Identifying a Kaluza Klein Treatment of a Graviton Permitting a Deceleration Parameter q ( z ) as an Alternative to Standard DE

The case for a four-dimensional graviton mass (non-zero) influencing reacceleration of the universe in five dimensions is stated, with particular emphasis on whether five-dimensional geometries as given below give us new physical insights as to cosmological evolution. A comparison with the quantum gas hypothesis of Glinka shows how stochastic GW/gravitons may emerge in vacuum-nucleated space, with emphasis on comparing their number in phase space with different strain values. The final question is, can DM/DE be explained by a Kaluza Klein particle construction? i.e., the author presents a Kaluza Klein particle representation of a graviton mass with the first term to the right equal to a DM contribution and with the 2 term to the right being effective DE. We propose obtaining the rate of production of relic universe Kaluza Klein gravitons, based on an analogy to the production of axions from the Sun over a wide range of frequencies. This last statement is a work in progress being developed by the author, which is being discussed with colleagues of the author in Chongqing University.


Introduction
The idea behind this article has been to investigate how a speed up of cosmological expansion could occur, i.e. involving the de celeration parameter as defined below [1] is a way to look for a mechanism to explain how a speed up of expansion is possible.Settling upon the lowly graviton is done, since if the graviton has tion of gravitons would be in narrow frequency ranges.as predicted by Grishchuk [4] Gravitational wave density will be presented as necessary background.
We will start with a first-principle introduction to determination of gravitational wave energy density gw Ω using the definition given by Maggiore [5] gw Ω is defined as a way to measure gravitational wave strength as a function of general random energy fluctuation backgrounds.Equation (1) measures the strength of the gravitational wave signal.
where f n is the frequency-based numerical count of gravitons per unit phase space.Since gw Ω is usually an extremely small fraction of very noisy relic cosmological background conditions, the representation of gw Ω is based on scaling its relative strength.i.e., Figure 1  tied in with analysis of the plots in Figure 1.Note, these models as not consistent with each other.
Grishchuk [7] states the relation between gw Ω (which he refers to as g Ω ) and the frequency spectrum ( ) , g h v τ as given in Equation ( 2).The models brought up in Figure 1 need to be compared with each other, as part of experimental inquiry.( ) We will address the divergences between models presented by Maggiore, Abott, and Grishchuk and will develop candidate discriminating criteria for a number count f n based on Glinka's [8] graviton gas work: If we assume final ã a , and substituting Glinka's [8] approximation of a gra- viton "gas" by treating gravitons as boson "particles" in the early universepermitting a rapid accumulation of "gravitons" in the initial phase of the big bang-then for very high ( ) Frequently, a researcher will be looking at  as bosons, will permit up to 10 6 particles as entropy units being manufactured within Planck length values of space-time volume, i.e., roughly 10 120 times smaller As will be explained in the Appendix, there is a way to relate graviton count and entropy, so then the numbers associated with f n are a de facto counting algorithm for entropy per unit phase space.Note that the highest counting numbers for entropy are associated with , which, according to Figure 1, is associated with pre-big-bang GW/graviton production.
is associated with usual inflation, as given in Figure 1.i.e., if one is looking for standard creation of entropy paradigms associated with the early universe, a typical phase transition argument for early entropy production is given by Tawfik [9] (2008), which for QCD regimes is shown in Equation ( 6).
where S = entropy.We assume here that 58 total ~10 S may be associated with gravitons/GW at the very end of inflation, and not the beginning (where one would have a far lower initial count, i.e., S ~ 10 7 ) and with frequencies initially on the order of 10 8 to 10 10 in the beginning of cosmological evolution.Such a huge burst of graviton production for temperatures on the order of ~174 MeV T would lead to measurable consequences.

Gravitons with a Non Zero Rest Mass. The KK Treatment
Consider if there is then also a small graviton mass, i.e., as stated by Beckwith [1] [2]: Note that Rubakov (2002) works with KK gravitons, without the tiny mass term for a 4 dimensional rest mass included in Equation (7).To obtain the KK graviton/DM candidate representation along RS dS brane world, Rubakov obtains his values for graviton mass and graviton physical states in space-time after using the following normalization: , , , J J N N which are different forms of Bessel functions.
His representation of a graviton state is given by Equation (8), which is almost completely acceptable for our problem, since the rest mass of a graviton in four dimensions is so small.If so, then the wave function for a graviton with a tiny 4 Journal of High Energy Physics, Gravitation and Cosmology dimensional space time rest mass can be written as [10].
Equation ( 8) is for KK gravitons having a TeV magnitude mass for mass values at 0.5 TeV to above 1 TeV) on a negative tension RS brane.It would be useful to relate this KK graviton, which is moving with a speed proportional to 1 H − with regards to the negative tension brane with ( ) as an initial starting value for the KK graviton mass.
If Equation ( 8) is for a "massive" graviton with a small 4 dimensional gravition rest mass and if ( ) [11], who argue for a graviton mass, using CMBR measurements, of 20 KK Graviton ~10 eV M − .Also, Equation ( 9) will be the starting point used for a KK tower version of Equation (9).So from Maartens [12], Maartens [12] also gives a 2 nd Friedman equation:.
[ ] Also, an observer is in the low redshift regime for cosmology, for which P ρ ≅ − , for red-shift values z from zero to 1.0 -1.5.One obtains exact equality, P ρ = − , for z between zero to 0.5.The net effect will be to obtain, based on Eq- uation (10), assuming 0 K Λ = = and using to get a deceleration parameter q as given in Equation (11).
[ ] ( ) ( ) ( ) These set values, along with a revised Equation ( 10) allow a graviton-based substitute for DE.0 K Λ = = plus a small rest mass for a graviton in four dimensions allows for "massive gravitons" that behave the same as DE.Setting 0 K Λ = = , while having a modified behavior for the density expression, for a Friedman equation with small 4 dimensional graviton mass, means that dark energy is being replaced by a small 4 dimensional rest mass for a graviton.

Consequences of Small Graviton Mass for Reacceleration of the Universe
In a revision of Alves et al. [13], Beckwith [1] [2] used a higher-dimensional model of the brane world combined with KK graviton towers per Maartens [12].
The energy density ρ of the brane world in the Friedman equation is used in a form similar to Alves et al. [13] by Beckwith [1] [2] for a non-zero graviton: Beckwith [6] [7] suggests that at z ~ 4, a billion years ago, acceleration of the universe increased, as shown in Figure 1. Figure 1 is, if confirmed a good verification of the Ng [12] hypothesis, and would be a starting point to investigate the role of gravitons in cosmology.The author notes that Buonnano [13] assumes a much lower range of initial frequencies for relic GW than the author.
Beckwith [6], [7] obtained a re-acceleration of the universe result as given in Figure 2. The contribution of a low rest mass for 4 dimensional gravitons, as given in Equation ( 7) leads to a speed-up of acceleration of the expansion of the universe a billion years ago, i.e. for a red shift slightly smaller than 0.5.Figure 2 below is predicated upon a small 4 dimensional rest mass (stated in Equation (7) for a graviton behaving the same as dark energy)...We will state in our discussions section as to what is needed to give experimental confirmation as to what is a current for a "massive" graviton which is appropriate for explaining in part, Figure 2 below.

Comparison with Axion Flux Results from the Sun: What Would be Needed to Measure Dm Flux for a New Model of Dm/De?
This section is intended to explain a rate of particle production as given for solar axions, and to determine what may be necessary to adapt such an approach for Figure 2. Re-acceleration of the universe based on Beckwith [1]; (note that q(z) < 0 if z < 0.423).Journal of High Energy Physics, Gravitation and Cosmology Kaluza-Klein gravitons in the onset of inflation.Figure 3 is a redo by the author, using the Dimpoulou et al. [16] value of axion flux, while noting that it is similar to the axion flux from the sun, per Lazaruth et al. [17].It would be appropriate to do the same with the DM implied by Equation (7). Figure 3 is based on Equation (13), given by Buoanno [15], for applied frequency ω vs. ( ) r ω , axion flow in KeV values of solar axions: Equation ( 13) models what happens for axions in the sun, and we hope to eventually obtain a similar rate expression of graviton production for Kaluza Klein gravitons as given in Equation ( 7) versus energy, similar to what is shown in Figure 3, in the case of solar axions.
The rate of production of solar axions ( ) r ω is plotted against frequency, ω .
Beckwith's goal is to eventually duplicate Figure 3 for relic gravitons, using data collection for Kaluza-Klein gravitons.This rate equation plotted in Figure 3, as given in Equation ( 13) as used by Beckwith, is dominated by (ϖ a plasma interaction effect that is to be determined, and ω is part of the expression for permitted solar axion energy values.ω ϖ in most cases.The author suggests finding grounds for a similar energy plot of DM values from a suitably modified version of Equation ( 7) for KK dark-matter candidates.Doing so would mean understanding how a rate equation based upon Equation ( 13) for DM production could commence using a model of KK DM production/evolution.
The author suggests that it would be appropriate to use an early universe counterpart to the known model of axion production in stars to illustrate the correspondence of axion production in the sun with relic particle production in the early universe.For Equation ( 14), Z n the number density of Z atoms with an ionized K shell, e n the number density of free electrons, ρ a general density of states for the axion-producing background, and σ the axio-recombination (free-bound) cross-section is given by Dimopoulous [16].However, this leaves open the question of whether the cross section σ for the LHC values of mas- sive gravitonsis similar to what is done in stars.The usual assumption is that the Figure 3. Beckwith's revision of axion flux from the Sun, in terms of (frequency=energy, if 1 = ), with the plot of the nmber of axions produced by the Sun in terms of KeV val- ues of solar axions.definitions correspond.Equation ( 14) is a rate of production of axions.Equation ( 14) has corresponds partially with Equation ( 13), but differs due to the density value ρ , which may be derived experimentally.In the early un- iverse, for a KK dark matter counterpart, would still have a density value ρ to consider and a possible σ for cross section for some interaction of KK DM production.However, due to early universe conditions, there would be no counterpart to Z n or e n .Durrer, Marozzi , and Rinaldi [18] have used very early universe plasmas, going back to the electroweak transition and turbulence, as a model for early-universe GW production.One would need to specify how to obtain σ for some interaction of KK DM production, which is why observa- tion of the mass and width (or cross section) of one or more KK gravitons, as part of a DM candidate, at the LHC, as remarked by Grzadkowski et al. [19] may be the only way to obtain experimental inputs into a graviton production/KK DM version of Equation ( 14).However, this leaves open the question of whether the cross section σ for the LHC values of massive gravitons, etc, would be the same as what would occur for early universe conditions.So far, the only known theoretical calculations of the above are along the lines of σ arising from pho- ton and neutrino annihilation rates, as given by Hewett [20].The author is attempting to obtain suitable values of σ to put in the given calculation at the start of the inflationary era.

Findings, Discussion of Results
We can use Figure 3 as an idea of how to identify rate ofgraviton collection opportunities against frequency for a detector along the lines of reference given in reference [3].What is expected though is that instead of the smoothly varying curve that if gravitons were matched against rates, in a similar manner, that there would be spikes, instead of smooth variations.Once GW astronomy becomes a fact, Figure 3 for gravitons will emerge.The sharpness of the spikes, if analyzed properly will say much about the supposition given in Equation (7) about a KK decomposition of the mass of a graviton into dark matter and dark energy contributions.
What is most intriguing, is the possibility of Equation ( 11) having a non uniform frontier of re acceleration of the universe, a billion years ago.Not a "perfect sphere" of re acceleration, but one with a jagged edge of moving space time regions.i.e. a complicated structure.In part, adding more details to supposed "cosmic voids" and regions of space time distrutions of galaxies in, say, a fractal geometric manner, as given in reference [21].If and when GW astronomy becomes a fact, suppositions as to how galaxies are distributed through space time may obtain a phenomenological descriptive rationale, which we hope leads to falsifiable experimental measurements.[22].The main point we wish to emphasize is that to do all of this, the following current behavior in a GW/graviton detector would have to be verified for massive gravitons.This is the geometry of space time which may be confirmed by appropriate analysis of Equation (11) The 1 st term to the right hand side of Equation ( 16) is the energy -momentum tensor of the back ground electromagnetic field, and the 2 nd term to the right hand side of Equation ( 16) is the first order perturbation of an electromagnetic field due to the presence of gravitational waves.The above Equation Eventually, with GW affecting the above two equations, we have a way to isolate ( ) T .If one looks at if a four dimensional graviton with a very small rest mass included [24] we can write where for 0 The claim which A. Beckwith made [24] is that As stated by Beckwith, in [24] shows the magnitude of gw Ω for different cosmological models.Here, gw Ω is the same as of Figure 1 below.Combining experimental confirmation of Equation (1) with observations using different values of a

Figure 1 .
Figure 1.Abbott et al. [6] (2009) shows the relation between GW frequency and GW energy density for different cosmological models.
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.This rest mass of a graviton is ( ) 48 graviton 4-Dim GR ~10 eV m the range of the graviton mass, it may be a way to make sense of what was presented by Dubovsky et al.

( 15 )
and Equation(16) will eventually lead to a curved space version of the Maxwell equation.As was given in[23]

65 4 D
Gravition ~10 grams m − , while count n is the number of gravitons which may be in the detector sample.What researchers intend to do is to try to isolate out an appropriate ( ) 1 uv T assuming a non zero

Table 2 .
If one assumes

Table 3 .
If one assumes