A Revision of the Friedmann Cosmology

1. 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 ρcritical = 3Ho2/8πG 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 Riemann 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 of the 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 8πG/c4=3Ho2Vcritical.Mcritical.c2/Mcritical.c4 = 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 ΛEinstein.
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 cycle time n=Hot and where then time t is the 4-vector time-space of Minkowski light-path 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.

ΛEinstein = GoMo/R(n)2 - 2cHo/(n+1)3 = Cosmological Acceleration - Intrinsic Universal Milgröm Deceleration as: gμνΛ = 8πG/c4 Tμν - Gμν

then becomes Gμν + gμνΛ = 8πG/c4 Tμν and restated in a mass independent form for an encompassment of the curvature fine structures.

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} = 4πa2Ro3.{da/dt}

dE/dt = d(mc2)/dt = c2.d{ρV}/dt = (4πRo3.c2/3){a3.dρ/dt + 3a2ρ.da/dt}

dE + PdV = (4πRo3.a2){ρc2.da/dt + [ac2/3].dρ/dt + P.da/dt} = 0 for the cosmic fluid energy-pressure continuity equation:

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

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

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

{(d2a/dt2)/a} = -4πG/3{ρ+ 3P/c2} + Λ/3 ..................................................................................(3)

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

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

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

(2(da/dt)/a).{(d2a/dt2).a - (da/dt)2}/a2 = (8πG/3){-3(da/dt)/a}.{ρ + P/c2} + 2{(da/dt)/a}.(kc2/a2) +0
2{(da/dt)/a}.{(d2a/dt2).a - (da/dt)2}/a2 = 2{(da/dt)/a}{-4πG.{ρ + P/c2} + (kc2/a2)} +0 with kc2/a2= 8πGρ/3 + Λ/3 - {(da/dt)/a}2

d{H2}/dt = 2H.dH/dt = 2{(da/dt)/a}.dH/dt
dH/dt = {[d2a/dt2]/a - H2} = {-4πG.(ρ+ P/c2) + 8πGρ/3 + Λ/3 -H2} = -4πG/3(ρ + 3P/c2) + Λ/3 - H2} = -4πG/3(ρ + 3P/c2) + Λ/3 - 8πGρ/3 + kc2/a2 - Λ/3} = -4πG.(ρ + P/c2) + kc2/a2

dH/dt = -4πG{ρ+P/c2} as the Time derivative for the Hubble parameter H for flat Minkowski space-time with curvature k=0

{(d2a/dt2).a - (da/dt)2}/a2 = -4πG{ρ+ P/c2} + (kc2/a2) + 0 = -4πG{ρ + P/c2} + 8πGρ/3 - {(da/dt)/a}2 + Λ/3

{(d2a/dt2)/a} = (-4πG/3){3ρ + 3P/c2- 2ρ} = (-4πG/3){ρ + 3P/c2} + Λ/3 = dH/dt + H2

For a scale factor a=n/[n+1] = {1-1/[n+1]} = 1/{1+1/n}​


dH/dt + 4πGρ = - 4πGP/c2 .... (for V4/10D=[4π/3]RH3 and V5/11D=2π2RH3 in factor 3π/2)

areset = Rk(n)AdS/Rk(n)dS + ½ = n-∑∏nk-1+∏nk

Scale factor modulation at Nk={[n-∑∏nk-1]/Πnk } = 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 + 4πGρ = - 4πGP/c2

-Ho2(2n+1)(n+3/2)/T(n)2 + GoMo/{RH3(n/[n+1])3}{4π} = Λ(n)/{RH(n/[n+1])} + Λ/3
-2Ho2{[n+1]2-¼}/T[n]2 + GoMo/RH3(n/[n+1])3{4π} = Λ(n)/RH(n/[n+1]) + Λ/3
-2Ho2{[n+1]2-¼}/T(n)2 + 4π.GoMo/RH3(n/[n+1])3 = Λ(n)/RH(n/[n+1]) + Λ/3

For a scale factor a=n/[n+1] = {1-1/[n+1]} = 1/{1+1/n}

Λ(n)/RH(n/[n+1]) = - 4πGP/c2 = GoMo/RH3(n/[n+1])3 -2Ho2/(n[n+1]2)

and Λ = 0
for -P(n) = Λ(n)c2[n+1]/4πGonRH = Λ(n)Hoc[n+1]/4πGon = Moc2[n+1]3/4πn3RH3 - Ho2c2/2πGon[n+1]2

For n=1.13271:............ - (+6.696373x10-11 J/m3)* = (2.126056x10-11 J/m3)* + (-8.8224295x10-11 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=-0.5586, the DE Lambda calculates as -6.696x10-11 (N/m2=J/m3)*, albeit as a positive pressure within the negative quintessence.

2. An expanding multi-dimensional super-membraned open and closed Universe

The expansion of the universe can be revisited in a reformulation of the standard cosmology model Lambda-Cold-Dark-Matter or LCDM in terms of a parametrization of the standard expansion parameters derived from the Friedmann equation, itself a solution for the Einstein Field Equations (EFE) applied to the universe itself.
A measured and observed flat universe in de Sitter (dS) 4D-spacetime with curvature k=0, emerges as the result of a topological mirror symmetry between two Calabi Yau manifolds encompassing the de Sitter space time in a multi timed connector dimension.
The resulting multiverse or brane world so defines a singular universe with varying but interdependent time cyclicities.

It is proposed, that the multiverse initiates cyclic periods of hyper acceleration or inflation to correlate and reset particular initial and boundary conditions related to a baryonic mass seedling proportional to a closure or Hubble mass to ensure an overall flatness of zero curvature for every such universe parallel in a membrane time space but co-local in its lower dimensional Minkowski space-time.

On completion of a 'matter evolved' Hubble cycle, defined in characteristic Hubble parameters; the older or first universal configuration quantum tunnels from its asymptotic Hubble Event horizon into its new inflaton defined universal configuration bounded by a new Hubble node.
The multidimensional dynamics of this quantum tunneling derives from the mirror symmetry and topological duality of the 11-dimensional membrane space connecting two Calabi Yau manifolds as the respective Hubble nodes for the first and the second universal configurations.

Parallel universes synchronise in a quantized protoverse as a function of the original lightpath of the Instanton, following not preceding a common boundary condition, defined as the Inflaton.
The initial conditions of the Inflaton so change as a function of the imposed cyclicity by the boundary conditions of the paired Calabi Yau mirror duality; where the end of a Instanton cycle assumes the new initial conditions for the next cycle of the Instanton in a sequence of Quantum Big Bangs.

The outer boundary of the second Calabi Yau manifold forms an open dS space-time in 12-dimensional brane space (F-Vafa 'bulk' Omnispace) with positive spheroidal curvature k=+1 and cancels with its inner boundary as a negatively curved k=-1 hyperbolic AdS space-time in 11 dimensions to form the observed 4D/10-dimensional zero curvature dS space-time, encompassed by the first Calabi Yau manifold.

A bounded (sub) 4D/10D dS space-time then is embedded in a Anti de Sitter (AdS) 11D-space-time of curvature k=-1 and where 4D dS space-time is compactified by a 6D Calabi Yau manifold as a 3-torus and parametrized as a 3-sphere or Riemann hypersphere.
The outer boundary of the 6D Calabi Yau manifold then forms a mirror duality with the inner boundary of the 11D Calabi Yau event horizon and images the positive curvature in 12D-F-Vafa space in a 'convex lense' effect of 11-dimensional M-Witten spacetime.


Rps = λps/2π as the wormhole radius of the Instanton as a conformally transformed Planck-Length Lp = √{Goh/2πc3} from the Inflaton.
The Schwarzschild metric for 2Lp = 2GoMp/c2 transforms a 3D Planck-length in the Planck-mass Mp = √{hc/2πGo} from the Planck-boson gravitational fine structure constant 1 = 2πGoMp2/hc.

The Schwarzschild metric for the Weyl-wormhole radius Rps then defines a hypermass Mhyper as the conformal mapping of the Planck-mass Mp as Mhyper = ½{Rps/Lp}Mp = ½{Rps/Lp}2.Mps and where Mps = Eps/c2 = hfps/c2 = kTps/c2 in fundamental expressions for the energy of Abba-Eps as one part of the super membrane Eps.Ess in physical quantities of mass m, frequency f and temperature T.

c2 and h and k are fundamental constants of nature obtained from the initializing algorithm of the Mathimatia and are labeled as the 'square of lightspeed c' and 'Planck's constant h' and 'Stefan-Boltzmann's constant k' respectively. The complementary part of super membrane EpsEss is EssBaab. Eps-Abba is renamed as 'Energy of the Primary Source-Sink' and Ess-Baab is renamed as 'Energy of the Secondary Sink-Source'. The primary source-sink and the primary sink-source are coupled under a mode of mirror-inversion duality with Eps describing a vibratory and high energy micro-quantum quantum entanglement with Ess as a winding and low energy macro quantum energy. It is this quantum entanglement, which allows Abba to become part of Universe in the encompassing energy quantum of physicalized consciousness, defined in the magnetopolar charge.

The combined effect of the applied Schwarzschild metric then defines a Compton Constant to characterize the conformal transformation as: Compton Constant h/2πc = MpLp = MpsRps. Quantum gravitation now manifests the mass differences between Planck-mass Mp and Weyl mass Mps. The Black Hole physics had transformed Mp from the definition of Lp; but this transformation did not generate Mps from Rps, but rather hypermass Mhyper, differing from Mps by a factor of ½{Rps/Lp}2.

Every Inflaton defines three Hubble nodes or timespace mirrors; the first being the 'singularity - wormhole' configuration; the second the nodal boundary for the 4D/10D dS space-time and the third the dynamic lightpath bound for the Hubble Event horizon in 5D/11D AdS time-space.
The completion of a 'de Broglie wave matter' evolution cycle triggers the Hubble Event Horizon as the inner boundary of the time-space mirrored Calabi Yau manifold to quantum tunnel onto the outer boundary of the space-time mirrored Calabi Yau manifold in a second universe; whose inflaton was initiated when the light-path in the first universe reached its second Hubble node.

For the first universe, the three nodes are set in time-space as {3.3x10-31 s; 16.88 Gy; 3.96 Ty} and the second universe, time shifted in t1=to+t with to=1/Ho has a nodal configuration {to+1.4x10-33; to+3,957 Gy; to+972.7 Ty}; the latter emerging from the time-space as the instanton at time marker to.
A third universe would initiate at a time coordinate t2=to+t1+t as {1/Ho+234.472/Ho +5.8x10-36 s; to+t1+972.7 Ty; to+t1+250,223 Ty}; but as the second node in the second universe cannot be activated by the lightpath until the first universe has reached its 3.96 trillion year marker (and at a time for a supposed 'heat death' of the first universe due to exhaustion of the nuclear matter sources); the third and following nested universes cannot be activated until the first universe reaches its n=1+234.472=235.472 time-space coordinate at 3,974.8 billion years from the time instanton aka the Quantum Big Bang.

For a present time-space coordinate of npresent=1.13271 however; all information in the first universe is being mirrored by the time-space of the AdS space-time into the dS space-time of the second universe at a time frame of t = t1-to = 19.12 - 16.88 = 2.24 billion years and a multi-dimensional time interval characterizing the apparent acceleration observed and measured in the first universe of the Calabi Yau manifold compressed or compactified flat dS Minkowski cosmology. The solution to the Dark Energy and Dark Matter question of a 'missing mass' cosmology is described in this discourse and rests on the evolution of a multiverse in matter.




Yn = RHubble/rWeyl = 2πRHubbleWeyl = ωWeyl/Ho = 2πnWeyl = nps/2π = 1.003849x1049

2nd Inflaton/Quantum Big Bang redefines for k=1: RHubble(1) = n1RHubble = c/Ho(1) = (234.472)RHubble = 3.746x1028 m* in 3.957 Trillion Years for critical nk
3rd Inflaton/Quantum Big Bang redefines for k=2: RHubble(2) = n1n2RHubble = c/Ho(2) = (234.472)(245.813)RHubble = 9.208x1030 m* in 972.63 Trillion Years for critical nk
4th Inflaton/Quantum Big Bang redefines for k=3: RHubble(3) = n1n2n3RHubble = c/Ho(3) = (57,636.27)(257.252)RHubble = 2.369x1033 m* in 250.24 Quadrillion Years for critical nk
5th Inflaton/Quantum Big Bang redefines for k=4: RHubble(4) = n1n2n3n4RHubble = c/Ho(4) = (14,827,044.63)(268.785)RHubble = 6.367x1035 m* in 67.26 Quintillion Years for critical nk
(k+1)th Inflaton/Quantum Big Bang redefines for k=k: RHubble(k) = RHubble Π nk = c/Ho Π nk

nk = ln{ωWeylRHubble(k)/c}/lnY = ln{ωWeyl/Ho(k)}/lnY

n1 = 234.471606...
n2 = 245.812422...
n3 = 257.251394...
n4 = 268.784888...

Dark Energy DE-Quintessence Λk Parameters:

A general dark energy equation for the kth universe (k=0,1,2,3,...) in terms of the parametrized Milgröm acceleration A(n); comoving recession speed V(n) and scale factored curvature radius R(n):

Λk (n) = GoMo/Rk(n)2 - 2cHo(Πnk)2/{n-ΣΠnk-1+Πnk)3} for negative Pressure Pk = -Λk(n)c2/4πGoRk

= {GoMo(n-ΣΠnk-1+Πnk)2/{(Πnk)2.RH2(n-ΣΠnk-1)2} - 2cHo(Πnk)2/{n-ΣΠnk-1+Πnk)3}

Λo = GoMo(n+1)2/RH2(n)2 - 2cHo/(n+1)3
Λ1 = GoMo(n-1+n1)2/n12RH2(n-1)2 - 2cHon12/(n-1+n1)3
Λ2 = GoMo(n-1-n1+n1n2)2/n12n22RH2(n-1-n1)2 - 2cHon12n22/(n-1-n1+n1n2)3

Lambda-DE-Quintessence Derivatives:

Λk'(n) = d{Λk}/dn =
{GoMo/Πnk2RH2}{2(n-ΣΠnk-1+Πnk).(n-ΣΠnk-1)2 - 2(n-ΣΠnk-1).(n-ΣΠnk-1+Πnk)2}/{(n-ΣΠnk-1)4} - {-6cHo(Πnk)2}/(n-ΣΠnk-1+Πnk)4

= {-2GoMo/ΠnkRH2}(n-ΣΠnk-1+Πnk)/(n-ΣΠnk-1)3 + {6cHo(Πnk)2}/(n-ΣΠnk-1+Πnk)4

= {6cHo(1)2}/{(n-0+1)4} - {2GoMo/1.RH2}{(n-0+1)/(n-0)3}........................................ for k=0
= {6cHo(1.n1)2}/{(n-1+n1)4} - {2GoMo/n1.RH2}{(n-1+n1)/(n-1)3}................................ for k=1
= {6cHo(1.n1.n2)2}/{(n-1-n1+n1.n2)4} - {2GoMo/n1n2.RH2}{(n-1-n1+n1n2)/(n-1-n1)3}...... for k=2

For k=0; {GoMo/3c2RH} = constant = n3/[n+1]5
for roots nΛmin = 0.23890175.. and nΛmax = 11.97186...
{GoMo/2c2RH} = constant = [n]2/[n+1]5

for Λo-DE roots: n+/- = 0.1082331... and n-/+ = 3.40055... for asymptote Λ0∞ = GoMo/RH2 = 7.894940128...x10-12 (m/s2)*

For k=1; {GoMo/3n13c2RH} = constant = [n-1]3/[n-1+n1]5 = [n-1]3/[n+233.472]5
for roots nΛmin = 7.66028... and nΛmax = 51,941.9..
{GoMo/2n14c2RH} = constant = [n-1]2/[n-1+n1]5 = [n-1]2/[n+233.472]5
for Λ1-DE roots: n+/- = 2.29966... and n-/+ = 7,161.518... for asymptote Λ1∞ = GoMo/n12RH2 = 1.43604108...x10-16 (m/s2)*

For k=2; {GoMo/3n13n23c2RH} = constant = [n-1-n1]3/[n-1-n1+n1n2]5 = [n-235.472]3/[n+57,400.794]5
for roots nΛmin = 486.7205 and nΛmax = 2.0230105x108
{GoMo/2n14n24c2RH} = constant = [n-1-n1]2/[n-1-n1+n1n2]5 = [n-235.472]2/[n+57,400.794]5
for Λ2-DE roots: n+/- = 255.5865... and n-/+ = 1.15382943...x107 for asymptote Λ2∞ = GoMo/n12n22RH2 = 2.37660590...x10-21 (m/s2)*

For k=3; {GoMo/3n13n23n33c2RH} = constant = [n-1-n1-n1n2]3/[n-1-n1-n1n2+n1n2n3]5 = [n-57,871.74]3/[n+1.47691729x107]5
for roots nΛmin = 67,972.496 and nΛmax = 8.3526797...x1011
{GoMo/2n14n24n34c2RH} = constant = [n-1-n1-n1n2]2/[n-1-n1-n1n2+n1n2n3]5 = [n-57,871.74]2/[n+1.47691729x107]5
for Λ3-DE roots: n+/- = 58,194.1... and n-/+ = 1.9010262...x1010 for asymptote Λ3∞ = GoMo/n12n22n32RH2 = 3.59120049...x10-26 (m/s2)*

and where

Πnk=1=no and Πnk-1=0 for k=0
with Instanton/Inflaton resetting for initial boundary parameters

Λo/adeBroglie = {GoMo/Rk(n)2}/ΠnkRHfps2
= {GoMo(n-ΣΠnk-1+Πnk)2}/{[Πnk]2.RH2(n-ΣΠnk-1)2(ΠnkRHfps2)} = (Πnk)½Ωo

for Instanton-Inflaton Baryon Seed Constant Ωo = Mo*/MH* = 0.02803 for the kth universal matter evolution

k=0 for Reset n=nps=Hot and Λo/adeBroglie = GoMo(nps+1)2/{RH3nps2(fps2)} = GoMo/RHc2 = Mo/2MH = ½Ωo
k=1 for Reset n=1+nps and Λo/adeBroglie = GoMo(1+nps-1+n1)2/{[n1]2.RH3(1+nps-1)2(n1fps2)} = Mo/2n1MH = Mo/2MH* = ½Ωo*
k=2 for Reset n=n1+1+nps and Λo/adeBroglie = GoMo(n1+1+nps-1-n1+n1n2)2/{[n1n2]2.RH3(n1+1+nps-1-n1)2(n1n2fps2)} = ½Ωo**
k=3 for Reset n=n1n2+n1+1+nps and Λo/adeBroglie = GoMo(n1n2+n1+1+nps-1-n1-n1n2+n1n2n3)2/{[n1n2n3]2.RH3(n1n2+n1+1+nps-1-n1-n1n2)2(n1n2n3fps2)} = ½Ωo***

with nps = 2πΠnk-1.Xnkps/RH = Hotps = Ho/fps = ctps/RH and RH=2GoMH/c2


tk = t - (1/Ho)ΣΠnk-1 for no=1 and No=n



Rk(n) = ΠnkRH(n-ΣΠnk-1)/{n-ΣΠnk-1+Πnk}

.....= RH(n/[n+1]) = n1RH(N1/[N1+1]) = n1n2RH(N2/[N2+1]) =.....

Vk(n) = dRk(n)/dt = c{Πnk}2/{n-ΣΠnk-1+Πnk}2

.....= c/[n+1]2 = c/[N1+1]2 = c/[N2+1]2 =.....
.....= c/[n+1]2 = c(n1)2/[n-1+n1]2 = c(n1n2)2/[n-1-n1+n12n22]2 =.....

Ak(n) = d2Rk(n)/dt2 = -2cHo(Πnk)2/(n-ΣΠnk-1+Πnk)3

.....= -2cHo/(n+1)3 = -2cHo/n1(N1+1)3 = -2cHo/n1n2(N2+1)3=.....
..... = -2cHo/[n+1]3 = -2cHo{n1}2/[n-1+n1]3 = -2cHo(n1n2)2/[n-1-n1+n1n2]3 =.....

GoMo is the Gravitational Parameter for the Baryon mass seed; Curvature Radius RH = c/Ho in the nodal Hubble parameter Ho and c is the speed of light

Hubble Parameters:

H(n)|dS = {Vk(n)}/{Rk(n)} = {c[Πnk]2/[n-ΣΠnk-1+Πnk]2}/{Πnk.RH[n-ΣΠnk-1]/(n-ΣPnk-1+Πnk)} = ΠnkHo/{[n-ΣΠnk-1][n-ΣΠnk-1+Πnk]}

H(n)|dS = ΠnkHo/{[n-ΣΠnk-1][n-ΣΠnk-1+Πnk]}

.....= Ho/{[n][n+1]}=Ho/T(n) = n1Ho/{[n-1][n-1+n1]} = n1n2Ho/{[n-1-n1][n-1-n1+n1n2]} =..... for dS

H(n)'|dS = Ho/[n-ΣΠnk-1] for oscillating H'(n) parameter between nodes k and k+1 ||nps+ΣΠnk-1 - ΣΠnk||

H(n)|AdS = H(n)'|AdS = {Vk(n)}/{Rk(n)} = c/{RH(n-ΣΠnk-1)}

H(n)|AdS = H(n)' = Ho/(n-ΣΠnk-1)

.....= Ho/n = Ho/(n-1) = Ho/(n-1-n1) =..... for AdS

For initializing scale modulation Rk(n)Ads/Rk(n)dS + ½ = ΠnkRH(n-ΣΠnk-1)/{ΠnkRH(n-ΣΠnk-1)/(n-ΣΠnk-1+Πnk)} + ½Πnk = {n - ΣΠnk-1 + Πnk + ½} reset coordinate

dH/dt = (dH/dn)(dn/dt) = -Πnk.Ho2{(2n-2ΣΠnk-1+Πnk)(n-ΣΠnk-1+Πnk+½Πnk)}/{n2-2nΣΠnk-1+(ΣΠnk-1)2+Πnk[n-ΣΠnk]}2
= -2ΠnkHo2{[n - ΣΠnk-1 + Πnk]2 - ¼ΣΠnk2}/{(n-ΣΠnk-1)(n-ΣΠnk-1+Πnk)}2

dH/dt|dS = -2ΠnkHo2{[n - ΣΠnk-1 + Πnk]2 - ¼(ΣΠnk)2}/{(n-ΣΠnk-1)(n-ΣΠnk-1+Πnk)}2

.....= -2Ho2([n+1]2-¼)/{n[n+1]}2 = -2n1Ho2{[n-1+n1]2-¼n12}/{[n-1][n-1+n1]}2 = -2n1n2Ho2{[n-1-n1+n1n2]2-¼n12n22}/{[n-1-n1][n-1-n1+n1n2]}2 =.....

dH/dt = (dH/dn)(dn/dt) = -Hoc/{(RH(n-ΣΠnk-1)2} = -Ho2/{n-ΣΠnk-1}2 for AdS

dH/dt|AdS = -Ho2/{n-ΣΠnk-1}2

.....= -Ho2/n2 = Ho2/(n-1)2 = -Ho2/(n-1-n1)2 = .....

dH/dt + 4πGoρ = - 4πGoP/c2

dH/dt + 4πGoMo/Rk(n)3 = Λk(n)/Rk(n) = - 4πGoP/c2 = GoMo/Rk(n)3 - 2(Πnk)Ho2/{(n-ΣΠnk-1)(n-ΣΠnk-1+Πnk)2} for dS with

{-4π}P(n)|dS = Moc2/Rk(n)3 - 2Πnk(Hoc)2/{Go(n-ΣΠnk-1)(n-ΣΠnk-1+Πnk)2} = Moc2(n-ΣΠnk-1+Πnk)3/{Πnk.RH(n-ΣΠnk-1)}3 - 2ΠnkHo2c2/{Go(n-ΣΠnk-1)(n-ΣΠnk-1+Πnk)2}

Λk(n)/Rk(n) = -4πGoP/c2 = GoMo/Rk(n)3 - dH/dt = GoMo/{RH(n-ΣΠnk-1)}3 - Ho2/{n-ΣΠnk-1}2 for AdS with

{-4π}P(n)|AdS = Moc2/Rk(n)3 - (Hoc)2/{Go(n-ΣΠnk-1)2} = Moc2/{RH(n-ΣΠnk-1)}3 - Ho2c2/{Go(n-ΣΠnk-1)2}

Deceleration Parameters:

qAdS(n) = -Ak(n)Rk(n)/Vk(n)2 = -{(-2cHo[Πnk]2)/(n-ΣΠnk-1+Πnk)3}{ΠnkRH(n-ΣΠnk-1)/(n-ΣΠnk-1+Πnk)}/{[Πnk]2c/(n-ΣΠnk-1+Πnk)}2 = 2(n-ΣΠnk-1)/Πnk

qAdS+dS(n) = 2(n-ΣΠnk-1)/Πnk

qdS(n) = 1/qAdS+dS(n) - 1 = Πnk/{2[n-ΣΠnk-1} - 1

with Ak(n)=0 for AdS in areset = Rk(n)AdS/Rk(n)dS + ½ = {RH(n-ΣΠnk-1)}/{RH(n-ΣΠnk-1)/(n-ΣΠnk-1+1)} + ½ = n-ΣΠnk-1+1+½

Scalefactor modulation at Nk = {n-ΣΠnk-1}/Πnk = ½ reset coordinate

.....= 2n = 2(n-1)/n1 = 2(n-1-n1)/(n1n2) = 2(n-1-n1-n1n2)/(n1n2n3) = ..... for AdS

.....= 1/{2n} -1 = n1/{2[n-1]} -1 = n1n2/{2(n-1-n1)} -1 = n1n2n3/{2(n-1-n1-n1n2)} -1 = ..... for dS

Dark Energy Initiation for qdS=1 with qAdS=1

k=0 for n = ½ = 0.50000 for qdS=0 with qAdS=1
k=1 for n = ½n1+1 = 118.236.. for qdS=0 with qAdS=1
k=2 for n = ½n1n2+n1+1 = 29,053.605.. qdS=0 with qAdS=1
k=3 for n = ½n1n2n3+n1n2+n1+1 = 7,471,394.054.. qdS=0 with qAdS=1


T(n) ={Moc2/(1100σπ2.Rk(n)2.tk)} and for tk = (n-ΣΠnk-1)/Ho

Tk(n) ={HoMoc2(n-ΣΠnk-1+Πnk)2/[1100σπ2.RH2.(n-ΣΠnk-1)3]}
={(Ho3Mo(n-ΣΠnk-1+Πnk)2)/[1100σπ2(n-ΣΠnk-1)3]} = {18.199(n-ΣΠnk-1+Πnk)2/(n-ΣΠnk-1)3}

T(n) .....= {18.2[n+1]2/n3} ={18.2[n-1+n1]2/(n-1)3} = {18.2[n-1-n1+n1n2]2/(n-1-n1)3} =.....

Comoving Redshift:

z + 1 = √{(1+v/c)/(1-v/c)} = √{([n-ΣΠnk-1+Πnk]2+[Πnk]2)/([n-ΣΠnk-1+Πnk]2-[Πnk]2)} =
√{([n-ΣΠnk-1]2+2Πnk(n-ΣΠnk-1)+2(Πnk)2)/([n-ΣΠnk-1]2+2Πnk(n-ΣΠnk-1)} = √{1 + 2(Πnk)2/{(n-ΣΠnk-1)(n-ΣΠnk-1+2Πnk)}

z+1 = √{ 1 + 2/{[n2-2nΣΠnk-1 +(ΣΠnk-1)2+2n-2ΣΠnk-1} = √{1+2/{n(n+2-2ΣΠnk-1) + ΣΠnk-1(ΣΠnk-1-2)}}

....= √{1+2/(n[n+2])} = √{1+2/([n-1][n-1+2n1])} = √{1+2/([n-1-n1][n-1-n1+2n1n2])} =......

Baryon-Dark Matter Saturation:

ΩDM = 1-ΩBM until Saturation for BM-DM and Dark Energy Separation

ρBM+DM/ρcritical = ΩoY{[n-ΣΠnk-1]/Πnk}/{(n-ΣΠnk-1)/(n-ΣΠnk-1+Πnk)}3 = MoY{[n-ΣΠnk-1]/Πnk}/{ρcriticalRk(n)3}

Baryon Matter Fraction ΩBM = ΩoY{Nk} = Ωo.Y{[n-ΣΠnk-1]/Πnk}

Dark Matter Fraction ΩDM = ΩoY{[n-ΣΠnk-1]/Πnk}{1-{(n-ΣΠnk-1)/(n-ΣΠnk-1+Πnk)}3/{(n-ΣΠnk-1)/(n-ΣΠnk-1+Πnk)}3 = ΩoY{[n-ΣΠnk-1]/Πnk}{(n-ΣΠnk-1+Πnk)3-(n-ΣΠnk-1)3}/{n-ΣΠnk-1}3
= ΩoY{[n-ΣΠnk-1]/Πnk}{(1+Πnk/[n-ΣΠnk-1])3 -1} = ΩBM{(1+Πnk/[n-ΣΠnk-1])3 -1}

Dark Energy Fraction ΩDE = 1- ΩDM - ΩBM = 1 - ΩBM{(1+Πnk/[n-ΣΠnk-1])3}

ΩBM=constant=0.0553575 from Saturation to Intersection with Dark Energy Fraction

ΩoY{[n-ΣΠnk-1]/Πnk} = ρBM+DMRk(n)3/MH = [Nk]3/[Nk+1]3 = {(n-ΣΠnk-1)/(n-ΣΠnk-1+Πnk)}3 = Rk(n)3/VH = VdS/VAdS
for ρBM+DM = MH/RH3 = ρcritical and for Saturation at Ni = 6.541188... = constant ∀ Ni

(Mo/MH).Y{[n-ΣΠnk-1]/Πnk} = {(n-ΣΠnk-1)/(n-ΣΠnk-1+Πnk)}3 with a Solution for f(n) in Newton-Raphson Root Iteration and first Approximation x0

xk+1 = xk - f(n)/f'(n) = xk - {(Mo/MH).Y{[n-∑∏nk-1]/Πnk} - (n-∑∏nk-1)/(n-ΣΠnk-1+Πnk)3}/{(Mo/MH).[lnY]Y{[n-ΣΠnk-1]/Πnk} - 3(n-ΣΠnk-1)2/(n-ΣΠnk-1+Πnk)4}

x1 = x0 - {(Mo/MH).Y[n] - (n/n+1)3}/{(Mo/MH).[lnY]Y[n] - 3n2/[n+1]4}
= x0 - {(Mo/MH).Y{N0} - (N0)3/(N0+1)3}/{(Mo/MH).[lnY]Y{N0} - 3(N0)2/1(N0+1)4}
x1 = x0 - {(Mo/MH).Y{[n-1]/n1} - (n-1)3/(n-1+n1)3}/{(Mo/MH).[lnY]Y{[n-1]/n1} - 3(n-1)2/(n-1+n1)4}
= x0 - {(Mo/MH).Y{N1} - (N1)3/(N1+1)3}/{(Mo/MH).[lnY]Y{N1} - 3(N1)2/n1(N1+1)4}
x1 = x0 - {(Mo/MH).Y{[n-1-n1]/n1n2} - (n-1-n1)3/(n-1-n1+n1n2)3}/{(Mo/MH).[lnY]Y{[n-1-n1]/n1n2} - 3(n-1-n1)2/(n-1-n1+n1n2)4}
= x0 - {(Mo/MH).Y{N2} - (N2)3/(N2+1)3}/{(Mo/MH).[lnY]Y{N2} - 3(N1)2/n1n2(N2+1)4}

n = 1.N0 = Ni = 6.541188....⇒ Ni ∀i for ∏nk = n0 = 1
n = n1N1+1 = (234.472)(6.541188...)+1 = 1534.725.... for ∏nk = n0n1 = n1
n = n1n2N2+1+n1 = (234.472x245.813)(6.541172)+1+234.472 = 377,244.12.... for ∏nk = n0n1n2 = n1n2
n = n1n2n3N3+1+n1+n1n2 = (234.472x245.813x257.252)(6.541172)+1+234.472+(234.472x245.813) = 97,044,120.93.... for ∏nk = n0n1n2n3= n1n2n3

Baryon-Dark Matter Intersection:

Nk=√2 for n = √2.Πnk + ΣΠnk-1
n0 = 1.√2 + 0 = no
n1 = n1√2 + 1 = 332.593 = n1√2 + 1
n2 = n1n2√2 + 1 + n1 = 81,745.461
n3 = n1n2n3√2 + 1 + n1 +n1n2 = 21,026,479.35



Hypermass Evolution:

Yk{(n-ΣΠnk-1)/Πnk} = 2πΠnk.RHps = Πnk.RH/rps = ΠnkMH*k/mH*k for MH = c2RH/2Go and mH = c2rps/2Go

Hypermass MHyper = mH.Yk{(n-ΣΠnk-1)/Πnk}

.....= Yn = Y([n-1]/n1) = Y([n-1-n1]/n1n2) =.....

k=0 for MHyper = MH = 1.MH = mH.Y{(n)} with n = 1.{ln(2π/nps)/lnY} = n1
= 234.472

k=1 for MHyper = n1.MH = MH* = mH.Y{(n-1)/n1} with n = [1] + n1.{ln(2πn1/nps)/lnY} = [1] + n1n2
= 1 + 234.472x245.812 = 57,637.03

k=2 for MHyper = n1n2.MH = MH** = mH.Y{(n-1-n1)/n1n2} with n = [1 + n1] + n1n2.{ln(2πn1n2/nps)/lnY} = [1 + n1] + n1n2n3
= 235.472 + 234.472x245.812x257.251 = 14,827,185.4

k=3 for MHyper = n1n2n3.MH = MH*** = mH.Y{(n-1-n1-n1n2)/n1n2n3} with n = with n = [1 + n1 + n1n2] + n1n2n3.{ln(2πn1n2n3/nps)/lnY} = [1 + n1 + n1n2] + n1n2n3n4
= 57,871.74 + 234.472x245.812x257.251x268.785 = 3,985,817,947.8

The Friedmann's acceleration equation and its form for the Hubble time derivative from the Hubble expansion equation substitutes a curvature k=1 and a potential cosmological constant term; absorbing the curvature term and the cosmological constant term, which can however be set to zero if the resulting formulation incorporates a natural pressure term applicable to all times in the evolvement of the cosmology.

Deriving the Instanton of the 4D-dS Einstein cosmology for the Quantum Big Bang (QBB) from the initial-boundary conditions of the de Broglie matterwave hyper expansion of the Inflaton in 11D AdS then enables a cosmic evolution for those boundary parameters in cycle time n=Hot for a nodal 'Hubble Constant' Ho=dn/dt as a function for a time dependent expansion parameter H(n)=Ho/T(n)=Ho/T(Hot).

It is found, that the Dark Matter (DM) component of the universe evolves as a function of a density parameter for the coupling between the inflaton of AdS and the instanton of dS space times. It then is the coupling strength between the inflationary AdS brane epoch and the QBB dS boundary condition, which determines the time evolution of the Dark Energy (DE).
Parametrization of the expansion parameter H(n) then allows the cosmological constant term in the Friedmann equation to be merged with the scalar curvature term to effectively set an intrinsic density parameter at time instantenuity equal to Λ(n) for ΛpsQBB=GoMo/λps2 and where the wavelength of the de Broglie matter wave of the inflaton λps decouples as the Quantum Field Energy of the Planck Boson String in AdS and manifests as the measured mass density of the universe in the flatness of 4D Minkowski spacetime.

3. Temperature Evolution in the Multiverse

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 = λps/2π, the critical modulated Schwarzschild radius rss = 2πλss = 2πx1022 m* for λps = 1/λss
and for an applied scalefactor a = n/[n+1] =λss/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) = 6π2λps2.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/(2π2σRH2[550n3/[n+1]2]) for
T(n)4 = {[n+1]2/n3}HoMoc2/(2π2σRH2[550]) = 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:


Applying the actual VPE at the Instanton to this temperature gradient:

ρVPEEMR = {4πEpsps3}/{8π5Eps4/15h3c3} = 15/2π4 = 0.07599486.. = 1/12.9878.. indicating the proportionality EVPE/EEMR = kTps/kTEMR = 2Tps/Tpotential at the Instanton from the Inflaton as a original form of the virial theorem, stating 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 phase transition 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 the 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

ρcritical = 3Ho2/8πGo {Sphere} and Ho2/4π2Go {Hypersphere-Torus in factor 3π/2} (constant for all n per Hubble cycle)
ρcritical = 3.78782x10-27 [kg/m3]* and 8.038003x10-28 [kg/m3]*

ρdSVdS = ρdS'VdS' = ρAdSVAdS = ρcriticalVHubble = MHubble = c2RH/2Go = 6.47061227x1052 kg*