Using Barbour’s Ephemeris Time, and Padmanabhan’s Inflaton Value, plus Will’s Massive Graviton Velocity to Isolate Rest Energy of Massive Graviton as Compared to Racetrack Inflation Results of Graviton Physics and Modified Wheeler de Witt Results of Wormhole Physics

The idea is to identify via ephemeris time as given by Barbour and an inflaton field as given by Padmanabhan, for scale factor proportional to time to the alpha power and a velocity given by Will for massive gravitons, an initial energy for a massive graviton in space-time. The spatial values for the graviton production could be from the Planckian to Electro weak regime, with a nod to using a worm hole from a prior to a present universe as a delivery font for gravitational energy, as an information carrying bridge from prior universe “information settings” to the present space-time. The number of Gravitons will be set as N, and the initial time, as a tie in with Barbour’s ephemeris time, a constant times Planck time. In setting up the positions, as input into the positions and distributions of gravitons in our model, we will compare results as could be generated by Racetrack inflation, for presumed position of relic gravitons when just produced in the universe, as compared with results given by an adaptation of an argument presented by Crowell, in a modifica-tion of the Wheeler de Witt equation he gave germane to worm hole physics. In addition, with this presentation we will discuss entropy generation via graviton production. And compare that with semi classical arguments, as well as Brane-anti in the limit when a massive graviton mass approaches zero{\displaystyle m\to 0}. In particular, while at small scales, Newton’s gravitational law is recovered, the bending of light is only three quarters of the result Albert Einstein obtained in general relativity


Introduction
One of the inquiries as to graviton physics, is to ascertain how to gauge the real actual energy of a "massive" graviton. The reason for doing this, is due to the well known physics problem of how the bending of light by massive gravitons via the Planet Mercury is 3/4th that of the actual results seen in GR i.e. In the 1970s, van Dam and Veltman [1] and Zakharov [2] discovered a property of Fierz-Pauli massive gravity. Its predictions do not match those of general relativity in the limit when a massive graviton mass approaches zero{\displaystyle m\to 0}. In particular, while at small scales, Newton's gravitational law is recovered, the bending of light is only three quarters of the result Albert Einstein obtained in general relativity. This is known as the vDVZ discontinuity. [3] [4] [5] gives a summary on page 94 as to the details of the Vainstein solution which in the limit of non-linearized gravity, in its Equation (2.184) give a partial solution via a solution with a screening Yukawa type of potential as to what happens, when the mass of a graviton, approaches zero.
We will try to avoid using Yukawa style screening, and our start will be to ascertain an actual "rest energy" of a "massive" graviton, where we may be able to recover the limit behavior we want as 0 g m → . To do this, we will be using [6] by Barbour, but not in the sense of [7] [8]. In addition, [9] will be employed to obtain a velocity for a massive graviton, which has the energy E term we will attempt to isolate. [10] has the inflaton, we will be using which we will utilize for early universe kinetic energy contributions.
Afterwards, in 2 nd part of the manuscript we will briefly state some phenomenological consequences of what we have derived, and then detail those findings with possible consequences to the problem of early universe graviton generation and of an average energy, for a graviton, resulting from early universe production of gravitons.
The 3 rd part of the manuscript introduces in a general sense the problem of the position of gravitons, as assumed to be evaluated.
In the 4 th part of the manuscript, we will allude to racetrack inflation [11] as far as its connections to graviton physics, as well as non standard treatments of the WdW equation which were written up by Crowell, in 2005 [12].
The 5 th part of this manuscript will be a discussion between different choices of entropy.
The 6 th part of this manuscript is a review of applications of non-standard treatments of the WdW equation which were written up by Crowell, in 2005 [12].
Our conclusion will be a wrap up of our findings plus a prospectus as we see it as to what to possibly expect next, and to ascertain what may be fruitful lines of inquiry. As to the originally stated problem of fixing massive gravitons, only ¾ of the angular deviation of light about the planet mercury is given.

Barbour's Ephemeris Time, and Padmanabhan's Inflaton Value, Plus Will's Massive Graviton Velocity to Isolate Rest Energy of Massive Graviton
From the use of [6], we have a statement of Ephemeris Time which is ( ) E V − is kinetic energy of the system we are analyzing. We will use the construction given in [10] to construct the relevant kinetic energy of the system we are trying to analyze to make our point.
Hence, we get a kinetic energy value of Thereby leading to Here is where we will use the reduced speed of the massive graviton. [9] gives us A. W. Beckwith Journal of High Energy Physics, Gravitation and Cosmology Secondly set the early-universe i g m m  → , in the early universe, with N the number of gravitons.
If we make the following approximation, i.e.
This is the net energy associated with a graviton and we will spend the rest of our article analyzing the consequences of such for our questions as what is to known as the vDVZ discontinuity. And its possible resolution.

The Possible
In the case of black body radiation, this would be for a random distribution of "gravitons" in a closed thermal box.
If true, i.e. that assumption. We would likely then be able to generate some version of Bose-Einstein statistics, here, for a graviton "gas" i.e. along the lines of N for the number of n. assumed gravitons, roughly In this case, we would be revisiting the Solvay conference arguments as of 1927 with respect to [13] [14] [15] [16] [17]. Note that a variant of Equation (9) has also been approved by Weinberg [18] ( ) Note that both [19] and [20] in different ways, mean that the neat blackbody radiation approximation assumed in Equation (10) would need huge re adjustments. I.e. [19] would if a model of filament or structural turbulence leading to non-uniform in space-time graviton production, whereas [20] pretty explicitly rules out the idea of a blackbody cavity as far as containment of gravitons.
Hence, we will have to, if either [19] or [20] hold, consider something other than the traditional quantum thermal excitation of say even gravitons within axion walls [21], as has been thought of as possible by this author, and which then may lead to the author positing ways to come up with cosmological dynamics for entries of the terms i d δ in Equation (7). Journal of High Energy Physics, Gravitation and Cosmology To do that, we will consider, racetrack inflation, and also some of the ideas of what Crowell wrote up in [12]. For entries of the terms i d δ in Equation (7).
Afterwards, in making some assumptions, as to this set of entries into terms i d δ in Equation (7) we will go to what we mentioned earlier, which is how to recast the problem of massive gravity in a way which may avoid the vDVZ discontinuity. [3] [4], which will require a long discussion of its own. I.e. mind you this is not meant to be a complete resolution of that problem, but an indication of what our formulation of E, energy, portends to. Once the slow-roll conditions break down, the scalar field switches from being overdamped to being underdamped and begins to move rapidly on the Hubble timescale, oscillating at the bottom of the potential. As it does so, it decays into conventional matter.

First Review of the Racetrack Inflation Scenario and
End of quote I.e. this is well after the onset of inflation. [24] indicates that there is a detail of the spectrum which is significant in the initial phases of inflation, as given in [25] [26] [27], and which is given a spectrum value as stated in [24] as Specifically, if 2 k T       is less than one, due to elevated temperatures, which is what occurs in inflation. Hence, as by [24] the relic condition for gravitational waves cannot be ignored, and [24] states that there is a thermal vacuum state which is given as Notwithstanding what was said about [22] and [23], which appears to rule out significant contributions to relic gravitational waves, due to racetrack we will focus upon what could lead to a thermal vacuum state via racetrack, with comments.
In [28], on page 2 of the article Quote Hubble scale during inflation is bounded by the present value of the gravitino mass, i.e., 3 2 H m < . This relation, which ties the amplitude of primordial gravi-tational waves to the scale of supersymmetry breaking, appears to be rather generic.
End of quote What this says, is that the racetrack though, in common with other string theory cosmology, has, at the point of symmetry breaking, of the racetrack, a regime where gravitinos, when produced, are giving bounding behavior to the Hubble scale, which in turn [29] [30] The mass of a graviton, massive, is of the order of [31], so then we use the following [32] [33] Using this, we would have a numerical factor of N, and a time factor of t δ put in Equation (7) 36 Due to the uncertainty of the exact commencement of the relative distance of the radii of the universe in the electroweak era, [34], we will say then that this relationship will have to be speculated on, in the next section. And this will also incorporate comments on [35] thermal-cavity semi-classical

Estimating a Range of Values for δdi in Cosmology up to the Electro Weak Era Using Electro Weak Era as the Hot Spot for Relic Graviton Production
An e fold of 65 in inflation [36] is 10 28 magnification of an initial radius, and so if we consider an electro weak magnification at the end of inflation, for a radii of 10 −35 meters start to a magnified initial radius of about 1 meter at the very end of inflation, tops, with an initial radii of say 10 −7 meters at the start of the electro weak era, to about 1 meter at the close of the electro weak era. Meaning 42 10 N gravitons, in a spatial regime of say a ring in between a distance of 10 −7 meters to 1 meter from the 'center' of inflation in a time regime of roughly 36 7 electro-weak~1 0 s~10 P t t t δ − ∝ . This would be, if we use the idea of racetrack inflation, and of 1 gravitino roughly equivalent to 42 10 N gravitons, input into Equation (7). Keep in mind, that Guth, on page 135, of [37] estimates that the probable total Journal of High Energy Physics, Gravitation and Cosmology reach of inflation is an expansion of up to or more than 10 75 in volume for inflation, i.e. this is then giving us the following inputs, put into Equation (7) For the sake of convenience, in this first approximation model we will be initially assuming the rest mass of a graviton is about 10 −65 grams, in line with [31] We next will, if we assume that there is a correlation between entropy, due to S -N with the number N = (count of particles) [38] next comment upon what this may be saying about entropy, in the early universe. If there are no units of entropy, at the start of expansion of space time, we will choose the methodology of the Racetrack which implies that entropy production and graviton production, and gravity waves would have to await at a minimum, going to the electro weak regime of space time, I.e. That space time expands 10 million times past an initial starting point.

Review of Different Models of Entropy to Choose from
I.e. both the semi classical picture and brane picture tend to support the idea of graviton production starting at the electro-weak era, but if the graviton is a carrier of entropy, and if the radii of the initial configuration of the universe, is not zero, then we will be reason to bring up some of the issues the author raised in [42], which then leads to, if [42] is not wrong, leading to the idea of non-zero initial energy, perhaps recycled from a prior universe, as a starting point for our cosmology. This has implicitly raised the issue of [43], i.e. if there is a H = 0 initial starting point, of a possible reflection of this, as a causal barrier which may have CMBR overtones. Note that Beckwith, in [41], generalized a version of the Penrose cyclic conformal cosmology, to multiverses, which may be a way to ascertain if there is, as mentioned earlier, a recycling of space-time, at the start of the universe. In doing so, the author states that this necessitates either a proof, or a counter example to what is given in the traditional [45] with Penrose's supposition as to if there is a A. W. Beckwith Journal of High Energy Physics, Gravitation and Cosmology mandatory singularity at the start of cosmological expansion, as the supposition to either prove or disprove, and with [46] as the non linear electrodynamics speculation to either confirm or falsify as well.
[47] has a lucid competing theory as to non-zero initial radii of the universe speculations, but again, if that is not your favorite, you can peruse the idea brought up, as to using what is given in Crowell, 2005 [12] as to worm hole physics. As asked by the referee, what guarantee that one could use a worm hole as a start to cosmology? Go then to Appendix D, as a start to our discussion. I.e. this is to ascertain if we can say something cogent as to the scale of gravity effects, as either classical or quantum, and then afterwards, go to Appendix E, as to the Crowell-Beckwith suppositions as to worm holes, and the early universe. This is briefly alluded to in Appendix E, which fills in some details as to Appendix F, gives a statement largely based upon Mukhanov [48], i.e. how an energy flux from a prior universe may lead to release of entropy in the present universe, and it is in sync with the idea of graviton generation, of early universe entropy after traversing the H = 0 barrier in line with the work done in [42] [43] [44]. Appendix G and Appendix H give qualitative descriptions as to the behavior of the scalar field, presumably like an inflaton, which may be zero in the initial phases of entry into the "bubble" before a presumed causal barrier at H = 0, and Appendix G gives an interpretation of the largeness of a presumed energy flux which would go out of the cosmological "bubble" of initial space time.
Note that the end effect of all this is to argue for very different dynamics, of space time, i.e. for the entropy being generated just past the H = 0 barrier of space time, with a radii of say 10 −35 meters, and all that, the answers we will get out of Equation (7) will look profoundly different than say, entropy and gravitons, and GW produced at 10 −7 meters to 1 meter in radii "distance" from the start of presumed space-time. Either choice will have profound implications for interpreting Equation (7) of our text. What is given below is for what we would have for Equation (7) inputs if we have entropy produced well "before" the electro weak regime

Conclusion, Now Back to Our Treatment of the Bending of Light by Massive Gravity. What We Can Say about What We Have So Far
In [49], in Equation (12) of [49], there is an expressed equation of the form for a light ray hitting, say the Sun θ ϑ A. W. Beckwith Journal of High Energy Physics, Gravitation and Cosmology The impact parameter, b of the "photons", i.e. light ray, with the sun, and the Schwartzshield radius s r of the sun.
This is the first item to discuss, and the last term is the one which should be minimized, whereas the first two terms are in sync with [50] [51] [52] whereas the third term, which can be written up in exact parameterization, is too small to contribute much of anything to the problem. I.e. the 2 nd term is a post Newtonian contribution and the third term is a quantum correction largely based upon the Born approximation and can be seen in [50], Chapter 21 of that reference.
This derivation is part of a manuscript with the following deviation of the potential system put in, i.e.

( )
If m is the mass of a graviton, almost 10 −65 grams, whereas M is the mass of a planet, say Mercury, and that Equation (21) Our task would be to look at a total energy, say making this deduction, of 10 It would be a lot of work, but it would also be more direct than what De Rahm and other tried in [53].
What we have done, is to find a basis for a different way to address the issue of if we have relic gravitational waves at just the electro weak regime, as quantified in this paper, or if we have earlier based processes and/or the influence of recycled earlier universes, which may influence the transmission of gravitons, and possibly pre universe information to our present universe.
Do we have a repeating universe, with shared from the prior cosmos information? The logical extension of the inquiry so presented may allow for answering this question. In the meantime, the touch of using Barbour's version of time, in-A. W. Beckwith Journal of High Energy Physics, Gravitation and Cosmology itially was put in to ascertain, a working benchmark for the twinning of a definite time step, with graviton production, and also then, if graviton production, i.e. the number of gravitons, is proportional to entropy, what has been done is in essence vetting the start of times arrow, via entropy production in the universe.
Equation (7) is by necessity very preliminary and we expect to revisit it with greater precision later on.
Finally we have presented a different way to start an inquiry as to working to a solution to the vDVZ discontinuity.
See Appendix I, as to the remarks made as to the foundations of gravitational astronomy. The document so presented is expected to be in fidelity with respect to these observations and guidelines. as Cool down from Initial Expansion Commences P. Brax, A. Davis et al. [54] devised a way to describe racetrack inflation as a way to look at how super gravity directly simplifies implementing how one can have inflation with only three T (scalar) fields. The benefit to what we work with is that we may obtain two gaugino condensates and look at inflation with a potential given by [54] ( ) ( ) ( ) This has scalar fields , X φ as relatively constant and we can look at an effective kinetic energy term along the lines of ( ) ( ) − must be small so that 1 X  in a race track potential system when we analyze how to fit Equation (1) for flat potential behavior modeling inflation. This assumes that we are working with a spectra index of the form so that if the scalar field power spectrum is

Appendix B: Semi Classical Models of Entropy Generation
Kolb and Turner [56] have a temperature T related entropy density which can be treated as being written as: This pre supposes when we do it that we are able to state a total entropy as the entropy density times space time volume 4 In this situation we are writing for initial conditions with a temperature 32 10 K T ≈ for the initiation of quantum effects for quantum gravity as given by Weinberg (1972) [18] which is further elaborated upon by Padmanabhan, in [57]. 32 28 19 10 K 1.3 10 eV~1.3 10 GeV T ≈ ≈ × × . This gives us the option of comparing what we get in entropy with Seth Lloyds [58] [ ] We will examine if or not the following is actually true in terms of time, i.e.
can we write ? This is assuming that the density 00 vacuum-energỹ T ρ ≡ Λ which is initially enormous, and which will be due in terms of a transfer of energy density from a prior universe to our present universe, which will be elaborated upon later in this document.
We can if we take the absolute value of Equation (b3) and (b2) above get for small volume values good estimates as to the relative volume of the phase space in early universe cosmology where Equation (b2) and Equation (b3) are congruent with each other. For our purposes, we will take time as greater than (or equal) to a Planck time interval, in line with the temperature dependence of entropy density mentioned in Equation (b1) above.
We can compare this with Thanu Padamanadan's [59] treatment of entropy which is with regards to micro canonical ensemble as defined via The This has when we do it Total E as in Equation(c1) above, and proportional to the cosmological vacuum energy parameter. Of course, in string theory, the energy is also defined via  Furthermore we also claim that the interaction of the branes and anti-branes will form an instanton structure, which is implicit in the treatment outlined in Equation (c4), and that the numerical counting given in Equation (c6) merely reflects that branes and anti-branes, even if charge conjugates of each other have the same "wrapping number" i n .
, H Λ by 2 initial initial , H Λ . In addition we may look at inputs from the initial value of the Hubble parameter to get the necessary e folding needed for inflation, according to ( )