Our plan is as follows. We state the Modified HUP results, as a Pre-Octonionic space-time result, and then we will specify that we are transitioning to Octonionic space time. The transition to Octonionic space time will then preserve one key result, that we have, due to the earlier pre-Octonionic space-time, a minimum time step.

In a word, this is the setup of the new physics, plus our resolution

1) In Pre-Octonionic (Pre-Planckian) Space-time there exist conditions for which we form an initial smallest time step, and that the Pre-Planckian Space-time is where we specify initially a modified HUP (Heisenberg Uncertainty principle).

2) In Octonionic (Planckian) Space-time, we recover QM and the usual HUP, but also, we have the benefits of keeping the minimum time step as to what is given from the Pre-Octonionic structure.

3) The Octonionic structure, as mentioned below, is U (1) cross SU (2) cross SU (3). In itself, the Octonionic structure only allows for the standard model physics, and so we will describe it below

4) To get “instants” of time, we need to go beyond the standard Model. After having said this, let us go to the construction of Octonionic non commutative geometry.

5) We will conclude with how specific properties of octonion numbers algebra influence the structure and behavior of the early-cosmology model. What one of the referees reviewing this document did not realize is that the Octonionic space time closely conforms to the Standard Model of physics, as has been stated repeatedly, and that the Pre-Octonionic state is when the Standard Model of physics does not apply. I.e. the division line between the Pre Octonionic Model and the Octonionic model directly correlates a transformation from Pre-Planckian physics to Planckian physics. This is thoroughly discussed in Section 14 of this manuscript.

Keep in mind one basic fact. If we restrict ourselves solely to Octonionic geometry, we are embedded deeply in only what the Standard Model of physics allows. We should though understand what is implied by the physics of the Octonionic structure and so the rest of this first discussion is devoted to it.

In [1] Wilson gives a generalized structure as to Octonionic geometry, and it is a generalized way to introduce higher level geometry into the formation of standard model physics. Crowell, in [2] examines its applications as to presumed space-time structure. Also note what is said in [3] the take away from it, is that as quoted from [3] , that there exists

Quote:

(A linkage to the) mathematics of the division algebras and the Standard Model of quarks and leptons with U (1) × SU (2) × SU (3) gauge fields

End of quote:

Once again, if we have only U (1) × SU (2) × SU (3) gauge fields, we have only the standard model, and that if we wish to have a minimum time step, we need to go beyond the standard model.

The division algebras are linked to octonionic structure in a way which is touched upon by Crowell [2] , but the main take away is that in the Pre-Planckian space-time regime, that there was specific non commutative structures, as reflected in the document below, which in Pre-Planckian space time would eventually become commutative. This development is illustrated in the text below.

The entire transition from Pre-Planckian space-time to Planckian space-time would be in tandem with findings by Beckwith, in [4] , and [5] as to the physics, as given in both [4] [5] that kinetic energy would be greater than potential energy in the Pre-Planckian space-time regime, and also to the possibility of a causal discontinuity, as given in [6] which may be linked to the odd situation of which slow roll physics, as usually delineated by [7] becomes dominant,. It is also the considered opinion of the author that E 8 as referenced in [1] as well as [8] in a classical setting which may be linkable to the Octonionic structure, as well as an extension of issues brought up by Lisi in [9] . This is elaborated in greater detail in terms of Octonionic math in [10] by Baez.

Now that we have made note of the geometry, it is time to look at the metric tensor based fluctuations of space time which may be the bridge between the Pre-Planckian space-time behavior, and standard Planckian space-time

First of all, why would we have a different version of the HUP, in Pre-Octonionic geometry? So as to answer this question we will look at a Proto SUSY potential, and the inflaton, if ϕ ~ ξ + ≪ M Planck which is what we assert we work with. This step, next then will allow us to reference an initial time step, which is non zero. We state that the HUP is modified, due to the existence of ϕ ~ ξ + ≪ M Planck for an inflaton, and we outline what this deviance from the Standard model of physics says about the formation of an alternative statement of the HUP. From there we will then go to the use of the modified HUP to the formation of a minimum time step.

Now, start with the HUP as given in Pre-Planckian space-time Physics. As given in [5] we have that the following fluctuation may be germane to our problem, namely as given by a

Quote from [5] as

If we look at the Susy potential as given by [11]

V ( ϕ ) = μ 4 ⋅ [ b ⋅ ln ( ϕ m Planck ) + ( 1 − ( ϕ m Planck ) 2 ) 2 ] (1)

We will be looking at the value of Equation (1) if ϕ ~ ξ + ≪ M Planck . In short, we have then that

( Δ l ) i j = δ g i j g i j ⋅ l 2 ( Δ p ) i j = Δ T i j ⋅ δ t ⋅ Δ A ϕ ~ ξ + ≪ M Planck (1a)

If we use the following, from the Roberson-Walker metric [11] [12] [13] .

g t t = 1 g r r = − a 2 ( t ) 1 − k ⋅ r 2 g θ θ = − a 2 ( t ) ⋅ r 2 g ϕ ϕ = − a 2 ( t ) ⋅ sin 2 θ ⋅ d ϕ 2 (2)

Following Unruth [14] [15] , write then, an uncertainty of metric tensor as, with the following inputs

a 2 ( t ) min ~ 10 − 110 , r ∼ l P ∼ 10 − 35 meters (3)

Then, the surviving version of Equation (1) and Equation (2) is, then, if Δ T t t ~ Δ ρ [11] [14] [15]

V ( 4 ) = δ t ⋅ Δ A ⋅ r δ g t t ⋅ Δ T t t ⋅ δ t ⋅ Δ A ⋅ r 2 ≥ ℏ 2 ⇔ δ g t t ⋅ Δ T t t ≥ ℏ V ( 4 ) (4)

This Equation (4) is such that we can extract, up to a point the HUP principle for uncertainty in time and energy, with one very large caveat added, namely if we use the fluid approximation of space-time [16]

T i i = diag ( ρ , − p , − p , − p ) (5)

Then by [11]

Δ T t t ~ Δ ρ ~ Δ E V ( 3 ) (6)

Then,

δ t Δ E ≥ ℏ δ g t t ≠ ℏ 2 Unless     δ g t t ~ O ( 1 ) (7)

This Change in the HUP, as outlined above, will be part and parcel of the transformation from Pre-Octonionic space time, to Octonionic, i.e. from Pre- Planckian to Planckian physics, with all the resulting consequences, which will be outlined below

Before doing so, we say something about the introduction of what is meant by a metric tensor to begin with.

See the next mini session as to why the issue of the minimum fluctuation of the metric tensor is so important.

Having said this, we will be referring to Equation (7a) in our document as far as specifics, in the rest of this paper.

The basic issue is, given as follows

δ t Δ E ≥ ℏ / δ g t t ⇔ δ t g t t ≥ ℏ δ g t t Δ E (7a)

The change in energy, as given in Δ E is enormous, i.e. almost equivalent to the entire energy budget of the Universe, at the start of the big bang, hence, to keep the minimum time step as larger than or equal to zero, it will require specific analysis of the fluctuation of the quantity δ g t t , but before doing this we need to understand what the metric tensor is physically, before initiating a description of what we are doing in Equation (7a) as to δ g t t .

In general relativity the metric g_ab(x, t) is a set of numbers associated with each point which gives the distance to neighboring points. I.e. general relativity is a classical theory. The problem is that in quantum mechanics physical variables, either as in (QED) electric and magnetic fields have uncertainty as to their values. As is well known if one makes an arbitrary, high accuracy position measurement of a quantum particle, one has lack of specific momentum values. I.e. its velocity. In Octonionic geometry, the commutation relationships are well defined. There is through a bridge between the classical regime of space time and its synthesis leading to a quantum result. It would be appropriate to put in specific constraints. Note that as an example in gauge theories, the idea is to use ‘gauge fixing’ to remove the extra degrees of freedom. The problem is though that in quantum theory, the resulting theory, (i.e. a quantum gravity theory) may not be independent of the choice of gauge. Secondly

In GR, it is possible to extract a time for each solution to the Einstein equations by DE parametrizing GR. Then the problem is, in quantum versions of cosmology that if space-time is quantized along these lines, the assumption (of evolving then quantizing) does not make sense in anything but an approximate way. That is, the resulting evolution does not generate a classical space-time! Rather, solutions will be wave-functions (solutions of some Schrödinger-type equation). What is being attempted HERE is to describe the limits of the quantum process so as to avoid having space time wave functions mandated to be Schrodinger clones. I.e. to restore quantum behavior as the geometric limit of specialized space time conditions.

Here is a problem. (In some approaches to canonical gravity, one fixes a time before quantizing, and quantizes the spatial portions of the metric only). Frankly fixing time before quantizing and then applying QM to just the spatial part is missing the point. If Quantum gravity is valid, then the commutation relationships in a definite geometric limit must hold. The paper refers to these regimes of space time where the octonionic commutation relations DO hold. The assertion made, is that before Planck temperature is reached, i.e. there is a natural embedding of space time geometry with the octonionic structure reached as the initial conditions for expansion of the present universe.

The premise followed in the paper is that before the Planckian regime, there are complex geometrical relationships involving quantum processes, but that the quantum processes are “hidden from view”, due to their combination. The quantum processes are not measurable, in terms of specific quantum mechanical commutation relations until Planck temperature values (very high) are reached in terms of a buildup of temperature from an initially much lower temperature regime. Appendix A describes an embedding multi verse in terms of the present universe.

Rovelli [17] notes (2007, p. 1304), that modeling the gravitational field as an emergent, collective variable does not imply an absence of quantum effects, and it is possible that collective variables too are governed by quantum theory. Our re statement of this idea is to say that one has quantum effects emerging in highly specialized circumstances, with collective variables behaving like squeezed states of space time matter. The octonionic gravity regime, obeying quantum commutation behavior has its analog in simplification of collective variable treatment of a gravitational field, which becomes very quantum commutation like in its behavior in the Planck temperature limit. This paper will endeavor as to describe the emergent collective treatment of the gravitational field appropriately so octonionic gravity is a definite limiting structure emerging in extreme temperatures and state density.

What we are considering is the following transformation, simply put. And this will be hopefully detected by a change in phase, given by phase δ 0

δ t Δ E ≥ ℏ δ g t t | Pre-Octonionic → δ 0   phasechange δ t Δ E ≥ ℏ | Octonionic with   δ t ≥ ℏ δ g t t Δ E   fixed (7b)

This paper examines geometric changes that occurred in the earliest phase of the universe, leading to values for data collection of information for phase δ 0 , and explores how those geometric changes may be measured through gravitational wave data. The change in geometry is occurring when we have first a pre-quan- tum space time state, in which, in commutation relations [2] (Crowell, 2005) in the pre Octionic space time regime no approach to QM commutations is possible as seen by.

[ x j , p k ] ≠ − β ⋅ ( l Planck / l ) ⋅ ℏ ⋅ T i j k and   does   not → i ℏ δ i , j (8)

Equation (8) is such that even if one is in flat Euclidian space, and i = j, if there is no phase shift then there is no way to move beyond a flat space representation of

[ x j , p k = j ] ≠ i ℏ | Pre-Octonionic (8a)

If one does not have the phase transition, then one observes that without the Pre-Octonionic to Octonionic phase shift that there is a permament stuck at the inequality given by Equation (8a) above.

In the situation when we approach quantum “octonionic gravity applicable” geometry, Equation (8) becomes

[ x i , p j ] = − β ⋅ ( l Planck / l ) ⋅ ℏ ⋅ T i j k → approachingflatspaceafter δ 0 i ⋅ ℏ ⋅ δ i , j (9)

Equation (9) is such that even if one is in flat Euclidian space, and i= j, then if the phase transition from Pre-Octonionic to Octonionic has occurred,

[ x j , p k = j ] = i ℏ | Octonionic   flatspaceOctonionic (9a)

.Also the phase change in gravitational wave data due to a change in the physics and geometry between regions where Equation (8) and Equation (9) hold will be given by a change in phase of GW, which may be measured inside a GW detector.

The simplest way to consider what may be involved in alterations of geometry is seen in the fact that in pre-octonionic space time regime (which is Pre-Planckian), one would have [2] (Crowell, 2005)

This Pre-Octonionic space-time behavior should be seen to be separate from the flatness condition as referred to in [18] . But retuning to [2] we have that, in Pre-Planckian space-time, that

[ x j , x i ] ≠ 0 Under ANY circumstances, with low to high temperatures, or flat or curved space. (10)

Whereas in the octonionic gravity space time regime where one would have Equation (9) hold that for enormous temperature increases (9), then by [2] (Crowell, 2005)

[ x j , x i ] = i   Θ j , i → Temp → ∞ 0 (11)

Here,

Θ j , i ~ Λ N C − 2 ~ Λ 4-dimensional − 2 ∝ 1 / T 2 β → T → ∞ 0 (11a)

Specifically Equation (8) and Equation (10) will undergo physical geometry changes which will show up in δ 0 .The space time shift from pre Planck to the Planck epoch has gravity wave background radiation containing the imprint of the very earliest event. Next, is to consider what happens if Quantum (octonionic geometry) conditions hold. The supposition as given by in [19] (Lee, 2010)

Considering all these recent developments, it is plausible that quantum mechanics and gravity has information as a common ingredient, and information is the key to explain the strange connection between two.

When quantum geometry holds, as seen by Equation (9) and Equation (11), GW information is loaded into the octonion space time regime, and then transmitted to the present via relic GW which identified via the phase shift in GW as measured in a GW detector. This phase shift is δ 0 . The following flow chart is a bridge between the two regimes of [2] (Crowell, 2005) the case where the commutators for QM

[ x i , p j ] ≠ − β ⋅ ( l Planck / l ) ⋅ ℏ ⋅ T i j k → TransitiontoPlanckianspace [ x i , p j ] = − β ⋅ ( l Planck / l ) ⋅ ℏ ⋅ T i j k (12)

Equation (12) above represents the transition from pre-Planckian to Planckian geometry.

Also questions relating to how pre and post Planckian geometries evolve can be answered by a comparison of how entropy, in flat space geometry is linked with quantum mechanics [19] (Lee, 2010). Once Equation (12) happens, Beckwith hopes to look at the signals in phase shift δ 0

[ x i , p j ] = − β ⋅ ( l Planck / l ) ⋅ ℏ ⋅ T i j k → Transitiontoreleaseofrelicgravitationalwavesinflatspace Planckian   era   generatedgravitationalwave (13)

Lee’s paper [19] (Lee, 2010) gives the details of information theory transfer of information from initially curved space geometry to flat space. When one gets to flat space, then, by Equation (13) one then has a release of relic GW. The readers are referred to appendix A summarizing the relevant aspects of [19] (Lee, 2010) in connecting space time geometry (initially curved space, of low initial degrees of freedom) to Rindler geometry for the flat space regime occurring when degrees of freedom approach a maxima, initially from t > 0 s up to about t < 1 s as outlined in an argument given in Equation (14). One of the primary results is reconciling the difference in degrees of freedom versus a discussion of dimensions. Also, as Equation (12) occurs, there will be a buildup in the number of degrees of freedom, from a very low initial level to a higher one, as in the Gaussian mapping [20] (Beckwith, 2010)

x i + 1 = exp [ − α ˜ ⋅ x i ] + β ˜ (14)

The feed in of temperature from a low level, to a higher level is in the pre- Planckian to Planckian thermal energy input as by (Beckwith, 2010a) [21]

E thermal ≈ k B 2 ⋅ T temperature ∝ Ω 0 ⋅ T temperature ∼ β ˜ (15)

Equation (14) would have low numbers of degrees of freedom, with an eventual Gauss mapping up to 100 to 1000 degrees of freedom, as described by (Kolb and Turner, 1990) [21] . The rest of this paper will be in describing an extension of an idea by [22] (Beckwith, 2011c) which may give multiple universes as put into Equation (15). And [22] about multiple universes uses [23] explicitly, by Penrose. In reality, what we are doing is equivalent to [24] , which has the useful caveat that

Quote

We propose that in time dependent backgrounds the holographic principle should be replaced by the generalized second law of thermodynamics.

End of quote.

As there have been numerous ways to add in an active time component into Pre-Planckian space time physics, to Planckian, this substitution of a generalized 2^nd law of thermodynamics is equivalent to the transformation from Pre Plankian to Planckian space-time, which again is in a 1 - 1 correlation to when we are doing which is to go to Pre-Octonionic to Octonionic structures, and we will be elaborating upon this point in the next several parts of this manuscript. Starting off with the sequential development of VeV (Vacuum expectation values) and emergent space-time physics.

Further elaboration is tied in with a summary of properties of a mutually unbiased basis (MUB), [25] (Chaturvedi, 2007) which is topologically adjusted to properties of flat space Rindler geometry.

δ 0 . The key point is an inter relationship between a change in MUB, from initial highly complex flat space time, as a new way to quantify a phase transition, for experimentally verifiable detection of δ 0 .The values of δ 0 are set by the difference between Renyi entropy [26] Salvail, 2009), and a particle count version of entropy, i.e. S ~ 〈 n 〉 . The topological transition is due to a change in basis/geometry from the regime of Renyi entropy to entropy in a particle count version of entropy, i.e. S ~ 〈 n 〉 [27] (Ng, 2008). As by [28] (Beckwith and Glinka. 2010) (assuming a vacuum energy ρ Vacuum = [ Λ / 8 π ⋅ G ] initially), with Λ part of a closed FRW Friedman Equation solution.

a ( t ) = 1 Λ / 3 cosh [ Λ / 3 ⋅ t ] (16)

To flat space FRW equation of the form (Beckwith and Glinka, 2010) [28]

( a ˙ ( t ) / a ( t ) ) 2 + ( 1 / a ( t ) ) 2 = Λ / 3 (17)

Beckwith tried inputs into the initial value of Λ as high energy fluctuations, this ρ vaccum = Λ / 8 π G links initial vacuum expectation value (VeV) behavior with the following diagram. Note that cosmology models have to be consistent with the following Figure 1.

As stated by [30] (Crowell, 2010), the way to delineate the evolution of the VeV is to consider an initially huge VeV, due to inflationary geometry. Note by Equation (18), [31] (Poplawski, 2011):

ρ Λ = H ⋅ λ QCD (18)

Where λ QCD is 200MeV and similar to the QCD scale parameter of the SU (3) gauge coupling constant, where H a Hubble parameter. Here if there is a relationship between Equation (18) above and ρ vaccum = Λ / 8 π G then the formation of inputs into our vacuum expectation values V ~ V VEV ~ 3 〈 H 〉 4 / 16 π 2 , and equating V ~ V VEV ~ 3 〈 H 〉 4 / 16 π 2 with V ( ϕ ) ~ ϕ 2 / 2 would be consistent with an inflaton treatment of inflation which has similarities to [32] (Kuchiev and Yu, 2008). Then equate vacuum potential with vacuum expectation values as:

ρ vaccum = Λ / 8 π G = ρ Λ = H ⋅ λ QCD ⇔ V ~ V VEV ~ 3 〈 H 〉 4 / 16 π 2 ≈ V ( ϕ ) ~ ϕ 2 (19)

Different models for the Hubble parameter, H exist, and are linked to how one forms the inflaton. The author presently explore what happens to the relations as given in Equation (14) before, during, and after inflation. Table 1 below. Is how to obtain inflation?

We will briefly allude to temperature drivers which may say something about how thermal energy will be introduced into the onset of a universe. Begin first with [33] (Beckwith, 2008)

| Λ 5-dim | ≈ c 1 / T α (20)

In contrast with the traditional four-dimensional version of the same, as given by [34] (Park, 2002)

Λ 4-dim ≈ c 2 ⋅ T β (21)

If one looks at the range of allowed upper bounds of the cosmological constant, the difference between what [35] (Barvinsky, 2006) predicted, and [34] (Park, 2002) is:

Λ 4-dim ≈ c 2 ⋅ T β → Graviton   production   for   time   t > t ( Planck ) 360 ⋅ m Planck 2 ≪ c 2 ⋅ ( T ≈ 10 32 Kelvin ) β (22)

Right after the gravitons are released, one sees a drop-off of temperature contributions to the cosmological constant. Then for time values t ≈ δ 1 ⋅ t Planck ; 0 < δ 1 ≤ 1 and integer n [33] (Beckwith, 2008)

Λ 4-dim | Λ 5-dim | − 1 ≈ n − 1 (23)

Initial phases of the big bang, with large vacuum energy ≠ ∞ and a ( t min ) ≠ 0 , a ( t min ) ≪ 1 then

Table 1 may suggest a discontinuity in the pre-Planckian regime, for scale factors [35] (Beckwith, 2008).

[ a ( t ∗ + δ t ) a ( t ∗ ) ] − 1 < ( value ) ≈ ε + ≪ 1 (24)