_{1}

^{*}

We examine through the lens of dynamical systems a “one dimensional” time mapping of emergent VEV from Pre-Planckian space time conditions. As it is, we will from first principles examine what adding acceleration does as to the HUP previously derived. In doing so, we will be trying it in our discussion with the earlier work done on the HUP. not equal to zero, constant, but large would frequently imply which would have three dissimilar real valued roots. And the situation with not equal to zero yields more tractable result for which will have implications for the HUP inequality in Pre-Planckian space-time, and buttresses an analysis of a 1 dimensional “time” mapping for emergent VEV (vacuum expectation values).

We will be following a first principle investigation of initial equations of state for energy density in space-time as given by B. Hu [

The second line of the above is making the approximation that the insides of the first line, are averaged out to a constant, which is defensible in the situation of a Pre- Planckian space-time condition. Secondly, we are assuming in all of this that ^{nd} line of Equation (2) referencing an averaged out value for the number of created particles which we then identify as

If so, then we could define having a net energy as given by [

We have several different ways to address what is meant by this energy. Our supposition is that we could make a reference, here, to, if c (speed of light) = 1, to have, here, initially, a transfer of gravitons, as an information carrier, from a prior universe to our present universe so that as a result of a match up in Pre-Planckian space-time to Planckian space time we would have Equation (2) as rendered by, using Hu again, [

And a graviton count, in the Pre-Planckian era we would give as [

Here, we would have that

Then there would be the rough equivalence given of, say:

We will from here, state that this initial graviton production say for a Planck instant of time would be of the order of 10^{5}, so as to have, then if the temperature becomes

If

Then, the above reduces to the form of equivalencies which we will write up as follows, which will be accessed toward the end of this article.

Becomes

If one has a grasp of the number of VeV quasi particles where

Our assumptions are that then we would have a way to, get bounds on

Our last part of information, using Hu again [

Also the mass of

We are then ready after some additional work to apply our HUP for Pre-Planckian metric tensor and to determine admissible

We will be examining a Friedmann equation for the evolution of the scale factor, using explicitly one case being when the acceleration of expansion of the scale factor is kept in, and the intermediate cases of when the acceleration factor, and the scale factor is important but not dominant. In doing so we will be tying it in our discussion with the earlier work done on the HUP but from the context of how the acceleration term will affect the HUP, and making sense of [

Namely we will be working with [

i.e. the fluctuation

In short, we would require an enormous “inflation” style

Then by [

Begin looking at material from page 483-485 of [

Then, consider two cases of what to do with the ration of

1) Solutions for Equation (15), in Cubic form for Equation (15) gained by NOT abandoning

Following [

Our approximation is, to set

as a non-dimensional but very large quantity. Then a solution exists as given as for a reduced cubic version of Equation (15) which can be given by modifications as presented in this document. i.e. we are using material as given in [

Our approximation is, to set

as a non-dimensional but very large quantity. Then a solution exists as given as for a reduced cubic version of Equation (15) which can be given by [

And

And when

If so then

If

Here, with very large constant initial

This means that in terms of Equation (21) especially if we have three unequal roots, for Equation (19) that the choice is, in acceleration for a chaotic environment [

Using [

In short, we would require an enormous “inflation” style

Going to Kolb, Pi, and Raby, [

With a minimum value for Equation (23) according to the first derivative,

With a minimization of a SUSY style Equation (23), and Equation (26) below if

i.e. this is still, with some tweaking a commonly accepted SUSY VeV model, with a minimum if

We will first of all, look at the inner dynamics of the metric tensor fluctuation. To do this we encompass the following background. 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 [

The reasons for saying this set of values for the variation of the non ^{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 [

We will be using the approximation given by Unruh [

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

Following Unruh [

Then, if

This Equation (32) 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 [

Then [

Then, Equation (32) and Equation (33) and Equation (34) together yield

How likely is

To begin this process, we will break it down into the following coordinates. In therr, θθ and

If as an example, we have negative pressure, with T_{rr}, T_{θθ} and

We will next be considering the role of a possible dynamical systems mapping upon this problem. To begin with, we will be looking at the role of

What is in Equation (37) shows from inspection that there is, defacto a 1 dimensional mapping for an initially three dimensional process, which is furthermore reflected in what is written up as of the frequency via the following, namely look at if

This can be put into Equation (37). A more conservative treatment of the above,

would be to write a constant,

in Pre-Planckian space to write.

If so, then there would be an iterative map looking like

Given this iterative mapping, we can then state clearly its relationship to the Alexan- drov theorem, 1942, which the author was able to ascertain on January 29^{th} at the Stony brook University weekly talk on Dynamical systems. What we heard is, simply, is that from this talk, that if we ask the following, namely:

Consider a

Theorem: Alexandrov, 1942:

Suppose

i.e. see http://fillastre.u-cergy.fr/wp-content/uploads/2011/04/moscow-fillastre.pdf

The long and short of it is, that if we look at a quantum bounce “ball” of infinitely small radii, but not a point in space, that the relation given in Equation (39a) will define a metric fluctuation,

A singular manifold Calabi-Yau determines the physical characteristics of the topological soliton states that are interpreted as particles in high energy physics. i.e. what we are doing is when considering the graviton as a particle wave duality, in the formation of Equation (39a) and in doing so, we have to face up to the fact, that the gravitons, in string theory, and the Chalabi Yau setting are almost always massless. i.e. in addition, it is next to impossible for there to be any massive gravitons, since gravitons in this setting as given by [

In order to start this approximation, we will be using Barbour’s value of emergent time [

Initially, as postulated by Barbour [

If

Key to Equation (41) will be identification of the kinetic energy which is written as

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 [

If we are where we are in a very small Planckian regime of space-time, we could, then say write Equation (43) as proportional to

A way of solidifying the approach given here, in terms of early universe GR theory is to refer to Einstein spaces, via [

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 [

So as to recover, via the Einstein spaces, the seemingly heuristic argument is given above. Furthermore when we refer to the Kinetic energy space as an inflation

In the case of the general elliptic operator K if we are using the Fulling reference, [

Then, according to [

Then

If the frequency, of say, Gravitons is of the order of Planck frequency, then this term, would likely dominate Equation (49). More of the details of this will be worked out, and also candidates for the

The details of the elliptic operator K will be gleaned from [

It is important to note, that the proper evaluation of Equation (49) will permit, once the role of gravitons in the changing of an inflation contribution is thoroughly experimentally vetted for us to analyze if the criteria raised in [

i.e. quote:

Binary black hole systems at larger distances contribute to a stochastic background of gravitational waves from the superposition of unresolved systems.

End of quote.

Thoroughly understanding the role of Equation (49) has to be done as to avoid a similar issue here, especially when the emergence of the inflation, as presupposed may significantly add to stochastic noise.

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

Beckwith, A.W. (2016) Gedanken Experiment for Looking at dg_{tt} for Initial Expansion of the Universe and Influence on HUP via Dynamical Systems, with Positive Pre-Planckian Acceleration. Journal of High Energy Physics, Gravitation and Cosmology, 2, 531-545. http://dx.doi.org/10.4236/jhepgc.2016.24046