^{*}

We present what is relevant to squeezed states of initial space time and how that affects both the composition of relic GW, and also gravitons. A side issue to consider is if gravitons can be configured as semi classical "particles", which is akin to the Pilot model of Quantum Mechanics as embedded in a larger non linear "deterministic" background.

Gravitons may be de composed via an instanton-anti instanton structure. i.e. that the structure of SO(4) gauge theory is initially broken due to the introduction of vacuum energy [_{(4)} gauge theory is able to describe gravity provided the gauge field possesses a specific polarized vacuum state. In this vacuum the instantons and anti-instantons have a preferred direction of orientation”, and furthermore “Gravitons appear as the mode describing propagation of the gauge field which strongly interacts with the oriented instantons” Furthermore, as given by Ivan Andrić, Larisa Jonke and Danijel Jurman [

1) Modeling of entropy, generally, as kink-anti-kinks pairs with the number of the kink-anti-kink pairs. This number, is, initially in tandem with entropy production, as will be explained later2) The tie in with entropy and gravitons is this: the two structures are related to each other in terms of kinks and anti-kinks. It is asserted that how they form and break up is due to the same phenomenon: a large insertion of vacuum energy leads to an initial breakup of both entropy levels and gravitons. When a second-order phase transition occurs, there is a burst of relic gravitons. Similarly, there is an initial breakup of net entropy levels, and after a second-order phase transition, another rapid increase in entropy.

The supposition we are making here is that the value of N so obtained is actually proportional to a numerical graviton density we will refer to as , provided that there is a bias toward HFGW, which would mandate a very small value for . Furthermore, structure formation arguments, as given by Perkins [

where we assume due to inflation

At the very onset of inflation, , and if (assuming) is due to inputs from a prior universe, we have a wide range of parameter space as to ascertain where comes from and plays a role as to the development of entropy in cosmological evolution In the next Chapter , we will discuss if or not it is feasible/reasonable to have data compression of prior universe “information”. It suffices to say that if is transferred from a prior universe to our own universe at the onset of inflation, at times less than Planck time seconds, that enough information MAY exit for the preservation of the prior universe’s cosmological constants, i.e. (fine structure constant) and the like. Confirmation of this hypothesis depends upon models of how much “information” actually require to be set in place, at the onset of our universe’s inflation, a topic which we currently have no experimental way of testing at this current time.

It is useful to state this convention for analyzing the resulting entropy calculations, because it is a way to explain how and why the number of instanton-anti instanton pairs, and their formulation and break up can be linked to the growth of entropy. If, as an example, there is a linkage between quantum energy level components of the quantum gas as brought up by Glinka [4,5] and a number of instanton-anti instanton pairs, then it is possible to ascertain a linkage between a Wheeler De Witt worm hole introduction of vacuum energy from a prior universe to our present universe, and the resulting brane-anti brane (instanton-anti instanton) units of entropy. Such an approach may permit asking how information is transferred from a prior to the present universe .What would be ideal would be to make an equivalence between a quantum number, n, say of a quantum graviton gas, as entering a worm hole, i.e. going back to the Energy (quantum gas), and the number of pairs of brane-anti brane pairs showing up in an entropy count, and the growth of entropy. We are fortunate that Dr. Jack Ng’s research into entropy [

Furthermore, finding out if or not it is either a drop in viscosity [7,9,10], then

or a major increase in entropy density may tell us how much information is, indeed, transferred from a prior universe to our present. If it is, for all effective purposes, at the moment after the pre big bang configuration , likely then there will be a high degree of “information” from a prior universe exchanged to our present universe. If on the other hand, due to restriction of ‘information from four dimensional “geometry” to a variable fifth dimension, so as to indicate almost infinite collisions with a closure of a fourth dimensional “portal” for information flow, then it is likely that significant data compression has occurred. While stating this, it is note worthy to state that the Penrose-Hawking singularity theorems do not give precise answers as to information flow from a prior to the present universe. Hawking’s singularity theorem is for the whole universe, and works backwards-in-time: it guarantees that the big-bang has infinite density. This theorem is more restricted, it only holds when matter obeys a stronger energy condition, called the dominant energy condition, which means that the energy is bigger than the pressure. All ordinary matter, with the exception of a vacuum expectation value of a scalar field, obeys this condition.

This leaves open the question of if or not there is “infinite” density of ordinary matter, or if or not there is a fifth dimensional leakage of “information” from a prior universe to our present. If there is merely infinite “density”, and possibly infinite entropy density/disorder at the origin, then perhaps no information from a prior universe is transferred to our present universe. On the other hand, having, or at least be very small may indicate that data compression is a de rigor way of treating how information for cosmological parameters, such as, G, and the fine structure constant. arose, and may have been recycled from a prior universe. Details about this have to be worked out, and this because that as of present one of the few tools which is left to formulation and proof of the singularity theorems is the Raychaudhuri equation, which describes the divergence θ of a congruence (family) of geodesics, which has a lot of assumptions behind it, as stated by Naresh Dadhich [

of the order of being within an order of magnitude of the Planck length value, as implied by Beckwith [

This suggests that entropy scaling is proportional to a power of the vacuum energy, i.e., entropy ~ vacuum energy, if is interpreted as a total net energy proportional to vacuum energy, as given below. Conventional brane theory actually enables this instanton structure analysis, as can be seen in the following. This is adapted from a lecture given at the ICGC-07 conference by Beckwith [

The approximation we are making, in this treatment initially is that where we are looking at a potential energy term [

What we are paying attention to, here is the datum that for an exponential potential (potential energy) [

De facto, what we come up with pre, and post Planckian space time regimes, when looking at consistency of the emergent structure is the following. Namely, [

for (4a)

Also, we would have

for (4b)

The switch between Equations (4a) and (4b) is not justified analytically. i.e. it breaks down. Beckwith, et al. [

Now according to Weinberg [

so that one has a scale factor behaving as

Then, if

there are no quantum gravity effects worth speaking of. i.e., if one uses an exponential potential a scalar field could take the value of, when there is a drop in a field from to for flat space geometry and times to [

Then the scale factors, from Planckian time scale as [

The more then the less likely there is a tie in with quantum gravity. Note those that the way this potential is defined is for a flat, Roberson-Walker geometry, and that if and when then what is done in Equation (8) no longer applies, and that one is no longer having any connection with even an octonionic Gravity regime. If so, as indicated by Beckwith, et al. [

Here is a quick review of how to have an instaton-anti instanton construction for entropy, and then proposing a similar construction for gravitons. Afterwards, we will analyze squeezed states. It is the authors conviction that semi classical treatment of Gravitons, if gravitons are in an instanton-anti instanton paring is equivalent to the break down of the “thin wall approximation” used in density wave physics. In what may be by some peoples visualization, an outrageous simplication, the issue of squeezing of graviton states is similar to what happens with the break down of the purely quantum mechanical analogy done for initially non squeezed states, which when squeezed have their own non quantum mechanical flavor.

We will start first looking at entropy, as an instanton-anti instanton construction and go from there:

Traditionally, minimum length for space-time benchmarking has been via the quantum gravity modification of a minimum Planck length for a grid of space-time of Planck length, whereas this grid is changed to something bigger

.

So far, we this only covers a typical string gas model for entropy. is assigned as the as numerical density of brains and anti-branes. A brane-antibrane pair corresponds to solitons and anti-solitons in density wave physics. The branes are equivalent to instanton kinks in density wave physics, whereas the antibranes are an anti-instanton structure. First, a similar pairing in both black hole models and models of the early universe is examined, and a counting regime for the number of instanton and anti-instanton structures in both black holes and in early universe models is employed as a way to get a net entropy-information count value. One can observe this in the work of Gilad Lifschytz [

electron charge and for the corresponding anti-kink

positron charge. Here, in the bottom expression, is the number of kink-anti-kink charge pairs, which is analogous to the simpler CDW structure [

This expression for entropy (based on the number of brane-anti-brane pairs) has a net energy value of as expressed in Equation (9) above, where is proportional to the cosmological vacuum energy parameter; in string theory, is also defined via

Equation (10) can be changed and rescaled to treating the mass and the energy of the brane contribution along the lines of Mathur’s^{ }CQG article [

is the tension of the ith brane, and are spatial dimensions of a complex torus structure). The toroidal structure is to first approximation equivalent dimensionally to the minimum effective length of

times Planck length centimeters

The windings of a string are given by Becker et al. [

Now, how do we make sense of the following entropy values? Note the following:

As an example of present confusion, please consider the following discussion where leading cosmologists, i.e. Sean Carroll [^{88 }in non dimensional units for observable non dimensional entropy units for the observable universe. Assume that there are over one billion spiral galaxies, with massive black holes in their center, each with entropy, and then there is due to spiral galaxy entropy contributions entropy units to contend with, vs. entropy units to contend with for the observed universe. i.e. at least a ten to the eight order difference in entropy magnitude to contend with. The author is convinced after trial and error that the standard which should be used is that of talking of information, in the Shannon sense, for entropy, and to find ways to make a relationship between quantum computing operations, and Shannon information. Making the identification of entropy as being written as. This is Shannon information theory with regards to entropy, and the convention will be the core of this text. What is chosen as a partition function will vary with our chosen model of how to input energy into our present universe. This idea as to an input of energy, and picking different models of how to do so leading to partition functions models is what motivated research in entropy generation. From now on, there will be an effort made to identify different procedural representations of the partiton function, and the log of the partion function with both string theory representations, i.e. the particle count algorithm of Y. Jack Ng [

A further datum to consider is that Equation (8) with its variance of density fluctuations may eventually be linkable to Kolmogrov theory as far as structure formation. If we look at R. M. S. Rosa [

Equation (13) above can be linked to an eddy break down process, which leads to energy dissipated by viscosity. If applied appropriately to structures transmitted through a “worm hole” from a prior to a present universe, it can explain 1）How there could be a break up of “encapsulating” structure which may initially suppress additional entropy beyond, in the onset of inflation 2) Provide a “release” mechanism for [

, with

perhaps a starting point for increase in entropy in

, rising to for times up to 1000 seconds after the big bang.

Here is, in a nutshell the template for the Gravitons which will examine, and eventually link to Gravitational waves, and entropy.

As mentioned above, there is a question of what frequency range of GW is dominant during the onset of the big bang. To begin with let us look at frequency range of GW from relic conditions. As given by for a peak amplitude as stated by Tina Kahniashvili [

The equation, as given by Kahniashvili [

(15)

Here, in this choice of magnitude h of a GW today, and frequency f detected today, as presumed by using a factor given by Kahniashvili [

(16)

Why? The factor is due to complicated physics which gives a tensor/scalar ratio As well as

Why? Equation (17) is a two correlation point function, much in the spirit of calculations of two point correlation functions, i.e. greens functions of Quantum field theory. See [

This is where, commonly, we have a way to interpret in terms of via

As well as a wave equation we can write as

What is above, is a way for making sense of GW “density” as given by the formula

Here, the temperature for the onset of a phase transition, i.e. usually interpreted as a 2nd order phase transition plays a major role as to if or not the frequency, f, for today is very low, or higher, and if or not energy density is high, or low, as well as the attendant amplitude of a GW, as given by Equation (19) above is important. Furthermore appropriate calculations of Equation (21) very much depend upon the correlation function as given by Equation (17) is correctly done, allowing for a minimization of sources of noise, of the sort alluded to by [

If the frequency is much lower, we will see, if the particle-wave duality has large, for DM candidates

This graviton counting as given in Equation (22) will next be connected to information counting which will be a necessary and sufficient condition for information exchanged from a prior to the present universe.

A. K. Avessian’s [

(24)

The idea is that we are assuming a granular, discrete nature of space time. Futhermore, after a time we will state as t ~ t_{Planck} there is a transition to a present value of space time, which is then probably going to be held constant. It is easy to, in this situation, to get an inter relationship of what is with respect to the other physical parameters, i.e. having the values of written as, as well as note how little the fine structure constant actually varies. Note that if we assume an unchanging Planck’s mass

this means that G has a time variance, too. This leads to us asking what can be done to get a starting value of recycled from a prior universe, to our present universe value. What is the initial value, and how does one insure its existence? We obtain a minimum value as far as “information” via appealing to Hogans [

and this can be compared with A. K. Avessian’s article [

i.e. a choice as to how has an initial value, and entropy as scale valued by gives us an estimate as to compressed values of which would be transferred from a prior universe, to todays universe. If, this would mean an incredibly small value for the INITIAL H parameter, i.e. in pre inflation, we would have practically NO increase in expansion, just before the introduction vacuum energy, or emergent field energy from a prior universe, to our present universe. Typically though, the value of the Hubble parameter, during inflation itself is HUGE, i.e. H is many times larger than 1, leading to initially very small entropy values. This means that we have to assume, initially, for a minimum transfer of entropy/information from a prior universe, that H is neligible. If we look at Hogan’s holographic model, this is consistent with a non finite event horizon [

This is tied in with a temperature as given by

Nearly infinite temperatures are associated with tiny event horizon values, which in turn are linked to huge Hubble parameters of expansion. Whereas initially nearly zero values of temperature can be arguably linked to nearly non existent H values, which in term would be consistent with as a starting point to entropy. We next then must consider how the values of initial entropy are linkable to other physical models. i.e. can there be a transfer of entropy/information from a pre inflation state to the present universe. Doing this will require that we keep in mind, as Hogan writes, that the number of distinguishable states is writable as [

If, in this situation, that N is proportional to entropy, i.e. N as ~ number of entropy states to consider, then as H drops in size, as would happen in pre inflation conditions, we will have opportunities for N ~ 10^{5}. ^{}

Camp and Cornish [_{ij} with a typically traceless value summed as and off diagonal elements of on each side of the diagnonal to mix with a value of

This assumes is the distance to the source of gravitational radiation, with the retarded designation on the Equation (30) denoting replaced by a retarded time derivative, while TT means take the transverse projections and substract the trace. Here, we call the quadrupole moment, with a density measurement. Now, the following value of the as given gives a luminosity function, where is the “characteristic size” of a gravitational wave source. Note that if is the mass of the gravitating system

After certain considerations reported by Camp and Cornish [

This last equation requires that

gravitational radius of a system, with a black hole resulting if one sets

Note that when

we are at an indeterminate boundary where one may pick our system as having black hole properties.

Now for stars, Camp and Cornish [

frequency(38)

As well as a mean time for half of gravitational wave potential energy to be radiated away as

The assumption we make is that if we model

for a sufficiently well posed net mass M that the star formulas roughly hold for early universe conditions, provided that we can have a temperature T for which we can use the approximation

that we also have or higher, so, that at a minimum we recover Grishchuck’s [

Equation (40) places, for a specified value of R, which can be done experimentally, an upper bound as far as far as what a mass M would be. Can this be exploited to answer the question of if or not there is a minimum value for the Graviton mass?

The key to the following discussion will be that

, or larger.

P. Tinyakov [

If we use LISA values for the Pulsar Gravitational wave frequencies, this may mean that the massive graviton is ruled out. On the other hand

leads to looking at, if

If the radius is of the order of 10 billion light-years ~4300 Mpc or much greater, so then we have, as an example

This Equation (45) is in units where.

If grams per graviton, and 1 electron volt is in rest mass, so. Then

Then, exist

If each photon, as stated above is grams per photon, [

initially transmitted photons.(47)

Futhermore, if there are, today for a back ground CMBR temperature of 2.7 degrees Kelvin, with a wave length specified as. This is for a numerical density of photons per cubic meter given by

As a rough rule of thumb, if, as given by Weinberg [

then we have to work with a de facto initial volume. i.e. the numerical value for the number of photons at, if we have a per unit volume area based upon Planck length, in stead of meters, cubed is photons for a cubic area with sides at Kelvin However,

initially transmitted photons! Either the minimum distance, i.e. the grid is larger, or Kelvin We have, now, so far linked entropy, gravitons, and also information with certain qualifications. Next, we will attempt to quantify the treatment of gravitons, as given in

The next part of our discussion will be in linking sequeezed states, with a break down of the purely quantum mechanical modeling of gravitons.

In the quantum theory of light (quantum electrodynamics) and other bosonic quantum field theories, coherent states were introduced by the work of [

To put it mildly, if we are looking at a solution to minimize graviton position uncertainty, we will likely be out of luck if string theory is the only tool we have for early universe conditions. Mainly, the momentum will not be small, and uncertainty in momentum will not be small either. Either way, most likely, In addition, it is likely, as Klaus Kieffer [

Putting it mildly, the string theory case is far more difficult. And that is the problem, with regards to string theory, what is an appropriate vacuum expectation value for treating a template of how to nucleate gravitons into a coherent state with respect to relic conditions [

Here, the value has yet to be specified, and that actually for energy values approximately of the order of which may be the mean temperature for the expanding universe mid way, to the end of inflation, which does not equal current even smaller string theory estimates as presented by Li et al. [

string theory values for inflationary Gravitational amplitudes. i.e. the more modern treatments are predicting almost infinitesimal GW fluctuations. It is not clear from Ford’s [

if we consider experimental conditions for obtaining. Note that this would put severe restrictions upon the variations in momentum. A subject which will be referenced in whether or not the Li-Baker detector can suitably obtain such small values of in detection capacity. To do so will require an investigation into extreme sensitivity requirements, for this very low value of. Fanguy Li. et al. [

would require up to 10^{5} seconds in evaluative time for a clean signal, for GW. What will be asked in further sections is if or not the 10^{5 }seconds in evaluative time for a clean signal can evaluate additional data. i.e. what if one would have to do to distinguish if or not coherent states of gravitons which merge to form GW may be measured via the protocols brought up by Li et al. [

However, what one sees in string theory, is a situation where a vacuum state as a template for graviton nucleation is built out of an initial vacuum state,. To do this though, as Venkatartnam, and Suresh [

where combining the with (55) leads to a single mode squeezed coherent state, as they [

The right hand side of Equation (56) given above becomes a highly non classical operator, i.e. in the limit that the super position of states occurs, there is a many particle version of a “vacuum state” which has highly non classical properties. Squeezed states, for what it is worth, are thought to occur at the onset of vacuum nucleation, but what is noted for being a super position of vacuum states, means that classical analog is extremely difficult to recover in the case of squeezing, and general non classical behavior of squeezed states. Can one, in any case, faced with do a better job of constructing coherent graviton states, in relic conditions, which may not involve squeezing? Note L. Grishchuk [

If is picked, and a Schrodinger equation is made out of the Lagrangian used to formulate Equation (50) above, with, and,

and an arbitrary function.. Also, we have a finite volume.

Then the Lagrangian for deriving Equation (58) is (and leads to a Hamiltonian which can be also derived from the Wheeler De Witt equation), with for zero point subtraction of energy

then there are two possible solutions to the S. E. Grushchuk [

The non squeezed state has a parameter

where is an initial time, for which the Hamiltonian given in Equation (60) in terms of raising/lowering operators is “diagnonal”, and then the rest of the time for, the squeezed state for is given via a parameter B for squeezing which when looking at a squeeze parameter r, for which, then Equation (53) has, instead of

Taking Grishchuck’s formalism [

A reasonable research task would be to determine, whether or not

would correspond to a vacuum state being initially formed right after the point of nucleation, with at time with an initial cosmological time some order of magnitude of a Planck interval of time

seconds The next section will be to answer whether or not there could be a point of no squeezing, as Grishchuck implied, for initial times, and initial frequencies, and an immediate transition to times, and frequencies afterwards, where squeezing was mandatory. Note that Grischchuk [

.

Having with a possible displacement operator, seems to be in common with, whereas which is highly non classical seems to be in common with a solution for which This leads us to the next section, i.e. does when of time seconds, and then what are the initial conditions for forming “frequency”?

A curved spacetime is a coherent background of gravitons, and therefore in string theory is a coherent state Joseph Gerard Polchinski [

becomes

(66)

Polochinski [_{c}10^{8} GeV, where the symmetry partners of the graviton would evolve from an ultrahard fluid to pressureless dark matter. indicates m10 MeV for the massive components of the graviton multiplet”. This has a counter part in a presentation made by Berkenstein [

Carlo Rovelli [

(67)

Rovelli [

Furthermore, Bojowald [49,50] specified a criteria as to how to use an updated version of and in his GRG manuscript on what could constitute grounds for the existence of generalized squeezed initial (graviton?) states. Bojowald [

and is a “volume” operator where the “ volume” is set as, Note also, that Bojowald has, in his initial Friedman equation, density values

so that when the Friedman equation is quantized, with an initial internal time given by, with becoming a more general evolution of state variable than “internal time”. If so, Bojowald writes, when there are squeezed states [50,51]

for his Equation (26), which is incidently when links to classical behavior break down, and when the bounce from a universe contracting goes to an expanding present universe. Bojowald [

This discussion is to present a not so well known but useful derivation of how instanton structure from a prior universe may be transferred from a prior to the present universe. This discussion is partly rendered in [

1) The solution as taken from L. Crowell’s [

2) Left unsaid is what embedding structure is assumed.

3) A final exercise for the reader. Would a WKB style solution as far as transfer of “material” from a prior to a present universe constitute procedural injection of non compressed states from a prior to a present universe? Also if uncompressed, coherent states are possible, how long would they last in introduction to a new universe?

This is the Wheeler-de-Witt equation with pseudo time component added. From Crowell [

This has when we do it, and frequently constant, so then we can consider

In order to do this, we can write out the following for the solutions to Equation (72) above.

and

This is where and refer to inte grals of the form

and.

Next, we should consider whether or not the instanton so formed is stable under evolution of space-time leading up to inflation. To model this, we use results from Crowell [

This has:

This assumes that the cosmological vacuum energy parameter has a temperature dependence, leading to

as a wave functional solution to a Wheeler-de-Witt equation bridging two space-times, similar to two space-times with “instantaneous” transfer of thermal heat, as given by Crowell [

This has as a pseudo cyclic and evolving function in terms of frequency, time, and spatial function. This also applies to the second cyclical wave function, where Equation (74) and Equation (75) then we get that Equation (79) is a solution to the pseudo time WDM equation.

The question which will be investigated is if Equation (79) is a way to present either a squeezed or un squeezed state. A way forward is to note that Prado Martin-Moruno, Pedro F. Gonzalez-Diaz in July [

leading to the sort of formalism as attributed to Luis J. Garay’s [

Now in the case of what can be done with the worm hole used by Crowell [

, , and a kinetic energy value as given of the form . The supposition which we have the worm hole wave functional may be like, so, use the wave functional looking like

where the for the WeinerNordstrom metric will be

As far back as 1982, Linde [

This is when the “mass” has the form, (here M is the bare mass term of the field in de Sitter space, which does not take into account quantum fluctuations)

Specified non linearity of at a time from the big bang, of the form

The question raised repeatedly in whether or not 1) if higher dimensions are necessary, and whether or not 2) mass gravitons are playing a role as far as the introduction of DE speed up of cosmological expansion may lead to an improvement over what was specified for density fluctuations and structure formation (the galaxy hierarchy problem) of density fluctuations given as

Equation (74) is for four space, a defining moment as to what sort of model would lead to density fluctuations. It totally fails as to give useful information as to the galaxy hierarchy problem as given in

Usual experimental values of density fluctuations experimentally are

, instead of,(89)

and this is assuming that is extremely small. In addition, Linde [

inside a false vacuum bubble. If something other than the Klein Gordon relationship

occurs, then different models of how density fluctuation may have to be devised. A popular model of density fluctuations with regards to the horizon is

(91)

where, and and to first order,. The values, typically of If working with

and with a density value

where grams, and is usually picked to avoid over production of black holes, a very complex picture emerges. Furthermore, if working with and

The above equation gives inter relationships between the time evolution of a pop up inflaton field, and a Hubble expansion parameter H, and a wave length parameter for a mode given as. What should be considered is the inter relation ship of the constituent components of Equation (94) and. What the author thinks is of particular import is to look at whether or not the more general expression, as given by the below equation also holds [

To first order, variations of and, should be compared with admissible values of

which would closely correspond to

and. i.e. the precise values of this may help us out in determining how to unravel what is going on in the galaxy formation i.e. how can we have earlier than expected galaxy formation?

One of the aspects of early universe topology we need to consider is how to introduce a de facto break down of quantization in curved space time geometries, and this is a problem which would permit a curved space treatment of. i.e. as R gets of the order of

, say that the spatial geometry of early universe expansion is within a few orders of magnitude of Planck length, then how can we recover a field theory quantization condition for in terms of path integrals. We claim that deformation quantization, if applied successfully will eventually lead to a great refinement of the above Wheeler De Witt wave functional value, as well as allow a more through match up of a time independent solution of the Wheeler de Witt equation, with the more subtle pseudo time dependent evolution of the wave functional as Crowell wrote up. i.e. the linkage between time independent treatments of the wave functional of the universe, with what Lawrence Crowell [

Resolution of which add more detail to. Having said this, it is now important to consider what can be said about how relic gravitons/ information can pass through minimum vales of.

We shall reference what A. W. Beckwith presented [

This expression of power should be compared with the one presented by Giovannini on averaging of the energy-momentum pseudo tensor to get his version of a gravitational power energy density expression, namely [

Giovannini states that should the mass scale be picked such that, that there are doubts that we could even have inflation. However, it is clear that gravitational wave density is faint, even if we make the approximation that

as stated by [

What we would like to do for future development of entropy would be to consider a way to ascertain if or not the following is really true, and to quantify it by an improvement of a supposition advanced by [

2) Goldstone gravitons would arise in the beginning due to a violation of Lorentz invariance. i.e. we have a causal break, and merely having the above condition does not qualify for a Lorentz invariance breakdown Kiefer, Polarski, and Starobinsky [

If the phase spaces can be quantified, as a starting point of say, with being part of how to form the “dimensions” of, and

part of how to form the dimensions ofand being, for a given, and in certain cases, then avoiding having dS/dt = ∞ at S = 0 will be straight forward We hope to come up with an emergent structure for gravitational fields which is congruent with obtaining naturally, so this sort of procedure is non controversial, and linked to falsifiable experimental measurement protocol, so quantum gravity becomes a de facto experimental science. This will mean looking at Appendix B, fully. Appendix C, and Appendix D give further issues we describe later on. In future publications. We give them as pertinent information for the future development of this project.

This work is supported in part by National Nature Science Foundation of China grant No. 11075224. The author thanks Dr. Fangyu Li for conversations as to the physics of GW and Graviton physics, and also has a debt of gratitude to Stuart Allen, CEO for his efforts to permit the author to do physics work

The author presents a post Newtonian approximation based upon an earlier argument/paper by Clifford Will as to Yukawa revisions of gravitational potentials in part initiated by gravitons with explicit mass dependence in their Compton wave length.

Post Newtonian approximations to General relativity have given physicists a view as to how and why inflationary dynamics can be measured via deviation from simple gravitational potentials. One of the simplest deviations from the Newtonian inverse power law gravitational potential being a Yukawa potential modification of gravitational potentials. So happens that the mass of a graviton would factor directly into the Yukawa exponential term modification of gravity. This appendix indicates how a smart experimentalist could use the Li-Baker detector as a way to obtain more realistic upper bounds as to the mass of a graviton and to use it as a template to investigate modifications of gravity along the lines of a Yukawa potential modification as given by Will [

The easiest way to ascertain the mass of a graviton is to investigate if or not there is a slight difference in the speed of graviton ‘particle’ propagation and of HFGW in transit from a “source” to the detector. Visser’s [

, and Hertz in line with L. Grischuck’s treatment of relic HFGW’s [

But Equation (1) above is an approximation of a much more general result which may be rendered as

The terms and E refers to the graviton rest mass and energy, respectively. Now specifically in line with applying the Li Baker detector, [

The above formula depends upon

, with where and are the differences in arrival time and emission time of the two signals (HFGW and Graviton propagation ), respectively, and Z is the redshift of the source. Z is meant to be the red shift. Specifically, the situation for HFGW is that for early universe conditions, that, in fact for very early universe conditions in the first few mili seconds after the big bang, that. An enormous number.

The first question which needs to be asked is, if or not the Visser [

](A4)

The closer the emission times for production of the HFGW and Gravitons are to the time of the initial nucleation of vacuum energy of the big bang, the closer we can be to experimentally using Equation (4) above as to give experimental criteria for stating to very high accuracy the following.

More exactly this will lead to the following relationship which will be used to ascertain a value for the mass of a graviton. By necessity, this will push the speed of graviton propagation very close to the speed of light. In this, we are assuming an enormous value for D

This Equation (A6) relationship should be placed into with a way to relate this above value of

with an estimated value of E coming from the LiBaker detector [

A suitable numerical treatment of this above equation, with data sets could lead to a range of bounds for, as a refinement of the result given by Will [

(A8)

The above Equation (A8) gives an upper bound to the mass as given by

Needless to say that an estimation of the bound for the graviton mass, and the resulting Compton wavelength would be important to get values of the following formula, namely

Clifford Will [

Eric Davis, quoting Pisen Chen’s article [

The immediate consequence of the prior discussion would be to obtain a more realistic set of bounds for the graviton mass, which could considerably refine the estimate of gravitons produced per year at the LHC, with realistically 365 × 86400 seconds = 31536000 seconds in a year, leading to gravitons produced per second. Refining an actual permitted value of bounds for the accepted graviton mass, m, as given above, while keeping 1.2209 × 10^{19} GeV/c^{2 }would allow for a more precise set of gravitons per second which would significantly enhance the chance of actual detection, since right now for the LHC there is too much general uncertainty as to the likelihood of where to place a detector for actually capturing/detecting a graviton.

The physics community now has an opportunity to experimentally infer the existence of gravitons as a knowable and verifiable experimental datum with the onset of the LHC as an operating system. Even if the LHC is not used, Pisen Chens parameterization of inputs from his table [

The Li-Baker detector [

Begin first of all looking at

This leads to consider what to do with

Samtleben et al. [

is the fraction of the sky covered in the measurement , and is a measurement of the total experimental sensitivity of the apparatus used. Also is the width of a beam, while we have a minimum value of which is one over the fluctuation of the angular extent of the experimental survey.

i.e. contributions to uncertainty from sample variance is equal to contributions to uncertainty from noise. The end result is

Durrer [

and

Here we are interpreting amplitude of metric perturbations at horizon scale, and we set, where is the conformal time, according to physical time, where we have as the scale factor. Then for, and, and a pure power law given by

We get for tensor fluctuation, i.e. gravity waves and a scale invariant spectrum with

This is a re capitulation of what is written by S. Capoziello, et al. [

Which in turn will lead to, with qualifications, for thin shell approximations,

so that is a spherical Bessel equation for which we can write

Similarly, leads to

Also, when

Realistically, in terms of applications, we will be considering very small values, consistent with conditions near a singularity/worm hole bridge between a prior to our present universe. This is for.