_{1}

^{*}

First of all, we will reiterate a mechanism for relic early universe gravitons. This is the precursor of the main result of this paper and involves the decay of micro black holes, with relic graviton production. For the 2
^{nd} part of graviton physics, we invoke reacceleration of the universe. The case for a four-dimensional graviton mass (non-zero) influencing reacceleration of the universe in five dimensions is stated, with particular emphasis on whether five-dimensional geometries as given below give us new physical insights as to cosmological evolution. The final question is, can DM/DE be explained by a Kaluza Klein particle construction?
i.e., the author presents a Kaluza Klein particle representation of a graviton mass with the first term to the right equal to a DM contribution and with the 2
^{nd} term to the right being effective DE. This is the 2
^{nd} era of DE production. Finally, we in the third part invoke the methodology of a modification of 1/r potentials, and the reacceleration of the universe as a bridge between the first and second parts of this document, in terms of the physics of DE and gravitons. This is compared with an equation of state given as a confirming basis for perhaps rationalizing a linkage between early universe production of gravitons, and then a subsequent Equation of state, for DE which may be able to predict reasons for quintessence.

1^{st} part briefly refers to production of DE due to as an example graviton release from 100 or more black holes, due to relic conditions: (source for this paper is provided). It is already accepted by JHEPGC and is summed up in [

2^{nd} part refers to DE as created by Gravitons creating re acceleration of the Universe one billion years Ago (jerk calculation). This is also an accepted JHEPGC paper which will be published later [

3^{rd} part is a modification of gravity which may produce DE. This involves modification of 1/r potentials for the reason brought up in this document. This is also an accepted JHEPGC paper, which will be published later [

The conclusion of this document compares both part 1 and part 2, while asking if there exists a bridge between the initial production of gravitons, possibly by the mechanism (Black hole in a micro scale, leading to their rapid decay and delivery of subsequent gravitons) brought up in the 1^{st} part, with the subsequent asking of pertinent links to the reinflation era, which is z = 0.43, i.e. one billion years ago. This is also compared to a result given by [^{st} and the 2^{nd} part mechanisms may entail answering and fine tuning the details given in part 3, which will be attempted later.

i.e. look at what is given in [

Quote,

In Corda’s recent work [^{2} black holes of mass 10^{2} times Planck mass (will set Planck mass, as (1)

M ≈ 10 2 m Planck Δ V n − 1 → n | Total ≈ 10 2 × Δ V n − 1 → n ≈ 10 2 × 16 π | ( 10 2 m Planck ) 2 − n 2 − ( 10 2 m Planck ) 2 − n − 1 2 | (1)

Here we make the following simplification of Equation (47) to read as

Δ V n − 1 → n | Total ≈ 10 2 × Δ V n − 1 → n ≈ 10 3 × 16 π ⋅ ( n 4 × 10 4 ) ⋅ | 1 − ( n − 1 n ) | (2)

Our supposition is that there are 10^{2} mini black holes, and a mass of 10^{2} times Planck mass, per each black hole, so that we perform the following normalization, i.e. find n for quantum number, so that to first approximation

16 π ⋅ ( n 4 × 10 4 ) ⋅ | 1 − ( n − 1 n ) | ≈ 10 6 (3)

i.e. that say 1000 times Planck length, we have the beginning of say creation of 100 mini black holes, each of mass about 100 times Planck mass which would put a huge restriction upon the admissible value, n, whereas giving a quantum value, n, for the enhanced quantum perturbation, used for penetration of the initial quantum state so assumed in this document as we go from Pre Planckian to Planckian physics, by emergent field construction. Equation (3) could be used to ascertain a quantum value, n, which would be for quantization level used to penetrate beyond the shell used to create the cosmological constant

End of quote.

What we have, is in Equation (3) the initial production of say 10^{6} relic gravitons, and we are specifying here that relic graviton production is due, maybe even before the electroweak regime of spacetime, and that this may serve as a template as to initial relic graviton production. If we have graviton production this way, we do have a way to ascertain if we have a new candidate to form DE, as we will state next.

Consider if there is then also a small graviton mass, i.e., as stated by Beckwith [

m n ( Graviton ) = n 2 L 2 + ( m graviton rest mass = 10 − 65 grams ) 2 = n L + 10 − 65 grams (4)

Note that Rubakov (2002) works with KK gravitons, without the tiny mass term for a 4 dimensional rest mass included in Equation (7). To obtain the KK graviton/DM candidate representation along RS dS brane world, Rubakov obtains his values for graviton mass and graviton physical states in space-time after

using the following normalization: ∫ d z a ( z ) ⋅ [ h m ( z ) ⋅ h m ˜ ( z ) ] ≡ δ ( m − m ˜ ) . Rubakov

[

h m ( z ) = m / k ⋅ J 1 ( m / k ) ⋅ N 2 ( [ m / k ] ⋅ exp ( k ⋅ z ) ) − N 1 ( m / k ) ⋅ J 2 ( [ m / k ] ⋅ exp ( k ⋅ z ) ) [ J 1 ( m / k ) ] 2 + [ N 1 ( m / k ) ] 2 (5)

Equation (5) is for KK gravitons having a TeV magnitude mass M Z ~ k (i.e., for mass values at 0.5 TeV to above 1 TeV) on a negative tension RS brane. It would be useful to relate this KK graviton, which is moving with a speed proportional to H − 1 with regards to the negative tension brane with

h ≡ h m ( z → 0 ) = const ⋅ m k as an initial starting value for the KK graviton mass. If Equation (5) is for a “massive” graviton with a small 4 dimensional graviton rest mass and if h ≡ h m ( z → 0 ) = const ⋅ m k represents an initial state, then

one may relate the mass of the KK graviton moving at high speed with the initial rest mass of the graviton. This rest mass of a graviton is m graviton ( 4-Dim GR ) ~ 10 − 48 eV , opposed to M X ~ M KKGraviton ~ 0.5 × 10 9 eV . Whatever the range of the graviton mass, it may be a way to make sense of what was presented by Dubovsky et al. [

a ˙ 2 = [ ( κ ˜ 2 3 [ ρ + ρ 2 2 λ ] ) a 2 + Λ ⋅ a 2 3 + m a 2 − K ] (6)

Maartens [^{nd} Friedman equation:.

H ˙ 2 = [ − ( κ ˜ 2 2 ⋅ [ p + ρ ] ⋅ [ 1 + ρ 2 λ ] ) + Λ ⋅ a 2 3 − 2 m a 4 + K a 2 ] (7)

Also, an observer is in the low redshift regime for cosmology, for which ρ ≅ − P , for red-shift values z from zero to 1.0 - 1.5. One obtains exact equality, ρ = − P , for z between zero to 0.5. The net effect will be to obtain, based on Equation (7), assuming Λ = 0 = K and using a ≡ [ a 0 = 1 ] / ( 1 + z ) to get a deceleration parameter q as given in Equation (8).

q = − a ¨ a a ˙ 2 = − 1 + 2 1 + κ ˜ 2 [ ρ / m ] ⋅ ( 1 + z ) 4 ⋅ ( 1 + ρ / 2 λ ) ≈ − 1 + 2 2 + δ ( z ) (8)

These set values, allow a graviton-based substitute for DE. Λ = 0 = K plus a small rest mass for a graviton in four dimensions allows for “massive gravitons” that behave the same as DE. Setting Λ = 0 = K , while having a modified behavior for the density expression, for a Friedman equation with small 4 dimensional graviton mass, means that dark energy is being replaced by a small 4 dimensional rest mass for a graviton.

2^{nd} part, continued. Consequences of small graviton mass for reacceleration of the universe

In a revision of Alves et. al. [

ρ ≡ ρ 0 ⋅ ( 1 + z ) 3 − [ m g ⋅ ( c = 1 ) 6 8 π G ( ℏ = 1 ) 2 ] ⋅ ( 1 14 ⋅ ( 1 + z ) 3 + 2 5 ⋅ ( 1 + z ) 2 − 1 2 ) (9)

Beckwith [

is a current for a “massive” graviton which is appropriate for explaining in part,

From [

Chaotic inflation would be using the approximation that

V ( φ ) ≈ k 2 a 2 ⋅ φ 2 (10)

Use the approximation that the time derivative is d / d τ , and φ ≡ φ k , and if so, then

φ ˙ k 2 2 = ( 3 8 π G ⋅ κ a 2 + k 2 a 2 ⋅ φ k 2 − ρ ) & φ ˙ k 2 2 = ( 3 8 π G ⋅ κ a 2 + k 2 a 2 ⋅ φ k 2 − 16 3 ⋅ c 1 ⋅ B 4 ) (11)

The last line of Equation (11) states that, if we apply it to the Pre Planckian to Planckian regime, that there will be a change in the energy, which we will call

Δ E ≈ ( 3 8 π G ⋅ κ a 2 + k 2 a 2 ⋅ φ k 2 − 16 3 ⋅ c 1 ⋅ B 4 ) (12)

We then will call this shift in energy, as equivalent to a change in KINETIC energy, and then reference the Virial theorem which in a general form, will be interpreted as

〈 ψ | Kinetic Energy | ψ 〉 = 〈 ψ | r ⋅ ∇ V ( Potential energy ) | ψ 〉 (13)

Leading to

〈 ψ | [ Kinetic Energy ≈ ( 3 8 π G ⋅ κ a 2 + k 2 a 2 ⋅ φ k 2 − 16 3 ⋅ c 1 ⋅ B 4 ) ] | ψ 〉 ≈ 〈 ψ | ( r ⋅ ∇ [ V ( Potential energy ) ≈ c 2 / r α ] ) | ψ 〉 (14)

In the Pre Planckian to Planckian space time, we will approximate, in the instant before time is initialized, formally, the mean value theorem as to the computed values of both the Left and right hand sides of Equation (13) and Equation (14) with the results that we obtain

( 3 8 π G ⋅ κ a 2 + k 2 a 2 ⋅ φ k 2 − 16 3 ⋅ c 1 ⋅ B 4 ) ≈ − α / r α ≡ − α / ( Planck length ) α ⇔ [ α / ( Planck length ) α ] ≈ 16 3 ⋅ c 1 ⋅ B 4 − ( 3 8 π G ⋅ κ a 2 + k 2 a 2 ⋅ φ k 2 ) (15)

This variant of dark energy should be compared with the main scheme which has re acceleration of the Universe as its main constituent point. I.e what is equivalent to DE is given below.

This shows in part that α is no longer strictly real valued but is strongly influence by the input from φ k , i.e. which has real and imaginary components

To put it mildly, we are concerned as to making a bridge between the initial and second physics realms as to graviton physics and cosmology. In a nutshell, we are proposing a modification of the 1/r gravitational potential for gravitational waves which in themselves will have gravitons, which is a way to not only get at the DE puzzle, but to answer specifics as to the interaction of gravitons with gravitational waves. i.e. if the 2^{nd} part of our presentation is accurate, we are setting the grounds for examining quantization of gravity, if we confirm both part 2 and part 3 of our inquiry. The gravitational field g u v also has field equations, sort of like those of electromagnetism (the gravitational equations are much more complicated, but they’re morally pretty similar). When we quantize the gravitational field as an effective field theory, we find that it too comes in set “quanta,” called gravitons. In short, we argue that to come up with a graviton based model of DE with reacceleration of the universe, and to have it commensurate with the modification of the 1/r potential, we are really coming up with a program of finding out if gravity can be quantized. In addition what we have done is also to specify a relic black hole genesis for early universe gravitons, which compliments the picture of turbulence in the electroweak era which in turn is relatable to the question of if micro black holes, could contribute to cosmology. This in turn means comparison with [

A Development to Look into, to Avoid Incompatible Models

At first glance, this looks hopeless. i.e. the models are incommensurate with each other and we do have a huge problem. A way of having reconciliation may be to consider what is brought up on pages 114 to 115 of Li, Li, Wang, and Wang, [

ω equation of state DE = ϕ ˙ 2 − 2 V ( ϕ ) ϕ ˙ 2 + 2 V ( ϕ ) ≈ ϕ ˙ 2 − 2 V 0 exp ( − 2 m ˜ ϕ m Plank ) ϕ ˙ 2 + 2 V 0 exp ( − 2 m ˜ ϕ m Plank ) for acceleration of universe m ˜ > 1 (16)

Again, this looks like a very hard problem, but in a nutshell, what we need to accomplish is having the following identification made, which may allow for us to make a concrete bridge between formalisms which otherwise look like they have no linkage to each other, i.e. do the following, namely is there a way to link ϕ of Equation (16) with φ k of Equation (11) and Equation (12) so as to come up with an acceptable form of ϕ which would satisfy the following equation, given in Equation (18) below, while keeping in mind that [

Here , we need for acceleration of universe m ˜ > 1 V ( ϕ ) = V 0 exp ( − 2 m ˜ ϕ m Plank ) (17)

Then, we should try to reconcile the following, if is there a way to link ϕ of Equation (16) with φ k of Equation (11) and Equation (12)

ω equation of state DE = ϕ ˙ 2 − 2 V ( ϕ ) ϕ ˙ 2 + 2 V ( ϕ ) ≈ ϕ ˙ 2 − 2 V 0 exp ( − 2 m ˜ ϕ m Plank ) ϕ ˙ 2 + 2 V 0 exp ( − 2 m ˜ ϕ m Plank ) for acceleration of universe m ˜ > 1 reconcile above with ( 3 8 π G ⋅ κ a 2 + k 2 a 2 ⋅ φ k 2 − 16 3 ⋅ c 1 ⋅ B 4 ) ≈ − α / r α ≡ − α / ( Planck length ) α ⇔ [ α / ( Planck length ) α ] ≈ 16 3 ⋅ c 1 ⋅ B 4 − ( 3 8 π G ⋅ κ a 2 + k 2 a 2 ⋅ φ k 2 ) (18)

In doing so, we would, after a great deal of work perhaps be able to make linkage to the scale factor a which shows up in Equation (15) above. i.e. having results in terms of the scale factor, and ϕ while making full identification with the Equation (18) above, may be a way to reformulate the linkage between early DE, which we want. In addition, such restraints upon scale factor a may also begin to explain some of the scale factor behavior which may be implied by Equation (7) and Equation (8) above. Also, if we do this, the final equation of state constraint we have to keep in mind is also given in [

− 2 ⋅ ( m Plank ) 2 ⋅ H ˙ = ϕ ˙ 2 (19)

This should be compared against Equation (7) and justified, if possible, as part of the bridge between early universe expressions of DE and the re acceleration of the universe.

It is important to note that LIGO as noted in references [

Furthermore, Corda in [

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.W. (2019) A Kaluza Klein Treatment of a Graviton and Deceleration Parameter Q(Z) as an Alternative to Standard DE and Its Possible Link to Standard DE Equation of State as Given by Li, Li, Wang and Wang, in 2017. Journal of High Energy Physics, Gravitation and Cosmology, 5, 208-217. https://doi.org/10.4236/jhepgc.2019.51012