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Boltzmann's Brains and Eternal De Sitter Space
In 2002 Dyson, Kleban and Susskind (DKS) wrote a land mark paper which
investigated the problem associated with an eternal De Sitter space. (
Hep-th/0208013]
Based on current astronomical data and the predictions of Holographic
Dark energy model (Which I have posted on in detail) our own Universe is on a
straight trajectory to just such a condition. Therefore understanding the
implications of an eternal De Sitter space is important. Even accepting that
a non zero vacuum energy condition itself is not eternal does not give us
a surefire way to avoid an eternal De Sitter space condition. Only if the
decay constant were relatively large with respect to the efold time of the
De Sitter space could we avoid this state. Given the age of Universe of 13.7
billion years, it seems unlikely any possible decay constant is large
enough to prevent an Eternal De Sitter space.
In their study DKS investigated the relative probabilities of a Quantum
fluctuation that produced an inflating Universe and a Quantum fluctuation that
produced a Universe similar in broad outline to our Universe. This
calculation also relates to the Boltzmann Brain problem, the concern here being
the relative probabilities of a Quantum fluctuation producing an isolated
brain and an inflating Universe.
The problem deals with the relative decrease in entropy for each
possibility. A tunneling even to inflation requires a fluctuation to a very low
entropy state, while the two aforementioned events can occur at higher entropy
states. In this post I will look at a simplified version of the DKS argument
and then relate, what I think, is a satisfactory solution proposed by
Andreas Albrecht and Lorenzo Sorbo in 2004. [hep-th/0405270v2]
To analyze a De Sitter space we must divide it up into causal patches.
While De Sitter space is infinite, each casual patch is finite allowing to
perform calculations which give us the probability we are after. Therefore we
define each causal patch by the De Sitter Horizon.
We get
R_ds= sqrt[3/lambda]
And a Hawking Gibbons temperature
T_gh= 1/(2*pi*R_ds)
We define the entropy of the De Sitter space (as Vic does in the Fallacy
of Fine Tuning) as
S_ds= pi*R_ds^2/L_plk^2
DKS assume that De Sitter space as essentially ergodic based on its
equilibrium state, that is each possible state persist over the same interval.
Therefore the number of the number of micro states is given by
N_ds= exp[S_ds]
and the number of states associated with a given fluctuation is
N_f= exp[S_f]
Therefore the probability of a given fluctuation is
P_f = N_f/N_ds= exp[S_f-S_ds]
DKS estimate that S_f for a non inflating Universe to fluctuate into
existence about 1E85 while the entropy needed for a fluctuation into an
inflating Universe is about 1E10
> From this we can
P_unv/P_inf= exp[1E85-S_ds]/exp[1E10-S_ds] = exp[ 1E85-1E10] = exp[1E85]
The probability for an inflating Universe is a tiny fraction of that for a
whole Universe fluctuation into existence, likewise for a Boltzmann brain.
Reality should be full of Boltzmann's brains and very few inflating
Universes.
But is this reasoning correct. Albrecht and Sorbo say no and I think they
make a good argument. The flaw in the DKS argument is that rather than look
at the whole causal patch of De Sitter space, they consider each Quantum
fluctuation as an isolated event. By doing this they fail to see the overall
effect that each Quantum fluctuation has. In fact it can be shown that
local low entropy fluctuations are far more probable than local higher entropy
fluctuations once they are analyzed globally. To see this we consider the
number of states of the De Sitter space before and after the fluctuation.
Therefore;
P_f= N_ds/N_ds(0) = exp[ S_ds-S_f]/exp[S_ds]= exp[ -S_f]
This gives us
P_unv/P_inf = exp[-1E85]/ exp[-1E10] = exp[-1E85]
Here the probability of a whole Universe fluctuating into existence (or a
Boltzmann brain) is but a tiny fraction of the probability of a fluctuation
producing an inflating Universe. Consequently based on the logic used by
Albrecht and Sorbo inflation events are by many orders of magnitude more
numerous than fluctuations in non inflating Universes and Boltzmann's brains.
Bob Zannelli