Leach and Lesame, 2005) [

Also there exists an “impact parameter”

This leads to, practically, a condition of “accessibility” via R so defined with respect to “bulk dimensions”

Here, k = 0 for flat space, k = −1 for hyperbolic three space, and k = 1 for a three sphere, while an radius of curvature

This assumes a negative bulk cosmological constant

This leads to an upper bound with respect to release of a graviton from an anti De Sitter brane (Leach and Lesame, 2005) [

In the language of general relativity, anti de Sitter space is the maximally symmetric, vacuum solution of Einstein's field equation with a negative cosmological constant Λ.

How do we link this to our problem with respect to di quark contributions to a cosmological constant? Here we make several claims. These are all from [

Claim 1: We reset

Proof of Claim 1: We are noting that the end result is that

And we take the value of the five dimensional value to be in magnitude presentable as

Note how

value of the Cosmological Constant. Then

Here, we define

The upshot is that we have an approaching to the 3 Kelvin background temperature, as in present day Space-time.

This, for potential

Claim 2:

where L is the physical distance

This claim 2 breaks down completely when one has a strongly curved space, which is what we would expect in the first instant of less than Planck time evolution of the nucleation of a new universe.

Claim 3: Equation (74) has a first potential which tends to be for a di quark nucleation procedure which just before a defined Planck’s time^{120} time greater. i.e.

Which 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

Proof of Claim 4: This comes as a result of temperature changes after the initiation of inflation and changes in value of [

After this, we need to discuss how this thermal input into the axion wall occurs, leading to these results.

Abbay Ashtekar’s quantum bounce results as given in [

As well as energy term

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

This has

And

Ashtekar [^{P}, but will have a discrete geometry. This may permit an early universe “quantum bounce” and an outline of an earlier universe collapsing, and then being recycled to match present day inflationary expansion parameters. The main idea behind the quantum theory of a (big) quantum bounce is that, as density approaches infinity, so the behavior of the quantum foam changes. The foam is a qualitative description of the turbulence that the phenomenon creates at extremely small distances of the order of the Planck length. Here ^{P} structure as implied below. This will entail either confirming or falsifying the structure given to

This is for a crucial “momentum” value [

And

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

With each

The existence of gravitons in itself would be able to either confirm or falsify the existence of non L^{P} structure in the early universe. This structure was seen as crucial to Ashtekar, A, Pawlowski, T. and Singh, in their arXIV article [

With the advent of this re definition of momentum we are seeing what Ashtekar [

This in itself would permit confirmation of if or not a quantum bounce condition existed in early universe geometry, according to what Ashtekar’s two articles predict [

This creates problems, so we look for other ways to get what we want. Grushchuk writes that the energy density of relic gravitons is expressible as [

where the subscripts i and f refer to initial and final states of the scale factor, and Hubble parameter. This expression though is meaningless in situations when we do not have enough data to define either the scale factor, or Hubble parameter at the onset of inflation. How can we tie in with the Gaussian wave functional

An appropriate value for a Gaussian representation of an instanton awaits more detailed study. But for whatever it is worth we can refer to the known spaleraton value for a multi dimensional instanton via the following procedure. We wish to have a finite time for the emergence of this instanton from a pre inflation state.

If we have this, we are well on our way toward fixing a range of values for

in order to get a value for

If so, then, most likely, the question we need to ask though is the temperature of the “pre inflationary” universe and its link to graviton production. This will be because the relic graviton production would be occurring before the nucleation of a scalar field. We claim, as beforehand that this temperature would be initially quite low, but then rising to a value at or near 10^{12} degrees Kelvin after the dissolving of the axion wall contribution given in the dominant value of Equation (21) leading to Equation (22) for a chaotic inflationary potential. And now we shall consider why we need to look at relic graviton production, anyway.

To briefly review what we can say now about standard graviton detection schemes, as mentioned above, Rothman wrote that Dyson doubts we will be able to detect gravitons via present detector technology [

Here, at best, we usually can set

This in part is why we are looking at relic graviton production for early universe models, usually detectable via the criteria developed for white dwarf stars of one graviton for

Furthermore,

We should state that we will generally be referring to a cross section which is frequently the size of the square of Planck ’s length

Note that we obtainnn upper bound to the cross section

M-Planck scale in 4 + n dimensions

iverse extra dimension “square” volume ≈ 10 - 15 mm per side. So, having this limitation, Chongqing University is looking at objects like Neutron stars, as Graviton detection sites, and the like. It is a serious problem, and one which mandates serious choices of GR/ Graviton detection, to be thoroughly vetted by appropriate astrophysical models.

Note that if the Rothman analysis [^{12} Kelvin as pre Big Bang temperature up to Plank temperatures for observing a graviton burst, i.e. the supposition of Caroll and Chen [

We can point to the following as tentative successes of our model which need further elaboration.

Gravitons would appear to be produced in great number in the

A Randall-Sundrum effective potential, as outlined herein, would give a structure for embedding an earlier axion potential, which would be a primary candidate for an initial configuration of dark energy. This structure would, by baryogenesis, be a shift to dark energy. We need to determine if the sets of JDEM space observations have data which could be configured to determine if WIMPS are in any way tied into the supposed dark energy released after a

In doing this, we should note the following. We have reference multiple reasons for an initial burst of graviton activity, i.e. if we wish to answer Freeman Dyson’s question about the existence of gravitons in a relic graviton stand point [

We have already found it necessary to avoid the methodology of [

A lot of work can and should be done to update the power law for graviton production along the lines of an update to the graviton power spectra expression via taking into account a per solid angle expression.

As I was asked about earlier, this does have a directional component which was given by Weinberg in 1972 [

(102)

where we can write

Getting realistic values of

Also, the issues raised in [

Doing all of this will enable us, once we understand early universe conditions, to be able to initiate de facto engineering work pertinent to power source engineering. We will initiate engineering work so as to allow this concept to become the basis of new space craft technology. Also in order to do this engineering work properly we need to understand the issues raised in [

As to reference [

This work is supported in part by National Nature Science Foundation of China grant No. 11375279. I also wish to thank Dr. Christian Corda and Dr. Fangyu Li for motivation, as well as Dr. Xi Yang of Brookhaven laboratory who in the early 2000s helped clean my thinking for my PhD defense as to the intricacies of False vacuum nucleation used in my PhD defense. That methodology has been present for years afterwards.

Andrew Walcott Beckwith, (2016) Can Thermal Input from a Prior Universe Account for Relic Graviton Production and Imply Usage of the Bogomolnyi Inequality, as a Bridge between Brane World Models and Loop Quantum Gravity in Early Universe Conditions?. Journal of High Energy Physics, Gravitation and Cosmology,02,412-431. doi: 10.4236/jhepgc.2016.23036