In this document, we are revisiting the following statement made earlier [1][2]

Quote

Usingthefollowing

Then

Then, Equation (1) and Equation (2) together yield

What we are going to do is to, in the initial variation of the GUP is to look hard at the initial idea given in Equation (3) is to make the following treatment at the start of expansion of the Universe [1][2]

If this is effectively almost zero, the effect would be to embedd Quantum mechanics within a 5 dimensional structure.

Snip

i.e. this deterministic embedding is in part in spirit similar to what is given by Wesson [3]

EndofQuote

What we will be doing is to add more context to this is to use the Wesson result directly in our own work and use it to in effect prove a deterministic contrubution in line with Equation (3) and Equation (4) of this document.

Before proceeding we should state that the inflaton field so used in Equation (3) and Equation (4) satisfies the following [2][4]

In the spirit of use of the inflaton field what we will propose is that [3]

i.e. assume that if the initial time step is near Planck time which is normalized to 1 that

In addition we will go to Wesson [5] and to make the following adjustments.

where we will define l and L as follows

First, define L in terms of the cosmological “constant” by [5]

Also

Also, 5 dimensional wave number is defined via [5]

In the case of Pre Planckian space-time the idea is to do the following [5]

Making use of all this leads to [3][4]

Our approximation is to set G = 1 = h (Planck units) with Planck time normalized to 1. Then

Then we have that at Planck time, normalized to 1 we look at

We assume that we have an emergent space-time. If so, and in the spirit of [6], we state

This concludes having a linkage as to [6] in terms of a huge energy flux into the early universe

Having said this, what can we now say about graviton mass, in the early universe?

When initial radius R initial → 0 if Stoica [7] actually derived Einstein equations in a formalism which remove the big bang singularity pathology, then the reason for Planck length no longer holds. This incidently is not too dissimilar to what was brought up in [8][9] However, we present entanglement entropy in the early universe with a shrinking scale factor, due to Muller and Lousto [10], and show that there are consequences due to initial entangled S Entropy = 0.3 r H 2 / a 2 for a time dependent horizon radius r H in cosmology, with (flat space conditions) r H = η for conformal time. Even if the 3 dimensional spatial length goes to zero. This construction preserves a minimum non zero Λ vacuum energy, and in doing so keep the bits, for computational bits cosmological evolution even if R initial → 0 . We state that the presence of computational bits is necessary for cosmological evolution to commence. Note that τ time is with regards to In the time domain, the usual choice to explore the time response is through the step response to a step input , or the impulse response to a Dirac delta function input In the frequency domain. Here the step input is with regards to initial conditions of the early universe. Also, conformal time d η = d t / a ( t ) , where what is called time t, is in this case identical to τ time.

This article is to investigate what happens physically if there is a non pathological singularity in terms of Einsteins equations at the start of space-time. This eliminates the necessity of having then put in the Planck length since then ther would be no reason to have a minimum non zero length. The reasons for such a proposal come from [7] by Stoica who may have removed the reason for the development of Planck’s length as a minimum safety net to remove what appears to be basic pathologies at the start of applying the Einstein equations at a space-time singularity, and are commented upon in this article. ρ ~ H 2 / G ⇔ H ≈ a − 1 in particular is remarked upon. The idea is that entanglement entropy will help generate bits, due to the presence of a vacuum energy, as derived at the end of the article, and the presence of a vacuum energy non zero value, is necessary for cosmological evolution. Before we get to that creation of what is a necessary creation of vacuum energy conditions we refer to constructions leading to extremely pathological problems which could lead to minus the presence of initial non zero vacuum energy. [6] also adds more elaboration on this. Note a change in entropy formula given by Lee [9] about the inter relationship between energy, entropy and temperature as given by

As a reviewer has asked about Equation (17) and the inter relationship of a mass m, and acceleration, the key point of this review is to look at if gravitons have a mass, m, in the beginning, and if Equation (17) is used, which the mass of a graviton is proportional to the following

The reason why the mass of a graviton is stated as given by Equation (1a) is to presume that the relation ship given by Lee [9], as to any mass, is given by Equation (17) and Equation (18) so we can relate any presumed mass linked to gravitons to change in entropy. As to acceleration appearing, the acceleration, a ^ ≅ c 2 Δ x was part of the formula given by Equation (17) and by default Equation (18). and also by thermodynamic reasoning the generalized temperature

If we assume, in the onset of expansion of the universe, that Equation (19) holds, then we can review the application of Equation (18) to graviton mass, m, as m = Δ E c 2 = T U ⋅ Δ S c 2 , and to have acceleration, given by a ^ ≅ c 2 Δ x as part of a definition of generalized temperature, given by Equation (19) with the rate of operations < E / ℏ ⇒ # operations < E / ℏ × time = M c 2 ℏ ⋅ l c . Ng wishes to avoid black-hole formation ⇒ M ≤ l c 2 G . This last step is not important to our view point, Note the interesting interplay between entropy in Equation (18) and the Lousto result of the universe as given in [10]i.e. entangled S Entropy = 0.3 r H 2 / a 2 for a time dependent horizon radius r H in cosmology, with (flat space conditions) r H = η for conformal time. Even if the 3 dimensional spatial length goes to zero. This construction preserves a minimum non zero Λ vacuum energy, and in doing so keep the bits, for computational bits cosmological evolution even if R initial → 0 . We state that the presence of computational bits is necessary for cosmological evolution to commence. And this can lead to the following, i.e. Review of Ng, [11]-[13] with comments. First of all, Ng refers to the Margolus-Levitin theorem but we refer to it to keep fidelity to what Ng brought up in his presentation. Later on, Ng refers to the # operations ≤ ( R H / l P ) 2 ~ 10 123 with R H the Hubble radius. Next Ng refers to the # bits ∝ [ # operations ] 3 / 4 . Each bit energy is 1 / R H with R H ~ l P ⋅ 10 123 / 2

ThekeypointasseenbyNg [11]-[13]andtheauthorisin

AssumingthattheinitialenergyEoftheuniverseisnotsetequaltozero, whichtheauthorviewsasimpossible, theaboveequationsaysthatthenumberofavailablebitsgoesdowndramaticallyifonesets R initial ~ 1 # ℓ N g < l Planck ? AlsoNgwritesentropySasproportionaltoaparticlecountviaN.

Werescale R H tobe

Theupshotisthattheentropy, intermsofthenumberofavailableparticlesdropsdramaticallyif # becomeslarger.

So, as R initial ~ 1 # ℓ N g < l Planck growssmaller, as # becomeslargerbutaswestate, wedoNOThaveaworkingalmostsingularstart, i.e. closebutnosingularityatthestartofexpansion. Havingsaidthis, wewillconflatethisdiscussionwithGravitons, andblackholes, withaventureintoTorsionanddarkenergy [14][15].

One, of the things to check into, is do we have Equation (19) going up say close to Planck temperature? IfsowecanthentototheideaofaBECscalingargumentastotheformationofearlyniversestructure,andthelinktoprimordialgravitons.

To do this review how Torsion may allow for understanding a quantum number n? And Primordial black holes and the cosmological constant.

Following [14][15], we do the introduction of black hole physics in terms of a quantum number n.

And then a BEC condensate given by [14]-[17] as to

This is promising but needs to utilize [4][14] in which we make use of the following. First, a time step. Here, Equation (25) to Equation (28) are with respect to Planck era physics

By use of the HUP [14][15], we use Equation (25) for energy [14][15] for radiation of a particle pair from a black hole,

Here we assert that the spatial variation goes as

This is of a Plank length, whereas we assume in Equation (28) that the mass is a Planck sized black hole [14][15]

This mass of primordial black holes is part of the first table, i.e. keep in mind that is with regards to a Plank length value as given in Equation (27) where likely Equation (27) as Planck length was formed in the pre Plank era and holds with respect to the Planck physics domain.

Table 1. From [14][15] assuming Penrose recycling of the Universe as stated in that document.

As to Table 1, we obtain, due to the quantum number n, per black hole

The Table 1 data will be connected to the following given consideration of spin density, as to Planck sized black holes.

In [14][15] we have the following, i.e., we have a spin density term of [14]-[16]. And this will be what we input black hole physics into as to forming a spin density term from primordial black holes.

And, also, the initial energy, E. from [18] per black hole given as

We then can use for a Black hole the scaling,

We then reference Equation (24) to observe the following.

This is a stunning result. We interpret as the mass of a Black hole becomes smaller that the black hole heats up. As the heat up occurs, this generates more energy. More energy leads to a higher count of n gravitons i.e. Equation (24) is BEC theory, but due to micro sized black holes, that we assume that the number of the quantum number, n associated goes way UP. Is this implying that corresponding increases in quantum number, per black hole, n, are commensurate with increasing temperature? We start off with Table 1 for conditions with the entropy as given in Equation (23) and Equation (24), for primordial black holes as brought up in Table 1. Whereas for the Tokamaks, we eventually have [14][15]

This value of Equation (33) as to the number of gravitons, would be then related to the quantum number N (gravitons) as related to a quantum number n i.e. only in the very onset of the operation of the Tokamak. i.e. we would have the number of gravitons go UP as we would have a shrinking graviton wavelength for a massive graviton. i.e. more on this later. However, the wave length of the massive graviton as in Equation (33) as related to GW frequency and Tokamaks will be described when we conclude our document with respect to the Wavefunction of the Universe, i.e. a work partly drawing upon Kieffer, and also Weber. The wavefunction of the universe condition heavily is influenced by the similarities as to Equation (33) with the quantum number n, per black hole, and the number N, of black holes, as brought up in Table 1, initially presented. To do so we consider Table 1 as giving a template as to a wormhole connecting a prior universe to the present universe. This Equation (33) as far as quantum number, n, and its relationship to gravitons, is contingent upon Equation (11) yielding Planck temperature. i.e. note the following, i.e. the HUP for universe contributions all come from the radial equation, which greatly simplifies our picture as far as nucleation of early universe black holes, i.e. see the following treatment of the HUP which enables this alteration.

To begin this process, we will break it down into the following coordinates [1] and keep in mind this is when a ( t ) is a scale factor, as specified in [1]. In the rr, θ θ and ϕ ϕ coordinates, we will use the Fluid approximation, T i i = d i a g ( ρ , − p , − p , − p ) [1] with

If as an example, we have negative pressure, with T r r , T θ θ and T ϕ ϕ < 0 , and p = − ρ , then the only choice we have, then is to set δ g r r ~ δ g θ θ ~ δ g ϕ ϕ ~ 0 + , since there is no way that p = − ρ is zero valued.

i.e. this is a semi classical embedding via a use of the modification of the HUP as given, as to how we could have a semi classical embedding of QM within a “higher dimensional” structure. This is also linked to the blend of semi classical and quantum structure seen in Table 1, above. i.e. after we do this we have to determine if our use of black holes, gravitons and all that will lead to GW generation, as to the formation of Table 1. i.e.

In the quantum theory of light (quantum electrodynamics) and other bosonicquantum field theories, coherent states were introduced by the work of Glauber (1963) [19]. Now, it is well appreciated that Gravitons are NOT similar to light. So what is appropriate for presenting gravitons as coherent states? Coherent states, to first approximation are retrievable as minimum uncertainty states. If one takes string theory as a reference, the minimum value of uncertainty becomes part of a minimum uncertainty which can be written as given by Venziano (1993) [20][21], where l S ≅ 10 α ⋅ l Planck , with α > 0 , and l Planck ≈ 10 − 33 centimeters

To put it mildly, if the author is looking at a solution to minimize graviton position uncertainty, the author, will likely be out of luck if string theory is the only tool the author has for early universe conditions. Mainly, the momentum will not be small, and uncertainty in momentum will not be small either. Either way, most likely, Δ x > l S ≅ 10 α ⋅ l Planck In addition, it is likely, as Klaus Kieffer (2008) [22] in his book “Quantum Gravity” that if gravitons are excitations of closed strings, then one will have to look for conditions for which a coherent state of gravitons, as stated by [23] Mohaupt (2003) occurs. What Mohaupt is referring to is a string theory way to reproduce what [24] Ford gave in 1995, i.e. conditions for how Gravitons in a squeezed vacuum state, the natural result of quantum creation in the early universe will introduce metric fluctuations. Ford’s (1995) treatment is to have a metric averaged retarded Green’s function for a mass less field becoming a Gaussian. The condition of Gaussianity is how to obtain semi classical, minimal uncertainty wave states, in this case de rigor for coherent wave function states to form. [24] Ford uses gravitons in a so called ‘squeezed vacuum state’ as a natural template for relic gravitons. i.e. the squeezed vacuum state (a squeezed coherent state) is any state such that the uncertainty principle is saturated. In QM coherence would be when Δ x Δ p = ℏ / 2 . In the case of string theory it would have to be

Begin with noting that if one is not using string theory, the author, Beckwith, merely set the term l S → non string 0 , but that the author is still considering a variant of the example given by Glauber (1963) [19] with string theory replacing Glaubler’s stated (1963) example. [19]. However, in string theory, the author, Beckwith observes a situation where a vacuum state as a template for graviton nucleation is built out of an initial vacuum state, | 0 〉 . To do this though, as [25] Venkatartnam, and Suresh did, involved using a squeezing operator Z [ r , ϑ ] defining via use of a squeezing parameter r as a strength of squeezing interaction term, with 0 ≤ r ≤ ∞ , and also an angle of squeezing, − π ≤ ϑ ≤ π as used in Z [ r , ϑ ] = exp [ r 2 ⋅ ( [ exp ( − i ϑ ) ] ⋅ a 2 − [ exp ( i ϑ ) ] ⋅ a + 2 ) ] , where combining the Z [ r , ϑ ] with

Equation (37) leads to a single mode squeezed coherent state, as they define it via

The right hand side of Equation (38) given above becomes a highly non classical operator, i.e. in the limit that the superposition of states | ς 〉 → α → 0 Z [ r , ϑ ] ⋅ | 0 〉 occurs, there is a many particle version of a ‘vacuum state’ which has highly non classical properties. Squeezed states, for what it is worth, are thought to occur at the onset of vacuum nucleation, but what is noted for | ς 〉 → α → 0 Z [ r , ϑ ] ⋅ | 0 〉 being a super position of vacuum states, means that classical analog is extremely difficult to recover in the case of squeezing, and general non classical behavior of squeezed states. Can one, in any case, faced with | α 〉 = D ( α ) ⋅ | 0 〉 ≠ Z [ r , ϑ ] ⋅ | 0 〉 do a better job of constructing coherent graviton states, in relic conditions, which may not involve squeezing?. Note L. Grishchuk [26] wrote in (1989) in “On the quantum state of relic gravitons”, where he claimed in his abstract that ‘It is shown that relic gravitons created from zero-point quantum fluctuations in the course of cosmological expansion should now exist in the squeezed quantum state. The authors have determined the parameters of the squeezed state generated in a simple cosmological model which includes a stage of inflationary expansion. It is pointed out that, in principle, these parameters can be measured experimentally’. Grishchuk, etal., [26] (1989) reference their version of a cosmological perturbation h n l m via the following argument. How the author works with the argument will affect what is said about the necessity, or lack of, of squeezed states in early universe cosmology. From [27] Class. Quantum Gravity: 6 (1989), page 161-165, where h n l m has a component μ n l m ( η ) obeying a parametric oscillator equation, where K is a measure of curvature which is = ± 1 , 0 , a ( η ) is a scale factor of a FRW metric, and n = 2 π ⋅ [ a ( η ) / λ ] is a way to scale a wavelength, λ , with n, and with a ( η )

If y ( η ) = μ ( η ) a ( η ) is picked, and a Schrodinger equation is made out of the Lagrangian used to formulate the above Equation (11) above, with P ^ y = − i ∂ y , and M = a 3 ( η ) , Ω = n 2 − K 2 a ( η ) , a ⌣ = [ a ( η ) / l Planck ] ⋅ σ , and F ( η ) an arbitrary function. y ′ = ∂ y / ∂ η . Also, the author is working with an example which has a finite volume V finite = ∫ g ( 3 ) d 3 x

Then the Lagrangian for deriving Equation (40) is (and leads to a Hamiltonian which can be also derived from the Wheeler De Witt equation), with ς = 1 for zero point subtraction of energy

Then there are two possible solutions to the S.E. Grishchuk [27][28] created in 1989, one a non squeezed state, and another squeezed state. So in general the author works with [27]-[29]

The nonsqueezedstate has a parameter B | η → η → η b B ( η b ) ≡ ω b / 2 where η b is an initial time, for which the Hamiltonian given in (40) in terms of raising/lowering operators is “diagonal”, and then the rest of the time for η ≠ η b , the squeezedstate for y ( η ) is given via a parameter B for squeezing which when looking at a squeeze parameter r, for which 0 ≤ r ≤ ∞ , then Equation (43) has, instead of B ( η b ) ≡ ω b / 2

Taking Grishchuck’s formalism literally, a state for a graviton/GW is not affected by squeezing when the author is looking at an initial frequency, so that ω ≡ ω b initially corresponds to a non squeezed state which may have coherence, but then right afterwards, if ω ≠ ω b which appears to occur whenever the time evolution, η ≠ η b ⇒ ω ≠ ω b ⇒ B ( ω , η ≠ η b ) ≡ i 2 ⋅ ( μ / a ( η ) ) ′ ( μ / a ( η ) ) ≠ ω b 2 A reasonable research task would be to determine, whether or not B ( ω , η ≠ η b ) ≠ ω b 2 would correspond to a vacuum state being initially formed right after the point of nucleation, with ω ≡ ω b at time η ≡ η b with an initial cosmological time some order of magnitude of a Planck interval of time t ≈ t Planck ∝ 10 − 44 seconds, we argue for reviewing the idea that coherent states are directly linked to the viability of applying Table 1 to our physics problem.

If so, then we have that frequency is proportional to 1/t, where t is time. i.e. hence if there is a value of n = 0 and making use of the frequency, we then would be able to write as given by Kieffer, 2025 as a dust model of the universe, with [22]

Or,

With, say.

This is t the so called dust approximation as to the formation of the Universe, with the result that if r goes to zero, we avoid a singularity, which then also increases the likelihood of Table 1 holding up.

Secondly is the open question of if or not we can form gravitational solitons [30] as well as [31] phenomenology being either proven or disproven.

Having given this formulation of a wavefunction of the Universe, what can we say about the before the Planckian regime of space time, to the post Planckian regime of space time implied by Table 1? i.e. we need to specify if we have h bar, leading to the possibility of quantum processes, in our analysis.

We will now give a first order estimate as to calculation of h bar, i.e. isolate the actual spatial length, for the creation of a present day h bar Planck’s constant. To do this look at

Then THE FOLLOWING ARE EQUIVLENT

The idea would be that the Planck constant, h bar would be formulated as of the present day value,. Also, the modification for the string length, would have Δ x | min ~ 10 β l Planck , so then

Then,

This should be greater than a Plank length, mainly due to the situation of

We assume, here that this will be occurring in an interval of time approximately the value of Planck time given by

Here, the number, N, is given as the number of gravitons, and the important factor is that Equation (6) is non zero.

How likely is δ g t t ~ O ( 1 ) ? Not going to happen. Why? The homogeneity of the early universe will keep

In fact, we have that from Giovannini [32], that if ϕ is a scalar function, and a 2 ( t ) ~ 10 − 110 , then if

We see further confirmation of this in reference [33].

Then, there is no way that an early universe HUP is going to come close to δ t Δ E ≥ ℏ 2 . i.e. It depends assuming time is for all purposes fixed at about Planck time to isolate V 0 .

i.e. for the sake of argument, in the near Planckian regime, we can figure that Equation (54) will have as far as evaluation of the argument the following configuration, i.e.

Given this we will be looking at, if we do the set up [34]

Then eventually we obtain

So then we are now doing an Evaluation of Equation (81) if we are near Planck time. Two limits

1^st, what if we have expansion of the scale factor initially at greater than the speed of light?

Set ν ≈ 10 88 and then we can obtain if we are just starting off inflation say a min 2 ≈ 10 − 44 . Then

If we wish to have a Planck energy magnitude of the V 0 term, we will then be observing

i.e. the system complexity will become effectively almost infinite, and this will be explained in the conclusion by use of

On the other hand, if there is a very small value for 2 γ ˜ ∂ C ∂ V we can see the following behavior for Equation (58), namely

i.e. low complexity in the measurement process will then imply an enormous initial inflaton potential energy

2^nd, now what if we have instead v ≈ 1

The threshold if 2 γ ˜ ∂ C ∂ V ≈ [ 10 88 ] i.e. a huge value for initial complexity would be effectively made insignificant in cutting down the initial inflaton lead to

i.e. we come to the seemingly counter Intuitive expression that the initial inflaton potential would still be infinite if we used Equation (62) in Equation (58). We now get to our next important idea and result. Having said that does it make sense to ascertain the following as far as early universe geometry? i.e. we say its not so simple. i.e. here is an early idea which we do not follow.

[33] Abhay Ashtekar (2006) wrote a simple treatment of the Bounce causing Wheeler De Witt equation along the lines of, for ρ ∗ ≈ c o n s t ⋅ ( 1 / 8 π G Δ ) as a critical density, and Δ the eignvalue of a minimum area operator. Small values of Δ imply that gravity is a repulsive force, leading to a bounce effect.

Furthermore, [31] Bojowald (2008) specified criteria as to how to use an updated version of Δ and ρ ∗ ≈ c o n s t ⋅ ( 1 / 8 π G Δ ) in his GRG manuscript on what could constitute grounds for the existence of generalized squeezed initial (graviton?) states. [31] Bojowald (2008) was referring to the existence of squeezed states, as either being necessarily, or NOT necessarily a consequence of the quantum bounce. As Bojowald (2008) wrote it up, in both his Equation (26) which has a quantum Hamiltonian 〈 V ^ 〉 ≈ H , with

and V ^ is a “volume” operator where the “volume” is set as V , Note also, that [31] Bojowald has, in his initial Friedman equation, density values ρ ≡ H matter ( a ) a 3 , so that when the Friedman equation is quantized, with an initial internal time given by ϕ , with ϕ becoming a more general evolution of state variable than ‘internal time’. If so, [31] Bojowald (2008) writes, when there are squeezed states

We think that this is simply not realistic and we offer a different interpretation, which is heavily using the HUP in a different fashion

There are three regimes of black hole numbers given in Table 1. From Pre Planckian, to Planckian and then to post Planckian physics regimes. This is all assuming CCC cosmology. To start to make sense of this, we need to examine how one could achieve the complexity as indicated by Figure 1 in the Planckian era. To do this at a start, we will pay attention to a datum in [34], namely a Horizon, like a Schwarzschild black hole construction with [14][15][34]

In what one deems as a corpuscular gravity, one would have a “kinetic energy term” per graviton

And the mass of a black hole, scaling as [14][15][34]

so then we have N ˜ = N and furthermore [14][15][34] also has

If so for Black holes, we have the following

Now as to what is given in [14][15][34] as to Torsion that we can do some relevant dimensional scaling.

First look at numbers provided by [14][15][34] as to inputs, i.e. these are very revealing, i.e. we go back to the arguments as to the beginning of the document, namely Λ P l c 2 ≈ 10 87 .

This is the number for the vacuum energy and this enormous value is 10¹²² times larger than the observed cosmological constant. Torsion physics, as given by [14][15][34] is solely to remove this giant number.

Our timing is to unleash a Planck time interval t about 10⁻⁴³ seconds. Also the creation of the torsion term is due to a presumed “graviton” particle density of n P l ≈ 10 98   cm − 3 .

This particle density is directly relevant to the basic assumption of how to have relevant Gravitons initially created as to obtain the huge increase in complexity alluded to, in order to obtain the number of micro black holes in the Planckian era [14][15][34].

i.e. assume that there are, then say initially up to 10⁹⁸ gravitons, initially, and then from there, go to Table 1 to assume what number of micro sized black holes are available, i.e.Table 1 has say a figure of 10⁴⁵ to at most 10⁵⁰ micro sized black holes, presumably for 10⁹⁸ gravitons being released, and this is meaning we have say 10⁵⁰ black holes of say of Planck mass, to work with.

Eventually, in the case of an unpolarized spinning fluid in the immediate aftermath of the big bang, we would see a Roberson Walker universe given as, if σ is a torsion spin term added due to [14][15][34] as

Here the term k ˜ is a measure of if one has positive, negative, or zero curvature. In all of this, the values of k ˜ are usually linked to questions of if we have an open or closed universe. It has three possible values: 1 for a closed positive curvature universe (like a sphere), 0 for a flat Euclidian Universe, and 01 for an open negative-curvature universe. The question, which we need to address is as follows.

What [14][15][34] does as to Equation (72) versus what we would do and why? In the case of [14][15][34] we would see σ be identified as due to torsion so that we obtain

The claim is made in [14][15][34] that this is due to spinning particles which remain invariant so the cosmological vacuum energy, or cosmological constant is always cancelled. Our approach instead will yield

i.e. the observed cosmological constant Λ 0 bserved is 10⁻¹²² times smaller than the initial vacuum energy. The main reason for the difference in Equation (72) and Equation (74) is in the following observation.

Mainly that the reason for the existence of σ 2 is due to the dynamics of spinning black holes in the precursor to the big bang, to the Planckian regime, of space time, whereas in the aftermath of the big bang, we would have a vanishing of the torsion spin term. i.e. the Table 1 dynamics in the aftermath of the Planckian regime of space time would largely eliminate the σ 2 term.

First look at numbers provided by [14][15][34] as to inputs, i.e. these are very revealing

This is the number for the vacuum energy and this enormous value is 10¹²² times larger than the observed cosmological constant. Torsion physics, as given by [14][15][34] is solely to remove this giant number. In order to remove it, the reference [14][15][34] proceeds to make the following identification, namely

What we are arguing is that instead, one is seeing, instead

Our timing as to Equation (77) is to unleash a Planck time interval t about 10⁻⁴³ seconds. As to Equation (76) versus Equation (77) the creation of the torsion term is due to a presumed particle density of

Finally, we have a spin density term of σ P l = n P l ℏ ≈ 10 71 which is due to innumerable black holes initially. This treatment of torsion and the spin density term, i.e. of black holes, with a particle density term of Equation (78) is due solely to the HUP employed, as to arguments given in the first part of the paper. And gives substance to Table 1 estimates about the number of black holes in the prior universe to the present universe.

Why I find this very interesting are the questions raised in [35][36]i.e. what is special about the Plank mass as an example. We submit that a through examination of Torsion may allow an answer to this issue in terms of early universe cosmology.

Go first as to AppendixA as to the details as for Equation (79) in this document.

What this means in terms of phenomenology, is as follows, i.e. if we look at the section given by [34] as elaborated upon in the fifth force argument as given we can say the following.

First of all is the old standby namely in the onset of inflation, there would be a huge speed of inflationary expansion with the coefficient of scale factor given as [14][15][34]. i.e. this is looking at the coefficient showing up in a ( t ) ≈ a 0 t ν scale factor expansion, that if we go to Equation

For mass greater than Planck mass, namely M m a s s ≈ ζ m p , with m p for Planck Mass. We refer to [34], in that this is for the mass of a physical system, i.e. M mass of an object which in its physical configuration is generating gravitational waves, ω g w and we find that in the Planckian regime, β ˜ is a coefficient connected to a fifth force argument due to reasoning from [34].

Thisleadstothefollowingi.e.in [34]whichisreproducedhere, Inadditionafterapproximating 〈 r 2 〉 2 ≈ ℓ p 4 , i.e. Planck length to the fourth power and r 〈 r 2 〉 2 ≈ ℓ p 5

We find then we have at the immediate beginning of inflation, an almost Planck frequency value of 1.855 times 10⁴³ Hertz, we would need ν be 10⁵⁰² which would be factored into Equation (1a) and the scale factor value for the term ν. This would mean for the fifth force argument that we would have an almost infinitely quick expansion in the neighborhood of Planck length for the start of inflation.

We find then we have at the immediate beginning of inflation, an almost Planck frequency value of 1.855 times 10⁴³ Hertz, we would need ν be 10⁵⁰² which would be factored into the coefficient of time which shows up in a scale factor argument and the scale factor value for the term ν. This would mean for the fifth force argument that we would have an almost infinitely quick expansion in the neighborhood of Planck length for the start of inflation.

We refer the reader to AppendixA, where two of the coefficients of Equation (76), ζ is connected to how many times a GW generating system has a mass greater than Planck mass, namely M mass ≈ ζ m p , with m p for Planck Mass. We refer to AppendixA, in that this is for the mass of a physical system, i.e. M mass of an object which in its physical configuration is generating gravitational waves, ω g w and we find that in the Planckian regime, β ˜ is a coefficient connected to a fifth force argument which is elaborated upon in AppendixA. We also state that the coefficient given as Equation (11) is such that in the Planckian regime, ν → value ≈ ( ω g w ) 12 . The meaning of this is that at the start of inflation we have that Equation (79) will likely from Pre Planckian to Planckian space time be enormous corresponding to ν → value ≈ ( ω g w ) 12 .

This is all defined in [4] in an article written by the author for Intech, for our convenience which is in AppendixA, in full

If so, by Novello [37] we then have a bridge to the cosmological constant as given by

Consider first the relationship between vacuum energy and the cosmological constant. Namely ρ Λ ≈ ℏ k max 4 where we have that

where we define the mass of a graviton as in the numerator given by Equation (80), and then we can also use the following.

This is useful in terms of determining conditions for a cosmological constant [14][15][34][38].

This means shifting the energy level of the Equation (81) downward by 10⁻³⁰, i.e. the top value energy becomes a down scale of Planck energy times 10⁻³⁰.

Gravitons are used as the back bone of how to reduce Equation (82) to the vacuum energy of today.

In addition there is one line of reasoning, which I believe bears mentioning.

What would be a way to determine necessary and sufficient conditions for a massive graviton to exist. To do so, we will look first at [39] Linde (Les Houches, 2013), whom wrote of the probability of creation of a closed universe as given by

The potential energy, so identified in Equation (83) is none other than the one used by Padmanbhan [3] in which the H so identified is the Hubble ‘constant’ parameter, which actually changes over time. In this case, the potential so identified in Equation (83) is given by

Here, if N is an integer number for dimensionality of space-time, and [3]

Applying the HUP uncertainty principle be looking at a minimum uncertainty principle situation of time.

If so, then if we have V as proportional to an energy E, then we can by the Heisenberg uncertainty principle

Then, if Δ t = t (minimum), and Δ E = E initial ≡ c 4 2 G ⋅ r critial ~ c 4 L P 2 G n initial π

Now, by Valev, [40] at the start of inflation, and this is before massive red shifting

Inflation would reduce the frequency by 26 orders or so of magnitude (massive red shifting) [40]

A clever argument but it clashes with the Mukhanov approach as to structural inhomogenity.

We look at what Mukhanov [41] writes as far as structure formation. Mainly that there is a formulation of what is called self reproduction of inhomogeneity in terms of early universe conditions [27][41]. In this, the starting point is if one used the meme of chaotic inflation, i.e. inflation generated by a potential of the form as given by Guth [26] as well as Mukhanov [41]

In this, Mukhanov [41] write that one can look at a scalar field at the end of (chaotic) inflation, with an amplitude given by, with ϕ i for the initial value of the inflaton such that (where m will be determined by inputs to be brought up later.)

In terms of the initial inflaton, inhomogenities do not form if the initial inflaton is bounded [41]as given by

This leads to (low?) inhomogeneity in the space-time generated by inflation. Inflation is eternal [27] if. There is only the inequality. i.e. we think that this is dubious. i.e. this is suggesting that there is eternal inflation solely due to this inequality. i.e. I doubt it.

This idea supports the Figure 1diagram given below.

Figure 1. Which is from [41] for the onset of eternal inflation.

Clever but way too simple

We would like to have some connection to the reference [42]. So this leads to us looking at the Penrose treatment of Universe behavior and our modification of it.

Next, let us view the Penrose suggestion as to Black holes from a prior universe.

In order to see this, consider a suggestion as to black holes, being the template for a start to the present universe given by [43], which has the Penrose suggestion of an imprint of a prior Universe black holes having an effect upon the CMBR spectrum. The CMBR spectrum is a real datum, but the worth of getting this information would be in terms of having what was said in [43] as to the “ghost” of prior universe black hole radiation.

Furthermore, this means accessing the material given in [44][45] before we discuss developing the modified Penrose CCC material.

Figure 2. As given in [43] which has competing black hole radiation, and can we see this today in the CMBR?

From the 2nd page of reference [43] we have the following quote.

Quote (from page 2 of reference[43])

Aconformaldiagramrepresentingtheeffectofahighlyenergeticeventoccurringatthespace-timepointH. InCCC, HistakentobeaHawkingpoint, wherevirtuallytheentireHawkingradiationofaprevious-aeonsupermassiveblackholeisconcentratedatHbytheconformalcompressionofthehole’sradiatingfuture. ThehorizontallineatthebottomstandsforthecrossoversurfacedividingthepreviouscosmicaeonfromourownanddescribesourconformallystretchedBigBang. Inconventionalinflationarycosmology, Xwouldrepresentthegracefulexitturn-offofinflation. Ineachcase, thefuturelightconeofHrepresentstheoutercausalboundaryofphysicaleffectsinitiatedatH, andsucheffectscanreachDonlywithintheroughly 0.08 radianspreadindicatedatthetopofthediagram.

EndofQuote

A) What can we expect from the transition from a Prior universe, to the Planckian regime of micro black holes? A transition from initially gigantic black holes to micro black holes.

B) In a word, we would likely have in the prior universe MASSIVE black hole, which would be broken up into millions (billions?) of Planck sized.

In a word the GW radiation and thermal/photonic input would have to fight through a thicket of pairs of micro black holes which would be in binary configuration generating their OWN GW background.

We first will discuss this “binary black holes” signal background which the Planckian early Universe stars would have to impinge upon, in order to come to our attention.

Now for the discussion of the millions (more than that) of micro sized black hole pairs which would create a generalized GW signature.

To evaluate the above in terms of our model, we need to refer to a formula given in [45], on a change in power from rotating Planck sized black holes separated say by a Planck length

This then leads to the following question which we cite below i.e.

So, then what are the number of gravitons emitted via a spinning Planck sized black holes component binary in terms of gravitons [44][45]?

Likely from the situation in [44][45] for items as of about a Planck length, and involving Planck sized masses, we would see the Equation (94). Formula we are looking at is the formula which is for Luminosity from a black hole and the two black holes emit GW with a wave frequency 2 times the rotation frequency of the orbit of the two black holes to each other.

If we assume that we are still using this approximation above, from [46] we can see support for our choice of Planck length as the minimum separation distance between the two black holes via using Plank units normalized to 1 as yielding

Going to Clifford Will, [47] we see on page 252 a loss or shrinkage of the period for the rotating black hole pair defined by P

Whereas, with the Mechanics version of P for a sphere to be defined by, where M is a mass of a star, and we assume a binary system with two masses of equal mass M, so that, if R is the separation between the two masses would be [48]

The frequency of rotation would be half that of the GW emitted by these two Planck mass black holes which would collapse into each other. So what we would see from the beginning of expansion of the Universe, to today is that the Primordial Black holes, of about Planck mass, all of say 10³¹ of them initially, which would due to turbulence, and ergodic mixing of space-time form binary pairs which would collapse into each other. This process would continue so that 10³¹ primordial black holes of Planck mass would re combine, via pairs collapsing into larger black holes to we having say 2 times 10¹¹ super massive Black holes of mass about 10⁹, times the mass of the Sun today, in the center of galaxies.

Not only this was going on, we also had dark stars forming, initially from Dark Matter, which would also form massive black holes. See Karen Freeze, in [49]-[51], whereas we also are assuming a non singular bounce, as to the start of the universe, as given in loop quantum gravity [31].

Our final concluding point to this chapter is to review the physics of Figure 2, and then to ask, can we ascertain the GW radiation of pre Universe stars getting into the present universe? i.e. keep in mind any suggestion as to cosmology will have to satisfy the following diagram.

Figure 3. According to the physics of the CMB, as given in [49] Abhay Ashtekar in Zeldovich4. On September 7, 2020 [11].

In our Figure 3, we copy what was done by Ashtekar, in Zelsovich4 as to what was part of anisotropic fits to the E and B polarization, as given.

We argue that this realistically means quantum entanglement, and this leads to our multiverse generalization of the Penrose suggestion.

We now outline the generalization for Penrose CCC (Cosmology) before inflation which we state we are extending Penrose’s suggestion of cyclic universes, black hole evaporation, and the embedding structure our universe is contained within, This multiverse has BHs and may resolve what appears to be an impossible dichotomy. The following is largely from [52][53] and has serious relevance to the final part of the conclusion. That there are N universes undergoing Penrose ‘infinite expansion’ (Penrose) [52] contained in a mega universe structure [53] Furthermore, each of the N universes has black hole evaporation, which is with Hawking radiation from decaying black holes. If each of the N universes is defined by a partition function, called { Ξ i } i ≡ 1 i ≡ N , then there exist an information ensemble of mixed minimum information correlated about 10⁷ - 10⁸ bits of information per partition function in the set { Ξ i } i ≡ 1 i ≡ N | before , so minimum information is conserved between a set of partition functions per universe [53]