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[Phys-l] Holographic Dark Energy in Induced Gravity



Holographic Dark Energy in Induced Gravity


Ever since the discovery of Dark Energy Physicists have been confronted by
two problems, the fine tuning problem and the coincidence problem. The fine
tuning problem is simply the need to understand why the observed vacuum
energy is 1E-60 times smaller than it's predicted value based on summing over
the Zero point energy of all the Quantum fields up to the SUSY breaking
scale. The coincidence problem is the need to explain why the total dark
energy in the Universe is the same order of magnitude as the density of matter.
They evolve very differently over the range of the scale factor, why are
they not at very different values in the Universe we live in?

A very elegant solution is suggested by the Holographic principle. In the
Holographic model Entropy scales not with the volume of space time but with
its surface area. This would allow us to scale the Dark Energy at the
cosmological scale. Looking at the Einstein equation we would have;

G_mu,nu= R_mu,nu -(1/2)*R*g_mu,nu = 8*pi*G*T_mu,nu/c^4 - Lambda*g_mu,nu

Lambda is proportional to l^(-2) a scale at the same order as the
Cosmological scale.

If the vacuum energy were proportional to the inverse Hubble scale the fine
tuning problem would be solved which also solves the coincidence problem
given the Holographic bound on total energy in any O region of the Universe.
Unfortunately this attractive idea produces an equation of state value
which predicts no acceleration of the expansion rate.

Rho(a) = rho(0)*(a/a(0))^-3*(1+w)

1/a^3= a^-3(1+w)

-2*lna=-3(*1+w)*lna


w= -1/3


Given


d^2a/dt^2= - 4*pi*G * ( rho+ 3*P/c^2)*a

w= P/rho*c^2

d^2a/dt^2= - 4*pi*G * ( rho+ (3/c^2)*w*rho*c^2)*a


d^2a/dt^2= - 4*pi*G * ( rho-rho)*a =0




Another Holographic Dark Energy approach is to scale the ZPE with the
future Horizon scale. However, this would required backward causation and a
non local effect. In addition this involves obvious circular reasoning. In
order to interpret the cosmic acceleration the Holographic dark energy model
has presumed the Acceleration.


A different approach that avoids these pitfalls is the idea of Induced
Gravity Dark Energy proposed by Zu-Yao Sun and You-Gen Shen. Induced Gravity
models can be used as an effective theories of Gravity in the Super String
model or a Pre Geometric canonical Quantum theory of Gravity. Shen and Sun,
by applying this model to the ZPE of Quantum fields, offer a possible
Induced Gravity Dark Energy model. This is closely related to the various
proposals which incorporate the extended set of solutions of the Relativistic
equations to the ZPE calculation, the models proposed by Klauber, THooft,
Moffat, Sundrum, Randel and others.

Based on the above we can write the equation


T_mu,nu= zeta*{ chi_a*int Dw L(+) + chi^a*DwL(-)}*g_mu,nu

Where Zeta is a constant, chi_a and chi^a are Killing vector fields and
the integration is summing over the positive and negative energy vacuum
fluctuations.

The action for the Induced Gravity model is


S= int d^4*sqrt[-g]{ -(1/2)*zeta*phi^2*R - (1/2)*g^mu,nu*pd_mu*pd_nu-
V(phi) +L_sm}


Where R is the scalar curvature, phi is an effective scalar
field and L_sm is the standard model Lagrangian.


We get the metric

ds^2= chi^a*chi_a dt^2c^2- (chi^a*chi_a^-1)*( dx^2 +dy^2 +dz^2)

In terms of electromagnetic property of the vacuum we have


mu= mu(0)*{ chi^a*chi_a)^(-1/2)

and

epsilon = epsilon(0)* { chi^a*chi_a)^(-1/2)


Based on the Induced Gravity version of the Holographic dark energy
the length scale is given by the average radius of the of the Ricci scalar
curvature, R^(-1/2)


Looking at the Ricci curvature tensor we have


R_00= -6*[ dH/dt + 2*H^2 + k/a^2]


R_11 =R_22 = R_33 = -3*[dH/dt+ 2*k/a^2]


Where H is the Hubble parameter. We assume

dH/dt=0 and k= 0 based on current observational evidence.

R=TR[R_mu,nu] = 12*H^2


Rho_de = c^2*R/(16*pi*G)= 3*H^2*c^2/(4*pi*G)= 1.28 E -9 J/m^3


Which is on the order of the observed value of vacuum energy density.


We can plug this back into the Einstein equation.


R_mu,nu -(1/2)R*g_mu,nu + Lambda*g_mu,nu= (8*pi*G?c^2)*T_mu,nu




R_mu,nu -(1/2)R*g_mu,nu + 8*pi*G/c^4*rho_de*g_mu,nu=
(8*pi*G?c^2)*T_mu,nu


R_mu,nu -(1/2)R*g_mu,nu + (8*pi*G/c^4)*c^2/16*pi*G)*R*g_mu,nu=
(8*pi*G?c^2)*T_mu,nu



R_mu,nu -(1/2)R*g_mu,nu + (1/2)*c^2R*g_mu,nu= (8*pi*G?c^2)*T_mu,nu




R_mu,nu -(1/2)R*g_mu,nu + (mu(0)*epsilon(0)/2)*R*g_mu,nu=
(8*pi*G?c^2)*T_mu,nu



So here we see that the dark energy term is

(mu(0)*epsilon(0)/2)*R*g_mu,nu


Using this model we have a formalism which produces the observed vacuum
energy density within an order of magnitude of the observed value. In
addition, this model equates the vacuum energy density with the electromagnetic
parameters of the vacuum using an Induced Gravity model of gravity.

Bob Zannelli