The first matter of business will be to introduce a framework of the speed of gravitons in “heavy gravity”. Heavy Gravity is the situation where a graviton has a small rest mass and is not a zero mass particle, and this existence of “heavy gravity” is important since eventually, as illustrated by Will [1] [2] gravitons having a small mass could possibly be observed via their macroscopic effects upon astrophysical events. Secondly, our manuscript’s inquiry also will involve an upper bound to the rest mass of a graviton. The second aspect of the inquiry of our manuscript will be to come up with a variant of the Heisenberg Uncertainty principle (HUP), involving a metric tensor, as well as the Stress energy tensor, which will in time allow us to establish a lower bound to the mass of a graviton, preferably at the start of cosmological evolution. The article concludes in its last section as to why a statement by Mukhanov in Marcel Grossman 14, 2015, Rome, that a multiverse contribution to a new universe would have a causal barrier averaging of time contributions even if there were contributions from a multiverse, so there was only one space-time contribution is possibly indefensible.

We reference what was done by Will in his living reviews of relativity article as to the “Confrontation between GR and experiment”. Specifically we make use of his experimentally based formula of [1] [2] , with the speed of a graviton, and the rest mass of a graviton, and in the inertial rest frame given as:

Furthermore, using [2] , if the rest mass of a graviton is very small we can make a clear statement of

Here, is the difference in arrival time, and is the difference in emission time/in the case of the early Universe, i.e. near the big bang, then if in the beginning of time, one has, if we assume that there is an average, and

Then, and if, so one can set

And if one sets the mass of a graviton [3] into Equation (1), then we have in the present era, that if we look at primordial time generated gravitons, that if one uses the

Note that the above frequency, for the graviton is for the present era, but that it starts assuming genesis from an initial inflationary starting point which is not a space-time singularity.

Note this comes from a scale factor, if, i.e. 55 orders of magnitude smaller than what would normally consider, but here note that the scale factor is not zero, so we do not have a space-time singularity.

We will next discuss the implications of this point in the next section, of a non-zero smallest scale factor. Secondly the fact we are working with a massive graviton, as given will be given some credence as to when we obtain a lower bound, as will come up in our derivation of modification of the values [3]

The reasons for saying this set of values for the variation of the other metric components will be in the 3^rd section and it is due to the smallness of the square of the scale factor in the vicinity of Planck time interval.

Begin with the starting point of [4] [5]

We will be using the approximation given by Unruh [4] [5] , of a generalization we will write as

If we use the following, from the Roberson-Walker metric [6] .

Following Unruh [4] [5] , write then, an uncertainty of metric tensor as, with the following inputs

Then, the surviving version of Equation (7) and Equation (8) is, then, if

This Equation (11) is such that we can extract, up to a point the HUP principle for uncertainty in time and energy, with one very large caveat added, namely if we use the fluid approximation of space-time [6] for the stress energy tensor as given in Equation (12) below.

Then

Then, Equations (11)-(13) together yield

How likely is? Not going to happen. Why? The homogeneity of the early universe will keep

In fact, we have that from Giovannini [6] , that if is a scalar function, and, then if

Then, there is no way that Equation (14) is going to come close to. Hence, the Mukhanov sugges-

tion as will be discussed toward the end of this article, is not feasible. Finally, we will discuss a lower bound to the mass of the graviton.

To begin this process, we will break it down into the following co ordinates

In the rr, and coordinates, we will use the Fluid approximation, [7] with

If as an example, we have negative pressure, with, and < 0, and, then the only choice we have, then is to set, since there is no way that is zero valued.

Having said this, the value of being non zero, will be part of how we will be looking at a lower bound to the graviton mass which is not zero.

In order to start this approximation, we will be using Barbour’s value of emergent time [8] [9] restricted to the Plank spatial interval and massive gravitons, with a massive graviton [10]

Initially, as postulated by Babour [8] [9] , this set of masses, given in the emergent time structure could be for say the planetary masses of each contribution of the solar system. Our identification is to have an initial mass value, at the start of creation, for an individual graviton.

If in Equation (11), using Equation (11) and Equation (18) we can arrive at the identification of

Key to Equation (19) will be identification of the kinetic energy which is written as. This identification will be the key point raised in this manuscript. Note that [11 raises the distinct possibility of an initial state, just before the “big bang” of a kinetic energy dominated “pre inflationary” universe. i.e. in terms of an inflaton [7] . The key finding which is in [11] is, that, if the kinetic energy is dominated by the “inflaton” that

This is done with the proviso that w < −1, in effect, what we are saying is that during the period of the “Planckian regime” we can seriously consider an initial density proportional to Kinetic energy, and call this K.E. as proportional to [7]

If we are where we are in a very small Planckian regime of space-time, we could, then say write Equation (21) as proportional to [7] , with initial degrees of freedom, and T the initial temperature as just before the onset of inflation. The question to ask, then is, what is the value of the initial degrees of freedom, and what is the temperature, T, at the start of expansion? For what it is worth, the starting supposition, is that there would then be a likelihood for an initial low temperature regime

Here, the initial, or so and so the density in Equation (21) at Planck time would, be proportional to the Planck Frequency [7]

This is at the instant of Planck time. We can then ask what would be an initial time contribution before the onset of Planck time. i.e. does Equation (22) represent the initial value of graviton frequency?

This value of the frequency of a graviton, which would be red shifted enormously would be in tandem with an initial time step of as given by [12]

This value for the initial time step would be probably lead to Pre Planckian time, i.e. smaller than 10^-43 seconds, which then leads us to consider, what would happen if a multi verse contributed to initial space-time conditions as seen in Equation (11) above. If the cosmic fluid approximation as given by Equation (12) were legitimate, and one could also look at Equation (13), then

But, then if one is looking at a multiverse, we first will start at the Penrose hypothesis for a cyclic conformal universe, starting with [13]

However, in the multiverse contribution to Equation (12) above, we would have, that

So, does something like this hold? In a general sense?

If the fluid approximation as given in Equation (12) and Equation (13) hold, then Equation (27) conceivably could be identifiable as linkable to.

If we could write, say

Then, if each j is the j^th contribution of N “multiverse” contributions to a new single universe being nucleated, one could say that there was, indeed, likely an “averaging” and that the causal barrier which Mukhanov spoke of, as to each, and actually to each graviton entering into the present universe, one could mathematically average out the results of a sum up of each of the contributions from each prior to a present universe, according to

If Equation (30) held, then we could then write

Instead, we have, Equation (28), and that it is safe to say that for each collapsing universe which might contribute to a re cycled universe that the following inequality is significant.

Hence, the absence of an averaging procedure, due to a multiverse, would then rule against a causal barrier, as was maintained by Mukhanov, in his discussion with the author, in Marcel Grossman 14, in Italy. Then the possible approximation says of

Would not hold, and that in itself may lead to a breakdown of the Causal barrier hypothesis of Mukhanov, which the author emphatically disagreed with.

A way of solidifying the approach given here, in terms of early universe GR theory is to refer to Einstein spaces, via [14] as well as to make certain of the Stress energy tensor [15] as we can write it as a modified Einstein field equation. With, then as a constant.

Here, the term in the Left hand side of the metric tensor is a constant, so then if we write, with R also a constant [15]

The terms, if we use the fluid approximation given by Equation (12) as well as the metric given in Equation (9) will then tend to a constant energy term on the RHS of Equation (35) as well as restricting i, and j, to t and t.

So as to recover, via the Einstein spaces, the seemingly heuristic argument given above. Furthermore when we refer to the Kinetic energy space as an inflaton where we assume that the potential energy is proportional to V, so as to allow us to write [7] , we can also then utilize the following operator equation for the generation of an “inflaton field” given by the following set of equations

In the case of the general elliptic operator K if we are using the Fulling reference, [16] in the case of the above Roberson-Walker metric, with the results that the elliptic operator, in this case become,

Then, according to [16] , if R above, in Equation (37) is initially a constant, we will see then, if m is the inflation mass, that

Then as an unspecified, for now constant will lead to a first approximation of a Kinetic energy dominated initial configuration, with details to be gleaned from [16] - [18] to give more details to the following equation, R here is linked to curvature of space-time, and m is an inflaton mass, connected with the field with the result that

If the frequency, of say, Gravitons is of the order of Planck frequency as in Equation (22), then this term, would likely dominate Equation (39). More of the details of this will be worked out, and also candidates for the will be ascertained, most likely, we will be looking the Rindler Vacuum as specified in [19] as well as also details of what is relevant to maintain local covariance in the initial space-time fields as given in [20] .

Why is a refinement of Equation (39) necessary?

The details of the elliptic operator K will be gleaned from [16] - [18] whereas the details of inflaton [7] are important to get a refinement on the lower mass of the graviton as given by the left hand side of Equation (24). We hope to do this in the coming year. The mass, m, in Equation (37) for the inflaton, not the Graviton, so as to have links to the beginning of the expansion of the universe. We look to what Corda did, in [21] for guidance as to picking values of m relevant to early universe conditions.

Finally, as far as Equation (39) is concerned, there is one serious linkage issue to classical and quantum mechanics, which should be the bridge between classical and quantum regimes, as far as space time applicability. Namely, from Wald (19), if we look at first of all arbitrary operators, A and B

As we can anticipate, the Pre Planckian regime may the place to use classical mechanics, and then to bridge that to the Planckian regime, which would be quantum mechanical. Taking [19] again, this would lead to a sympletic structure via the following modification of the Hamilton equations of motion, namely we will from (19) get the following re write,

Then there exists a re formulation of the Poisson brackets, as seen by

So, then the following, for classical observables, f, and g, we could write, by [19]

Then, we could write, say Equation (40) and Equation (43) as

If so, then we can set, in the interconnection between the Planck regime, and just before the Planck regime, say, by setting classical variables, as given by

Then by utilization of Equation (44) we may be able to effect more precision in our early universe derivation, especially making use of derivational work, in addition as to what is given here, as to understand how to construct a very early universe partition function Z based upon the inter relationship between Equation (44) and Equation (45) so as to write up an entropy based upon, as given in [19]

If this program were affected, with a first principle construction of a partition function, we may be able to answer if Entropy were zero in the Planck regime, or something else, which would give us more motivation to examine the sort of partition functions as stated in [22] [23] . See Appendix A as to possible scenarios. Here keep in mind that in the Planck regime we have nonstandard physics. Appendix A indicates that due to the variation we have worked out in the Planckian regime of space-time that the initial entropy is not zero. The consequences of this show up in this paper’s Appendix B, as to a specific formulation of the Ricci scalar. The consequences of Appendix A and Appendix B may be for a small cosmological constant, and large “ Hubble expansion” that there would be an initially large magnitude of cosmological pressure, even if negative, which would give credence to a non-zero cosmological entropy, that if large negative pressure, even in the Pre Planckian regime will lead to a large terms which would show up in Equation (1A), even if we used a partition function based upon Lattice Hamiltonians, as on page 135 of [26] which would usually in a lattice gauge arrangement would have considerably smaller contributions than. Note the conditions of flat space, are that Equation (B9) almost vanishes due to the behavior of the numerator, no matter how small is. The supposition is that the numerator becomes far smaller than The initiation of conditions of flat space, is also the regime in which we think that non zero entropy is started, and Appendix C gives an initial estimate of what we think Entropy would be in the aftermath of the uncertainty relationship we have outlined in this article. i.e. to first order,. We finalize our treatment as of space-time fluctuations and geometry by considering the applications of Appendix D to graviton mass, and Appendix E to the Riemann-Penrose inequality for conditions as to a minimum frequency, as a consequence of cosmological evolution, and what it portrays as consequences for Electromagnetic fields. Appendix D and E give varying initial graviton masses as a starting point, with Appendix D giving a higher initial graviton mass than what is assumed as of today. Finally, Appendix F states a pre Planckian kinetic energy so the inflaton [7] . This last step, so important to our development will be considerably refined in future document.

We start the process of understanding the consequences of choosing the inflaton [7] as given in part in Appendix G and Appendix H.

The consequences of the above mentioned appendix entries are, mainly that if we wish to avoid the problems given in Appendix G and Appendix H that we really need to keep in mind the following:

1) Our uncertainty principle is fundamentally different from the Black hole commensurate uncertainty principles cited in Appendix G. They do not take into consideration the possibility that there may be Pre Planckian time, which may immensely impact the fluctuations in the metric tensor.

2) As an exercise, Appendix G shows that a highly restricted parameter space is required if we insist upon making our Pre Planckian uncertainty principle commensurate with the possibility that our metric Heisenberg Uncertainty principle (HUP) is in fact, giving us the flat space result which was brought up by Mukhanov, in Marcel Grossman 14. But it is so restrictive that we doubt it is actually mathematically a useful development

3) Appendix H gives us Equation (H1) which is the Pre Planckian Inflaton, which is of foundational importance in determination of if we have general relativity or some other gravitational theory, i.e. the issue of if there is an additional polarization. But to do that, we have to for reasons given in Appendix G, choose our parameter space, wisely. It is still not clear if there is a connection between Black hole physics, and avoiding the catastrophe of Bicep 2. For that much additional experimental work has to be done.

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

Andrew Walcott Beckwith, (2016) Gedanken Experiment for Refining the Unruh Metric Tensor Uncertainty Principle via Schwarzschild Geometry and Planckian Space-Time with Initial Nonzero Entropy and Applying the Riemannian-Penrose Inequality and Initial Kinetic Energy for a Lower Bound to Graviton Mass (Massive Gravity). Journal of High Energy Physics, Gravitation and Cosmology,02,106-124. doi: 10.4236/jhepgc.2016.21012

We will be looking at inputs from page 290 of [23] so that if

And using Ng’s infinite quantum statistics, we have to first approximation [24] [25]

This is due to a very small but non vanishing with the partition functions covered by [23] , and also due to [24] [25] with a non-zero number of initial “particle” or information states, about the Planck regime of space-time, so that the initial entropy is non zero.

We begin with Kolb and Turner [7] discussion of the Roberson-Walker metric, say page 49 with, if R is the Ricci scalar, and k the measurement of if we have a close, open, or flat universe, that if

Then by [7]

Leading to

If [7] , then with a bit of algebra

Next, using [27] , on page 47, at the boundary between Pre Planckian to Planckian space-time we will find

Then, we can obtain

Right at the start of the Planckian era,

The consequences of this would be that right after the entry into Planckian space time, that there would be the following change of pressure

Then, the change in the k term would be like, say, from Pre Planckian to Planckian space time

This goes almost to zero if the numerator shrinks far more than the denominator, even if the initial scale factor is of the order of 10⁻⁵⁵ or so.

We are making use of the Padmanabhan publication of [28] [29] where we will make use of

Then, if is for the energy of the Universe after the initiation of Equation (11) as a bridge between Pre Planckian, to Planckian physics regimes we could write, then

The value of initial entropy, should be contrasted with the entropy for the entire Universe as given in [30] below.

Here we can view the possibility of considering the following, namely [31] is extended by [32] so we can we make the following identification?

Should the N above, be related to entropy, and Equation (8) this supposition has to be balanced against the following identification, namely, as given by T. Padmanabhan [28] [29]

But should the energy in the numerator in Equation (D2) be given as say by (C2), of Appendix C, we have quintessence. then there would have been quintessence, i.e. variation in the “Einstein constant”, which would have a large impact upon mass of the graviton, with a sharp decrease in being consistent with an evolution to the ultra-light value of the Graviton, with initial frequencies of the order of say for wavelength values initially the size of an atom,

The final value of the frequency would be of a magnitude smaller than one Hertz, so as to have value of the mass of the graviton would be then of the order of 10⁻⁶² grams [10] , due to Equation (D2) approaching [31] below, namely

Leading to the upper bound of the Graviton mass of about 10⁻⁶² grams [31] [32] in the present era

Equation (D5) has a different value if the entropy/particle count is lower, as has been postulated in this note. But the value of Equation (D5) becomes the Graviton mass of about 10⁻⁶² grams [10] in the present era which is in line with the entropy being far larger in the present era [30]

If from Giovannini [33] we can write

Refining the inputs from Equation (E1) means more study as to the possibility of a non-zero minimum scale factor [34] , as well as the nature of as specified by Giovannini [33] . We hope that this can be done as to give quantifiable estimates and may link the non-zero initial entropy to either Loop quantum gravity “quantum bounce” considerations [35] and/or other models which may presage modification of the sort of initial singularities of the sort given in [1] . Furthermore if the non-zero scale factor is correct, it may give us opportunities as to fine tune the parameters given in [34] below.

where the following is possibly linkable to minimum frequencies linked to E and M fields [34] , and possibly relic Gravitons

So, now we investigate the question of applicability of the Riemann Penrose inequality which is [36] , p431, which is stated as

Riemann Penrose Inequality: Let (M, g) be a complete, asymptotically flat 3-manifold with Non negative-scalar curvature, and total mass m, whose outermost horizon has total surface area A. Then

And the equality holds, if (M, g) is isometric to the spatial isometric spatial Schwartzshield manifold M of mass m outside their respective horizons.

Assume that the frequency, say using the frequency of Equation (E3), and of Equation (E4) is employed. So then say we have, if we use dimensional analysis appropriately, that

Assume that we also set the input frequency as to Equation (E3) as according to i.e. does

Our supposition is that Equation (E6) should give the same frequency as of Equation (D3) above. So if we have in

In doing this, this is a frequency input into Equation (E3) above where we are safely assuming a graviton mass of about [10]

Does the following make sense? i.e. look at, when

We claim that if this is an initial frequency and that it is connected with relic graviton production, that the minimum frequency would be relevant to Equation (E3), and may play a part as to admissible B fields

Note, if Appendix D is used, this makes a re do of Equation (E8) which is a way of saying that the graviton mass given by [10] no longer holds.

In either case, Equation (E8) and Equation (E3) in some configuration may argue for implementation of work the author did in reference [37] as to relic cylindrical GW, i.e. their allowed frequency and magnitude, so considered.

We give this as a plausibility argument which undoubtedly will be considerably refined, but its importance cannot be overstated. i.e. this is for Pre inflationary, Pre Planckian physics, so as to get a lower bound to the Graviton mass. To do this, we look at what [7] is saying and also we will be enlisting a new reference, [38] , by Bojowald, and also Padmanbhan [39] as to details to put in, so as to confirm a dominance of Kinetic energy. Start with a Friedman equation of

We will treat, then the Hubble parameter, as

Now from Padmanabhan, [39] , we can write density, in terms of flux according to

Then using 463 of [39] , if T is temperature, here, then if N is the particle count in the flux region per unit time (say Planck time), as well as using the “ideal gas law” approximation, for superhot conditions

Next, according to [38] , we can make the following substitution.

Therefore, if

If the scale factor is very small, say of the order of, then no matter how fall the initial volume is, in four space (it cancels out in the first part of the brackets), it’s easy to see then that [7] .

We will in the future add more structure to this calculation so as to confirm via a precise calculation that the lower bound to the graviton mass, is about 10⁻⁷⁰ grams. This value of 10⁻⁷⁰ grams is an approximation, via dimensional analysis and will be improved, by more exact calculations.

We are looking here at what was done in [40] [41] and noting that in particular that the [40] calculation of fluctuations in energy as given by bounds given by Black hole physics, such that, if we pick Planck’s constant

Compare that with our given value of

This should be compared with our value of equivalence between these two equations which demands

The collapse to a situation with ourselves recovering the standard Heisenberg Uncertainty relationship for fluctuations of energy is seen in, if Equation (G1) and Equation (G2) are both correct having then that

Here, we want the situation for which we would have any time situation with the fluctuation of time, going to a very small number, and that the inverse fluctuation in time going to infinity would be, trivially due to, if is of Planck length, obtaining for which.

It’s an equation for, with a vanishingly small contribution for. i.e. we would have, to first order, i.e. being very small. But that in turn would require, to first order

This would be equivalent to, then setting

Then by necessity, we would want to have a situation for which to have a more general situation as given in our document for a

In fact, to reconcile Equation (G1) and Equation (G2) in the case of recovering a

That not only would obey Equation (G7) that it would likely be fairly large.

The situation as given by L. Crowell in [41] as it is attuned to dimensional analysis, as given in

Here, R is the radius of a sphere for the origins of an emitted wave, which is in turn requiring R to be extraordinarily small. i.e. we recover the inputs for our analysis of [40] as it applies to our document but only if we have extremely sharp restraints upon R, if we wish to have fidelity with Equation (G4) and Equation (G5) in the sense of recovery of the traditional Heisenberg relations. is a Planck time interval as given in [41] It is extremely small, commensurate with Equation (G9) being approximately Planck Length in value.

The problem with Equation (G9) is that there is no provision given as to Pre Planckian length values, and that it is restricted, dimensionally to Planckian Length and temperature, with no clue given as to what happens before a Planck length.

Quoting from the Authors’ recent publication [42] .

From [43] we have the following to consider, namely trying to determine restraints upon the nature of gravity, i.e. is it consistent with General relativity or do we have an alternative situation as given in the following quote. We hope that getting a consistent model of inflaton physics will help clarify the following alternatives

Quote, in [42] of the result given in [43] :

This fact rules out the possibility of treating gravitation like other quantum theories, and precludes the unification of gravity with other interactions. At the present time, it is not possible to realize a consistent Quantum Gravity Theory which leads to the unification of gravitation with the other forces [17] [18] . On the other hand, one can define Extended Theories of Gravity those semi classical theories where the Lagrangian is modified, in respect to the standard Einstein-Hilbert gravitational Lagrangian, adding high-order terms in the curvature invariants (terms like R2, etc…) or terms with scalar fields non minimally coupled to geometry (terms like φ2R) [17] [18] .

End of quote from [43] .

We then will cite what is in [42] i.e. namely that our uncertainty relationship leads to inflaton physics, as given in the following quote.

Quote, from [42]

Needless to say we will require careful analysis of the result as given in reference [42] that

This enormous value for the inflaton, initially, needs to be examined further. It further should be linked to Corda’s pioneering work with “gravity’s breath”, i.e. traces of the inflaton as given by [21] [44] and is the justification of Equation (H1) above. We can use this to determine what to make of the stochastic background of pre space time physics.

Next, Avoiding the Bicep 2 mistake. What we can do with Equation (H1)

Following [42] [43] what we are doing is examining the stochastic regime of space-time where the following holds.

Omni-directional gravitational wave background radiation could arise from fundamental processes in the early Universe, or from the superposition of a large number of signals with a point-like origin. Examples of the former include parametric amplification of gravitational vacuum fluctuations during the inflationary era, termination of inflation through axion decay or resonant preheating, Pre-Big Bang models inspired by string theory, and phase transitions in the early Universe; the observation of a primordial background would give access to energy scales of 10 to the 9 power, up to 10 to the 10 power GeV, well beyond the reach of particle accelerators on Earth

Needless to say though, we need above all to avoid getting many multiple stochastic signals, in what we process for primordial gravitational waves, and to use, instead tests to avoid getting dust signals which is what doomed Bicep 2, i.e. as was made very clear in [42] [45] [46] .

i.e. the problem is in avoiding multiple stochastic signals, and this is explained in the conclusion of [42] . But to obtain what is in [42] , Equation (H1) has to be thoroughly understood, and Equation (H1) is commensurate with the details as cited in Equation (G3) to Equation (G7) which have to be vetted experimentally. i.e. the uncertainty principle as cited in Equation (H1) leads to an inflaton which will allow us to determine if a third Polarization exists, as in scalar-tensor gravity, or the more traditional considerations given in [42] [43] .

This in turn may allow understanding if our document is commensurate with the considerations given in [47] .