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3.3 The Parametrisation of the Friedmann Equation

It is well known, that the Radius of Curvature in the Field Equations of General Relativity relates to the Energy-Mass Tensor in the form of the critical density rcritical = 3Ho2/8pG and the Hubble Constant Ho as the square of frequency or alternatively as the time differential of frequency df/dt as a cosmically applicable angular acceleration independent on the radial displacement.

The scientific nomenclature (language) then describes this curved space in differential equations relating the positions of the 'points' in both space and time in a 4-dimensional description called Riemannian Tensor Space or similar.
This then leads mathematically, to the formulation of General Relativity in Einstein's field Equations: for the Einstein-Riemann tensor and is built upon ten so-called nonlinear coupled hyperbolic-elliptic partial differential equations, which needless to say, are mathematically rather complex and often cannot be solved analytically without simplifying the geometries of the parametric constituents (say objects interacting in so called tensor-fields of stress-energy {Tμν} and curvatures in the Riemann-Einstein tensor {Gμν}, either changing the volume in reduction Ricci tensor {Rij} with scalar curvature R as {Rgμν} for the metric tensor {gμν} or keeping the volume of considered space invariant to volume change in a Tidal Weyl tensor {Rμν}).

The Einstein-Riemann tensor then relates Curvature Radius R to the Energy-Mass tensor E=Mc2 via the critical density as 8pG/c4=3Ho2VcriticalMcriticalc2/Mcriticalc4 = 3Ho2Vcritical/c2 = 3Vcritical/R2 as Curvature Radius R by the Hubble Law applicable say to a nodal Hubble Constant Ho = c/RHubble.

The cosmological field equations then can be expressed as the square of the nodal Hubble Constant and inclusive of a 'dark energy' terms often identified with the Cosmological Constant of Albert Einstein, here denoted LEinstein. Substituting the Einstein Lambda with the time differential for the square of nodal Hubble frequency as the angular acceleration acting on a quantized volume of space however; naturally and universally replaces the enigma of the 'dark energy' with a space inherent angular acceleration component, which can be identified as the 'universal consciousness quantum' directly from the standard cosmology itself.

The field equations so can be generalised in a parametrization of the Hubble Constant assuming a cyclic form, oscillating between a minimum and maximum value given by Ho=dn/dt for cycletime n=Hot and where then time t is the 4-vector timespace of Minkowski lightpath x=ct.
The Einstein Lambda then becomes then the energy-acceleration difference between the baryonic mass content of the universe and an inherent mass energy related to the initial condition of the oscillation parameters for the nodal Hubble Constant.

LEinstein = GoMo/R(n)2 - 2cHo/(n+1)3 = Cosmological Acceleration - Intrinsic Universal Milgröm Deceleration as: gmnL = 8pG/c4 Tmn - Gmn
then becomes Gmn + gmnL = 8pG/c4 Tmn and restated in a mass independent form for an encompassment of the curvature finestructures.

Energy Conservation and Continuity:

dE + PdV = TdS =0 (First Law of Thermodynamics) for a cosmic fluid and scaled Radius R=a.Ro; dR/dt = da/dt.Ro and d2R/dt2 = d2a/dt2.Ro

dV/dt = {dV/dR}.{dR/dt} = 4pa2Ro3.{da/dt}

dE/dt = d(mc2)/dt = c2.d{rV}/dt = (4pRo3.c2/3){a3.dr/dt + 3a2r.da/dt}

dE + PdV = (4pRo3.a2){rc2.da/dt + [ac2/3].dr/dt + P.da/dt} = 0 for the cosmic fluid energy-pressure continuity equation:

dr/dt = -3{(da/dt)/a.{r + P/c2}} .........................................................................................(1)

The independent Einstein Field Equations of the Robertson-Walker metric reduce to the Friedmann equations:

H2 = {(da/dt)/a}2 = 8pGr/3 - kc2/a2 + L/3 ...................................................................................(2)

{(d2a/dt2)/a} = -4pG/3{r+ 3P/c2} + L/3 ..................................................................................(3)

for scale radius a=R/Ro; Hubble parameter H = {da/dt)/a}; Gravitational Constant G; Density r; Curvature k ; light speed c and Cosmological Constant L.

Differentiating (2) and substituting (1) with (2) gives (3):

{2(da/dt).(d2a/dt2).a2 - 2a.(da/dt).(da/dt)2}/a4 = 8pG.(dr/dt)/3 + 2kc2.(da/dt)/a3 + 0 = (8pG/3)(-3{(da/dt)/a.{r + P/c2}} + 2kc2.(da/dt)/a3 + 0

(2(da/dt)/a).{(d2a/dt2).a - (da/dt)2}/a2 = (8pG/3){-3(da/dt)/a}.{r + P/c2} + 2{(da/dt)/a}.(kc2/a2) +0 2{(da/dt)/a}.{(d2a/dt2).a - (da/dt)2}/a2 = 2{(da/dt)/a}{-4pG.{r + P/c2} + (kc2/a2)} +0 with kc2/a2= 8pGr/3 +L/3 - {(da/dt)/a}2

d{H2}/dt = 2H.dH/dt = 2{(da/dt)/a}.dH/dt dH/dt = {[d2a/dt2]/a - H2} = {-4pG.(r + P/c2) + 8pGr/3 + L/3 -H2} = -4pG/3(r + 3P/c2) + L/3 - H2} = -4pG/3(r + 3P/c2) + L/3 - 8pGr/3 + kc2/a2 - L/3} = -4pG.(r + P/c2) + kc2/a2

dH/dt = -4pG{r+P/c2} as the Time derivative for the Hubble parameter H for flat Minkowski spacetime with curvature k=0

{(d2a/dt2).a - (da/dt)2}/a2 = -4pG{r + P/c2} + (kc2/a2) + 0 = -4pG{r + P/c2} + 8pGr/3 - {(da/dt)/a}2 + L/3
{(d2a/dt2)/a} = (-4pG/3){3r + 3P/c2- 2r} = (-4pG/3){r + 3P/c2} + L/3 = dH/dt + H2

dH/dt + 4pGr = - 4pGP/c2 .... (for V4/10D=[4p/3]RH3 and V5/11D=2p2RH3 in factor 3p/2)

areset = Rk(n)AdS/Rk(n)dS + ½ = n-SPnk-1+Pnk +½ Scalefactor modulation at Nk={[n-SPnk-1]/Pnk } = ½ reset coordinate
{dH/dt} = areset .d{Ho/T(n)}/dt = - Ho2(2n+1)(n+3/2)/T(n)2 for k=0

dH/dt + 4pGr = - 4pGP/c2

-Ho2(2n+1)(n+3/2)/T(n)2 + GoMo/{RH3(n/[n+1])3}{4p} = L(n)/{RH(n/[n+1])} + L/3 -2Ho2{[n+1]2-¼}/T[n]2 + GoMo/RH3(n/[n+1])3{4p} = L(n)/RH(n/[n+1]) + L/3 -2Ho2{[n+1]2-¼}/T(n)2 + 4p.GoMo/RH3(n/[n+1])3 = L(n)/RH(n/[n+1]) + L/3
For a scalefactor a=n/[n+1] = {1-1/[n+1]} = 1/{1+1/n}

L(n)/RH(n/[n+1]) = - 4pGP/c2 = GoMo/RH3(n/[n+1])3 -2Ho2/(n[n+1]2)
and L = 0
for -P(n) = L(n)c2[n+1]/4pGonRH =L(n)Hoc[n+1]/4pGon = Moc2[n+1]3/4pn3RH3 - Ho2c2/2pGon[n+1]2

For n=1.13242:............ -(+6.7003x10-11 J/m3)* = (2.12682x10-11 J/m3)* + (-8.82709x10-11 J/m3)*
Negative Dark Energy Pressure = Positive Matter Energy + Negative Inherent Milgröm Deceleration(cHo/Go)

The Dark Energy and the 'Cosmological Constant' exhibiting the nature of an intrinsic negative pressure in the cosmology become defined in the overall critical deceleration and density parameters. The pressure term in the Friedmann equations being a quintessence of function n and changing sign from positive to negative to positive as indicated. For a present measured deceleration parameter qdS=-0.5585, the DE Lambda calculates as -6.700x10-11 (N/m2=J/m3)*, albeit as a positive pressure within the negative quintessence.

In the early radiation dominated cosmology; the quintessence was positive and the matter energy dominated the intrinsic Milgröm deceleration from the Instanton n=nps to n=0.18023 (about 3.04 Billion years) when the quintessence vanished and including a Recombination epoch when the hitherto opaque universe became transparent in the formation of the first hydrogen atoms from the quark-lepton plasma transmuted from the X-L Boson string class HO(32) of the Inflaton epoch preceding the Quantum Big Bang aka the Instanton.

From the modular membrane duality for wormhole radius rps = lps/2p, the critical modulated Schwarzschild radius rss = 2plss = 2px1022 m* for lps = 1/lss and for an applied scalefactor a = n/[n+1] = lss/RH = {1-1/[n+1]}
for a n=Hot coordinate nrecombination = 6.259485x10-5 or about 6.259485x10-5(16.88 Gy) = 1.056601 Million years attenuated by exp{-hf/kT} = e-1 = 0.367879 to a characteristic cosmological time coordinate of 0.36788x1.056601 = 388,702 years after the Instanton nps.

The attenuation of the recombination coordinate then gives the cosmic temperature background for this epoch in the coordinate interval for the curvature radius R(n=2.302736x10-5) = 3.67894x1021 m* to R(n=6.259485x10-5) = 1022 m*. This radial displacement scale represents the size of a typical major galaxy in the cosmology; a galactic structure, which became potentialised in the Schwarzschild matter evolution and its manifestation in the ylemic prototypical first generation magnetar-neutron stars, whose emergence was solely dependent on the experienced cosmic temperature background and not on their mass distributions.

The temperature evolution of the Instanton can be written as a function of the luminosity L(n,T) with R(n)=RH(n/[n+1]) as the radius of the luminating surface L(nps,T(nps) = 6π2lps2.σ.Tnps4 = 2.6711043034x1096 Watts*, where σ = Stefan's Constant = 2π5k4/15h3c2 and as a product of the defined 'master constants' k, h, c2, π and 'e'.

L(n,T) = 3HoMo.c2/550n and for Temperature T(nps) ----------- T(nps) = 2.93515511x1036 Kelvin*.

T(n)4 = HoMoc2/(2p2sRH2[550n3/[n+1]2]) for T(n)4 = {[n+1]2/n3}HoMoc2/(2p2sRH2) = 18.1995{[n+1]2/n3} (K4/V)* for a temperature interval in using the recombination epoch coordinates T(n1=6.2302736x10-5) = 2945.42 K* to T(n2=6.259485x10-5) = 2935.11 K*

This manifests as a 'false vacuum' and as a temperature gradient, as a causation of the Big Bang Instanton on physical grounds. The metaphysical ground is the symmetry breaking from the source parity violation described in the birth and necessity of the Graviton to resymmetrize the UFoQR.

T(nps) of the singularity is 0.0389 or 3.89% of the pre-singularity.

So the POTENTIAL Temperature manifests as 3.89% in the KINETIC Temperature' which doubles in the Virial Theorem to 7.78% as 2KE + PE = 0: TEMPERATURE/T(nps)=7.544808988..x1037/2.93515511x1036=25.705=1/0.03890... Applying the actual VPE at the Instanton to this temperature gradient:

rVPE/rEMR = {4pEps/lps3}/{8p5Eps4/15h3c3} = 15/2p4 = 0.07599486.. = 1/12.9878.. indicating the proportionality EVPE/EEMR = 2Tps/Tpotential at the Instanton from the Inflaton as a original form of the vbirial theorem, staing the Kinetic Energy of the Instanton and the QBB Lambda to be twice the Potential Energy of the de Broglie wave matter Inflaton, then manifesting as the Mo/2MHubble = rHyper/2RHubble Schwarzschild mass cosmo-evolution.

Now reducing the timeinstanton tps=nps/Ho of 3.33x10-31 seconds by the Temperature Gradient in the Luminosity Function gives you the scalar Higgs Potential Maximum at a pre-singularity time of tHiggsPE=tps.T(nps)/TEMPERATURE=1.297x10-32 seconds.

This then extrapolates the Big Bang singularity backwards in Time to harmonise the equations and to establish the 'driving force of the vacuum' as potential scalar Higgs Temperature Field. All the further evolvement of the universe so becomes a function of Temperature and not of mass. The next big phasetransition is the attunement of the BOSONIC UNIFICATION, namely the 'singularity' temperature Tps=1.41x1020 K with the Luminosity function. This occurs at a normal time of 1.9 nanoseconds into the cosmology.

It is then that the universe as a unity has this temperature and so allows BOSONIC differentiation between particles. The INDIVIDUATED PHOTON of the mass was born then and not before, as the entire universe was a PHOTON as a macroquantised superstring up to then. The size of the universe at that time was that of being 1.14 metres across. Next came th

e electroweak symmetry breaking at 1/365 seconds and at a temperature of so 1015 Kelvin* and so it continued.
The lower dimensional lightpath x=ct in lightspeed invariance c=lf so becomes modular dualised in the higher dimensional lightpath of the tachyonic de Broglie Inflaton-Instanton Vdebroglie=c/nps of the Inflaton. {(2-n)(n+1)}3/n3 = VdS'/VdS ......(4.36038 for npresent) in the first completing Hubble cycle n3/(2-n)3 =VAdS/VdS' ................. (2.22379 for npresent) in the first completing Hubble cycle (n+1)3 = VAdS/VdS .....................(9.69657 for npresent) in the first completing Hubble cycle

rcritical = 3Ho2/8pGo {Sphere} and Ho2/4p2Go {Hypersphere-Torus in factor 3p/2} (constant for all n per Hubble cycle) rcritical = 3.78782x10-27 [kg/m3]* and 8.038003x10-28 [kg/m3]*
rdSVdS = rdS'VdS' = rAdSVAdS = rcriticalVHubble = MHubble = c2RH/2Go = 6.47061227x1052 kg* 