Reformulation of the Hubble Law in General Relativity

Reanalysis of the Hubble Law in General Relativity solves the cosmic acceleration phenomenon.

I shall outline the demetrification of General Relativity as an example for demistification of theoretical physics.

Now this simply means that the complexities of say the mathematics of tensors, string theory or the determinants of particle physics become encompassed in the same mathematics upon whom they are themselves built and constructed upon.

So the multidimensional approach of the tensor is reduced to the common vector and the modern physics they represent mathematically becomes reborn in the classical , say Newtonian approach of their own historical development.

What I am saying here, is that it is the nature of space and time themselves, which allows their demetrification.

The Einstein Universe evolved from the Newton Universe and much of Newton's physics is simply rewritten in the developed form of the classical approach.

I am saying that the ever growing and evolving complexity in the attempt to model reality in terms of space and time must eventually reach a point of simplified space and time, namely the origins of space and time themselves as physical parameters describing reality.

It is this point of decomplexification, which I have termed demetrication.

It develops somewhat like this.

The Einstein Tensor and associated metrics describe the curvature of space in a multidimensional system of partial differential equations, which relate the parameters of the variables or coordinates of space and time through the presence of inertia, described by density and mass=densityxvolume.

The metrics then lead to equations of motion in terms of not space and time as commonly understood, but as fractional change or percentages of them.

This is why the redshifts and the dimensionless quantities like the (velocity v/lightspeed c) ratio formulations are so important in the relativities.

But these fractional-change-ratios themselves are analysed in terms of Newtonian dynamics, who in a sense swallow up the percentage changes in the more familiar evolution of displacement vecors, velocities and accelerations.

But the complexity in say General Relativity to model physical reality, is the interpretation of the percentages in a physically meaningful way and methodology.

Now the great realisation relative to me, is that the percentages are dimensionless and thus a similar approach to dimensionless space and time allows the demetrification of both space and time in the percentages of Quantum Relativity.

Quantum Relativity was constructed just on the premise of space and time being dimensionless and thus naturally must encompass into whatever complexities the metrics might evolve into.

It is like the future physics having linked to its own past.

Because my scientific experience and intuition must by necessity be rooted in that past, I, like all of you find myself in a continuing process of self-obscuration, trying to decode the past meeting its own future.

But this future is subject to imagination and scientific dreaming or intuition and that is what created the passion to construct Quantum Relativity.

So I shall illustrate how the dynamical equations derived from the metrics and solved as the percentages (called expansion parameter, deceleration parameter and redshift based on the Hubble Constant and proportional rates of change) by fundamentally classical methods of calculus, can be encompassed by the demetricated approach.

This demetricated approach then does not change the solutions of the dynamical equations as such, but swallows up the metrics of say a cosmological constant related to pressure in percentage changes in one form of variables, to variables which specify the percentage changes in other terms.

So you have an asymptotic (static) solution for the dynamics of the Einstein-de Sitter universe which geometrically defines the curvature evolution as function of Newtonian time t mapped as the geometric solution of the same Einstein-de Sitter universe as a function of demetricated time t=n/Ho in Quantum Relativity, with n=Hot being dimensionless.

Because the metric t-evolution engages possible additional parameters, such as the cosmological constant or quintessence and the pressure also as functions of time t, which complicate the dynamical equations to be solved; there are a number of possible cosmologies of say varying curvatures, resulting in the dynamical solutions.

So I shall indicate those conventional dynamical solutions say in the Einstein-de Sitter cosmology and the Friedmann-Le Maitre cosmology as the most appropriate and this will be necessarily somewhat laborious as it must use the standard cosmological approach.

But the present dilemmas in cosmology will also become apparent.

Is there a cosmological constant or a quintessence as the vacuum 'dark energy'?

Or is it or the pressure of matter term 0 and is the Omega 1 and twice the deceleration parameter?

Those questions are answered by Quantum Relativity in the mapping of the geometries of the metrics onto the nonmetrics of the percentage changes.

This assimilates the relationships between the Omega and the Lambda and the Hubble Constant in an unified approach and naturally crystallises the evolution of the universe in a given parametric definition of the scalefactors described by General Relativity as a function of metric time.

In other words the big question in Quantum Relativity is not about beginnings and endings, but where are we NOW.

The birth and projected death of the universe are given as initial boundary conditions in the superbrane parametres from which the subsequent cosmology develops.

I shall not describe this here, but this is the Quantum Relativity, not searching for unification, but being unified, allowing its symmetry breaking and deunification as underpinning mathematical and theoretical premise.

So why am I doing this?

Daily I ask myself, why am I doing this? Have I not got something better to do?

The answer is no.

Despite the misgivings of some and the appreciation of others, I am doing this for all of us.

I simply feel this is a work that must be done and perhaps I must do it alone; despite my continuing call and ask for help of assistance, noone appears so far to have grasped the nature of this sufficiently to make enough sense of it to meaninful critisise it in contribution and coauthorship.

And yes, I know that some of you have grasped the philosophy underpinning Quantum Relativity wonderfully and are resonating with this work in fraternity.

So the cosmic family is growing, there is joy and exhuberance in the spirit of the metaphysics which connects us all.

But here I am talking of a scientific paradigm change on the greatest possible scale and that is not achievable by the philosopy which is the basis for all true religion, which is gnosis by poetry and beautiful literature and words of depth and meaningful essence.

So the hard work is to present the mathematics of the demetrication to the world and these forums allow me and us to do just that.

Can you see what we are doing here?

We are a spiritual family of, well, 'spirits' finding ourselves in embodiment, and being ultimately rather uncomfortable in this situation, we are here struggling, arguing and attempting to recognise ourselves and to find recognition.

This search for recognition must be understood for what it truly is.

It is not some egocentricity running wild, but as my friend Cisco has said, it is necessary to be or appear selfish and self-centred to recognise oneself first.

For how can one recognise the family of the spirit, if one doesn't know oneself?

And as my friend Pythagoras has said: 'Man know thyself, then shall you know the Universe and God".

So it doesn't matter at all if someone reading all this and understanding the 'beautiful physics' can somehow break through the elitism of the old entrenched science orthodoxy to begin its own metamorphosis from the human science of the earthbound caterpillar to the starhuman science of the cosmic butterfly.

What matters is that we, and all of you through me and me through all of you recognise this work as our contribution towards this metamorphosis.

Without any of you this work would not be what it is, and the family of spirits knows this very well.

The ones attuned to that are the Philosophers of Quantum Relativity.

With these words I'll end the introduction and begin the revision process.

Tony B.

......Revision under engagement.....the following is under construction.....

So I shall present a decisive little paper which shall shed light on the cosmological conundrums of the 'missing mass' and the accelerating universe, using the field equations of General Relativity directly.

I shall derive the following:

1. The reformulation of Hubble's Law regarding the Hubble-Constant as the ratio of recessional velocity over displacement, underpinning modern cosmology and often subject to controversy.

2. The precise cosmological redshift, where the cosmic acceleration begins to take over from the previous deceleration and a fact which is accentuated in the scientific literature.

3. The demetricated expression for the evolution of the universe in parametric terms.

I shall try to derive those expressions from first principles and add commentary for the mathematically inexperienced reader however.

There are a couple of references at the end, which should provide enough background data to the topics discussed.

The references do not show photograps in this rtf-file here, but can be found on the web as indicated.

(1) http://www.pnas.org/cgi/content/full/96/8/4224

(2) http://www.physicstoday.org/pt/vol-54/iss-6/p17.html

(3) http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

(4) http://astronomyonline.org/default.asp?Cate=Home

(5) http://www.talkorigins.org/faqs/nri.html

(6) http://users.rcn.com/wcri/wcri/Big%20Bang%20Text.htm

(7) http://users.rcn.com/wcri/wcri/index.htm

1. The Reformulation of the Hubble Law

Let us state the Einstein Field Equation underpinning all of cosmology.

The Einstein-Riemann Tensor Guv=Ruv-guvR/2 = -8pG*Tuv/c^2 relates the Riemann-Metric guv with scalar tensor R to the Ricci-Tensor Ruv for a stress-energy density tensor Tuv. The Weyl Curvature in Ruv 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 Wolford-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: lps=2prps.

The Einstein-Riemann Tensor then represents 16 partial nonlinear differential equations, which as a system lead to the most basic and general solution for a gravitational field in the standard-isotropic-metric: dt^2=B(r)dt^2-A(r)dr^2-r^2dq^2-r^2sin^2qdf^2, where A(n) and B(n) are the functions (of r) to be determined for the static solutions of the metric.

The metric then leads to the differential equations, which describe the evolution of the universe in dynamical equations with respect to the scalefactor R=aRo as the 'size' or radius of the universe.

In the Friedmann-Lemaitre cosmological model the universe is homogenous and isotropic, that is it doesn't change in its uniform composition in any directional sense. Here the cosmological constant Lambda (L) is taken as positive for various curvatures k and a nonzero pressure p.

In the Einstein-de Sitter cosmological model this universe has also constant curvature k and a zero cosmological constant (Lambda L) and pressure p.

The equations of motion for the two models can be derived as follows:

Density r(r)=M(r)/V(r); M(r)=4pr(r).r^3/3; dM=4pr(r)r^2.dr

r(doubledot)=-GM(r)/r^2 + Lr/3 for a(doubledot)ro=-4pr(r)G.aro/3 +Laro/3 and a(doubledot)/a=-4pGr/3 + L/3

Since acceleration r(doubledot)=d(r(dot)^2/2)/dr, integrating r(doubledot) gives:

r(dot)^2/2=(a(dot).ro)^2/2=GM(r)/r +L(aro)^2/6 + constant with constant of integration given in r(dot)=c for r=ro. and so constant=-c^2 for c^2=c^2+c^2+constant .

{This will give L=3Ho^2 in a de Sitter cosmology of a massless universe with Ho=c/Rmax in demetrication}.

So r(dot)^2=(a(dot).ro)^2=2GM(r)/r + L(aro)^2/3 - c^2.

This becomes: {a(dot)/a}^2=2GM(r)/r(aro)2 - c^2/(aro)^2 + L/3=8pGr/3-c^2/(aro)^2+L/3............. (FLM1)

Now consider the universe's expansion to be adiabatic, that is thermodynamically closed. Energy E and the pressure (P) variation with respect to Volume V sum to 0 change in the 'heat content Q' (or enthalpy H=U+PV for internal heat content U).

dQ=dE+PdV=0; dE/dt+PdV/dt=0 for E=M(r)c^2=4pSrR^3c^2/3 and total density Sr=rmatter +pressure

d{SrR^3}/dt=-(3P.R^2/c^2).dR/dt =R^3.r(dot)+3R^2.r.R(dot) for

r(dot)+3r.(a(dot)/a)=-(3P/c^2).(a(dot)/a) and the dynamical equation:

r(dot)+3(a(dot)/a){r+P/c^2}=0....... (FLM2)

The combined Friemann-Lemaitre equation of motion for matter density r then is:

{a(dot)/a}^2 = 8pG/3{r+3P/c^2} - c^2/(aRo)^2 + L/3..........................................................(1)

The Equation of motion in the Einstein-de Sitter cosmology then sets P=L=0 and a constant curvature k=1/Ro^2=0 in the Friedmann-Lemaitre model for:

(a(dot)/a)^2=8pGr/3 = H^2 with R=aRo=aRmax

Solving for a(dot)^2=2GM/aRmax^3 via Sqrt(a).da=Sqrt(2GM/Rmax^3)dt leads to

a.Rmax=Cuberoot{9GM/2}.t^[2/3] for a limiting boundary condition ao=0 for to=0; (which we shall see is actually n=nps=Ho.tps for a=1/(1+Rmax/lps).

Using H=a(dot)/a, 9GM/2=(aRmax)^3/t^2 for H=Sqrt(4/9t^2)=(2/3t) and the Hubble Time becomes

1/H=3t/2 as the age of the universe for time t.

We shall show that this Hubble Time actually represents the completion of a Hubble-Oscillation and so the LIGHTPATH R(n)=ct where Rmax necessarily represents a semiwavelength as the distance between the even nodes 0,2,4,6.. and the odd nodes 1,3,5,7,....

For define R(n)=Rmax(n/(n+1)) with n=Hot and a=(n/(n+1)) for a lightpath of 2Rmax =ct*=2c/Ho, then this time t* given in c-invariance in say 11D/5D hyperspace will be reduced in n=2 for R(2)=Rmax(2/3)=2c/3Ho for the matter dominated cosmology in 10D/4D.

This then maps the 'nodally corrected' Hubble Law as Ho=2c/3R(2) onto the old H=2/3t.

The Friedmann-Lemaitre cosmology, incorporating the pressure and lambda terms solves in terms of the curvature k=1/Ro^2. We use R=aRo and R(dot)=a(dot)Ro and a(dot)/a=R(dot)/R=H and the definition of Omega (W)=r/rcritical=8pGr/3Ho^2 andwrite (1) as a COSMOLOGICAL EQUATION:

R(dot)^2=8pGrR^2/3+LR^2/3-kc^2 .

So curvature kc^2=R^2{Ho^2W+L/3-Ho^2}=(aRo)^2{Ho^2(W-1)+L/3}..............(Curvature*)

Then for (R/c)^2=(aRo/c)^2=(R(n)/c)^2=(Rmax(n/(n+1))/c)^2=(a/Ho)^2, k=0 iff (a/Ho)^2{Ho^2(W-1)+L/3}=0

which is the case for W=1 and L=0 OR for L(W-1)=-3Ho^2 (as frequency squared cycle units).

In the demetricated scenario W=0.028 with a varying L as quintessence in the mass parametric and open cosmology, however encompassed by W=1 in the electromagnetic oscillatory closure.

Then as Ho=1.877728045x10^-18 1/s* , 'L'=constant=3Ho^2/0.972=1.0882292x10^-35 and of the order of the Planck Scale.

This indeed is the experimental observation of the 'cosmological constancy'.

The curvature is 1 for 'L'=3Ho^2{2-W+2/n+1/n^2} and a function of cyclenumber n however.

Then for the initial condition of the inflaton and the instanton, n=nps=lps/Rmax=6.259..x10^-49 and 'L' is upper bounded in 3c^2/lps^2=2.7x10^61 frequency units.

This becomes simply 3fps^2 as the source frequency in the demetrication for the constant 'L', expressed in the quintessence of the variable L in the de Broglie inflaton of the 0-node discussed later.

For n=1 and the first odd node at the semi-wavelength for the Hubble Oscillation 'L'=5.26x10^-35 frequency units for the 0 approximation.

At the completion of the Hubble Oscillation n=2 and 'L'=3.41x10^-35 frequency units and decreasing towards 0 after infinite time t=n/Ho.

This then solves the cosmological constant dilemma in the superpositioning of spacetimes.

Next we define 1+z=a/ao specifying the deceleration parameter q.

q=-[a(doubledot)/a]/[a(dot)/a]^2=-(a(doubledot)a)/(a(dot))^2n=-(a(doubledot)/a)/H^2 in say Taylor-Expansion a(t) about t=to, that is some initial time to where a=ao and H=Ho, which in the demetrication is a double value Rmin=lps=c/fps,Ro=Rmax=c/Ho for k=0,1 for no=nps,infinity limit and ao=no/(no+1) or

lps/Rmax and 1 respectively.

a(t)=a(to)+a(dot)(to)[t-to]+(a(doubledot)(to)/2)[t-to]^2+...=ao{1+Ho[t-to]-(qo/2)Ho^2[t-to]^2+...}.

So 1+z=1+Ho[t-to]+(-qo/2)Ho^2[t-to]^2+...=a/ao=fo/f=Ho/H.

Then density ro=(a/ao)^3.W.rcritical=(a/ao)^3.W.3H^2/8pG...........(boundary density).

This becomes (1+z)^2={(1+v/c)/(1-v/c)} in demetrication with v=c/(1+n)^2=R(dot).

Then (1+z)^2=(n^2+2n+2)/(n^2+2n)=1+2/{(n+1)^2-1} and

a/ao=1+z=Sqrt{1 + 2/[(n+1)^2-1]}=Sqrt{1+2/[(c/v)-1]}..............................Expansion-Redshift-Parameter

For the boundary conditions , given by ao then, (1) is written with k=(a.Ho/c)^2{W-1+L/3Ho^2}:

(ao(dot))^2=8pGroao^2/3-kc^2+L.ao^2/3=8pGroao^2/3-(aHo)^2.(W-1)+(ao^2-a^2)L/3}

So (a(dot)/a)^2=Ho^2{W.a/ao-(W-1)}+((ao/a)^2-1)L/3.................Friedmann-Lemaitre Equation of motion

For W=1=2q and L=0 (a(dot)/a)^2=Ho^2(1+z)

For W=0.028 and k=1, L=3Ho^2( Rmax/a+0.972) for the curvature limit k=1/Rmax^2=1.

2a(doubledot)/a=-{8pG/3c^2}S[ri*c^2+3pi]......................(1) (Raychaudhuri Equation)

Here G is the Gravitational Constant, c the lightspeed r(rho) is the mass-density and p is the pressure, both summed over the universe (Si).

I shall attempt to simplify the Raychaudhuri formula in demetrication.

It uses a radius-scalefactor aR(to)=R(t), a velocity a(dot)Ro=v(t)=dR(t)/dt and an acceleration a(doubledot)Ro=a(t)=dv(t)/dt=d^2(R(t))/dt^2.

It then defines the Hubble-Constant as the ratio of 'Ho'=v(t)/R(t)=a(dot)/a as the Hubble-Law.

We shall show, that this ratio, which is termed epoch-dependent, is indeed not a constant, but varies as a function of inverse square relative to a true NODAL HUBBLE CONSTANT, which we here denote as Ho to differentiate it from the changing Hubble Constant as H(t)=v(t)/R(t).

Now this scalefactor R(t)=a.Ro means that some 'size' of the universe at time to was characterised by the scale Ro, which has a, called the expansion parameter, naturally getting bigger in time.

But what about this original size of the universe Ro?

Contemporary cosmology says that Ro was the size of a 'grapefruit' and has now grown to the size of the Hubble radius of magnitude between 10 and 20 Billion lightyears.

Now the expansion parameter a does indeed increase as we shall show, but the Ro is actually a maximum size, which R(t) is increasing towards asymptotically in an oscillating series expressible in the sequence 0/1, 1/2, 2/3, 3/4, 4/5,...,n/(n+1) thus eternally approaching but never reaching unity 1.

So the contemporary model is mathematically correct, but then begins to assume the Hubble Relation to DECREASE in the H=a(dot)/a definition as a constancy, albeit epoch dependent.

If a increases, as it must, then a(dot) must increase in the same proportion for constant H(t).

Equation (1) then says that a(doubledot)/a=-4pG/3{r+3p/c^2}/H^2=deceleration parameter q........(1*),

because differentiating a as dimensionless fraction for the expansion of scalefactor R(t) with respect to time t is a(dot)H and differentiating again for acceleration is a(doubledot)H^2 for the correct units for acceleration in displacement per time squared.

q=-(a(doubledot)/a)/(a(dot)/a)^2=-a(doubledot)a/a(dot)^2=L(nps)/(Rmax.fps^2)=GoMo/(c^2.Rmax^2) and qo=Mo/(2Mcritical)=ro.Vo/(2rcritical.Vmax)={4pGoro/3Ho^2}(Ro/Rmax)^3..........(14)

Then the standard cosmology reappears in distributing the density ro(Ro/Rmax)^3=r+3P/c^2, that is

Pressure P=(c^2/3){ro[Ro/Rmax]^3-r},

P=0 for ro=r given by the mass seedling Mo and the Omega of 0.028=2qo which assumes Ro=Rmax in the superposed curvatures of the boundary conditions k=0,1.

So we see that the Lamda Quintessence is incorporated into standard cosmology as the evolution of the universe bounded in its curvatures.

This means, that the evolving cosmology uses a negative hyperbolic curvature for a open universe, yet bounded in its asymptote of k=0 for the expansion parameter a=n/(n+1) approaching 1 in infinite time t=n/Ho.

This superposes the fixed W=2qo for the required Euclidean flatness onto the quintessential L from the inflaton nps=lps/Rmax.

Curvature k=R^2{Ho^2W+L/3-Ho^2}=(aRo/c)^2{Ho^2(W-1)+L/3} then becomes naturally -1 for W='L'=0 and Ho=c/Rmax=c/Ro, implying that a=1 that is the limit for k=0 and k=1.

But this cosmological constant 'L' is NOT the same as the quintessence and can be set equal to 0 just as originally proposed by Albert Einstein.

The demetricated expansion parameters then become:

a=n/(n+1); a(dot)=Ho/(n+1)^2 and a(doubledot)=-2Ho^2/(n+1)^3.

Now solving qo=Wo/2=Mo/2Mcritical=0.01405.. DEFINES no=qo at redshift z=7.477 and gives in the definition of (14) q(n)=2n; (generally for n=a/(1-a), (aq(n)=2 for a=X for the acceleration redshift discussed later).

So q(no)=2qo=2no=Wo for a radiation dominated universe with r=3P/c^2 and as the boundary condition for the superposed curvatures.

The deceleration parameter of the standard cosmology so defines this limiting threshold between a radiation dominated universe with qo=Wo and a matter dominated universe for which qo=Wo/2.

q(n)=2n and q(a)=2a/(1-a) the latter function increasing q by 2 for any half-cycle Hubble Oscillation.

q(0)=0; q(1/2)=2; q(2/3)=4; q(3/4)=6; q(a=n/(n+1))=2/(1/n)=2n.

Now this shows that for any n >1/4 the universe exceeds the threshold q(n=0.25)=1/2 and the standard cosmological parameter begins to diverge from its stationary value qo.

n=0.25 defines a universe of age 1/4Ho of so 4.225 billion years and at a recessional velocity of (16/25)c for which a 64% (v/c) ratio implies a cosmological redshift of z=Sqrt(1.64/0.36)-1=1.13437...

This indicates the 'acceleration' scenario manifested at n=0.618..so 6.23 billion years from the qo-threshold and so 8.67 billion years in the past.

The standard cosmological equations then are solved in writing R(doubledot)=a(doubledot)Rmax

or R(doubledot)=-GM(R)/R^2 +Q(n), where Q(n)=Q(R/(Rmax-R)) via R(n)=Rmax(n/n+1).

-2cHo/(n+1)^3=a(doubledot)Rmax=-(4pr(n)G(n)/3)R(n)+Q(n), considering the distribution of densities and limiting value for the deceleration parameter q as discussed with Q(n) forming the quintessence as Q(n)=Q(a/(1-a)).

This reincorporates q but not as function of the Hubble-Law.

Rather it uses its limit in the curvatures and in W and incorporates L in the form of a(doubledot).

-2cHo/(n+1)^3= -(4pG(n)/3)(3M(n)/4pR(n)^3)(R(n))+Q(n)=-G(n)M(n)/R(n)^2+Q(n).

It is not required to solve for the expansion parameter as a function of n, but one can solve for the Quintessence Q(n).

Q(n)=G(n)M(n)/R(n)^2-2cHo/(n+1)^3 = Omega-Factor + Deceleration-Milgrom-Factor=Lambda-Factor

It can be shown, that M(n)=Mo.Sqrt{Y^n} and that G(n)=Go.X^n with X=1/Y as pentagonal symmetry parameters of the cosmogenesis.

It is then evident, that the demetrication of the General Relativity parameters in dimensionless cycletime n not ony defines the expansion parameter a, but also redefines the Hubble Law in its boundary conditions.

Now in Quantum Relativity, Ro=Rmax and a=n/(n+1) defining the series above as a sequence of the expansion parameter over time, which becomes demetricated in parameter n=Hot.

Then R(n)=Rmax(n/(n+1)) and a(dot)=Ho.Rmax/(n+1)^2 with a(doubledot)=-2Ho^2.Rmax/(n+1)^3.

But HoRmax=c and the metricated acceleration parameter in General Relativity becomes reformulated as demetricated a(doubledot)=-2cHo/(n+1)^3 in Quantum Relativity.

The equation for dynamical motion for the universe then becomes in General Relativity:

a(doubledot)/a = -4pG/3{r+3p/c^2} and -2cHo/n(n+1)^2 in Quantum Relativity for a zero cosmological constant.

Integrating a(doubledot)/a via acceleration A=vdv/dx=d(v^2/2)/dx with respect to time and for r=aro and r(dot)=a(dot)ro and r(doubledot)=a(doubledot)ro and M=rV=4prr^3/3 gives:

Sr(doubledot)/(r)dr=S(GM(r)/r^3)dr and d(r(dot)^2)/dt=2r(dot)r(doubledot)=-2GM(r).d(1/r)/dt=2GM(r)r(dot)/r^2.

As r(dot)^2=a(dot)^2ro^2=2GM(r)/r -c^2, with constant of integration ro(dot)^2=a(dot)^2ro^2=-c^2.

Hence [a(dot)/a]^2=2GM(r)/(ra^2ro^2)-c^2/(a^2ro^2)=8prG(r/aro)^2/3 -{c/aro}^2=8prG/3-{c/aro}^2...(9).

Now deceleration parameter q=(a(doubledot)/a)/H^2=(4pG/3)(r+3P/c^2)/H^2 with H^2=8prcriticalG/3 and so as r=rcriticalW q=(W/2)(1+3p/rc^2), showing that for a flat universe, W=1 and q=1/2 for p=0.

Equation (9) simply adds L/3 as the 'cosmological constant', which if large overpowers the other terms in

a(dot)/a=Sqrt(L/3) for a(dot)=a.Sqrt(L/3), which on integrating da/a=Sqrt(L/3) yields ln(a)=Sqrt(L/3) and the exponential evolution of a(t)=exp[Sqrt(L/3)t.

(-8p

We shall then find that the relationship between Ho as the nodal Hubble Constant becomes the proportionality constant for the changing Hubble-Constant as a function of parameter n.

Specifically the timederivative of n is Ho in dn/dt=Ho and n can be said to be the dimensionless tau-time in General Relativity (GR), which allows the partitioning of the derivatives as chains.

Example (actually used in GR):

Acceleration a=dv/dt=dv/dx*dx/dt=v*dv/dx=d(v^2/2)/dx.

This eliminates time in the acceleration parameter and represents acceleration as a function of displacement x.

Considering the universe to be a summation of density states of pressure p and density r then we rewrite the Raychaudhuri equation:

Acceleration dR^2(t)/dt^2 = -(4pGR(t)/3)S{r+3p/c^2}.

Now we can relate the 'Equation of State' in the relationship between mass-density r and pressure p via the summation of the parameters enclosed in {} as say dimensionless integer states.

r=Mass/Volume and p=Force/Area=Energy/Volume so (M)ass*p=(E)nergy*r or Mp=Mc^2*r.

The Equation of State is then w=p/r*c^2.

For relativistic matter and radiation w=1/3 and for nonrelativistic matter and radiation w=0. This indicates that the pressure p=0 for any density r considered in the invariance of lightspeed c, that is demetrication.

What does this mean?

It means that an universe subject to its own boundary condition (the lightmatrix as the aether of continuity) is nonrelativistic as a 'cosmos of light'.

This is the modular duality of the 11th dimension of the superbranes.

The universe in 11D then never changes its own invariant matrix parameter of invariant c.

This is precisely the postulate of Special Relativity and can so be seen as a direct consequence of Quantum Relativity as the Demetrication of General Relativity using Special Relativity in the process.

The 11D universe is 'Steady State' in encompassment of the lower dimensional universe, subject to the pressure variations and the velocities of inertial objects upper bounded in the invariant 'c'.

The modular duality then allows a minimum frequency to be be in c-proportionality to a maximum frequency.

This minimum frequency is the nodal Hubble Constant Ho and the maximum frequency is the frequency as embodied in the volume-invariant Weyl-Tensor and denoted as fps.

The invariance of the relativistic cosmos is then given in the identity (of superbrane parameters):

c=lps.fps=Rmax.Ho or Ho=c/Rmax...............................................(4).

Now Rmax is simply the limiting R(t) in GR as the scalefactor said to be taken as unity for the Hubble-Epoch.

So GR uses the ratio R(t)/Rmax or a/ao as the scale factor for cosmological distances.

But the 11D cosmic 'steady state' represents the electromagnetic universe, always travelling with c-speed in cycles of expansion and contraction.

That means, that the universe oscillates between even nodes as defined in maximum frequency fps and odd nodes defined in minimum frequency Ho.

But we shall now return to the metric universe as the complexified 10D universe which does not oscillate but expands asymptotically as required by the critical density rcritical for the Euclidean flatness experimentally observed.

The informed reader can then witness, how the metric universe in the public realm is simply the consequence of a demetricated universe as indicated in the above.

We begin our demetrication process in proposing the socalled critical density rcritical for a perfectly flat Euclidean universe to be embodied in the equation of state.

We then rewrite (1) as -(4pGR(t)*rcritical/3)S(1+3p/rcritical*c^2).

Conventional (nondemetricated GR) then derives this critical density in differentiating between the summation of density- and prressure states in partitioning the Raychaudhuri expression into its density part (2) and pressure part (3) and introducing curvature k= (0,+1,-1).

v(n)^2/c^2 +k = 8pGrR(t)^2/3c^2.......................................................(2) and

2a(t)/a + {v(t)^2+kc^2}/R(t)^2 = -8pGp/c^2..................................(3).

Standard Cosmology now introduces the Omega Factor as the ratio between density ri and rcritical or W=ri/rcritical as alternative to the summation states of matter, radiation and the lambda (L) 'cosmological constant' now said to be 'dark energy'; i.e. rtotal=rMass+rRadiation+rL.

The important point here is, that Standard Cosmology assumes the present scalefactor to be ao as the reference of 'looking back in time' for a Earth-based 'present time'.

The correct scenario is that ao=Rmax IS the fixed scalefactor for all cosmic times given in (linear) time t=n/Ho. Then of course dn/dt=Ho as the odd-nodal and true Hubble constant.

I should say here, that this misinterpreted application of the GR scale factor leads to the 'problems in cosmology' associated with space expansion and the redshift scenarios.

Then the critical density derives from the acceleration coefficient and the incorrect Hubble-Constant definition as 'Ho'=v(t)/R(t).

8prcritical*GR(t)^2/3c^2 =1 and rcritical=3c^2/(8pGR(t)^2)=3'Ho'^2/8pG in assuming that v(t)=c for R(t)=Rmax for a 'present Hubble epoch'.

But this is precisely what I have described earlier as the invariance of the 11D universe.

So in the demetricated scenario we use (2) to show that the 11D universe is simply an encompassing 'Mother Black Hole' of Schwarzschild solution with scale Rmax and critical Volume Vcritical for a critical Mass Mcritical, which AS the critical density renders the universe flat with k=0 and W=1; just as required in the Standard Big Bang Model.

Then rcritical derives classically as Rmax=2GMcritical/c^2 for rcritical=Mcritical/Vcritical=Rmaxc^2/(2G*[4pRmax^3/3])=3c^2/(8pRmax^2)=3Ho^2/8pG by definition of (4).

Equation ( 1*) now becomes deceleration parameter q=(1/2rcritical)(r+3p/c^2)=(W/2){1+3p/rc^2}.

This means that q=1/2 for a matter dominated universe with pressure p=0 (or small) and that q=1 for a radiation dominated universe r=3p/c^2 because the equation of state defines w=1/3.

The basic Cosmological Equation incorporating the curvature k becomes then:

R(dot)^2 = 8prGR^2/3 + LR^2/3 - kc^2

k=Ho^2[Wo^2-1]+L/3]/c^2.

So k=0 for W=1 and L=0 and k=1 for W=1 and L=3 and k=-1 for W=1 and L=-3 for a mass dominated universe of the Friedmann-Lemaitre solutions for the Einstein Metric

ds^2=

Next the flatness of the universe is the ASYMPTOTIC boundary of 11D, allowing the oscillating electromagnetic universe to form the 'aether' for the 10D universe.

Please note, that a 10D universe is a 4D universe with 6D congruent with the spacetime of the relativistic Minkowski-Einstein-Riemann spacetime.

It is known as the Riemann Hypersphere or the Poncare 3-Sphere topologically.

So while I often mention the 10D universe, there is nothing 'hidden' just multivalued parameters because of the freedom degrees represented by volumes containing scaled subvolumes and the like.

3 Degrees of translation become multivalued in 3 degrees of rotation (each about one of the translations, say XYZ) and then in 3 degrees of vibration for a 9D 'spacial'or volumed cosmos.

Then linear time becomes the 10th dimension for the metric universum.

Modular duality introduces nondimensional cycletime n and 3 degrees of quantisation for a 12D continuum in which the 12th dimension accomodates the EXPANSION of the Rmax Limit into the undefined 'void' becoming defined and is imaged in the 11D of the 'Witten Mirror' then being the 10D of the asymptotic spacetime which does not exceed Rmax because of the c-invariance.

Furthermore the curvature k is now found to be multivalued.

Euclidean flatness implies k=0 for the asymptotic expansion and the W=1.

But we also now know, that the universe is closed in 11D and open in 12D, 12D (the Father-Space for the Mother-Space Black Hole to expand into).

So the positive curvature k=1 depicts the electromagnetic universe superposed onto the 10D asymptoticone and imaged in the negative curvature of k=-1 as the 12D/10D mirror mapping.

We now simplify (2) to formalise this as the reformulation of the Hubble Law.

The asymptotic expansion is given by the demetricated scale factor R(n) as a function of the cosmic nodal frequencies or Hubble Constants from maximum fps= to minimum Ho=c/Rmax.

So scalefactor a=R(n) describes a decreasing curvature from metric maximum warpspace given by wormhole wavelength lps to the asymptotic bound of Rmax.

An expression for R(n)can be derived a number of ways, but is based on the definition for the natural exponent 'e'=limit[n->Infinity] {1+1/n}^n; the derivative of the basic exponential function being itself of course in d(e^x)/dx=e^x.

R(n)=Rmax{1-n/(1+n)}=Rmax(n/(n+1)) is the result.

Now recall the dimensionless nature of n=Hot which so demetricates all of GR in substitution R(n=Hot).

We have elimninated linear time in analysing the universe as an oscillating wave in 11 dimensions.

Then the position of the evolving 10-dimensional universe is a function of the nodal frequencies and the scenario of the expanding universe into the void becomes one of an oscillating universe with precise boundary parameters.

The n-derivative for this displacement of the scalefactor is simply V(n)=dR(n)/dn=Rmax/(n+1)^2.

To reclaim the linear t-parameter, we partition the derivative as V(n)=dR(n)/dt*dt/dn for v(t)=dR(t)/dt.

Then V(n).Ho=v(t)=Ho.Rmax/(n+1)^2=c/(n+1)^2, as required by the definitions.

In (2) we so write the Special Relativity ratio v(t)^2/c^2=(dR/dn*dn/dt)^2/c^2=(dR/dn)^2*Ho^2/c^2=(dR/dn)^2/Rmax^2.

So the reader sees, that the velocity units depending on time have become swallowed up in the demetrication of cycletime n in scale factor R(n).

We now proceed to relate v(t) to v(n) in finestructure and definition of R(n), thus finding explicit expressions for the Hubble Constant as function H(n).

H(t)=v(t)/R(t)=Ho*Rmax/{(n+1)2*R(n)}=Ho*(n+1)/{n(n+1)^2}=Ho/T(n) where T(n)=n(n+1)............(5).

Thus the function T(n)=Ho/H(t) redefines the Hubble Law and is bounded in the nodal frequencies.

T(n) is simply the EXPANSION PARAMETER denoted as 'a' in the standard cosmology and a(dot) there becomes a(dot)=(da/dn)(dn/dt)=Ho/(1+n)^2 with the Hubble Law as v/(R(n)=a(dot)/a=[Ho(n+1)]/[n(n+1)^2].

At the Beginning of SpaceTime, the time to (say) specifies the maximum Hubble-Constant as fps.

This is termed the timeinstanton of Inflation and also defines the eigenfrequencies for all mass/inertia parameters in modular duality as the inverse time-frequency modulation.

To derive H(to=1/fps), we simply use our definitions n=Hot and R(n).

H(to)=Ho/[nps(nps+1)] where nps=min/max=lps/Rmax ratio of R(n) in the nodes.

So H(to)=Ho.Rmax/[lps(lps/Rmax+1)]=c/lps=fps in the limit of nps->0.

This limit condition is satisfied in superbrane parameters as it is found that Rmax=1.596..x10^26 metres* and lps=10^-22 metres* for nps=6.265...x10^-49 dimensionless.

This eliminates the 'dreaded singularity' in General Relativity for the infinity reduction of physical quantities, such as pressure and density and temperature.

The infinite asymptotic approach for the 10D universe would of course diminish H(t) linearly to 0 for n approaching Infinity in T(n)=n(n+1).

This is simply the modular duality between the mathematical inversion properties relating 0,1 and Infinity.

Our velocity v(t) relationship is then simply v(t)/c=R(t)/(Rmax*T(n)), that is the linear recessional velocity v(t) defined in the relativistic redshift (z) formulation (6) is v(t)=c*R(n)/Rmax*T(n)=c/(n+1)^2 as required.

The cosmological redshift relation is: 1+z=Sqrt{(1+v/c)/(1-v/c)}....................(6)

2. The cosmic acceleration redshift

Using (6) and the insights of section 1., we can see that the universe is a scaled universe of basic simplicity.

Whilst the mathematical formalisms of the partitioned universe are formidable, its basic geometrical nature should be comprehensible by the scientific novice.

We particularly note the importance of the v(t)/c velocity ratio of Special Relativity.

It is ubiquitous in cosmology and is shown to be a binomially distributed in Quantum Relativity as parent theory for both General and Special Relativity.

Our demetricated velocity V(n=Hot)=Ho.v(t)=Ho.c/(n+1)^2=Rmax/(n+1)^2.

In (6) we use v(t)/c=1/(n+1)^2 to calculate the cosmological redshifts.

We see that NO HUBBLE CONSTANT is required to calculate cosmological distances, once the COSMOLOGICAL REDSHIFT, describing the expansion of the 10D universe within its own 11D boundary is known.

The redshift must be z-cosmological and NOT any superposed motion of galaxy, quasar or cosmological object relative to its surroundings however.

The cosmological redshift is a pure interdimensional 'space' parameter, bounded in H(t)=Ho/T(n) as described.

So the simple consequence of the oscillating higher dimensionality is that the asymptotic bound of the n=1 odd node has already been reached 'presently' and defines the half-cyclic oscillation of the Hubble-Universe.

But WHEN is or was the n=1 nodal universe EQUAL to itself?

When did the 10D asymptotic limit COINCIDE with the first full 11D cycle?

This cyclic NOW-Time is defined as about 16.9 Billion years of sidereal measurement.

So it takes LIGHT to travel Rmax in that linear time interval from n=0 (nps) to n=1.

So in 11D the asymptotic limit in 10D is attained in 16.9 Billion years from timeinstantenuity or the 'Quantum Relativistic Big Bang'.

Now the nodal Hubble Constant is Ho=58.04 km/Mpc.s (defined via Ho=c/Rmax) for the linear time t(1)=n(1)/Ho as the minimum frequency and increases for increasing fractional n towards the maximum frequency fps at the odd nodes again.

So the H(t) function oscillates between its stationary values with a present H(tpresent) calculated in extrapolation as so 66.9 km/Mpc.s and supported as average by measurements (which presume the 'constant' Hubble Law and hence show a large spread in the error bars).

But the observed cosmic acceleration has shown that there is something amiss with the standard interpretation and the discrepancy is the relationship between the space-volume expansion as hypothesised and a PREEXISTING higher D 'space' which is independent on the inertial variables, being bounded by the critical density parameters of part 1.

What we find is that the linearity of the asymptotic expansion given in R(n)=Rmax(n^2/T(n)) imposes a restriction on the cycle time n; namely the condition of actually IDENTIFYING H(t)=Ho.

So there will be a special moment in the linear evolution for the universe when the two cosmic eigenfrequency values of maximum fps and minimum Ho become modularly 'averaged' in a Mean Hubble-Constant Haverage.

Recall v(t)/c=R(n)/{Rmax*T(n)} or rewriting T(n)=R(n)*c/(Rmax*v(t)).

So the asymptotic scale factor or 'size of the universe' always less than its maximum becomes 'harmonised' in c always exceeding the 'group'-velocities of the partitioned wavestates and measured as the velocities of the particles or inertial objects.

Also solving v(t) for n gives you the demetricated cosmological redshift relation n=Sqrt(c/v(t))-1...(7).

As v(t)<=c for all (groupvelocities) v(t); n is always positive but allows for a special value somewhere in between the two nodes, i.e. for the n-interval (0,1), mapped as -(1,2) in the full Hubble Oscillation.

H(n=0)=fps and H(n=1)=Ho with H(X)=Ho/T(X)=Ho/(X(X+1)).

Next we unitise H(t)=Ho/T(n) that is we identify H(X)=Ho; a condition requiring T(n)=1=n(n+1).

Solving the quadratic n^2+n-1=0=(n+1/2)^2-1/4 shows n1=(Sqrt(5)-1)/2 as a positive root and n2=-(Sqrt(5)+1)/2 as a negative root with the property that n1+n2=n1*n2=-1.

We so identify X=n1 and Y=n2; with X setting the REAL TIME tX=X/Ho in the 10D asymptotic mass-parametric cosmolology and Y setting the IMAGINARY TIME tY=Y/Ho in its 12D mirror-image in 'OmniSpace'.

We find in Quantum Relativity, that a cosmic wavefunction descriptive of this 'omnispace' is perfectly reflective about a functional Riemann Bound of n=-1/2 and as consequence of normalising the quadratic root identity from above.

But now we know what the special n-coordinate is, namely n=X=0.618033...

Using (6) with (7) we can calculate the corresponding cosmological redshift, which superposes the nodal limits for the 10D universe as its linearised oscillation of the 11D 'envelope'.

v(n1)/c=1/(n1+1)^2=1/(-Y)^2=X^2=(3-Sqrt(5))/2=0.381966011...., meaning that the 'special moment' in the cosmic evolution occurred when the universe of mass and inertia receeded from its own singularity (which is nps as '0' finitised) at 38.2% of matrix lightspeed.

(6) now calculates the PRECISE cosmological redshift as z=Sqrt{(1+X^2)/(1-X^2)}-1=0.49534878...

Setting the displacement of the extrapolated Hubble Constant at coordinate x in (0,1) gives

Haverage=Ho=/(1-X)=58.04 km/Mpc.s/0.381966011..=151.95...km/Mpc.s.

This corresponds to a linear Hubble-Time of taverage=n1/Ho~10.4 Billion years after timeinstantenuity or (19.1-10.4) =8.7 Billion years in the past.

This is the approximation in the standard cosmology for the cosmic acceleration onset, using the incorrect version of the Hubble Law.

It also shows, that the universe is consistently measured as 'too young' with H(t) too big, compared to say nuclear synthesis in the oldest stellar populations. Edwin Hubble himself calculated a Hubble-Constant in the hundreds, using his basic 'galaxies as nebulae' data.

Should the reader consult countless references as to WHEN the universe appears to have begun changing from deceleration to acceleration (observed by analysis of supernova type Ia luminosity curves), then a redshift of 0.5 always becomes the approximate parameter.

Here then I have shown the limit for any such approximations and the point to which all cosmological measurements of increasing precision will converge to.

3. The cosmic evolution of the 10D-asymptotic universe in demetrication

We now show the Schwarzschild solution to be a direct consequence of the Raychaudhuri equation in the Ricci Tensor of Mass-Density and show how this solution combines the curvatures as described in part 1.

(2) is written as: v(t)^2/c^2 +k=8pGr*R(n)^2/3c^2 and as the basic solution for the Riemann-Einstein metric.

We now substitute a 3D volume for the density r(n)=3M/4pR(n)^3 into the RHS of this equation to get 2GM*R(n)^2/R(n)^3c^2=2GM/R(n)c^2.

Then we multiply both LHS and RHS by scalefactor R(n) to get the RHS as the Schwarzschild metric and Radius for a Black Hole (nonrotating and uncharged and as applicable for the universe as a whole).

The LHS becomes R(n){[v(t)/c]^2 +k}=2GM/c^2=RHS...........(2*).

We have seen in part 2. that the velocity ratio of Special Relativity is a key component for the cosmology and can be rewritten as R(n)/RmaxT(n)=1/(1+n)^2.

This allows us to engage the modular duality in the inversion property, say of the curvature R(n) for the scale factor and its reciprocal 1/R(n) as the curvaturre of the surface.

Equality of R(n)=Rmax invokes the 'singularity' condition at the time instanton nps=lps/Rmax to become the modular dual for the infinite asymptotic expansion in n/(n+1)=1, i.e. n=1+n for curvature k=1 mapping curvature k=0 in the 1<->0 mapping Rmax at infinite linear time onto lps at tps time.

We so map the scalefactor R(n) onto the expression Rmax^2/R(n) in (2*), effectively transforming the velocity ratio term from R(n)^2(/Rmax.T(n)) into R(n)/T(n)^2.

(2*) then becomes R(n){1/T(n)^2 +k}=2GM/c^2..............................(2**)

and incorporates the multiplicity of curvatures.

This transformation then uses the inversion of R(n) in the reciprocal of n/(n+1) as 1+1/n which is the exponential generator.

Our previous redefinition for the Hubble Law at cosmological redshift zHubble=0.49534.. reappears in the form of the 1/T(n)^2 factor requiring to be 1 for the R(n)=Rmax condition of the modular mapping.

Write {v(t)/c}*{Rmax/R(n)}=v(t)/(Ho.R(t)) which, if it equals 1 leads to the 'Restricted' Hubble Law

Ho(t)=v(t)/R(t)={c/(n+1)^2}/{Rmax(1-1/(n+1))}=Ho/T(n).

So the 'restriction' is, that T(n)=1 is the requirement for the Hubble Law to hold.

T(n)=1 for n=X at the redshift zHubble=0.49534.. and Ho=c/Rmax at the even nodes in 33.8 Billion year intervals and beginning at a linear time of n=1 at 16.9 billion years.

SO INDEED, the universe is HUBBLE-ACCELERATING for all epoch (t) dependent Hubble-Constants, using Ho(t)=v(t)/R(t).

The Hubble Redshift implies the Hubble Constant as the HAverage=151.95 in extrapolation for a true recessional velocity of 0.382 c and a true 10D-scalefactor of 0.382 Rmax, which gives the nodal Hubble Constant Ho in unique constancy.

This then shows the 'meeting' between the expanding universe and the recessional 'lightpath'.

Mathematically this is the fact of X^2=(1-X) in the fivefolded cosmological symmetry.

For the Schwarzschild Solution to represent the entire universe we require the Radius R(n)=Rmax(n/(n+1)) to be Rmax and we recall (2**) as R(n){1/T(n)^2 + k}=2GM/c^2...................................(8).

The universe is Euclidean flat for k=0 and Omega=1 which would imply a critical mass density as given in the universe having a total mass of Mcritical=Rmax*c^2/2G=6.47058...x10^52 kg (in superbrane units which also use a varying G constant initialised in Go for WM=0 in terms of electric permittivity eo, i.e. a quasi-charged but massless universe).

This initialisation defines a corresponding M-seed Mo=1.81371...x10^51 kg.

And not surprisingly, the ratio Mo/Mcritical=0.02803.. which is the Omega for mass as ri/rcritical from part 1.

So anyone can see, that the universe can be closed and Euclidean flat, yet appear to be deficient in mass to ensure the asymptotic cosmos.

This mass deficiency is set by Omega as proportionality and grows in a DIM-Factor defined in the ratio of Volumes in the two Riemann-Spheres describing the 10D/11D interaction.

Since the universe is a 4D Hypersphere of toroidal volume 2p^2R(n)^3 in asymptotic 10D as a 3D-Surface and open in 12D with volume n.Rmax^3; this DIM-Factor becomes:

DIM=n(n+1)^3/n^3=(n+1)^3/n^2, calculating for the present n-time as about 7.56.

Then 7.56 10D Hyperspheres 'fit'into the 11D one and the 'missing' mass' or dark matter becomes Omega*DIM=(2.803%)(7.56)=21.20% incorporating the Baryonic Mass-Seed Mo.

Superposed onto Mo is however its 'missing mass' evolution, related by a decrease of the initialising gravitational G-constant value, keeping the productation G(n)mi(n)mj(n) constant for all n via the Euler Identity and the T(n) definitions.

G(n)=X^n.Go and the protonucleonic mass is upper bounded by mc.(Y^n). The Mo/mc ratio describing the number of elementary particles will be addressed shortly.

Then the Baryonic Mass Mo has grown by a factor of Sqrt(Y^n) or Y^ [n/2]~1.3132 for npresent~1.1324..

Omega*DIM*Growth-Factor so becomes the experiemntally observed 21.20%(1.3132)=27.84% of Mcritical.

What is required, is that this Mo itself represents a subset of the Rmax Hubble-Universe in Black Hole equivalence with a Schwarzschild Radius also forming a boundary condition for the asymptotically expanding 10D encompassed by the Rmax in 11D.

I call this sublimit for the demetricated General Relativity the Sarkar Radius, after the Oxford cosmologist Subir Sarkar, who first measured and analysed the large scale supercluster distribution of galaxies and who found a typical scale of about 300 Million Lightyears for them in the late 1990's.

The WM=0.02803...then 'closes' the universe as a Sarkar Black Hole with radius RSarkar=2GoMo/c^2 or so 4.4783x10^24 metres, which are so 473.4 Million lightyears.

WM=2*Deceleration-Paramerter qo for qo=0.0140 and a Sarkar-'Half-Scale' of 236.7 Million lightyears.

This 'half-scale' relates to the inflation scenario of the Big Bang cosmology, where a phasechange occurred to transform a preGR setting of five demetricated superstring classes into the Einstein Field Equation relating the volume invariant and volume changing (via pressure exerted by mass in the equations of state) parts of part 1.

The precursive energy directly from the superstrings was one of temperature and not of mass.

A 'false vacuum' quantum tunneled as a Temperature Gradient of superstring-tension from the well known Planck-Scale to the Weyl-Tensor scale as given by the maximum frequency fps defined in part 1.

Describing the 'Scalar Higgs Temperature Field' for this superstring tension then shows that the deceleration parameter qo=Mo/2Mcritical=Wo/2=GoMo/c^2Rmax and thus half the ordinary Schwarzschild metric.

That this is a natural consequence for demetrification can be seen classically in simply equating the Zero-Point-Energy of the absolute Temperature (which is arbitrarily definable) AS a MINIMUM FREQUENCY OSCILLATOR (which the universe IS as a microquantised superstring) with (E)nergy E=hf/2 and setting this as Gravitational Potential Energy E=GMminmmin/rmin for the superstring at the Weyl-Limit referenced by the mass Mmin for the universe.

Then hfps/2=mpsc^2/2=GoMmin*mps/rps and since 2prps=lps=c/fps we have rps=GoMmin/c^2 as the minimum scale for the Weyl-wormhole perimeter and relating a maximum superstring mass mps to a minimum 'macro-Black Hole mass' Mmin (which is half of 12,891.6 kg or so 6445.8 kg).

(There is a minimum superstring mass defined in, what else, but a Planck-Scale oscillation).

The mps-mass is very important too, transforming into the base nucleon mass (proton or neutron) from a later superstring of a different, yet related 'heterotic' class (HO(32)), the Weyl-class being HE(8x8).

This generally becomes the supersymmetric Higgs-Mass-Induction of the lepton-quark families described in the standard model of particle physics. It, needless to say, is the other major part in Quantum Relativity in tandem with the cosmology described here.

But the rps Schwarzschild Solution is the minimum scale for the Sarkar cosmic oscillation, just as it is the minimum scale for the Hubble oscillation.

So for the prudent reader, the asymptotic approach of the 10D universe can be seen to become infiltrated by the Sarkar Oscillation, preventing the Heat-Death' of the entropic universe in a 'running out' of nuclear fuel for the stellar generations of the birth and death of the stars (in the supernovae discussed in the references).

The 'recharge cycle' is found to be so 7.6 trillion years, coinciding with the projected 'heat death' by the way and is a function of the rps-wormhole and Sarkar Radius evolution (described by Strominger branes of extremal Black Holes which become massless as the elementary particles as transformed superbranes).

So the whole universe works in a harmony of modular duality relating its smallest constituents of the superstrings transforming into elementary particles (basically neutrons decaying into protons, electrons and antineutrinos) and its largest constituents, which are superclustered galaxy structures.

But this potential for energy to metamorphose into different forms was born in the gravitational potential of the zero-point-oscillator which is the EpsEss superbrane manifesting the physical universe in the birthing of General Relativity.

Because the superstrings are physical 'singularities', manifesting in the base nucleon, called an ylemic neutron (mass denoted as mc), which is a bosonic dineutron with superconductive properties.

As transformed mps-mass quantum, it gives rise to all the members of the 'particle-zoo' except the gauge-bopsons of the fields for the four fundamental interactions, which derive from a superinteraction uniting the superstring classes.

But relevant to our discussion of GR's demetrication; the Mo/mc mass ratio becomes an important cosmological parameter, because it defines the 'number of elementary particle' making up the mass-seed Mo, thus relating to definitions for the G-constant evolution, which is related to the evolution of the Einstein-Lambda (L) and so the 'dark energy' for cosmic closure under the curvatures k.

The Raychaudhuri equation (2) is called an 'acceleration' equation and we describe the earliest cosmology in terms of acceleration.

It should be almost self evident, that the decisive parameters for the inflation must engage the 'hyperacceleration' as a function of the Weyl-Frequency fps, acting on the Hubble-Radius Rmax.

This 'de Broglie' matter-wave inflation is simply Rmax.fps=4.793026....x10^56 metres per second as the inflation speed for a 'de Broglie' matter-wave acceleration' of Rmax.fps^2=1.437907...x10^87 metres per second squared.

But this is not the Einstein-Lambda of the 'dark energy', which is L(nps)=GoMo/lps^2=2.015..x10^85 acceleration units for R(nps)=Rmax(lps/Rmax)=lps in the 'singularity limit' (see part 1).

So now calculate the ratio of primordial 'dark energy' to 'de Broglie phase acceleration'.

It is L(nps)/Rmax.fps^2=0.0140.... WHICH IS PRECISELY THE DECELERATION PARAMETER qo and the half of the Mo/Mcritical mass ratio defining the mass content for the universe for the curvatures.

What does this mean?

It means that the 'false vacuum' for the universe became established in the Sarkar architecture of the sub Black Hole evolution of the superclusters, which REPRESENTS the deceleration parameter and hence the density evolution of the cosmos.

The corresponding cosmological redshift for n=0.0140.. is z=7.477 and represents the largest such redshift any cosmological body (say quasar or Gamma Ray Burster (GRB) can attain.

(8) describes just this.

Rmax{1/T(n)^2+k} for k=0 in (2) must give the Schwarzschild solution Rmax=2GMcritical/c^2.

So { } must equate to unity 1, which it does for our condition T(n)=1 and H(t)=Ho from part 1.

But this could also result in a nonzero k for the asymptotic approach as n tends to the infinity limit, which renders 1/T(n)^2 as limited by 0 for positive curvature k=1.

So the k=1 ellipsoidal curvature in 11D is equivalent to the k=0 curvature of the asymptotic 10D universe by definition of 8 via 2* and 2 and 1.

The hyperbolic curvature k=-1 for openness is simply the imaged complex solution for the k=1 scenario mirroring 10D-Omnispace in 12D-Omnispace.

One can see this in the half-scale and the definition process for the beforementioned cosmic wavefunction describing the cosmic wavefunction {B(n)=(2e/hA).exp[-Alpha.T(n)]} which is the n=-1/2 Functional-Riemann-Bound (FRB) of the Riemann-Hypothesis in pure number theory.

B(n) is a Gaussian Function of the form f(x)=A*exp{-[x-B]^2)/C^2} with constants A=(2e*exp[Alpha/4]/hA) and B=-1/2 and C=1/Sqrt(Alpha).

B(n) so integrates to Sqrt(p) as antiderivative or the error function as the standard normal distribution function with Standard Deviation s=1/Sqrt(2Alpha) and Mean m=-1/2. from substitution Z=(n+1/2)/(1/Sqrt(2Alpha) and dZ/dn=1/s=1/(1/Sqrt(2Alpha)).

The constant coefficient for the normalisation is then A*s.Sqrt(2p)=(2e/hA)*exp(Alpha/4)*Sqrt(p/Alpha).

To analyse B(n) (from the differential equation dB(n)/dT(n)+Alpha.B(n)=0), one uses the mirror property of OmniSpace in solving T(n)=1 and T(n)=-1.

T(n)=1 yields our Fibonacci Roots X and Y and the Euler Identity: XY=X+Y=-1=i^2=exp(ip).

But T(n)=-1 changes the sign of the constant in the generating quadratic n^2+n (+/-)1=0; which is just the curvature k in General Relativity.

The solution for T(n)=-1 as n^2+n+1=0 changes the exponential term in the B(n) to its inverse in modular duality and gives complex roots about the FRB as -1/2(+/-) iSqrt(3).

Now assume a 'doubling' of {} in (2**) to reattain the Schwarzschild solution from the 'de Broglie' inflationary preepoch of the superbranes.

Then {1/T(n)^2+k}=2, which for k=0 and the asymptote requires T(n)^2=1/2, namely the reflected FRB from the B(n) and the Riemann Hypothesis.

Alternatively, we can consider T(n)=1=k for n=X, again combining the curvatures for the asymptotic approach and the higher dimensional encompassment.

A finestructure for this curvature unity is our (R(n)/Rmax)*(Rmax/R(n))=(n/(n+1))*(1+1/n)=1 in the

modular duality which is prevalent in the pentagonal symmetry of omnispace.

Expressing a fractional curvature then finestructures k in say 2=1+k(1+Y) with k=1/(1+Y)=1/(X+2)=X^2.

(Actually the pentagonal symmetry in omnispace derives precisely the (X+2) mapping for the gravitational interaction to unify the symmetry broken in the four fundamental force interactions with invariant X mapping Alpha as the Electromagnetic Interaction and (X+2)=1/X^2 mapping X^3 in the Unification Polynomial U(C)=C^4+2C^3-C^2-2C+1=0 or [1-C][C][C+1][C+2]=1 as the G-Alpha=Alpha^18.)

The Standard Deviation for normalised B(n) is 1/Sqrt(2Alpha) and so the variance converges to n^2=1/2 with mean the FRB (say in Chi^2 Probability-Density-Distribution of the central limit theorem, which says that a normal distribution of means results for normally distributed individual means).

R(n) can also be written as R(n)=Rmax - ct/T(n) for T(n)=ct/(Rmax - R(n)).

The Radius of Curvature of a sphere say is the Radius R with its reciprocal 1/R the Radius of Curvature for the Surface.

This is just the applied premise of modular duality which is more elementary in allowing operational definitions for this inversion (in Fourier transforms of displacement and momentum vectors).

But if Rmax is the Hubble Radius for a 4D-Volume as a 3D-Surface, then 1/Rmax is the curvature of the

3D-Volume as a 2D-Surface that is Poincare's 3-Sphere as Riemann's Hypersphere.

Geometrically, the convergence of the squared standard distribution or variance to n^2=1/2 is the intersection of the curvatures in R(n)=Rmax(1-n/(n+1)) and the curvature function C(n)=Rmax/(2n+1).

n/(n+1)=1/(2n+1) gives 2n+1=1+1/n for 2n^2=1.

But the derivative of T(n) is T'(n)=dT(n)/dn=2n+1 which as stationary value is 0 for n=-1/2, that is the FRB.

So C(n)=Rmax/T'(n)=R(n)(1+1/n)/T'(n).

Recall that the derivative of the Logarithmic Functions are Inverse Functions, say if y=ln(x), then dy/dx=1/x and so integrating the inverse functions as the curvatures results in logarithmic functions or their expanded infinite series. This relates of course to the fact that if y=e^x, then its derivative and integal remains as itself in dy/dx=e^x.

The important generalisation is that the ratio f'(x)/f(x) integrates to ln(f(x)).

For example integrating 2x/x^2=2/x gives ln(x^2)=2ln(x) and integrating tan(x)=sin(x)/cos(x) gives -ln(cosx).

Another interesting aside in "" follows to show the intrinsic nature of the parameters in regards to the quantum physics which demetricate General Relativity.

""Further analysis of this produces the Bohr Atom from finestructure T(n)=1.

Let T(n)=1 map Rmax^2-R(n)^2=a^2-b^2=(a+b)(a-b)=T(n)=(n+1/2)^2-1/4 for a=(n+1/2) and b=1/2.

Form the ratio (a+b)/(a-b)=1+1/n as the definition basis for exponential and logarithmic functions.

a/b=2n+1 as T'(n) and so a/b=Rmax/R(n) reflects Radius R(n) in Curvature 1/R(n) (in analogy of superstring EpsEss coupling via constant h^2 (Planck's Constant squared) in partition Eps*Ess).

So for T(n)=1, T(n)=constant K^2=Rmax^2{1-n^2/(n+1)^2}=(2n+1).Rmax^2/(n+1)^2=(nRmax)^2.L, where constant L=T'(n)T(n)^2.

Convergence is established by: L=S[1/n^2 - 1/(n+1)^2] in the limit (n->Infinity){1/1-1/4+1/4-...-1/(n+1)^2}=1.

Then 1=K^2=L(nRmax)^2=(nRmax)^2*(1/n^2 -1/(n+1)^2).

But we already know that nRmax describes the expansion of the 11D universe into 12D F-space and that L represents the Rydberg Finestructure for the Bohr Atom.

{An electron of mass me and (de Broglie wavelength) le is defined in mec^2=hf=hc/le and for an electron to be absorbed or emitted in light matter interaction, its wavelength must be proportional to the electromagnetic finestructure constant Alpha in the Compton wavelength lCompton=Re/Alpha via 1/le=T(n)*mec^2/hc=T(n).Alpha.mec/(60pe^2) for proportionality constant T(n) and h=60pe^2/Alpha (from superbrane parameters and e the electron charge quantum).

But Re, the classical electron radius defining the quantum scale for all particles as the superstring quantisation of lps in Re=10^10*lps/360 can also be written in terms of the electron mass as Re=30e^2/mec, which sets 1/le=T(n).Alpha/2p.Re.='Constant'{1/n^2 - 1/(n+1)^2}.

The atomic quantisation number Z^2=2n^2 as the variance n^2=1/2 and the Rydberg Constant (at infinity) is Ry=Alpha^3/(4pRe) as 'Constant'=Z^2.Ry for finestructure T(n)=1=K^2, with (nRmax)^2 mapping (Alpha.Z)^2 /2 in the variance definition for Alpha as Alpha= G^2(1/2)/(2p.Variance).

The Gamma-Function G(1/2)=Sqrt(p) as intrinsic parameter of the normal statistical distribution and describes the Factorial Function as recurrence relation in analogy to T(n)=n(n+1) via G(x+1)=xG(x) and obtained in integrating the analytic Gamma-Function, defined analytically as {G(x)=Integral{ t^[x-1].e^ -t} dt in the interval [0,Infinity] by parts.

Then for x an integer G(n+1)=n!G(1) and the Gamma Function, starts the same way as the Fibonacci Series of Numbers (0,1,1,2,3,5,8,13,...) in G(0)=1=G(1)=G(2), G(3)=2G(2)=2, G(4)=3G(3)=6, etc. and as the factorials of the successive natural number count.

Use substitution t=u^2 for dt=2u.du in the integrand for x=1/2 to evaluate G(1/2).

Then the integrand changes to (u^2)^[x-1]*e^-u^2*2u.du=2u^[2x-1]*e^ [- u^2]du=2e^[-u^2], which is the error function or normal distribution function in the doubled infinity interval for x=1/2 and so Sqrt(p).

The first eigenstate (n=1) for the hydrogen atom becomes 1/ln=-Alpha^3.Z^2./4pRe with Rydberg Energy Ryhc=13.856 eV and first Bohr Radius RB1=2h^2eo^2.e*/e^4=5.217x10^-11 metres, where e* is the primordial magnetocharge defining superstring energy Eps=1/e*=lps.fps"".

Before continuing with part 3; I shall post this for your due consideration and commentary.

Best of Science from the gnosis to you all. Tony B.