Gedanken Experiment for Degree of Flatness, or Lack of, in Early Universe Conditions

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Received 19 October 2015; accepted 4 January 2016; published 7 January 2016

1. Introduction

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 the aftermath of a quantum bounce implies [1] , and what we should be looking forward in terms of structure formation afterwards. The relevant equation we will be working with is from the time component of the Stress energy Tensor which we will write up as, if is a statement of volume and if for the cosmological “constant”.

(1)

In picking this, we are using Ng infinite quantum statistics [1] as a counting factor and likely for have a Planck length cubed, volume as a starting point, if so then, the mass of the graviton, will be important as well as some considerations given if stays the same, to the present era, or if it has quintessence [2] [3] , a topic we will bring up. In delineating Equation (1) above, we will examine the following from first principle, while keeping in mind that [4]

(2)

Then

(3)

If we make the substitution of. The consequences afterwards follow for Equation (1).

Our supposition is, that if [1] [5] is used, as well as [5] , and, and grams [6] then we have an almost but not zero negative value for the value, we will from here discuss its implications and what it says physically.

2. Implications as to Choosing for our Problem: Where it Comes from

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 [5] [7] but it says more than that. In reality the very small value for the in the aftermath of the quantum bounce, with, has some very interesting implications for information transfer from a prior to a present universe which we will be brought up next. We start with what Turok [8] wrote up as to the initial starting point of analysis, as to where he described the cosmological evolution to describe a perfect bounce,” in which the universe passes smoothly through the initial singularity. In what we analyze four our purposes, we have that the 2^{nd} order perturbative term of for cosmological perturbations obey, here with a 2^{nd} order contribution we can set as

(4)

Which is a 2^{nd} order perturbative term for the equation for the evolution of h, if is nonlinear [8]

(5)

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

(6)

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 [8] to obtain

(7)

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

(8)

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 [7]

(9)

Then, we would have, the initiation of quantum effects as of about [8]

(10)

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

(11)

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

(12)

Depending upon, this will then lead to a condition for which Equation (4) vanishes, which is in turn due to

(13)

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

3. Considerations of What Could Lead to Equation (4), i.e. 2^{nd} Order Perturbation to Cosmological Evolution, Vanishing

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

(14)

If Equation (4) is zero due to and we want Equation (4) to vanish, it leads to the following for the vanishing of the 2^{nd} order perturbative effect, with the critical value of wavelength for which Equation (4) vanishes, i.e. hence,

(15)

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

(16)

4. Comparing the Variance in Position Given in Equation (16) with Modified HUP

Note this very small value of x comes from a scale factor, if [9] , i.e. 55 orders of magnitude smaller than what would normally consider, but here note that the scale factor is not zero, so we do not have a space-time singularity. Then

(17)

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 [10] [11] , of a generalization we will write as

(18)

If we use the following, from the Roberson-Walker metric [3] [4] .

(19)

Following Unruh [5] , write then, an uncertainty of metric tensor as, with the following inputs

(20)

Then, if [3] - [5]

(21)

5. The Questions of Nonstandard Cosmologies, before Delving into What Our GW Generation Implies

It is noteworthy that, there have been numerous attempts to vet and prove a modification [12] for the Friedman Walker cosmology, which has been correction to the large scale inhomogeneity raised as a possibility by [13] - [17] , as the approach taken by Wald, and Green, a summary can be seen in the statement that as given by [14] that,

“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 [12] the statement is that

“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 that the x component may be defining a situation through Equation (16) with, and with the r in re defined as x as in Equation (16) which may be validating [12] (especially in the above quote from [12] given above), whereas Equation (21) may be in fact satisfying what is quoted in [14] .

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 [18] in the quotation that

“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…) or terms with scalar ﬁelds non minimally coupled to geometry (terms like)”, allows for conditions giving more structure to the terms in the Pre Planckian possible quantization of GR we give as

(22)

In Equation (22), inputs into the terms, and may determine if the quote taken about the admissibility of adding in higher order terms in the curvature as alluded to in [18] above is accurate, and that the definition of classical versions of inputs eventually quantized and put into, and, reflects a purely minimum contribution to terms in the curvature as given in Equation (2), which will in turn affect the magnitude of Equation (1). We state that adding more detail as to how the curvature affects the magnitude of Equation (1) which in turn may seriously affect semi classical input into the terms, and. If Equation (1) is small, it is likely that the higher order terms as in the quote from [18]

“the curvature invariants (terms like…) or terms with scalar ﬁelds non minimally coupled to geometry (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

6. The Matter of GW, as Ascertained through Reference [19] plus Distinguishing between More Cosmologies than Just Extended Gravity

We start off with a quote from [19] which neatly summarizes up the interesting issues of GW research we should keep in mind;

“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, well beyond the reach of particle accelerators on Earth.”

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 [20] has, in page 66 the datum that if there exist an inflaton field or fields, that as stated

“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 [20] this leads to a rapid increase in turbulent interacting scalar waves. i.e. one could, unless we are very, very careful still, even if we have a primordial signal, have through the turbulence, due to preheating a stochastic background, and we go through the needle in a haystack problem with a vengeance. The author in [21] wrote also that the problem is complicated by the following

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 [21]

An important, direct connection between the strain of relic gravitational waves and the inflaton field has been released by Dr. Corda [22] as far as the formula he derived for an inflaton and inputs of strain upon the inflaton field. This was given by Dr. Corda as [22]

(25)

Here, H is given as the evolving Hubble parameter, and represents the averaged amplitude of the perturbations of the RSBGWs, where RSBGWs is an abbreviation for relic stochastic background of gravitational waves (RSBGWs) which is proposed by the Pre-Big-Bang Theory. Below we work with an. amplitude, as compared to for a frequency range Corda gave as for when one has for the Hubble parameter when setting for a narrower frequency band width given. The upshot as claimed by Corda is for that range of GW that as a lower bound for the inflaton field. If so, then the inflaton field may have a different lower bound if, as an example one looksat, even if one looks at. The lower bound of the inflaton field becomes especially significant, if as an example inflaton fields are connected with initial entropy conditions which Beckwith picked as.

The upshot with the frequency, to this range, will affect the size of the initial scale factor, admissible to the perturbation of the term which will be important as to the

7. Conclusions

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 [23] we can write

(26)

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 as specified by Giovannini [23] . Then we will assert that if r = x then if we

use and then the volume, as used in [3] - [5]

(27)

This Equation (19) will be put into, if, it means that that this is defined for all x as to where and when holds, with the lower value for x sig-

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 in

. Furthermore we have, if there is an eventual weak field approximation according to Katti [4]

gravitational spin off according to, with a gravitational wave signal according to, if [4]

(28)

If the contribution from Pre-Planckian to Planckian is due to the stress energy tensor as given in form [5] , it means that the relevant relic GW signal will be of the form, with a small quadrupole tensor. This with space-time which is almost flat according to Equation (1) initially as the genesis of the GW which may be analyzed with a dominant contribution coming from [4]

(29)

The importance of Equation (29) is in giving a compliment to [23] as to the problem of relic gravitational waves, which the author views, is extremely important. A correct rendering of Equation (25) would be to determine additional experimental constraints which may determine if detection of early universe gravitational waves is feasible with LIGO technology, or if there is a requirement for other detectors. Certainly, what is wished by this inquiry is to avoid the problems associated with BICEP 2, through a refinement of methods as given in [24] . In addition, of special note would be to avoid picking up interstellar dust effects upon Gravitational waves, which has been a primary reason for the development of methodologies as given in [25] . Note also that BICEP2 only observed in one wavelength, which made it difficult for them to prove the B-modes they saw were truly from gravitational waves. Ascertaining Equation (29) properly, may help alleviate that problem.

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 [5] .

(30)

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, for say right may, with some additional fine tuning help us find a relic GW frequency range, so we avoid the BICEP 2 problem of getting signals from presumed relic GW producing conditions the same as what would be for dust, as has been stated clearly as the problem destroying the BICEP 2 findings as of 2014.

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

As [27] has brought up, the presence of massive gravity, which is a byproduct of the derivation of this paper, and in particular as stated in [26] “the solution for the massive mode arising from the Starobinsky’s high order gravity theory” may be akin to the problems of not only the round off done in Equation (26) but also of giving further meaning to the physics of Equation (4) and Equation (5), as brought up by the author. This is a topic which the author will investigate on his own in the next sequels to this document. Note that [28] has a distinct nonlinear component as to gravity, and so does this document, especially in the spin offs of Equation (4) and Equation (5).

Further refinements may be due to [5] [29] where we consider as given in [30] details of a quantum bounce which may give more information, as well as additional investigations into what Turok brought up [3] . The issues as to Equation (30) and what they imply are quite different from [30] for reasons we will go into in a future publication.

Acknowledgements

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

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