Dark Energy 2

Dark Energy
Richard Bradford
Apr 8, 2023

My thanks to my friend Gordon Rogers for his effort in collaboration toward the solution.
This paper is dedicated to Ward Rogers who helped pioneer the first particle accelerator at Lawrence Berkeley.

The vacuum is composed of positive and negative energy space-time as described by the Stückleberg-Feynman interpretation. Constituents of the vacuum includes positive energy forward time and negative energy backward time electron-antielectron as well as Planck anti Planck mass that along with gravitational wave states form space-time, where all vacuum constituents exist in the ground state. Development of transient electrically charged electron-antielectron dipoles as half-wave dipole EM quanta emitters results in the numerical value of the measured vacuum energy density within its uncertainty. A coupling between gravitational fields emanating from transient uncharged Planck anti Planck dipoles and electromagnetic interactions of electrically charged electron-antielectron dipoles within the vacuum via a Feynman process is responsible for space time expansion. Shown is the natural exponential dependency based on Planck dipole configurations.

Table of Contents.

Introducing ground state dynamics. Page 3.
Definition of total charge, energy, time, spatial displacement, positive-negative ST, and Algebra. Page 3.
Development of the dipole as a dipole antenna. Page 5.
Vacuum density of dipole antennas #dipoles/(+-m3). Page 7.
Subtended Angle of Dipoles. Page 7.
Non-absorbed emitted energy EREDEMIT. Page 8.
Mass independence of dipoles. Page 9.
Total Vacuum Energy Density. Page 10.
Non-Energy Contribution to the Vacuum Energy from Planck-Schwarzschild Dipoles. Page 10.
Effect of Vacuum Planck Dipole Distribution on Positive ST Hubble Constant, H. Page 11.
Development of Time Dependent Hubble Constant. Page 25.
References. Page 33.

A) Introducing ground state dynamics.
The relativistic equations describe on-shell spin 0, ½, and 1 particles with wave nature. They are the Klein-Gordon equation for spin 0, (∂2/∂t2 – ∂2/x2)φ(x,t) + m2φ(x,t) = 0, the Dirac equation i∂ψ(x,t) = -iα*∂ψ(x,t) + βmψ(x,t) describing up and down spin ½ fermions, and the vector potential, A, in electromagnetic (EM) theory describing spin 1 photons. These equations contain positive and negative energy-frequency components describing forward and backward time particles or waves. If time, t, in the negative energy state becomes –t, then it looks like a positive energy state, the basis of the Stückleberg-Feynman interpretation.

In EM theory, the harmonic oscillator is quantized with discrete energy levels having positive and negative energy levels given as +-En = +-ħω(n + ½), where n > 0 correspond to real positive and negative energy-time and n = 0 are in vacuum ground state mode +-E0 = +-(ħω0)/2. Positive and negative energy states are mirrored and separated by a potential. Changing the potential allows energy exchange.

Scattering in the real particle field of e- and e+ is described by M1 + M2, where M1 is given by the relation ∫d4x1d4x2[exp-ip1.x1]expip2.x2[d4q/(2π)4]iexp(x1 – x2)gμν/q2exp(ip3.x1)exp(ip4.x2)(without spinors). Amplitude M2 is similar. g2 = α is the fine structure constant. A real inelastic interaction with e- and e+ with emission or absorption coupling to an EM wave has amplitude α ≈ 7.2968X10-3.

Coupling an annihilation of a transient ground state e-e+ dipole into an energy conserving ground state EM wave in the vacuum is dependent upon the fine structure constant α. Likewise coupling a ground state EM wave to creation of a transient e-e+ ground state dipole is governed by the fine structure constant. If there are 1/α – 1 = 137 – 1 = 136 number of transient EM ground states creating-annihilating in a sequence from a ground state electron dipole annihilation, then there is creation of a transient e-e+ ground state dipole with certainty on the 1/α = 137th creation sequence. This is the forward time process. Concurrently, there is a backward time process occurring between the created transient dipole backward in time to the annihilated dipole. This forms a forward-backward time link between the backward and forward time components of e-e+ dipoles and EM ground states. Since the process is controlled by the fine structure constant if there is energy conservation between the forward and backward components, then the process ensures that the fine structure constant is maintained in terms of its fundamental constants. If the system has deficits in energy then the fine structure constant is not in balance. This condition allows energy exchange between positive and negative energy states until balance is restored.

B) Definition of total charge, energy, time, spatial displacement, positive-negative Space-time (ST), and Algebra.
In Stückleberg-Feynman interpretation, SFI, a forward time, displacement, positive energy and positive (negative) charged antiparticle (particle) is equivalent to a backward time, displacement, negative energy, and negatively charged antiparticle implying existence of positive and negative ST along with associated mathematical operations. Subtraction results in negative ST -X – (+Y) = -Z, where -X is a negative ST quantity and +Y a positive ST quantity. In positive ST, +Y – (-X) = +Z. Using SFI, a negative ST antiparticle becomes a positive ST antiparticle. Both exist in positive ST. The subtraction operation becomes addition given as +X + (+Y) = +Z, where +X, +Y are particle and antiparticle. If the positive antiparticle is changed to a negative particle while the negative particle remains, then both exist in negative ST. The subtraction and -X + (-Y) = -Z. Using SFI, subtraction becomes addition, while negative ST quantities are replaced by positive ST and vice versa. Quantities in negative and positive ST are mirror images. The operation of division is modeled upon the mirrored properties of positive and negative ST. Division is given by -X/-Y = +X/+Y = -Z in negative ST and +X/+Y = -X/-Y = +Z in positive ST. Division, -X/+Y = +X/-Y = +Z in negative ST while division -X/+Y = -Z = +X/-Y = -Z in positive ST. Multiplication in negative ST gives -X-Y = +X+Y = -Z and in positive ST +X+Y = -X-Y = +Z. In negative ST -X+Y = +X-Y = +Z and in positive ST -X+Y = +X-Y = -Z. In positive ST square root (+-X)1/2 = +Z and +Zi and in negative ST (-+X)1/2 = -Z and –Zi.

In positive and negative ST, there is an operation of negation (-), Which changes quantities from positive to negative ST and negative to positive ST. Negation of the quantity -X in negative ST becomes +X in positive ST and vice versa. Negation changes a positive ST relation +X – (-Y) = +Z into a negative ST relation –[+X – (-Y)] = -X – +Y = -Z. Relation -Y – (+X) = -Z in negative ST negates to –[-Y – +X] = +Y – -X = +Z in positive ST. Negation does not change the sign of the mathematical operation. The quantity +X/-Y = -Z in positive ST is negated to -X/+Y = +Z in negative ST. In positive ST +X+Y = +Z negates to -X-Y = -Z and -X+Y = -Z in positive ST is negated into –[-X+Y = +Z] = +X*-Y = +Z in negative ST. Also, to find solution in negative ST, perform operations in positive ST and negate.
At t = -1, creation of positive energy and annihilation of negative energy raises energy +ΔEE – (-ΔEE) = +2ΔEE, while at t = 0, annihilation of positive energy and creation of negative energy lowers energy –ΔEE – (+ΔEE) = -2ΔEE conserving energy. Total energy, +-2ΔEE has components +ΔEE and –ΔEE between t = -1 and 0. Electron dipole total energy equals +mE(c2) – (-mE(c2)) = +2mEc2, where mE is rest mass. Total positive and negative ST energy is +-ΔEE – (-+ΔEE) = +-2ΔEE. Time-energy relate as +-ΔT = h/(+-ΔE). At t = -1, creation of an electron and annihilation of an anti-electron raises time +h/ΔEE – (-h/ΔEE) = +2h/ΔEE or +ΔT – (-ΔT) = +2ΔT. At t = 0, electron annihilation and anti-electron creation lowers time –ΔT – (+ΔT) = -2ΔT or -h/ΔEE – (+h/ΔEE) = -2h/ΔEE conserving time. Total time, +-2ΔT, bifurcates into +ΔT and –ΔT between time t = -1 and 0. In positive ST total time is, +ΔT – (-ΔT) = +2ΔT and -ΔT – (+ΔT) = -2ΔT in negative ST.

Displacement relates to time +-ΔTc = +-lΔQl. At t = -1, creation of an electron and annihilation of an anti-electron raises displacement by +lΔQl – (-lΔQl) = +2lΔQl. At t = 0, annihilation of an electron and creation of an anti-electron lowers displacement by -lΔQl – (+lΔQl) = -2lΔQl conserving displacement. +-2lΔQl, bifurcates into +lΔQl and -lΔQl between t = -1 and 0. In positive ST total displacement is +lΔQl – (-lΔQl) = +2lΔQl and in negative ST -lΔQl – (+lΔQl) = -2lΔQl.
Total area and total volume are defined as +-A – (-+A) = +-2A and +-V – (-+V) = +-2V, where +A and +V are positive ST components and –A and –V are negative ST components.

The Compton wavelengths +-λCE = h/[+-mEc] = +-h/mEc correspond to the electron and anti-electron with total displacement given as +-λCE – (-+λCE) = +-2λCE in positive and negative ST. Time, +-2ΔT = +-2h/mEc2, relates to displacement as +-2ΔT = +-2λC/c = +-2h/mec2. Total time of existence is +-ΔT – (-+ΔT) = +-2ΔT or +-h/ΔmEc2 – (-+h/ΔmEc2) = +-2h/ΔmEc2 = +-2λCE/c. Total distance covered travelling at the speed of light in time +-2ΔT is +-2ΔTc, equaling total dipole displacement +-2λCE.
Consider two objects at some distance apart. In regards to forward and backward time states object 1 forward time emission is simultaneous with backward time reception from object 2. Then, the relative positions of object 1 and 2 is not unique or any relative position produce the same result. The components of total states can establish the distance between object 1 and 2. The forward time component of the EM wave state propagates from object 1 to object 2 in forward time +t establishing a distance +d and the backward time component propagates from object 2 to 1 in time –t establishing a distance –d, where l+tl = l-tl and l+dl = l-dl. In terms of components, absolute distance and time are equal. Total time and distance is time and distance composed of propagation from object 1 to object 2 in forward time together with propagation in backward time from object 2 to object 1 resulting in +d – (-d) = +2d and +t – (-t) = +2t. Since the absolute distance between the two objects in terms of forward or backward time and distance is the same in both components, the total time and distance does not change the relative distance in forward or backward time and distance but only adds to it a round trip in backward time. Relationship between total distance and time to the components is (1/2)(+-d – + (-+d) = +-(1/2)2d = +-d and (1/2)(+-t – (-+t)) = (1/2)2t = t, where the objects remain in the same relative position in dipole or constituent particle cases.
C) Development of the dipole as a dipole antenna. Trajectories of separated constituent charged particles inside a Compton region are unknowable and untraceable as well as the relative configuration of their EM fields. Also unknown, are the dynamics of dipole creation and annihilation pertaining to charge separation and recombining. The charged particles and their EM fields are in a superposition state of all possible orientations within the Compton wavelength region. In the Compton wavelength term +-h/mEc, the momentum term +-mEc, contains a constant velocity and mass implying no emission of electromagnetic radiation from the dipole. Creation and annihilation of separating and recombining of opposite charges particles does generate EM emission.
A dipole antenna, in a particular coordinate orientation, separates positive and negative electrical charge with movement of negative charge to one end of the dipole antenna and an equivalent movement of positive charge to the other end due to a voltage-current generator. The current then reverses direction, where the charges momentarily recombine during the reversal process, and then move onward to opposite polarity at the ends of the antenna. The current again reverses and at the second charge recombination and the antenna has undergone one cycle and so on. During this process, EM ground state radiation in the form of waves are emitted having a wavelength inversely proportional to frequency, which depends upon cycle time period.
Assume that creation and annihilation of separated opposite charges in a transient dipole is equivalent to one-half cycle of sinusoidal separation-recombination of opposite charges of a dipole antenna in the Compton region having total diameter +-2λCE, and one-half cycle time equal to the total time of existence of the dipole, +-2λCE/c. The dipole antenna emits total EM energy, +-ERE. The argument equally applies to the backward time negatively charged electric current due to its forward time positively charged current equivalence. A dipole antenna emitter, according to the strong reciprocity theorem, is also a receiver implying that an emitted quanta may be absorbed from another dipole. In the superimposed state, a dipole antenna has all orientations throughout 4π spherical radians and total energy output of 4πERE. The superposed dipole antenna, during its time of creation in the Compton region, is affected by a forward-backward time emission-absorption of EM waves from dipoles emitting in the past and future resulting in reduction of the superposition by a factor of 1/4π to a particular dipole antenna orientation. Figure below right illustrates the quantum clock process of dipole creation-annihilation from and to EM wave packets, where there are two co-existing time and energy channels in positive and negative ST.

The SFI of the ground state EM state is used to link an emitter and absorber. An EM wave has both forward and backward time components within its own waveform as well as in the emitter, Emit, and absorber, Absorb. An emitter that is also an absorber, EmitPRESENT and AbsorbPRESENT, emits both a forward time +1/2E positive energy and backward time -1/2E negative energy to a future and past absorber, AbsorbFUTURE and AbsorbPAST. In forward time, total energy of emission is +(1/2)E – -(1/2)E = +1E and in backward time, total energy of emission is -(1/2)E – +(1/2)E = -1E. The process of emission at EmitFUTURE is the same at EmitPRESENT, where positive energy +(1/2)E propagates forward in time to a future absorber, AbsorbFUTURE2, and negative energy -(1/2)E propagates backward in time to AbsorbPRESENT. Absorption process, AbsorbPRESENT absorbs negative energy, -(1/2)E from EmitFUTURE and at the same time absorbs positive energy +(1/2)E from EmitPAST increasing the energy level of AbsorbPRESENT by +(1/2)E – -(1/2)E = +E in positive ST restoring emitter to its rest energy value. See figures above.
Total average energy output ERE of the dipole antenna is determined by power output multiplied by the total time of dipole existence. Peak current +-I0 is defined as (number of positive or negative charges per volume)(number of charges passing through area associated with the volume)(speed of the charge, v)(unit of charge, q). In the dipole antenna, the speed of an electron current does not equal that of the speed of light but moves at a speed less than. To obtain a sub light speed, a relativistic relation for momentum is used to express the momentum +-mEc in terms of a relativistic current +-mEγv resulting in +-v = +-c/+-21/2 or +-0.707c. The dipole antenna current is separation and recombination of 1 electrical unit of positive and negative charge, in total volume +-(4π/3)λCE3 passing through total area given as +-πλCE2, where radius is +-½(2λCE) = +-λCE. The peak electric current is, +-I0 = +-[(3/4πλCE3)(πλCE2)(c/(2)1/2)q] = +-[3cq/((4)21/2λCE)]. Then, +-I0 = +-[3(2.997925X108m/s)(1.602177X10-19C)/((4)21/2 (2.426316X10-12m))] = +-10.498570C/s. Let the electric current from charge separation and recombination be sinusoidal. The RMS or time effective current is then given as +-IRMS = +-I0/21/2 = +-[3cq/((4)21/2λCE)][1/21/2] giving +-3cq/(8λCE) = +-7.423610 Ampere. Dipole antenna length, +-L, equals the total dipole displacement of +-2λCE = +-4.852632X10-12m. One cycle time of 2π is 2(+-2TλCE ) = +-4TλCE, where +-2TλCE = 2λCE/c is the 1/2 cycle time π of wave emission analogous to the separation and recombination of charge in the transient dipole. Antenna frequency is given as, +-f = +-1/4TλCE = +-1/4h/mEc2. Emitted wavelength is +-λ = +-c/f = +-c/1/4TλCE = +-4cTλCE = +-4(2.997925X108m/s) (1.618664X10-20s) = +-1.941053X10-11m = +-2(2λCE). Antenna length +-L = +-4.8526232X10-12m = +-2λCE. Thus, +-L/λ = +-½ and the antenna is a half-wave dipole. The transient dipole has no loss of energy as heat due to electrical resistance and is due solely to EM vacuum resistance about +-73.312 Ohm(Ω) for a one-half wave antenna. RMS total power is given by the relationship +-PRMS = +-73.312ΩIRMS2/2 = 9c2q273.312Ω/128λCE2 = +-73.312Ω(+-7.423610 Amp)2/2 = +-73.312Ω(27.554995) = +-2020.119928J/s. Emitted energy is +-EEMIT = +-PRMS(+-2TλCE) = 3.269885X10-17J. The analysis was done using total quantities. Total emitted EM energy requires both e- and anti e-. Component ST considers only e- or anti e-. Then no proper EM dipole emission would occur, but if the calculation is carried out with one particle component introducing a factor of ½ along with component Compton region λCE and radius λCE/2, the calculated component power PCOMP = PRMS, the total power. The normal component energy output giving the total would be +-(PCOMP)(TλCE) – -+(PCOMP)(TλCE) = +-2(PCOMP)(TλCE) = +-(PRMS)(2TλCE), where (PRMS)(2TλCE) is proper since the component case PCOMP is non-physical. Since both e- and anti e- are required in each positive and negative ST, the total energy output appears in each ST. Thus, energy emitted in positive or negative ST is PPRMS(+-2TλCE). Then, by taking into account +-2PRMS(TλCE) in positive and negative ST, +-2PRMS(TλCE) becomes +-4PRMS(TλCE) total.
A created e- and anti e- dipole exhibiting forward and backward time charge separation and recombination existing in its total Compton wavelength 2λCE annihilates creating an EM ground wave state of equal energy. Time of state existence is defined as (21/2)2λCE/(21/2)c = 2λCE/c, since e- and anti e- have an effective traversal speed c/(21/2) and the RMS value of a wave amplitude 2λCE is given as 2λCE/(21/2). The state existence time is then, 1.6186636X10-20s. Thus, the state creates and annihilates in 2λCE/c s into creation of an EM ground wave state which annihilates creating another and so on until creation of a new dipole. Total creation and annihilation succession is 1/α = 136 EM ground states plus 1 dipole giving of 137 states.
1) Total radiated dipole energy in Positive-Negative ST is: +-EEMIT = +-PRMS(+-2T) giving the total emitted energy of +-3.269885X10-17J in positive or negative total ST.
D) Vacuum density of dipole antennas #dipoles/(+-m3): The succession of 136 electromagnetic ground states plus dipole for a total of 137 states, 1/α, are composed of one dimensional randomly oriented segments, where each state or segment has total displacement +-2λCE. Distribution of dipoles throughout space-time is random in large numbers giving maximum entropy, isotropy, and homogeneity. At any one time, a random distribution of dipole and electromagnetic states throughout space implies independence of density with respect to any volume encompassing dipoles. To enclose all dipoles, there must be close packing of volumes without gaps, eliminating spherical spheres. Let the cube be the standard volume. Consider a cube where the edges having 5 states are aligned along the x-y-z axis in a Cartesian coordinate system. The total number of states in the cube is 53 = 125. Let the five states be comprised of 1 dipole and 4 electromagnetic ground states. The total number of states along one row on the x-axis is 5 and there is 1 dipole given by 5/5 = 1. The number of rows composing a face on the x-y coordinate plane is 5 giving a total of 51 = 5 dipole states or equivalently 52 = 25 total number of states and 5 = 52/5 dipoles. The number of faces along the z-axis, completing the cube, is 5. Then, there are 52 = 25 dipoles or equivalently 53/5 = 25. In terms of dipoles the general formula is then m3 dipole states/n EM plus 1 dipole state. Volume density is fundamentally calculated with total displacement of dipoles. In terms of total displacement of states +-2λCE, the density of single dipoles or number of dipoles per +-m3 is (2λCE)-3. Let there be 136 electromagnetic states and 1 dipole for a total of 137 states (1/α ≈ 137). Then, the total number of dipole states comprising the total cube is ((2λCE-1))3 and density of e-e+ dipoles is ((2λCE)-3/137) = [(1/8)(λCE)-3]/137 = 6.387742X1031/m3 in total forward and backward time or in component form of dipole constituents, where dipoles or constituents are separated by invariant relative positions. Dipole density is invariant in component or total ST viewpoint, where number of dipoles is 2X or 1 in positive ST and 1 in negative ST. Total volume is +V – V = +-2V and so 2X/2V = X/V.
2) # Dipoles/m3 = 6.387789X1031/m3, invariant in total ST and component.
E) Subtended Angle of Dipoles. Dipoles emit EM wave energy then undergo annihilation creating EM wave states during the latter stage of dipole existence, which then propagate through a succession of annihilation and creations until on the 136th step a dipole creates, which absorbs and reduces the EM wave state during the beginning of the dipole’s existence. See figure 5 above. The dipole during its time of existence is contained inside the Compton region and the emission occurs during latter stage implying the emission occurs at the outer region of the Compton region and EM absorption occurs at the outer region of the Compton region. The two outer regions face each other due the forward and backward time nature of the EM and dipole states. Emission energy ERE with an angular distribution of probability and energy or power is given by ERE = [cosθ((π/2)cosθ)/sinθ]2, where θ = 00, the axis of the electron dipole antenna, implies E = 0 and θ = 900 gives E = 1. Assuming energy out is near 1, average separation of e-e+ dipoles is [((2λCE)-3/137)]-1/3 = [(1/8)(λCE)-3/137)]-1/3 = 2.501X10-11m in total or component distance of dipoles or its constituents from center to center. The distance from outer region of the emitting dipole to the outer region of the absorbing dipole is then (2.501X10-11m) – 2(2.426X10-12) = 2.016X10-11m. There is an angular solid conic field of view from one dipole to the other with max subtended angle tan-1[2.426X10-12m/2.016X10-11m] = tan-1 (0.12) = 6.860 ≈ 70. The subtended angles bring in other probabilities besides 1 further reducing energy output. The total angular view is +-70 measured from 900. Using ERE = [cosθ((π/2)cosθ)/sinθ]2 the range is enumerated as 900 = 1, 890 = 0.99955, 880 = 0.99821, 870 = 0.99598, 860 = 0.99287, 850 = 0.98889, 840 = 0.98403, and 830 = 0.97833.
Average of probabilities from 830 to 900 to 830 in one degree increments along one cross section parallel to the axis of the solid cone at 900 is given by the relation (2∑8389P + 1)/(14 + 1) = 14.87572/15 = 0.99171, where P is the probability at a particular angle. Increasing the number of cross sections around the solid cone’s axis of revolution gives greater accuracy of the superposition composed of degrees with probabilities and angular position around the axis of the solid cone. The general expression for the limit approximating average reduction in energy output is limn → ∞[(2∑8389P)10n + 1]/[(14)10n + 1], where n = 0, 1, 2 to infinity. To approximate the limit, 2∑8389P = 13.87572:
Let n = 0, then, [(2∑8389P)1 + 1]/[(14)1 + 1] = 14.87572/15 = 0.99171,
n = 100, then [(2∑8389P)10100 + 1]/[(14)10100 + 1] = 0.99113, n = 106 average = 0.99112, n = 109 average = 0.99112, and n = 1012 average = 0.99112. The limit is 0.99112.
Using Brownian motion or random walk in three dimensions on EM states that create in random directions upon annihilation of the previous EM state gives the ending relative position relation given by distance from origin in total or in constituent form, 2d = 2λCE0.932×0.4975 or d = λCE0.932×0.4975 x is number of steps. For x = 136 EM wave states, d = (2.426X10-12m)(0.932)[136]0.4975 = 2.6046X10-11m. Comparing this result to the average relative separation of dipoles or constituents it can be seen that the emission occurs at one dipole and is received at a later total time of 136(1.618X10-20s) = 2.201X10-18s at its nearest neighbor. Energy, EEMIT, is reduced by a factor of 0.99112.
3) Total EEMIT is reduced by a factor 0.99112.
F) Tot non-absorbed emitted energy EREDEMIT. Total emitted energy is EEMIT(#dipoles/m3)(1m3). Classically, any reception over a smaller region than the total region of the EM emission field has less energy. Energy emitted is treated quantum mechanically, where the dipole emission is described by a change in energy levels emitting a quanta equal to the total emission energy, EEMIT. Overwhelmingly, due to a large number of dipoles, energy quanta emitted are absorbed but there is a chance that energy EEMIT is not absorbed. Energy absorbed by a large distribution of dipole antennas is EEMIT(#dipoles/m3)(1m3) – (emitted energy not absorbed). Consider that energy not absorbed has to do with the conditions for absorption. Absorption from an emission implies conditions, where dipole orientations favor absorption such as orientation of dipoles over volume or 4π steradians and 4π for orientation of the emitting dipole. If the axis of the antenna is filliped across the perpendicular plane the emission and absorption is unchanged giving a 2X symmetry. Also, as a condition for absorption is the subtended angle and cross section of the antenna Compton region. Cross section is given by πλCE2, which when compared to 1m2 gives πλCE2/1m2. The chance of non-absorption is related to having nearly exact orientation similar to average conditions of absorption, where absorption alignment tolerances given above decrease with distance. The probability of energy non-absorption is, then, (1/4π)(1/4π)(2)(0.99112)(πλCE2/1m2) = (1/8π)(0.99112) (λCE2/1m2) and the non-absorbed energy available in 1m3 is determined by the number of number of dipoles present or (1/8π)(0.99112)(λCE2/1m2)(EEMIT)(#dipoles/1m3)(1m3). This energy released into the vacuum has an effect on vacuum dynamics as seen later in the paper.
4) Total EREDEMIT = (1/8π)(0.99112)(λCE2/1m2)(EEMIT)(#dipoles/1m3)(1m3).
G) Mass independence of dipoles. Some species of electrically charged particles are composed of quarks and in total are colorless so that quarks do not appear individually in the vacuum, they are isolated. Thus, a given particle is independent of its composition of quarks. Leptons are not composed of quarks. A dipole creates and annihilates as an elementary or composite particle and its antiparticle. An aspect of all particles that manifests is mass. The mass of a constituent particle is approximately nmE where n ≥ 1 is a natural number, and mE is the mass of the electron, which is taken as the reference since it is the least massive charged particle having n = 1. If there are two particles, one with mass nmE and the other mmE n ≠ m, then they count as two different particles regardless of electric charge. If n = m, then it counts as one since they are indistinguishable particles with respect to mass. If a particle group each with mass nmE has x different electric charges, then there is equal probability 1/x for any one appearing.
Assume that the fraction of appearance or relative probability of appearance of a particle with mass nmE is 1/n independent of its electric charge. This can also be written as a space or time frequency of appearance compared to the electron. As an example: Let a particle have mass nmE. Using the fraction of appearance as a time based rate of appearance, the relative rate of the particle’s appearance is 1/n per second. The mass per second is then nmE*1/n = mE per second equaling that of the electron having n = 1. Density of vacuum dipoles composed of an electron-antielectron is #/m3E = [(2λCE)3]-1/137. Density of dipoles having constituent particles with mass nmE is #/m3n = [(2λCE/n)3]-1/137. Particles with mass nmE increase the density by a factor of n3 when present but the probability of existence of any dipole is reduced by 1/n resulting in (n3/n)[(2λCE)3]-1/137 = (n2)[(2λCE)3]-1/137. Average emitted electromagnetic energy is P(2T), where power P is proportional to current squared I2 = (3cq/8λn)2 = n2(3cq/8λCE)2 and 2T is proportional to 1/nmE. Then, emitted energy depends as (n2mE2/nmE) = nmE = nERE. The subtend angle depends upon (1/n)2λCE. The probability with cross sectional area πλCE2 depends as 1/n2. It can be seen that the entire relation giving EREDEMIT occurs with (1/n)th probability. Combining factors of n, where the factor associated with density of dipoles is n3, subtend angle 1/n, probability 1/n2, emitted energy n, and overall probability of occurring 1/n gives the result (n)(1/n)(1/n2)(n3)(1/n) = 1, which is electron mass. Thus, calculations can be done equivalently in terms of electron parameters.
Of all known particles, whose existence are established or all those established plus those seen with a lower confidence level, the vast majority carry electric charge of 1│e│ but there are some carrying charge 2│e│. Since current is linearly proportional to charge, doubling the charge doubles the current and squared produces four times the emitted average electromagnetic energy than that of a single charge. Thus, the electron dipole density must be separated into fractions of single and double charged particles multiplied by the average emitted energy, EAVE. Each group containing a double charged particle also contains single charge particles having as a close approximation the same mass and, thus, are treated as the same particle with a probability of obtaining a particular charge. The double charged particles comes as a triplet or doublet group of 3 or 2 with probability of appearing 1/3 and 1/2. An accurate measure for y1/3 and y1/2 is the mean value taken from probabilities of various numbers of double charges appearing from a total number of possibilities or y1/3 or y1/2 = ∑k=1Nk[N!/k!(N – k)!]pk(1 – p)N – k, N is total number of double charged particles, k from 1 to N are number of double charged particles present, and p is the probability of appearance. Let there be a total of x + y particles where x particles are single and y = y1/3 + y1/2 are double charged groups. The fraction of single charged particles is x/(x + y) and for double charge particles it is y/(x + y). Average emitted energy is EAVE = [(x + 4y)/(x + y)]EREDEMIT, where [(x + 4y)/(x + y)] is the particle fraction, invariant in total ST and component.
5) Average emitted energy is Total EAVE = {[x/(x + y)] + [y/(x + y)]4}EREDEMIT.
H) Total Vacuum Energy Density, ρVAC. As of 2016 particle data, the number of accepted particles present is 129 = 124.17 (1e) + 4.83 (2e), 3.33 (2e) are from the groups of 10 triplets and 1.5 (2e) are from the 3 doublets as shown. The mean number of 2e in a group of 10 triplets is ∑k=1Nk[10!/k!(10 – k)!]0.333k0.66710-k = 3.33, k = 1 to 10 and mean for 2e in 3 doublets is ∑k=1Nk[3!/k!(3 – k)!]0.510 = 1.50, k = 1, 2, 3 with total 3.33 + 1.50 = 4.83 2e on average. Number of accepted particles plus number of particles with a low or poor confidence level is 247 = 238.1735 (1e) + 8.8265 (2e) There are 22 groups of triplets and 3 doublet groups. Mean number 2e using ∑k=1Nk[22!/k!(22 – k)!]0.333k0.66722 – k is 7.3265, k = 1 to 22. Mean number in 3 doublets is 1.5000, k = 1 to 3 giving 7.3265 + 1.5000 = 8.8265 doubly charged particles.
The particle factor is {[x/(x + y)] + [y/(x + y)]4} and for 129 particles having a mean of 4.8300 double charge {124.1700/129 + (4.8300/129)4} = 1.1123. The particle factor for 247 particles with a mean of 8.8265 double charged particles is {238.1735/247 + (8.8265/247)4} = 1.1072.
In positive ST, total energy density, ρVAC = +(1/8π)(0.99112)(x + 4y)/(x + y)(EEMIT){[(2λCE)3]-1/137} = (3.70948X10-4Ω/1m2)[(x + 4y)/(x + y)]c3q2mE2/h2.
Total vacuum energy density in positive ST, +ρVAC, from 2015 Planck satellite data is 5.367X10-10)j/m3.
247 particles gives the result (4.107136X10-4Ω/1m2)c3q2mE2/h2 = 5.368930X10-10J/ m3 with 0.036% error.
6) In positive ST, total energy density, ρVAC = +(3.70948X10-4Ω/1m2)[(x + 4y)/(x + y)]c3q2mE2/h2.
I) Non-Energy Contribution to the Vacuum Energy from Planck Dipoles. A succession of gravitational ground wave states generated from Planck dipole annihilation to Planck dipole creation is similar to the succession of EM ground wave states from electron dipole annihilation to dipole creation. In the electron case, the number of EM ground states is given by the inverse of electron coupling constant 1/α =137.
Minimum Planck mass, time, and displacement are mP = [hc/2πG]1/2 = 2.17671X10-8kg, TP =[ћG/c5]1/2 = 5.39106X10-44s and LP = [ћG/c3]1/2 = 1.6162X10-35m, where lP/TP = c. The Compton wave length is λP = h/mPc = 1.015395X10-34m and Schwarzschild diameter is 4GmP/c2 = 6.464181X10-35m. Compton wavelength traversal time, TλP = 3.386993X10-43s. Then, LP < Schwarzschild diameter < λP. The total Planck Compton wave length is 2h/mPc with total traversal time 2TλP. Using the quantum clock, the positive time associated with mass creation +mP and annihilation of +mP, along with negative time creation and annihilation of mass -mP, where positive time creation of +mP corresponds to negative time annihilation of –mP and annihilation in positive time of +mP corresponds in negative time to the creation of –mP. The system, where +mP and –mP occur in opposite time directions operate within the total Planck Compton wavelength, 2λP. The created dipole produces separation and recombination of +mP and –mP at annihilation. Due to opposite time dynamics, the system at any orientation is assumed to have a binary orbit about their center point emitting quadrupole gravitational radiation. The binary orbit has an angular frequency associated with the movement. Analogous to the EM case, sinusoidal behavior is exhibited having an associated RMS value, so that the total region of orbit is 2λP/21/2 with traversal time of 2TλP/21/2. In component form having one Planck particle, one has λP/21/2 with traversal time of TλP/21/2. The total angular frequency, ωTOT is total radians taking into consideration the full quantum clock composed of both Planck particles, where each has one-half orbit π giving π – -π = 2π divided by total time or 2π/2TλP/21/2 = π/TλP/21/2 = ωCOMP. Thus, total angular frequency equals the component angular frequency, ωTOT = ωCOMP. Since the creation and annihilation of a Planck dipole is sudden, in component form ωCOMP can be described by the positive square wave having width λP/21/2 and duration TλP/21/2. Then, ωTOT is described as two total positive amplitude squares/total time, where total squares is 1sq – -1sq = 2sq, 2 having total width 2λP/21/2 given as 2sq/2TλP/21/2 or 2/2TλP/21/2, which equals 1sq/TλP/21/2 or 1/TλP/21/2in component form and invariant. The amplitude of the square wave reflects the RMS value of the sinusoidal wave nature. Gravitational wave luminosity from two masses orbiting in the x-y plane having quadrupole tensor IXX is LGW = dE/dt = (G/5c5)<∂3IXX/dt3∂3IXX/dt3>, where IXX = m1x12 + m2x22 = [[μ2a2m1/m12] + [μ2a2m2/m22]]cos2ωt (1). Two Planck mass, 2mP, is found in total Planck Compton region having an orbit 2λP/21/2 = a. Planck dipole mass components are equal, m1 = m2, so reduced mass μ = m1m2/(m1 + m2) = mP/2 and (1) becomes 2[(mP/4)(4λP2/2)]cos2(ωt) = (mP)(λP2)cos2(ωt). Then, the relation is transformed to [mPλP2/2](1 + cos(2ωt). Now, ∂3IXX/dt3 = 4mPλP2ω3sin(2ωt) and squared 16[mP2λP4ω6] = 8mP2λP4ω6 where = ½. Using the definition of the square wave above, luminosity, LGW = (8G/5c5)23mP2λP4(2sq)6/[2TλP6/(21/2)6] = (64/5c5)mP2λP416/TλP6. Now, λP/TλP = c so LGW = (64G/5c5)mP2c4/TλP2 or (64G/5c)mP2/TλP2. The total positive or negative ST emitted energy is then given as ETOT = +-LGWTOT(2TλP) = +-(64G/5c)mP2/TλP = (64G/5c)mP2/TλP = +-3.985387X109J. Total dipole energy is +-2EP = +-2(2.17671X10-8kg)(c2) = 3.91266X109J.
In component form, there is one Planck mass, but cannot orbit about another Planck mass and thus no center of orbit without proper energy emission. But assuming an orbit center gives μ2 = mP2 and λ2/(21/2)2 giving [mPλP2/2](1 + cos(2ωt)). Then due to ωTOT = ωCOMP and component square wave equal to total square wave, LGWTOT = LGWCOMP, but LGWCOMP is unphysical, and thus requires both components so that emitted energy is +-LGWCOMP(TλP) – -+LGWCOMP(TλP) = +-2LGWCOMP(TλP) = +-LGWTOT(2TλP) = +-3.985387X109J. One Planck dipole contains energy 2mPc2 = 3.9126599X109J.
As can be seen the total or component emitted energy is greater than the dipole mass-energy. Then, the process is creation of a Planck mass dipole with energy that is emitted by quadrupole gravitational radiation into a gravitational wave energy state that then undergoes a succession of annihilations and recreations until creation of another Planck dipole governed by the gravitational coupling 1/αG =[Gme/(137.03599kq2)]-1 = 5.7098437X1044 = S. Creation to annihilation total life time of a Planck dipole and gravitational wave state is defined with RMS of the total region of orbit or 2λP/21/2 with traversal time (2λP/21/2)/c = 4.78993097X10-43s.
J) Effect of Vacuum Planck Dipole Distribution on Positive ST Hubble Constant, H. Space-time is a succession of created and annihilated gravitational wave states from created and annihilated PS dipoles, where the number of successive ground states from Planck dipole annihilation to creation is given by 1/αG = [Gme/(137.03599kq2)]-1 = 5.7098437X1044 = S. In the EM case, the number of successive states is given by the inverse of the fine structure constant, 1/α = 137.
Transient gravitational dipoles are point-like attractive sources of gravity affecting other dipoles locally while gravitational wave states are non-point like gravitational fields dispersed throughout the vacuum resulting in non-directional gravitational attraction. Gravitational force between positive energy mass is attractive and force between negative energy and time mass is also attractive. In positive ST, the gravitational vector force is F21 = -[G(+m)(+m)/r2]r12/lrl. Negation interchanges the direction of the unit displacement and force vector or, 21 and 12 becomes 12 and 21, and F21 = -[G(+m)(+m)/+r2]r12/lrl becomes -F12 = [G(-m)(-m)/ -r2]r21/lrl = -[G(+m)(+m)/+r2]r21/lrl and F12 = [G(+m)(+m)/+r2]r21/lrl, where distance is a negative ST quantity, -r. In both cases gravitational force is attractive. In positive (negative) ST the negative (positive) mass backward (forward) time anti-particle is seen as a positive (negative) mass forward (backward) time anti-particle and in both cases they are attractive. The case of opposite mass given as F21 = -[G(+m)(-m)/ +r2]r12/lrl then does not occur resulting in no paradoxes.
The potential energy between two Planck dipoles is given as ΔU = 4Gm2P(1/rb – 1/ra). The potential energy is defined where at infinity it equals 0 or rb = ∞ and 1/rb = 0. Then, the potential energy is negative and is given as ΔU = -4Gm2P /r. Taking the derivative with respect to r gives a force term as dU/dr = 4Gm2P/r2 and change in potential energy as dU = [4Gm2P/r2]dr. Gravitational potential is U = -Gm/r in positive ST and upon negation –U = G(-m)/(-r) or U = Gm/r in negative ST. In positive ST as r increases, (+Δr), the potential energy U becomes less negative or increases at the expense of a decrease in positive EM vacuum energy. In negative ST as –r increases, (-Δr), the potential energy –U that is positive becomes less positive or decreases at the expense of an increase in EM negative energy. Adding vacuum energy gives expansion in positive ST and adding negative vacuum energy gives expansion in negative ST.
Define the smallest 3 dimensional expanding volume unit a quantum cube, which is composed of an array of 27 fundamental sub-cubes each with volume [(2λP)-3/S]-1 = r3 = 4.7821077X10-57m3/Planck dipole. The quantum cube has 3 fundamental volumes along the x, y, z axis containing 27 Planck dipoles in different arrangements, where some fundamental cubes contain 1 or more Planck dipoles, while some may contain zero. The quantum cube is the smallest 3-D array allowing for expansion in all three spatial dimensions. The number of quantum cubes per 1m3 is then, (2λP)-3/27S = 7.744919X1054qc/m3. The term given as [X/(2λP)-3/27S] gives the volume, per quantum cube group, where X is number of quantum cubes/group.
Adding vacuum energy to the vacuum causes an expansion of quantum cubes containing Planck dipoles while being opposed by the attractions from the sum total of all the pairwise gravitational forces between Planck dipoles in the quantum cubes. Three dimensional volume expansion, given by dU = -pdV from thermodynamics, written in terms of vacuum energy density based on a standard volume given as a 1 meter cubic volume, dU, gravitational pressure given as gravitational force divided by 6 times the area of a cubic face or pressure, p, and change in volume, dV, is incorrect as illustrated below. A similar relation, where the only changes are substituting area of a cubic meter or 6m2 with 6 times the volume of a cubic meter, 6m3, and gravitational force multiplied by 1 meter resulting in units of gravitational energy designated as standard gravitational energy per 6 times the volume gives the correct equation relating vacuum energy to volume expansion. The equation reflecting total volume expansion, where 2ΔρVAC is derived from combining total positive emission energy from total positive energy with total negative energy in positive and negative ST or +2PRMS(TλCE.) – -2PRMS(TλCE.) = +4PRMS(TλCE.), which is used towards the derivation of vacuum energy as a combination of total energy +2PRMS(TλCE.) in positive minus total energy -2PRMS(TλCE.) is given by total relation 2ΔρVAC = [2[(2λP)-3/27S]/2][((2)4(1m)GmP2)/(2)6m3r2]FactordV, where 2ΔρVAC is total vacuum energy, [(2λP)-3/27S] is total number of quantum cubes by combing number of quantum cubes in positive and negative ST per twice the volume and quantum cube density in each ST is invariant, [((2)4(1m)GmP2)/r2] is total gravitational energy over twice the 6m3 volume, [Factor] is invariant, where [Factor] is defined below, Planck dipole separation squared r2 = [(2λP)-3/S]-2/3 = 2.838439X10-38m2, and 2dV is total volume expansion. The relation reduces to 2ΔρVAC = [(2λP)-3/27S] [4((1m)GmP2)/6m3r2][Factor]2dV, where 2dV = 1.382088X10-78m3. To obtain the expression in total positive ST only, divide by 2 giving the relation ΔρVAC = [(2λP)-3/27S][4((1m)GmP2)/6m3r2][Factor]dV giving dV = 6.9104394X10-79m3 in total positive ST and -6.9104394X10-79m3 in total negative ST.
The relation derived from dU = pdV given as ΔρVAC = 2[(2λP)-3/27S][4(GmP2)/6m2r2][Factor]dV is similar but incorrect. To illustrate the correct equation, consider the usual thermodynamic relation, where total area for pressure is 6rX2 and rX2 = [X/[(2λP)-3/ 27S]]2/3 and X is the number of quantum cubes per group, given as ΔρVAC(rX3) = (2λP)-3/27S][(4GmP2)/6(rX2)r2][Factor]dVgr or ΔρVAC = [(2λP)-3/27S][(4(GmP2)/6(rX2)r2] [Factor]dVgr, where r2 is given as [(2λP)-3/27S]-2/3 = 2.838439X10-38m2 and [Factor] = 135.0468 as derived below. Let X = 1X1032qc/gr. Then, rX2 = 5.5037084X10-16m2 and dVgr = 3.8033043X10-94m3. Multiply by number of groups in 1m3 or 7.744919X1022gr gives 2.945628X10-71m3 ≠ 6.9104394X10-79m3, which has area 6m2 for a 1m3 volume. Now let ΔρVAC(rX3) = [(4GmP2)/6(rX3)r2(2λP)-3/27S][Factor]dVgr, where area, rX2 is now replaced by volume, rX3 = X/[(2λP)-3/27S],= 1.291169X10-23m3. Then, dVgr = 8.922545X10-102 and dVgr(7.744919X1022gr) = 6.9104394X10-79m3. As can be seen, replacing gravitational force/area, p, in dU = pdV with standard gravitational energy/volume renders the correct relation, dU = edv, where e is standard gravitational energy per volume. For 1m3, dVgr = dV and rX2 = rX3.
Working in total positive ST, the total change in vacuum energy in the relation rendering total volume change is dU = ΔρVAC(1m3) = 5.367X10-10J and occurs during each electric dipole energy emission step in time given as 1.61856X10-20s. The energy goes into increasing total volume +dV through an addition of energy that is absorbed by groups of quantum cubes increasing mutual potential energy between Planck dipoles, which increases their mutual separation. Increasing or less negative potential energy due to mutual gravitational attraction between Planck dipoles that make up the configuration of a quantum cube requires an input of external gravitational energy. Upon reduction and releasing EM surplus energy at e-e+ dipoles, EM quanta are found throughout the 1m3. When two EM surplus quanta interact, a transformation occurs, where two EM quanta transform into two gravitational wave states via a Feynman process, where energy is conserved. The outgoing gravitational wave states are absorbed by a group of quantum cubes raising their potential energy, which increases Planck dipole separation in a quantum cube through expansion. Converting EM quanta to gravitational wave states is very likely to occur since the density of quantum cubes in 1m3 is very large compared to density of e-e+ dipoles, where emissions are non-absorbed, in 1m3 or 7.744919X1054 qc compared to (1/8π)(0.99112)(λCE2/1m2)(#dipoles/1m3)(1m3) = 1.48258X107e-e+ dipoles giving a ratio of (7.744919X1054qc)/(1.48258X107e-e dipoles) = 5.223942X1047 qc/e-e+dipole. Gravitational coupling is αGR = 1.7513614X10-45. Thus, due to 5.223942X1047qc/e-e+dipole being available, this value is compatible with (1.7513614X10-45)-1 = 5.7098437X1044 and energy conversion is likely. One surplus quanta per e-e+ dipole given as (1.1072)(3.269885X10-17J) = 3.620417X10-17J on average converts to gravitational energy resulting in expansion of 5.223942X1047qc. Each quantum cube absorbs vacuum energy in amount given as 5.367X10-10J/m3/[7.744191X1054quantum cubes in 1m3] = 6.9297045X10-65J/qc. The number of quantum cubes required in order to absorb the gravitational energy, 3.620417X10-17J is (3.620417X10-17J/e-e+)/(6.9297045X10-65J/qc) = 5.2244897X1047qc/group-e-e+ dipole energy. Due to quantization of energy emitted, energy absorption by a quantum cube group changes the potential energy state of pairs of Planck dipoles in the entire group.
Let X = [(2λP)-3(1m3)/27S], be the number of quantum cubes/m3 in 1m3. Then, (X-1)dV = dVqc, where dV is given by 6.9104394X10-79m3 with [Factor] reflecting uniformity equal to 135.0648 and dVqc is the expansion of the quantum cube. Expansion of 1 qc is dVqc = [27S/(2λP)-3]dV = (1.291169X10-55) (6.9104394X10-79m3) = 8.922545X10-134m3/qc. This implies possible existence beyond Planck parameters. Linear expansion distance is [8.922545X10-134m3/qc]1/3 = dr = 4.468512X10-45m/qc, which is 10 orders of magnitude smaller than Planck length lP = 1.616X10-35m. This implies that groups of quantum cubes expand from a unit absorption of converted gravitational energy.
As stated, the vacuum energy converts to volume expansion of quantum cube groups and hence the 1m3 volume. This places ST in an unnatural state, where fundamental constants and their relationships are altered or strained. In response to the altered state, Planck dipoles or gravitational wave states or simply states with density (2λP)-3 are created. The energy involved in creating positive and negative states is much greater than the energy available from EM vacuum energy. There is no apparent single independent energy source imputing required positive and negative energy but linking positive and negative ST kept separated without energy exchange due to a balance in fundamental parameters or constants now becomes imbalanced allowing energy exchange between positive and negative ST creating positive and negative energy states resulting in restoration of an imbalance to a balance of fundamental physical parameters. There is transfer of positive energy +mP from positive to negative ST creating +mP anti-state in negative ST and simultaneously a transfer of negative energy -mP from negative to positive ST creating -mP anti-state in positive ST. This results in a state +mP – -mP = +2mP in positive ST and state –mP – +mP = -2mP in negative -2mP in negative ST. This is also true of the e- anti e- dipoles emissions that are not reabsorbed by dipoles. The energy not reabsorbed causes the successive creation and annihilation process of e- and anti e- dipoles with creation and annihilation of EM wave states to be unsustainable. Thus, to preserve the process, energy is transferred from positive and negative ST in order to restore a sustained process. There is energy exchange removing energy deficits of -2EAVEE│λCE│2N seen in positive ST and energy deficits +2EAVEE│λCE│2N seen in negative ST. Similarly for electron-antielectron dipole creation during volume expansion with exchange of energy +-mEc2. It is the fundamental constants that determine energy flow between positive and negative ST. Energy transfers ceases when fundamental constants involved in the process are returned to normalcy. Having volume increase dV = 6.9104394X10-79m3 creates (2λP)-3dV = 8.251078X1022states, which are Planck dipoles due to initial creation.
The quantum cubes present absorbs the energy during its created lifetime of 4.78993097X10-43s and new dipoles are created in a volume as a function of the amount of expansion resulting in the determination of the amount of energy transferred between positive and negative ST. Upon Planck dipole annihilation to gravitational wave states, new Planck dipoles comprising new quantum cubes appear but no energy from e-e+ dipole emission is available since those emissions occur every 1.61856X10-20s. Energy is initially absorbed by a group of quantum cubes is transferred to a new generation of quantum cubes until complete expanded state is achieved. When a subsequent e-e+ emission occurs at a time increment of 1.61856X10-20s the set of previously expanded quantum cubes are available, which absorbs energy and further expands with creation of a new set of Planck dipoles. The number of Planck dipoles continue to grow. Planck dipoles created in a previous expansion, have since annihilated to a succession of gravitational wave states until their recreation in (5.7098437X1044states)(4.78993097X10-43s/state) = 273.498s. At a point in time 273.498s after a particular e-e+ dipole emission, there will be newly created Planck dipoles present possibly co-existent with recreated Planck dipoles. Volume increases are due to e-e+ dipole emissions and are filled with newly created Planck dipoles along with previous volume increases associated with recreated dipoles. Volume expansion and recreations ensure uniformity of Planck dipole density throughout volume expansion.
Excess vacuum energy ρVAC in a 1m3 volume is not uniformly distributed throughout the 1m3 volume but instead deposited discrete throughout the entire 1m3 volume. Energy surplus from e-e+ reduction occurs in steps separated in time by 2T = 1.6185X10-20s or frequency 1/2T. All quantum cubes and groups within the 1m3 cube are affected simultaneously by energy surplus from EM dipole emissions. Once energy is deposited throughout various locations in the 1m3 volume, energy is spread throughout causing group expansions to interconnect over space and time establishing causality, which results in the overall 1m3 or any volume expansion. Interconnecting the physical influences from all sub-expansions resulting in causal expansion of the 1m3 volume occurs per X amount of time. Instead of expansion causality being established at the maximum rate given as the speed of light, c, or expansion per time 3.335640X10-9s, the transfer of energy causing expansion takes into account the geometry that affects the rate of causality or expansion per time X. Looking at geometric effects, establishing causality over a cubic or any symmetric volume, geometric factors, which take into account varying distances in different directions throughout any volume, established by quantum cubes, must be taken into consideration. The maximum rate to establish causality is at the speed of light in the vacuum, c. Thus, the geometric factor modifies the rate of establishing causality throughout a volume. A causality factor, which reflects the rate of causality, is multiplied to c. The greatest rate of causality is along an edge while two other basic geometric directions distances are greater than the edge, which are diagonal across a face, and from vertex to vertex. Lines connecting vertices give a slower effective causal speed or c traversing a greater distance. A weighted average accounts for all three effects. Let a cube be normalized by having an edge equal to 1. Then, diagonally across the face gives 21/2 and from vertex to vertex 31/2. There are 12 edges, 12 face diagonals, and 4 vertex to vertex connections. Then, the weighted average is [4(31/2) + 12(21/2) + 12]/28 = 1.2821 and is greater than 1 indicating that the rate of causality is less than maximum. Inverse gives the geometric factor multiplying c, (1.2821)-1c = 0.7800c = 2.33838X108m/s giving causal linear expansion dr in 1m3 per causal time 4.27643X10-9s = [0.7800c/1m]-1 or causal expansion rate dr(0.78c/1m) = dr(2.33838X108/s). On-shell EM radiation propagating in the vacuum releases energy expanding quantum cubes but it is retrieved by their contraction since the EM radiation is not transient in the same way as that from e-e+ dipoles, thus continuing propagation. Transient e-e+ dipoles emit one-time EM radiation that is absorbed by quantum cubes but is not retrievable. 7.3591Tev gamma rays have wavelengths comparable to Planck dipole separation, which interacts with the vacuum lattice producing a lagging in the propagation process.
Causal linear expansion dr established per time 4.27643X10-9s occurs in 3 dimensions x, y, and z simultaneously and so causal volume expansion, dV, also occurs per time 4.27643X10-9s. The linear causal expansion rate dr(0.7800c/1m) is given as linear expansion per second of 1m3 volumes each exhibiting causal linear expansion. Causal volume expansion rate, then, is composed of causal linear expansion rates in 3 dimensions of space simultaneously or [dr(0.7800c/1m)]3 = dV(1.278634X1025/s) causally connecting volume expansions of 1m3 groups each exhibiting causal volume expansion. Thus, expansion per causal time between the linear and volume case is invariant but the causal rate depends upon the number of dimensions, where the linear case is fundamental. The linear case is fundamental since observations of a causally expanding location in the universe from an observer’s view point is linear in all directions. A three dimensional causal rate mapping is multidimensional so that a 3 dimensional mapping is composed of a simultaneous set of linear casual rate viewpoints. Frequency of deposited energy surplus in a 1m3 cube is given by the inverse of time to deposit energy or [2T = 1.6185X10-20s]-1 = 6.17856X1019/s. The frequency of deposited energy surplus determines state creation rate, where each energy deposit creating the states manifests after each causal establishment period. Causal expansion rate does not determine the rate of states created. Linear expansion dr per X second establishes causality rate while at each dV, states create, and causality is established per X time both occurring at frequency (2T)-1. The Hubble rate measured at some distance is dependent upon linear causal expansion rate and linear expansion dr = (dV)1/3 since linear and volume expansion per time is invariant. Thus, for a particular linear expansion dr and causal expansion rate, dr/causal time, both appearing at frequency (2T)-1, the Hubble rate for a given distance occurs with a particular value but at frequency (2T)-1 = 6.17856X1019/s.
Expanding quantum cubes, due to vacuum energy, everywhere are in contact with surrounding quantum cubes, where they share Planck dipoles on a common face, edge, and vertex. As will be seen, this determines the probability of Planck dipole creation from state creation due to volume change. The relation that will determine the number of Planck dipoles required in creating a subsequent 1m3 volume from linear causal expansion rate is determined by fundamental constants and the interconnectedness of the quantum cubes throughout the volume. This implies that there will be Planck dipole creation along with the creation of gravitational wave state taken from general Planck states (2λP)-3. Due to the interconnection of Planck dipoles in quantum cubes, space-time vacuum as a whole is expanding in the same way from every location in space-time. There is no one spatially privileged reference location or center over all other locations in determining spatial expansion and, therefore, must occur at all uniformly distributed spatial locations occupied by quantum cubes. The expansion of one quantum cube multiplied by the number of quantum cubes in a 1m3 volume gives the expansion of the 1m3 volume.
[Factor] represents mutual gravitational attractions between all Planck dipoles in a quantum cube that modifies the gravitational energy term, which is written in terms of two Planck dipoles. The general 1m3 relation with quantum cube Planck dipole arrangement defining [Factor] gives dV as ΔρVAC = [4(1m)GmP2(2λP)-3dV)/S(27)6m3r2][[(#Adipoles)(#Bdipoles)(#occurrences)/g2] + …]. In [Factor], an arrangement of dipoles have #Adipoles multiplying #Bdipoles, multiplying #occurrences, and dividing by the geometric factor squared g2 since r is squared, which can also be represented as a sum given by: ∑[(#Adipoles)(#Bdipoles)(#occurrences)/g2] and multiplies [4(1m)GmP2(2λP)-3(dV)/S(27)6rX3r2]. Change in the component volume dV is [ΔρVACS(27)6m3r2/[4(1m)GmP2(2λP)-3[Factor]]] = 9.3323271X10-77m3/[Factor].
The Hubble constant in terms of fundamental constants is developed as follows, where [Factor] = ∑((#Adipoles)(#Bdipoles)(#occurrences)/g2) and ΔρVAC = (3.70948X10-4Ω/1m2)[(x + 4y)/(x + y)]c3q2mE2/h2. With 247 particle count the result is (4.10714X10-4Ω/1m2)c3q2mE2/h2 = 5.3689X10-10J/m3. Ω has units of kg-m2/s-C2. The time independent Hubble constant, H, is derived from change of volume dV. The relation for dV is dV = [ΔρVACS(27)6m3r2/[4(1m)GmP2(2λP)-3[Factor]]] and meters in (Ω/1m2)(6m3)(1/(1m)G) cancel giving 6Ω/G. In deriving Hubble constant, dimensionless expansion is required and denoted dV/1m3. The Hubble constant directly depends upon linear expansion given by [dV/1m3]1/3 = dr/1m. Then, dr/1m = (1/1m)[ΔρVACS(27)6m3r2/[4(1m)GmP2(2λP)-3[Factor]]]1/3. Hubble constant is given by H = [dr/1m]geometric factor(3.084X1022m/Mpc) or [dr/1m]0.7800c/1m.
The fundamental constants to place into the Hubble constant equation are: mP2 = hc/2πG, S = kq2/Gme2α, 2λP = 2h/mPc = 5.013257h1/2G1/2/c3/2, #qc/m3 = (2λP)-3/27S = c9/2mE2α/[3.401917X103h3/2G1/2kq2] and r2 = [(2λP)-3/S)]-2/3 = 25.13274hG1/3k2/3q4/3/[c3mE4/3α2/3], fundamental constants in ρVAC, and numerical values including geometric factor, 0.7800 giving 0.7800c, 1/1m2 (1/1m for dimensionless linear expansion and 1/1m so that c/1m gives expansion per time).
Then, H = [1.608391X1020/1m2]Ω1/3[(x + 4y)/(x + y)]1/3[q16/9k5/9G5/18/[h1/6c5/6α5/9mE4/9[∑((#Adipoles) (#Bdipoles)(#occurrences)/g2)]]km/Mpc. H = (316.229)[Ω1/3/1m2][(x = 4y)/(x + y)]1/3/[Factor]1/3km/s-Mpc. For [(x = 4y)/(x + y)] = 1.1072 and [Factor] = 135.0468, H is given as 63.765km/s-Mpc.
Hubble expansion has dependency upon distances between nearest surrounding dipoles in different quantum cube arrangements denoted [Factor]. Let one fundamental cube contain 1 dipole. The dipole is anywhere in the cubic volume since 4.7821077X10-57m3/dipole and is larger the Planck dipole (2λP)3 = 8.375197X10-102m3. Maximum uniformity implies that each fundamental cube contains one dipole at the center. Define a geometric factor, g, reflecting the modifier of distances between dipoles that are in geometric arrangements in the quantum volume. There are 26 single fundamental cubic volumes joined with a central single volume, X = 27, where 6 are joined at a face, 12 are joined at an edge, and 8 are joined at a vertex. Normalize the length of each single volume side to 1. At uniformity, face joined cubes have g = 1, edge joined cubes g = 21/2, and cubes joined at vertices g = 31/2. The distance r between two dipoles is given by the cube root of the inverse number of dipoles per m3. The actual separation is then gr and g2r2 is involved with the gravitational force between two Planck dipoles developed earlier. Dipoles that are separated by face joined cubes have gr = 1r, edge 21/2r, and vertex 31/3r. All possible mutual g factors in this cubic arrangement are 11/2, 21/2, 31/2, 41/2, 51/2, 61/2, 81/2, 91/2, 121/2. Note: There are no g factors equaling 71/2, 101/2, and 111/2. Given maximum uniformity in 27 fundamental cubic volumes, see Figure 4, #Adipoles and #Bdipoles equal 1. There are 54 occurrences of g2 = 1, 72 occurrences of g2 = 2, 32 occurrences of g2 = 3, 27 occurrences of g2 = 4, 72 occurrences of g2 = 5, 48 occurrences of g2 = 6, 18 occurrences of g2 = 8, 24 occurrences of g2 = 9, and 4 occurrences of g2 = 12 for a total of 351 pairings in a quantum cube. Figure 4.
[Factor] = ∑19(1 dipole)2(#occurrences)/(g2).
54.000 = 12(54)/(1) 2) 35.9991 = 12(72)/(2) 3) 10.6677 = 12(32)/(3)
4) 6.7500 = 12(27)/(4) 5) 14.3991 = 12(72)/(5) 6) 8.0001 = 12(48)/(6)
7) 2.2491 = 12(18)/(8) 8) 2.6676 = 12(24)/(9) 9) 0.3321 = 12(4)/(12)
SUM = 135.0468 is [Factor]. H = [327.148km/s-Mpc]/[135.0468]1/3 = 63.765km/s-Mpc with particle factor of 1.1072. Number of Planck dipoles filling volume dV = 6.9104394X10-79m3 is ((2λPS)-3states/m3)dV = (1.1940X10101#/m3) (6.9104394X10-79m3) = 8.25106X1022 Planck dipoles. Positive creation energy required is (8.25106X1022Planck dipoles)(1.53901X10-8kg/Planck dipole)(2.992925X108m/s)2 = 1.61418X1032J. This is clearly greater than vacuum energy ρVAC(1m3) = 5.367X10-10J resulting in energy exchange between positive and negative ST in order to create the necessary number of Planck dipoles in order to restore the vacuum ST governed by fundamental constants. In Figure 4, the Hubble constant for Planck dipole uniformity is 63.765km/s-Mpc and is close to the Hubble constant CMB value 67.4km/s-Mpc. Thus, a small variation from Planck dipole uniformity resulting in groups of quantum cubes having density variations manifested in small variations of CMB density may result in the CMB Hubble constant. Below, are deviations from uniformity having clumping of Planck dipoles in fundamental volumes, where there may be 2 or more Planck dipoles in one fundamental volume and 0 in others due to fluctuations away from uniformity. The variation in dipole density increases with time. In Figures 4 through 13, starting on the next page, various dipole and dipole cluster arrangements are depicted illustrating an increasing trend in density variations and with it increasing values of the Hubble Constant possibly explaining early universe verses late universe value trend. An example for deriving the Hubble constant in Figure 7 is analyzed in Figure 11. All figures are based upon particle factor equaling 1.1072.
The experimental values for the late universe taken from various surveys are given below:
1) SHOES 74.0 +-1.4km/s-Mpc. 2) Freedman et all 69.8 km/s-Mpc.
3) Megamasars 74.8 km/s-Mpc. 4) Miras 73.6 km/s-Mpc.
5) Galaxy Brightness 76.5 km/s-Mpc. 6) HOLICOW 73.3 km/s-Mpc
** Average of 1 through 6 = 73.7km/s-Mps. Early CMB from Planck 2015 Survey: 67.4km/s-Mpc.

Figure 4. (Uniformity)

Figure 5.

Figure 6.

Figure 7.

Figure 8.

Figure 9.

Figure 10.

Figure 11.

An example for deriving the Hubble Constant 69.447km/s-Mpc in Figure 7 is shown below. Format in deriving the factor is [(#dipoles A)(#dipoles B)(#occurrences)/(g2)].

(1)(1)(15)/(1) = 15.0012 (1)(2)(3)/(1) = 5.9994 (2)(2)(1)/(6) = 0.6669
(1)(1)(16)/(2) = 8.0001 (1)(2)(7)/(2) = 7.0011 (4)(4)(1)/(9) = 1.7766
(1)(1)(7)/(3) = 2.3328 (1)(2)(3)/(3) = 2.0007 (2)(3)(1)/(2) = 2.9997
(1)(1)(3)/(4) = 0.7506 (1)(2)(1)/(4) = 0.4995 (2)(3)(1)/(12) = 0.4995
(1)(1)(9)/(5) = 1.8009 (1)(2)(5)/(5) = 2.0007 (2)(4)(1)/(2) = 3.9987
(1)(1)(8)/(6) = 1.3338 (1)(2)(2)/(6) = 0.6669 (2)(4)(1)/(4) = 2.0007
(1)(1)(2)/(8) = 0.2511 (1)(2)(2)/(8) = 0.4995 (2)(4)(2)/(5) = 3.1995
(1)(1)(5)/(9) = 0.5562 (1)(2)(1)/(9) = 0.2214 (3)(4)(1)/(5) = 2.4003
(1)(1)(1)/(12) = 0.0837 (1)(2)(0)/(12) = 0 (3)(4)(1)/(8) = 1.5012
SUM 30.1104 SUM 18.8892 SUM 19.0431
(1)(3)(1)/(1) = 2.9997 (1)(4)(0)/(1) = 0
(1)(3)(1)/(3) = 0.9990 (1)(4)(3)/(3) = 3.9987
(1)(3)(1)/(2) = 1.5012 (1)(4)(5)/(2) = 10.0008 SUM ALL TALLIES OR FACTOR
(1)(3)(2)/(4) = 1.5012 (1)(4)(3)/(4) = 2.9997 30.1104
(1)(3)(3)/(5) = 1.8009 (1)(4)(7)/(5) = 5.5998 +18.8892
(1)(3)(2)/(6) = 0.9990 (1)(4)(3)/(6) = 2.0007 +19.0431
(1)(3)(1)/(8) = 0.3726 (1)(4)(1)/(8) = 0.4995 +10.5057
(1)(3)(1)/(9) = 0.3321 (1)(4)(2)/(9) = 0.8883 +25.9875
(1)(3)(0)/(12) = 0 (1)(4)(0)/(12) = 0 104.5359 = [Factor]
SUM 10.5057 SUM 25.9875
Sum of all tallies = 104.536 is [Factor]. Given particle count of 1.1072, the time independent Hubble constant H = [327.148km/s-Mpc]/[104.536]1/3 = 69.447km/s-Mpc.
Judging from Figures above, clusters of dipoles containing greater than one dipole along with approximate uniformly results in an increased Hubble Constant. Dipole clusters accumulate dipoles from regions nearby resulting in approximate uniformity of single dipoles, regions with zero dipoles, and clusters. Also, during volume expansion, new Planck dipoles are created in various regions throughout the volume resulting in changes to Planck dipole configurations in quantum cubes. Below, Figures 12, 13, are two examples of added slowing due to placement of 1 dipole at position 14.

Figure 12.
Figure 13.
Adding 1 dipole to the quantum cube center slows the expansion rate and in greater amount when there is uniformity of large dipole clumping.

Figure 14.

Figure 15.

Figure 16.
In the three figures above, in reference to Figure 15, only one dipole was moved out of a large cluster and one dipole was added to a larger cluster having slight variation in dipole numerical arrangements, producing three Hubble constant increases. Dipoles clustering in a fundamental volume may have in some cases equal effect as Planck dipoles simply moving closer together producing greater density while others move apart decreasing density but in both cases remaining in their fundamental volumes. Matter has effect on ST while ST has an effect on matter and energy. Thus, when ST expands matter of low density expands along with ST but only in cases where their mutual gravitational or other force attractions are small so as not to overcome ST expansion.
K) Development of Time Dependent Hubble Constant.
The Hubble constant depends upon fundamental parameters such as 2λCE = h/mEc. Implement dipole mass change by nmE. It was shown in the section on mass independence that changing the mass away from that of the electron has no greater or smaller effect on the vacuum energy density and thus gives the same results as for the e-e+ dipole. However, changing the fundamental constants, h and c, while the mass term is constant in 2λCE, results in changes of the Hubble constant. If the linear meter is divided into number of increments governed by the linear dimension of the e-e+ dipole Compton wavelength or 1m/2λCE, or of one changed, the causal time of light traversing a meter is given by (1m/2λCE)(2λCE/c) = 1m/c = (1m/2λN)(2λN/c) and constant. Since the magnitude of the Hubble constant depends on causal volume change governed by the semi-thermodynamic relation containing the mutual gravitational attractions of Planck dipoles, vacuum energy density, and the causal time, the Hubble constant would remain constant unless the vacuum energy changed by changing the magnitude of constant h, which also changes the magnitude of the mutual gravitational attractions between Planck dipoles. Thus, a Hubble constant with a different magnitude would occur by changing fundamental constants. Fundamental constants play a number of roles as explained here and a role of energy exchange between positive-negative ST.
Based upon fundamental constants, the density of quantum cubes is 7.7449185X1054qc/m3 and so multiplying by 27 gives the density of Planck dipoles 2.091128X1056Planck dipoles/m3. Time to double 1m3 volume, where particle factor is 1.1072 and having condition of Planck dipole uniformity or [Factor] = 135.0468 yielding a Hubble constant 63.765km/s-Mpc, is [dr(0.7800c/1m)]-1 = 4.837074X1017s. During each expansion dV of the initial 1m3 cubic volume the number of Planck dipoles created is given by the relation (2λP)-3dV = (1.194001X10101quantum states/m3)(6.9104394X10-79m3) = 8.251078X1022Planck dipoles. Planck dipoles are created since they initiate upon annihilation gravitational wave states. If a gravitational wave state is created, then there would be no standard governed by fundamental parameters where along the sequence of annihilation successions a Planck dipole is created. The states created upon expansion, dV, are Planck dipoles and the number of Planck dipoles created after doubling the 1m3 cubic volume is then 2.091128X1056Planck dipoles. Consider the minimum time for light to traverse dr or (dr/c) = (6.9104394X10-79m3)1/3/c = 2.9490431X10-35s. The number of Planck dipoles annihilating to gravitational wave states during this time in component form is given by (2dr/c)/2TP = (dr/c)/TP, where TP is the time of Planck dipole existence = 3.386993X10-43s. Result is (2.9490431X10-35s)/ (3.386993X10-43s/Planck dipole) = 8.70697X107Planck dipole to wave states or simply Planck wave states. The ratio given as (8.251078X1022Planck dipole)/(8.70697X107 Planck wave state) = 9.476406X1014Planck dipole/Planck wave state. Then, (9.476406X1014Planck dipole/Planck wave state)(273.498wave state factor)(6.17856X1019/s)(4.837074X1017s) = 7.745827X1054Planck dipole, where wave state factor is ratio given by number of wave states from Planck dipole annihilation to its recreation to number of total Planck Compton wave lengths in one second or (5.7098437X1044 wave states)/(2.0877128X1042total Planck Compton wavelengths) = 273.498 wave state factor and (6.17856X1019/s) is frequency of expansions based on e-e+ EM emission rate, 1/(2Te-e+), where 2Te-e+ = 1.6185001X10-20s. The result equals the number of quantum cubes, qc, in 1m3, however. Thus, the relation has to be converted to units of quantum cubes. Then, (8.251078X1022Planck dipole) becomes (1qc/27Planck dipole)(8.251078X1022Planck dipole) and (8.70697X107wave state) becomes (1qc wave state/27Planck wave state)(8.70697X107Planck wave state) implying that the quantum cube with 27 Planck dipoles can annihilate simultaneously to wave states along with others in the 1m3 volume. Then, the ratio (1qc/27Planck dipole)(8.251078X1022Planck dipole)/(1qc wave state/27 Planck wave state)(8.70697X107Planck wave state) = (3.0559548X1021quantum cube)/(3.224803X106qc wave states) = 9.4764056X1014qc/wave state, the same as above. The result is 7.745827X1054 quantum cubes in 1m3. Multiply by 27Planck dipoles/qc to obtain the number of Planck dipoles in 1m3 or (27Planck dipoles/qc) (7.745827X1054quantum cube = 2.091373X1056Planck dipoles. The relation is then, 27(9.4764056X1014 Planck dipole/Planck wave state)(273.498wave state factor)(6.17856X1019/s)(4.395354X1017s) = 2.091373X1056Planck dipoles or 27(2λP)-3dV/((dV1/3)/c)/ 2TP(1/(2Te-e+))[(dV)1/3(0.7800c/1m)]-1. Compare with 2.091128X1056 Planck dipoles in 1m3 calculated by fundamental constants resulting in +0.012% error. Uniformity will be taken as the standard.
Consider a Hubble constant of 70.2km/s-Mpc having [Factor] = 101.36. Given 70.2km/s-Mpc, dV = 9.21027X10-79m3 and dr = 9.729506X10-27m. The volume change dV creates (1.194001X10101quantum states/m3)(9.21027X10-79m3) = 1.09971X1023Planck dipoles. (dr/c)/TP = 9.58199X107 wave states so that 1.09971X1023 Planck dipoles/9.58199X107 wave states = 1.14768X1015 Planck dipoles/wave state. Inverse of causal rate is [9.729506X10-27(0.78)c]-1 = [2.275130X10-18m/s]-1 = 4.395354X1017s/m(1m) = 13.93Byr, which is the Hubble time to double the 1m3 volume. Then, by the same calculation as above, the relation becomes 27(1.14768X1015Planck dipole/Planck wave state)(273.498wave state factor)(6.17856X1019/s) (4.395354X1017s) = 2.30156X1056Planck dipoles.
Based upon causal volume expansion rate, which is based upon linear expansion rate, a greater number of Planck dipoles are created even though there is a shorter Hubble time. The slowest causal expansion rate, uniformity, or 2.067153X10-18m/s compared to 2.275130X10-18m/s is related by the ratio (2.275130X10-18m/s)/(2.067153X10-18m/s) = [(135.0468/101.36]1/3 = 1.1004. Volume increase dV is related as (6.9104394X10-79m3)[135.0468/101.36] = 9.210167X10-79m3. Linear increase dr then is related as (8.8410101X10-27m)[135.0468/101.36]1/3 = 9.729470X10-27m. Comparing the time to double the 1m3 volume, it takes less time to establish dipole increase or 4.83707X1017s/m(1m)[135.0468/101.36]-1/3 = 4.395354X1017s. The number of Planck dipoles created are related as (2.09162X1056Planck dipoles) [135.0468/101.36]1/3 = 2.30156X1056Planck dipoles. In general, (X at uniformity)[Factor]U/[Factor]Y = (Y at non-uniformity). In terms of steps, the relation is generalized (X created)[Factor]X/[Factor]Y[Factor]W/[Factor]Z = (Z created) in terms of sub steps and (X created)[Factor]X/[Factor]Z = (Z created) in terms of the two limits, where the product of individual [[Factor]1/[Factor]2] steps equals the overall step. The Hubble rate (70.2km/s-Mpc)/(63.765km/s-Mpc) = 1.10092 and 135.0468/101.36 = 1.3323 = (1.10092)3.
In general dVUNIFORMITY[135.0468/[Factor]X] = dVX, drUNIFORMITY[135.0468/[Factor]X]1/3 = drX, (#dipoles created during volume expansion at uniformity)[135.0468/[FactorX]] = (#dipoles created)X, (#wave states at uniformity)[135.0468/[Factor]X]1/3 = (#wave states)X, (Hubble time at uniformity)[135.0468/[Factor]X]-1/3 = (Hubble time)X. The number of Planck dipoles created over Hubble time is, using uniformity [135.0468] and 70.2km/s-Mpc with [Factor] = 101.36 is (27){[(#Pl dip)[135.0468/101.36]]/[#wave states [135.0468/101.36]1/3}(273.498Wave factor)(6.17856X1019/s)(4.837074X1017s)[135.0468/101.36]-1/3 = (27)(1.14768X1015Planck dipoles/Planck wave state)(273.498wave state factor)(6.17856X1019/s) (4.395354X1017s) = 2.301547X1056Planck dipoles. Hubble doubling time for uniformity 4.837074X1017s.
In terms of e-e+ dipole creation, the number created during 1m3 doubling is [(2)(2.42631X10-12m)]-3/137 = 6.387796X1031 m3. There are 2.091128X1056 Planck dipoles/m3 and so the number of Planck dipoles per e-e+ dipole is [(2)(2.42631X10-12m)] 3 = (1.142691X10-34m3/e-e+ dipole)(2.091128X1056 Planck dipoles/m3) = 2.389511X1022Planck dipoles/e-e+ dipoles. Then, there are (2.091128X1056 Planck dipoles/m3)/ (2.389511X1022Planck dipoles/e-e+ dipoles = 8.751280X1033e-e+ dipoles in 1m3. Divide by 137 EM wave states to achieve e-e+ dipoles without EM wave states or 8.751280X1033e-e+ dipoles/137 = 6.387796X1031e-e+ dipoles. Equivalently the result can be arrived by (1.142691X10-34 m3/e-e+ dipole)-1/ 137. As Planck dipoles are created during volume expansion an e-e+ dipole is created when 2.389511X1022Planck dipoles create. Thus, upon creation of an additional 1m3 volume, 6.387796X1031e-e+ dipoles create.
A method for deriving number of e-e+ dipoles in 1m3 with Hubble rate 63.765km/s-Mpc is given. Time to double 1m3 volume, where particle factor is 1.1072 and having condition of Planck dipole uniformity or [Factor] = 135.0468, is [dr(0.7800c/1m)]-1 = 4.837074X1017s. Frequency of expansion is (6.17875X1019/s) and based upon e-e+ EM emission rate in component form, 1/(2Te-e+), where 2Te-e+ = 1.61845X10-20s. In the same way as for calculating the number of Planck dipoles in a newly created 1m3 volume (dr/c) = (6.9104394X10-79m3)1/3/c = 2.9490431X10-35s. Instead of Planck dipole existence time the existence time of e-e+ dipole is (2.9490431X10-35s)/(1.61849X10-20s/e-e+ wave state) = 1.822095X10-15 wave state. The number of e-e+ dipoles created upon (dV)1/3 = dr expansion is given by dr/2λe-e+ = 8.8410101X10-27m/ [(2)(2.42631X10-12m/e-e+ dipole) = 1.821904X10-15e-e+ dipole. The ratio gives (1.821904X10-15e-e+ dipole)/(1.822095X10-15wave state) = 0.99990e-e+ dipole/wave state. The wave factor in this case is 137wave state/e-e+ dipole As determined prior, the number of e-e+ dipole emissions not absorbed amongst 6.387789X1031 e-e+ dipoles in a 1m3 volume is 1.48258X107emission/1m3 e-e+ dipoles. Lastly, the factor, 4.36, derived below has units of average g2/emission. Combining the terms gives [(137wave state/e-e+ dipole)(0.99990e-e+ dipole/wave state)(6.17876X1019/s)(4.837074X1017s)]/[(1.48258X107 emission/1m3 e-e+ dipoles)(4.36average g2/emission)] = 6.333676X1031e-e+ dipoles/1m3. The number of e-e+ dipoles in 1m3 volume calculated previously was 6.387789X1031e-e+ dipoles/1m3 giving -0.8% error.
Given Hubble rate 70.2 having [Factor] = 101.36, Hubble time to create 1m3 volume is 4.395354X1017s. The factors above in the first formulation above except the Hubble time are constants. Thus the Product of (137)(0.9999)(6.17856X1019)/[(4.36)(1.48258X107)] is a constant, where g2 for uniformity is explained in the following paragraph. Multiplying by the Hubble time 4.395354X1017s gives 5.75511X1031 e-e+ dipoles however, which is smaller. A normalization factor for 1m3 creation could operate on the Hubble time by multiplying the cube root of the ratio of [Factor] for uniformity versus [Factor] = 101.36 in the case of smaller Hubble time to get (5.75511X1031 e-e+ dipoles)[135.0468/101.36]1/3 resulting in 6.33276X1031e-e+ dipoles. A more plausible explanation is to operate on the g2 factor by [101.36/ 135.0468]1/3, where g2 reflects non-uniformity and is shown below.
The factor g2/emission = 4.36 affects not a causality rate but is composed of the average g2 factor, which in this case is generated for uniformity or near uniformity in the quantum cube. The factor reflects the e-e+ emission expanding quantum cubes, where on the average g2 is reflective of the mutual gravitational forces between the Planck dipoles. Let the Planck dipole distribution be uniform. Since the dipole and quantum cube distribution have boundary conditions, the g2 factors associated with a boundary are eliminated in order to isolate the quantum cube. Consider a quantum cube that is encompassed by quantum cubes. Then, the boundary condition and associated g2 factors are eliminated by placing a unit quantum cube around the boundary and subtracting the g2 factors and number of occurrences, which are 12(1), 12(2), and 4(3) formatted as X(Y), where X is the number of occurrences and Y is a particular g2 value. The g2 factors for a uniform distribution in an isolated quantum cube are 54(1), 72(2), 32(3), 27(4), 72(5), 48(6), 18(8), 24(9), 4(12). The g2 factors for the unit cube are 12(1), 12(2), and 4(3). The average g2 factor is then, [54(1) + 72(2) + 32(3) + 27(4) + 72(5) + 48(6) + 18(8) + 24(9) + 4(12) – 12(1) – 12(2) – 4(3)]/ [54 + 72 + 32 + 27 + 72 + 48 + 18 + 24 + 4 – 12 – 12 – 4] = 1410/323 = 4.36.
Operating on the g2 factor as alluded to above using the cube root of the ratio [101.36/135.0468]1/3 = 0.9088 results in a g2 factor of 4.36(0.9088) = 3.96. Then, the number of e-e+ dipoles created is (137)(0.9999)(6.17856X1019)(4.395354X1017s)/[(3.96)(1.48258X107)] = 6.336434X1031e-e+ dipoles.
Consider Figure 7 on page 22, where dipole clumping in the quantum cube is taken into consideration so all dipole products, composed of g2 factor and average gravitational effects due to non-uniformity from all dipole combinations, are taken into consideration. The Hubble rate is 69.447km/s-Mpc and [Factor] = 104.536. The average g2 factor taken from the product of #occurrences and g2 divided by number of occurrences is given as average g2 = (516)/(108) = 4.778. The average of the product given by #dipoleA and #dipoleB is 157/41 = 3.829. The total average taking into account the g2 factor and the gravitational effects due to Planck dipole clumping and non-clumping is (4.778 + 3.829)/2 = 4.30. The ratio 4.3(104.536/135.0468)1/3 = 4.3(0.92) = 3.96. The Hubble time is 4.440780X1017s The number of e-e+ dipoles created during volume doubling is (137)(0.9999 e-e+ dipoles)(6.17856X1019/s)(4.44078X1017s)/ [(3.96)(1.48258X107)] = 6.40192X1031 e-e+ dipoles.
Consider the case where there is pronounced clumping and many unoccupied fundamental volumes in the quantum cube as in Figure 11 having Hubble rate 79.425km/s-Mpc and [Factor] = 69.881. The sum of Planck dipole products excluding products of one Planck dipole is (4)(4) + (4)(4) + (4)(4) + (4)(4) + (1)(4) + (1)(4). The total occurrences including products of one Planck dipole or (1)(1), which there are two is 8. The average is given as 72 products/8 occurrences = 9 Planck dipole products. The product of number of occurrences and g2 factor are (8)(5) + (2)(4) + (2)(12) + (2)(1) + (1)(4) + (3)(8) + (10)(3) + (6)(2) having a sum 144. The number of occurrences is 34. As given with a unit cube taken into consideration and along in this case the number of many unoccupied fundamental volumes between faces and vertices in the quantum cube, the total product of occurrences and g2 factor is 144 – (1)(2) – (2)(2) – (3)(2) – 12 end zero’s = 120 and 34 – 12 end zeros – 2 – 2 – 2 = 34 – 18 = 16. The average g2 factor along with occurrences is 120/16 = 7.50. The average of the sum of average Planck dipole products plus average g2 factor and occurrences is (9 + 7.50)/2 = 8.25. Then, 8.25[69.881/135.0468]1/3 = 8.25(0.8028) = 6.62. Hubble time is given as 3.882908X1017s. Then, the number of e-e+ dipoles upon addition of 1m3 volume is given by (137)(0.9999e-e+ dipole)(6.17856X1019/s)(3.8829084X1017s)/[(6.62)(1.48258X107)] = 3.348455X1031e-e+ dipoles.
Given an initial 1m3 volume and quantum cubes populated uniformly with Planck dipoles, there exists time evolutionary process occurring before steady state exponential growth. As seen, Hubble constant depends upon various constants but varies with time dependency upon the configuration of Planck dipoles in the quantum cube. In particular, the lowest Hubble rate occurs at Planck dipole uniformity and in general increases with time as Planck dipole clumping increases. See Figures 4 through 11, where the Hubble rate increases from 63.765km/s-Mpc to 79.425km/s-Mpc. As mentioned, initially, quantum cubes are composed mainly of Planck dipoles in uniformity. During volume expansion over time, clumping of Planck dipoles occur along with a mixing of newly created ones. Nominal clumping is a balance that depends upon time and the creation of Planck dipoles, which are placed in quantum cubes during their clumping. The creation of Planck dipoles, which find themselves added to quantum cubes results in a smaller rate of Hubble expansion, Figures 12 and 13. Thus, the overall dynamics are an interplay between clumping and insertion of created dipoles.
The uniform distribution of Planck dipoles in the quantum cube having a Hubble rate 63.765km/s-Mpc occurs early in the initial stage of universe expansion around the time of CMB and the measurement in the latter stage of universe evolution particularly during when the universe began accelerated expansion is in the vicinity of 72km/s-Mpc and 5Byr ago. The value of the Hubble rate due to Planck dipole clumping only, Figure 11, is 79.425km/s-Mpc up from 63.765km/s-Mpc at time of CMB. There is also slowing of the Hubble rate due to Planck dipole creation and insertion into quantum cubes. In Figures 12 and 13, adding one Planck dipole to the center of the quantum cube results in slowing of (69.447km/s-Mpc – 66.930km/s-Mpc) = 2.517km/s-Mpc/Planck dipole/qc in Figure 12 and (76.682km/s-Mpc – 75.615km/s-Mpc) = 1.067km/s-Mpc/Planck dipole/qc in Figure 13. The average is (2.517km/s-Mpc/Planck dipole/qc + 1.067km/s-Mpc/Planck dipole/qc)/2 = 1.792km/s/Planck dipole/qc. The number of Planck dipoles/qc required in order to initiate the slowing from the largest to the smallest Hubble rate is (79.425km/s-Mpc – 63.765km/s-Mpc)/(1.792km/s-Mpc/Planck dipole/qc) = (15.660km/s-Mpc)/(1.792km/s-Mpc/Planck dipole/qc) = 8.739. A measure of the increasing quantum cube expansion along with Hubble rate is given by the average, (79.425km/s-Mpc + 63.765km/s-Mpc)/2 = 71.6km/s-Mpc.
The quantum cube, besides the number of Planck dipoles at the center has 26 fundamental volumes, which are affected by other quantum cubes in the group. This has an effect of altering the total effectiveness of the remaining 26 Planck dipoles on the central added Planck dipoles since they are shared amongst other quantum cubes. The shared Planck dipoles are as follows: 8 vertex Planck dipoles shared 8 ways for a ratio of 1, 6 face centered Planck dipoles shared 2 ways for a ratio 6/2 = 3, and 12 middle edge Planck dipoles shared 4 ways giving a ratio 12/4 = 3. The average ratio is (1 + 3 + 3)/3 = 2.33. This results in 26/2.33 = 11.14 effective existing Planck dipoles in the quantum cube, which is added to the number of created Planck dipoles added to affect slowing, which in this case is (11.14 + 8.739)Planck dipoles/qc = 19.8Planck dipoles/qc. Then, the total number of dipoles is given as (19.8Planck dipoles/qc) (7.744919X1054quantum cubes) = 1.533X1056Planck dipoles.
Having a Hubble rate 71.6km/s-Mpc the linear expansion dr is 9.92849X10-27m and change in volume dV = 9.787006X10-79m3. The number of Planck dipoles created from expansion is 1.16857X1023Planck dipoles. The number of wave states or (dr/c)/TP = 9.7780X107wave states. The number of Planck dipoles per wave state is (1.16857X1023Planck dipoles/9.7780X107wave states) = 1.1951X1015Planck dipoles/wave state. Then, (27)(1.1951X1015Planck dipoles/wave state)(273.498wave state factor)(6.17856X1019/s)(Xs) = 1.533X1056Planck dipoles. The time to produce, Xs = 1.533X1056Planck dipoles/5.45267X1038Planck dipoles/s = 2.81147X1017s = 8.9Byr. Hubble time for 71.6km/s-Mpc is given as [(71.6km/s-Mpc)/ (3.084X1019Km/Mpc]-1 = 4.30726X1017s = 13.7Byr. Then time from the present to time of 71.6km/s-Mpc Hubble rate and the start of accelerated expansion is 13.7Byr – 8.9Byr = 4.8Byr ago.
Hubble rate 63.765km/s-Mpc reflects a uniform distribution of Planck dipoles. Upon volume expansion and Planck dipole creation, an additional 1m3 volume and 2.091373X1056Planck dipoles are created. Given 71.6km/s-Mpc Hubble rate and [Factor] = 95.388, the time to expand into an additional volume was 4.30726X1017s = 13.7Byr. The number of Planck dipoles created, 2.34861X1056, exceeds the nominal number to fill a 1m3 volume. Compare the number of Planck dipoles created at Hubble rate 71.6km/s-Mpc to that of 63.765km/s-Mpc or 2.34861X1056 to 2.091373X1056. In terms of e-e+ dipoles, the relation is [(137wave state/e-e+ dipole)(0.9999e-e+ dipole/wave state)(6.17856X1019/s)(4.395354X1017s)]/ [(1.48258X107emission/1m3 e-e+ dipoles)(4.36average g2/ emission)[95.388/135.0468]1/3] = 6.32970X1031e-e+ dipoles/1m3, where the clumping factor operates on g2.
Consider the equation ΔρVAC = [(2λP)-3/27S][4((1m)GmP2)/6(vol)r2][Factor]dV, where [Factor] = 95.388 and vol is the volume under consideration associated with gravitational energy. Volume applied to ΔρVAC and [(2λP)-3/27S] cancel leaving vol as the only variable. The equation becomes 9.781643X10-79(vol) = dV. At time = 0, the vacuum is composed entirely of 1m3 volumes. Consider vol = 1m3 for the first doubling of volume at Hubble time (1)4.30726X1017s. Then, dV = 9.781643X10-79m3(1) and dr = 9.92849X10-27m. The number of Planck dipoles created upon expansion is 1.16793X1023. The number of wave states is 9.77796X107 giving 1.194452X1015Planck dipoles/wave state. Then, the number of dipoles created after Hubble time is (27)(1.194452X1015Planck dipoles/wave state)(273.498wave factor)(6.17856X1019/s) (4.30726X1017s) = 2.34733X1056Planck dipoles. The volume containing the dipoles is (2.34733X1056Planck dipoles)/(2.091128Planck dipoles/m3) = 1.1225m3. Then, after 1 Hubble time, the universe originally composed of 1m3 volumes is added to the expanded volume 1.1225m3 volumes to give 2.1225m3
Now the expanded volume in excess of a simple doubling or 2m3 creates additional Planck dipoles and volume. The excess volume created above simple doubling is 0.1225m3. The excess volume starts from 0m3 initially and terminates at 0.1225m3. The average is (0m3 + 0.1225m3)/2 = 0.06125m3. The volume increase is then given as 9.781643X10-79(0.06125m3) = 5.99126X10-80m3. The linear displacement dr = 3.91297X10-27m. The number of Planck dipoles created upon volume expansion is 7.15357X1021 and the number of wave states is 3.85364X107 giving 1.856314X1014Planck dipoles/wave state. The number of dipoles created after one Hubble time is 3.648024X1055 and the excess volume created is (3.64802X1055Planck dipoles)/(2.091128Planck dipoles/ m3) = 0.1745m3. The total volume is then 2.1225m3 + 0.1745m3 = 2.2970m3 after one Hubble expansion time and found everywhere in the universe.
During the time after the first Hubble time and towards the second Hubble time, the initial vol is 2.2970m3. Then, 9.781643X10-79(2.2970m3) = 2.2246843X10-78m3. Linear expansion dr = 1.309757X10-26m. Number of Planck dipoles created upon volume expansion is 2.682733X1023 and the number of wave states is 1.289899X108. The number Planck dipoles/wave state = 2.079801X1015Planck dipoles/wave state and so the number of Planck dipoles created after an additional Hubble time is 4.08722X1056Planck dipoles. The volume increase is (4.08722X1056Planck dipoles)/(2.091128Planck dipoles/m3) = 1.9546m3 for an expansion of 2.2970m3 + 1.9546m3 = 4.2516m3. The excess volume created itself contributes to further expansion.
Average excess volume is 1.9546m3/2 = 0.9773m3. Then, 9.781643X10-79(0.9773m3) = 9.55937X10-79m3 and dr = 9.85091X10-27m. Number of Planck dipoles created from volume expansion is 1.14139X1023 and the number of wave states is 9.70155X108. Then, the number of Planck dipoles/wave state = 1.176502X1015. The number of Planck dipoles created after Hubble time is 2.312060X1056Planck dipoles and the volume associated with the created dipoles is (2.312060X1056Planck dipoles)/(2.091128Planck dipoles/m3) = 1.1057m3. The total volume after two Hubble time periods is 2.2970m3 + 1.9546m3 + 1.1057m3 = 5.3573m3.
During the time after the second Hubble time and towards the third Hubble time, the initial vol is 5.3573m3. Then, 9.781643X10-79(5.3573m3) =5.2403196X10-78m3. Linear expansion is given as dr = 1.7369444X10-26m. Number of Planck dipoles created upon volume expansion is 6.25695X1023 and the number of wave states is 1.7106094X108. The number Planck dipoles/wave state = 3.6577316X1015Planck dipoles/wave state and so the number of Planck dipoles created after an additional Hubble time is 7.1881661X1056Planck dipoles. The volume increase is (7.1881661X1056Planck dipoles)/(2.091128Planck dipoles/m3) = 3.4375m3 for total expansion of 5.3573m3 + 3.4375m3 = 8.7948m3. The excess volume created, 3.4375m3, itself contributes to further expansion.
Average excess volume is 3.4375m3/2 = 1.71873m3. Then, 9.781643X10-79(1.71873m3) = 1.6811997X10-78 m3 and dr = 1.189067X10-26m. Number of Planck dipoles created from volume expansion is 2.00735X1023 and the number of wave states is 1.117139X108. Then, the number of Planck dipoles/wave state = 1.7969233X1015. The number of Planck dipoles created after Hubble time is 3.5313097X1056Planck dipoles and the volume associated with the created dipoles is (3.5313097X1056Planck dipoles)/(2.091128Planck dipoles/m3) = 1.6887m3. The total volume is 5.3573m3 + 3.4375m3 + 1.6887m3 = 10.4835m3.
Again the creation of excess expansion creates additional excess volume. Average excess volume is 1.6887m3/2 = 0.84435m3. Then, 9.781643X10-79(0.84435m3) = 8.2591303X10-79 m3 and dr = 9.38235X10-27 m. Number of Planck dipoles created from volume expansion is 9.861410X1022 and the number of wave states is 9.240094X107. Then, the number of Planck dipoles/wave state = 1.0672413X1015. The number of Planck dipoles created after Hubble time is 2.0973403X1056Planck dipoles and the volume associated with the created dipoles is (2.0973403X1056Planck dipoles)/(2.091128Planck dipoles/m3) = 1.0030m3. The total volume so far is 5.3573m3 + 3.4375m3 + 1.6887m3 + 1.0030m3 = 11.4865m3.
Finally for illustration, the creation of excess expansion creates additional excess volume. Average excess volume is 1.0030m3/2 = 0.5015m3. Then, 9.781643X10-79(0.5015m3) = 4.905494X10-79 m3 and dr = 7.886681X10-27 m. Number of Planck dipoles created from volume expansion is 5.857165X1022 and the number of wave states is 7.76711X107. Then, the number of Planck dipoles/wave state = 7.540984X1014. The number of Planck dipoles created after Hubble time is 1.4819525X1056Planck dipoles and the volume associated with the created dipoles is (1.481925X1056Planck dipoles)/(2.091128Planck dipoles/m3) = 0.7087m3. The total volume after 3 Hubble times is 5.3573m3 + 3.4375m3 + 1.6887m3 + 1.0030m3 + 0.7087 = 12.1952m3.
To summarize, initially at time = 0 the universe contains 1m3 volumes and 1m3exp[(n)(0.8366)] with n = 0 = 1m3exp[(n)(0.8366) = 1m3.
After 1 Hubble time, total volume is then 2.1225m3 + 0.1745m3 = 2.2970m3. With n = 1, 1m3exp[(1)(0.8366] = 2.3086.
After 2 Hubble time, total volume is 2.2970m3 + 1.9546m3 + 1.1057m3 = 5.3573m3. With n = 2, 1m3exp[(2)(0.8366)] = 5.3298m3.
After 3 Hubble time, total volume is 5.3573m3 + 3.4375m3 + 1.6887m3 + 1.0030m3 + 0.7087m3 = 12.1952m3. With n = 3, 1m3exp[(3)(0.8366] = 12.3047. With further iteration of excess volume, the calculated value approaches the natural exponential value.
This suggests that the correct formulation is a natural exponential of the form 1m3exp[tHO(0.8366], where t is time, Hubble rate HO = 71.6km/s-Mpc/3.084X1019km/Mpc = 2.32166X10-18/s and Hubble time is the inverse of Hubble rate = 4.30726X1017s. The value 0.8366 provides for a good fit to the calculations.
The density parameter Ω is defined as the ratio of the observed density ρ to the critical density of the Friedmann universe. The first Friedmann equation can be expressed in terms of present day values of the density parameters or H2/HO2 = ΩR,0a-4 + ΩM,0a-3 + ΩK,0a-2 + ΩΛ,0.
With H = (da/dt)/a, H = H0(ΩR,0a-4 + ΩM,0a-3 + ΩK,0a-2 + ΩΛ,0)1/2. Then, da/dt = = H0(ΩR,0a-2 + ΩM,0a-1 + ΩK,0a + ΩΛ,0a2 )1/2. Integrating and using variables a* and t* gives tH0 = ∫aia da/(ΩR,0a-2 + ΩM,0a-1 + ΩK,0a + ΩΛ,0a2 )1/2 having initial time ti = 0s. For a universe dominated by dark energy Λ, gives tH0 = ∫aia da/(ΩΛ,0a*2)1/2. Integrating gives Then, tH0(ΩΛ,0)1/2 = ln(aaia) and a = aiexp(tH0(ΩΛ,0)1/2). Let ai = 1m3, then Xm3 of volume expansion verses time is Xm3 = 1m3 iexp(tH0(ΩΛ,0)1/2).
Some values for ΩΛ,0 are:
1) 0.73 +-0.04 Carroll and Ostlie.
2) 0.685+0.017-0.016 Table of Astrophysical constants and parameters in K.A. Olive et al.
3) 0.691 Planck cosmology probe.
0.83662 = 0.6989 ≈ 0.7. Then, 1m3exp[(3)(0.8366] goes to 1m3exp[tH00.71/2] ΩΛ,0 = 0.7, t = 3.
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