1Papers on LCQ at the 12th Marcell Grossman Meeting in 2009 (http://www.icra.it/MG/mg12/en/)
Recently, a big bounce has been proposed1 as an alternative to singularity conditions that Hawking’s, Ellis  , and others use. The 1st problem is that there appears to exist no fundamental argument presented in either traditional Friedman metric GR or LQG for preservation of the same value for Planck’s constant or the fine structure constant from prior universes (before ours) and the present universe. Ashtekar  , in conversations with the author in the inaugural opening of the Penn State gravity center (2007) told the author that the universe preserves most of its “memory” in cosmological cycles, but the proof of this assertion does not show up in Rovelli’s  reference on Quantum Gravity. The driving force for this present investigation is due to a conversation the author had with Steinhardt and ‘tHooft at the meeting “Fundamental Frontiers of Physics” in a parallel session about LQG, and new developments in it.
2. What Are Necessary First Principles to Consider in Graviton/GW Detection?
Modeling how much information may be carried by an individual graviton can be achieved by measuring the graviton via instrumentation. Normalized energy density of gravitational waves, as given by Maggiore  is
where is a frequency-based count of gravitons per unit cell of phase space?
Is Equation (1.1) above fundamental physics? And what is the significance of the and terms with regards to if gravitons could have been cycled from a prior to the present universe? The rest of the document will attempt to answer the question of what ultra high frequency inputs into the as well as term are relevant to, assuming that the quantum bounce model of a “recycled” universe is in part, correct.
2.1. Estimating the Size of Contribution to Energy in , Assuming a Frequency Hertz for Relic Gravitons, If the Standard Chemical Potential Is Effectively at the Onset of Creation
As suggested earlier by Beckwith  , gravitons may have contributed to the re-acceleration of the universe one billion years ago. When q becomes negative, the rate of acceleration of the universe is actually increasing, rather than slowing down   . The suggestion Beckwith made for implementing re acceleration involves correct use of the de celebration parameter, and also looking at the behavior of gravitons. The use of Equation (1.2) below to have re acceleration in the application Beckwith made is dependent upon “heavy gravity” and the rest mass of gravitons in four dimensions having a small mass term.
We wish next to consider what happens not a billion years ago, but at the onset of creation itself. If a correct understanding of initial graviton conditions is presented, it may add more credence to the idea of a small graviton mass, in a rest frame, which may give backing, in part, to Beckwith’s use of Equation (1.2) for re acceleration of the universe, in a manner usually associated with Dark Energy. Here, we are making use of refining the following estimates. In what follows, we will have even stricter bounds upon the energy value (as well as the mass) of the graviton based upon the geometry of the quantum bounce, with a radii of the quantum bounce on the order of meters   .
For looking at the onset of creation, with a bounce; if we look at for the quantum bounce with a value put in for when , where
Then, taking note of this, one is obtaining having a scaled entropy of when one has an initial Planck temperature . One needs, then to consider, if the energy per given graviton is, if a frequency and , then
Having said that, the is 1022 greater than the rest mass energy of a graviton if grams is taken when applied to Equation (1.2) above.
2.2. Making Sense of the Factor of 1038 in Equation (1.5). i.e. How to Reconcile Equation (1.5) with Used by Y. Jack Ng for DM Particles in His Entropy/Particle Counting Algorithm?
Note that J. Y. Ng uses the following.  i.e. for DM, , but this is for DM particles, presumably of the order of mass of a WIMP, i.e. , as opposed to a relic graviton
If one drops the effective energy contribution to , as has been suggested, then the relic graviton mass-energy relationship is:
Finally, if one is looking at the mass of a graviton a billion years ago, with
i.e., if one is looking at the mass of a graviton, in terms of its possible value as of a billion years ago, one gets the factor of needing to multiply by 1038 in order to obtain WIMP level energy-mass values, congruent with Y. Jack Ng’s counting algorithm, i.e., the equivalence relationship for entropy and “particle count” may work out well for the WIMP sized DM candidates, and may break down for the graviton mass-energy problem.
2.3. The Electro Weak Generation Regime of Space Time for Entropy and Early Universe Graviton Production before Eletro Weak Transitions
A typical value and relationship between an inflation potential , and a Hubble parameter value, H is
Also, if we look at the temperature occurring about the time of the Electro weak transition, if when was a critical value, (of which we can write , where denotes the Higgs vacuum expectation value at the critical temperature Tc., i.e. according to C. Balazc et al. (2005)  and denotes that the electro weak transition was a “strongly first order phase transition”) then one can write, by conventional theory that
Here, the factor put in, of is the number of degrees of freedom. Kolb and Turner  put a ceiling of about in the early universe as of about the electro weak transition. If, however, or higher for earlier than that, i.e. up to the onset of inflation for temperatures up to , it may be a way to write, if we also state that that if
Should the degrees of freedom hold, for temperatures much greater than , and with at the onset of inflation, for temperatures, rising up to, say T-1019 GeV, from initially a very low level, pre inflation, then this may be enough to explain how and why certain particle may arise in a nucleated state, without necessarily being transferred from a prior to a present universe.
I.e. the suggestion being presented is that a more standard thermodynamic dependence of entropy upon temperature, i.e. for values of degrees of freedom may be envisioned if one has when or even higher even if is envisioned, in place of if , and assuming that , i.e. that an upper limit of in degrees of freedom is all that is permitted.
Furthermore, if one assumes that  when or even higher even if , then there is the possibility that when could also hold, if there was in pre inflationary states very LOW initial temperatures, which rapidly built up in an interval of time, as could be given by seconds which has the following “vacuum nucleation” interpretation which will be given below, as exemplified by the example of a harmonic system having, in a time interval an infusion of energy, into what is otherwise a typical harmonic oscillatory system. Observe the following argument as given by V. F. Mukhanov, and Swinitzki  as to additional particles being “created” due to an infusion of energy in an oscillator, obeying the following equations of motion
Given , with a starting solution of if , Mukhanov state that for ;
The Mukhanov et al. argument leads to an exercise which Mukhanov et.al.  claims is solutions to the exercise yields an increase in number count, as can be given by first setting the oscillator in the ground state with , with the number of particles linked to amplitude by , leading to
I.e. for non zero , Equation (1.14) leads to exponential expansion of the numerical state. For sufficiently large , Equation (1.12) and Equation (1.14) are equivalent to placing of energy into a system, leading to vacuum nucleation. A further step in this direction is given by Mukhanov  on page 82 of his book leading to a Bogoluybov particle number density of becoming exponentially large
Equation (1.12) to Equation (1.14) are, for sufficiently large a way to quantify what happens if initial thermal energy are placed in a harmonic system, leading to vacuum particle “creation” Equation (1.15) is the formal Bogolyubov coefficient limit of particle creation. Note that , for corresponds to a thermal flux of energy into a time interval . If or some multiple of and if , then Equation (1.12), and Equation (1.14) plus its generalization as given in Equation (1.15) may be a way to imply either vacuum nucleation, or transport of gravitons from a prior to the present universe. Having said that, the problem of Heavy Gravity raises its ugly head in the following field theory example.
3. Massive Graviton Field Theories and the Limit
As given by M. Maggiore  , and further elaborated upon in  the massless equation of the graviton evolution equation takes the form
When , the above becomes
The mismatch between these two equations, when is largely due to as , which in turn is due to setting
. The mismatch between these two expressions is one of several reasons for exploring what happens for semi-classical models when , grams, noting that in QM, a spin 2 only two
has five degrees of freedom, whereas the gram case has helicity states. Note that string theory treats gravitons as “excitations” of a closed string, as given by Keifer  , with a term added to a space-time metric, , such that with a linkage to coherent states of gravitons. This is partly in relation to the Venziano  expression of
, where . Kieffer  gives a correction due to quantum gravity in page 179 of the order of If the mass,
, it will be hard to measure as an individual “particle.” But, if exists, as a macro effect one billion years ago, i.e. as a substitute for DE, it also would be potentially relevant toward information exchange between a prior to the present universe, provided that there was no cosmic singularity and that the LQG quantum bounce hypothesis has some validity., Note that the author has been informed by J. Dickau of research by  de Rham and Gabadadze which in the authors opinion clears up the problem of ghosts and heavy gravity (massive Gravitons). However, the issue of if a graviton could survive a quantum bounce in LQG  stands alone as a problem which the author believes has been removed from being impossible to entertain, to one which cannot be ruled out.
A way to obtain traces of information exchange, from prior to present universe cycles is finding a linkage between information and entropy. If such a parameterization can be found and analyzed, then Seth Lloyd’s  shorthand for entropy,
could be utilized as a way to represent information which can be transferred from a prior to the present universe. The question to ask, if Equation (1.18) does permit a linkage of gravitons as information carriers, and there can be a linkage of information, in terms of the appearance of gravitons in the time interval of, say either by vacuum nucleation of gravitons/information packets along the lines of Equation (1.12) and Equation (1.14) or by reconciling the counting algorithm questions put up in Section 2.2.
4.1. Further Research Questions for Investigative Inquiry and How to Link Our Inquiry to the Overall Geometry of the Universe
The problem of reconciling the existence of a graviton mass with quantum mechanics, in spin two particles usually having zero mass appears to be resolvable, and may imply a linkage between DE and DM  Furthermore, the radius of the universe problem, as presented by Roos  , will yield rich applications of the Friedmann equations used in this document, once there are falsifiable experimental criteria for determining both the Hubble Parameter on the basis of choices of Friedman equations, and , using variables chosen and described in this present paper. Both are pertinent to the problem of re-acceleration of the universe parameter set in Equation (1.2), 
Specifically, the author is convinced that analyzing Equation (1.19) will be tied in, with appropriate analysis of the following Figure 1.
The relation between and the spectrum is often expressed as written by Grishchuk,  , as
Figure 1. This figure from B. P. Abbott et al.  (2009) shows the relation between and frequency.
The importance of understanding the radius of the universe question, and making sense of Equation (1.19) lies in reconciling the conflicting estimates put in Section 2.2 above. If one can get an answer to reconciling the estimates put in Section 2.2, one has gone a long way toward answering, or laying the ground work to answering the question as to the classical nature of gravitons, or if they have a semi classical interpretation.
t’Hooft  , writes, that gravitons can attain mass by spontaneous local symmetry breaking. “The question is whether this can happen in a Lorentz-invariant way”. The author submits that when t’Hooft writes that “These arguments should not be regarded as opposed to, but rather complimentary to the AdS/CFT approach to solve QCD using superstring theory  , where the 3 + 1 dimensional theory is mapped onto a 5 dimensional AdS theory. There, the massless graviton in 5 dimensions is mapped onto a massive graviton in 4 dimensions”, that one is actually considering, as an example, mapping of higher dimensional contributions of gravitons before the electro weak transition, into the time space which may lead to the construction of newly nucleated graviton states, contributing to, in one sense or another to the different scenarios as given in Section 2.2 above. If a quantum bounce, is the only way, without higher dimensions to answer the issues in Section 2.2, then one has to ask if enough experimental evidence exists to confirm if Equation (1.12) to Equation (1.15) are implying nucleation of gravitons in a relic sense AFTER a LQG big bounce regime of energy transfer, or of an actual transmission of gravitons from a prior universe.
4.2. Further Inquiry as to If the Chemical Potential, as Given by Approaches Zero in the Onset of Inflation/Super Inflation
Beckwith has very deliberately set with approaching zero. Note that L. Glinka  in his quantum rendition of a graviton gas has , and calculates entropy based upon a partition function, explicitly with results. Beckwith thinks that Glinka’s work is sound, but has decided to set to for the following reason. The main benefit of chemical potentials is in applications of BBN and/or neutrino physics, i.e. a good example of such is given by Raffelt  as of neutrino physics, BBN, and cosmology. At the very onset of inflation which is where the analysis of setting occurs, Beckwith is very deliberately setting initial nucleation at or before the BBN/neutrino physics contributions to cosmology.
If the author, Beckwith, is wrong, he will be quite happy to amend his work along the lines given by L. Glinka’s 2007 work  . However, in lieu of the absence of either a neutrino physics/BBN contribution, he is attempting to come up with falsifiable experimental results using initially and also attempting to make sense of if there is a way to distinguish between the criteria given in Section 2 of this document.
Note that Appendix A below summarizes some of the methods used by the author in terms of counting of gravitons and initial entropy assumed in this document. The reader should also review  as well which places the idea of infinite quantum statistics in context.
In addition, in Appendix B, the author gives a summary as to some emerging trends in gravitational wave astronomy which are extremely important. References  ,  and  are extraordinarily relevant to the ideas brought up and shared here, i.e. that Corda in  has outlined protocol as to the emerging issues concerning interferometry and the tests for the nature of gravity is undeniably relevant to our manuscript, and not to mention the speculations on extra dimensions in  .
This work is supported in part by National Nature Science Foundation of China grant No. 11375279
Appendix A: Entropy Generation via Ng’s Infinite Quantum Statistics
The author brings up entropy development as given by  ,  . Furthermore, information counts, as discussed in this appendix tie in with information packing as brought up in the conclusion of the present paper. How do relic high frequency gravitational waves inter relate experimentally with the nucleation of short wave length relic gravitons? A small graviton creation volume, V, for relic gravitons of a high frequency ( short wave length ) right after the big bang would be consistent Graviton volume V for nucleation is tiny, well inside inflation. So the log factor drops out of entropy S if V is chosen properly for both Equation (C.1) and Equation (C.2). Ng’s  result begins with a modification of the entropy/partition function Ng used in an approximation of temperature and its variation with respect to a spatial parameter, starting with temperature ( can be thought of as a representation of the region of space of the particles in question). Furthermore, assume that the volume of space to be analyzed is of the form and look at a preliminary numerical factor we shall call , where the denominator is Planck’s length (on the order of centimeters). We also specify a “wavelength” parameter . So the value of and of are approximately of the same order of magnitude. Now this is how Jack Ng  changes conventional statistics: he outlines how to get , which with additional arguments we refine to be (where is graviton density). Begin with
This, according to Ng, leads to entropy of the limiting value of, if
But , so unless N in Equation (A.2) above is about 1, S (entropy) would be < 0, which is a contradiction. Now Equation (A.2) is where  introduces removing the N! term in Equation (A.1) above, removing the expression of N inside the Log expression in Equation (A.2).
Appendix B: Essential Gravitational Wave, and Experimental Gravity Constraints
Keep in mind as well that there has been recent confirmation by Abbot  as to the existence of gravitational waves, which further extends what was brought up by Abbot, et al. of the LIGO observational team, which is in terms of black hole binaries, which further confirms the solutions of the issues, brought up by Abbot in  as well as also room to explore the insights brought up by Corda in  which await further investigation. In addition the speculation that extra dimensions  are involved becomes a testable datum, especially with the emerging science of gravitational wave astronomy. Which is of foundational import.