Analyzing if a graviton gas acts like a cosmological vacuum state and ‘ cosmological ’ constant parameter

1.Introduction: Volovik’s [1] book as of 2003 has a chapter on how a Bose gas can be used to obtain a vacuum energy. We extrapolate from this idea, and link it to what was done by Glinka [2], as to Wheeler De Witt (WdW) treatment of semi classical style physics in his boson treatment of a ‘graviton gas’ in order to make a similar analogy to what is done by Park [3], namely his so called version of a temperature sensitive cosmological constant parameter. Then, afterwards, links of how entropy may be connected with an evolution of the resulting cosmological vacuum energy expression, for a graviton gas are explored.

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 Eq. E1 0 ≠ when n is very small, and E1 = 0 as 10 10 → n at the onset of inflation.
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 ' tHoofts 4 non linearity embedding of quantum systems for gravitons would be in that the conditions for non linear embedding are likely in place as a pre cursor to graviton production. What we are observing is right at the start of the production of gravitons, i.e. the moment emergence of graviton states occurs, we have extinguishment of a contribution of classical embedding, but the pre cursor to that would mean graviton production would be initially 'framed' by a non linear contribution. To quantify this, it would be to have ( ) an additional, ' tHooft 4 style embedding of a usual Q.M. treatment of a spin two particle. In what is stated later about emergence, the author claims that, in analogy to CDW, with emergence of CDW particles, that if there is emergence, that  ( 1 n E would be equivalent to the degree of 'slope' of a emergent 'instanton' and/ or instanton-anti instanton structure, which is written in CDW as S-S ' The statement as to emergence, if it occurs is, in both ) ( 0 x φ moving from the 'floor' of figure 3, as it rises above, is in sync with moving toward the 'thin wall approximation' of minimization of classical contributions to the emergence state φ . I.e. if Figure   3 were a rectangular block moving upward, with no contributions other than the block itself moving 'upward' it would represent a pure 'QM' contribution to emergence. Deviations from this block shape represent a non linear semi classical embedding state, with different, continuum of ) ( 0 x φ being continuum states and part of 't Hooft 4 equivalence classes as seen in the CDW wave function below 5 There exist a 'regularization term' we identify with regularization term' which will be seen in Eq. (1.5) below, and which has a functional dependence in a fashion which will be derived in the future as plays a role, albeit in nearly a nearly non existent fashion, for tiny graviton mass, then the existence of this second term is in sync with ' tHoofts 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 * f f~ with * f an initial frequency which can be picked up in GW / Graviton detectors. We shall now consider how to model emergent structure as given in Fig 1, Fig 2,

Review of Y. J. Ng's entropy hypothesis
As used by Ng 6 This, according to Ng,6 Eventually, the author hopes to put on a sound foundation what 'tHooft 4 is doing with respect to. ' tHooft 4 deterministic quantum mechanics and equivalence classes embedding quantum particle structures. Our supposition is that the sample space, V is extraordinarily small, putting an emphasis upon λ being quite small, leading to high frequency behavior for the resulting generated N. For extremely small valumes for nucleation of a particle, in initial space, this leads to looking at an inter relationship between a term for initial entropy, of the order of 10 10 , and if the following expression for detectable frequency , with * f = initial frequency ~ 1/ λ , * a an initial scale factor, and 0 a today's scale factor behavior, as given by As written up by Buoanno 7 , even if initial frequencies are enormous, the present day frequencies should be , tops of the order of 100 Hz for initial gravitational waves. I.e. the factor , [ ] 0 a a * would be almost non existent. On the other hand, if the embedding structure containing the initial vacuum energy formation has an initially undisturbed character, with minimum breakage of an instanton formation of composite particles, then the frequency would be, instead closer to * f f~ with * f an initial frequency ~ 1/ λ .We Conditions to test for experimentally to determine if * f f~ exist in the present era.
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 8 "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 Eq. (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 9 article Another more familiar example of extra anisotropic stress is that of a primordial magnetic field (PMF).
The amplitude of the energy density ∝ a .We doubt that such anisotropic stress would be pertinent to HFGW production. Our supposition is that relic graviton production , not just eddies, as speculated by Durrer also play a role as far as detection, Durrer's 8 write up exclusively focuses upon eddies, and turbulence in initial GW production.
Wei-Tou Ni 10 in has a very direct statement that DECIGO [11] and Big Bang Observer [12] look for GWs in the higher frequency range, which may give * f f~ measurements, especially if * f is not low frequency. Ni also writes, for stochastic backgrounds, that "The minimum detectable intensity of a stochastic GW background" h0 ΩGW min(f) ~ const.  11 . Having said that, then the issue is , are relic conditions for gravitons and GW are linked to entropy, and an initial entropy values of ~ 10 10 . Before saying this, we need to consider the role degrees of freedom, * g is in the initial phases of inflation. * g is in the initial phases of inflation.

Difficulty in visualizing what
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 10 10 at the start of the big bang to about 90 10 to 100 10 today.
One such way to conflate entropy with an initial cosmological constant may be of some help, i.e. if

( )
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 By conventional cosmological theory, limits of * g are at the upper limit of 100-120, at most, according to Kolb and Turner 13 (1991).   15 (1990): that inflation will terminate with observable frequencies in the range of 100 or so Hertz. The problem is though, that after several years of LIGO, no one has observed such a GW signal from the early universe, from black holes, or any other source, yet. About the only way one may be able to observe a signal for GW and/or gravitons may be to consider how to obtain a numerical count of gravitons and/or neutrinos for (1.21) . And this leads to the question of how to account for a possible mass/ information content to the graviton.

Break down of Quark -Gluon models for generation of entropy
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 deconfinement 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 , or less in value, it puts major restrictions upon viscosity, as well as entropy. A drop in viscosity, which can lead to major deviations from π 4 1 in typical models may be due to more collisions.
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 . Either the viscosity drops nearly to zero, or else the entropy 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, perturbatively 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 We assert that at a minimum, we can write, the following. Namely that to begin a reasonable inquiry, that , the above effect is to put restrictions upon stochastic treatments of S n (f) for frequencies at or above 10 6 Hertz. Note here that S n (f) spectral density is , in some cases allowing for substitution of the spectral density function via the sort of arguments given in Appendix B below.

Conclusion. A graviton gas inevitably has semi classical features. Cosmological constant parameter initially may be accounted for via graviton release initially?
The author is fully aware of how Durrer 8 and others use turbulence in early universe conditions, as a way to , at the time of the electro weak transition to account for relic graviton production . The electro weak transition, as noted by Rubakov 21 , and others 22 is a candidate for computing the gravity waves induced by anisotropic stresses of stochastic primordial magnetic fields. I.e. a specified magnetic field in the onset of early universe conditions. The author suggests that earlier generation, requiring increased sensitivity of GW detectors, perhaps of 25 24 10 10 − − h may be necessary as to be able to reach higher frequency GW created by graviton production at the onset of inflation. Note that L. Grishchuk 23 , in 2007 specified relic GW production as up to 10 GHz which is far in excess of the values Durrer and others 22 propose. Indeed, Durrer, Marozzi, and Rinaldi 24 are convinced that any relic conditions for GW much be much lower, with no relic GW observable as they specify it on alleged practical grounds. If one is unable to obtain detector sensitivities of the order of 25 24 10 10 − − h in the foreseeable future, Durrer, Marozzi, and Rinaldi 24 may be right by default. It is worth noting though that physics should be considering if relic GW occur at all, and the author , and L. Grishchuk 23 have presented mechanisms which may account for their existence in regions of space time evolution well before the electro weak transition, and not necessarily due to conditions linked to anisotropic stress of magnetic fields.
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 10 10 S in initial Planck time and of what dimensional embedding is needed to do so. Furthermore, what is obtained should be reconciled with an additional constraint which will be put in the next page.
Note that Corda 25 has modeled adiabatically-amplified zero-point fluctuations processes in order to show how the standard inflationary scenario for the early universe can provide a distinctive spectrum of relic gravitational waves. De Laurentis, and Capozziello 26 (2009) have further extended this idea to give a qualified estimate of GW from relic conditions which will be re produced here. Begin with De Laurentis's idea of a gravitational wave spectrum 0 2 / 1 2 ) 1 ( 1 9 16 H is today's Hubble parameter, while f is GW frequency, and eq z is the red shift value of when the universe became matter dominated. I.e. red shift z = 1.55 with an estimated age of 3.5 Giga year, or larger, would be a good starting point. I.e. this is for larger than 3. 100 eq z , which should be investigated.

Appendix A : Looking at situations when the mass of a graviton is not zero 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 24 (2009) , state that there would be probably negligible for this case ( practically non existent ) graviton production in cosmological eras after the big bang.. In fact, they state that they investigate the creation of massless particles in a Universe which transits from a radiation-dominated era to any other (via an) expansion law . "We calculate in detail the generation of gravitons during the transition to a matter dominated era. We show that the resulting gravitons generated in the standard radiation/matter transition are negligible" This indicated to the author, Beckwith, that it is appropriate to look at the onset of relic GW/ Graviton production.. One of the way to delineating the evolution of GW is the super adiabatic approximation, done for when 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 number of Kaluza Klein gravitons per unit time per unit volume Note this production rate is for a formula assuming mass for which T * > M X , and that we are assuming that the temperature * T T~ . Furthermore, we also are looking at total production rate of KK gravitons of the form Where R is the assumed higher dimension 'size' and , d is the number of dimensions above 4, and typically we obtain T >>1/R. I.e. we can typically assume tiny higher dimensional 'dimensions', very high temperatures, and also a wave length for the resulting KK graviton for a DM candidate looking like 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 for massive graviton evolution as KK gravitons along 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 (A.10) 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 (A.11) Here the expression = NL f variations from Gaussianity, while the statements as to what contributes, or does not contribute will be stated in our presentation. Furthermore, is a linear Gaussian potential, and the over all gravitational potential is altered by inputs from the term, presented, NL f . The author discussed inputs into variations from Gaussianity, which were admittedly done from a highly theoretical perspective with Sabino Matarre 30   The following section is to improve upon the range of GW detected, as can be presented below.
The curve of the pre-big-bang models shows that g Ω of the relic GWs is almost constant 6 6.9 10 − × from 10 Hz to 10 10 Hz. g Ω of the cosmic string models is about 10 -8 in the region 1Hz 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 7  By conventional cosmological theory, limits of * g as given by Kolb and Turner 13 (1991) are at the upper limit of 100-120. In addition according to Kolb and Turner 13 (1991) . , which is what Dr. Fangyu Li 38 disputes: The following notes are also in response to a referee quote which Fangyu answered the following query, which is re produced Quote:

GeV T
"The most serious is that a background strain 30 10 − h at 10GHz corresponds to a g Ω (total) 3 10 − which violates the baryon nuclei-synthesis epoch limit for either GWs or EMWs. g Ω (Total) needs to be smaller than 10 -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 39,40   correspond to the GWs of the same energy flux density. This means that the EM detection schemes with the sensitivity of h=10 -30 , (or better) g ν 1GHz-10GHz in the future should not be surprise .
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 astro physics community values as told to the author by Dr. Weiss 41 , the estimate, for the upper limit of g Ω F on relic GWs should be smaller than 5 10 − , while recent data analysis (B.P. GHz v such a GW signal from the early universe, from black holes, or any other source, yet. About the only way