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[Phys-l] The Cosmological Constant from Ricci Dark Energy



The Cosmological Constant from Ricci Dark Energy


About 70% of the mass energy in the Universe consist of a strange substance
called Dark Energy. One possibility for what this Dark Energy might be is
nothing less than Einstein's cosmological constant (CC). A promising
approach to explaining the origins of this Dark Energy is based on what is known
as the Holographic principle (HP) . This approach is supported by String
Theory, which is in essence a Holographic model of the Universe.

Based on the HP the entropy of a system (hence its total energy content)
scales not with the volume of that system as might be expected, but with the
surface area. To see how this could help solve the cosmological constant
problem we note that in the Einstein equation

G_-mu,nu= (8*pi*G/c^2)*T_mu,nu - lambda*g_mu,nu

The lambda term is the inverse of some length squared L^-2 , and to be
consistent with observation L must be on the order of the present cosmological
scale. In other words, the observed vacuum energy density relates to the
inverse of the Hubble scale in some way.

Unfortunately the most straight forward application of this idea has severe
problems. Given a naive application of this principle the equation of
state for the vacuum energy predicts a de accelerating Universe. In addition,
for a spatially flat Friedmann- Robertson -Walker Universe it is well known
that the future event horizon exists, if and only if, the Universe is
accelerating. So in order to interpret the cosmic acceleration, the Holographic
Dark Energy (HDE) model itself has presumed the acceleration, a case of
circular reasoning.

However, a HDE model which avoids these problems is to consider the length
scale L for the vacuum energy as given by the average radius of the Ricci
scalar curvature, R^-(1/2). This a straightforward application of an Induced
Gravity model version of the HP.

We can write the action of this HP induced gravity model as


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


Giving the effective value of Newton's constant as


G= 1/(*pi*alpha*phi^2)


Here phi is the emergent scalar field resulting from the collective
excitation of Zero Point Energy (ZPE) with alpha being considered as some
constant.

However, as related by me in a recent post, there seems to be an intrinsic
relationship between the entropy bound and the geometry of space, that is,
an intrinsic relationship between the geometry of space time and the
vacuum degrees of freedom of any given Hubble volume. The most straightforward
assumption, though not the only one possible, is to assume an equation of
state for the vacuum energy of-1, Einstein's cosmological constant.

This assumption has been given a more rigorous justification by Kyoung Yee
Kim, Hyung Won Lee and Yun Soo Myung of the Inge University in Gimhae Korea
in their paper "On the Ricci Dark Energy Model" In this paper they
demonstrate that the Ricci Dark Energy model is nothing more than a prediction of
a cosmological constant based on the HP applied to a Hubble volume. Given
this constrain a very simple Dark energy model is possible.

In the Ricci Dark energy model the vacuum energy is given by

rho_de= -alpha* R/(16*pi*G)

( Holographic dark energy models : a comparison from the latest
observational data" Li, Li, Wang and Zhang.)

Where R is the scalar curvature.

Based on the model proposed by Kim, Lee, and Myung we must assert that

alpha*phi^2 = constant value.

This means that the scalar field vev and coupling "constant" are in
fact dynamic and are tuned by the geometry of space time. And given the
assertions of Kim, Lee and Myung the relationship of these parameters in the
Ricci scalar equation is easy to determine.


In the Ricci tensor we have

R_mu,nu= (1/c^2)*M_mu,nu

Where

M_00 = 3*A

M_11=M_22=M_33= 2*B+A

Where

A= H^2+dH/dt= (d^2a/dt^2)*(1/a) =
Lambda*c^2/3=(4*pi*G/(3*c^2))*(pho +3*P)

B= H^2+k*c^2/a^2=(da/dt)^2*1/a^2)= lambda*c^2/3+8*pi*G*rho/3*c^2)


Therefore

R=TR[R_mu,nu] = -(6/c^2)*(A+B)= -(6/c^2)*2*H^2+dH/dt+k*c^2/a^2]


Given that


Rho_de= -alpha*R/(16*pi*G)

and based on the assertions of Kim, Lee and Myung we see that


Alpha= Omega_de*c^2/2

Where omega_de is the density parameter for dark energy.

So that

rho_de= -omega_de*c^2*R/(32*pi*G) =- Omega_de*c^2*alpha*phi^2

Based on the relationship between geometry and the energy density is any
given Hubble volume we have

Omega_m+ Omega_de=1

Which gives us


H= sqrt[ 8*pi*G*(rho_m+rho_de)/3] ( we ignore radiation here).


Giving us a Hubble radius of

R_H= c*sqrt[3/(8*pi*G*(rho_m+rho_de)]


The De Sitter radius is given by

R_ds= c*sqrt[3/(8*pi*G*ho_de]


Given that we have

rho_de= constant and rho_m scales like 1/a^3


We see that the Hubble radius is expanding into the De Sitter Horizon and
that under these simple assumptions the end state for any Hubble volume is
a timeless De Sitter Universe. Of course whether such a state is stable or
not is an open question. This prediction has several problems including the
issue of Boltzmann brains.

Bob Zannelli