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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.

We first state the summary findings of the de Vega, Sanchez, and Comes [

The rest of the manuscript, borrows from Doldov’s [

In [

Z = ∬ ℘ ϕ exp ( − S ( ϕ ) ) = ∬ ℘ ϕ exp ( − 1 T e f f ∫ d 3 x [ ( ∇ ϕ ) 2 − 2 μ 2 exp ( ϕ ( x ) ) 2 ] ) (1)

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

T e f f = 4 π G m 2 T − 1 ; μ 2 = 2 π − 1 ⋅ z ⋅ G ⋅ m 7 / 2 ⋅ T (2)

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

〈 ρ ( r ) 〉 = μ 2 T e f f − 1 ⋅ 〈 exp ( ϕ ( r ) ) 〉 (3)

If we use the following from Padmanabhan, [

ϕ ( r ( t ) ) ~ ϕ ( t ) ≈ 2 n ˜ ⋅ m P l ⋅ ln ( t ) ≃ ln ( t 2 n ˜ ⋅ m P l ) (4)

〈 ρ ( r ) 〉 = μ 2 T e f f − 1 ⋅ t 2 n ˜ ⋅ m P l ∝ μ 2 T e f f − 1 ⋅ t 2 n ˜ (5)

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

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

m ~ ( λ 2 π 3 T 3 / 2 t 2 n ˜ ) 2 / 5 (6)

Dolgov, in [

ρ Vacuum ~ m 4 2 λ ~ λ 8 / 5 2 λ ⋅ ( 2 π 3 ) 8 / 5 ⋅ ( T 3 / 2 ⋅ t 2 n ˜ ) 8 / 5 (7)

Then the given by [

Λ cos .const ~ 8 π ⋅ ρ Vacuum / m Planck 2 ~ [ 4 π ⋅ λ 3 / 5 ( 2 π 3 ) 8 / 5 ⋅ m Planck 2 ] ⋅ ( T 3 / 2 ⋅ t 2 n ˜ ) 8 / 5 (8)

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

Λ cos .const | today's ~ ( 2.4 × 10 − 11 GeV / c 2 ) 4 (9)

Comparison of Equations (8) and (9) leads to n ˜ ~ 25 / 32 , and

( 2.4 × 10 − 11 GeV / c 2 ) 4 ⋅ ( 1.2009 2 × 10 38 ( GeV ) 2 / c 4 ) ~ [ 4 π ⋅ λ 3 / 5 ( 2 π 3 ) 8 / 5 ] ⋅ ( T 3 / 2 ⋅ t 50 / 32 ) 8 / 5 (10)

And using [

ϕ ( r ( t ) ) ~ ϕ ( t ) ≈ ( 50 / 32 ) ⋅ m P l ⋅ ln ( t ) (11)

Then according to [

U ( ϕ ) ~ − m 2 ϕ 2 + λ ϕ 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 λ

( 2.4 × 10 − 11 GeV / c 2 ) 4 ⋅ ( 1.2009 2 × 10 38 ( GeV ) 2 / c 4 ) ~ [ 4 π ⋅ λ 3 / 5 ( 2 π 3 ) 8 / 5 ] ⋅ ( T Planck 3 / 2 ⋅ t Planck 50 / 32 ) 8 / 5 (13)

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

We acknowledge that Glinka, [^{−62} grams as given in [

m ~ ( λ 2 π 3 T Planck 3 / 2 t Planck 50 / 32 ) 2 / 5 ~ N graviton ⋅ m graviton ⇒ N graviton ~ S ( Initialentropy ) ~ ( λ 2 π 3 T Planck 3 / 2 t Planck 50 / 32 ) 2 / 5 / m graviton (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 [

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.

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

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. https://doi.org/10.4236/jhepgc.2023.91015

First of all, Ng [

The key point as seen by Ng [

# bits ~ [ E ℏ ⋅ l c ] 3 / 4 ≈ [ M c 2 ℏ ⋅ l c ] 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 initial ~ 1 # l Ng < l Planck ? Also Ng writesentropy S as proportional to a particle count via N.

S ~ N ≅ [ R H / l P ] 2 (2)

We rescale R H to be

R H | rescale ~ l Ng # ⋅ 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 initial ~ 1 # l Ng < l 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?