A Newtonian-Einstein-De Sitter Universe in Cosmological Mirror-Supersymmetry

Progress-Status: 35%

by Hans Schatten

{For Deborah Martina, (from 8th January 1982 to 2009-01-08)}
On behalf of TonyB. http://www.tonyb.freeyellow.com/

Abstract:

The introduction of a characteristic length scale λ_o into the kinematics of the Poincare-Lorentz group of a descriptive flat Minkowski spacetime renders the latter a curved de Sitter spacetime with a cosmological lambda parameter Λ_o relating this length scale as a Planck-Length (L_P=√[hG/2πc³] ) and where  λ_o becomes a transformed minimum configuration for the local description of classically geometric General Relativity and its derived dynamics.

Expressed as energy density, the lambda function Λ(n=H_ot) then naturally supplements the gravitational energy-stress-momentum tensor in the standard FRLW-cosmology to reformulate the Hubble Law H(t) as a function of the expansion parameter a(t) in comoving reference frames R(t)=a(t)R_o relative to a cosmological displacement scale given by the Hubble horizon R_o=R_Hubble.

The characteristic length scale L in Minkowski space of a local subrealm of de Sitter space then assumes a local lambda parameter Λ_L as the conformal mapping from the de Sitter curvature space into the local Minkowski flatness in the form of L=λ²/L_P with Λ_L~1/L² for  λ relating the local energy density, say in a de Broglie lambda  λ_dB=h/mv or a Compton wavelength  λ_C= h/mc=hc/E.

The Hubble radius R_o so depicts a limiting curvature scale for the Friedmann cosmology, being conformal to de Sitter space in the factor L=R_o=λ²/L_P with λ=√[R_oL_P] ~ 1.2x10^-4 m, that is a bacterial-microbial scale on the biological phenomenological level and characterised by a vanishing local curvature of order 10^-52~1/R_o².

The so called 'Dark Energy' then manifests as intrinsic part of the spacetime geometry as the ratio of the trace of the energy-momentum tensor of the Friedmann cosmology to its static Schwarzschild metric in curved de Sitter spacetime encompassing it.

One major consequence for the intrinsic de Sitter curvature becomes the 'Dark Energy' manifesting in a differential of acceleration between inertial and noninertial frames of references. The local solar system is a comoving part of the Friedmann expansion into de Sitter spacetime and so becomes a non-inertial comoving reference frame relative to the inertial and static reference frame of de Sitter spacetime. 
This then leads to a logical explanation for the Pioneer anomaly measured for the last decade or so.

As the flat Friedmann universe requires a critical density for its flatness, which is supplied in the 'Dark Energy' in the form of the de Sitter Lambda encompassing the Minkowski universe in positive curvature and closure; the 'missing mass' in the open Friedmann cosmology readjusts the critical mass in the formulations describing the 'Dark Energy'.
This manifests in the 'higher dimensional' curved de Sitter spacetime forming an acceleration gradient relative to the 'lower dimensional' flat Minkowski spacetime.
Considering the de Sitter cosmology to be 'background'-inertial then results in the Minkowski spacetime to be rendered noninertial by the experience of  a 'de Sitter' force or pressure.
It shall be shown, that the present de Sitter lambda in Minkowski spacetime is 2.8% its value in de Sitter spacetime, resulting in the formulations: Omega+Milgrom=Lambda to become {2.807x10^-11 -1.162x10^-10 = -8.812x10^-11  } for the Friedmann cosmology and {9.989x10^-10 -1.162x10^-10 =+8.827x10^-10} for the de Sitter cosmology in acceleration units for the present time.
The flat Friedmann universe for a zero cosmological constant so balances the Omega deceleration with the Intrinsic (Milgrom) acceleration and omits the de Sitter component, which acts in addition to the Omega for the present time to balance the Milgrom in opposite direction. 

Introduction:
This paper shall attempt to refine the contemporary standard model of the Big Bang cosmology in examining various initial- and boundary conditions of its cosmogony.

The model for this cosmogony - the universe's absolute beginnings within parameters of space and time and matter - described in this paper; shall enable the prevailing paradigm to dramatically revise and finetune its cosmological parameters and solve the 'riddles' of the 'horizon problem'; the 'lambda problem'; the 'monopole problem' and the 'flatness problem' in the determination of the responsible boundary- and initial conditions for the stated cosmogony.

The great 'culprit' in the present Friedmann-Robertson-LeMaitre-Walker (FRLW) cosmological description and solution for Einstein's Field equations, is the Hubble-Law in the form of its interpretation through the cosmological expansion parameter 'a'.

The expansion parameter is defined to describe the dynamical evolution of the universe in the rate of its expansion and as a dimensionless quantity scaling a curvature radius R to some comoving reference scale R_o  in defining a(t)=R(t)/R_o.

From this the Hubble-Law develops in the ratio of the generalised Hubble-Frequency: H(t)=(dR/dt)/R.

The FRLW-Equation then inserts this expression:  [H(t)]²=8πGρ(t)/3 - kc² + Λc²/3   into the Einsteinian Tensor equation of General Relativity:
 


and seeks to describe the equations of motion for the universe in terms of H(t) and so the expansion parameter a(t), and where the density ρ(t) incorporates all energy densities, including radiation pressure and neutrino distributions.

The gravitational effects produced by a given mass are described in General Relativity by 16 coupled hyperbolic-elliptic nonlinear partial differential equations, called the Einstein field equations. As result of the symmetry of tensorial components, the actual number of equations reduces to 10, although there are an additional four differential identities (the Bianchi identities), which must be satisfied to render the coordinate descriptions selfconsistent.
The nonlinearity of the Einstein field equations stems from the fact that masses affect the very geometry of the space in which they dwell.  Mass is said to fundamentally curve the geometry of spacetime, and the geometry of spacetime in turn 'forces' mass how to move. 

The Tensor components are the Ricci curvature tensor R_μν; the scalar curvature R; the metric tensor g_μν and the stress-momentum-energy tensor T_μν and where the Einstein-Riemann tensor G_μν= R_μν - ½g_μν, so describes a curvature inherent in spacetime independent of its inertia given by T_μν . 
is the  cosmological constant, G is the gravitational constant and  c is the speed of light.

The Einstein-Riemann Tensor G_uv= R_uv - ½ g_uvR  for G_uv +  g_uvΛ = 8πGT_uv/c⁴ relating the Riemann-Metric g_uv with scalar tensor R to the Ricci-Tensor R_uv for a stress-energy density tensor T_uv. The Weyl Curvature in R_uv preserves volume as a tidal shear effect, whilst the Ricci Curvature acts on the density and changes the density and so the volumes.
The Weyl Curvature Nullification hypothesis of Roger Penrose (Oxford University, UK) shows, that the Weyl Curvature must become 0 at the threshold between General Relativity's metrics and the 'singularity' of quantum mechanics for the selfconsistency of the physical universe to hold in its inertial parameters.

A 0 Weyl curvature means that the Lorentz Contraction of a tangential displacement vector travelling around a 'wormhole singularity' or Weyl-Centre as Black Hole event horizon must dewarp itself at that wormhole perimeter in accompanying invariance of the scalar orthogonal radius vector not subject to the Lorentz contraction of Special Relativity in say a rotating system.
We shall describe this Weyl-Limit as a superbrane parameter negating the mathematical singularity of General Relativity in a minimum superstring condition:  λ_o=2πr_o.

It shall be shown, that the FRLW-Equation above should be applied for a de Sitter universe with a curvature k=1 and a cosmological 'constant' Λ=Λ_E=Λ_Einstein, where this 'constant' is however not constant at all, but describes an intrinsic parameter for the dynamical universe as Λ_o.

In particular, Λ_o=Λ_E describes the evolutionary dynamics of the universe as a superposed curvature radius 1/L²=1/f(n).R_o² and where the positive curvature k=1 remains constant in the expression H_o=c/R_o  for all time relative to the de Sitter cosmology, but accomodates the zero curvature of the Friedmann universe, simulating an infinite asymptotic expansion in that time.

The overall curvature of the universe so becomes asymptotically defined in a quasi-flat Minkowski spacetime, but only attains k=0 in superposition with the k=1 de Sitter curvature in the function f(n)=[T(n)]²=[n(n+1)]², that is as a summation integral of oscillatory spacetimes described in a reformulated expansion parameter
a=n/(n+1) or equivalently n=a/(1-a).

The boundary condition for the universe so becomes set at a de Broglie inflaton-instanton, which ascertains R_o as the reference scale of the universe at any time of its linear evolution, but as a nodal boundary condition for the minimum Hubble-Frequency in the asymptotic dynamics and as H_o=dn/dt=c/R_o, so giving a temporality parameter of proper time t=n/H_o.

The de Sitter curvature so redefines k=1 after the completion of the first semicycle of the cosmic dynamics for a_∞=1 and for an electromagnetic expansion of the de Sitter space superposed onto the inertial and mass-parametric expansion of the Friedmann space, the latter carrying the expansion factor a(1)=½.

In other words, whilst a c-constant noninertial expansion has travelled a lightpath x=ct=R_o; a c-limited inertial expansion has travelled only half that lightpath x=½R_o in the same linear time interval n=H_ot=1.

This context serves the universe for the transmission of information between the nodes in the form of the 'Holographic Universe' and instigated at the de Broglie instanton.

The semicycle of the electromagnetic de Sitter cosmology so becomes a full cycle, if the de Broglie instanton is set to define a mirror space; which immediatly after the inflaton defined the wavefunction of the universe to reflect itself in its own nodal boundary conditions.

There so exists a total de Sitter lightpath 2x=2R_o superimposed onto the Friedmann lightpath x=½R_o.

It can be shown, that this 'doubling' of the de Sitter universe sets the choice for the expansion parameter to reflect the mathematical supersymmetry between the two cosmologies in the form of a maximised efficiency for the cosmic dynamics.

Briefly, the function f(n)=eⁿ=f'(n)=df(n)/dt carries a 'doubling' symmetry in its limiting series approximations for the summation of terms in arithmetic progressions (A.P's) via 2Σ(1+2+3+4+...n)=n(n+1)=T(n) and as a kind of 'sums over histories' Feynmanian path integral.

The mathematical definition for the transcendental number 'e' as the limit of {1+1/n}ⁿ as n approaches infinity so becomes appropriate.

The function g(n)=(n+1)/n=1+1/n is summed as: (1+1/1)+(1+1/2)+(1+1/3)+(1+1/4)+(1+1/5)+...; for n>0; whilst the function

h(n)=1/g(n)=n/(n+1)=[1-1/(n+1)] is summed as: (1-1/1)+(1-1/2)+(1-1/3)+(1-1/4)+(1-1/5)+...; for n including 0.

Adding the function g(n) to its inverse, then yields (2)+(2)+(2)+(2)+... as the doubling effect of this supersymmetry, which is also embodied in the Euler identity: X+Y=XY=i²=-1= e^iπ  and as the product of f'(n)/f(n)=1=e⁰  and so the infinite summation of the doubling becomes finitised in the unity expressed in the Aleph-All cardinality: lim{n→X}(T(n))=1 and counting infinities as the units of the Aleph-Null cardinality: lim{n→∞}(T(n))=∞ (Infinity).

This approach also reflects the fundamental concept for displacement-momentum definitions in the T-Duality of superstring theory; where the encompassing principle can be defined as the Principle of Modular Duality; which relates the inversion property of a displacement parameter to modular energy-momentum expressions such as winded and vibratory superstrings describing the same physics at scales inversely proportional to each other.

The de Broglie inflaton-instanton can be defined to set a de Broglie phasespeed of V_dB=R_oc/λ_o and a de Broglie phase acceleration A_dB=R_oc²/λ_o² to define the maximum Hubble-Frequency as f_o=c/λ_o  in a direct (string parametric) coupling between displacement and frequency using the lightspeed invariance:
 c = f_o.λ_o = H_o.R_o and subsequently defining the crucial initialisation parameter n_o=H_ot_o = λ_o/R_o.

This 'normalisation' of the Hubble-Radius R_o in the closed de Sitter universe implies a 'quasi-displacement' R_o superimposed onto a quantum displacement λ_o as the characterisatic length parameter for the local Friedmann universe, yet coupled to the characteristic boundary of the Hubble radius R_o=R_H in the de Sitter cosmology.

This coupling is then attained in the time instanton and which can best be modeled on string parameters defining the de Broglie wave mechanics as a string-inflationary matter wave, which initialised the two displacement scales as minimal and maximal boundary conditions for the asymptotic dynamics of the de Sitter spacetime.

In other words, the universe was defined in a hierarchy of Planckian string classes and including a Planck-Length bounce of the order Sqrt(Alpha)=√[e²/2ε_ohc]=[ec]√(μ_o/2hc)={e/c².L_P} ~1/11.7 until a string-boson decoupling occurred (via a 'monopole mass' [ec]) at the instanton t_o=1/f_o=λ_o/c.
Five string classes transform into each other in energy gradients from the Planck class I (for open and closed string modalities) via the Monopole class (selfdual IIB) to the heterotic XL-Boson class (HO(32) bifurcating into quark-lepton templates) into the CosmicRay class (IIA open-ended but D-brane attached as all classes bar the I class) into the final heterotic Weyl-class (HE(64)) of the wormhole scale λ_o, which effectively quantizes all physical parameters in the modular duality definitions of the final Planck-Length transformation.

The conformal mapping between the de Sitter space of the closed HDU and the Friedmann open LDU then manifests the limiting scale parameter atomically and in the form of a localised L=λ_o²/L_P ~ 5x10^-10 meters with a maximised de Sitter curvature of Λ_L~1/L² ~4x10¹⁸ and where λ=λ_o=10^-22 meters as the Weyl-perimeter to map the typical scale of a star.
For the wormhole radius r_o=λ_o/2π, the contracted atomic scaling of 10^-11 meters then results in the conformal mapping for an extended astronomical scale at local curvature  Λ_L~1/L² ~ 6x10²¹.

Phenomenologically then, the Planck-Weyl-Coupling-String transformation is characterised in the atomic mapping of stellar scales from a Planck-Scale reduction C_PW={λ_o/L_P} of order 5x10¹², with the Planck-Weyl coupling leading to the emergence of the Big Bang inertia seedling M_o  and the energy-stress tensor in General Relativity T_uv traced in the expression 8πG/c⁴ with energy density ε=mc²/V.

The square of the inverse Planck-Time or Planck-Curvature {c/L_P}²=2πc⁵/hG~2x10⁸⁶ (s^-2)  is transformed into the Weyl-Curvature {c/λ_o}²=f_o² of the instanton of the maximum Hubble-Frequency and  to become the omnipresent cosmological parameter for the cosmogenesis and the universe's dynamics.

The gravitational inertia trace emerges from the Einstein-lambda via the zero-point of the Planck-Oscillator and as: E_omin=½hf_o=½m_oc² = ½Gm_oM_o/R*_S   for  2R*_S=λ*_S/π =2GM_o/c² 
 Λ_o(n_o)=GM_o/λ_o² =GM_of_o²/c² ~ 2x10⁸⁵  (m/s²)  to unitise the emergent static Schwarzschild perimeter as: λ*_S=2πR*_S = 4π{Λ_o(n_o)}{L_P/c}² =4π{GM_o/c²}{L_P/λ_o}²  ~1  for R*_S={2GM_o/c²}/C_PW².

The atomic scale of  λ=10^-10 meters conformally maps the scale of a typical solar system in L=10¹⁵ meters for a Λ_L~10^-30 (1/m²);

the nuclear-leptonic scale of  λ=10^-15 meters in Minkowski space conforms to the size of an astronomical orb in L=10⁵ meters for a local curvature of Λ_L ~10^-10 (m^-2) and the subnuclear-mesonic range at λ=10^-18 meters maps L=10 centimeters as a macroscopic scale with a de Sitter curvature of Λ_L ~100 (m^-2).

The maximum local scale of the Hubble horizon then gives R_o=λ²/L_P ~ for a curvature mapping of the Hubble scale onto the biovital boundary for the microbial realm at   λ~60 microns for the localised zero curvature of the Friedmann universe as Λ_L~10^-52 1/m²  within  the de Sitter encompassment.

But the asymptotic condition for a Euclidean flat universe of zero curvature no longer requires a Hubble-scale at infinity, but assigns the expansion parameter the asymptotic form for an oscillatory Hubble evolution and as a(n)=n/(n+1).

The de Sitter universe so behaves like a 'Standing Wave' bouncing the lightpath in semicycles given in the nodal Hubble-Time 1/H_o between its minimum value at the odd nodes as H_o and its maximum value at the instanton as f_o at the even nodes.

After the instanton t_o, the universe expands in dualistic fashion into the de Broglian de Sitter space created in the inflaton in timeinstantenuity, that is a 'higher dimensional' universe of hypersphere volume 2π²R_o³, enveloping a spherical volumar 4π[R(n)]³/3.

A thermodynamic Planckian Black Body radiator expands relativistically and classically and under the auspices of the de Sitter 'Lambda' Λ_o(n) aka the 'Dark Energy' and in terms of an intrinsic acceleration, here labeled the Milgrom-Deceleration A_Milgrom= -2cH_o/(n+1)³ = -2H_o²/R_o(n+1)³.

This Friedmann-Milgrom universe is the one described by inertial parameters, as it alone manifests the stress-energy tensor in General Relativity, coupled however to the de Sitter lambda Λ_o(n) in a form of quintessence.

This 'lower dimensional universe' (LDU) decelerates asymptotically, incorporating a 'Hookessence' Λ_o(n) and using a parametrisation of the expansion parameter a(t) in terms of the dimensionless cycletime n.

It has an intrinsic zero curvature, which however inflattens asymptotically because of the superposition of the positive de Sitter curvature.

Overall, the described multidimensional universe is given in perfect Euclidean flatness, due to the manifestation of the initial boundary conditions.

Superimposed onto the LDU, is however the expansion of the 'higher dimensional universe' HDU, here named the de Sitter universe; which juxtaposes the de Broglie inflaton and which exceeded lightspeed in hyperspace: {V_dB=λf=(h/mc)(mc²/h)=c²/v>c  "  v<c}.

The de Sitter universe so 'accelerates' relative to the Friedmann-Milgrom universe with constant lightspeed 'c' unaffected by the inertial parameters of the LDU and creates new spacetime in the Hubble-Oscillation bounded as the Standing Wave in the HDU.

The de Sitter universe so has an electromagnetic volume which always exceeds the inertial volume of the Friedmann universe, where however the electromagnetic HDU intersects the inertial LDU for all times n>½ or about 8.45 billion years following the Big Bang event. At that n-coordinate then, the collection of information gathered hitherto by the evolving universe electromagnetically, could begin to be (re)processed by the inertia carriers of the Friedmann cosmology through the intersection of the subjective de Sitter mirror space with the objective Friedmann mirror space.

But using the Hubble-Law as stated, results in the cyclicity of the universe to give the evolution of the expansion parameter in terms of a summation integral of the function T(n)=n(n+1).
 

The de Broglie Hubble-Node of the instanton has a 'returning' or 'reversing' lightparameter, which 'meets' its 'forwarding' counterpart at the precise median between the minimum and the maximum Hubble nodes.

So for n=1: [T(1)]=1.2=2=(1+1); for the 'doubling' of the lightpath x=ct to 2x=2ct;

for n=2: [T(2)]=2.3=6=(1+1)+(2+2); for the sum of the doubling and the double-doubling of the lightpath x=ct;

for n=3: [T(3)]=3.4=12=(1+1)+(2+2)+(3+3); as the sum of the tripling summation of the lightpath x=ct;

for n=4: [T(4)]=4.5=20=(1+1)+(2+2)+(3+3)+(4+4); as the summation of for the 4th doublecycle of the lightpath x=ct;

...

for n=n [T(n)]=n(n+1), counting the nested cycles of the light parameter, bouncing in between the two nodes.

This superposition of the HDU onto the LDU has consequences for the Hubble-Parameter H(n)=H(t=n/H_o); as it now gives a LDU displacement measurement superposed onto the projected displacement of the lightpath in the HDU.

The values obtained in the above are those of using the Hubble-Law in the Friedmann formulation as the function [H(t)]² and result in a series of diminishing Hubble-constants of the form H_o/T(n).

If applied to the expansion parameter a(t), the cosmological 'problems' eventuate in the nonharmonisation of displacement scales and the various density considerations.

In particular, it will be shown, that the LDU can be described in the static Schwarzschild solution as a Black Hole equivalence, nested within the HDU Black Hole equivalence.

This can best be described geometrically as a Strominger Brane as an extremal Black Hole, which can be considered massless in its boundary condition.

The de Broglie inflaton defined the critical density of General Relativity and the Friedmann formulation and with it, its dual curvature coupling between the closed de Sitter universe and the open Friedmann-Milgrom universe.

This coupling then gives the deceleration parameter q_o=½Ω_o=M_o/2M_critical=Λ_o(n_o)/A_dB~0.01405 and so the coupling between the curvature radius R(n) at a linear Friedmann time t in the LDU to its multiplication in the HDU.

This then is expressed as the present curvature radius R(n_present)~0.531R_o in the LDU to define the Friedmann universe as a 'Daughter-Black Hole' with a characteristic inertial mass seed
M(n_present)=R(n_present)c²/2G~3.43x10⁵² kg and as so 53.1% of the critical mass required for overall flatness closure with respect to the HDU as a 'Mother-Black Hole' of mass M_critical=R_oc²/2G.

The Friedmann equation should so be rewritten in the form:

f(n).H_o²=4πGρ(n)/3 + 0 + Λ_o(n) to directly incorporate the nodal Hubble constancy with the variations of the expansion parameter and the 'Dark Energy' as a function of n.

The particular formulation engages the minimum energy configuration for the Schwarzschild metric at the instanton in terms of a Zero-Point-Planckian Oscillator:

E_min=½hf=½mc²=½hc/λ=GmM_o/R for R=R_S=2GM_o/c² for the Hubble-Frequency maximum at
 f_o²=GM_o/R_S λ_o² =Λ_o(n_o)/R_S.

The parametrisations:
 R(n)=R_on/(n+1); V(n)=1/(n+1)² and A(n)=-2cH_o/(n+1)³ then give f(n)=-2R(n)/(n[n+1]²)=-2R_o/(n+1)³ as the 'Dark Energy' expressed in units of acceleration.

At the instanton then, Λ_o(n_o)=GM_o/λ_o² ~2x10⁸⁵ m/s², which becomes the maximum Hubble-Frequency f_o=Λ_o(n_o)/R_S(n_o)  for R_S(n_o)=2GM_o/c² and in using the baryon seed as the total mass content for the universe at the inflaton.
Using M_o then will give Λ_o(n) as the acceleration inherent in de Sitter space as the remnant of inflation; whilst using the Schwarzschild metric for R_S(n)=2GM(n)/c²=R(n) will map the R(n) equivalent Schwarzschild horizon of the Friedmann cosmology onto its evolutionary scale R(n).

This gives Λ_o(n)/R_S(n)=GM_o/[R(n)]³, which for the present time becomes 3.308..x10^-37 (s^-2) squared Hubble units or the  'Dark Energy' component represented by the energy-stress-momentum trace
8πGρ(n)/3=2GM_o/[R(n)]³.
As the Friedmann equation now reads: -2R_oH_o²/(n+1)³ = 4πGρ(n)/3 + Λ_o(n) = Ω_E + Λ_E = A_Milgrom.

As the factor R_oH_o²=cH_o=c²/R_o  is of order G~10 ^-10, the intrinsic Milgrom deceleration of the universe maximises at the instanton with a value A(n_o)=-2cH_o=-1.12664x10^-9 m/s²  and is insignificant in comparison to the large Omega ~ 2x10⁸⁵ m/s² at the birth of the inertial universe from its string-epoched predecessor.

The 'Dark Lambda' must so slightly exceed the 'Dark Omega' in absolute value to give the Milgrom deceleration as the vector differential and as the Omega is always positive in this formulation and always acting opposite the Milgrom deceleration.

The 'Dark Lambda' so is:  Λ_E(n) = {Ω_E(n) + A_Milgrom(n)} = GM_o/[R(n)]²-2cH_o/(n+1)³.
For the present time this reads:
Λ_E(n_present) = {Ω_E(n_present) + A_Milgrom(n_present)} = GM_o/[R(n_present)]²-2cH_o /(n_present+1)^3_;

Λ_E(1.1324..) = {Ω_E(1.1324..)- A_Milgrom(1.1324..)}
                            = (2.807x10^-11 - 1.162x10^-10) m/s² = -8.812x10^-11 m/s².

The intrinsic Milgrom deceleration so specifies the Curvature Acceleration as the sum of the 'Dark Omega' of inertia and the 'Dark Lambda' of the gravita, the latter labeling the gravitational mass equivalence from the de Sitter HDU relative to the inertial mass equivalence of the Friedmann LDU.

The Omega acts always opposite the Milgrom, resulting in the Lambda to become the 'Dark Energy' balance, which can be either positive, negative or zero, depending on the cosmic dynamics as function of cycletime n.

As the flat Friedmann universe requires a critical density for its flatness, which is supplied in the 'Dark Energy' in the form of the de Sitter Lambda encompassing the Minkowski universe in positive curvature and closure; the 'missing mass' in the open Friedmann cosmology readjusts the critical mass in the formulations describing the 'Dark Energy'.
This manifests in the 'higher dimensional' curved de Sitter spacetime forming an acceleration gradient relative to the 'lower dimensional' flat Minkowski spacetime.
Considering the de Sitter cosmology to be inertial as a static- meaning c-invariant - HDU, then results in the Minkowski spacetime to be rendered noninertial by the experience of  a 'de Sitter' force or pressure.

As the deceleration parameter q_o=½Ω_o=M_o/2M_critical=Λ_o(n_o)/A_dB~0.01405 defines the Omega relative to the Friedmann spacetime; replacing the baryon seed M_o by the critical mass M_critical will adjust the Friedmann lambda in the factor Ω_o=0.0282 in the de Sitter lambda.

Then, Omega+Milgrom=Lambda becomes {2.807x10^-11 -1.162x10^-10 = -8.812x10^-11  } for the Friedmann cosmology and {9.989x10^-10 -1.162x10^-10 =+8.827x10^-10} for the de Sitter cosmology in acceleration units for the present time.
The flat Friedmann universe for a zero cosmological constant so balances the Omega deceleration with the Intrinsic (Milgrom) acceleration and omits the de Sitter component, which acts in addition to the Omega for the present time to balance the Milgrom in opposite direction. 
This then is the reason behind the Pioneer anomaly detailed in a following agenda.

Dark Energy  Λ_E(n) = G_oM_o/R(n)² - 2cH_o/(n+1)³ .

The derivative
Λ_E'(n) = -2G_oM_o R'(n)/R(n)³ + 6cH_o/(n+1)⁴ = 6cH_o/(n+1)⁴ - 2G_oM_o H_o²(n+1)/c²n³

Then the absolute minimum for Λ_E'(n)=0 and for n=0.2389... becomes:
Λ_E(0.2389) =2.12319...x10^-10 - 5.92482...x10^-10 = -3.80163...x10^-10 (m/s²)*.

The roots for Λ_E(n)=0 are calculated via 2c³/G_oM_oH_o=(n+1)⁵/n² as n₁=0.10823... and  n₂=3.40055...
This corresponds then to the Dark Energy beginning at the very high positive value of 2x10⁸⁵  (m/s²)* at the instanton and reaching its first zero for the galaxy formation in the HDU after 1.83 billion years.
This process of galaxy formation then peaks at the minimum so 4.04 billion years after the Big Bang and in tandem with the galaxy evolution in the LDU and peaking (0.2389..-0.2352..=0.0037) or so 62.5 million years earlier.

The Λ_E(n→∞) = G_oM_o/R_o²  then as the asymptotic 'Cosmological Constant';
 Λ_E∞= 7.9136027..x10^-12 m/s²   or as  Λ_E∞/R_o = 4.9532x10^-38  s^-2  for an asymptotic 'Hubble-Constant'
H_asymptotic= 2.2256x10^-19  or about 6.88 km/Mpc.s.
The curvature radius  for H_asymptotic then is Λ_E∞/c²R_o = 5.5035..x10^-55  m^-2  for a projected Hubble horizon of 1.347x10²⁷ meters or 142.6 billion lightyears.


The Dark Matter epoch begins 1.83 billion years after the instanton-inflaton and ends so 3.4 cycles afterwards at a 'oscillation coordinate' of 3.4R_Hubble or about 57.5 billion years.  

As the cosmological principle demands isotropy and homogeneity at an observed scale of galactic superclusters and at about 100 Mpc or 326 million lightyears; this (Sarkar) scale can now become ascertained in defining a inertial restmass seedling M_o  at that scale.

The conformal ratio for the Sarkar scale is L~√(R_Sarkar.L_P) ~ 10^-6  meters at the micron level.

The precise formulation for this baryon seed derives from string parameters and is:
M_o=√E.m_P.m_c/m_e ~ 1.818x10⁵¹ kg and where m_c and m_e represent the protonucleonic and protoelectronic masses from the string epoch (the XL-HO(32)-bifurcation) preceding the de Broglie inflaton and where E describes a spacetime quanta counter, say applicable as quantum loops of the no initialisation parameter.

This description then implies, that for a curvature radius of R_Sarkar=2GM_o/c² ~ 4.49x10²⁴ meters or about 475 million lightyears, the LDU displays isotropy and homogeneity in ceasing gravitational interaction between constituent parts in regard to dynamical motion.

It is here then, that the LDU intersects the HDU in terms of the Black Hole equivalents of the Schwarzschild metrics.

But this also means, that after about 475 million years of the Big Bang, the LDU has 'caught' the 'coordinate' defining the cosmological principle set by the HDU and from that time onwards, the inertial mass content of the LDU would be 'growing' from its seeded M_o to also 'catch' its 'boundary' or saturation-critical value of the 'Motherly' Mcritical as 100%.

For a present time then, the initial baryonic mass seed has grown from 2.81% to 53.1% in the HDU, but remains at 2.83% in the LDU, subject to a 'mass evolution' engaging the coupling of a string gravitational Constant G_o=1/k (Stoney-Units) to a so called 'Dark Matter' particle, here termed the RestMass-Photon (RMP) as a gauge unifier in Supersymmetry and as a massless derivative of the Higgs Boson template (see references at linked website).

In this paper, the 'stringed' G_o is used for calculations and as the derivative from the Planck-Mass:
1=2πG_om_P²/hc.

The Equivalence Principle of General Relativity thus derives from the equality between the inertial mass in the LDU and the gravitational mass in the HDU, where the latter can also be expressed as part of the intrinsic curvature in a purely electromagnetic de Sitter universe where G_o=4πε_o=4π/μ_oc² (in Stoney-Units unified with Planck-units).

The linear scale of the present universe is 53.1% of R_o in the LDU, but is increased in a factor of [T(n)]=[2.415] in terms of {(da/dt)/ca}² = {H_o/cT(n)}² ~ 6.72x10^-54 m^-2 and which 'stretches' the Hubble-Horizon in the LDU from 0.531Ro to 2.415Ro or from 8.97 billion lightyears to 40.81 billion lightyears.

But this is the 'distorted' value of the T(n)=H_o/H(t) application, which should be 'corrected' to the curvature radius 1/(nR_o)²,  that is the factor (n+1)². The true projected curvature radius for the present time so is n_present.R_o ~19.11 billion lightyears for a projected curvature of [H(n_present)]²~2.75x10^-36=[H_o/n_present]².
This projected Hubble-Constant for the present epoch of 1.6582x10^-18 1/s is about 88.3% of the nodal H_o and describes the scale of the open Friedmann universe with zero curvature under utility of the expansion parameter and the Hubble Law as presently employed.
 

The electromagnetic 'return' for the present time so incorporates a 'duplication interval' of 2(2.2) billion years and the light parametric measurements result in a projected age for the universe too young by 4.4 billion years.

Using the 'corrected' Hubble-Constant for the present time in the Friedmann universe then gives H(n_present)~67 (km/Mpc.s) as the LDU coordinate, extrapolated from the nodal constancy of 58 (km/Mpc.s).

The present value for H(t_present)~71 (km/Mpc.s) so if extrapolated, will increase the age of the universe to 14.7 billion years from the 'Hubbled' 13.7 billion years publisized.

The HDU Hubble coordinate is of course H_o/n_present ~ 51.2 (km/Mpc.s) for the projected true age of the universe of 19.1 billion years.

The flatness of the Friedmann universe so becomes accomodated in the increase of the 'refracted' de Sitter curvature radius into R⁴-space, reflected however into the R³-space of the Friedmannian 'Hubble-Bubble.

But the postulate of an ever increasing Hubble horizon and a dispersation of the Friedmann universe can be abandoned.

Since the expansion of the universe is cyclic in projecting the growth in linear displacement inwards in the Hubble-Oscillations; the distance measurements are simplified, whilst becoming multivalued.

The Hubble-Law becomes unneccessary to calculate the distances to cosmological objects, as the expansion of the Friedmann universe obeys the postulates of the relativities and remains lightspeed invariant within the de Sitter universe.

There is no spacetime expansion exceeding lightspeed relative to the Big Bang observer, however the expanding Big Bang wavefront in the form of the Hubble horizon of the LDU is itself doppler shifted relative to that Big Bang observer.

A distinction between a "Local Flow' and a 'Hubble Flow' must so be made as the effect of the 'duplication interval' bounded by the cosmological redshift of the Hubble wavefront. This 'Arpian' redshift is calculated for the present time to be about z_Arp~ 0.25053.

The measurement of cosmological redshift so suffices to determine the corresponding n-coordinate for the time the light became emitted by the object; as then the recession velocity of the universe itself was v(n) and as incorporated in the relativistic redshift formulations.

The oscillating lightpath parameter does indeed also expand into the volume of the 4-sphere, which defines the 3-sphere as its 3-dimensional boundary of Riemann's hypersphere as a manifold in 3 dimensions: {V₄=½π²R⁴ for dV₄/dR=2π²R³=V₃} and this continuation allows the described de Sitter cosmology to become defined as a multiversal seed of phaseshifted protoverses for the omniverse.
 

For an overall multidimensional age of the universe of 19.11 billion years then, it it found, that for 2.2 billion years the first Hubble-Node has been passed to allow the 'refracted' lightpath into the 4-sphere to become itself 'reflected'.

This indicates the possibility for the 4-dimensional spacetime described by this de Sitter cosmology to transform into a 5-dimensional Kaluza-Klein cosmology guided by the Holographic Principle in an extension of this model.
The possibility for the 'tearing' of spacetime and its subsequent 'regluing' under certain mirror symmetries in conventional string theory was rigorously established by Greene, Aspinwall and Morrison at Princeton in 1992. It is this, which is envisaged by the author for the not too distant future, say a post 2012 scenario. Entirely new physical possibilities, will then eventuate, the latter being described in many works describing the properties of hyperspace comprised of 4 spacial dimensions with one temporal dimension.

The doppler shift expression for relativistic cosmological redshift becomes appropriate in the extraposed blending of the closed de Sitter universe onto the open Friedmann universe and the refinements of the Hubble-Law as presented in this paper.

Lorentzian relativity under Poincare symmetry remains unviolated in the quantum relativistic approach of this model and the quantum gravitational parameters crystallize from the string parameters of the Planckian pre-Big Bang epoch from Planck-time to Weyl-time, defining the de Broglie instanton-inflaton, which ended the string epoch and began the classical quantum relativity.

Agenda:
0. The Parable of Hans Schatten.
1. Historical Introduction to Newton's Apple-Seeded Classical Universe
2. The Newtonian Cosmology as the basis for the FRLW-Einstein-Riemann cosmology of 
    General Relativity in De Sitter Spacetime.


a) The Pioneer Anomaly as Milgrom Deceleration Effect of de Sitter Curvature
b) The De Sitter Universe
c) The De Sitter Spacetime
d) The Cosmological Hookessence
e) The 'Cosmological Problems'

3. The Einsteinian Field Equations in General Relativity and the Friedmann Equation
4. tba
5. The Guth-Inflaton and the Monopole from the Planck-String-Monopole coupling
6. tba
7. tba
8.





0. The Parable of Hans Schatten

There once lived a gardener in a place not known in part but in all. The gardener so could not plant anything somewhere in particular, but could only plant where he himself was as being nowhere and everywhere.

The gardener wished to plant an apple-seed he had found to be part of himself in the place he was and so the gardener thought of himself as not being nowhere anymore, but to be right in the place of the apple seed.

So the apple seed became real and occupied a real space, but caused the gardener to disappear from that real space into an unreal space. The gardener so became the unreal image in unreal space of the real image of the apple seed in the real space.

And so the apple seed was born as real space to occupy, but being surrounded by an unreal space and there where the gardener still followed his dream to see the apple seed grow and blossom into a full apple tree after he had planted it.

For the dream of the gardener was to grow the apple seed into a full apple tree and after reaching maturity, the apple tree would blossom and yield its fruit of apples which carried their own apple seeds within.
Because this was the plan of the gardener as the unreal image of the real image. Should the single apple seed become two apple seeds, then any two apple seeds could image each other and the real images in the real space, would enable the gardener to use the real image of one of the apple seeds to mirror himself in the realness of one of the apple seeds in the gardener's unreality becoming real in the reality.

Then the single seed of the gardener could multiply and the single apple tree could become a forest of apple trees and so on ad infinitum.

But there would always have to be the first single apple seed which the gardener had become in real space as the image of himself in the unreal space. There could not have been two seeds of the one gardener, because two seeds would have meant that the gardener divided itself into two and that was not the plan of the planter.

This initial apple seed would always remain to be the Seed of all Seeds around which the other seeds and apple trees and apple forests could grow, multiply and reproduce.

Then the gardener would find himself in the real space too and leave his exile in the unreal space. The gardener would become reborn as the image of the image and all other apple seeds would similarly become gardeners themselves, as this was the nature of all things and the beginning of it all.

Hans Schatten;  for my apple seeded daughter in the unreal space to become real.

1. Historical Introduction to Newton's Apple-Seeded Classical Universe

The 21st century shall rediscover the old perennial philosophy of antiquity, and which experienced its climax in the works of Isaac Newton.
The scientific thinkers of this century will emerge from a sense of remembrance as from a deep slumber and attune their accumulated wisdoms of facts and information to the 'inner knowing' of the wisdom keepers of old, the Egyptians, the Greeks, the Gnostics and the Alexandrites of all the ages.
Isaac Newton is considered the 'last of the alchemists', the last of the 'scientific' thinkers, who pondered the universe from the position of a 'natural philosopher' and the agenda of an intuitive understanding what the nature of reality 'should be' - logical, yet mysterious and hidden, but subject to the human mind in comprehension and worthy of the most serious of investigative endeavours.
In other words, Isaac was not afraid to FEEL the universe mentally and its cosmic reality. After he had felt IT, he thought about IT and so gave great purpose and meaning to the cosmology so thought about.
This essay so shall construct the cosmology just as Isaac Newton would construct it in the 21st century.
The Newtonian universe is a Unity and as such this Unity is described in a Newtonian mechanics applicable to this unity.
It must so be necessarily be limited in scope to an overall description of this Unity and just as the famous Newtonian Laws serve as approximations to a rather more detailed description of the subunities from that holism by the postulates of contemporary theoretical physics.

To describe the approximate motion of planetary orbits around a common center of gravity, the mathematics of Newton and Kepler suffices; but to compute the detailed dynamics, the extended formalism of Riemann's curvilinear coordinate systems becomes necessary.
Applying boundary and initial conditions to that more detailed dynamics then will recrystallise the basic Newtonianism as say a first approximation.

The following points of agenda will so be addressed to render description for the parable above in mathematical and in scientific logistical terminologies.

2. The Newtonian Cosmology as the basis for the FRLW-Einstein-Riemann cosmology of General Relativity refined in a De Sitter universe of Closure

(Basic wikipedia or similar references are shaded as common introduction with nonshaded commentary interspersed).


(a) The Pioneer Anomaly as Milgrom Deceleration Effect of  de Sitter Curvature

What causes the apparent residual sunward acceleration of the Pioneer spacecraft?  

The Pioneer anomaly or Pioneer effect is the observed deviation from predicted trajectories and velocities of various unmanned spacecraft visiting the outer solar system, most notably Pioneer 10 and Pioneer 11.

Both Pioneer spacecraft are escaping from the solar system, and are slowing down under the influence of the Sun's gravity. Upon very close examination, however, they are slowing down slightly more than expected. The effect can be modeled as a slight additional acceleration towards the Sun.

At present, there is no universally accepted explanation for this phenomenon. The explanation may be mundane, such as measurement error or thrust from gas leakage or uneven radiation of heat. However, it is also possible that current physical theory does not correctly explain the behaviour of the craft relative to the sun.

Initial indications

The effect is seen in radio Doppler and ranging data, yielding information on the velocity and distance of the spacecraft. When all known forces acting on the spacecraft are taken into consideration, a very small but unexplained force remains. It appears to cause a constant sunward acceleration of (8.74 ± 1.33) × 10^-10 m/s² for both spacecraft. If the positions of the spacecraft are predicted one year in advance based on measured velocity and known forces (mostly gravity), they are actually found to be some 400 km closer to the sun at the end of the year. The magnitude of the Pioneer effect is numerically quite close to the product of the speed of light and the Hubble constant, but the significance of this, if any, is unknown. Gravitationally bound objects such as the solar system, or even the galaxy, do not partake of the expansion of the universe - this is known both from theory and by direct measurement.

Data from the Galileo and Ulysses spacecraft indicate a similar effect, although for various reasons (such as their relative proximity to the Sun) firm conclusions cannot be drawn from these sources. These spacecraft are all partially or fully spin-stabilised.

The effect is much harder to measure accurately with craft that use thrusters for attitude control. These spacecraft, such as the Voyagers, acquire small and unpredictable changes in speed as a side effect of the frequent attitude control firings. This 'noise' makes it impractical to measure small accelerations such as the Pioneer effect.

The Cassini mission also had reaction wheels for altitude control, thus avoiding this particular problem, but also had radioisotope thermoelectric generators (RTGs) mounted close to the spacecraft body, radiating kilowatts of heat in hard-to-predict directions. The measured value of unmodelled acceleration for Cassini is (26.7 ± 1.1) × 10^-10 m/s². Unfortunately, this is the sum of the uncertain thermal effects and the possible anomaly. Therefore the Cassini measurements neither conclusively confirm nor refute the existence of the anomaly.

(From Wikipedia, the free encyclopedia)

One major consequence for the intrinsic de Sitter curvature becomes the 'Dark Energy' manifesting in a differential of acceleration between inertial and noninertial frames of references. The local solar system is a comoving part of the Friedmann expansion into de Sitter spacetime and so becomes a non-inertial comoving reference frame relative to the inertial and static reference frame of de Sitter spacetime. 
This then leads to a logical explanation for the Pioneer anomaly measured for the last decade or so.

As the flat Friedmann universe requires a critical density for its flatness, which is supplied in the 'Dark Energy' in the form of the de Sitter Lambda encompassing the Minkowski universe in positive curvature and closure; the 'missing mass' in the open Friedmann cosmology readjusts the critical mass in the formulations describing the 'Dark Energy'.
This manifests in the 'higher dimensional' curved de Sitter spacetime forming an acceleration gradient relative to the 'lower dimensional' flat Minkowski spacetime.
Considering the de Sitter cosmology to be 'background'-inertial then results in the Minkowski spacetime to be rendered noninertial by the experience of  a 'de Sitter' force or pressure.

As the deceleration parameter q_o=½Ω_o=M_o/2M_critical=Λ_o(n_o)/A_dB~0.01405 defines the Omega relative to the Friedmann spacetime; replacing the baryon seed M_o by the critical mass M_critical will adjust the Friedmann lambda in the factor Ω_o=0.0282 in the de Sitter lambda.

Then the formulation:  Omega+Milgrom=Lambda  becomes {2.807x10^-11 -1.162x10^-10 = -8.812x10^-11  } for the Friedmann cosmology and {9.989x10^-10 -1.162x10^-10 =+8.827x10^-10} for the de Sitter cosmology in acceleration units for the present time.
The flat Friedmann universe for a zero cosmological constant so balances the Omega deceleration with the Intrinsic (Milgrom) acceleration and omits the de Sitter component, which acts in addition to the Omega for the present time to balance the Milgrom in opposite direction. 

The Pioneer anomaly then becomes quantifiable in the Planck-Action applied to the de Sitter spacetime and manifesting in the Minkowski spacetime.

The Planck-Action is the product of Planck-Momentum p_P=m_Pc and the Hubble-Radius R_o for m_PcR_o=m_Pc²/H_o to indicate the Heisenberg Uncertainty Principle in px=Et.
The general energy operator then is p²/m for an acceleration  a=p²/m²x=(p².p)/(m²xp_P)=(mv³)/(xm_Pc).
Applying the de Sitter referential lambda acceleration as R_oH_o²=c²/R_o=(mv³)/(xm_Pc) yields a proportional relationship between the de Sitter Hubble-Radius as reference for a subscale x in the Friedmann universe in:
(x/R_o)=(v/c)³   and relative to the Planck-Mass standard.

Applied to the Pioneer anomaly then, a characteristic displacement of x=100 AU (1.5x10¹³ meters), will infer a characteristic velocity of  v=c.Cuberoot(x/R_o) ~ 13.6 km/s or 0.0045% of lightspeed.

This is indeed the order of velocity measured for the Pioneer probbes at such distances.

b) The De Sitter Universe

A de Sitter universe is a solution to Einstein's field equations of General Relativity which is named after Willem de Sitter. It models the universe as spatially flat and neglects ordinary matter, so the dynamics of the universe are dominated by the cosmological constant, thought to correspond to dark energy.
A de Sitter universe has no ordinary matter content but with a positive cosmological constant which sets the expansion rate, H. A larger cosmological constant leads to a larger expansion rate:

,

where the constants of proportionality depend on conventions. The cosmological constant is Λ and M_pl is the Planck mass.

This corresponds to solving the Friedmann equation for Ω=0, which renders the Einstein formulation
 G_uv +  g_uvΛ = 0  traced in the form [H(t)]² = f(n)Λ, with the function f(n) describing the curvature of the flat spacetime intrinsic to the Einstein-Riemann tensor G_uv.
Expressing the supposedly constant Λ term in the form of energy density Mc²/V for a inertia-gravita equivalence m_P=hf_P/c²=h/λ_Pc then gives a local energy density ε=mc²/V proportional to the Planck-Energy density ε_P= m_Pc²/V_P  for  f(n)=ε/ε_P  as some unity reference incorporating the Planckian gravita as the Planck-mass reference.
H²/Λ=1 then gives the equation of motion of the expansion parameter a(t)=R(t)/R_o and becomes
(da/dt) = a(t)√Λ, solving as ∫da/a = √Λt = ln(a/a_o) or  R(t)=R_oe^t√Λ = R_oe^tH.

,

where the constant H is the Hubble expansion rate and t is time. As in all FLRW spaces, a(t), the scale factor, describes the expansion of physical spatial distances.

Here, the 'flaw' in the physical interpretation of the mathematics is the assumption of a(t) expanding without limit in the flat Minkowski spacetime of Poincare-Lorentz.
This is required for zero curvature, as only then the curvature radius approaches infinity in the curvature expression c²/R(t)² for infinite time t.

But in de Sitter spacetime, the intrinsic curvature of the flat Minkowski spacetime becomes asymptotic and so the expression a(t)=e^Ht    with t=n/H_o  and t_o=n_o/H_o=1/f_o becomes  n/(n+1)=1=e^Ht,  requiring
H(t_o)=f_o=1/t_o, which is the instanton-inflaton of the de Broglie matter wave.

For the initialisation by n_o=λ_o/R_o ~ 6x10^-49 ~ 0   then; a_o= n_o=H_ot_o and the Friedmann flatness a_o=0 mirrored in the de Sitter curvature of  a_∞=1 by the inversion property of the natural exponent 'e'.
e → {1 + 1/n}=1/a  and 1/e → {n/(n+1)}={1- 1/(n+1)}=a.



Our universe is becoming like de Sitter universe?!

Because our Universe has entered the Dark Energy Dominated Era a few billion years ago, our universe is probably approaching a de Sitter universe in the infinite future. If the current acceleration of our universe is due to a cosmological constant then as the universe continues to expand all of the matter and radiation will be diluted. Eventually there will be almost nothing left but the cosmological constant, and our universe will have become a de Sitter universe.

This is a misinterpretation of the 'Dark Energy', as the 'Dark Energy' is everpresent as intrinsic part of the de Sitter curvature superposed onto the Minkowski flatness.
The universe always was a de Sitter universe and will always be amenable to be described by de Sitter cosmology.
The dilution of matter relates to the manifestation of the baryon seedling M_o, which simply distributes the seed inertia as a Friedmann LDU 'Daughter-Black Hole' within the 'Mother-Black Hole' of the de Sitter HDU.
As this dilution is fixed in the critical density and the Omega of the flat cosmology, the expansion of the universe is dual in an oscillatory mapping of the HDU into the LDU, accompanied by the expansion of the R³-spacetime of this Minkowskian multiverse into the R⁴ hyperspace of the HDU boundary.


Modelling Cosmic Inflation?!
Another application of de Sitter space is in the early universe during cosmic inflation. Many inflationary models are approximately de Sitter space and can be modeled by giving the Hubble parameter a mild time dependence. For simplicity, some calculations involving inflation in the early universe can be performed in de Sitter space rather than a more realistic inflationary universe. By using the de Sitter universe instead, where the expansion is truly exponential, there are many simplifications.