Several earlier posts explored the idea that of the Holographic dark energy
proposals modeled in terms of some form of induced gravity model. In this
post I would like to flesh out this idea in a little more detail. The basic
idea in induced gravity is that gravity is in fact not a fundamental force
but rather an induced effect of the collective action of all the Quantum
fields. Specifically that a gravity field is really a modification of the
action density of the vacuum states in the vicinity of collections of matter
energy. As Sakharov writes "Vacuum Quantum Fluctuations in Curved Space and
the Theory of Gravitation"
"Considering the density of the vacuum Lagrange function in a "
simplified" model of the theory of non interacting free fields with particles m=aprox
k_0 shows that fixed ratios of the masses of real particles and "ghost"
particles (i.e., hypothetical particles which give an opposite contribution
to that of the real particles to the R dependent action)."
In notion previously introduced we might write then;
T_nu,mu = g_mu,nu* { Chi(-) Intergral Dw L(+) + Chi (+) Integral Dw L( -) }
Here we can define the factoring functions chi+-) as
chi(+-) = 1/gamma_g (+-)^2
gamma_g (+ -) =1/sqrt [ 1 +- 2*G*M/R*c^2]
Based on this approach given the Einstein equation
Where rho_global*g_mu,nu represents the dark energy contribution to the
Einstein Tensor.
So how might we write the factoring function for the global vacuum energy
contribution? It would seem this function must be in terms of the Hubble
parameter. So we might propose
chi ( +- )= 1/gamma(+-)^2
gamma( +-)= 1/sqrt [ 1 +- a_lambda^2/ a_planck^2]
Where a_lambda is the acceleration due to dark energy and a_plk is the
Planck scale acceleration