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Gedanken Experiment for B Field Contributions to Initial Conditions for Relic Graviton Production Based upon an Initial Inflaton Value

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1. Introduction

This article starts with updating what was done in [1] , which is symbolized by, if the scale factor is very small, metric variance [2] [3]

$\begin{array}{l}\langle {\left(\delta {g}_{uv}\right)}^{2}{\left({\stackrel{^}{T}}_{uv}\right)}^{2}\rangle \ge \frac{{\hslash}^{2}}{{V}_{\text{Volume}}^{2}}\\ \underset{uv\to tt}{\to}\langle {\left(\delta {g}_{tt}\right)}^{2}{\left({\stackrel{^}{T}}_{tt}\right)}^{2}\rangle \ge \frac{{\hslash}^{2}}{{V}_{\text{Volume}}^{2}}\\ \&\text{\hspace{0.17em}}\delta {g}_{rr}~\delta {g}_{\theta \theta}~\delta {g}_{\varphi \varphi}~{0}^{+}\end{array}$ (1)

In [4] this lead to

$\begin{array}{l}\delta t\Delta E\ge \frac{\hslash}{\delta {g}_{tt}}\ne \frac{\hslash}{2}\\ \text{Unless}\text{\hspace{0.17em}}\delta {g}_{tt}~O\left(1\right)\end{array}$ (2)

We assume $\delta {g}_{tt}$ is a small perturbation and look at $\delta t\Delta E=\frac{\hslash}{\delta {g}_{tt}}$ with

$\Delta {t}_{\text{time}}\left(\text{initial}\right)=\hslash /\left(\delta {g}_{tt}{E}_{\text{initial}}\right)=\frac{2\hslash}{\delta {g}_{tt}\cdot {g}_{\ast s}\left(\text{initial}\right)\cdot {T}_{\text{initial}}}$ (3)

This would put a requirement upon a very large initial temperature ${T}_{\text{initial}}$ and so then, if

$S\left(\text{initial}\right)~n\left(\text{particle-count}\right)\approx {g}_{\ast s}\left(\text{initial}\right)\cdot {V}_{\text{volume}}\cdot \left(\frac{2{\text{\pi}}^{2}}{45}\right)\cdot {\left({T}_{\text{initial}}\right)}^{3}$ [5]

$S\left(\text{initial}\right)~n\left(\text{particle-count}\right)\approx \frac{{V}_{\text{volume}}}{{g}_{\ast s}^{2}\left(\text{initial}\right)}\cdot \left(\frac{2{\text{\pi}}^{2}}{45}\right)\cdot {\left(\frac{\hslash}{\Delta {t}_{\text{initial}}\cdot \delta {g}_{tt}}\right)}^{3}$ (4)

And if we can write as given in [2] [3]

${V}_{\text{volume}\left(\text{initial}\right)}~{V}^{\left(4\right)}=\delta t\cdot \Delta {A}_{\text{surface-area}}\cdot \left(r\le {l}_{\text{Planck}}\right)$ (5)

The volume in the Pre Planckian regime would be extremely small, i.e. if we are using the convention that Equation (4) holds, then it argues for a very large ${g}_{s}^{\ast}$ beyond the value of 102, as given in [5] . In any case, our boundary between the Pre Planckian regime and Planckian, as far as the use of Equation (4) yields a preliminary value of, for a distance less than or equal to Planck Length, of non-zero value, with

${10}^{20}\le S\left(\text{initial}\right)~{n\left(\text{particle-count}\right)|}_{r\le {l}_{P}}\le {10}^{37}$ (6)

This is also assuming a $\delta {t}_{\text{initial}}\approx \Delta {t}_{\text{initial}}\propto \text{Plank-time}$ , i.e. at or smaller than the usual Planck time interval.

2. Counter Pose Hypothesis, by String Theory, for Equation (6)

The author is aware of the String theory minimum length and minimum time which is different from the usual Planck lengths, but are avoiding these, mainly due to a change in the assumed entropy formulae to read as the square root of the above results, namely [6] [7] [8]

${{10}^{10}\le {S\left(\text{initial}\right)|}_{\text{String-Theory}}~\sqrt{n\left(\text{particle-count}\right)}|}_{r\le {l}_{P}}\le {10}^{16}$ (7)

The above is still non-zero, but it cannot be exactly posited as in the Pre Planckian regime of Space-time, since the minimum length may be larger than Planck Length, i.e. as of the sort given in [8] .

3. Questions as to Refining Both Equation (6) and Equation (7) for More Precise Entropy Bounds

If from Giovannini [9] we can write

$\delta {g}_{tt}~{a}^{2}\left(t\right)\cdot \varphi \ll 1$ (8)

Refining the inputs from Equation (8) means more study as to the possibility of a non-zero minimum scale factor [10] , as well as the nature of $\varphi $ as specified by Giovannini [9] . We hope that this can be done as to give quantifiable estimates and may link the non-zero initial entropy to either Loop quantum gravity “quantum bounce” considerations [11] and/or other models which may presage modification of the sort of initial singularities of the sort given in [12] . Furthermore if the non-zero scale factor is correct, it may give us opportunities as to fine tune the parameters given in [10] below:

$\begin{array}{l}{\alpha}_{0}=\sqrt{\frac{4\text{\pi}G}{3{\mu}_{0}c}}{B}_{0}\\ \stackrel{\u2322}{\lambda}\left(\text{defined}\right)=\Lambda {c}^{2}/3\\ {a}_{\mathrm{min}}={a}_{0}\cdot {\left[\frac{{\alpha}_{0}}{2\stackrel{\u2322}{\lambda}\left(\text{defined}\right)}\left(\sqrt{{\alpha}_{0}^{2}+32\stackrel{\u2322}{\lambda}\left(\text{defined}\right)\cdot {\mu}_{0}\omega \cdot {B}_{0}^{2}}-{\alpha}_{0}\right)\right]}^{1/4}\end{array}$ (9)

where the following is possibly linkable to minimum frequencies linked to E and M fields [10] , and possibly relic Gravitons are generated if

$B>\frac{1}{2\cdot \sqrt{10{\mu}_{0}\cdot \omega}}$ (10)

This can be contrasted with looking at what happens if [13]

$\begin{array}{l}a\approx {a}_{\mathrm{min}}{t}^{\gamma}\\ \iff \varphi \approx \sqrt{\frac{\gamma}{4\text{\pi}G}}\cdot \mathrm{ln}\left\{\sqrt{\frac{8\text{\pi}G{V}_{0}}{\gamma \cdot \left(3\gamma -1\right)}}\cdot t\right\}\\ \iff V\approx {V}_{0}\cdot \mathrm{exp}\left\{-\sqrt{\frac{16\text{\pi}G}{\gamma}}\cdot \varphi \left(t\right)\right\}\end{array}$ (11)

So as talked about with [14] setting a minimum energy density given by

${\rho}_{\Lambda}\approx \frac{G{\left(E/{c}^{2}\right)}^{2}}{l}{l}^{-3}=\frac{G{E}^{6}}{{c}^{8}{\hslash}^{4}}$ (12)

And with the following substitution of

$E\underset{\text{Pre-Planckian}\to \text{Planckian}}{\to}\Delta E~\frac{\hslash}{\Delta t\cdot \left(\delta {g}_{tt}\approx {a}_{\mathrm{min}}^{2}{\varphi}_{\text{initial}}\right)}$ (13)

Then to first order we would be looking at Equation (12) re written as leading to

${\rho}_{\Lambda}~\frac{G}{{c}^{8}{\hslash}^{4}}\cdot {\left(\frac{\hslash}{\Delta t\cdot \left(\delta {g}_{tt}\approx {a}_{\mathrm{min}}^{2}{\varphi}_{\text{initial}}\right)}\right)}^{6}$ (14)

And if Equation (15) holds,

${\Lambda}_{\text{initial}}\cdot {H}_{\text{initial}}^{-2}\approx o\left(1\right)$ (15)

we would have by [15]

$\begin{array}{l}{\Lambda}_{\text{initial}}\approx {H}_{\text{initial}}^{2}~{\gamma}^{2}/{t}^{2}\\ {\Lambda}_{\text{initial}}\cdot {L}_{p}^{2}\approx {10}^{-123}\end{array}$ (16)

So

${10}^{-123}~{\gamma}^{2}{L}_{P}^{2}\cdot {\left(\Delta E\cdot \delta {g}_{tt}\right)}^{2}/{\hslash}^{2}$ (17)

Equation (17) would be key to the entire business, i.e. using this, we would have if

$\Delta E\sim \hslash {\omega}_{\text{graviton}}$ (18)

Then

${10}^{-123}~{\gamma}^{2}{L}_{P}^{2}\cdot {\left(\hslash {\omega}_{\text{graviton}}\cdot \delta {g}_{tt}\right)}^{2}/{\hslash}^{2}~{\gamma}^{2}{L}_{P}^{2}\cdot {\left({\omega}_{\text{graviton}}\cdot \delta {g}_{tt}\right)}^{2}$ (19)

Then if we go to Equation (10) we have a threshold magnetic field for the production of gravitons which looks like if we apply the minimum scale factor condition [16] , that

$\begin{array}{l}{B}_{\mathrm{min}}\ge \frac{1}{2\cdot \sqrt{10{\mu}_{0}\cdot \omega}}\approx \frac{\sqrt{\gamma {L}_{P}\delta {g}_{tt}}{10}^{123/4}}{2\cdot \sqrt{10{\mu}_{0}}}\approx \frac{\sqrt{\gamma {L}_{P}{\varphi}_{\text{initial}}}{a}_{\mathrm{min}}{10}^{123/4}}{2\cdot \sqrt{10{\mu}_{0}}}\\ \&\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{\mathrm{min}}~\left({10}^{-123/4}\right)\Rightarrow {B}_{\mathrm{min}}\ge \frac{\sqrt{\gamma {L}_{P}{\varphi}_{\text{initial}}}}{2\cdot \sqrt{10{\mu}_{0}}}\end{array}$ (20)

i.e. we get graviton production if the last line of Equation (20) is satisfied, which means that the initial value of the inflaton, in this case is crucially important.

With that initial inflaton value determined in part by Equation (11).

4. Conclusions: The Inflaton Minimum Helps Determine a Lower Bound for a Cosmological Initial Production of Gravitons

The last line of Equation (20) helps establish a minimum magnetic field for the production of relic gravitons, with a magnetic field established through Equation (10) and subsequently modified by Equation (20).

This adds substance to what was brought up by Beckwith in [16] namely that we have a minimum scale factor of

$\begin{array}{l}{a}_{\mathrm{min}}~\left({10}^{-123/4}\right)~{\left(\Delta E/{E}_{P}\right)}^{3/2}\\ \&\text{\hspace{0.17em}}{\varphi}_{\text{initial}}^{2}~o\left(\sqrt{\Delta E\cdot \gamma \cdot {L}_{P}}/{\hslash}^{2}\right)\end{array}$ (21)

But Equation (21) and Equation (20) interplay and also give more substances to the use of Equation (19) with our guess of Equation (18) for the determination of the initial graviton frequency, which has to be at least of the order of 10^45 Hertz due to the fantastically small initial bubble of space time considered.

In doing so, we need to consider initial conditions so considered that Equation (20) and Equation (21) should be consistent with the inflaton and “gravity’s breath” document by Corda [17] . In addition, Freeze’s statement of initial conditions for inflaton, as given by [18] should be adhered to. It is also extremely important that the LIGO results, even if this is of relic gravitational waves, as seen by Abbott in [19] [20] [21] , should not be contravened.

We conclude also with the hope that interpolating between the results of Equation (19), Equation (20) and Equation (21) will also in time confer answers as to the initial evaluative conditions for gravity as given in [22] by Corda. This also may in time with further analysis tie in with minimum values of entropy by further analysis of Equation (4) in further future analysis of this problem.

Fund

Work partially supported by National Nature Science Foundation of China grant No. 11375279.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

*Journal of High Energy Physics, Gravitation and Cosmology*,

**3**, 651-656. doi: 10.4236/jhepgc.2017.34049.

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