_{1}

^{*}

This document will from first principles delineate the degree of flatness, or deviations from, in early universe models. We will, afterwards, make comparison with recent results we have looked at concerning metric tensor fluctuations and comment upon the role of what early universe gravitational energy may play a role in the presumed deviation from flat space results. Note that N~S
_{initial(graviton)}~10
^{37 }will be tied into the presumed results for initial state density, in ways we will comment upon, leading to observations which are supporting the physics given by Equation (26) of this document as with regards to Gravitational waves, from relic conditions. The deviations from flat space may help confirm the conclusions given by Buchert, Carfora, Kolb, and Wiltshire allegedly refuting the claim by Green and Wald that “the standard FLRW model approximates our Universe extremely well on all scales, except close to strong field astrophysical objects”, as well as give additional analysis appropriate for adding detail to expanding experimental procedures for investigating non FLRW models such as the Polynomial Inflation models as given by Kobayashi, and Seto, as well as other nonstandard cosmologies, as brought up by Corda, and other researchers. As well as improve upon post Bicep 2 measurements which will avoid GW signatures from interstellar dust, as opposed to relic GW. We hope that our approach may help in the differentiation between different cosmology models. Most importantly, our procedure may help, with refinement of admissible frequency range, avoid the problem of BICEP 2, which had its presumed GW signals from presumed relic conditions identical to dust induced frequencies, as so identified by the Planck collaboration in reference [25] which we comment upon in the conclusion.

We will start off first, with a description of the following equation which we will derive in the next section. We discuss the implications of a deviation from flat space, with a description of what

In picking this, we are using Ng infinite quantum statistics [

Then

If we make the substitution of

Our supposition is, that if

First of all, this non zero initial value of the entropy is consistent with a quantum bounce, as can be postulated through LQG, as by [^{nd} order perturbative term of ^{nd} order contribution we can set as

Which is a 2^{nd} order perturbative term for the equation for the evolution of h, if

Then setting a conformal time as approaching early universe conditions requires that

Our supposition is, then that we have the following for well behaved GW and early cosmological perturbations being viable, in the face of cosmological evolution with modifying the formalism of Turok [

In practical terms near the initial expansion point it would mean that near the beginning of cosmological expansion we would have an initial energy density of the order of

If so then, if we assume that gravitons, of initial mass about 10^{−62} grams, i.e. and that we have Planck mass of about 10^{−}^{5} grams, if gravitons were the only “information” passed into a new universe, making use of the following expression for the initiation of quantum effects, i.e. by Haggard and Rovelli [

Then, we would have, the initiation of quantum effects as of about [

Then by making use of Equation (10) we could, by dimensional analysis, start the comparison by setting values from Equation (7) and Equation (10) to obtain

So that to first order, a graviton count, for a radii of about the order of

Depending upon

Equation (13) would put restrictions upon the following, namely

The simple short course as to the radius achieving its starting point to being quantum mechanical in its effects, from the big bang initiating from a quantum bounce is to have the following threshold for quantum effects to be in action, to the vanishing of Equation (1). Here the quantum effects start with a value of

If Equation (4) is zero due to ^{nd} order perturbative effect, with

It means that there is the following interval may be our best Quantum Mechanical perturbative indicator in terms of Equation (4), that is

Note this very small value of x comes from a scale factor, if [

We will next discuss the implications of this point in the next section, of a nonzero smallest scale factor

We will be using the approximation given by Unruh [

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

Following Unruh [

Then, if

It is noteworthy that, there have been numerous attempts to vet and prove a modification [

“We develop a new, mathematically precise framework for treating the effects of nonlinear phenomena occurring on small scales in general relativity. Our approach is an adaptation of Burnett’s formulation of the “shortwave approximation”, which we generalize to analyze the effects of matter inhomogeneities as well as gravitational radiation. Our framework requires the metric to be close to a “background metric”, but allows arbitrarily large stress-energy fluctuations on small scales”.

In the case of [

“The large-scale homogeneity and isotropy of the universe is generally thought to imply a well defined background cosmological model. It may not. Smoothing over structure adds in an extra contribution, transferring power from small scales up to large. Second-order perturbation theory implies that the effect is small, but suggests that formally the perturbation series may not converge”.

We have the situation in defining

Having said that, the issues of the nature of determining if there is or not if there are conditions allowing for quantization in the genesis of GR, as given by [

“On the other hand, one can deﬁne Extended Theories of Gravity those semiclassical 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

In Equation (22), inputs into the terms

“the curvature invariants (terms like

do not play a large role, and that we do not have to talk about extended gravity. If Equation (1) is not small, then it is likely that extended gravity will have to be taken seriously with contributions to the Curvature, and the Lagrangian as seen for Equation (2) have to be painstakingly calculated. Having said that, let us now consider the matter of Gravity waves, and their implications

We start off with a quote from [

“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

First off, are we considering contribution from a multitude of point like origins for GW generation, or do we have fundamental processes in the early universe to consider? We are trying to obtain GW which are Primordial, in origin. Not only is the above, as alluded to in the quote, there is one other complication which is in the next paragraph, namely [

“The bubble like structure of the Fields is reflected in their statistics. Perhaps more surprisingly, the statistics of both fields remain non-gaussian for a long time after preheating. At the end of our simulation, at t = 300 m the fields were noticeably non Gaussian.”

As stated in [

In the section called “technical problems which need to be addressed in order to improve the quality of research for relic signals” the author wrote in [

An important, direct connection between the strain of relic gravitational waves and the inflaton field has been released by Dr. Corda [

Here, H is given as the evolving Hubble parameter, and

The upshot with the frequency, to this range,

Equation (21) may, with refinements of r = x, in the four-dimensional Volume, give the new HUP, in our problem, its impact upon GW generation and its relevance to Bicep 2, the search for validation of nonstandard cosmologies, and GW searches.

If from Massimo Giovannini [

Refining the inputs from Equation (26) means more study as to the possibility of a non zero minimum scale factor, as well as the nature of

use

This Equation (19) will be put into

nifying the spatial range of x for which quantum mechanics is valid, with three times that value connected as to when the perturbative methods break down. Thereby influencing the range of values for

gravitational spin off according to

If the contribution from Pre-Planckian to Planckian is due to the stress energy tensor as given in

The importance of Equation (29) is in giving a compliment to [

Note that also the value of a correct rendering of Equation (29) would be to ascertain the axial tilt as would be expected in early universe cosmology, and relic Gravitational waves, with greater precision than which showed up in the BICEP 2 results.

This value of Equation (29) would have, as its origins, the near flat space physics given by Equation (1) as its genesis with this to consider, as the start [

Equation (30), if confirmed experimentally, is of potential decisive importance to the problem of discriminating between different cosmology models. Note in the case of Bicep 2 the Planck collaboration had that the frequency of dust signals was about the same as what was reported by Bicep 2 presumed gravitational waves.

Hence, the conclusion is inescapable. The value of the flatness calculation as of Equation (1) and of getting a range of

There are some other issues to consider in addition to avoiding having presumed GW frequency being the same as for dust induced signals.

As [

Further refinements may be due to [

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

Andrew WalcottBeckwith, (2016) Gedanken Experiment for Degree of Flatness, or Lack of, in Early Universe Conditions. Journal of High Energy Physics, Gravitation and Cosmology,02,57-65. doi: 10.4236/jhepgc.2016.21006