[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.

Acknowledgements

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

Appendix A: Scenarios as to the Value of Entropy in the Beginning of Space-Time Nucleation

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

(1A)

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

(2A)

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.

Appendix B: Calculation of the Ricci Tensor for a Roberson-Walker Space-Time, with Its Effect upon the Measurement of If or Not a Space Time, Is Open, Closed or Flat

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

(B1)

Then by [7]

(B2)

(B3)

Leading to

(B4)

If [7] , then with a bit of algebra

(B5)

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

(B6)

Then, we can obtain

Right at the start of the Planckian era,

(B7)

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

(B8)

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

(B9)

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 1055 or so.

Appendix C: Initial Entropy, from First Principles

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

(C1)

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

(C2)

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

Appendix D: Information Flow, Gravitons, and Also Upper Bounds to Graviton Mass

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

(D1)

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]

(D2)

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,

(D3)

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

(D4)

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

(D5)

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]

Appendix E: Applying the Riemannian Penrose Inequality with Applications in Our Fluctuation

If from Giovannini [33] we can write

(E1)

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.

(E2)

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

(E3)

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

(E4)

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

(E5)

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

(E6)

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]

(E7)

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

(E8)

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.

Appendix F: First Principle Treatment of Pre Planckian Kinetic Energy So the Inflaton [7]

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

(F1)

We will treat, then the Hubble parameter, as

(F2)

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

(F3)

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

(F4)

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

(F5)

Therefore, if

(F6)

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.

Appendix G: The Generalized Uncertainty Principle in Quantum Gravity Compared with Our Heisenberg Uncertainty Principle for a Metric in Pre Planckian Space-Time

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

(G1)

Compare that with our given value of

(G2)

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

(G3)

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

(G3)

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.

(G4)

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

(G5)

This would be equivalent to, then setting

(G6)

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

(G7)

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

(G8)

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

(G9)

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.

Appendix H: Considerations as to Bicep 2, the Matter of Scalar-Tensor Polarizations as an Alternative to General Relativity and Alternate Gravitational Theories. And Experimental Tests of General Relativity via Interferometric Methods

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

(H1)

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] .

Conflicts of Interest

The authors declare no conflicts of interest.

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