Gedankenexperiment for Contributions to Cosmological Constant from Kinematic Viscosity Assuming Self Reproduction of the Universe with Non-Zero Initial Entropy

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1. Introduction

We use Mukhanov’s “self-reproduction of the universe” criteria [1] plus Padmahan’s inflaton value [2] in the case of $a\left(t\right)~{a}_{\text{starting-point}}\cdot {t}^{\alpha}$ in order to come up with a criterion as to initial mass. We also will be examining the influence of [3]

$\frac{\Delta \rho}{\Delta t}~\left(\text{visc}\right)\times \left({H}_{\mathrm{int}}^{2}\right)\times {a}^{4}$ . (1)

With the initial Hubble parameter, in this situation a constant value in the Pre Planckian regime of space-time, instead of the usual

${H}_{\text{Hubble}}=\stackrel{\dot{}}{a}/a$ . (2)

Also, $\text{visc}$ in Equation (1) is for a viscous “fluid” approximation in a non-singular regime of space-time , where we are using [4] explicitly, namely, that we have initially due to [5] and the proportionality of energy to Boltzman’s constant times temperature [6]

$\Delta {t}_{\text{initial}}~\frac{\hslash}{\delta {g}_{tt}{E}_{\text{initial}}}~\frac{2\hslash}{\delta {g}_{tt}{k}_{B}{T}_{\text{initial}}}$ . (3)

If so, then Equation (1) and Equation (3) will give, to first approximation

$\begin{array}{c}\Delta \rho ~\left(\text{visc}\right)\times \left({H}_{\mathrm{int}}^{2}\right)\times {a}^{4}\times \frac{2\hslash}{\delta {g}_{tt}{k}_{B}{T}_{\text{initial}}}\\ ~\left(\text{visc}\right)\times \left({H}_{\mathrm{int}}^{2}\right)\times {a}_{\text{init}}^{2}\times \frac{2\hslash}{{\varphi}_{\mathrm{inf}}{k}_{B}{T}_{\text{initial}}}\end{array}$ . (4)

Here, we will be using an inflaton given by [2]

$\begin{array}{l}a\approx {a}_{\mathrm{min}}{t}^{\gamma}\\ \iff \varphi \approx \sqrt{\frac{\gamma}{4\text{\pi}G}}\cdot \mathrm{ln}\left\{\sqrt{\frac{8\text{\pi}G{V}_{0}}{\gamma \cdot \left(3\gamma -1\right)}}\cdot t\right\}\end{array}$ . (5)

Which is also in tandem with a Potential term given by

$V\approx {V}_{0}\cdot \mathrm{exp}\left\{-\sqrt{\frac{16\text{\pi}G}{\gamma}}\cdot \varphi \left(t\right)\right\}$ . (6)

These will be used in the rest of this paper, for our derivations, which will be in tandem with an emergent Cosmological Constant parameter which is in tandem with an alteration of the initial Penrose singularity theorem as brought up in [7] [8] , which would also be in tandem with an emergent cosmological constant parameter which we bring up next.

2. Emergent Cosmological Parameter, in the Pre Planckian to Planckian Space-Time Regime

Start off with a definition of

$\Delta \Lambda ~\left({\alpha}^{2}~\frac{1}{{\left(137\right)}^{2}}\right)\times {M}_{\text{initial}}^{4}$ . (7)

With volume defined in four space by

${V}_{\text{volume}\left(\text{initial}\right)}~{V}^{\left(4\right)}=\delta t\cdot \Delta {A}_{\text{surface-area}}\cdot \left(r\le {l}_{\text{Planck}}\right)$ . (8)

And with the “three volume” defined above, with the time factored out. So then we will be looking at initial mass given by

${M}_{\text{initial}}~\Delta {A}_{\text{surface-area}}\cdot \left(r\le {l}_{\text{Planck}}\right)\times \left(\text{visc}\right)\times \left({H}_{\mathrm{int}}^{2}\right)\times {a}_{\text{init}}^{2}\times \frac{2\hslash}{{\varphi}_{\mathrm{inf}}{k}_{B}{T}_{\text{initial}}}$ . (9)

And initial scale factor given as [4]

$\begin{array}{l}{\alpha}_{0}=\sqrt{\frac{4\text{\pi}G}{3{\mu}_{0}c}}{B}_{0}\\ \stackrel{\u2322}{\lambda}\left(\text{defined}\right)=\Lambda {c}^{2}/3\\ {a}_{\mathrm{min}}={a}_{0}\cdot {\left[\frac{{\alpha}_{0}}{2\stackrel{\u2322}{\lambda}\left(\text{defined}\right)}\left(\sqrt{{\alpha}_{0}^{2}+32\stackrel{\u2322}{\lambda}\left(\text{defined}\right)\cdot {\mu}_{0}\omega \cdot {B}_{0}^{2}}-{\alpha}_{0}\right)\right]}^{1/4}\end{array}$ . (10)

Here, the minimum scale factor has a factor of $\Lambda $ which we interpret as today’s value of the cosmological constant. B is the early cosmological B field, the Frequency of the order of 10 ^ 40 Hz, and ${a}_{\mathrm{min}}~{a}_{\text{initial}}~{10}^{-55}$ .

We will be combining the above into a commentary on Equation (7) to Equation (10) next.

3. The Emergent Cosmological Constant Parameter Is Affected, Immediately by a Visc (Viscous) Fluid Parameter

We also can restate the above behavior with an initial mass density we can give as

$\Delta \rho ~{\left[{V}_{3}\left(\text{volume}\right)\right]}^{-1}\times {N}_{\text{initial-count}}\times {m}_{\text{graviton}}$ . (11)

And, after Using Ng. Infinite quantum statistics [9] and a massive graviton [10]

$\begin{array}{l}{M}_{\text{initial}}~\Delta {A}_{\text{surface-area}}\cdot \left(r\le {l}_{\text{Planck}}\right)\times \left(\text{visc}\right)\times \left({H}_{\mathrm{int}}^{2}\right)\times {a}_{\text{init}}^{2}\times \frac{2\hslash}{{\varphi}_{\mathrm{inf}}{k}_{B}{T}_{\text{initial}}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}~{N}_{\text{gravitons}}\cdot {m}_{\text{gravitons}}\\ \iff {N}_{\text{gravitons}}~\frac{\Delta {A}_{\text{surface-area}}}{{m}_{\text{gravitons}}}\cdot \left(r\le {l}_{\text{Planck}}\right)\times \left(\text{visc}\right)\times \left({H}_{\mathrm{int}}^{2}\right)\times {a}_{\text{init}}^{2}\times \frac{2\hslash}{{\varphi}_{\mathrm{inf}}{k}_{B}{T}_{\text{initial}}}\end{array}$ . (12)

We found that the above would yield an N~10^{4} or so, which is not zero, but is only nonzero if we have the visc term not equal to zero in the initial bubble of space-time, and also that we observe having

$\begin{array}{l}a\approx {a}_{\mathrm{min}}{t}^{\gamma}\\ \iff \varphi \approx \sqrt{\frac{\gamma}{4\text{\pi}G}}\cdot \mathrm{ln}\left\{\sqrt{\frac{8\text{\pi}G{V}_{0}}{\gamma \cdot \left(3\gamma -1\right)}}\cdot t\right\}\\ \&\text{\hspace{0.17em}}\varphi >0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{iff}\sqrt{\frac{8\text{\pi}G{V}_{0}}{\gamma \cdot \left(3\gamma -1\right)}}\cdot \delta t>1\end{array}$ . (13)

This puts a major restriction upon admissible ${V}_{0}$ and $\delta t$ terms, for our problem.

In addition we postulate that the existende of massive gravitons is syominous with the classical-quantum mechanics linkage as given in [11] , i.e. on page 121 of [11] the authors manage to convert a D’ Alembert wave equation is converted to a Schrodinger equation, if the group velocity is included as having the form

${v}_{\text{group}}\propto E/\sqrt{2\left(E-V\right)}<c$ . (14)

Presumably in [11] the normalization of c =1 means that (14) is if it refers to an equation like the D’Alembert equation one for which it has classical behavior, to the Schrodinger equation.

Note that as given in [4] that massive gravitons travel at less than the speed of light. Our suggestion is that by inference, those massive gravitons, then would be commensurate with the HUP which we brought up in this document and that the formulation is consistent.

All this should also be tied into an investigation of how the viscosity of Equation (4) would also tie into the results above, with the interplay of Equation (3) and Equation (4) maybe giving by default some information as to condition for which the quantization condition linkage in the classical regime (represented by the Classical De Alembert equation) and the quantum Schrodinger equation have analogies in our model.

4. Conclusions: And Now for the Use of the Idea of Mukhanov’s Self Reproduction of the Universe

And now back to [1] . We have defined the initial mass, and done it with an eye toward a constraint condition upon the entries into Equation (12). It is now time to go to the problem of if evolution of the inflaton will allow us to have classical and quantum contributions to the inflaton which if we follow Mukhanov’s will lead to the inflaton growing if and only if

$\begin{array}{l}\Delta {\varphi}_{\text{total}}=\Delta {\varphi}_{\text{classical}}+\Delta {\varphi}_{\text{quantum}}~-{\varphi}_{\mathrm{inf}}^{-1}+{M}_{\text{total}}{\varphi}_{\mathrm{inf}}>0\\ \iff {\varphi}_{\mathrm{inf}}>{\left(1/{M}_{\text{total}}\right)}^{1/2}\end{array}$ . (15)

We need to satisfy Equation (11) and Equation (12) in order to make sense out of Equation (15).

Furthermore, though, the number N, of Equation (12) will be nonzero and well behaved with a nonzero real value for positive N and entropy only if we have

$\varphi >0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{iff}\text{\hspace{0.17em}}\sqrt{\frac{8\text{\pi}G{V}_{0}}{\gamma \cdot \left(3\gamma -1\right)}}\cdot \delta t>1$ . (16)

This also is the same condition for which we would have to have visc, i.e. the viscosity of the initial spherical starting point for expansion, nonzero as well as reviewing the issues as of [12] - [18] .

We argue that the formulation of Equation (15) and Equation (16) would be, with the inclusion of the mass of the graviton, especially as in ${\varphi}_{\mathrm{inf}}>{\left(1/{M}_{\text{total}}\right)}^{1/2}$ as a precondition for steady growth of the inflaton, in cosmological expansion as out lined in [1] will provide a template for a non zero initial entropy as that N, in the denominator of ${\varphi}_{\mathrm{inf}}>{\left(1/{M}_{\text{total}}\right)}^{1/2}$ if set equal to zero would imply far more stringent conditions than what is given in Equation (16) above, i.e. Infinite initial values of the inflaton, initially even in the Pre Planckian regime. We find that this is doubtful. Also issues of the graviton mass as given in [12] [13] may prove to be extremely important. Next, confirmation of the usefulness of the restriction of the ${\varphi}_{\mathrm{inf}}>{\left(1/{M}_{\text{total}}\right)}^{1/2}$ limit needs to be reconciled with [14] as to the initial conditions of inflaton. Whereas how we do it may allow for the Corda references [15] [17] to be experimentally investigated. Finally the Abbot articles of [16] [18] must be adhered to as far as the formation of Gravitational signatures even from the early universe [19] [20] .

Acknowledgements

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

References

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