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[Phys-l] Dark Energy as an Entropic Force ( No Fine Tuning Required )

Dark Energy as an Entropic Force

If any discovery has revolutionized our understanding of the Universe it's
the Holographic principle. Now due to the potentially important proposal by
Erik Verlinde we may possibly be close to a new paradigm for gravity. If
this proposal turns out to be valid this should have implications for our
understanding of the nature of Dark Energy. In this post I will build on my
many previous posts on this topic to make the case that Dark Energy is a
natural result of what we might call the Entropic principle.

Physical Systems over time evolve to a state of maximum entropy.


The idea of Gravity as an entropic force is closely related to the proposal
that Gravity is not a fundamental force, but the result of summing over
all the effects of Quantum Fields i.e. that class of gravity theories which
postulate gravity as an emergent property of the Universe. Based on the
Weinberg-Witten theorem any emergent gravity theory must also involve space
time as emergent. As shown by Verlinde this paradigm is consistent with String
Theory and as shown by Smolin consistent with Loop Quantum Gravity.

The idea of gravity as an induced force was first introduced by Sakharov in
an effort to unite gravity with Quantum theory. This was expanded by
Zeldovich who showed that Induced gravity predicts a non zero value for the
cosmological constant.

The basic principle in Sakharov model is that mass energy affects the
action density of the zero point fluctuations of the vacuum state. Sakharov's
model postulates a ghost sector which" give opposite contribution from that
of real particles to the R (Ricci scalar) dependent action." The idea is
that local vacuum energy density is the result of a balance between this
"ghost sector" and real Quantum fields. This will be important in understanding
the role that Induced Gravity is proposed to play in the existence of Dark

This idea of a ghost sector as an explanation for the observed value of the
vacuum energy density has been proposed in several papers. A sampling of
these papers are below.


Mechanism for Vanishing Zero-Point Energy
Authors: Robert D. Klauber
(Submitted on 24 Sep 2003 (_v1_ (
), last revised 19 Jul 2007 (this version, v3))

Abstract: In addition to the two standard solutions of the quantum field
equations having the form e^{+/-(iwt-ikx)}, there exist two additional
solutions of the form e^{+/-(iwt+ikx). By incorporating these latter solutions,
deemed "supplemental solutions", into the development of quantum field
theory, one finds a simple and natural cancellation of terms that results in an
energy VEV, and a cosmological constant, of zero. This fundamental, and
previously unrecognized, inherent symmetry in quantum field theory shows
promise for providing a resolution of the large vacuum energy problem, simply
and directly, with little modification or extension to the extant
mathematics of the theory. In certain scenarios, slight asymmetries could give rise
to dark energy.


Invariance under complex transformations, and its relevance to the
cosmological constant problem
Authors: Gerard 't Hooft, Stefan Nobbenhuis
(Submitted on 20 Feb 2006 (_v1_ ( ),
last revised 4 Apr 2006 (this version, v2))

Abstract: In this paper we study a new symmetry argument that results in a
vacuum state with strictly vanishing vacuum energy. This argument exploits
the well-known feature that de Sitter and Anti- de Sitter space are
related by analytic continuation. When we drop boundary and hermiticity
conditions on quantum fields, we get as many negative as positive energy states,
which are related by transformations to complex space. The paper does not
directly solve the cosmological constant problem, but explores a new direction
that appears worthwhile.


Positive and Negative Energy Symmetry and the Cosmological Constant
Authors: J. W. Moffat
(Submitted on 13 Oct 2006)

Abstract: The action for gravity and the standard model includes, as well
as the positive energy fermion and boson fields, negative energy fields.
The Hamiltonian for the action leads through a positive and negative energy
symmetry of the vacuum to a cancellation of the zero-point vacuum energy
and a vanishing cosmological constant in the presence of a gravitational
field solving the cosmological constant problem. To guarantee the
quasi-stability of the vacuum, we postulate a positive energy sector and a negative
energy sector in the universe which are identical copies of the standard model.
They interact only weakly through gravity. As in the case of antimatter,
the negative energy matter is not found naturally on Earth or in the
universe. A positive energy spectrum and a consistent unitary field theory for a
pseudo-Hermitian Hamiltonian is obtained by demanding that the
pseudo-Hamiltonian is ${\cal P}{\cal T}$ symmetric. The quadratic divergences in the
two-point vacuum fluctuations and the self-energy of a scalar field are rem
oved. The finite scalar field self-energy can avoid the Higgs hierarchy
problem in the standard model.


A Symmetry for the Cosmological Constant
Authors: David E. Kaplan, Raman Sundrum
(Submitted on 31 May 2005 (_v1_ ( ),
last revised 2 Jun 2005 (this version, v2))

Abstract: We study a symmetry, schematically Energy -> - Energy, which
suppresses matter contributions to the cosmological constant. The requisite
negative energy fluctuations are identified with a "ghost" copy of the
Standard Model. Gravity explicitly, but weakly, violates the symmetry, and
naturalness requires General Relativity to break down at short distances with
testable consequences. If this breakdown is accompanied by gravitational
Lorentz-violation, the decay of flat spacetime by ghost production is
acceptably slow. We show that inflation works in our scenario and can lead to the
initial conditions required for standard Big Bang cosmology.



Based on Sakharov's model the effective gravity coupling is a function of a
scalar field which emerges from the collective excitation of the zero
point Quantum fields. This gives us'

G= - 1/(16*pi*A*int kdk)

Where A is a constant.

If we do a naive calculation of the Quantum field Hamiltonian we get a
prediction of a divergent vacuum energy density. We have;

We get

H_bos= w*(a^dag*a+1/2)

H_ferm w*(a^dag*a-1/2)

Where a^dag and a are the creation and annihilation operators in QFT. So
this given us a total energy in any given volume of space for a given Quantum
field as

E_bos= (N + !/2)*hbar*w


Where N is the number operator

So for a vacuum state energy density we have to integrate over all
possible Quantum fields Which gives us.

int dw F(,w,m)_bos= (N_b/4*pi^2)*int{0to w_c} w^2*sqrt[w^2-m^2]dw =



int dw F(w,m)_ferm = -(N_b/4*pi^2)*int{0to w_c} w^2*sqrt[w^2-m^2]dw =


Using a short hand notation we write

int Dw L(+)= int dw F(,w,m)_bos + int dw F(w,m)_ferm

Based on the mass split between the fermions and bosons, and with a
cutoff presumed at the SUSY scale, we get a vacuum energy prediction of
about +1E60 J/m^3. This is of course impossible, such a large vacuum energy
density gives us a hugely repulsive cosmological constant.

However, based on the papers listed above we must include the ghost sector
in our Quantum field calculation. We can equate this ghost sector with the
so called unphysical solutions of the relativistic equations. This doubles
the number of particle states. We have;

The Physical Solutions

Matter (Positive energy flow into the future)

Antimatter (Negative Energy Flow into the past.) Reinterpreted as charge
conjugated positive energy into the future.

The Unphysical Solutions

Negative Matter (Negative energy flow into the future)

Negative anti matter (positive energy flow into the past) Reinterpreted as
charge conjugated negative energy into the future.

When these "supplemental" states are integrated over we get

int Dw L(-)= - int dw F(,w,m)_bos + int dw F(w,m)_ferm

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

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

So that

int Dw L(+) + int Dw L(-) =0

at all energy scales.

Of course there can't be any real negative energy particles and the
various papers above attempt to deal with this problems in different ways. I
will not go into this topic here.

Based on the Induced Gravity model the presence of mass energy shifts the
balance of the action density of these two sectors the real particle and
the ghost sector. We can model this by decomposing the Killing vector

chi^a*chi_a=- g_tt= 1/(-2*G*M/R*c^2)

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

This lays the ground work for understanding Dark Energy.


The Zero Point energy fluctuations are not the only predicted source of
vacuum energy. Associated with every symmetry breaking event in the Universe
is an effective scalar field created by the associated Fermion condensate.
We get the action;

S= int d^4x sqrt[-g]*{ (1/2)*g^mu,nu pd_mu (psi) pd_nu(psi)-V(psi)}

Where g is the determinate of the metric tensor and pd stands for
partial derivative. We get the stress energy tensor

T_mu,nu={ (1/2)*g^mu,nu pd_mu (psi) pd_nu(psi) -V(psi)}*g_mu,nu

These fields will settle into their lowest energy state giving no
contribution from the kinetic energy terms. However, the potential energy gives

rho_vac= approx. hbar*k_max^4

Where k_max is the energy cutoff scale where the symmetry breaking
occurs. Based on our best understanding we have;

Quark -anti Quark condensate 1E44

Weak Superconducting condensate 1E56

Unified superconducting condensate (strong force) 1E112

ZPE fluctuations (SUSY cutoff, no ghost sector) 1E60

And there be certainly be more. Based on this we can define the vacuum
energy density as

rho_vac= rho_con +rho_zpe

Obviously something is badly wrong here. We shall show on modeling
gravity as an entropic force may make sense of what we observe for vacuum


If gravity is truly an entropic force than there must be a clear
connection between gravity and the geometry of space time. And in fact this is
the case. It is a well known fact, based on CMB data that our causal patch of
space is flat. The reason invoked for this is cosmic inflation, and there
is a great deal of evidence that an inflation event occurred for our
Universe. But if gravity is an entropic force then we have a new way to think
about inflation. Inflation becomes the natural result of the Entropic
Principle. Physical Systems over time evolve to a state of maximum entropy. Such a
connection between entropy and inflation has been proposed by Victor H


Protecting the Holographic Principle: Inflation
Authors: Victor H. Cardenas
(Submitted on 16 May 2002 (_v1_ ( ),
last revised 9 Apr 2003 (this version, v2))

Abstract: A scenario where inflation emerges as a response to protect the
holographic principle is described. A two fluid model in a closed universe
inflation picture is assumed, and a possible explanation for secondary
exponential expansion phases as those currently observed is given.

_ (


Inflation as a response to protect the Holographic Principle
Authors: Victor H. Cardenas
(Submitted on 3 Aug 2009)

Abstract: A model where the inflationary phase emerges as a response to
protect the Fischler-Susskind holographic bound is described. A two fluid
model in a closed universe inflation picture is assumed, and a discussion of
conditions under which is possible to obtain an additional exponential
expansion phase as those currently observed is given.



As can be seen by a simple calculation a comoving patch of space with a
flat geometry is at maximum entropy. Universes where K=1 violate the
Bekenstein entropy bounds and are therefore unstable. We can postulate that
Inflation results to protect this bound.


S= 2*pi*R*E/( hbar*c) =pi*R^2*c^3/(hbar*G)

Where R= c/H

Where H is the Hubble parameter

We get

Rho_bek= 3*H*c^2/(8*pi*G)

Which is the critical density needed to have a flat space geometry. We can
also see that the entropy of our comoving patch of space time is;

S_H= pi*c^5/(hbar*G*H^2)

Our Hubble Volume entropy is a function of the Hubble parameter.
Therefore the cosmic AOT is associated with dH/dt<0 evolution. Based on our current
best evidence, the final state of the Universe will be a timeless De
Sitter Space where dH/dt=0 and H is at its minimum value. It is an open question
whether this is a stable state or will tunnel into a Minkowski space or
create, via Quantum fluctuations, new inflation bubbles.


The connection between the Holographic principle and Dark Energy now
becomes obvious based on the above. This basic idea has been proposed in a
slightly different from then I will do here by Zu-Yao Sun and Yiu-Gen Shen Paper
not in eprint archive and. Changjun Gao, Fengquan Wu and Xuelei Chen.


Holographic Dark Energy Model from Ricci Scalar Curvature
Authors: Changjun Gao, Fengquan Wu, Xuelei Chen, You-Gen Shen
(Submitted on 10 Dec 2007 (_v1_ ( ), last
revised 25 Dec 2008 (this version, v4))

Abstract: Motivated by the holographic principle, it has been suggested
that the dark energy density may be inversely proportional to the area of the
event horizon of the Universe. However, such a model would have a
causality problem. In this paper, we propose to replace the future event horizon
area with the inverse of the Ricci scalar curvature. We show that this model
does not only avoid the causality problem and is phenomenologically viable,
but also naturally solves the coincidence problem of dark energy. Our
analysis of the evolution of density perturbations show that the matter power
spectra and CMB temperature anisotropy is only slightly affected by such

_ (


In the Induced Gravity model we see that presence of mass energy shifts
the action density between the real and ghost sector. This is illustrated by
the equation;

T_mu,nu= A*{ chi_a*int Dw L(+) + chi_^a int Dw L(-)}*g_mu,nu


Chi^a*chi_a= 1/( 1-2*G*M/R*c^2)

However, as suggested by Zeldovich there is a causal connection
between space time curvature and a shift in the action density which effects the
balance of energy contribution between the real and ghost sector. The
measure of this curvature is a function of the average radius of the Ricci
scalar curvature;


We can therefore suggest the equation'

Rho_ZPE= A*{ chi^a int Dw L(+) + chi_a int Dw L(-)}


Chi^a*chi_a= 1-R

Where R is the Ricci scalar curvature.


Based on Induced Gravity we can write;

rho_zpe= {z*c^2/(16*pi*G)}*R

z being a term I will expand on shortly.

Given that

R_00= -3*dH/dt

R_11=R_22=R_33= -( 2*H^2+dH/dt+k*c^2/a^2)

R= -6*[2*H^2+dH/dt+k*c^2/(2*a^2)]


Rho_zpe= z*[ 3*H^2*c^2/8*pi*G) + 3*c^2/(8*pi*G)*dH/dt+ 3

Which gives us approximately

Rho_zpe=z*[ 3*H^2*c^2/(8*pi*G)]= z*rho_crit

Based on the Entropic principle each Hubble volume must drive its parameter
s to the Bekenstein entropy bound

This requires that the density parameters

O_tot= O_matter+O_radiation +O_condensate +O_zpe=1

O_matterscales likes the inverse third power of scale factor,
O_radiation scales to the inverse forth power of scale factor while O_condensate
remains constant. Based on a paper by Kyoung Yee Kim, Hyung Won Lee and Yun
Soo Myung , the Ricci scalar dark energy density is also constant.


On the Ricci dark energy model
Authors: Kyoung Yee Kim, Hyung Won Lee, Yun Soo Myung
(Submitted on 22 Dec 2008)

Abstract: We study the Ricci dark energy model (RDE) which was introduced
as an alternative to the holographic dark energy model. We point out that
an accelerating phase of the RDE is that of a constant dark energy model.
This implies that the RDE may not be a new model of explaining the present
accelerating universe.



Therefore we can say that

rho_vac + Rho_zpe =constant (Baring any further symmetry breaking


Rho_zpe is constant. Based on the Friedmann equation

rho= H^2*c^2/(8*pi*G)

Where dH/dt<0 we can see that that the critical energy value will
decrease over time. However,

z*rho_crit =constant.

Therefore z is not a constant but an evolving parameter.


O_tot= O_matter + O_radiation +O_condensate +O_zpe

We can see that that

z= O_zpe= 1- ( O_matter + O_radiation +O_condensate)

Since O_matter and O_radaition are driving to zero so we might postulate
the final state of the Universe will be


O_zpe +O_condenste = Rho_vac* 8*pi*G/( 3*H_final^2*c^2) = 1

Where dH/dt= 0. A timeless De Sitter space.

If this model is correct the existence of Dark Energy is no mystery, it's
the natural result of the Entropic Principle and the accidental values which
resulted from the decay of the Inflaton field and energy scales of the
various symmetry breaking events. No fine tuning is required at all.

Bob Zannelli