Re: [Phys-l] Geometry, Energy, Entropy and the Evolution of the Hubble Parameter

[Spinozalens@aol.com]

The paper by Victor H. Cardenas of the University of Valparaisa in  Chile 
referenced below  represents the core ideas of what  I proposed in my earlier 
post. . I just located this paper a few hours ago.  I quote 
 
 Inflation as a Response to Protect the Holographic Principle (  Gr-qc 
arXiv: 0908.0287v1) 
 
 
   In general, holography has been used mainly in an attempts  to 
adequately formulate the HP in cosmology. However, because the HP must be  considered 
a Universal principle of a higher status than inflation, in this work  I 
look for a solution to the problem considering a different point of view: an  
attempt to derive inflation from the HP. This is so because we expect the 
HP, to  come from a (so far unknown) GUT theory, and then I will analyze if 
this new  point of view may constrain any inflationary model. From the 
declaration of  principles, it is clear that we do not find the current beliefs 
very informative  because inflation and the HP are considered (almost) 
independent  processes.
 
A first approach to this goal was presented in [25] where I proposed a  
model where inflation emerges from the saturation of the HP in a closed FRW  
universe. The analyze was made taken care of being thermodynamically 
consistent,  as was stressed in [22] in connection to the work of Rama[26]. Similar 
thoughts  has been suggested recently in [27]
 
 
 In this work, I study a model where inflation can interpreted as  a 
response to protect the violation of the HP a la FS, taking care of the role  of 
the curvature, and considering the non-adiabatic process of re heating after  
inflation. ...
 
At early times I found that an imminent violation of the HP forces the  
system to saturate it through a period of exponential growth of the scale  
factor. 
 
End quote 
 
 
The suggest that inflation rather than being an accident due to a scalar  
field being in a slow roll state is really an inevitable outcome of any 
causal  patch of space fluctuating into geometry where K=1. This state is 
unstable  causing the inflation process to occur inflating the space to a saturated 
 entropy state. Of course the exact mechanism for this needs to be worked 
out in  detail, this paper by Cardenas is just such an attempt.
 
 
This scenario if correct has obvious implications for the question of Dark  
Energy and the future evolution of the Universe. 
 
 Bob Zannelli 
 
 
 *****************************
 
 
In a message dated 11/1/2009 5:13:00 PM Eastern Standard Time,  
RBZannelli@AOL.COM writes:

I am reposting this with additions. 
 
 
Geometry, Energy, Entropy and the Evolution of the Hubble Parameter  




 
 
Holography and  Cosmology  
Where I  "prove" closed Universes are impossible.  
1)  The  Issue of the Curvature parameter K  
As is well known  there are three possible basic geometric conditions any 
Universe based on the  total positive energy density. It has been asserted 
incorrectly that K, the  normalized curvature parameter must be 1 in our 
Universe and does not  change. This assertion is based on a confusion between the 
curvature  parameters between what Tegmark calls levels of the multiverse. 
In Tegmark's  model there are four levels in the multiverse structure. These 
are  
Level 1   
Beyond our Cosmic Horizon. These are the 1E26 Hubble Volumes in the Level 1 
 Multiverse. Our oown Universe is therefore 1E-26 of the space time  
volume. 
Level 2  
Other post  inflation Bubbles which are the natural result of inflationary 
theory, the  fundamental multiverse paradigm.  
Level 3  
Quantum Many  Worlds 
This is  multiverse structure based on the Unitarity of the Quantum wave 
function of  the Universe. Support for this Unitarity for Quantum gravity 
comes from the  AdS/CFT correspondence  
Level 4  
Other Mathematical  Universes. We will not be dealing in any way with the 
existence of this layer  of multiverse structure in this post. The existence 
or non existence of this  Multiverse structure is irrelevant for the points 
made here. 
At this point it  might be well to mention that based on top down Quantum 
cosmology levels 2 and  3 may in fact be the same level. This has been 
asserted by Susskind where he  writes  
"The Many  Worlds of Everett seems at first sight to be quite a different 
conception  than the eternally inflating mega-verse. However, I think the two 
may really  be the same thing. I have emphasized several times that Quantum 
Mechanics is  not a theory that predicts the future from the past, but 
rather it determines  the probabilities for the possible alternate outcomes of 
an observation. These  probabilities are summarized in the basic mathematical 
object of Quantum  mechanics - the wave function" 
The Cosmic  Landscape, String Theory and the Illusion of Design"  
While this may well  be true, this assertion also has no relevance for the 
purpose of this post. I  include it for completeness. 
In any case, looking  at this multiverse structure we can get a clearer 
picture about how to think  about K,  the curvature parameter. When the 
assertion is made that K is a  constant as the Universe expands this refers to the 
level 1 multiverse, the  1E26 Hubble volumes. With regard to each Hubble 
volume since K/a^2 <<  H^2 we may correctly put K to zero in the Friedmann 
equations. The inflation  process inflates away curvature in each Hubble volume. 
In addition , it should  be mentioned, that concepts such as  energy and 
entropy are meaningless in describing the level 1 multiverse.   
2) Geometry  and Total Energy in the Universe(s)  
. We will look  at the three possible cases, K= -1, 0, and +1.  I will  
also briefly relate these energy conditions to the full set of solutions  of 
the relativistic equations. These are the physical solutions, Positive  energy 
flowing forward in time (matter) Negative energy flowing backward in  time 
(anti matter) and the two unphysical solutions Positive energy flowing  
backward in time (negative anti matter) and negative energy flowing  forward in 
time (negative matter). This will related in passing to the BIVERSE  model, 
the creations of Universe in CPT inverse pairs as postulated by many  
cosmological models such as Sakharov's Multisheet model, the Carroll Chen  Model, 
the Hawking Hartle no boundary model and Linde's twin Universe  hypothesis.  
We take a look at a  simple calculation we can use to define the energy 
condition of the Universe  as a function of geometry. Based on the Friedmann 
equations we  have; 
Rho_crit=  3*H^2/(8*pi*G)                     
Where,  H is the Hubble parameter, G is the gravity constant and Rho_crit 
is the  critical density in a given Hubble Volume to achieve K=0, a Euclidean 
metric.   
Therefore  here we have;  
Rho_Hubble=Rho_crit  
And  
K=0 and Omega=1  
Where K  is the curvature parameter and Omega is the energy density 
parameter.   
When we have K=1 we have a non Euclidean spherical metric where    
Rho_Hubble > Rho  Crit     
And  when  we have K=-1 we have a non Euclidean hyperbolic metric where;   
Rho_Hubble < Rho_Crit  
Therefore  
Rho_crit= 3*H^2/(8*pi*G)  
M_crit= Rho_crit*V_Hubble   
V_hubble = 4*pi*R_Hubble^3/3   
Since R_hubble = c/H  
V_Hubble= 4*pi*c^3/(3*H^3)  
M_crit= c^3/(2*H*G)  
Therefore  
E_mass= SUM {all i} gamma_i*m_i*c^2 
Here we are summing over all relativistic masses in the Hubble Volume.   
However the negative gravitational energy must be included  therefore  
E_grav= - G*gamma_i*m_i*M_unv/ R_i  
Given a homogenous and isotropic Universe we can set   
R_i= c/(2*H)  
and  
M_unv=Omega*M_crit  
Therefore  
E_unv = E_mass + E_grav = ( 1-Omega)*SUM {all i} gamma_i*m_i*c^2   
E_unv= (1-Omega)*E_mass  
Therefore  
K=-1    E_unv>0   
K=0     E_unv =0  
K=+!     E_unv < 0  
3) Geometry  and Entropy  
Now we can relate each of these geometric conditions to the Bekenstein  
Bound.  
S_Bek= <  2*pi*R*E/(hbar*c) = A_Hubble/(4*L_plk^2)   
Where S_Bek is the maximum entropy allowed, A_Hubble is the surface area of 
 the Hubble volume and L_plk is the Planck length. 
Therefore  
S_Bek= pi*R^2*c^2/(hbar*G)  
E_Bek=R*c^4/(2*G)  
Rho_Bek( energy) = 3*H*c^2/(8*pi*G)  
Rho_Bek(mass)= 3*H^2/(8*pi*G)  
> From this we can see that the critical mass energy density condition  for 
any Hubble Volume saturates the entropy of that volume based on the  
Bekenstein Bound. 
Looking at  the Friedmann equations we can see that 
K=(a^2/c^2)*(8*pi*G*rho(mass)/3-H^2)  
Where a is the scale factor 
For K= -1  E_unv>0 we have 
Rho(mass) = 3*(  H^2-c^2/a^2)  
Therefore  
S_Hubble<  S_Bek  
For   K=0 E_unv=0  we have 
Rho(mass)=  3*H^2/(8*pi*G)  
S_Hubble= S_Bek 
For K=+1 E_unv < 0 we have 
Rho(mass)=  3*(H^2+c^2/a^2)   
S_Hubble> S_Bek  
Therefore we find that for a closed  Universe the Bekenstein bound is 
violated for a given Hubble  Volume. We can relate this to the BIVERSE model by 
noting that K=1  is the unphysical relativistic solution, negative energy  
flowing forward in time. ( and its CPT partner a positive energy Universe  
flowing backward in time). 
This fact was discovered in a more rigorous treatment by W. Fischler and  
L. Susskind.   
Holography and  Cosmology  
Authors: W. Fischler, L.  Susskind 
(Submitted on 4 Jun  1998 (v1), last revised 11 Jun 1998 (this version, 
v2)) 
Abstract: A cosmological  version of the holographic principle is proposed. 
Various consequences are  discussed including bounds on equation of state 
and the requirement that the  universe be infinite.  
_http://arxiv.org/PS_cache/hep-th/pdf/9806/9806039v2.pdf_ 
(http://arxiv.org/PS_cache/hep-th/pdf/9806/9806039v2.pdf)  
Fischler and  Susskind write with regard to closed Universe K=1  
"Depending on the equation of state, the bound  will reached either while 
the Universe is still growing, for example when the  energy density is 
dominated by the non relativistic matter or during  recollaspse like in a 
radiation dominated Universe. This seems to indicate  that positively curved closed 
Universes are inconsistent with the Holographic  principle. We do not know 
what new behavior sets in to accommodate the  Holographic principle or if 
this violation of the Holographic principle just  excludes these Universes as 
inconsistent?" 
End  quote. 
Based on this  I would offer the following conjecture.  
Under  the assumption that the creation of Universes is properly described 
by the  general inflationary paradigm the only Universes ever created are 
zero energy  Universes in which each Hubble volume saturates the Bekenstein 
entropy bound.  Strictly speaking open Universe without dark energy are not 
precluded from  existing based on this analysis. However, based on some string 
 theory formulations a non zero CC is needed to break SUSY.  Therefore 
assuming that the general Super String SUSY paradigm is  correct we might 
exclude such Universes from existing too. It might  further be noted that this 
postulated zero sum Energy conditions for the  creations of Universes does not 
violate energy conservation even when the  "seed" energy needed to launch 
inflation is included in the  various BIVERSE models.  
4) Evolution of the  Hubble Parameter  
Current cosmological observations do not allow anything more than a 
conjecture  about the future evolution of our Universe. But since the fundamental  
conjecture that inspired this post is the existence of some deep  connection 
between the Zero Sum Universe, inflation, spatial flatness and Dark  Energy 
we shall carry this conjecture forward. The current equation of  state for 
Dark energy is bound between-1.48 and 0.72 with a mid value  of-1.1 at the 
95% confidence level. " Can We be Tricked into Thinking  that W is less 
than-1" Carroll, Felice and Trodden. 
Therefore for the purpose of this post I will assume an equation of state 
for  Dark Energy equal to-1. Based on this we get the equations; 
Rho_vac=  3*Omega_lambda*H^2/(8*pi*G) 
Rho_mass=  3*(1-omega_lambda)*H^2/8*pi*G) 
Therefore   
H= sqrt[ 8*pi*G*rho_vac/(3*omega_lambda)] = K_1/sqrt[omega_lambda] 
H=sqrt[ 8*pi*G*rho_massc/(3*( 1-omega_lambda))] = K_2*(sqrt[  
Rho_mass/(1-omega)]  
Based on these  assumptions and current cosmological data the Hubble 
parameter will evolves  from its current value of 2.31E-18 to 1.94E-18 in 1E5 Byrs 
or giving a change  of 3.7E-33 % per year from its current value.  
Of course based on  the evidence this is only one possible scenario. In the 
future I hope to  expand on this conjecture by possibly incorporating the 
inflation process  into this scenario.   
Bob Zannelli