^{*}

If a non-zero graviton mass exists, the question arises if a release of gravitons, possibly as a “Graviton gas” at the onset of inflation could be an initial vacuum state. Pros and cons to this idea are raised, in part based upon Bose gases. The analysis starts with Volovik’s condensed matter treatment of GR, and ends with consequences, which the author sees, if the supposition is true.

Volovik’s [

The authors’ beliefs as to if this hypothesis can be tested will be the final part of the manuscript.

Volovik [

The integrand to be considered is, using a potential defined by

For the sake of argument, m, as given above will be called the mass of a graviton, n a numerical count of gravitons in a small region of space, and afterwards, adaptations as to what this expression means in terms of entropy generation will be subsequently raised. A simple graph of the 2^{nd} term of Equation (1.2) with comparatively large m and with

If we view this as having an indication of when the deviation from usual quantum linearity, the implication is that right at the start of the production of n “gravitons” that there is a cut off right at the start of graviton production, i.e. the implications for ‘tHooft’s [

To quantify this, it would be to have

An interesting datum to bring up for evaluation. ‘tHooft [

There exist a “regularization term” we identify with regularization term

Furthermore, if we take density of this initial state, as given by

Go to Appendix A as far as a description as to how and why

mass of a graviton is nearly zero, and if the term

albeit in nearly a nearly non-existent fashion, for tiny graviton mass, then the existence of this second term is in sync with ‘tHooft’s deterministic quantum mechanics. Volovik calls the 2^{nd} term a “regularization term”, and its importance can be seen as a way to quantify the affects of an embedding of initial quantum information within a larger structure, which is highly non linear. Doing so would help us determine if

As used by Ng [

This, according to Ng [

Eventually, the author hopes to put on a sound foundation what ‘tHooft [

lence classes embedding quantum particle structures. Our supposition is that the sample space,

As written up by Buoanno [

As an example we consider a first order phase transition in the early universe. This can lead to a period of turbulent motion in the broken phase fluid, giving rise to a GW signal. Using the results from Durrer [

“If turbulence is generated in the early universe during a first order phase transition, as discussed in the introduction, one has the formation of a cascade of eddies. The largest ones have a period comparable to the time duration of the turbulence itself (of the phase transition).According to Equation (16), these eddies generate GWs which inherit their wavenumber. Smaller eddies instead have much higher frequencies, and one might at first think that they imprint their frequency on the GW spectrum. However, since they are generated by a cascade from the larger eddies, they are correlated and cannot be considered as individual sources of GWs.” We have serious doubts about that last sentence.

Also brought up are GWs produced by the neutrino anisotropic stresses, which generate a turbulent phase. These would be weaker than E and M contributions to anisotropic stresses. For the record as stated in Kojima’s [

Another more familiar example of extra anisotropic stress is that of a primordial magnetic field (PMF). The amplitude of the energy density

Wei-Tou Ni [

i.e. Equation (1.9), and the primary difficulty is in accommodating ^{10}. Before saying this, we need to consider the role degrees of freedom,

Secondly, we look for a way to link initial energy states, which may be pertinent to entropy, in a way which permits an increase in entropy from about

We assert here, that Equation (1.10) is the same order of magnitude as Equation (1.4). To get this, we also look at how to get a suitable ^{ }

A linkage between energy and entropy, as seen in the construction, looking at what Kolb [

Here, the idea would be, to make the following equivalence, namely look at,

Note that in the case that quantum effects become highly significant, that the contribution as given by ^{−105} m^{3} = 4.22419 × 10^{−96} cm^{3} leads us to conclude that even with very high temperatures, as an input into the initial entropy, that

De La Vega [

where we will put in a candidate for the

This will lead to comparatively low values for

And, also,

An initially

By conventional cosmological theory, limits of

for peak frequencies with values of 10 MHz. The net affect of such thinking is to proclaim that all relic GW are inaccessible. If one looks at Figure, ^{6} Hertz, this counters what was declared by Turner and Wilzenk [

And this leads to the question of how to account for a possible mass/informa- tion content to the graviton.

It gets worse if one is asserting that there is, in any case, a quark gluon route to determine the role of entropy. To begin this analysis, let us look at what goes wrong in models of the early universe. The assertion made is that this is due to the quark―Gluon model of plasmas having major “counting algorithm” breaks with non counting algorithm conditions, i.e. when plasma physics conditions BEFORE the advent of the Quark gluon plasma existed. Here are some questions which need to be asked.

1) Is QGP strongly coupled or not? Note: Strong coupling is a natural explanation for the small (viscosity) Analogy to the RHIC: J/y survives DE confinement phase transition

2) What is the nature of viscosity in the early universe? What is the standard story? (Hint: AdS-CFT correspondence models). Question 2 comes up since

typically holds for liquid helium and most bosonic matter. However, this relation breaks down. At the beginning of the big bang. As follows i.e. if Gauss- Bonnet gravity is assumed, in order to still keep causality, one needs

This even if one writes for a viscosity over entropy ratio the following

A careful researcher may ask why this is so important. If a causal discontinuity as indicated means the

Then, more collisions due to WHAT physical process? Recall the argument put up earlier, i.e. the reference to causal discontinuity in four dimensions, and a restriction of information flow to a fifth dimension at the onset of the big bang/ transition from a prior universe? That process of a collision increase may be inherent in the restriction to a fifth dimension, just before the big bang singularity, in four dimensions, of information flow. In fact, it very well be true, that initially, during the process of restriction to a 5^{th} dimension, right before the big bang,

that

density may, partly due to restriction in geometric “sizing” may become effectively nearly infinite. It is due to the following qualifications put in about Quark ? Gluon plasmas which will be put up, here. Namely, more collisions imply less viscosity. More Deflections ALSO implies less viscosity. Finally, the more momentum transport is prevented, the less the viscosity value becomes. Say that a physics researcher is looking at viscosity due to turbulent fields. Also, perturbative calculated viscosities: due to collisions. This has been known as Anomalous Viscosity in plasma physics, (this is going nowhere, from pre-big bang to big bang cosmology). Appendix B gives some more details as far as the

So happens that RHIC models for viscosity assume

As Akazawa [

If the temperature T wildly varies, as it does at the onset of the big bang, this breaks down completely. This development is FRANKLY Mission impossible: AND why we need a different argument for entropy, i.e. Even for the RHIC, and in computational models of the viscosity for closed geometries―what goes wrong in computational models

・ Viscous Stress is NOT µ shear

・ Nonlinear response: impossible to obtain on lattice ( computationally speaking)

・ Bottom line: we DO NOT have a way to even define SHEAR in the vicinity of big bang!!!!

i.e. the quark gluon stage of production of entropy, and its connections to early universe conditions may lead to undefined conditions which, i.e. like shear in the beginning of the universe, cannot be explained. i.e. what does viscosity mean in the neighborhood of time where

V.A. Rubakov and, P.G. Tinyakov [

If we use LISA values for the Pulsar Gravitational wave frequencies, this may mean that the massive graviton is ruled out. On the other hand

If the radius is of the order of

This Equation (1.28) is in units where

If

Then, exist

If each photon, as stated above is

Furthermore, if there are, today for a back ground CMBR temperature of 2.7 degrees Kelvin

As a rough rule of thumb, if, as given by Weinberg [

We assert that at a minimum, we can write, the following. Namely that to begin a reasonable inquiry, that

If one has that^{6} Hertz. Note here that

The author is fully aware of how Durrer [

The authors supposition is, in line with what has been presented in the above, that graviton production and early universe entropy production of the order of

firming if

perature ^{32} Kelvin, and the issue would be, if this is true, of giving sufficient reasons for having a scaling argument from initial condition, as specified, of confirming if an analytical proof, backed up by measurements confirms

or 10^{−47} GeV^{4}, or 10^{−29} g/cm^{3} or about 10^{−120} in reduced Planck units.

I.e. what value of ^{−120} today?

If falsifiable experimental measurements for Equation (1.34) may be obtained, the next step would be perhaps in confirming what degree of information exchange such a scaling may imply. The information exchange from a prior to a present universe would be modeled on the template of what

Note that Corda [

We at the close refer the readers to Appendix C for crucial considerations as to the emergence of gravitational astronomy as this relates to a summary as to how to confirm the models so referenced in this paper, as to work by Corda, and the LIGO GW team which is of potentially revolutionary import as far as observational astronomy confirming these ideas so presented.

The author thanks Dr. Raymond Weiss, of MIT as of his interaction in explaining Advanced LIGO technology for the detection of GW for frequencies beyond 1000 Hertz and technology issues with the author in ADM 50, November 7^{th} 2009. Dr. Fangyu Li, of Chongqing University is thanked for lending his personal notes to give substance to the content of page 10 of this document.

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

Beckwith, A.W. (2017) Analyzing If a Graviton Gas Acts Like a Cosmological Vacuum State and “Cosmological” Constant Parameter. Journal of High Energy Physics, Gravitation and Cosmology, 3, 388-413. https://doi.org/10.4236/jhepgc.2017.32032

A1: Linkage of DM to gravitons and gravitational waves?

Let us state that the object of early universe GW astronomy would be to begin with confirmation of whether or not relic GW were obtainable, and then from there to ascertain is there is linkage which can be made to DM production ... Durrer, Massimiliano Rinaldi [

Which to first order when

This will be contrasted with a very similar evolution equation for gravitons, of (i.e. KK gravitons in higher dimensions)

One of the models of linkage between gravitons, and DM is the KK graviton, i.e. as a DM candidate. KK gravitons. Note that usual Randal Sundrum brane theory has a production rate of _{*} > M_{X}, and that we are assuming that the temperature

where R is the assumed higher dimension ‘size’ and, d is the number of dimensions above 4, and typically we obtain

If KK gravitons have the same wavelength as DM, this will support Jack Ng’s treatment of DM. All that needs to put this on firmer ground will be to make a de facto linkage of KK Gravitons, as a DM candidate, and more traditional treatments of gravitons, which would assume a steady drop in temperature from

this value of R being unmanageable for d < 2. V.A. Rubakov [

As well as being related to an overall wave functional which can be derived from a line element

With

tion equation for the KK gravitons is very similar to work done by Baumann, Daniel, Ichiki, Kiyotomo, Steinhardt, Paul J. Takahashi, Keitaro [

Ruth Gregory, Valery A. Ruvakov and Sergei M. Sibiryakov [

This is similar to what Baumann, Ichiki, Steinhardt, and Takahashi [

dS branes is similar to evolution of GW in more standard cosmology that the author, Beckwith, thinks that the main challenge in clarifying this picture will be in defining the relationship of dS geometry, in overall Randall Sundrum brane world to that of standard 4 space,. We need though, now to look at whether or not higher dimensions are even relevant to GR itself.

A2: How DM would be influenced by gravitons, in 4 dimensions

We will also discuss the inter relationship of structure of DM, with challenges to Gaussianity. The formula as given by

Will be gone into. The variation, so alluded to which we will link to a statement about the relative contribution of Gaussianity, via looking at the gravitational potential

Here the expression

It is note worthy to note that the question of DM/KK gravitons, and also the mass of the graviton not only has relevance to whether or not, higher dimensions are necessary/advisable in space time models, but also may be relevant to if massive gravitons may solve/partly fulfill the DE puzzle. To whit, \KK gravitons would have a combined sum of Bessel equations as a wave functional representation. In fact V. A Rubakov [

after using the following normalization

where

This allegedly is for KK gravitons having an order of TeV magnitude mass

negative tension brane with

mass of the KK graviton, moving at high speed, with the initial rest mass of the graviton, which in four space in a rest mass configuration would have a mass many times lower in value, i.e. of at least

The following section is to improve upon the range of GW detected, as can be presented below. We use

The relation between

The curve of the pre-big-bang models shows that ^{10} Hz. ^{−8} in the region 1 Hz to 10^{10} Hz; its peak value region is about 10^{−7} - 10^{−6} Hz. The reason for this section is to deal with the statement made by Buoanno [

Here, Buoanno [

The problem in this is that the ratio

By conventional cosmological theory, limits of

Quote:

“The most serious is that a background strain ^{−}^{5} otherwise the cosmological Helium/hydrogen abundance in the universe would be strongly affected ...”

The answer, which the author copied from Dr. Li, i.e., If

The following is Dr. Fangyu Li’s argument as given to the author in personal notes:

1) LIGO and our coupling electromagnetic system [

2) The minimal detectable amplitude of LIGO depends on [

where L is the interferometer length. Because detecting band of LIGO is limited in ~1 Hz - 1000 Hz, this is a very strong constraint for h_{min}. Thus, h_{min} of LIGO is about ~10^{−23} - 10^{−24} in this band.

3) The minimal detectable amplitude of cavity depends on arguments similar to the ones brought up in reference [

For the constant-amplitude HFGWs, and

for the stochastic relic HFGWs and considerations which were given to the author in a discussion he had with Dr. Weiss of MIT [

Because Q factor of superconducting cavity in the low-temperature condition can reach up to ~10^{10} - 10^{12}, if we assume Q = 10^{11}, ^{3}, then

for the constant-amplitude HFGW.

and

for the stochastic relic HFGW.

4) The CEMS [

The minimal detectable amplitude h depends on the relative standard quantum limit (SQL) (G.V. Stephenson 2008, 2009), [

for the stochastic relic HFGW, ^{31}, then

If we use fractal membranes, even if a conservative estimation, we have

Equation (B.11) is similar to Equation (B.6) and Equation (B.7). An important difference is that

5) LIGO and our scheme have quite different detecting mechanisms (the displacement effect and the EM parameter perturbation effect) and detecting bands (~1 Hz - 1000 Hz and 1 GHz ~ 10 GHz), their comparison should not be only the amplitude of GWs, but also the energy flux of GWs. In fact, the energy flux of any weak GW is proportional to^{−30}, ^{−22}, ^{−30}, (or better)

The SQL is a basic limitation. Any useful means and advanced models might give better sensitivity, but there is no change of order of magnitude in the SQL range. For example, if we use squeezed quantum states for a concrete detector, then the sensitivity would be improved 2 - 3 times than when the squeezed quantum state is absent in the detector, but it cannot improve one order of magnitude or more According to more accepted by the general astrophysics community values as told to the author by Dr. Weiss [

so

where

(a)

If

If

(b)

^{−31}, then

If

Such values of

Furthermore, the numerical summed up value of binned

Equation (1.23) is for a very narrow range of frequencies, that to first approximation, make a linkage between an integral representation of

and

Thus an obvious gap still exists between the theoretical estimation and detecting reality, but there are large rooms to advance and improve the CEMS. These are upper values of the spectrum, and should be considered as preliminary. Needed in this mix of calculations would be a way to ascertain a set of input values for

Here, Buoanno [

The problem in this is that the ratio

By conventional cosmological theory, limits of ^{6} Hertz, this counters what was declared by Turner and Wilzenk [

Abbot et al., in [

Submit or recommend next manuscript to SCIRP and we will provide best service for you:

Accepting pre-submission inquiries through Email, Facebook, LinkedIn, Twitter, etc.

A wide selection of journals (inclusive of 9 subjects, more than 200 journals)

Providing 24-hour high-quality service

User-friendly online submission system

Fair and swift peer-review system

Efficient typesetting and proofreading procedure

Display of the result of downloads and visits, as well as the number of cited articles

Maximum dissemination of your research work

Submit your manuscript at: http://papersubmission.scirp.org/

Or contact jhepgc@scirp.org