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[Phys-l] Dark Energy From Quantum Foam



Dark Energy From Quantum Foam

A proposal by Michele Arzano, Thomas W. Kephart and Y. Jack Ng provides an
interesting and promising explanation for the "mystery" of Dark Energy. The
purpose of this post will be to outline this proposal as well as relate it to
the expanded particle state models proposed by Klauber, T Hooft, Sundrum,
Moffat and others in various forms.

Before last century, spacetime was regarded as a passive and absolute
background in which events took place. However, thanks to the work of Einstein we
now know this view of space and time, more properly referred to as space-time
is incorrect. With the discovery of Quantum mechanics the implication of the
Einstein's model of space-time becomes even more counter intuitive. Following
the work of Wheeler some physicists think that space-time has an ever
changing geometric and topological structure. The basic form of these space-time
fluctuations is described by the equation;


delta[L] = > L^(1-alpha)*L_plk^alpha

Where L is the distance measured and L_plk is the Planck length. If we
assume a smooth distribution of space-time fluctuations we get

alpha= 1/2.

We can test this possibility by measuring the fluctuations in phase of
distant quasars or other distant bright sources of EM radiation. We find that any
value of alpha < 0.6 is ruled out by data from the Hubble Space Telescope.
This model of Quantum foam, known as the Random Walk model is therefore ruled
out by observation.


However, based on the work of Susskind, T hooft and others we would expect
the fluctuations to scale to a value of alpha =2/3 rather than 1/2. We can see
this by applying the Margolus-Levitin theorem to the maximum possible
computation rate for any given volume of space-time. We get;


dI/dt= (2*M*c^2/pi*hbar)*(2*L/c)

Where M is the mass in the volume and L is the space-time measure of
this volume. But M for any given volume must follow the relationship below;


M=< L*c^2/(2*G)


to prevent black hole formation.

Taken together this means that

dI/dt < = 2*(L_L_plk)^2/pi

We can get only computation one "click" over the time interval 2*L/c


This can be regarded as partitioning the space-time volume into "cells"
occupying a space-time volume of 2*pi^2*L_L_plk^2/3


Giving us a minimum uncertainty of


Delta[L]= L^1/3*L_plk^2/3


Where

Alpha=2/3

Which is within the observational bound of the HST.


A much simpler argument is possible by a direct application of the
Holographic information bound which gives us


L^3/delta[L]^3= L^2/L_plk^2


delta[L]= L^1/3*L_plk^2/3

Alpha=2/3


Therefore this model of space-time foam is known as the Holographic Foam
Cosmology. (HFC)


In this model the maximal spatial resolution is possible only if the
maximum energy density is available to map the geometry of any given
space-time region. And this energy is bound by the limit of mass for any given volume
based on black hole formation. Therefore;

rho= (L*L_plk)^2.

(Hereafter contents of order unity will be dropped in most cases)


However, the above argument applies to a static space time, i.e., a
space-time with a constant scale factor.


d/a/dt=0


Of course this is not the Universe we live in. Therefore we must generalize
this argument. We can do by observing the relationship;


L=/H

Where H is the Hubble parameter.

H=(da/dt)*(1/a)


In our Universe H seems to be a constant, or nearly a constant, due the
acceleration of the expansion rate of the scale factor. In fact we might argue
that the parameter H is bound the HFC to a constant value. This would bound
the equation of state parameter to-1, providing an addition observational test
of the HFC model, the other being the constrain on phase shifting of EM
radiation from distant sources in the Universe.


We can therefore see that


R=G*M

M=R/G =L/L_plk^2


Since

Rho= M/L^3

Rho= 1/L*L_plk^2=(L*L_plk)^-2 = L_plk/H)^-2 = H^2/G


The full equations with constants are


Rho_vac= 2*c^2*H^2/(8*G) = 3*hbar*H^2/(8*L_plk^2*c)


Which give a value of approximately


2E-9 Joules/m^3

Which falls within the observational bounds based on defining vacuum
energy density based on the Hubble parameter.


We can model this process using the expanded particle states proposed by
Klauber, Sundrum, etc. by seeing how the HFC might constrain vacuum energy from
the zero point fluctuations of the Quantum fields. We know based on Quantum
Field theory that we must sum over all possible Quantum modes. We know from
Quantum Field theory that (looking at the Hamiltonian)

H_ferm= (w/2)* {a,a^dag} and H_bos=(w/2)*[a,a^dag]

Where a and a^dag are the annihilation and creation operators


So that

H_bos= hbarw(*a*a^dag +1/2) H_fer= hbarw(*a*a^dag -1/2)


Giving the energy eigenvalues


E_bos= hbar*w*(N+1/2) E_fer=hbar*w*(N-1/2)

Where N is the number operator.


We have to integrate over all the field modes so that


Rho_vac= *N*Integral { 0 to w_c} w^28sqrt[w^2-m^2]dw = N*{
w_c/8*(w_c^2-m^2)*sqrt[w_c^2-m^2] - m^4/8*cosh^-1[W_c/m]


Where N are the degrees of freedom of the Quantum field modes.


We must sum over both Fermion and Boson modes which give us


Rho_vac= Integral Dw L(+) = N_b*Integral dw F(w,m_f)
-N_f*Integral dw F (w,m_b)

Due the mass splitting between the fermion and boson modes we get an
impossibly large predictions of vacuum energy density for any reasonable cutoff
value. Since


Lamda= (8*pi*G/c^2)*pho_vac

Where Lambda is Einstein's cosmological constant, this wildly incorrect
prediction is known as the Cosmological Constant problem.

However, as demonstrated by Klauber, Sundrum, Moffat and others we might
include the full set of particle solutions from the relativistic equations.
This gives us the modified annihilation and creation operators;

a(-)= -i*a^dag


a(-)^dag=-i*a


So that we get the energy eigenvalues;


E_bos= -hbar*w*(N+1/2) E_ferm=- hbar*w*(N-1/2)



This would give us;




Rho_vac(+) = Integral Dw L(+) = N_b*Integral dw F(w,m_f) -N_f*Integral
dw F (w,m_b)

Rho_vac(-) = Integral Dw L(-) = N_f*Integral dw F(w,m_f) -N_b*Integral
dw F (w,m_b)


So that



rho_vac= Integral Dw L(+) +IntegralDwL(-)= 0



However based on the HFC rho_vac = approx 2E-9 Joules/m^3


So we might speculate that we have a relationship such that



rho_vac= Chi(+)^2 Integral Dw L(+) +(chi(-)^2*IntegralDwL(-)= 0


Chi(+) = 1/sqrt [1-2*G*M/R^c^2) Chi(-)= 1/sqrt [1+2*G*M/R^c^2)


Which we can relate to the hubble parameter


2*G*M/R= approx (H/M)*(H/G)*(1/M) =H^2/G


Which we redefine as some normalized Hubble parameter such
that


0=< h >=1


So that


Rho_vac= rho_vac= {1/(1-h)} Integral Dw L(+) +{(1/1+h)}*Integral
DwL(-)= 0



This equation however, remains pure speculation and is offered as
here as such.


Bob Zannelli








































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