_{1}

^{*}

We use a root finder procedure to obtain and an inflaton value due to use of a scale factor if we furthermore use as the variation of the time component of the metric tensor in Pre-Planckian Space-time up to the Planckian space-time initial values. In doing so, we obtain, due to the very restricted values for which are of the order of less than Planck time, results leading to an enormous value for the initial Cosmological constant.

Volovik [

E V a c ( N ) = ∫ d 3 r ⋅ ε ( n ) (1)

The integrand to be considered is, using a potential defined by U = c 2 m n as

given by Volovik for weakly interacting Bose gas particles, as well as

ε ( n ) = 1 2 U ⋅ n 2 + 8 15 π 2 ℏ 3 m 3 / 2 U 5 / 2 n 5 / 2 = 1 2 ⋅ c 2 ⋅ [ n ⋅ m + 4 15 ⋅ ( m 5 ℏ 2 ⋅ c ) ⋅ 1 n 2 ] (2)

For the sake of argument, m, as given above will be called the mass of a graviton, n a numerical count of gravitons in a small region of space, and afterwards, adaptations as to what this expression means in terms of entropy generation which will be subsequently raised. A simple graph of the 2^{nd} term of Equation (2) with comparatively large m and with ℏ = c = 1 has the following qualitative behavior, namely for

E 1 = [ c 2 / 2 ] ⋅ [ 4 15 ⋅ ( m 5 ℏ 2 ⋅ c ) ⋅ 1 n 2 ] (3)

E 1 ≠ 0 when n is very small, and E 1 = 0 as n → 10 10 at the onset of inflation. This will tie directly with a linkage between energy and entropy, as seen in the construction, looking at what Kolb [

ρ = ρ radiation = ( 3 / 4 ) ⋅ [ 45 2 π 2 g ∗ ] 1 / 3 ⋅ S 4 / 3 ⋅ r − 4 (4)

Here, the idea would be, to make the following equivalence. Namely look at,

[ [ Λ Max r 4 8 π G ] ⋅ ( 4 / 3 ) ⋅ [ 2 π 2 g ∗ 45 ] 1 / 3 ] 3 / 4 ~ S initial (5)

We furthermore make the assumption of a minimum radius of

R initial ~ 1 # l N g < l Planck (6)

This Equation (6) will be put as the minimum value of r, in Equation (5), where we have in this situation [

# bits ~ [ E ℏ ⋅ l c ] 3 / 4 ≈ [ M c 2 ℏ ⋅ l c ] 3 / 4 (7)

And if M is the total space-time energy mass, for initial condition and E1 is the main fluctuation in energy we have to consider, if Δ E ~ E 1 , as well as [

S initial ~ n graviton ~ initial graviton count (8)

Then what can be said about the inter relationship of graviton counts, and the onset of Causal structure?

We initiate our work, citing [

Δ t ⋅ | ( 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) − ( 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) 2 2 + ( 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) 3 3 − ⋯ | ≈ ( γ π G ) − 1 48 π ℏ a min 2 ⋅ Λ (9)

From here, we then cited, in [

S Λ | Arrow-of-time = π ⋅ ( R c | initial ~ c ⋅ Δ t l Planck ) 2 ≠ 0 (10)

This leads to the following, namely in [

( R c | initial ~ c ⋅ Δ t l Planck ) ~ ϑ ( 1 ) (11)

The rest of this article will be contingent upon making the following assumptions. FTR

Δ E ~ E 1 S initial ~ n graviton ~ initial graviton count [ [ Λ Max r 4 8 π G ] ⋅ ( 4 / 3 ) ⋅ [ 2 π 2 g ∗ 45 ] 1 / 3 ] 3 / 4 ~ S initial ⇔ [ [ Λ Max r 4 8 π G ] ⋅ ( 4 / 3 ) ⋅ [ 2 π 2 g ∗ 45 ] 1 / 3 ] 3 / 4 ~ n graviton Δ E ~ E 1 ~ V 0 r ~ R initial ~ 1 # l N g < l Planck (12)

In short, our view is that the formation of a minimum time step, if it satisfies Equation (11) is a necessary and sufficient condition for the formation of an arrow of time, at the start of cosmological evolution we have a necessary and sufficient condition for the initiation of an arrow of time. With causal structure, along the lines of Dowker, as in [

Δ E Δ t Volume ~ [ ℏ / Volume ⋅ ( δ g t t ~ a min 2 ⋅ ϕ initial ) ] | Pre-Planckian → ( Pre-Planckian ) → ( Planckian ) Δ E Δ t ~ ℏ | Planckian (13)

i.e. the regime of where we have the initiation of causal structure, if allowed would be contingent upon the behavior of [

g t t ~ δ g t t ≈ a min 2 ϕ initial ≪ 1 → Pre-Planck → Planck δ g t t ≈ a min 2 ϕ Planck ~ 1 ⇔ ( R c | initial ~ c ⋅ Δ t l Planck ) ~ ϑ ( 1 ) | Planck (14)

i.e. the right hand side of Equation (14) is the square of the scale factor, which we assume is ~10^−110, due to [

So, the question well will be leading up to is what does Equation (9), Equation (12), and Equation (13), tell us about graviton production, and the causal foundation condition stated at Equation (14)?

Here our derivation result which satisfies Equation (14) is contingent upon initial R c | initial ~ c ⋅ Δ t as an initial event horizon. So, our bubble of space-time is of the order of magnitude of approximately one Planck Length,

Δ E Volume | Pre-Planck ~ [ ℏ / ( Δ t ⋅ ( Volume ≡ ( l N g # ) 3 ) ⋅ ( δ g t t ~ a min 2 ⋅ ϕ initial ) ) ] | Pre-Planckian ~ c 2 ⋅ m graviton 2 30 ℏ 2 c ⋅ ⋅ ( Volume ≡ ( l N g # ) 3 ) − 1 ⋅ ( 1 n boson = n graviton ) 2 → ( Pre-Planckian ) → ( Planckian ) Δ E | Planckian ~ ℏ Δ t | Planckian ⋅ ( Volume ≡ ( l Planck ) 3 ) − 1 ~ ℏ Δ t | Planckian (15)

A convenient normalization would be to have

r ~ R initial ~ 1 # l N g < l Planck & l N g ~ l Planck ≡ 1 & r ~ R initial ~ 1 # (16)

If so then, Equation (14) would read as a causal formation transformation we would give as

r ~ R initial ~ 1 # l N g < l Planck & l N g ~ l Planck ≡ 1 & r ~ R initial ~ 1 # & ℏ ≡ c ≡ 1 Δ E Volume | Pre-Planck ~ [ 1 / ( Δ t ⋅ ( Volume ≡ ( 1 # ) 3 ) ⋅ ( δ g t t ~ a min 2 ⋅ ϕ initial ) ) ] | Pre-Planckian ~ m graviton 2 30 ⋅ ( Volume ≡ ( 1 # ) 3 ) − 1 ⋅ ( 1 n boson = n graviton ) 2 → ( Pre-Planckian ) → ( Planckian ) Δ E | Planckian ~ 1 Δ t | Planckian (17)

And then we would have the following equation if we make the following further normalization, as to Planck Mass, and Graviton mass, namely Planck Mass ~ 2.17645e−5 grams, whereas M (graviton) ~ 2.1e−62 grams. i.e. If Planck Mass = 1 in normalization, then M (graviton) ~ 10^−57

Δ E Volume | Pre-Planck ~ [ 1 / ( Δ t ⋅ ( Volume ≡ ( 1 # ) 3 ) ⋅ ( δ g t t ~ a min 2 ⋅ ϕ initial ) ) ] | Pre-Planckian ~ 10 − 114 30 ⋅ ( # 3 ( n boson = n graviton ) 2 ) ~ 10 − 114 30 → ( Pre-Planckian ) → ( Planckian ) Δ E | Planckian ~ 1 Δ t | Planckian ~ Ο ( 1 ) (18)

i.e. we would roughly have

10 − 114 30 → ( Pre-Planckian ) → ( Planckian ) 1 Δ t | Planckian ~ Ο ( 1 ) (19)

This outlines the enormity of the change from Pre Planckian to Planckian physics. If this is true, it indicates the enormity of the Pre Planckian to Planckian transformation. If we assume that a min 2 remains invariant, it means that the contribution of the inflaton becomes almost infinitely larger, i.e. a min 2 ~ 10^−110 in size.

So, if we have

Δ E | Pre-Planckian ~ 10 − 114 30 ⋅ ( # 3 ( n boson = n graviton ) 2 ) ~ 10 − 114 30 ~ V 0 (20)

and if Δ E | Pre-Planckian ~ 10 − 114 30 , so that we have

Δ t ⋅ | ( 8 π ⋅ ( Δ E | Pre-Planckian ) γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) − ( 8 π ( Δ E | Pre-Planckian ) 0 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) 2 2 + ( 8 π ⋅ ( Δ E | Pr e − P l a n c k i a n ) γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) 3 3 − ⋯ | ≈ ( γ π ) − 1 48 π a min 2 ⋅ Λ (21)

As

Δ t ⋅ | ( 8 π ⋅ ( 10 − 114 30 ) γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) − ( 8 π ( 10 − 114 30 ) γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) 2 2 + ( 8 π ⋅ ( 10 − 114 30 ) γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) 3 3 − ⋯ | ≈ ( γ π ) − 1 48 π a min 2 ⋅ Λ (22)

Or more approximately as

Δ t ⋅ | ( 10 − 114 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) − ( 10 − 114 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) 2 2 + ( 10 − 114 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) 3 3 − ⋯ | ≈ ( γ π ) − 1 48 π a min 2 ⋅ Λ (23)

Now, set Λ initial = Λ

Δ t ⋅ | ( 10 − 114 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) − ( 10 − 114 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) 2 2 + ( 10 − 114 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) 3 3 − ⋯ | ≈ ( γ π ) − 1 48 π ( a min 2 ~ 10 − 110 ) ⋅ Λ initial ≈ Δ t ≤ t Planck ~ 1 ⇔ 48 π ( a min 2 ~ 10 − 110 ) ⋅ Λ initial ~ Δ t ≤ t Planck ~ 1 ⇔ Λ initial ≥ 10 112 (24)

This is on the order of the Cosmological constant, as computed by [

This so happens to be consistent with Equation (5) of our document. It also has some similarities with the ideas given in [

Finally this should be seen in the light of [

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

Beckwith, A.W. (2018) How a Minimum Time Step and Formation of Initial Causal Structure in Space-Time Is Linked to an Enormous Initial Cosmological Constant. Journal of High Energy Physics, Gravitation and Cosmology, 4, 541-548. https://doi.org/10.4236/jhepgc.2018.43032