Using “Particle Density” of “Graviton Gas”, to Obtain Value of Cosmological Constant ()

Andrew Beckwith^{}

Department of Physics, College of Physics, Chongqing University, Chongqing, China.

**DOI: **10.4236/jhepgc.2023.91015
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Department of Physics, College of Physics, Chongqing University, Chongqing, China.

We use the work of de Vega, Sanchez, and Comes (1997), to approximate the “particle density” of a “graviton gas”. This “particle density” derivation is compared with Dolgov’s (1997) expression of the Vacuum energy in terms of a phase transition. The idea is to have a quartic potential, and then to utilize the Bogomol’nyi inequality to refine what the phase transition states. We utilize Ng, Infinite quantum information procedures to link our work with initial entropy and other issues and close with a variation in the HUP: at the start of the expansion of the universe.

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Beckwith, A. (2023) Using “Particle Density” of “Graviton Gas”, to Obtain Value of Cosmological Constant. *Journal of High Energy Physics, Gravitation and Cosmology*, **9**, 168-173. doi: 10.4236/jhepgc.2023.91015.

1. Introduction

We first state the summary findings of the de Vega, Sanchez, and Comes [1] self interacting gravitational gas. The piece, authored in 1997 gives a partition function, then a net “particle density” argument. This construction will form the basis of the subsequent evaluation. We for the sake of the gas, reference a bosonic Spin 2 “graviton gas” similar in part to what was done by [1] but adopting the conventions of Infinite quantum statistics by Ng [2] to conflate particle count with entropy, makes the case that what we are doing is to conclusively argue for a nonzero initial entropy.

The rest of the manuscript, borrows from Doldov’s [3] 1997 discussion of the variation of the “cosmological constant” and its inter relationship to a potential congruent with the mechanism of symmetry breaking. What we do is to equate the variation of the “cosmological constant” and from there ask what it portends if there is no variation in the cosmological “constant” from its inception to [3] to its present value. We should also note that we use Padmanbhan’s arguments [4] as to scalar fields, which will be used to confirm some of the details in [3]. This is the plan of the manuscript. Now let us proceed.

2. Reviewing the Implementation of Reference [1]

In [1], there is the use of a partition function which was initially planned for a “cold interstellar gas” but which we apply for a bosonic graviton gas, partly in the spirit of [5] but assuming in conjunction with the authors work in applying [2], and the idea of massive gravitons as given in [6]. To begin, look at the partition function [1], as given by

$\begin{array}{c}Z={\displaystyle \iint \wp \varphi \mathrm{exp}\left(-S\left(\varphi \right)\right)}\\ ={\displaystyle \iint \wp \varphi \mathrm{exp}\left(-\frac{1}{{T}_{eff}}{\displaystyle \int {\text{d}}^{3}x\left[\frac{{\left(\nabla \varphi \right)}^{2}-2{\mu}^{2}\mathrm{exp}\left(\varphi \left(x\right)\right)}{2}\right]}\right)}\end{array}$ (1)

Then, if *T* is a temperature, and *z* is the fugacity, and *m* is the mass, which we will decompose:

${T}_{eff}=4\pi G{m}^{2}{T}^{-1};{\mu}^{2}=\sqrt{2{\pi}^{-1}}\cdot z\cdot G\cdot {m}^{7/2}\cdot \sqrt{T}$ (2)

The key element which we will be working with is, a particle density expression of [1] as

$\langle \rho \left(r\right)\rangle ={\mu}^{2}{T}_{eff}^{-1}\cdot \langle \mathrm{exp}\left(\varphi \left(r\right)\right)\rangle $ (3)

If we use the following from Padmanabhan, [4], using the approximation of $a\left(t\right)~{t}^{\stackrel{\u02dc}{n}}$, then

$\varphi \left(r\left(t\right)\right)~\varphi \left(t\right)\approx \sqrt{2\stackrel{\u02dc}{n}\cdot {m}_{Pl}}\cdot \mathrm{ln}\left(t\right)\simeq \mathrm{ln}\left({t}^{\sqrt{2\stackrel{\u02dc}{n}\cdot {m}_{Pl}}}\right)$ (4)

$\langle \rho \left(r\right)\rangle ={\mu}^{2}{T}_{eff}^{-1}\cdot {t}^{\sqrt{2\stackrel{\u02dc}{n}\cdot {m}_{Pl}}}\propto {\mu}^{2}{T}_{eff}^{-1}\cdot {t}^{\sqrt{2\stackrel{\u02dc}{n}}}$ (5)

We will be utilizing these first five equations, with Equation (5) compared against results from [3], next.

3. Isolating m Value in Equation (2) and Equation (5) and Its Relevance to Reference [3]

Comparing Equations (2) and (5) get us a mass term of the proportional value

$m~{\left(\frac{\lambda}{\sqrt{2{\pi}^{3}}}{T}^{3/2}{t}^{\sqrt{2\stackrel{\u02dc}{n}}}\right)}^{2/5}$ (6)

Dolgov, in [3] has an emergent value of the vacuum energy density which he gives as follows with our subsequent valuation.

${\rho}_{\text{Vacuum}}~\frac{{m}^{4}}{2\lambda}~\frac{{\lambda}^{8/5}}{2\lambda \cdot {\left(\sqrt{2{\pi}^{3}}\right)}^{8/5}}\cdot {\left({T}^{3/2}\cdot {t}^{\sqrt{2\stackrel{\u02dc}{n}}}\right)}^{8/5}$ (7)

Then the given by [3] value for subsequent emergent fluctuation of the “cosmological constant” is

${\Lambda}_{\text{cos}\text{.const}}~8\pi \cdot {\rho}_{\text{Vacuum}}/{m}_{\text{Planck}}^{2}~\left[\frac{4\pi \cdot {\lambda}^{3/5}}{{\left(\sqrt{2{\pi}^{3}}\right)}^{8/5}\cdot {m}_{\text{Planck}}^{2}}\right]\cdot {\left({T}^{3/2}\cdot {t}^{\sqrt{2\stackrel{\u02dc}{n}}}\right)}^{8/5}$ (8)

Our subsequent point of evaluation will compare Equation (8) with a present day value of the cosmological constant of

${{\Lambda}_{\text{cos}\text{.const}}|}_{\text{today's}}~{\left(2.4\times {10}^{-11}\text{\hspace{0.17em}}\text{GeV}/{c}^{2}\right)}^{4}$ (9)

Comparison of Equations (8) and (9) leads to $\stackrel{\u02dc}{n}~25/32$, and

$\begin{array}{l}{\left(2.4\times {10}^{-11}\text{\hspace{0.17em}}\text{GeV}/{c}^{2}\right)}^{4}\cdot \left({1.2009}^{2}\times {10}^{38}{\left(\text{GeV}\right)}^{2}/{c}^{4}\right)\\ ~\left[\frac{4\pi \cdot {\lambda}^{3/5}}{{\left(\sqrt{2{\pi}^{3}}\right)}^{8/5}}\right]\cdot {\left({T}^{3/2}\cdot {t}^{\sqrt{50/32}}\right)}^{8/5}\end{array}$ (10)

And using [2]

$\varphi \left(r\left(t\right)\right)~\varphi \left(t\right)\approx \sqrt{\left(50/32\right)\cdot {m}_{Pl}}\cdot \mathrm{ln}\left(t\right)$ (11)

Then according to [3] we should look at the spontaneous symmetry breaking potential, given by

$U\left(\varphi \right)~-{m}^{2}{\varphi}^{2}+\lambda {\varphi}^{4}$ (12)

Setting the temperature, *T*, and the time, *t* as Planck temperature and Planck time, and specifying we are still adhering to Equation (10) leads to a spontaneous symmetry breaking potential of the form which has
$\lambda $

$\begin{array}{l}{\left(2.4\times {10}^{-11}\text{\hspace{0.17em}}\text{GeV}/{c}^{2}\right)}^{4}\cdot \left({1.2009}^{2}\times {10}^{38}{\left(\text{GeV}\right)}^{2}/{c}^{4}\right)\\ ~\left[\frac{4\pi \cdot {\lambda}^{3/5}}{{\left(\sqrt{2{\pi}^{3}}\right)}^{8/5}}\right]\cdot {\left({T}_{\text{Planck}}^{3/2}\cdot {t}_{\text{Planck}}^{\sqrt{50/32}}\right)}^{8/5}\end{array}$ (13)

We shall next, then proceed to discuss the idea of a graviton gas (bosonic), and the spontaneous symmetry breaking potential.

4. Conclusion: Existence of Graviton Gas? Non Zero Initial Entropy?

We acknowledge that Glinka, [5] pursued this idea in 2007. Our approach is fundamentally different from his, and we make use of using Equation (13) to set the
$\lambda $. As well as specify the mass of a graviton as 10^{−62} grams as given in [6]. Following up upon the Ng “infinite quantum statitistics” as given by [2] so we then write, *S* (entropy) as ~ *N* (counting number), and we specify *N*, via

$\begin{array}{l}m~{\left(\frac{\lambda}{\sqrt{2{\pi}^{3}}}{T}_{\text{Planck}}^{3/2}{t}_{\text{Planck}}^{\sqrt{50/32}}\right)}^{2/5}~{N}_{\text{graviton}}\cdot {m}_{\text{graviton}}\\ \Rightarrow {N}_{\text{graviton}}~S\left(\text{Initialentropy}\right)~{\left(\frac{\lambda}{\sqrt{2{\pi}^{3}}}{T}_{\text{Planck}}^{3/2}{t}_{\text{Planck}}^{\sqrt{50/32}}\right)}^{2/5}/{m}_{\text{graviton}}\end{array}$ (14)

The value of the initial graviton mass is specified as being 10^{−62} grams, meaning that this puts a premium upon the fine tuning of the initial parameters in the numerator of Equation (14). We hope that, if this is conclusively non zero, that it will enable CMBR style studies as alluded to in [7] and [8], as well as looking at non zero vacuum energy as given by non linear electrodynamics as in [9] and also the issue of the nature of gravity as up by Corda [10], as far as future studies and investigations, See Appendix A as to further elaborations as to the Infinite Quantum statistics brought up in this document. In addition, we argue that further understanding of Equation (14) will add more definition to the fluctuations of the metric tensor as alluded to in [6] and [7] and [8] as a compliment to Equations (6) and (7), *i.e.* further developments should specifically investigate the symmetry breaking potential as written up as enabling the metric tensor approach given in [8].

Acknowledgements

Part of this presentation was also in lectures to Graduate Physics Students in Chongqing University, in November 2013.

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

Appendix, a Review of Ng [2] with Comments

First of all, Ng [2] refers to the Margolus-Levitin theorem with the rate of operations
$<E/\hslash $
$\Rightarrow $
$\#\text{operations}<E/\hslash \times \text{time}=\frac{M{c}^{2}}{\hslash}\cdot \frac{l}{c}$. Ng wishes to avoid black-hole formation
$\Rightarrow M\le \frac{l{c}^{2}}{G}$. This last step is not important to our view point, but we refer to it to keep fidelity to what Ng brought up in his presentation. Later on, Ng refers to the
$\#\text{operations}\le {\left({R}_{H}/{l}_{P}\right)}^{2}~{10}^{123}$ with*
${R}_{H}$ *the Hubble radius. Next Ng refers to the
$\#\text{bits}\propto {\left[\#\text{operations}\right]}^{3/4}$. Each bit energy is
$1/{R}_{H}$ with
${R}_{H}~{l}_{P}\cdot {10}^{123/2}$

The key point as seen by Ng [2] and the author is in, if *M* is the “space-time” mass

*
$\#\text{bits}~{\left[\frac{E}{\hslash}\cdot \frac{l}{c}\right]}^{3/4}\approx {\left[\frac{M{c}^{2}}{\hslash}\cdot \frac{l}{c}\right]}^{3/4}$ * (1)

Assuming that the initial energy *E* of the universe is not set equal to zero, which the author views as impossible, the above equation says that the number of available bits goes down dramatically if one sets
${R}_{\text{initial}}~\frac{1}{\#}{\mathcal{l}}_{\text{Ng}}<{l}_{\text{Planck}}$ *? *Also Ng writesentropy *S* as proportional to a particle count via *N*.

*
$S~N\cong {\left[{R}_{H}/{l}_{P}\right]}^{2}$ * (2)

We rescale ${R}_{H}$ to be

*
${{R}_{H}|}_{\text{rescale}}~\frac{{l}_{\text{Ng}}}{\#}\cdot {10}^{123/2}$ * (3)

The upshot is that the entropy, in terms of the number of available particles drops dramatically if $\#$ becomes larger.

So, as ${R}_{\text{initial}}~\frac{1}{\#}{\mathcal{l}}_{\text{Ng}}<{l}_{\text{Planck}}$ grows smaller, as $\#$ becomes larger.

1) The initial entropy drops.

2) The nunber of bits initially available also drops.

The limiting case of Equations (2) and (3) in a closed universe, with no higher dimensional embedding is that both would almost vanish, *i.e.* appear to go to zero if
$\#$ becomes very much larger. The question we have to ask is would the number of bits in computational evolution actually vanish?

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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