Fjortof ’ s Theorem does not Apply for Defining Instability for Early Universe Thermodynamic Potentials . Asking if Nucleated Particles Result at / before ElectroWeak Era due to Injection of Matter-Energy at the Big Bang ?

This paper uses the “Fjortoft theorem” for defining necessary conditions for instability. The point is that it does not apply in the vicinity of the big bang. We apply this theorem to what is called by T. Padmanabhan a thermodynamic potential which becomes would be unstable if conditions for the applications of “Fjortoft’s theorem” hold. In our case, there is no instability, so a different mechanism has to be appealed to. In the case of vacuum nucleation, we argue that conditions exist for the nucleation of particles as of the electroweak regime. Due to injecting material from a node point, in spacetime. This regime of early universe creation, coexits with the failure of applications of “Fjortoft” theorem in such a way as to give necessary and sufficient conditons for matter creation, in a way similar to the Higgs Boson . .


1.Introduction
We first start off with a review of the classicial Fjortoft theorem [1] and from there apply it to an early universe thermodynamic potential described by T. Padamanabhan [2] in Dice 2010. The objective will be to show that one can come up with a first principle creation of nucleated "particles", likely from a semi classical stand point which can be introducing the creation of mass without appealing directly to the Higgs Boson in the first place. That due to the fact that the Fjortoft theorem does not apply.There is an inflection point for the speed up of acceleration of the universe which exists one billion years ago for reasons which we will introduce in this manuscript. But no such inflection point at the origin of the big bang itself, or at the electroweak era either.

Describing the Fjortoft theorem
From [1] we have that the theorem to be considered should be written up as follows, namely, look at Fjortoft theorem: A necessary condition for instability is that if z  is a point in spacetime for which For the proof, see [1] and also consider that the main discussion is to find instability in a physical system which will be described by a given potential U . Next, we will construct in the boundary of the EW era, a way to come up with an optimal description for U 2. Constructing an appropriate potential for using Fjortoft theorem in cosmology for the early universe cannot be done. We show why To do this, we will look at Padamanabhan [2] and his construction of (in Dice 2010) of thermodynamic potentials he used to have another construction of the Einstein GR equations. To start, Padamanabhan [2] We now will look at () a a b matter ab What this is saying is that there is no unique point, using this a   for which (5) holds. Therefore, we say there is no official point of instability of a   due to (4). The Lagrangian structure of what can be built up by the potentials given in (4) with respect to a   mean that we cannot expect an inflection point with respect to a 2 nd derivative of a potential system. Such an inflection point designating a speed up of acceleration due to DE exists a billion years ago [3]. Also note that the reason for the failure for (5) to be congruent to (1) is due to To use Eq (6) properly, we use the material, in our reasoning from [4], [5], [6], [7], and [8]. How and why can we do this ?
We state that (6) tells us is that there is an embedding structure for early universe geometry, some of which may take the form of the following diagram. Figure 1, from [7,8] 3. Working with a way to achieve energy injection into the universe, without appealing to Fjortoft theorem for alleged instabilities starting from Padmanabhan thermodynamic potential terms Padmanabhan [2] introduced the following discussion as to entropy, namely starting with energy, we have And the n value as in (7) (9) and (11) we state that the change in number count given in (11) is really a holographic surface pheonmena, with N defined [2]   The upshot is that we can, as implied by Ng [ 4 ] easily reference a change in entropy via [4], [5], [6] Sn (13) While having a change in n as due to a change in the spatial surface of spacetime as given in (11), we have to realistically infer that the local acceleration temperature (10) is from another pre universe contruction and that local instability is ruled out by (5) and (6). This leads us to ask as to what would be an acceptable way to form the formation of mass, i.e. say the mass of a graviton, via external factors introduced into our universe prior to the Electroweak era, in cosmology. To do that, look at if there are two branes on the 5 AdS space-time so that with one moving and one stationary, we can look at Figure 1 as background as to introduce such external factors in our present space-time universe during its initial expansion phase 4. Fall out from adopting Figure 1 and that due to no instability in the Padamanabhan supplied potentials. i.e. a way to obtain graviton mass via a root finding method.
Using [7], what we find is that there are two branes on the 5 AdS space-time so that with one moving and one stationary, we can look at figure 1 which is part of the geometry used in the spatial decomposition of the differential operator acting upon the h • Fourier modes of the ij h operator [7] . As given by [7], [8] we have that Using [8] (and also [7]) the solution to (14) above takes the form of having ij e is a polarization tensor, and the function   2 J my is a 2 nd order Bessel function [14] . A generalization offered by Durrer et al. [7], [8] leads to With the factor of coming in due to a boundary condition upon the wall of a brane put in, i.e. looking at [7]. With the right hand side of (16) due to a domain wall tension of a brane.
This will be in our example set as not equal to zero, in the right hand side, but equal to an extremely small parameter, namely The right hand side of (19) represents very small brane tension, which is understandable. Then using [7], [8], [9] , i.e.
Then, (22) is acting much as in [7], and [8], whereas, one is recovering a simple numerical exercise as to obtain a suitable solution as given by (18), and (19) due to [7,8] where the domain tension of the brane vanishes. The novelty as to this approach given in (22) is to obtain a time dependent behavior of the mass of the graviton, In doing all of this, keep in mind the mathematical information given in [9] which is repeatedly used in [7] and [8]; Needless to say, (22) can only be solved for, numerically, i.e. fourth order polynomial solutions for quartic equations still give over simplified dynamics, especially if (24) holds, and makes things more complicated. This is all being done to keep fidelity with respect to [10], and [11] as a possible feature of brane world dynamics as reflected in [10], [11], as well as certain issues brought up in [12] , as to what is a semi classical argument can obtain a usually quantum result. It also would allow for eventually understanding if entropy can also be stated in terms of gravitons alone in early universe models as was proposed by Kiefer & Starobinsky, et. al. [13]. Finally, it would address if QM is embedded in a larger deterministic theory as advocated by t' Hooft [14], as well as degrees of freedom in early universe cosmology as brought up by Beckwith in Dice 2010 [15]. We argue that making this step is consistant with keeping the value of Planck's constant uniform in spite of Avessans theories suggesting it vary in time [16] . To do this, we make extensive use of [17] and [18].
It is now then time to do a re cap and to organize how such speculation can be vetted using experimental proceedures. To do this we re cap what can be said about traces massive gravitons can be detected, prior to our conclusion WHICH

MAY LEAD TO AN EXPLANTION OF THE FOLLOWING ENTROPY FORMULA[19]
This is a bridge to future projects which should be kept in mind. I.e, could our formulation of graviton physics lead to identification of gravitons, in the early universe as the main driver of graviton physics being the primary entropy generator as suggested by [19]?

. 5. CONCLUSION Semiclassical method of obtaining graviton mass procedure cannot be ruled out, and it impacts relic GW searches
First of all, review the details of a massive graviton imprint upon ij h , and then we will review the linkage between that and certain limits upon h • The bottom line is that the simple de composition with a basis in two polarization states, of ,   will have to be amended and adjusted, if one is looking at massive graviton states.
In addition further developments as to (31) could influence giving a semi classical interpretation as to entrophic origins of gravity, along the lines brought up by both t'Hooft , indirectly [14] , and Lee [22] directly.
The experimental gravity considerations are covered in [23] , [24] , [25], and [26] , and the idea should be especially to in our work to examine if [23] and [25] in terms of gravity are adhered to. As these are LIGO projects, we should be looking to see if what we are doing contravenes or backs the post Newtonian approximations of physics, so brought up.
Reference [25] is a must to review. In it, Corda reviews GR tests and our document must not contravene these basics. Can we obtain through our representation of gravitons, confirmation, or refutation of if the data sets are in adherence, or partially refute General Relativity.
As far as [26], in terms of quantum cosmology, it is another similar parallel development to the ideas raised here. I urge readers to investigate it.
Finally, in lieu of [27] the author urges readers to look at Appendix A, which is a summary of what the author views as to what would be foundational investigation of gravity, and to see if it can be made in adherence to GR, and possibly quantized.
The author urges readers to look at [27] , as well as [28] by Kieffer, and that when seeing Appendix A, that this is a schematic the author believes would be appropriate for an investigation to confirm if Gravity could be derived as having quantum roots. Which in turn would affecxt the viability of presumed quantum gravity following, if we wish to transfer to quantum systems, we need to do the following, i.e. add to the initial classical Hamiltonian, T Eq. (A1), in a Poisson bracket formulation, was used by Dirac to transform to a set of quantization conditions, in pages 25 to 43 of. The problem is, that it is difficult to come up with constraint equations, as given in the top level of Eq.(A1) The following is easy to do, if you ignore constraints  [27] So as given in [27], as stated by Kieffer, [28] The derivation of the acceleration equation for GR, using the two equations cited is in [30] , page 60 In addition we will derive the Fluid equation also used, which is the same form used in Eq.(A5) making a linkage to relativity and quantum mechanics, possible, if one uses the following steps, as given on page 59 of [30] The GR and classical physics forms of the fluid equation, so derived, in Eq.(A8) and the results at the bottom of Eq. (A7) would allow us to make connection, with a lot of work to the sort of reasoning used in Eq. (A5) above, but due to the difference in the Friedman equation, in classical and GR form, as noted in Eq.(A6) , it would be using the Solvay methods , extremely difficult to make connection between an acceleration equation, using scale factors, as given in Eq. (A6) and Eq. (A7) with the Eq. (A5)(58) connection between classical and quantum mechanics with respect to an acceleration of the universe acceptable in both GR and quantum form.
We can state though that a bridge to the Fluid equation, as given i [27]n Eq.(A8) and Eq.(A5) would at least in principle very doable. Having said that, let us now go to the ideas of Quantum Geometrodynamics, as far as their use and future prospects to the study of Solvay 1927 methods [31], and quantum gravity issues [28] End of our quote from [27] As a close to this, all this, in terms of quantum gravity should also keep in mind issues brought up in [31], and [32], in particular, quantum entanglement and how information is transferred in cosmology. i.e. the Geometrodynamics part of [28]