We begin with a brief model as to singular universe, versus a multiverse in terms of input into singularity construction. The singularity behavior envisioned in this document is given by the following argument, as given by Kauffman [1] and the author in [2] , with the case of when one has been reevaluating the question of a “near singularity” in a multiverse. The multiverse will assume Ergotic mixing of space-time as given by [3] . Massless gravitons corresponds to the physics described in the beginMassless gravitons corresponds to the physics described in the beginning of the 2nd part of this document, whereas if massive gravitons exist, the resulting alterations of the general relativistic equations will be linking us to review questions as to if singularities exist at the start of cosmological expansion. Or, the start of cosmological expansions, if massive gravitons exist, assumes the existence of a small non singular regime of space-time. This sharply differentiates us from the physics given in reference [4] , as we explain in our document.

We follow the recent work of Kauffmann [1] , which sets an upper bound to concentrations of energy, in terms of how he formulated the following equation put in below as Equation (1). Equation (1) specifies an inter-rela- tionship between an initial radius R for an expanding universe, and a “gravitationally based energy” expression we will call T_G(r) which lead to a lower bound to the radius of the universe at the start of the Universe’s initial expansion, with manipulations. The term T_G(r) is defined via (2) afterwards. We start off with Kauffmann’s expression [1]

Kauffmann [1] calls a “Planck force” which is relevant due to the fact we will employ (1) at the initial

instant of the universe, in the Planckian regime of space-time. Also, we make full use of setting for small r, the following:

i.e. what we are doing is to make the expression in the integrand proportional to information leaked by a past universe into our present universe, with Ng [5] - [18] style quantum infinite statistics use of

Then Equation (3) will lead to

Here, , , and we set Planck length as:

where we set with, and. Typically is about at

the outset, when the universe is the most compact. The value of const is chosen based on common assumptions about contributions from all sources of early universe entropy, and will be more rigorously defined in a later paper. We argue that the above methodology, giving a non zero initial starting point is made especially tend ible if one is using a low temperature start, allowing for the existence of prior recycling universes gravitons to play a role, i.e. that in the single universe repeated again and again, there would be real issues as to the survival of the graviton allowing for the conclusion as to Equation (4). What Equation (4) is doing is to help us determine if conditions exist for a massive graviton versus a massless graviton. If Equatuion (4) is consistent with the existence of massive gravitons, then our inflaton model contributes to models which have Dark Energy as due directly to the existence of massive gravitons in space-time.

We will start with a first-principle introduction to detection of gravitational wave density using the definition given by Maggiore [6]

where is the frequency-based numerical count of gravitons per unit phase space. The author suggests that may also depend upon the interaction of gravitons with neutrinos in plasma during early-universe nucleation, as modeled by M. Marklund et al. [7] . Having said that, the question is, what sort of mechanism is appropriate for considering macro effects of gravitons, and the author thinks that he has one, i.e. reacceleration of the universe, as far as a function of graviton mass, i.e. what Beck with is to modify is what was in reference [8] Assume Snyder geometry and look at use of the following inequality for a change in the HUP [8] ,

and that the mass of the graviton is partly due to the stretching alluded to by Fuller and Kishimoto [9] a supposition the author is investigating for a modification of a joint KK tower of gravitons, as given by Maartens [10] [11] for DM. Assume the stretching of early relic neutrinos that would lead to the KK tower of gravitons―for when, is,

Note that Rubakov [12] - [14] writes KK graviton representation as, after using the following normalization

where are different forms of Bessel functions, to obtain the

KK graviton/DM candidate representation along RS dS brane world [12]

This Equation (8) and Equation (9) is for KK gravitons having a TeV magnitude mass (i.e. for mass values at. 5 TeV to above a TeV in value) on a negative tension RS brane. What would be useful would be managing to relate this KK graviton, which is moving with a speed proportional to with regards to the

negative tension brane with as an initial starting value for the KK graviton mass,

before the KK graviton, as a “massive” graviton moves with velocity along the RS dS brane. If so, and if

represents an initial state, then one may relate the 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 lower in value, i.e. of, as opposed to. Whatever the range of the graviton mass, it may be a way to make sense of what was presented by Dubovsky et al. [15] who argue for graviton mass using CMBR measurements, of Dubosky et al. [15] results can be conflated with Alves et al. [16] arguing that non zero graviton mass may lead to an acceleration of our present universe, in a manner usually conflated with DE, i.e. their graviton mass would be about

. Also assume that to calculate the deceleration, the following

modification of the HUP is used: [2] , where the LQG condition is,

and brane worlds have, instead, Also (10) will be the starting point used for a KK tower version of (10) below. So from Maarten’s [10] [11] paper,

Maartens [10] [11] also gives a 2^nd Friedman equation, as

Also, we are in the regime for which for redshift values z between zero and 1.0 - 1.5 with exact equality of pressure being equal to the negative value of density, for redshift z between zero to 0.5. The net effect will be to obtain, due to (6), and use

. As given by Beckwith [17] [18]

Equation (12) assumes, and the net effect is to obtain, a substitute for DE, by presenting how gravitons with a small mass done with, even if curvature K = 0.

In a revision of Alves et al. [16] , Beckwith [17] [18] used a higher-dimensional model of the brane world and Marsden [10] [11] KK graviton towers. The density of the brane world in the Friedman equation as used by Alves et al. [16] is use by Beckwith for a non-zero graviton [17] [18]

i.e. Equation (12), and Equation (13) above is making a joint DM and DE model, with all of. (13) being for KK gravitons and DM, and grams being a 4 dimensional DE. (11) is part of a KK graviton presentation of DM/DE dynamics. Beckwith [17] [18] found at z ~ 0.4, a billion years ago, that acceleration of the universe increased, as shown in Figure 1 [17] [18] .

Padmanabhan [19] [20] has written up how the 2^nd Friedman equation as of (11), which for z ~ 0.423 may be simplified to read as [10] [11]

Equation (14) would lead to an inflaton value of, when put in, for scale factor behavior as given by

, of, for the inflatonand inflation of [19] [20]

Assuming a decline of, Equation (15) yields [19]

As the scale factor of had time of the value of roughly

have a power law relationship drop below, the inflaton took Equation (16)’s value which may have been a factor as to the increase in the rate of acceleration, as noted by the q factor, given in Figure 1. Note that there have been analytical work projects relating the inflaton, and its behavior to entropy via noting that inflation stopped when the inflaton field settled down into a lower lower energy state. The way to relate an energy state to the inflaton is, if, then in the early universe, one has a potential energy term of [19] [20]

A situation where both grows smaller, and, temporarily, takes on Equation (16)’s value, even if the time value gets large, and also, if acceleration of the cosmic expansion is taken into account, then there is infusion of energy by an amount dV. The entropy, will lead, if there is an increase in V, as given by Equation (17) a situation where there is an effective increase in entropy. If there is, as will be related to later, circumstances, where [5] number of graviton states [17] [18] as will be derived in Equation (17), then at least in higher dimensions, we have an argument that the re emergence of an inflaton, with a corresponding reduction of Equation (17) in magnitude may be part of gravitons playing a role in the re acceleration of the universe.

If a way to obtain a graviton mass in four dimensions is done which fits in with the as given higher 5 dimensions specified by a slight modification of brane theory, or Maarten’s cosmological evolution [10] [11] equations, what benefits could this approach accrue for other outstanding problems in cosmology? The author, Beckwith, claims that due to the Friedmann equations, it would result in deceleration parameter q(z) similar to Figure 1 above. Snyder geometry for the four dimensional case with would specify Friedmann equations along the lines of in Equation (2) above. If one follows, then the Friedmann equations appear as giving details to the following equation [21]

The construction done from sections 1 to 3 are for. When, the claim is that almost all the complexity is removed, and what is left is a [21] treatment of the Friedmann equations, where he obtains, to first order, if is a scalar field density,

and

The interpretation of as a scalar field density [21] , and if one does as Alves et al. [16] uses Equation (7) above. We need to interpret the role of. In the LQG version, Equation (20) may be rewritten as follows: If conjugate momentum is in many cases, “almost” or actually a constant, using

Beckwith [17] [18] claims that the deceleration parameter q(z) incorporating Equation (19), Equation (20) and. Equation (21) should give much the same behavior as Figure 1 above. If so, then if one is differentiating between four and five dimensions by what is gained, in cosmology, one needs having it done via other criteria. The following is a real problem. As given by Maggiore [6] , the massless equation of the graviton evolution equation takes the form

When, the above becomes [6]

The mismatch between these two equations, when, is due to as,

which is due to setting a value of The semi classical method by t’Hooft [22]

[23] , using Equation (22) and Equation (23) is the solution. We generalize to higher dimensions the following diagram as given by Beckwith [24] [25] . Use an instanton- anti instanton structure, and t’Hooft [22] [23] equivalence classes along the lines of (24) below with equivalence class structure in the below wave functional to be set by a family of admissible values [24] [25]

We state that the process of nucleation of a graviton at the initiation of space-time creation. is similar in part to what occurs in the instanton-anti instanton formulation of Figure 2, above. At the end of the document will be a supposition as to taking this analogy far more directly as to the nature of gravitons, as a future works project.

This discussion above, would be consistent upon having a graviton represented by not only Equation (24). If

one is adding the small mass of grams, with grams, then the

problem being worked with is a source term problem of the form given by Peskins [26] as of the type

This is, using the language Rubakov [12] put up equivalent to obtain,

If is a constant, then the expression (24b) has delta functions. This is the field theoretic identification. Another way is to consider an instanton-anti instanton treatment of individual gravitons, and to first start with the supposed stretch out of gravitons to enormous lengths. Assuming grams for gravitons in 4 dimensions, the supposition by Bashinsky [27] and Beckwith³ is that density fluctuations are influenced by a modification of cosmological density in the Friedmann equations by the proportionality

factor given by Bashinsky [27] ,. This proportionality factor for as

showing up in the Friedmann equations should be taken as an extension of results from Marklund et al. [7] , due to graviton-neutrino interactions as proposed by Marklund et al. [7] , where neutrinos interact with plasmons and plasmons interact with gravitons. Thereby implying neutrino-graviton interactions Also, graviton wavelengths have the same order of magnitude of neutrinos. Note, from Valev [28] ,

Extending M. Marklund et al. [7] and Valev [28] , some gravitons may become larger¹⁴, i.e.

A way to accommodate this wave length as to an instanton-anti instanton packaging of gravitons, was to start with an analogy between Giovannini, [29] from a least action version of the Einstein―Hilbert action for “quadratic” theories of gravity involving Euler-Gauss-Bonnet. Then Giovannini’s [28] Equation (6) corresponds to

Givannini [28] represents of Equation (26a) as a kink, and makes references to an anti-kink solution, in Figure 1

in Givannini [28] . Furthermore the similarity between Equation (26a) and

in Beckwith’s [24] [25] treatment with regards to density wave physics instantons is obvious. If

is part of representing a graviton as a kink-anti-kink combination, arising from a 5 dimensional line element, [28]

The end result of this would be to have an instaton-anti instanton structure as to emergence of a massive graviton if noting, that there is the possibility of using t’Hoofts [17] [22] [23] classical embedding of “deterministic quantum mechanics” as a way to embed a nearly four dimensional graviton as having almost zero mass, in a larger non linear theory.

The interrelationship of structure of the profile of a DM cluster, with any perturbations DM density profile [29]

As told to the author by Sabino Matarre [29] , in July, 2009, in Como Italy, the gravitational potential has, perturbatively speaking an additional term added to variations in the gravitational potential term which Matarre [29] gave as

It is suggested that the function is largely due to entropy variations, some of which occurred during relic GW/graviton production. Here the expression variations from gaussianity. Furthermore, is a linear Gaussian potential, and the overall gravitational potential is altered by inputs from. Note that neutrinos flavor physics oscillations are not very important in terms of, as specified in conversations. Beckwith had in September 23, 2009 in Erice with George Raffert [30] . Which leads to emphasizing the role of entropy pro- cesses due to graviton-neutrino physics, as as written up by Beckwith [31] .

The real start to this investigation is to explain how and why the star HE0107-5240 could form with so little lithium in the first place [31] . As stated by Fuller et al. [9] neutrinos could interact with DM potential wells in ways Beckwith thinks could influence deviations from standard galaxy hierarchy formation models which will also have a counter part in deviations in the BBN nucleosynthesis of light elements, by examining the role of temperature fluctuations modeled on Equation (29) below, leading to fluctuations affecting BBN element rarity [31] .

While Equation (29) above would have its maximum impact for regions as of about red shift, the impact of Equation (29) would be as of red shifts, with the corresponding influenced by Bashinsky’s [27] neutrino―gravition damping as stated by the coefficient of density fluctuation modified by

[27] . Note that would be larger than of Equation (28) and

would be dominated by neutrino-graviton interactions, whereas would be dominated by graviton generated entropy, with neutrinos at hitting DM directly. We submit that a graviton with a small rest mass may be more amendable to such interaction with neutrinos, and that in addition Equation (27), Equation (28) and Equation (29) may influence and affect structure formation as seen by the following diagram in Figure 1. Note that this is assuming that early universe interactions which we are talking about eventually play out and reach, with the re acceleration of the universe, as outlined in the 1st half of our document to also be indirectly responsible for the famous “halo merging tree diagram we call Figure 3 below. At or about when

begins to delineate the neutrino-GW interaction becoming a significant damping impact upon each other, one would be seeing variations from the usual structure formation, as given by the following diagram. [32] .

We should keep in mind that the following holds, i.e. for flat space. That one will have Figure 3 in both flat and in curved space. Also note that, M. Marklund, G. Brodin, and P. K. Shukla [7] posted their own version of not only neutrino mass, as given by, where the overall mass is set by Note, here, that the

potential for where the frequency comes from is, here, is, and, according to Eberle and Ringwald et al. [33] , may have lightest relic neutrino masses of the order of

as opposed to, as given by D. Valev [34]

where, is a dimensionless Hubble constant, Very roughly put, for relic early universe conditions, one may be seeing that the neutrino has the effective mass than a graviton. Furthermore, for a neutrino we have

This will tie in directly with a neutrino mass limit we state as [7]

If, as if often expected in inflation, space becomes abruptly flat at the onset of inflation, then for a neutrino mass, as the will then lead to the following inequality [7] [31]

Now, how would variation from the above “halo Merging history tree”, partly due to the modulation, via entropy, of DM structure formation, due to GW/gravitons affecting DM profile affect the concentration for lithium in stars, and perhaps lead to the famous “lithium problem” being resolved? We are investigating it. But we do think that having a graviton with mass is affecting the particulars of the “halo mixing tree” diagram [32] .

Beckwith [35] has concluded that the only way to give an advantage to higher dimensions as far as cosmology would be to look at if a fifth dimension may present a way of actual information exchange to give the following parameter input from a prior to a present universe, i.e. the fine structure constant, as given by [35]

Equation (35) above is in tandem, with examining if the following holds, i.e. for the consistency of physical law, namely from cycle to cycle is there a preservation of Planck’s constant? Namely

The wave length as may be chosen to do such an information exchange would be part of a graviton as being part of an information counting algorithm as can be put below, namely: Argue that when taking the log, that the 1/N term drops out. As used by Ng [5]

This, according to Ng [5] , leads to entropy of the limiting value of, if will be modified by having the following done, namely after his use of quantum infinite statistics,

Eventually, the author hopes to put on a sound foundation what t’Hooft [22] [23] is doing with respect to t’Hooft [22] [23] deterministic quantum mechanics and equivalence classes embedding quantum particle structures. Doing so will answer the questions Kay [36] ²⁹ raised about particle creation, and the limitations of the particle concept in curved and flat space, i.e. the global hyperbolic space time which is flat everywhere expect in a localized “bump” of curvature. Furthermore, if we have an initial universe count of gravitons as S (initial) ~10¹⁰ to at most S (initial) ~10²⁰, we are assuming the existence of one operation per graviton. This one operation per graviton in the early universe may correspond to at least one unit of information per graviton, i.e. one unit of “information” per graviton is correlated directly with one “operation” per graviton. The operation in this case is likely the creation of initial gravitons, in the early universe. This datum needs experimental confirmation and is important to astro physics linkage of DE with DM, in the future. Equation (14) to Equation (17) if confirmed for Z ~ 0.423 may prove that higher dimensions are necessary for cosmology.

Beckwith [37] , in his FQXi document outlined a procedure where a graviton with mass may be indicative of the existence of multiple universes co existing. The details of the mapping of that multiple universe picture involve a transition from an analog physics (discrete, i.e. classical world picture) to one where octonian gravity is formed, i.e. a quantum picture as a pre cursor to quantum gravity. The existence of a small mass may mean the extension of quantum physics to a larger embedding/extension of quantum physics. Furthermore, keep in mind that tandem to that step of semi classical embedding of a graviton, that eventually we want to make explicit an idea by, T. Padmanabhan in DICE 2010 [38] , as to finding “atoms of space time” permitting a thermodynamic treatment of emergent structure similar to Gibbs treatment of statistical physics. i.e. for finding out if the following is possible, i.e. can an ensemble of gravitons, be used to construct an “atom” of space time congruent with relic GW. That is our ultimate end, as to our research. That would make our inquiry of the nature of gravitons most worthwhile. This idea was presented at DICE 2010, [39] and we would like to refine it in our future research work. This would be in tandem of adapting the Kiefer, Polarski, and Starobinsky [40] presentation of the evolution of relic entropy via the evolution of phase spaces, with being the ratio of “final (future)”/“initial” phase space volume, for k modes of secondary GW background. From “atoms of space time” treatment of early universe space time geometry according to [40]

This lead to the author, Beckwith to derive the following a important for structure formation, note the following about what happens if

i.e. especially if the degrees of freedom rises above.

Note that at T ~ 100 KeV Unless the term for were absolutely enormous, and if, then could happen, which would be physically meaningless. The other situation is that there could be situations for which would be undefined, especially if were close to an equality. We state here unequivocally that Equation (38) and Equation (39) above are important, and that this has serious experimental import. Having said that, we will next go to what would be a way to determine if Gravitons can have mass (massive Gravitons). i.e. in the conclusions section, we radically extend the consequences if, with a speculation as to what could happen as to dark matter and dark energy contributions, which we think is important to the matter of singularities and their purported connection to a multiverse. But before we get to that matter, we will examine the role of partition functions, in terms of background which will lead to several pages later, to contributions, especially for the regime of values, say of 1100 to 1200, which we think has to be seriously looked at.

This section is to determine if gravitons have mass and backs the assertion made earlier that multiverse construction has massive gravitons. Note that this section is directly linked to the first part of this document, as to what was done by the author to extend Kauffman’s work [1] .

We assume that there are no fewer than N universes undergoing Penrose “infinite expansion” (Penrose, 2006) [41] - [43] contained in a mega universe structure. Furthermore, each of the N universes has black hole evaporation, with the Hawking radiation from decaying black holes. If each of the N universes is defined by a partition

function, called, then there exist an information ensemble of mixed minimum information correlated as about bits of information per partition function in the set, so minimum information is

conserved between a set of partition functions per universe [44]

However, there is non-uniqueness of information put into each partition function. Furthermore

Hawking radiation from the black holes is collated via a strange attractor collection in the mega universe struc-

ture to form a new big bang for each of the N universes represented by. Verification of this mega

structure compression and expansion of information with a non-uniqueness of information placed in each of the N universes favors ergodic mixing treatments of initial values for each of N universes expanding from a singularity beginning. The value, will be using. How to tie in this energy expression, as in Equation (40) will be to look at the formation of a nontrivial gravitational measure as a new big bang for each of the N universes as by the density of states at a given energy for a partition function. (Poplawski, 2011) [45]

Each of identified with Equation (41) above, are with the iteration for N universes (Penrose, 2006) [41] - [44] Then the following holds, namely [44]

Claim 1, [44]

For N number of universes, with each for j = 1 to N being the partition function of each

universe just before the blend into the RHS of Equation (42) above for our present universe. Also, each of the

independent universes given by are constructed by the absorption of one to ten million

black holes taking in energy. i.e. (Penrose, 2006) [41] - [44] . Furthermore, the main point is similar to what was done in [18] in terms of general ergodic mixing

Claim 2 [44]

Claim 3 The idea here is to use what is known as CCC cosmology [41] - [44] .

First. Have a big bang (initial expansion) for the universe. After red shift z = 10, a billion years ago, SMBH formation starts. Matter-energy is vacuumed up by the SMBHs, which at a much later date than today (present era) gather up all the matter-energy of the universe and recycles it in a cyclic conformal translation, as follows, namely

is a constant. Then the main methodology in the Penrose proposal has been in Equation (45) evaluating a change in the metric by a conformal mapping [43] [44] to

Penrose’s suggestion has been to utilize the following [43]

Infall into cosmic black hopes has been the main mechanism which the author asserts would be useful for the recycling apparent in Equation (47) above with the caveat that is kept constant from cycle to cycle as represented by a restatement of Equation (35a) as in the multiverse as

Equation (47) is to be generalized, as given by a weighing averaging as given by Equation (42) where the averaging is collated over perhaps thousands of universes, call that number N, with an ergodic mixing of all these universes, with the ergodic mixing represented by Equation (42) to generalize Equation (47) from cycle to cycle.

In this chapter, we are looking at a generalization of Kolb and Turner’s [46] gravitational radiation result which is given as

In the immediate aftermath of inflation, and just before inflation, we generalize as a constant,

as well as approximate as a constant, with also putting in [34]

Then we have that if “Before” is just before the formation of the present universe, and “Final” is just after the formation of the present universe

Claim 4, in the case of a single repeating Universe, the RHS of (51) is zero, leading to

implying that the mass of a graviton in a single repeating universe is zero.

Proof: We will use the following value of the net energy, i.e. if

Now define an average gravitational energy as given by having a single universe, denoted by N (fixed), i.e. one universe out of N of them [maybe infinite] given as

This is the single universe, repeated, i.e. in this case, we assume that the Volume, per single repeating universe, is the same for a regime of the BB immediately before and after the cosmic explosion. Hence, we have that.

In terms of equipartition function definitions, and to rewrite Equation (52) as in the case of a multiverse, i.e. one out of N “universes”

It so happens, then that there are r “states” per universe, and an infinite number of them. Then the average graviton radiation density would be, for r =1 to infinite number of energy states per Nth universe, with the label N (full-range) being the number of universe domains in a multiverse.

In terms of the averaging procedure of Equation (42), we then have the initial and final states for the multiverse as

This would be due to the behavior of before the big bang, which will lead to

Which should be compared to

Equation (58) and Equation (59) above are not the same value, hence the results given in Equation (56). Hence the masses of the gravitons would not be the same by Equation (51).

Note that Feynman and Hibbs [47] have a different way of writing a net energy as can be written using as the total energy of the ith universe, and energy of the rth sub domain of the ith universe i.e. two different energy expressions.

Then, using Feynman and Hibbs [47] , the net energy can be written as

The results as outlined above are, again then, more obvious.

First of all, the contributions of Gravitons to reacceleration of the universe are outlined as a consequence of massive gravitons. In addition, the graviton mass of a non zero value is central to the process of entropy generation which leads to our next comment which is a further research project in its own right. For what it is worth, we will address an extension of an entropy versus graviton production linkage implied in the first linkage. This entropy versus gravition linkage, as seen below, will imply a non zero initial radius for the universe. Before that is brought up, we should consider entropy generation with an initial cosmological “constant” (vacuum energy) at the start of inflation.

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

Andrew Beckwith, (2016) Deceleration Parameter Q(Z) and Examining If a Joint DM-DE Model Is Feasible, with a Revisit to the Question of Cosmic Singularities. Journal of High Energy Physics, Gravitation and Cosmology,02,362-382. doi: 10.4236/jhepgc.2016.23033