We use the Padmanabhan 1^st integral [1] [2] of the form, with the third entry of Equation (1) having a Ricci scalar defined via [9] and usually the curvature ℵ set as extremely small, with the general relativity [3] [4]

S 1 = 1 2 κ ∫ − g ⋅ d 4 x ⋅ ( ℜ − 2 Λ ) &   − g = − det g u v &   ℜ = 6 ⋅ ( a ¨ a + ( a ˙ a ) 2 + ℵ a 2 ) (1)

Also, the variation of δ g t t ≈ a min 2 ϕ as given by [5] [6] will have an inflaton, ϕ given by [6] [7] [9] leads to [7] [8] to the inflaton, which is combined into other procedures for a solution to the cosmological constant problem. Here, a min is a minimum value of the scale factor [8] [9].

We are going to go to page 78 by Klauder [4] of what he calls on page 78 a restricted Quantum action principle which he writes as: S 2 where we write a one to one equivalence as in [1], which is also seen in [3]

S 2 = ∫ 0 T d t ⋅ [ p ( t ) q ˙ ( t ) − H N ( p ( t ) , q ( t ) ) ] ≈ S 1 = 1 2 κ ∫ − g ⋅ d 4 x ⋅ ( ℜ − 2 Λ ) (2)

Our assumption is that Λ is a constant, hence we assume then the following approximation from [3]

p 0 2 2 = p 0 2 ( N ) 2 + N ;     for   0 < N ≤ ∞ ,   and   q = q 0 ± p 0 t V N ( x ) = 0 ;     for   0 < x < 1 V N ( x ) = N ;     otherwise H N ( p ( t ) , q ( t ) ) = p 0 2 2 + ( ℏ ⋅ π ) 2 2 + N ;     for   0 < N ≤ ∞ (3)

Our innovation is to then equate q = q 0 ± p 0 t ~ ϕ and to assume small time step values. Then

Λ ≈ − [ V 0 3 γ − 1 + 2 N + γ ⋅ ( 3 γ − 1 ) 8 π G ⋅ t ˜ 2 ] 1 κ ∫ − g ⋅ d 3 x + ( 6 ⋅ ( a ¨ a + ( a ˙ a ) 2 ) ) | t = t ˜ (4)

These are terms within the bubble of space-time given in [1] using the same inflaton potential. The scale factor is presumed here to obey the value of the scale factor in [9].

In order to do this, we will replace the term, N, as appearing in Equation (3) and Equation (4) by the Starobinsky early inflation potential.

From [10] pp 152 to153, where we assume q ~ 1

N ≈ V ( ϕ ) = V 0 ⋅ ( 1 − exp [ − q ⋅ ϕ / m P ] ) 2 (5)

Also, making use of [7] and

a ¨ a = H ˙ + H 2 (6)

And

a ¨ a = H ˙ + H 2 = − t ⋅ γ ⋅ ( 3 γ − 1 ) m P G ⋅ 1 8 π + 8 π G 3 ⋅ [ V 0 ⋅ ( 1 − exp [ − q ⋅ ϕ / m P ] ) 2 ] (7)

Where we have used

( a ˙ a ) 2 = H 2 ≡ 8 π G 3 ⋅ [ V 0 ⋅ ( 1 − exp [ − q ⋅ ϕ / m P ] ) 2 ] (8)

And so we obtain if we have a scale factor behaving as

Λ ≈ − [ V 0 3 γ − 1 + 2 N + γ ⋅ ( 3 γ − 1 ) 8 π G ⋅ t ˜ 2 ] 1 κ ∫ − g ⋅ d 3 x + ( 6 ⋅ ( a ¨ a + ( a ˙ a ) 2 ) ) | t = t ˜ ≈ − [ V 0 3 γ − 1 + 2 V 0 ⋅ ( 1 − exp [ − q ⋅ ϕ / m P ] ) 2 + γ ⋅ ( 3 γ − 1 ) 8 π G ⋅ t ˜ 2 ] 1 κ ∫ − g ⋅ d 3 x     + 6 ⋅ − t ⋅ γ ⋅ ( 3 γ − 1 ) m P G ⋅ 1 8 π + 48 π G 3 ⋅ [ V 0 ⋅ ( 1 − exp [ − q ⋅ ϕ / m P ] ) 2 ] (9)

The time interval we are specifying in Equation (9) is of about Planck time.

We can write an expression for V 0 from [10], page 153 taking the form of

V 0 1 / 4 = 0.022 m P q N e-folds (10)

This value will be important when we make a linkage to Gravitons in the end of this analysis.

To complete our analysis, we will go to [11], and an interesting piece of analysis, about the cosmological constant. Before doing that we will say a bit about evaluation of a term in one of the denominators of Equation (9).

Here –g as in 1 κ ∫ − g ⋅ d 3 x .

We will be assuming, also that as from [7] that

a ( t ) = a initial t ν ⇒ ϕ = ln ( 8 π G V 0 ν ⋅ ( 3 ν − 1 ) ⋅ t ) ν 16 π G ⇒ ϕ ˙ = ν 4 π G ⋅ t − 1 ⇒ H 2 ϕ ˙ ≈ 4 π G ν ⋅ t ⋅ T 4 ⋅ 1.66 2 ⋅ g ∗ m P 2 ≈ 10 − 5 (11)

The numbers here are extremely small, hence we will be considering a restricted version of g in 1 κ ∫ − g ⋅ d 3 x with regards to [11].

From [11], page 399, in what is called considering the dynamics of a non vanishing cosmological constant, namely if φ is the classical mechanics potential for how a particle travels along a geodestics of a metric so considered

g → − g 00 = − 1 − 2 φ / c 2 φ = − κ M r − 1 6 ⋅ Λ ⋅ c 2 r 2 (12)

The claim from page 399 of [11] is that then there would be a net acceleration of a test particle, we could write as

a radial = Λ 3 ⋅ c 2 r (13)

Our supposition that such a small initial acceleration would be pertinent to the production/acceleration of Gravitons from the initial part/place of expansion of the universe, i.e. what would be the starting gun for inflation.

If this were the initial nudge of space-time this would be commensurate with the following interpretation of 1 κ ∫ − g ⋅ d 3 x with regards to [11] will to a certain degree presuppose not paying attention to the following initial line metric

d S Schwartchild | Spatialpartlineelement ≃ d l 2 | M → 0 = ( 1 − Λ r 2 / 3 ) − 1 d r 2 + r 2 d θ 2 + r 2 sin 2 θ d ϑ 2 (14)

I am pre supposing that in the initial stages of space-time that the term g appearing in 1 κ ∫ − g ⋅ d 3 x with regards to [11] would be treatable as being a constant, so as to place the initial configuration of the cosmological “constant” parameter as a straight out inter-play between a positive set of terms, and a negative set of terms, so as to nearly cancel, which is such that if the initial acceleration as to gravitons would be, at its initial inception very small. However, once perturbed, i.e. a barrier of sorts breached, inflation would commence as a violent universe creating phenomenon.

For the record, the term r in Equation (13) would be assumed to be very small, of the magnitude of Plancks length, whereas the driving force to expansion would be doable as linked to Equation (10)

Note plank mass would be presumed to be about 2.1764 times 10^−5 grams whereas a common mass of the graviton, if it has mass would be about 10^−65 grams, whereas, if we have by [12] [13] compared with Equation (10) the mass-energy equivalent of 10^62 gravitons, which may be in the situation specified as to the cosmological constant

Klauder’s program [4] is to embed via Equation (6) as a quantum mechanical well for a Pre Planckian-system for inflaton physics as given by Equation (3). As given in Klauder’s treatment of the action integral as of page 87 of [4] where Klauder talks of the weak correspondence principle, where an enhanced classical Hamiltonian is given one to one correspondence with quantum effects, in a non-vanishing fashion. If so, by Novello [3] [12]

m g = ℏ ⋅ Λ c (15)

This formula needs to be confirmed via likely CMB radiation as could be ascertained by interaction of the CMB spectra, and gravitational waves. However, if Equation (15) were true, it may mean that the initial regime of cosmological constant creation may be able to give a description as to what exactly initiated the transition to inflation, to begin with. And that is an open question.

The author has presupposed all of this using the following Uncertainty principle [13] [14] [15] [16]

Δ t ≥ ℏ Δ E + γ t P 2 Δ E ℏ ⇒ ( Δ E ) 2 − ℏ Δ t γ t P 2 ( Δ E ) 1 + ℏ 2 γ t P 2 = 0 ⇒ Δ E = ℏ Δ t 2 γ t P 2 ⋅ ( 1 + 1 − 4 ℏ 2 γ t P 2 ⋅ ( ℏ Δ t 2 γ t P 2 ) 2 ) = ℏ Δ t 2 γ t P 2 ⋅ ( 1 ± 1 − 16 ℏ 2 γ t P 2 ( ℏ Δ t ) 2 ) (16)

Δ E ≈ ℏ Δ t 2 γ t P 2 ⋅ ( 1 ± ( 1 − 8 ℏ 2 γ t P 2 ( ℏ Δ t ) 2 ) ) ⇒ Δ E ≈ either   ℏ Δ t 2 γ t P 2 ⋅ 8 ℏ 2 γ t P 2 ( ℏ Δ t ) 2 ,   or   ℏ Δ t 2 γ t P 2 ⋅ ( 2 − 8 ℏ 2 γ t P 2 ( ℏ Δ t ) 2 ) (17)

Δ E ≈ ℏ Δ t 2 γ t P 2 ⋅ 8 ℏ 2 γ t P 2 ( ℏ Δ t ) 2 ≡ 4 ℏ Δ t (18)

Our supposition is that the initial radial distance regime of creation of space-time allowing for a cosmological constant at the beginning of the universe is commensurate with a time interval Δ t . Again, if say 10^62 gravitons may be produced as of the time when a cosmological constant is produced, and if some of the gravitons are created by the decay of mini black holes, this says a lot about initial conditions which may be verifiable later as to suitable data sets in gravitational wave astronomy

In particular, Equation (10) is directly from Starobinsky as reported in [10] and it would be of stunning import if the following is confirmed

V 0 1 / 4 ≈ 5.4 × 10 16     GeV q N efolds (10)

Also see reference [17] - [22].

The term q is of about 1 in magnitude, where as N efolds is the number of evolds in inflation. Tying this in, say if a graviton were of 10^−65 grams would be of huge importance as far as GW astronomy is concerned.

This work is supported in part by the 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. (2022) How to Use Starobinsky Inflationary Potential Plus Argument from Alder, Bazin, and Schiffer as Radial Acceleration to Obtain First Order Approximation as to Where/When Cosmological Constant May Form. Journal of High Energy Physics, Gravitation and Cosmology, 8, 1012-1018. https://doi.org/10.4236/jhepgc.2022.84071