Boltzmann brains and the scale-factor cutoff measure of the multiverse

Article (PDF Available)inPhysical review D: Particles and fields 82(6) · September 2010with75 Reads
DOI: 10.1103/PhysRevD.82.063520 · Source: arXiv
Abstract
To make predictions for an eternally inflating “multiverse,” one must adopt a procedure for regulating its divergent spacetime volume. Recently, a new test of such spacetime measures has emerged: normal observers—who evolve in pocket universes cooling from hot big bang conditions—must not be vastly outnumbered by “Boltzmann brains”—freak observers that pop in and out of existence as a result of rare quantum fluctuations. If the Boltzmann brains prevail, then a randomly chosen observer would be overwhelmingly likely to be surrounded by an empty world, where all but vacuum energy has redshifted away, rather than the rich structure that we observe. Using the scale-factor cutoff measure, we calculate the ratio of Boltzmann brains to normal observers. We find the ratio to be finite, and give an expression for it in terms of Boltzmann brain nucleation rates and vacuum decay rates. We discuss the conditions that these rates must obey for the ratio to be acceptable, and we discuss estimates of the rates under a variety of assumptions.
arXiv:0808.3778v3 [hep-th] 20 Aug 2010
MIT-CTP-3975 SU-ITP-08/20
Boltzmann brains and the scale-factor cutoff measure of the multiverse
Andrea De Simone,1Alan H. Guth,1Andrei Linde,2, 3
Mahdiyar Noorbala,2Michael P. Salem,4and Alexander Vilenkin4
1Center for Theoretical Physics, Laboratory for Nuclear Science, and Department of Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2Department of Physics, Stanford University, Stanford, CA 94305, USA
3Yukawa Institute of Theoretical Physics, Kyoto University, Kyoto, Japan
4Institute of Cosmology, Department of Physics and Astronomy,
Tufts University, Medford, MA 02155, USA
To make predictions for an eternally inflating “multiverse,” one must adopt a procedure for
regulating its divergent spacetime volume. Recently, a new test of such spacetime measures has
emerged: normal observers — who evolve in pocket universes cooling from hot big bang conditions
— must not be vastly outnumbered by “Boltzmann brains” — freak observers that pop in and out of
existence as a result of rare quantum fluctuations. If the Boltzmann brains prevail, then a randomly
chosen observer would be overwhelmingly likely to be surrounded by an empty world, where all
but vacuum energy has redshifted away, rather than the rich structure that we observe. Using the
scale-factor cutoff measure, we calculate the ratio of Boltzmann brains to normal observers. We
find the ratio to be finite, and give an expression for it in terms of Boltzmann brain nucleation rates
and vacuum decay rates. We discuss the conditions that these rates must obey for the ratio to be
acceptable, and we discuss estimates of the rates under a variety of assumptions.
I. INTRODUCTION
The simplest interpretation of the observed accelerat-
ing expansion of the universe is that it is driven by a
constant vacuum energy density ρΛ, which is about three
times greater than the present density of nonrelativistic
matter. While ordinary matter becomes more dilute as
the universe expands, the vacuum energy density remains
the same, and in another ten billion years or so the uni-
verse will be completely dominated by vacuum energy.
The subsequent evolution of the universe is accurately
described as de Sitter space.
It was shown by Gibbons and Hawking [1] that an ob-
server in de Sitter space would detect thermal radiation
with a characteristic temperature TdS =HΛ/2π, where
HΛ=r8π
3Λ(1)
is the de Sitter Hubble expansion rate. For the observed
value of ρΛ, the de Sitter temperature is extremely low,
TdS = 2.3×1030 K. Nevertheless, complex structures
will occasionally emerge from the vacuum as quantum
fluctuations, at a small but nonzero rate per unit space-
time volume. An intelligent observer, like a human, could
be one such structure. Or, short of a complete observer,
a disembodied brain may fluctuate into existence, with
a pattern of neuron firings creating a perception of be-
ing on Earth and, for example, observing the cosmic mi-
crowave background radiation. Such freak observers are
collectively referred to as “Boltzmann brains” [2, 3]. Of
course, the nucleation rate ΓBB of Boltzmann brains is
extremely small, its magnitude depending on how one de-
fines a Boltzmann brain. The important point, however,
is that ΓBB is always nonzero.
De Sitter space is eternal to the future. Thus, if the ac-
celerating expansion of the universe is truly driven by the
energy density of a stable vacuum state, then Boltzmann
brains will eventually outnumber normal observers, no
matter how small the value of ΓBB [4, 7, 5, 8, 9] might
be.
To define the problem more precisely, we use the term
“normal observers” to refer to those that evolve as a re-
sult of non-equilibrium processes that occur in the wake
of the hot big bang. If our universe is approaching a sta-
ble de Sitter spacetime, then the total number of normal
observers that will ever exist in a fixed comoving volume
of the universe is finite. On the other hand, the cumula-
tive number of Boltzmann brains grows without bound
over time, growing roughly as the volume, proportional
to e3HΛt. When extracting the predictions of this the-
ory, such an infinite preponderance of Boltzmann brains
cannot be ignored.
For example, suppose that some normal observer, at
some moment in her lifetime, tries to make a predic-
tion about her next observation. According to the theory
there would be an infinite number of Boltzmann brains,
distributed throughout the spacetime, that would happen
to share exactly all her memories and thought processes
at that moment. Since all her knowledge is shared with
this set of Boltzmann brains, for all she knows she could
equally likely be any member of the set. The probability
that she is a normal observer is then arbitrarily small, and
all predictions would be based on the proposition that she
is a Boltzmann brain. The theory would predict, there-
fore, that the next observations that she will make, if she
survives to make any at all, will be totally incoherent,
2
with no logical relationship to the world that she thought
she knew. (While it is of course true that some Boltz-
mann brains might experience coherent observations, for
example by living in a Boltzmann solar system, it is easy
to show that Boltzmann brains with such dressing would
be vastly outnumbered by Boltzmann brains without any
coherent environment.) Thus, the continued orderliness
of the world that we observe is distinctly at odds with the
predictions of a Boltzmann-brain-dominated cosmology.1
This problem was recently addressed by Page [8], who
concluded that the least unattractive way to produce
more normal observers than Boltzmann brains is to re-
quire that our vacuum should be rather unstable. More
specifically, it should decay within a few Hubble times of
vacuum energy domination; that is, in 20 billion years or
so.
In the context of inflationary cosmology, however, this
problem acquires a new twist. Inflation is generically
eternal, with the physical volume of false-vacuum in-
flating regions increasing exponentially with time and
“pocket universes” like ours constantly nucleating out of
the false vacuum. In an eternally inflating multiverse,
the numbers of normal observers and Boltzmann brains
produced over the course of eternal inflation are both in-
finite. They can be meaningfully compared only after
one adopts some prescription to regulate the infinities.
The problem of regulating the infinities in an eter-
nally inflating multiverse is known as the measure prob-
lem [10], and has been under discussion for some time.
It is crucially important in discussing predictions for any
kind of observation. Most of the discussion, including the
discussion in this paper, has been confined to the classi-
cal approximation. While one might hope that someday
there will be an answer to this question based on a fun-
damental principle [11], most of the work on this sub ject
has focussed on proposing plausible measures and explor-
ing their properties. Indeed, a number of measures have
been proposed [12–28], and some of them have already
been disqualified, as they make predictions that conflict
with observations.
In particular, if one uses the proper-time cutoff mea-
sure [12–16], one encounters the “youngness paradox,”
predicting that humans should have evolved at a very
1Here we are taking a completely mechanistic view of the brain,
treating it essentially as a highly sophisticated computer. Thus,
the normal observer and the Boltzmann brains can be thought
of as a set of logically equivalent computers running the same
program with the same data, and hence they behave identically
until they are affected by further input, which might be differ-
ent. Since the computer program cannot determine whether it
is running inside the brain of one of the normal observers or one
of the Boltzmann brains, any intelligent probabilistic prediction
that the program makes about the next observation would be
based on the assumption that it is equally likely to be running
on any member of that set.
early cosmic time, when the conditions for life were rather
hostile [29]. The youngness problem, as well as the Boltz-
mann brain problem, can be avoided in the stationary
measure [19, 28], which is an improved version of the
proper-time cutoff measure. However, the stationary
measure, as well as the pocket-based measure, is afflicted
with a runaway problem, suggesting that we should ob-
serve extreme values (either very small or very large) of
the primordial density contrast Q[31, 32] and the grav-
itational constant G[33], while these parameters appear
to sit comfortably in the middle of their respective an-
thropic ranges [34, 33]. Some suggestions to get around
this issue have been described in Refs. [32, 35–37]. In
addition, the pocket-based measure seems to suffer from
the Boltzmann brain problem. The comoving coordinate
measure [12, 38] and the causal-patch measures [24, 25]
are free from these problems, but have an unattractive
feature of depending sensitively on the initial state of
the multiverse. This does not seem to mix well with
the attractor nature of eternal inflation: the asymptotic
late-time evolution of an eternally inflating universe is
independent of the starting point, so it seems appeal-
ing for the measure to maintain this property. Since the
scale-factor cutoff measure2[13–15, 17, 18, 39] has been
shown to be free of all of the above issues [40], we con-
sider it to be a promising candidate for the measure of
the multiverse.
As we have indicated, the relative abundance of normal
observers and Boltzmann brains depends on the choice of
measure over the multiverse. This means the predicted
ratio of Boltzmann brains to normal observers can be
used as yet another criterion to evaluate a prescription
to regulate the diverging volume of the multiverse: regu-
lators that predict normal observers are greatly outnum-
bered by Boltzmann brains should be ruled out. This
criterion has been studied in the context of several mul-
tiverse measures, including a causal patch measure [9],
several measures associated with globally defined time
coordinates [18, 41, 19, 30, 28], and the pocket-based
measure [42]. In this work, we apply this criterion to the
scale-factor cutoff measure, extending the investigation
that was initiated in Ref. [18]. We show that the scale-
factor cutoff measure gives a finite ratio of Boltzmann
brains to normal observers; if certain assumptions about
the landscape are valid, the ratio can be small.3
2This measure is sometimes referred to as the volume-weighted
scale-factor cutoff measure, but we will define it below in terms of
the counting of events in spacetime, so the concept of weighting
will not be relevant. The term “volume-weighted” is relevant
when a measure is described as a prescription for defining the
probability distribution for the value of a field. In Ref. [18],
this measure is called the “pseudo-comoving volume-weighted
measure.”
3In a paper that appeared simultaneously with version 1 of this
paper, Raphael Bousso, Ben Freivogel, and I-Sheng Yang inde-
pendently analyzed the Boltzmann brain problem for the scale-
3
The remainder of this paper is organized as follows.
In Section II, we provide a brief description of the scale-
factor cutoff and describe salient features of the multi-
verse under the lens of this measure. In Section III we
calculate the ratio of Boltzmann brains to normal ob-
servers in terms of multiverse volume fractions and tran-
sition rates. The volume fractions are discussed in Sec-
tion IV, in the context of toy landscapes, and the section
ends with a general description of the conditions neces-
sary to avoid Boltzmann brain domination. The rate of
Boltzmann brain production and the rate of vacuum de-
cay play central roles in our calculations, and these are
estimated in Section V. Concluding remarks are provided
in Section VI.
II. THE SCALE FACTOR CUTOFF
Perhaps the simplest way to regulate the infinities of
eternal inflation is to impose a cutoff on a hypersurface
of constant global time [13–17]. One starts with a patch
of a spacelike hypersurface Σ somewhere in an inflating
region of spacetime, and follows its evolution along the
congruence of geodesics orthogonal to Σ. The scale-factor
time is defined as
t= ln a , (2)
where ais the expansion factor along the geodesics. The
scale-factor time is related to the proper time τby
dt =H dτ , (3)
where His the Hubble expansion rate of the congruence.
The spacetime region swept out by the congruence will
typically expand to unlimited size, generating an infinite
number of pockets. (If the patch does not grow with-
out limit, one chooses another initial patch Σ and starts
again.) The resulting four-volume is infinite, but we cut
it off at some fixed scale-factor time t=tc. To find the
relative probabilities of different events, one counts the
numbers of such events in the finite spacetime volume
between Σ and the t=tchypersurface, and then takes
the limit tc→ ∞.
The term “scale factor” is often used in the context
of homogeneous and isotropic geometries; yet on very
large and on very small scales the multiverse may be
very inhomogeneous. A simple way to deal with this is
to take the factor Hin Eq. (3) to be the local divergence
of the four-velocity vector field along the congruence of
geodesics orthogonal to Σ,
H(x)(1/3) uµ;µ.(4)
factor cutoff measure [43].
When more than one geodesic passes through a point,
the scale-factor time at that point may be taken to be
the smallest value among the set of geodesics. In col-
lapsing regions H(x) is negative, in which case the corre-
sponding geodesics are continued unless or until they hit
a singularity.
This “local” definition of scale-factor time has a sim-
ple geometric meaning. The congruence of geodesics can
be thought of as representing a “dust” of test particles
scattered uniformly on the initial hypersurface Σ. As one
moves along the geodesics, the density of the dust in the
orthogonal plane decreases. The expansion factor ain
Eq. (2) can then defined as aρ1/3, where ρis the
density of the dust, and the cutoff is triggered when ρ
drops below some specified level.
Although the local scale-factor time closely follows the
FRW scale factor in expanding spacetimes — such as
inflating regions and thermalized regions not long after
reheating — it differs dramatically from the FRW scale
factor as small-scale inhomogeneities develop during mat-
ter domination in universes like ours. In particular, the
local scale-factor time nearly grinds to a halt in regions
that have decoupled from the Hubble flow. It is not clear
whether we should impose this particular cutoff, which
would essentially include the entire lifetime of any non-
linear structure that forms before the cutoff, or impose a
cutoff on some nonlocal time variable that more closely
tracks the FRW scale factor.4
There are a number of nonlocal modifications of scale
factor time that both approximate our intuitive notion of
FRW averaging and also extend into more complicated
geometries. One drawback of the nonlocal approach is
that no single choice looks more plausible than the others.
For instance, one nonlocal method is to define the factor
Hin Eq. (3) by spatial averaging of the quantity H(x)
in Eq. (4). A complete implementation of this approach,
however, involves many seemingly arbitrary choices re-
garding how to define the hypersurfaces over which H(x)
should be averaged, so we here set this possibility aside.
A second, simpler method is to use the local scale-factor
time defined above, but to generate a new cutoff hyper-
surface by excluding the future lightcones of all points
on the original cutoff hypersurface. In regions with non-
linear inhomogeneities, the underdense regions will be
the first to reach the scale-factor cutoff, after which they
quickly trigger the cutoff elsewhere. The resulting cutoff
hypersurface will not be a surface of constant FRW scale
factor, but the fluctuations of the FRW scale factor on
this surface should be insignificant.
As a third and final example of a nonlocal modifica-
tion of scale factor time, we recall the description of the
4The distinction between these two forms of scale-factor time was
first pointed out by Bousso, Freivogel, and Yang in Ref. [43].
4
local scale-factor cutoff in terms the density ρof a dust
of test particles. Instead of such a dust, consider a set of
massless test particles, emanating uniformly in all direc-
tions from each point on the initial hypersurface Σ. We
can then construct the conserved number density current
Jµfor the gas of test particles, and we can define ρas
the rest frame number density, i.e. the value of J0in
the local Lorentz frame in which Ji= 0, or equivalently
ρ=J2. Defining aρ1/3, as we did for the dust
of test particles, we apply the cutoff when the number
density ρdrops below some specified level. Since null
geodesics are barely perturbed by structure formation,
the strong perturbations inherent in the local definition
of scale factor time are avoided. Nonetheless, we have
not studied the properties of this definition of scale fac-
tor time, and they may lead to complications. Large-scale
anisotropic flows in the gas of test particles can be gener-
ated as the particles stream into expanding bubbles from
outside. Since the null geodesics do not interact with
matter except gravitationally, these anisotropies will not
be damped in the same way as they would be for pho-
tons. The large-scale flow of the gas will not redshift in
the normal way, either; for example, if the test particles
in some region of an FRW universe have a nonzero mean
velocity relative to the comoving frame, the expansion
of the universe will merely reduce the energies of all the
test particles by the same factor, but will not cause the
mean velocity to decrease. Thus, the detailed predictions
for this definition of scale-factor cutoff measure remain a
matter for future study.
The local scale-factor cutoff and each of the three non-
local definitions correspond to different global-time pa-
rameterizations and thus to different spacetime measures.
In general they make different predictions for physical
observables; however with regard to the relative number
of normal observers and Boltzmann brains, their predic-
tions are essentially the same. For the remainder of this
paper we refer to the generic nonlocal definition of scale
factor time, for which we take FRW time as a suitable
approximation. Note that the use of local scale factor
time would make it slightly easier to avoid Boltzmann
brain domination, since it would increase the count of
normal observers while leaving the count of Boltzmann
brains essentially unchanged.
To facilitate later discussion, let us now describe some
general properties of the multiverse. The volume fraction
fioccupied by vacuum ion constant scale-factor time
slices can be found from the rate equation [44],
dfi
dt =X
j
Mij fj,(5)
where the transition matrix Mij is given by
Mij =κij δij X
r
κri ,(6)
and κij is the transition rate from vacuum jto vacuum i
per Hubble volume per Hubble time. This rate can also
be written
κij = (4π/3)H4
jΓij ,(7)
where Γij is the bubble nucleation rate per unit spacetime
volume and Hjis the Hubble expansion rate in vacuum j.
The solution of Eq. (5) can be written in terms of the
eigenvectors and eigenvalues of the transition matrix Mij .
It is easily verified that each terminal vacuum is an
eigenvector with eigenvalue zero. We here define “ter-
minal vacua” as those vacua jfor which κij = 0 for all
i. Thus the terminal vacua include both negative-energy
vacua, which collapse in a big crunch, and stable zero-
energy vacua. It was shown in Ref. [22] that all of the
other eigenvalues of Mij have negative real parts. More-
over, the eigenvalue with the smallest (by magnitude)
real part is pure real; we call it the “dominant eigen-
value” and denote it by q(with q > 0). Assuming that
the landscape is irreducible, the dominant eigenvalue is
nondegenerate. In that case the probabilities defined by
the scale-factor cutoff measure are independent of the ini-
tial state of the multiverse, since they are determined by
the dominant eigenvector.5
For an irreducible landscape, the late-time asymptotic
solution of Eq. (5) can be written in the form6
fj(t) = f(0)
j+sjeqt +. . . , (8)
where the constant term f(0)
jis nonzero only in terminal
vacua and sjis proportional to the eigenvector of Mij
corresponding to the dominant eigenvalue q, with the
constant of proportionality determined by the initial dis-
tribution of vacua on Σ. It was shown in Ref. [22] that
sj0 for terminal vacua, and sj>0 for nonterminal
vacua, as is needed for Eq. (8) to describe a nonnegative
5In this work we assume that the multiverse is irreducible; that
is, any metastable inflating vacuum is accessible from any other
such vacuum via a sequence of tunneling transitions. Our re-
sults, however, can still be applied when this condition fails. In
that case the dominant eigenvalue can be degenerate, in which
case the asymptotic future is dominated by a linear combina-
tion of dominant eigenvectors that is determined by the initial
state. If transitions that increase the vacuum energy density are
included, then the landscape can be reducible only if it splits
into several disconnected sectors. That situation was discussed
in Appendix A of Ref. [40], where two alternative prescriptions
were described. The first prescription (preferred by the authors)
leads to initial-state dependence only if two or more sectors have
the same dominant eigenvalue q, while the second prescription
always leads to initial-state dependence.
6Mij is not necessarily diagonalizable, but Eq. (8) applies in any
case. It is always possible to form a complete basis from eigenvec-
tors and generalized eigenvectors, where generalized eigenvectors
satisfy (MλI)ks= 0, for k > 1. The generalized eigenvectors
appear in the solution with a time dependence given by eλt times
a polynomial in t. These terms are associated with the nonlead-
ing eigenvalues omitted from Eq. (8), and the polynomials in t
will not change the fact that they are nonleading.
5
volume fraction, with a nondecreasing fraction assigned
to any terminal vacuum.
By inserting the asymptotic expansion (8) into the dif-
ferential equation (5) and extracting the leading asymp-
totic behavior for a nonterminal vacuum i, one can show
that
(κiq)si=X
j
κij sj,(9)
where κjis the total transition rate out of vacuum j,
κjX
i
κij .(10)
The positivity of sifor nonterminal vacua then implies
rigorously that qis less than the decay rate of the slowest-
decaying vacuum in the landscape:
qκmin min{κj}.(11)
Since “upward” transitions (those that increase the en-
ergy density) are generally suppressed, we can gain some
intuition by first considering the case in which all upward
transition rates are set to zero. (Such a landscape is re-
ducible, so the dominant eigenvector can be degenerate.)
In this case Mij is triangular, and the eigenvalues are
precisely the decay rates κiof the individual states. The
dominant eigenvalue qis then exactly equal to κmin .
If upward transitions are included but assumed to have
a very low rate, then the dominant eigenvalue qis approx-
imately equal to the decay rate of the slowest-decaying
vacuum [45],
qκmin .(12)
The slowest-decaying vacuum (assuming it is unique) is
the one that dominates the asymptotic late-time volume
of the multiverse, so we call it the dominant vacuum and
denote it by D. Hence,
qκD.(13)
The vacuum decay rate is typically exponentially sup-
pressed, so for the slowest-decaying vacuum we expect it
to be extremely small,
q1.(14)
Note that the corrections to Eq. (13) are comparable
to the upward transition rate from Dto higher-energy
vacua, but for large energy differences this transition rate
is suppressed by the factor exp(8π2/H 2
D) [46]. Here and
throughout the remainder of this paper we use reduced
Planck units, where 8πG =c=kB= 1. We shall argue
in Section V that the dominant vacuum is likely to have
a very low energy density, so the correction to Eq. (13)
is very small even compared to q.
A possible variant of this picture, with similar conse-
quences, could arise if one assumes that the landscape
includes states with nearby energy densities for which
the upward transition rate is not strongly suppressed. In
that case there could be a group of vacuum states that
undergo rapid transitions into each other, but very slow
transitions to states outside the group. The role of the
dominant vacuum could then be played by this group of
states, and qwould be approximately equal to some ap-
propriately averaged rate for the decay of these states
to states outside the group. Under these circumstances
qcould be much less than κmin. An example of such a
situation is described in Subsection IV E.
In the asymptotic limit of late scale-factor time t, the
physical volumes in the various nonterminal vacua are
given by
Vj(t) = V0sje(3q)t,(15)
where V0is the volume of the initial hypersurface Σ and
e3tis the volume expansion factor. The volume growth in
Eq. (15) is (very slightly) slower than e3tdue to the con-
stant loss of volume from transitions to terminal vacua.
Note that even though upward transitions from the dom-
inant vacuum are strongly suppressed, they play a crucial
role in populating the landscape [45]. Most of the volume
in the asymptotic solution of Eq. (15) originates in the
dominant vacuum D, and “trickles” to the other vacua
through a series of transitions starting with at least one
upward jump.
III. THE ABUNDANCE OF NORMAL
OBSERVERS AND BOLTZMANN BRAINS
Let us now calculate the relative abundances of Boltz-
mann brains and normal observers, in terms of the vac-
uum transition rates and the asymptotic volume frac-
tions.
Estimates for the numerical values of the Boltzmann
brain nucleation rates and vacuum decay rates will be
discussed in Section V, but it is important at this stage to
be aware of the kind of numbers that will be considered.
We will be able to give only rough estimates of these
rates, but the numbers that will be mentioned in Section
V will range from exp (10120) to exp (1016 ). Thus,
when we calculate the ratio NBB /NNO of Boltzmann
brains to normal observers, the natural logarithm of this
ratio will always include one term with a magnitude of at
least 1016. Consequently, the presence or absence of any
term in ln(NBB/NNO ) that is small compared to 1016 is
of no relevance. We therefore refer to any factor ffor
which
|ln f|<1014 (16)
as “roughly of order one.” In the calculation of
NBB/NNO such factors — although they may be minus-
cule or colossal by ordinary standards — can be ignored.
6
It will not be necessary to keep track of factors of 2, π,
or even 10108. Dimensionless coefficients, factors of H,
and factors coming from detailed aspects of the geome-
try are unimportant, and in the end all of these will be
ignored. We nonetheless include some of these factors in
the intermediate steps below simply to provide a clearer
description of the calculation.
We begin by estimating the number of normal ob-
servers that will be counted in the sample spacetime re-
gion specified by the scale-factor cutoff measure. Normal
observers arise during the big bang evolution in the af-
termath of slow-roll inflation and reheating. The details
of this evolution depend not only on the vacuum of the
pocket in question, but also on the parent vacuum from
which it nucleated [47]. That is, if we view each vacuum
as a local minimum in a multidimensional field space,
then the dynamics of inflation in general depend on the
direction from which the field tunneled into the local min-
imum. We therefore label pockets with two indices, ik,
indicating the pocket and parent vacua respectively.
To begin, we restrict our attention to a single “an-
thropic” pocket — i.e., one that produces normal ob-
servers — which nucleates at scale-factor time tnuc. The
internal geometry of the pocket is that of an open FRW
universe. We assume that, after a brief curvature-
dominated period ∆τH1
k, slow-roll inflation inside
the pocket gives Nee-folds of expansion before thermal-
ization. Furthermore, we assume that all normal ob-
servers arise at a fixed number NOof e-folds of expan-
sion after thermalization. (Note that Neand NOare
both measured along FRW comoving geodesics inside the
pocket, which do not initially coincide with, but rapidly
asymptote to, the global” geodesic congruence that orig-
inated outside the pocket.) We denote the fixed-internal-
time hypersurface on which normal observers arise by
ΣNO, and call the average density of observers on this
hypersurface nNO
ik .
The hypersurface ΣNO would have infinite volume, due
to the constant expansion of the pocket, but this diver-
gence is regulated by the scale-factor cutoff tc, because
the global scale-factor time tis not constant over the
ΣNO hypersurface. For the pocket described above, the
regulated physical volume of ΣNO can be written as
V(ik)
O(tnuc) = H3
ke3(Ne+NO)w(tctnuc NeNO),
(17)
where the exponential gives the volume expansion factor
coming from slow-roll inflation and big bang evolution to
the hypersurface ΣNO, and H3
kw(tctnuc NeNO)
describes the comoving volume of the part of the ΣNO
hypersurface that is underneath the cutoff. The function
w(t) was calculated, for example, in Refs. [48] and [30],
and is applied to scale-factor cutoff measure in Ref. [49].
Its detailed form will not be needed to determine the
answer up to a factor that is roughly of order one, but to
avoid mystery we mention that w(t) can be written as
w(t) = π
2Z¯
ξ(t)
0
sinh2(ξ)=π
8sinh2¯
ξ(t)2¯
ξ(t),
(18)
where ¯
ξ(tctnuc NeNO) is the maximum value of
the Robertson-Walker radial coordinate ξthat lies under
the cutoff. If the pocket universe begins with a moderate
period of inflation (exp(Ne)1) with the same vacuum
energy as outside, then
¯
ξ(t)2 cosh1et/2.(19)
Eq. (17) gives the physical volume on the ΣNO hyper-
surface for a single pocket of type ik, which nucleates
at time tnuc. The number of ik-pockets that nucleate
between time tnuc and tnuc +dtnuc is
dn(ik)
nuc (tnuc) = (3/4π)H3
kκik Vk(tnuc)dtnuc
= (3/4π)H3
kκikskV0e(3q)tnuc dtnuc ,(20)
where we use Eq. (15) to give Vk(tnuc). The total number
of normal observers in the sample region is then
NNO
ik =nNO
ik ZtcNeNO
V(ik)
O(tnuc)dn(ik)
nuc (tnuc)
nNO
ik κikskV0e(3q)tcZ
0
w(z)e(3q)zdz .(21)
In the first expression we have ignored the (very small)
probability that pockets of type ik may transition to
other vacua during slow-roll inflation or during the sub-
sequent period NOof big bang evolution. In the sec-
ond line, we have changed the integration variable to
z=tctnuc NeNO(reversing the direction of inte-
gration) and have dropped the O(1) prefactors, and also
the factor eq(Ne+NO), since qis expected to be extraordi-
narily small. We have kept eqtc, since we are interested
in the limit tc→ ∞. We have also kept the factor eqz
long enough to verify that the integral converges with or
without the factor, so we can carry out the integral using
the approximation q0, resulting in an O(1) prefactor
that we will drop.
Finally,
NNO
ik nNO
ik κik skV0e(3q)tc.(22)
Note that the expansion factor e3(Ne+NO)in Eq. (17) was
canceled when we integrated over nucleation times, illus-
trating the mild youngness bias of the scale-factor cutoff
measure. The expansion of the universe is canceled, so
objects that form at a certain density per physical vol-
ume in the early universe will have the same weight as
objects that form at the same density per physical vol-
ume at a later time, despite the naive expectation that
there is more volume at later times.
To compare, we now need to calculate the number of
Boltzmann brains that will be counted in the sample
7
spacetime region. Boltzmann brains can be produced
in any anthropic vacuum, and presumably in many non-
anthropic vacua as well. Suppose Boltzmann brains are
produced in vacuum jat a rate ΓBB
jper unit space-
time volume. The number of Boltzmann brains NBB
j
is then proportional to the total four-volume in that vac-
uum. Imposing the cutoff at scale-factor time tc, this
four-volume is
V(4)
j=Ztc
Vj(t)=H1
jZtc
Vj(t)dt
=1
3qH1
jsjV0e(3q)tc,(23)
where we have used Eq. (15) for the asymptotic volume
fraction. By setting =H1
jdt, we have ignored the
time-dependence of H(τ) in the earlier stages of cosmic
evolution, assuming that only the late-time de Sitter evo-
lution is relevant. In a similar spirit, we will assume that
the Boltzmann brain nucleation rate ΓBB
jcan be treated
as time-independent, so the total number of Boltzmann
brains nucleated in vacua of type j, within the sample
volume, is given by
NBB
jΓBB
jH1
jsjV0e(3q)tc,(24)
where we have dropped the O(1) numerical factor.
For completeness, we may want to consider the effects
of early universe evolution on Boltzmann brain produc-
tion, effects which were ignored in Eq. (24). We will
separate the effects into two categories: the effects of
slow-roll inflation at the beginning of a pocket universe,
and the effects of reheating.
To account for the effects of slow-roll inflation, we ar-
gue that, within the approximations used here, there is
no need for an extra calculation. Consider, for example,
a pocket universe Awhich begins with a period of slow-
roll inflation during which H(τ)Hslow roll = const.
Consider also a pocket universe B, which throughout its
evolution has H=Hslow roll, and which by hypothesis
has the same formation rate, Boltzmann brain nucle-
ation rate, and decay rates as pocket A. Then clearly
the number of Boltzmann brains formed in the slow roll
phase of pocket Awill be smaller than the number formed
throughout the lifetime of pocket B. Since we will require
that generic bubbles of type Bdo not overproduce Boltz-
mann brains, there will be no need to worry about the
slow-roll phase of bubbles of type A.
To estimate how many Boltzmann brains might form
as a consequence of reheating, we can make use of the
calculation for the production of normal observers de-
scribed above. We can assume that the Boltzmann brain
nucleation rate has a spike in the vicinity of some par-
ticular hypersurface in the early universe, peaking at
some value ΓBB
reheat,ik which persists roughly for some time
interval ∆τBB
reheat,ik, producing a density of Boltzmann
brains equal to ΓBB
reheat,ik τBB
reheat,ik. This spatial density
is converted into a total number for the sample volume in
exactly the same way that we did for normal observers,
leading to
NBB,reheat
ik ΓBB
reheat,ik τBB
reheat,ik κik skV0e(3q)tc.
(25)
Thus, the dominance of normal observers is assured if
X
i,k
ΓBB
reheat,ik τBB
reheat,ikκik skX
i,k
nNO
ik κik sk.(26)
If Eq. (26) did not hold, it seems likely that we would
suffer from Boltzmann brain problems regardless of our
measure. We leave numerical estimates for Section V,
but we will see that Boltzmann brain production during
reheating is not a danger.
Ignoring the Boltzmann brains that form during re-
heating, the ratio of Boltzmann brains to normal ob-
servers can be found by combining Eqs. (22) and (24),
giving
NBB
NNO PjH3
jκBB
jsj
Pi, k nNO
ik κik sk
,(27)
where the summation in the numerator covers only the
vacua in which Boltzmann brains can arise, the sum-
mation over iin the denominator covers only anthropic
vacua, and the summation over kincludes all of their
possible parent vacua. κBB
jis the dimensionless Boltz-
mann brain nucleation rate in vacuum j, related to ΓBB
j
by Eq. (7). The expression can be further simplified by
dropping the factors of Hjand nNO
i, which are roughly
of order one, as defined by Eq. (16). We can also replace
the sum over jin the numerator by the maximum over j,
since the sum is at least as large as the maximum term
and no larger than the maximum term times the num-
ber of vacua. Since the number of vacua (perhaps 10500 )
is roughly of order one, the sum over jis equal to the
maximum up to a factor that is roughly of order one.
We similarly replace the sum over iin the denominator
by its maximum, but we choose to leave the sum over k.
Thus we can write
NBB
NNO maxj{κBB
jsj}
maxi{Pkκik sk},(28)
where the sets of jand iare restricted as for Eq. (27).
In dropping nNO
i, we are assuming that nNO
iH3
iis
roughly of order one, as defined at the beginning of this
section. It is hard to know what a realistic value for
nNO
iH3
imight be, as the evolution of normal observers
may require some highly improbable events. For exam-
ple, it was argued in Ref. [50] that the probability for life
to evolve in a region of the size of our observable universe
per Hubble time may be as low as 101000 . But even
the most pessimistic estimates cannot compete with the
small numbers appearing in estimates of the Boltzmann
brain nucleation rate, and hence by our definition they
8
are roughly of order one. Nonetheless, it is possible to
imagine vacua for which nNO
imight be negligibly small,
but still nonzero. We shall ignore the normal observers
in these vacua; for the remainder of this paper we will
use the phrase “anthropic vacuum” to refer only to those
vacua for which nNO
iH3
iis roughly of order one.
For any landscape that satisfies Eq. (8), which includes
any irreducible landscape, Eq. (28) can be simplified by
using Eq. (9):
NBB
NNO maxj{κBB
jsj}
maxi{(κiq)si},(29)
where the numerator is maximized over all vacua jthat
support Boltzmann brains, and the denominator is max-
imized over all anthropic vacua i.
In order to learn more about the ratio of Boltzmann
brains to normal observers, we need to learn more about
the volume fractions sj, a topic that will be pursued in
the next section.
IV. MINI-LANDSCAPES AND THE GENERAL
CONDITIONS TO AVOID BOLTZMANN BRAIN
DOMINATION
In this section we study a number of simple models
of the landscape, in order to build intuition for the vol-
ume fractions that appear in Eqs. (28) and (29). The
reader uninterested in the details may skip the pedagog-
ical examples given in Subsections IV A–IV E, and con-
tinue with Subsection IV F, where we state the general
conditions that must be enforced in order to avoid Boltz-
mann brain domination.
A. The FIB Landscape
Let us first consider a very simple model of the land-
scape, described by the schematic
FIB , (30)
where Fis a high-energy false vacuum, Iis a positive-
energy anthropic vacuum, and Bis a terminal vacuum.
This model, which we call the FIB landscape, was ana-
lyzed in Ref. [22] and was discussed in relation to the
abundance of Boltzmann brains in Ref. [18]. As in
Ref. [18], we assume that both Boltzmann brains and
normal observers reside only in vacuum I.
Note that the FIB landscape ignores upward transi-
tions from Ito F. The model is constructed in this way
as an initial first step, and also in order to more clearly
relate our analysis to that of Ref. [18]. Although the rate
of upward transitions is exponentially suppressed rela-
tive the other rates, its inclusion is important for the
irreducibility of the landscape, and hence the nondegen-
eracy of the dominant eigenvalue and the independence
of the late-time asymptotic behavior from the initial con-
ditions of the multiverse. The results of this subsection
will therefore not always conform to the expectations out-
lined in Section II, but this shortcoming is corrected in
the next subsection and all subsequent work in this pa-
per.
We are interested in the eigenvectors and eigenvalues
of the rate equation, Eq. (5). In the FIB landscape the
rate equation gives
˙
fF=κIF fF
˙
fI=κBI fI+κI F fF.(31)
We ignore the volume fraction in the terminal vacuum
as it is not relevant to our analysis. Starting with the
ansatz,
f(t) = seqt ,(32)
we find two eigenvalues of Eqs. (31). These are, with
their corresponding eigenvectors,
q1=κIF ,s1= (1, C ),
q2=κBI ,s2= (0,1) ,(33)
where the eigenvectors are written in the basis s
(sF, sI) and
C=κIF
κBI κI F
.(34)
Suppose that we start in the false vacuum Fat t= 0,
i.e. f(t= 0) = (1,0). Then the solution of the FIB rate
equation, Eq. (31), is
fF(t) = eκIF t,
fI(t) = C(eκIF teκBI t).(35)
The asymptotic evolution depends on whether κI F <
κBI (case I) or not (case II). In case I,
f(t→ ∞) = s1eκIF t(κI F < κBI ),(36)
where s1is given in Eq. (33), while in case II
f(t→ ∞) = eκIF t,|C|eκBI t(κB I < κIF ).
(37)
In the latter case, the inequality of the rates of decay for
the two volume fractions arises from the reducibility of
the FIB landscape, stemming from our ignoring upward
transitions from Ito F.
For case I (κI F < κBI ), we find the ratio of Boltzmann
brains to normal observers by evaluating Eq. (28) for the
asymptotic behavior described by Eq. (36):
NBB
NNO κBBsI
κIF sFκBB
κIF
κIF
κBI κI F κBB
κBI
,(38)
9
where we drop κIF compared to κBI in the denomina-
tor, as we are only interested in the overall scale of the
solution. We find that the ratio of Boltzmann brains to
normal observers is finite, depending on the relative rate
of Boltzmann brain production to the rate of decay of
vacuum I. Meanwhile, in case II (where κB I < κIF ) we
find
NBB
NNO κBB
κIF
e(κIF κBI )t→ ∞ .(39)
In this situation, the number of Boltzmann brains over-
whelms the number of normal observers; in fact the ratio
diverges with time.
The unfavorable result of case II stems from the fact
that, in this case, the volume of vacuum Igrows faster
than that of vacuum F. Most of this I-volume is in large
pockets that formed very early; and this volume domi-
nates because the F-vacuum decays faster than I, and is
not replenished due to the absence of upward transitions.
This leads to Boltzmann brain domination, in agreement
with the conclusion reached in Ref. [18]. Thus, the FIB
landscape analysis suggests that Boltzmann brain dom-
ination can be avoided only if the decay rate of the an-
thropic vacuum is larger than both the decay rate of its
parent false vacuum Fand the rate of Boltzmann brain
production. Moreover, the FIB analysis suggests that
Boltzmann brain domination in the multiverse can be
avoided only if the first of these conditions is satisfied for
all vacua in which Boltzmann brains exist. This is a very
stringent requirement, since low-energy vacua like Ityp-
ically have lower decay rates than high-energy vacua (see
Section V). We shall see, however, that the above condi-
tions are substantially relaxed in more realistic landscape
models.
B. The FIB Landscape with Recycling
The FIB landscape of the preceding section is re-
ducible, since vacuum Fcannot be reached from vacuum
I. We can make it irreducible by simply allowing upward
transitions,
FIB . (40)
This “recycling FIB” landscape is more realistic than the
original FIB landscape, because upward transitions out of
positive-energy vacua are allowed in semi-classical quan-
tum gravity [46]. The rate equation of the recycling FIB
landscape gives the eigenvalue system,
qsF=κI F sF+κFI sI,
qsI=κIsI+κI F sF,(41)
where κIκBI +κF I is the total decay rate of vacuum
I, as defined in Eq. (10). Thus, the eigenvalues q1and
q2correspond to the roots of
(κIF q)(κIq) = κI F κFI .(42)
Further analysis is simplified if we note that upward
transitions from low-energy vacua like ours are very
strongly suppressed, even when compared to the other
exponentially suppressed transition rates, i.e. κF I
κIF , κB I . We are interested mostly in how this small
correction modifies the dominant eigenvector in the case
where κBI < κI F (case II), which led to an infinite ratio
of Boltzmann brains to normal observers. To the lowest
order in κF I , we find
qκIκIF κF I
κIF κI
,(43)
and
sIκIF κI
κF I
sFsF.(44)
The above equation is a consequence of the second of
Eqs. (41), but it also follows directly from Eq. (9), which
holds in any irreducible landscape. In this case fI(t)
and fF(t) have the same asymptotic time dependence,
eqt, so the ratio fI(t)/fF(t) approaches a constant
limit, sI/sFR. However, due to the smallness of κF I ,
this ratio is extremely large. Note that the ratio of Boltz-
mann brains to normal observers is proportional to R.
Although it is also proportional to the minuscule Boltz-
mann brain nucleation rate (estimated in Section V), the
typically huge value of Rwill still lead to Boltzmann
brain domination (again, see Section V for relevant de-
tails). But the story is not over, since the recycling FIB
landscape is still far from realistic.
C. A More Realistic Landscape
In the recycling model of the preceding section, the
anthropic vacuum Iwas also the dominant vacuum, while
in a realistic landscape this is not likely to be the case. To
see how it changes the situation to have a non-anthropic
vacuum as the dominant one, we consider the model
ADFIB , (45)
which we call the “ADFIB landscape.” Here, Dis the
dominant vacuum and Aand Bare both terminal vacua.
The vacuum Iis still an anthropic vacuum, and the vac-
uum Fhas large, positive vacuum energy. As explained
in Section V, the dominant vacuum is likely to have very
small vacuum energy; hence we consider that at least one
upward transition (here represented as the transition to
F) is required to reach an anthropic vacuum.
Note that the ADFIB landscape ignores the upward
transition rate from vacuum Ito F; however this is
exponentially suppressed relative the other transition
rates pertinent to Iand, unlike the situation in Subsec-
tion IV A, ignoring the upward transition does not signif-
icantly affect our results. The important property is that
10
all vacuum fractions have the same late-time asymptotic
behavior, and this property is assured whenever there is
a unique dominant vacuum, and all inflating vacua are
accessible from the dominant vacuum via a sequence of
tunneling transitions. The uniformity of asymptotic be-
haviors is sufficient to imply Eq. (9), which implies im-
mediately that
sI
sF
=κIF
κBI qκI F
κBI κDκI F
κBI
,(46)
where we used qκDκAD +κF D , and assumed that
κDκBI .
This holds even if the decay rate of the anthropic vac-
uum Iis smaller than that of the false vacuum F.
Even though the false vacuum Fmay decay rather
quickly, it is constantly being replenished by upward
transitions from the slowly-decaying vacuum D, which
overwhelmingly dominates the physical volume of the
multiverse. Note that, in light of these results, our
constraints on the landscape to avoid Boltzmann brain
domination are considerably relaxed. Specifically, it is
no longer required that the anthropic vacua decay at a
faster rate than their parent vacua. Using Eq. (46) with
Eq. (28), the ratio of Boltzmann brains to normal ob-
servers in vacuum Iis found to be
NBB
I
NNO
IκBB
IsI
κIF sFκBB
I
κBI
.(47)
If Boltzmann brains can also exist in the dominant
vacuum D, then they are a much more severe problem.
By applying Eq. (9) to the Fvacuum, we find
sF
sD
=κF D
κFqκF D
κFκDκF D
κF
,(48)
where κF=κIF +κD F , and where we have assumed that
κDκF. The ratio of Boltzmann brains in vacuum D
to normal observers in vacuum Iis then
NBB
D
NNO
IκBB
DsD
κIF sFκBB
D
κF D
κF
κIF
.(49)
Since we expect that the dominant vacuum has very small
vacuum energy, and hence a heavily suppressed upward
transition rate κF D , the requirement that NBB
D/NNO
Ibe
small could be a very stringent one. Note that compared
to sD, both sFand sIare suppressed by the small fac-
tor κF D; however the ratio sI/sFis independent of this
factor.
Since sDis so large, one should ask whether Boltzmann
brain domination can be more easily avoided by allowing
vacuum Dto be anthropic. The answer is no, because
the production of normal observers in vacuum Dis pro-
portional (see Eq. (22)) to the rate at which bubbles of
Dnucleate, which is not large. Ddominates the space-
time volume due to slow decay, not rapid nucleation. If
we assume that Dis anthropic and restrict Eq. (28) to
vacuum D, we find using Eq. (48) that
NBB
D
NNO
DκBB
DsD
κDF sFκBB
D
κF D
κF
κDF
,(50)
so again the ratio is enhanced by the extremely small
upward tunneling rate κF D in the denominator.
Thus, in order to avoid Boltzmann brain domination,
it seems we have to impose two requirements: (1) the
Boltzmann brain nucleation rate in the anthropic vac-
uum Imust be less than the decay rate of that vacuum,
and (2) the dominant vacuum Dmust either not support
Boltzmann brains at all, or must produce them with a
dimensionless rate κBB
Dthat is small even compared to
the upward tunneling rate κF D. If the vacuum Dis an-
thropic then it must support Boltzmann brains, so the
domination by Boltzmann brains could be avoided only
by the stringent requirement κBB
DκF D.
D. A Further Generalization
The conclusions of the last subsection are robust to
more general considerations. To illustrate, let us general-
ize the ADFIB landscape to one with many low-vacuum-
energy pockets, described by the schematic
ADFjIiB , (51)
where each high energy false vacuum Fjdecays into a set
of vacua {Ii}, all of which decay (for simplicity) to the
same terminal vacuum B. The vacua Iiare taken to be a
large set including both anthropic vacua and vacua that
host only Boltzmann brains. Eq. (9) continues to apply,
so Eqs. (46) and (48) are easily generalized to this case,
giving
sIi1
κIiX
j
κIiFjsFj(52)
and
sFj1
κFj
κFjDsD,(53)
where we have assumed that qκIi, κFj,as we ex-
pect for vacua other than the dominant one. Using these
results with Eq. (28), the ratio of Boltzmann brains in
11
vacua Iito normal observers in vacua Iiis given by
NBB
{Ii}
NNO
{Ii}maxiκBB
IisIi
maxinXjκIiFjsFjo
maxiκBB
Ii
1
κIiXjκIiFj
1
κFj
κFjDsD
maxiXjκIiFj
1
κFj
κFjDsD
maxi(κBB
Ii
κIiXj
κIiFj
κFj
κFjD)
maxiXj
κIiFj
κFj
κFjD,(54)
where the denominators are maximized over the re-
stricted set of anthropic vacua i(and the numerators are
maximized without restriction). The ratio of Boltzmann
brains in the dominant vacuum (vacuum D) to normal
observers in vacua Iiis given by
NBB
D
NNO
{Ii}κBB
DsD
maxinXjκIiFjsFjo
κBB
D
maxiXj
κIiFj
κFj
κFjD,(55)
and, if vacuum Dis anthropic, then the ratio of Boltz-
mann brains in vacuum Dto normal observers in vacuum
Dis given by
NBB
D
NNO
DκBB
D
Xj
κDFj
κFj
κFjD
.(56)
In this case our answers are complicated by the pres-
ence of many different vacua. We can in principle deter-
mine whether Boltzmann brains dominate by evaluating
Eqs. (54)–(56) for the correct values of the parameters,
but this gets rather complicated and model-dependent.
The evaluation of these expressions can be simplified sig-
nificantly, however, if we make some very plausible as-
sumptions.
For tunneling out of the high-energy vacua Fj, one
can expect the transition rates into different channels to
be roughly comparable, so that κIiFjκDFjκFj.
That is, we assume that the branching ratios κIiFjFj
and κDFjFjare roughly of order one in the sense of
Eq. (16). These factors (or their inverses) will therefore
be unimportant in the evaluation of NBB/NNO, and may
be dropped. Furthermore, the upward transition rates
from the dominant vacuum Dinto Fjare all comparable
to one another, as can be seen by writing [46]
κFjDeAFjDeSD,(57)
where AFjDis the action of the instanton responsible for
the transition and SDis the action of the Euclideanized
de Sitter 4-sphere,
SD=8π2
H2
D
.(58)
But generically |AFjD| ∼ 1Fj. If we assume that
1
ρFj1
ρFk
<1014 (59)
for every pair of vacua Fjand Fk, then κFjD=κFkDup
to a factor that can be ignored because it is roughly of
order one. Thus, up to subleading factors, the transition
rates κFjDcancel out7in the ratio NBB
{Ii}/NNO.
Returning to Eq. (54) and keeping only the leading
factors, we have
NBB
{Ii}
NNO max
i(κBB
Ii
κIi),(60)
where the index iruns over all (non-dominant) vacua in
which Boltzmann brains can nucleate. For the dominant
vacuum, our simplifying assumptions8convert Eqs. (55)
and (56) into
NBB
D
NNO κBB
D
κup κBB
DeSD,(61)
where κup PjκFjDis the upward transition rate out
of the dominant vacuum.
Thus, the conditions needed to avoid Boltzmann brain
domination are essentially the same as what we found in
Subsection IV C. In this case, however, we must require
that in any vacuum that can support Boltzmann brains,
the Boltzmann brain nucleation rate must be less than
the decay rate of that vacuum.
7Depending on the range of vacua Fjthat are considered, the
bound of Eq. (59) may or may not be valid. If it is not, then
the simplification of Eq. (60) below is not justified, and the orig-
inal Eq. (54) has to be used. Of course one should remember
that there was significant arbitrariness in the choice of 1014 in
the definition of “roughly of order one.” 1014 was chosen to ac-
commodate the largest estimate that we discuss in Sec. V for
the Boltzmann brain nucleation rate, ΓBB exp(1016 ). In
considering the other estimates of ΓBB, one could replace 1014
by a much larger number, thereby increasing the applicability of
Eq. (59).
8The dropping of the factor eAFjDis a more reliable approxima-
tion in this case than it was in Eq. (60) above. In this case the
factor eSDdoes not cancel between the numerator and denom-
inator, so the factor eAFjDcan be dropped if it is unimportant
compared to eSD. We of course do not know the value of Sfor
the dominant vacuum, but for our vacuum it is of order 10122 ,
and it is plausible that the value for the dominant vacuum is
similar or even larger. Thus as long as 1Fjis small compared
to 10122, it seems safe to drop the factor eAFjD.
12
E. A Dominant Vacuum System
In the next to last paragraph of Section II, we described
a scenario where the dominant vacuum was not the vac-
uum with the smallest decay rate. Let us now study a
simple landscape to illustrate this situation. Consider
the toy landscape
FjIiB
րւ տց
AD1D2A
տց րւ
SA ,
(62)
where as in Subsection IV D the vacua Iiare taken to in-
clude both anthropic vacua and vacua that support only
Boltzmann brains. Vacua Aand Bare terminal vacua
and the Fjhave large, positive vacuum energies. Assume
that vacuum Shas the smallest total decay rate.
We have in mind the situation in which D1and D2are
nearly degenerate, and transitions from D1to D2(and
vice versa) are rapid, even though the transition in one
direction is upward. With this in mind, we divide the
decay rates of D1and D2into two parts,
κ1=κ21 +κout
1(63)
κ2=κ12 +κout
2,(64)
with κ12, κ21 κout
1,2. We assume as in previous sections
that the rates for large upward transitions (Sto D1or
D2, and D1or D2to Fj) are extremely small, so that
we can ignore them in the calculation of q. The rate
equation, Eq. (9), then admits a solution with qκD,
but it also admits solutions with
q1
2hκ1+κ2±p(κ1κ2)2+ 4κ12κ21i.(65)
Expanding the smaller root to linear order in κout
1,2gives
qα1κout
1+α2κout
2,(66)
where
α1κ12
κ12 +κ21
, α2κ21
κ12 +κ21
.(67)
In principle this value for qcan be smaller than κD, which
is the case that we wish to explore.
In this case the vacua D1and D2dominate the volume
fraction of the multiverse, even if their total decay rates
κ1and κ2are not the smallest in the landscape. We can
therefore call the states D1and D2together a dominant
vacuum system, which we denote collectively as D. The
rate equation (Eq. (9)) shows that
sD1α1sD, sD2α2sD,(68)
where sDsD1+sD2, and the equations hold in the
approximation that κout
1,2and the upward transition rates
from D1and D2can be neglected. To see that these
vacua dominate the volume fraction, we calculate the
modified form of Eq. (53):
sFj
sDα1κFjD1+α2κFjD2
κFj
.(69)
Thus the volume fractions of the Fj, and hence also the
Ijand Bvacua, are suppressed by the very small rate
for large upward jumps from low energy vacua, namely
κFjD1and κFjD2. The volume fraction for Sdepends on
κAD1and κAD2, but it is maximized when these rates are
negligible, in which case it is given by
sS
sDq
κSq.(70)
This quantity can in principle be large, if qis just a little
smaller than κS, but that would seem to be a very special
case. Generically, we would expect that since qmust be
smaller than κS(see Eq. (11)), it would most likely be
many orders of magnitude smaller, and hence the ratio in
Eq. (70) would be much less than one. There is no reason,
however, to expect it to be as small as the ratios that
are suppressed by large upward jumps. For simplicity,
however, we will assume in what follows that sScan be
neglected.
To calculate the ratio of Boltzmann brains to normal
observers in this toy landscape, note that Eqs. (54) and
(55) are modified only by the substitution
κFjD¯κFjDα1κFjD1+α2κFjD2.(71)
Thus, the dominant vacuum transition rate is simply re-
placed by a weighted average of the dominant vacuum
transition rates. If we assume that neither of the vacua
D1nor D2are anthropic, and make the same assump-
tions about magnitudes used in Subsection IV D, then
Eqs. (60) and (61) continue to hold as well, where we
have redefined κup by κup Pj¯κFjD.
If, however, we allow D1or D2to be anthropic, then
new questions arise. Transitions between D1and D2
are by assumption rapid, so they copiously produce new
pockets and potentially new normal observers. We must
recall, however (as discussed in Section III), that the
properties of a pocket universe depend on both the cur-
rent vacuum and the parent vacuum. In this case, the
unusual feature is that the vacua within the Dsystem are
nearly degenerate, and hence very little energy is released
by tunnelings within D. For pocket universes created in
this way, the maximum particle energy density during
reheating will be only a small fraction of the vacuum en-
ergy density. Such a big bang is very different from the
one that took place in our pocket, and presumably much
less likely to produce life. We will call a vacuum in the
Dsystem “strongly anthropic” if normal observers are
produced by tunnelings from within D, and “mildly an-
thropic” if normal observers can be produced, but only
by tunnelings from higher energy vacua outside D.
13
If either of the vacua in Dwere strongly anthropic,
then the normal observers in Dwould dominate the nor-
mal observers in the multiverse. Normal observers in the
vacua Iiwould be less numerous by a factor proportional
to the extremely small rate ¯κFjDfor large upward transi-
tions . This situation would itself be a problem, however,
similar to the Boltzmann brain problem. It would mean
that observers like ourselves, who arose from a hot big
bang with energy densities much higher than our vacuum
energy density, would be extremely rare in the multiverse.
We conclude that if there are any models which give a
dominant vacuum system that contains a strongly an-
thropic vacuum, such models would be considered unac-
ceptable in the context of the scale-factor cutoff measure.
On the other hand, if the Dsystem included one or
more mildly anthropic vacua, then the situation is very
similar to that discussed in Subsections IV C and IV D.
In this case the normal observers in the Dsystem would
be comparable in number to the normal observers in the
vacua Ii, so they would have no significant effect on the
ratio of Boltzmann brains to normal observers in the mul-
tiverse. If any of the Dvacua were mildly anthropic, how-
ever, then the stringent requirement κBB
Dκup would
have to be satisfied without resort to the simple solution
κBB
D= 0.
Thus, we find that the existence of a dominant vacuum
system does not change our conclusions about the abun-
dance of Boltzmann brains, except insofar as the Boltz-
mann brain nucleation constraints that would apply to
the dominant vacuum must apply to every member of the
dominant vacuum system. Probably the most important
implication of this example is that the dominant vacuum
is not necessarily the vacuum with the lowest decay rate,
so the task of identifying the dominant vacuum could be
very difficult.
F. General Conditions to Avoid Boltzmann Brain
Domination
In constructing general conditions to avoid Boltzmann
brain domination, we are guided by the toy landscapes
discussed in the previous subsections. Our goal, however,
is to construct conditions that can be justified using only
the general equations of Sections II and III, assuming
that the landscape is irreducible, but without relying on
the properties of any particular toy landscape. We will be
especially cautious about the treatment of the dominant
vacuum and the possibility of small upward transitions,
which could be rapid. The behavior of the full landscape
of a realistic theory may deviate considerably from that
of the simplest toy models.
To discuss the general situation, it is useful to divide
vacuum states into four classes. We are only interested
in vacua that can support Boltzmann brains. These can
be
(1) anthropic vacua for which the total dimensionless
decay rate satisfies κiq,
(2) non-anthropic vacua that can transition to an-
thropic vacua via unsuppressed transitions,
(3) non-anthropic vacua that can transition to an-
thropic vacua only via suppressed transitions,
(4) anthropic vacua for which the total dimensionless
decay rate is κiq.
Here qis the smallest-magnitude eigenvalue of the rate
equation (see Eqs. (5)–(8)). We call a transition “unsup-
pressed” if its branching ratio is roughly of order one in
the sense of Eq. (16). If the branching ratio is smaller
than this, it is “suppressed.” As before, when calculat-
ing NBB/NNO we assume that factors that are roughly
of order one can be ignored. Note that Eq. (11) forbids
κifrom being less than q, so the above four cases are
exhaustive.
We first discuss conditions that are sufficient to guar-
antee that Boltzmann brains will not dominate, postpon-
ing until later the issue of what conditions are necessary.
We begin with the vacua in the first class. Very likely
all anthropic vacua belong to this class. For an anthropic
vacuum i, the Boltzmann brains produced in vacuum i
cannot dominate the multiverse if they do not dominate
the normal observers in vacuum i, so we can begin with
this comparison. Restricting Eq. (29) to this single vac-
uum, we obtain
NBB
i
NNO
iκBB
i
κi
,(72)
a ratio that has appeared in many of the simple examples.
If this ratio is small compared to one, then Boltzmann
brains created in vacuum iare negligible.
Let us now study a vacuum jin the second class. First
note that Eq. (9) implies the rigorous inequality
κisiκij sj(no sum on repeated indices) ,(73)
which holds for any two states iand j. (Intuitively,
Eq. (73) is the statement that, in steady state, the total
rate of loss of volume fraction must exceed the input rate
from any one channel.) To simplify what follows, it will
be useful to rewrite Eq. (73) as
(κisi)(κjsj)Bji,(74)
where Bjiκij jis the branching ratio for the tran-
sition ji.
Suppose that we are trying to bound the Boltzmann
brain production in vacuum j, and we know that it can
undergo unsuppressed transitions
jk1...kni , (75)
14
where iis an anthropic vacuum. We begin by using
Eqs. (22) and (24) to express NBB
j/NNO
i, dropping ir-
relevant factors as in Eq. (28), and then we can iterate
the above inequality:
NBB
j
NNO
iκBB
jsj
Pkκik skκBB
jsj
κisi
κBB
jsj
(κjsj)Bjk1Bk1k2···Bkni
=κBB
j
κj
1
Bjk1Bk1k2···Bkni
,(76)
where again there is no sum on repeated indices, and
Eq. (9) was used in the last step on the first line. Each
inverse branching ratio on the right of the last line is
greater than or equal to one, but by our assumptions can
be considered to be roughly of order one, and hence can
be dropped. Thus, the multiverse will avoid domination
by Boltzmann brains in vacuum jif κBB
jj1, the
same criterion found for the first class.
The third class — non-anthropic vacua that can only
transition to an anthropic state via at least one sup-
pressed transition — presumably includes many states
with very low vacuum energy density. The dominant vac-
uum of our toy landscape models certainly belongs to this
class, but we do not know of anything that completely
excludes the possibility that the dominant vacuum might
belong to the second or fourth classes. That is, perhaps
the dominant vacuum is anthropic, or decays to an an-
thropic vacuum. If there is a dominant vacuum system,
as described in Subsection IV E, then κiq, and the
dominant vacua could belong to the first class, as well as
to either of classes (2) and (3).
To bound the Boltzmann brain production in this class,
we consider two possible criteria. To formulate the first,
we can again use Eqs. (75) and (76), but this time the
sequence must include at least one suppressed transition,
presumably an upward jump. Let us therefore denote the
branching ratio for this suppressed transition as Bup , not-
ing that Bup will appear in the denominator of Eq. (76).
Of course, the sequence of Eq. (75) might involve more
than one suppressed transition, but in any case the prod-
uct of these very small branching ratios in the denomi-
nator can be called Bup , and all the other factors can be
taken as roughly of order one. Thus, a landscape con-
taining a vacuum jof the third class avoids Boltzmann
brain domination if
κBB
j
Bup κj1,(77)
in agreement with the results obtained for the dominant
vacua in the toy landscape models in the previous sub-
sections.
A few comments are in order. First, if the only sup-
pressed transition is the first, then Bup =κup j, and
the above criterion simplifies to κBB
jup 1. Second,
we should keep in mind that the sequence of Eq. (75)
is presumably not unique, so other sequences will pro-
duce other bounds. All the bounds will be valid, so the
strongest bound is the one of maximum interest. Fi-
nally, since the vacua under discussion are not anthropic,
a likely method for Eq. (77) to be satisfied would be for
κBB
jto vanish, as would happen if the vacuum jdid not
support the complex structures needed to form Boltz-
mann brains.
The criterion above can be summarized by saying that
if κBB
j/(Bupκj)1, then the Boltzmann brains in vac-
uum jwill be overwhelmingly outnumbered by the nor-
mal observers living in pocket universes that form in the
decay chain starting from vacuum j. We now describe a
second, alternative criterion, based on the idea that the
number of Boltzmann brains in vacuum jcan be com-
pared with the number of normal observers in vacuum i
if the two types of vacuum have a common ancestor.
Denoting the common ancestor vacuum as A, we as-
sume that it can decay to an anthropic vacuum iby a
chain of transitions
Ak1...kni , (78)
and also to a Boltzmann-brain-producing vacuum jby a
chain
A1...mj . (79)
From the sequence of Eq. (78) and the bound of Eq. (74),
we can infer that
(κisi)(kAsA)BAk1Bk1k2···Bkni.(80)
To make use of the sequence of Eq. (79) we will want a
bound that goes in the opposite direction, for which will
need to require additional assumptions. Starting with
Eq. (9), we first require qκi, which is plausible pro-
vided that vacuum iis not the dominant vacuum. Next
we look at the sum over jon the right-hand side, and we
call the transition ji“significant” if its contribution
to the sum is within a factor roughly of order one of the
entire sum. (The sum over jis the sum over sources for
vacuum i, so a transition jiis “significant” if pocket
universes of vacuum jare a significant source of pocket
universes of vacuum i.) It follows that for any significant
transition jifor which qκi,
(κisi)(κjsj)ZmaxBji(κjsj)Zmax ,(81)
where Zmax denotes the largest number that is roughly
of order one. By our conventions, Zmax = exp(1014). If
we assume now that all the transitions in the sequence of
Eq. (79) are significant, and that qis negligible in each
case, then
(κjsj)(kAsA)Zm+1
max .(82)
15
Using the bounds from Eqs. (80) and (82), the Boltzmann
brain ratio is bounded by
NBB
j
NNO
iκBB
jsj
Pkκik skκBB
jsj
κisi
Zm+1
max
BAk1Bk1k2···Bkni
κBB
j
κj
.(83)
But all the factors on the right are roughly of order one,
except that some of the branching ratios in the denomi-
nator might be smaller, if they correspond to suppressed
transitions. If Bup denotes the product of branching ra-
tios for all the suppressed transitions shown in the de-
nominator (i.e., all suppressed transitions in the sequence
of Eq. (78)), then the bound reduces to Eq. (77).9
To summarize, the Boltzmann brains in a non-
anthropic vacuum jcan be bounded if there is an an-
cestor vacuum Athat can decay to jthrough a chain
of significant transitions for which qκfor each
vacuum, as in the sequence of Eq. (79), and if the
same ancestor vacuum can decay to an anthropic vac-
uum through a sequence of transitions as in Eq. (78).
The Boltzmann brains will never dominate provided that
κBB
j/(Bup κj)1, where Bup is the product of all sup-
pressed branching ratios in the sequence of Eq. (78).
Finally, the fourth class of vacua consists of anthropic
vacua iwith decay rate κiq, a class which could be
empty. For this class Eq. (29) may not be very useful,
since the quantity (κiq) in the denominator could be
very small. Yet, as in the two previous classes, this class
can be treated by using Eq. (76), where in this case the
vacuum ican be the same as jor different, although the
case i=jrequires n1. Again, if the sequence con-
tains only unsuppressed transitions, then the multiverse
avoids domination by Boltzmann brains in vacuum iif
κBB
ii1. If upward jumps are needed to reach an
anthropic vacuum, whether it is the vacuum iagain or a
distinct vacuum j, then the Boltzmann brains in vacuum
iwill never dominate if κBB
i/(Bup κi)1.
The conditions described in the previous paragraph
are very difficult to meet, so if the fourth class is not
empty, Boltzmann brain domination is hard to avoid.
These vacua have the slowest decay rates in the land-
scape, κiq, so it seems plausible that they have very
low energy densities, precluding the possibility of decay-
ing to an anthropic vacuum via unsuppressed transitions;
9Note, however, that the argument breaks down if the sequences in
either of Eqs. (78) or (79) become too long. For the choices that
we have made, a factor of Zmax is unimportant in the calculation
of NBB/NNO , but Z100
max = exp(1016) can be significant. Thus,
for our choices we can justify the dropping of O(100) factors that
are roughly of order one, but not more than that. For choices
appropriate to smaller estimates of ΓBB , however, the number of
factors that can be dropped will be many orders of magnitude
larger.
in that case Boltzmann brain domination can be avoided
if
κBB
iBupκi.(84)
However, as pointed out in Ref. [43], Bup eSD(see
Eq. (57)) is comparable to the inverse of the recurrence
time, while in an anthropic vacuum one would expect the
Boltzmann brain nucleation rate to be much faster than
once per recurrence time.
To summarize, the domination of Boltzmann brains
can be avoided by first of all requiring that all vacuum
states in the landscape obey the relation
κBB
j
κj1.(85)
That is, the rate of nucleation of Boltzmann brains in
each vacuum must be less than the rate of nucleation,
in that same vacuum, of bubbles of other phases. For
anthropic vacua iwith κiq, this criterion is enough.
Otherwise, the Boltzmann brains that might be produced
in vacuum jmust be bounded by the normal observers
forming in some vacuum i, which must be related to j
through decay chains. Specifically, there must be a vac-
uum Athat can decay through a chain to an anthropic
vacuum i, i.e.
Ak1...kni , (86)
where either A=j, or else Acan decay to jthrough a
sequence
A1...mj . (87)
In the above sequence we insist that κjqand that
κlqfor each vacuum pin the chain, and that each
transition must be “significant,” in the sense that pockets
of type pmust be a significant source of pockets of type
p+1. (More precisely, a transition from vacuum jto i
is “significant” if it contributes a fraction that is roughly
of order one to Pjκij sjin Eq. (9).) For these cases, the
bound which ensures that the Boltzmann brains in vac-
uum jare dominated by the normal observers in vacuum
iis given by
κBB
j
Bup κj1,(88)
where Bup is the product of any suppressed branching
ratios in the sequence of Eq. (86). If all the transi-
tions in Eq. (86) are unsuppressed, this bound reduces
to Eq. (85). If jis anthropic, the case A=j=iis
allowed, provided that n1.
The conditions described above are sufficient to guar-
antee that Boltzmann brains do not dominate over nor-
mal observers in the multiverse, but without further as-
sumptions there is no way to know if they are necessary.
All of the conditions that we have discussed are quasi-
local, in the sense that they do not require any global
16
picture of the landscape of vacua. For each of the above
arguments, the Boltzmann brains in one type of vacuum j
are bounded by the normal observers in some type of vac-
uum ithat is either the same type, or directly related to
it through decay chains. Thus, there was no need to dis-
cuss the importance of the vacua jand icompared to the
rest of the landscape as a whole. The quasi-local nature
of these conditions, however, guarantees that they can-
not be necessary to avoid the domination by Boltzmann
brains. If two vacua jand iare both totally insignificant
in the multiverse, then it will always be possible for the
Boltzmann brains in vacuum jto overwhelm the normal
observers in vacuum i, while the multiverse as a whole
could still be dominated by normal observers in other
vacua.
We have so far avoided making global assumptions
about the landscape of vacua, because such assumptions
are generally hazardous. While it may be possible to
make statements that are true for the bulk of vacua in
the landscape, in this context the statements are not use-
ful unless they are true for all the vacua of the land-
scape. Although the number of vacua in the landscape,
often estimated at 10500 [51], is usually considered to be
incredibly large, the number is nonetheless roughly of
order one compared to the numbers involved in the esti-
mates of Boltzmann brain nucleation rates and vacuum
decay rates. Thus, if a single vacuum produces Boltz-
mann brains in excess of required bounds, the Boltzmann
brains from that vacuum could easily overwhelm all the
normal observers in the multiverse.
Recognizing that our conclusions could be faulty, we
can nonetheless adopt some reasonable assumptions to
see where they lead. We can assume that the multiverse
is sourced by either a single dominant vacuum, or by a
dominant vacuum system. We can further assume that
every anthropic and/or Boltzmann-brain-producing vac-
uum ican be reached from the dominant vacuum (or
dominant vacuum system) by a single significant upward
jump, with a rate proportional to eSD, followed by some
number of significant, unsuppressed transitions, all of
which have rates κkqand branching ratios that are
roughly of order one:
Dk1...kni . (89)
We will further assume that each non-dominant an-
thropic and/or Boltzmann-brain-producing vacuum ihas
a decay rate κiq, but we need not assume that all of
the κiare comparable to each other. With these assump-
tions, the estimate of NBB/NNO becomes very simple.
Applying Eq. (9) to the first transition of Eq. (89),
κk1sk1κk1DsDκupsD,(90)
where we use κup to denote the rate of a typical transition
Dk, assuming that they are all equal to each other
up to a factor roughly of order one. Here indicates
equality up to a factor that is roughly of order one. If
there is a dominant vacuum system, then κk1Dis replaced
by ¯κk1DPακk1D, where the Dare the components
of the dominant vacuum system, and the αare defined
by generalizing Eqs. (67) and (68).10 Applying Eq. (9)
to the next transition, k1k2we find
κk2sk2=Bk1k2κk1sk1+...κk1sk1,(91)
where we have used the fact that Bk1k2is roughly of
order one, and that the transition is significant. Iterating,
we have
κisiκknsknκup sD.(92)
Since the expression on the right is independent of i,
we conclude that under these assumptions any two non-
dominant anthropic and/or Boltzmann-brain-producing
vacua iand jhave equal values of κs, up to a factor that
is roughly of order one:
κjsjκisi.(93)
Using Eq. (22) and assuming as always that nNO
ik is
roughly of order one, Eq. (93) implies that any two non-
dominant anthropic vacua iand jhave comparable num-
bers of ordinary observers, up to a factor that is roughly
of order one:
NNO
j∼ NNO
i.(94)
The dominant vacuum could conceivably be anthropic,
but we begin by considering the case in which it is not.
In that case all anthropic vacua are equivalent, so the
Boltzmann brains produced in any vacuum jwill either
dominate the multiverse or not depending on whether
they dominate the normal observers in an arbitrary an-
thropic vacuum i. Combining Eqs. (22), (24), (9), and
(93), and omitting irrelevant factors, we find that for any
10 In more detail, the concept of a dominant vacuum system is rele-
vant when there is a set of vacua that can have rapid transitions
within the set, but only very slow transitions connecting these
vacua to the rest of the landscape. As a zeroth order approxi-
mation one can neglect all transitions connecting these vacua to
the rest of the landscape, and assume that κq, so Eq. (9)
takes the form
κs=X
Bℓℓκs.
Here Bℓℓκℓℓis the branching ratio within this restricted
subspace, where κ=Pκis summed only within the dom-
inant vacuum system, so PBℓℓ= 1 for all .Bℓℓis non-
negative, and if we assume also that it is irreducible, then the
Perron-Frobenius theorem guarantees that it has a nondegen-
erate eigenvector vof eigenvalue 1, with positive components.
From the above equation κsv, and then
α=s
Ps
=v
κX
v
κ
.
17
non-dominant vacuum j
NBB
j
NNO
iκBB
jsj
Pkκik skκBB
jsj
κisiκBB
j
κj
.(95)
Thus, given the assumptions described above, for any
non-dominant vacuum jthe necessary and sufficient con-
dition to avoid the domination of the multiverse by Boltz-
mann brains in vacuum jis given by
κBB
j
κj1.(96)
For Boltzmann brains formed in the dominant vacuum,
we can again find out if they dominate the multiverse
by determining whether they dominate the normal ob-
servers in an arbitrary anthropic vacuum i. Repeating
the above analysis for vacuum Dinstead of vacuum j,
using Eq. (92) to relate sito sD, we have
NBB
D
NNO
iκBB
DsD
Pkκik skκBB
DsD
κisiκBB
D
κup
.(97)
Thus, for a single dominant vacuum Dor a dominant
vacuum system with members Di, the necessary and suf-
ficient conditions to avoid the domination of the multi-
verse by these Boltzmann brains is given by
κBB
D
κup 1 or κBB
Di
κup 1.(98)
As discussed after Eq. (84), probably the only way to
satisfy this condition is to require that κBB
D= 0.
If the dominant vacuum is anthropic, then the con-
clusions are essentially the same, but the logic is more
involved. For the case of a dominant vacuum system,
we distinguish between the possibility of vacua being
“strongly” or “mildly” anthropic, as discussed in Sub-
section IV E. “Strongly anthropic” means that normal
observers are formed by tunneling within the dominant
vacuum system D, while “mildly anthropic” implies that
normal observers are formed by tunneling, but only from
outside D. Any model that leads to a strongly anthropic
dominant vacuum would be unacceptable, because al-
most all observers would live in pockets with a maximum
reheat energy density that is small compared to the vac-
uum energy density. With a single anthropic dominant
vacuum, or with one or more mildly anthropic vacua
within a dominant vacuum system, the normal observers
in the dominant vacuum would be comparable in num-
ber (up to factors roughly of order one) to those in other
anthropic vacua, so they would have no significant effect
on the ratio of Boltzmann brains to normal observers in
the multiverse. An anthropic vacuum would also pro-
duce Boltzmann brains, however, so Eq. (98) would have
to somehow be satisfied for κBB
D6= 0.
V. BOLTZMANN BRAIN NUCLEATION AND
VACUUM DECAY RATES
A. Boltzmann Brain Nucleation Rate
Boltzmann brains emerge from the vacuum as large
quantum fluctuations. In particular, they can be mod-
eled as localized fluctuations of some mass M, in the
thermal bath of a de Sitter vacuum with temperature
TdS =HΛ/2π[1]. The Boltzmann brain nucleation rate
is then roughly estimated by the Boltzmann suppression
factor [7, 9],
ΓBB eM/TdS ,(99)
where our goal is to estimate only the exponent, not the
prefactor. Eq. (99) gives an estimate for the nucleation
rate of a Boltzmann brain of mass Min any particular
quantum state, but we will normally describe the Boltz-
mann brain macroscopically. Thus ΓBB should be mul-
tiplied by the number of microstates eSBB corresponding
to the macroscopic description, where SBB is the entropy
of the Boltzmann brain. Thus we expect
ΓBB eM/TdS eSBB =eF/TdS ,(100)
where F=MTdS SBB is the free energy of the Boltz-
mann brain.
Eq. (100) should be accurate as long as the de Sit-
ter temperature is well-defined, which will be the case
as long as the Schwarzschild horizon is small compared
to the de Sitter horizon radius. Furthermore, we shall
neglect the effect of the gravitational potential energy of
de Sitter space on the Boltzmann brain, which requires
that the Boltzmann brain be small compared to the de
Sitter horizon. Thus we assume
M/4π < R H1
Λ,(101)
where the first inequality assumes that Boltzmann brains
cannot be black holes. The general situation, which al-
lows for MRH1
Λ, will be discussed in Appendix A
and in Ref. [52].
While the nucleation rate is proportional to eSBB , this
factor is negligible for any Boltzmann brain made of
atoms like those in our universe. The entropy of such
atoms is bounded by
S.3M/mn,(102)
where mnis the nucleon mass. Indeed, the actual value
of SBB is much smaller than this upper bound because of
the complex organization of the Boltzmann brain. Mean-
while, to prevent the Boltzmann brain from being de-
stroyed by pair production, we require that TdS mn.
Thus, for these Boltzmann brains the entropy factor eSBB
is irrelevant compared to the Boltzmann suppression fac-
tor.
18
To estimate the nucleation rate for Boltzmann brains,
we need at least a crude description of what constitutes
a Boltzmann brain. There are many possibilities. We
argued in the introduction to this paper that a theory
that predicts the domination of Boltzmann brains over
normal observers would be overwhelmingly disfavored by
our continued observation of an orderly world, in which
the events that we observe have a logical relationship
to the events that we remember. In making this argu-
ment, we considered a class of Boltzmann brains that
share exactly the memories and thought processes of a
particular normal observer at some chosen instant. For
these purposes the memory of the Boltzmann brain can
consist of random bits that just happen to match those
of the normal observer, so there are no requirements on
the history of the Boltzmann brain. Furthermore, the
Boltzmann brain need only survive long enough to regis-
ter one observation after the chosen instant, so it is not
required to live for more than about a second. We will
refer to Boltzmann brains that meet these requirements
as minimal Boltzmann brains.
While an overabundance of minimal Boltzmann brains
is enough to cause a theory to be discarded, we nonethe-
less find it interesting to discuss a wide range of Boltz-
mann brain possibilities. We will start with very large
Boltzmann brains, discussing the minimal Boltzmann
brains last.
We first consider Boltzmann brains much like us, who
evolved in stellar systems like ours, in vacua with low-
energy particle physics like ours, but allowing for a de
Sitter Hubble radius as small as a few astronomical units
or so. These Boltzmann brains evolved in their stellar
systems on a time scale similar to the evolution of life on
Earth, so they are in every way like us, except that, when
they perform cosmological observations, they find them-
selves in an empty, vacuum-dominated universe. These
“Boltzmann solar systems” nucleate at a rate of roughly
ΓBB exp(1085),(103)
where we have set M1030 kg and H1
Λ= (2πTdS)1
1012 m. This nucleation rate is fantastically small; we
found it, however, by considering the extravagant possi-
bility of nucleating an entire Boltzmann solar system.
Next, we can consider the nucleation of an isolated
brain, with a physical construction that is roughly similar
to our own brains. If we take M1 kg and H1
Λ=
(2πTdS)11 m, then the corresponding Boltzmann
brain nucleation rate is
ΓBB exp(1043).(104)
If the construction of the brain is similar to ours, however,
then it could not function if the tidal forces resulted in
a relative acceleration from one end to the other that is
much greater than the gravitational acceleration gon the
surface of the Earth. This requires H1
Λ&108m, giving
a Boltzmann brain nucleation rate
ΓBB exp(1051).(105)
Until now, we have concentrated on Boltzmann brains
that are very similar to human brains. However a com-
mon assumption in the philosophy of mind is that of
substrate-independence. Therefore, pressing onward, we
study the possibility that a Boltzmann brain can be any
device capable of emulating the thoughts of a human
brain. In other words, we treat the brain essentially
as a highly sophisticated computer, with logical opera-
tions that can be duplicated by many different systems
of hardware.11
With this in mind, from here out we drop the assump-
tion that Boltzmann brains are made of the same mate-
rials as human brains. Instead, we attempt to find an
upper bound on the probability of creation of a more
generalized computing device, specified by its informa-
tion content IBB, which is taken to be comparable to the
information content of a human brain.
To clarify the meaning of information content, we can
model an information storage device as a system with
Npossible microstates. Smax = ln Nis then the max-
imum entropy that the system can have, the entropy
corresponding to the state of complete uncertainty of
microstate. To store Bbits of information in the de-
vice, we can imagine a simple model in which 2Bdistin-
guishable macroscopic states of the system are specified,
each of which will be used to represent one assignment
of the bits. Each macroscopic state can be modeled as
a mixture of N/2Bmicrostates, and hence has entropy
S= ln(N/2B) = Smax Bln 2. Motivated by this simple
model, one defines the information content of any macro-
scopic state of entropy Sas the difference between Smax
and S, where Smax is the maximum entropy that the de-
vice can attain. Applying this definition to a Boltzmann
brain, we write
IBB =SBB,max SBB ,(106)
where IBB/ln 2 is the information content measured in
bits.
As discussed in Ref. [53], the only known substrate-
independent limit on the storage of information is the
Bekenstein bound. It states that, for an asymptotically
flat background, the entropy of any physical system of
11 Note that the validity of the assumption of substrate-
independence of mind is not entirely self-evident. Some of us
are skeptical of identifying human consciousness with operations
of a generic substrate-independent computer, but accept it as a
working hypothesis for the purpose of this paper.
19
size Rand energy Mis bounded by12
SSBek 2πM R . (107)
One can use this bound in de Sitter space as well if
the size of the system is sufficiently small, RH1
Λ,
so that the system does not “know” about the horizon.
A possible generalization of the Bekenstein bound for
R=O(H1
Λ) was proposed in Ref. [54]; we will study this
and other possibilities in Appendix A and in Ref. [52].
To begin, however, we will discuss the simplest case,
RH1
Λ, so that we can focus on the most impor-
tant issues before dealing with the complexities of more
general results.
Using Eq. (106), the Boltzmann brain nucleation rate
of Eq. (100) can be rewritten as
ΓBB exp 2πM
HΛ
+SBB,max IBB,(108)
which is clearly maximized by choosing Mas small as
possible. The Bekenstein bound, however, implies that
SBB,max SBek and therefore MSBB,max/(2πR).
Thus
ΓBB exp SBB,max
RHΛ
+SBB,max IBB.(109)
Since R < H1
Λ, the expression above is maximized by
taking SBB,max equal to its smallest possible value, which
is IBB. Finally, we have
ΓBB exp IBB
RHΛ.(110)
Thus, the Boltzmann brain production rate is maxi-
mized if the Boltzmann brain saturates the Bekenstein
bound, with IBB =SBB,max = 2πM R. Simultaneously,
we should make RHΛas large as possible, which means
12 In an earlier version of this paper we stated an incorrect form of
this bound, and from it derived some incorrect conclusions, such
as the statement that the largest Boltzmann brain nucleation
rate ΓBB consistent with the Bekenstein bound is attained only
when the radius Rapproaches the Schwarzschild radius RSch.
This in turn led to the conclusion that the maximum rate allowed
by the Bekenstein bound is e2IBB , which can be achieved only if
M2=IBB/(9πG) and H2
Λ=π/(3GIBB ). While these relations
hold in the regime we considered, they are not necessary in the
general case. With the corrected bound, we find that the maxi-
mum nucleation rate is independent of R/RSch if RHΛ(see
Eq. (110)), and otherwise grows with R/RSch (see Appendix A).
However, once one is forced to consider values of RRSch, then
other issues become relevant. How can the system be stabilized
against the de Sitter expansion? Can the Bekenstein bound re-
ally be saturated for a system with large entropy, especially if
it is dilute? In this version of the paper we have added a dis-
cussion of these issues. We thank R. Bousso, B. Freivogel, and
I. Yang for pointing out the error in our earlier statement of the
Bekenstein bound.
taking our assumption RH1
Λto the boundary of its
validity. Thus we write the Boltzmann brain production
rate
ΓBB eaIBB ,(111)
where a(RHΛ)1, the value of which is of order a
few. In Appendix A we explore the case in which the
Schwarzschild radius, the Boltzmann brain radius, and
the de Sitter horizon radius are all about equal, in which
case Eq. (111) holds with a= 2.
The bound of Eq. (111) can be compared to the es-
timate of the Boltzmann brain production rate, ΓBB
eSBB , which follows from Eq. (2.13) of Freivogel and
Lippert, in Ref. [55]. The authors of Ref. [55] explained
that by SBB they mean not the entropy, but the num-
ber of degrees of freedom, which is roughly equal to the
number of particles in a Boltzmann brain. This estimate
appears similar to our result, if one equates SBB to IBB,
or to a few times IBB . Freivogel and Lippert describe
this relation as a lower bound on the nucleation rate for
Boltzmann brains, commenting that it can be used as an
estimate of the nucleation rate for vacua with “reason-
ably cooperative particle physics.” Here we will explore
in some detail the question of whether this bound can be
used as an estimate of the nucleation rate. While we will
not settle this issue here, we will discuss evidence that
Eq. (111) is a valid estimate for at most a small fraction
of the vacua of the landscape, and possibly none at all.
So far, the conditions to reach the upper bound in
Eq. (111) are R= (aHΛ)1∼ O(H1
Λ) and IBB =
Smax,BB =SBek. However these are not enough to ensure
that a Boltzmann brain of size RH1
Λis stable and can
actually compute. Indeed, the time required for commu-
nication between two parts of a Boltzmann brain sepa-
rated by a distance O(H1
Λ) is at least comparable to the
Hubble time. If the Boltzmann brain can be stretched by
cosmological expansion, then after just a few operations
the different parts will no longer be able to communi-
cate. Therefore we need a stabilization mechanism by
which the brain is protected against expansion.
A potential mechanism to protect the Boltzmann brain
against de Sitter expansion is the self-gravity of the brain.
A simple example is a black hole, which does not expand
when the universe expands. It seems unlikely that black
holes can think,13 but one can consider objects of mass
approaching that of a black hole with radius R. This,
together with our goal to keep Ras close as possible to
13 The possibility of a black hole computer is not excluded, however,
and has been considered in Ref. [53]. Nonetheless, if black holes
can compute, our conclusions would not be changed, provided
that the Bekenstein bound can be saturated for the near-black
hole computers that we discuss. At this level of approximation,
there would be no significant difference between a black hole
computer and a near-black hole computer.
20
H1
Λ, leads to the following condition:
MRH1
Λ.(112)
If the Bekenstein bound is saturated, this leads to the
following relations between IBB,HΛ, and M:
IBB MR M H 1
ΛH2
Λ.(113)
A second potential mechanism of Boltzmann brain sta-
bilization is to surround it by a domain wall with a sur-
face tension σ, which would provide pressure preventing
the exponential expansion of the brain. An investiga-
tion of this situation reveals that one cannot saturate
the Bekenstein bound using this mechanism unless there
is a specific relation between IBB,HΛ, and σ[52]:
σIBB H3
Λ.(114)
If σis less than this magnitude, it cannot prevent the ex-
pansion, while a larger σincreases the mass and therefore
prevents saturation of the Bekenstein bound.
Regardless of the details leading to Eqs. (113) and
(114), the important point is that both of them lead to
constraints on the vacuum hosting the Boltzmann brain.
For example, the Boltzmann brain stabilized by grav-
itational attraction can be produced at a rate approach-
ing eaIBB only if IBB H2
Λ. For a given value of IBB ,
say IBB 1016 (see the discussion below), this result
applies only to vacua with a particular vacuum energy,
Λ1016. Similarly, according to Eq. (114), for Boltz-
mann brains with IBB 1016 contained inside a domain
wall in a vacuum with Λ 10120, the Bekenstein bound
on ΓBB cannot be reached unless the tension of the do-
main wall is incredibly small, σ10164. Thus, the
maximal Boltzmann brain production rate eaIBB sat-
urating the Bekenstein bound cannot be reached unless
Boltzmann brains are produced on a narrow hypersurface
in the landscape.
This conclusion by itself does not eliminate the dan-
ger of a rapid Boltzmann brain production rate, ΓBB
eaIBB . Given the vast number of vacua in the landscape,
it seems plausible that this bound could actually be met.
If this is the case, Eq. (111) offers a stunning increase
over previous estimates of ΓBB.
Setting aside the issue of Boltzmann brain stabil-
ity, one can also question the assumption of Bekenstein
bound saturation that is necessary to achieve the rather
high nucleation rate that is indicated by Eq. (111). Of
course black holes saturate this bound, but we assume
that a black hole cannot think. Even if a black hole
can think, it would still be an open question whether
this information processing could make use of a substan-
tial fraction of the degrees of freedom associated with
the black hole entropy. A variety of other physical sys-
tems are considered in Ref. [56], where the validity of
Smax(E)2πER is studied as a function of energy
E. In all cases, the bound is saturated in a limit where
Smax =O(1). Meanwhile, as we shall argue below, the
required value of Smax should be greater than 1016.
The present authors are aware of only one example of a
physical system that may saturate the Bekenstein bound
and at the same time store sufficient information Ito
emulate a human brain. This may happen if the total
number of particle species with mass smaller than HΛis
greater than IBB &1016. No realistic examples of such
theories are known to us, although some authors have
speculated about similar possibilities [57].
If Boltzmann brains cannot saturate the Bekenstein
bound, they will be more massive than indicated in
Eq. (110), and their rate of production will be smaller
than eaIBB .
To put another possible bound on the probability of
Boltzmann brain production, let us analyze a simple
model based on an ideal gas of massless particles. Drop-
ping all numerical factors, we consider a box of size R
filled with a gas with maximum entropy Smax = (RT )3
and energy E=R3T4=S4/3
max/R, where Tis the temper-
ature and we assume there is not an enormous number
of particle species. The probability of its creation can be
estimated as follows:
ΓBB eE/HΛeSBB exp S4/3
max
HΛR!,(115)
where we have neglected the Boltzmann brain entropy
factor, since SBB Smax S4/3
max. This probability is
maximized by taking RH1
Λ, which yields
ΓBB .eS4/3
max .(116)
In case the full information capacity of the gas is used,
one can also write
ΓBB .eI4/3
BB .(117)
For IBB 1, this estimate leads to a much stronger
suppression of Boltzmann brain production as compared
to our previous estimate, Eq. (111).
Of course, such a hot gas of massless particles can-
not think — indeed it is not stable in the sense outlined
below Eq. (111) — so we must add more parts to this
construction. Yet it seems likely that this will only de-
crease the Boltzmann brain production rate. As a par-
tial test of this conjecture, one can easily check that if
instead of a gas of massless particles we consider a gas
of massive particles, the resulting suppression of Boltz-
mann brain production will be stronger. Therefore in our
subsequent estimates we shall assume that Eq. (117) rep-
resents our next “line of defense” against the possibility
of Boltzmann brain domination, after the one given by
Eq. (111). One should note that this is a rather delicate
issue; see for example a discussion of several possibili-
ties to approach the Bekenstein bound in Ref. [58]. A
21
more detailed discussion of this issue will be provided in
Ref. [52].
Having related ΓBB to the information content IBB of
the brain, we now need to estimate IBB. How much in-
formation storage must a computer have to be able to
perform all the functions of the human brain? Since no
one can write a computer program that comes close to
imitating a human brain, this is not an easy question to
answer.
One way to proceed is to examine the human brain,
with the goal of estimating its capacities based on its
biological structure. The human brain contains 1014
synapses that may in principle connect to any of 1011
neurons [59], suggesting that its information content14
might be roughly IBB 1015 –1016. (We are assuming
here that the logical functions of the brain depend on the
connections among neurons, and not for example on their
precise locations, cellular structures, or other information
that might be necessary to actually construct a brain.)
A minimal Boltzmann brain is only required to simulate
the workings of a real brain for about a second, but with
neurons firing typically at 10 to 100 times a second, it is
plausible that a substantial fraction of the brain is needed
even for only one second of activity. Of course the actual
number of required bits might be somewhat less.
An alternative approach is to try to determine how
much information the brain processes, even if one does
not understand much about what the processing involves.
In Ref. [60], Landauer attempted to estimate the total
content of a person’s long-term memory, using a variety
of experiments. He concluded that a person remembers
only about 2 bits/second, for a lifetime total in the vicin-
ity of 109bits. In a subsequent paper [61], however, he
emphatically denied that this number is relevant to the
information requirements of a “real or theoretical cog-
nitive processor,” because such a device “would have so
much more to do than simply record new information.”
Besides long-term memory, one might be interested in
the total amount of information a person receives but
does not memorize. A substantial part of this informa-
tion is visual; it can be estimated by the information
stored on high definition DVDs watched continuously on
several monitors over the span of a hundred years. The
total information received would be about 1016 bits.
Since this number is similar to the number obtained
above by counting synapses, it is probably as good an
estimate as we can make for a minimal Boltzmann brain.
If the Bekenstein bound can be saturated, then the es-
timated Boltzmann brain nucleation rate for the most
favorable vacua in the landscape would be given by
14 Note that the specification of one out of 1011 neurons requires
log21011= 36.5 bits.
Eq. (111):
ΓBB .e1016 .(118)
If, however, the Bekenstein bound cannot be reached for
systems with IBB 1, then it might be more accurate
to use instead the ideal gas model of Eq. (117), yielding
ΓBB .e1021 .(119)
Obviously, there are many uncertainties involved in the
numerical estimates of the required value of IBB . Our es-
timate IBB 1016 concerns the information stored in
the human brain that appears to be relevant for cogni-
tion. It certainly does not include all the information
that would be needed to physically construct a human
brain, and it therefore does not allow for the information
that might be needed to physically construct a device
that could emulate the human brain. 15 It is also possi-
ble that extra mass might be required for the mechanical
structure of the emulator, to provide the analogues of a
computer’s wires, insulation, cooling systems, etc. On
the other hand, it is conceivable that a Boltzmann brain
can be relevant even if it has fewer capabilities than what
we called the minimal Boltzmann brain. In particular,
if our main requirement is that the Boltzmann brain is
to have the same “perceptions” as a human brain for
just one second, then one may argue that this can be
15 That is, the actual construction of a brain-like device would pre-
sumably require large amounts of information that are not part
of the schematic “circuit diagram” of the brain. Thus there may
be some significance to the fact that a billion years of evolu-
tion on Earth has not produced a human brain with fewer than
about 1027 particles, and hence of order 1027 units of entropy.
In counting the information in the synapses, for example, we
counted only the information needed to specify which neurons
are connected to which, but nothing about the actual path of
the axons and dendrites that complete the connections. These
are nothing like nearest-neighbor couplings, but instead axons
from a single neuron can traverse large fractions of the brain, re-
sulting in an extremely intertwined network [62]. To specify even
the topology of these connections, still ignoring the precise loca-
tions, could involve much more than 1016 bits. For example, the
synaptic “wiring” that connects the neurons will in many cases
form closed loops. A specification of the connections would pre-
sumably require a topological winding number for every pair of
closed loops in the network. The number of bits required to spec-
ify these winding numbers would be proportional to the square
of the number of closed loops, which would be proportional to
the square of the number of synapses. Thus, the structural in-
formation could be something like Istruct b×1028, where b
is a proportionality constant that is probably a few orders of
magnitude less than 1. In estimating the resulting suppression
of the nucleation rate, there is one further complication: since
structural information of this sort presumably has no influence
on brain function, these choices would contribute to the multi-
plicity of Boltzmann brain microstates, thereby multiplying the
nucleation rate by eIstruct. There would still be a net suppres-
sion, however, with Eq. (111) leading to ΓBB e(a1)Istruct,
where ais generically greater than 1. See Appendix A for further
discussion of the value of a.
22
achieved using much less than 1014 synapses. And if one
decreases the required time to a much smaller value re-
quired for a single computation to be performed by a
human brain, the required amount of information stored
in a Boltzmann brain may become many orders of mag-
nitude smaller than 1016.
We find that regardless of how one estimates the infor-
mation in a human brain, if Boltzmann brains can be con-
structed so as to come near the limit of Eq. (111), their
nucleation rate would provide stringent requirements on
vacuum decay rates in the landscape. On the other hand,
if no such physical construction exists, we are left with
the less dangerous bound of Eq. (117), perhaps even fur-
ther softened by the speculations described in Footnote
15. Note that none of these bounds is based upon a re-
alistic model of a Boltzmann brain. For example, the
nucleation of an actual human brain is estimated at the
vastly smaller rate of Eq. (105). The conclusions of this
paragraph apply to the causal patch measures [24, 25] as
well as the scale-factor cutoff measure.
In Section III we discussed the possibility of Boltzmann
brain production during reheating, stating that this pro-
cess would not be a danger. We postponed the numeri-
cal discussion, however, so we now return to that issue.
According to Eq. (26), the multiverse will be safe from
Boltzmann brains formed during reheating provided that
ΓBB
reheat,ik τBB
reheat,ik nNO
ik (120)
holds for every pair of vacua iand k, where ΓBB
reheat,ik
is the peak Boltzmann brain nucleation rate in a pocket
of vacuum ithat forms in a parent vacuum of type k,
τBB
reheat,ik is the proper time available for such nucle-
ation, and nNO
ik is the volume density of normal observers
in these pockets, working in the approximation that all
observers form at the same time.
Compared to the previous discussion about late-time
de Sitter space nucleation, here ΓBB
reheat,ik can be much
larger, since the temperature during reheating can be
much larger than HΛ. On the other hand, safety from
Boltzmann brains requires the late-time nucleation rate
to be small compared to the potentially very small vac-
uum decay rates, while in this case the quantity on the
right-hand side of Eq. (120) is not exceptionally small. In
discussing this issue, we will consider in sequence three
descriptions of the Boltzmann brain: a human-like brain,
a near-black hole computer, and a diffuse computer.
The nucleation of human-like Boltzmann brains during
reheating was discussed in Ref. [28], where it was pointed
out that such brains could not function at temperatures
much higher than 300 K, and that the nucleation rate
for a 100 kg ob ject at this temperature is exp(1040).
This suppression is clearly more than enough to ensure
that Eq. (120) is satisfied.
For a near-black hole computer with IBB SBB,max
1016, the minimum mass is 600 grams. If we assume
that the reheat temperature is no more than the reduced
Planck mass, mPlanck 1/8πG 2.4×1018 GeV
4.3×106gram, we find that ΓBB
reheat <exp 2IBB
exp(108). Although this is not nearly as much sup-
pression as in the previous case, it is clearly enough to
guarantee that Eq. (120) will be satisfied.
For the diffuse computer, we can consider an ideal gas
of massless particles, as discussed in Eqs. (115)–(117).
The system would have approximately Smax particles,
and a total energy of E=S4/3
max/R, so the Boltzmann sup-
pression factor is exp hS4/3
max/(R Treheat)i. The Boltz-
mann brain production can occur at any time during the
reheating process, so there is nothing wrong with consid-
ering Boltzmann brain production in our universe at the
present time. For Treheat = 2.7 K and Smax = 1016, this
formula implies that the exponent has magnitude 1 for
R=S4/3
maxT1
reheat 200 light-years. Thus, the formula
suggests that diffuse-gas-cloud Boltzmann brains of ra-
dius 200 light-years can be thermally produced in our
universe, at the present time, without suppression! If
this estimate were valid, then Boltzmann brains would
almost certainly dominate the universe.
We argue, however, that the gas clouds described above
would have no possibility of computing, because the
thermal noise would preclude any storage or transfer
of information. The entire device has energy of order
ETreheat, which is divided among approximately 1016
massless particles. The mean particle energy is there-
fore 1016 times smaller than that of the thermal particles
in the background radiation, and the density of Boltz-
mann brain particles is 1048 times smaller than the back-
ground. To function, it seems reasonable that the diffuse
computer needs an energy per particle that is at least
comparable to the background, which means that the
suppression factor is exp(1016) or smaller. Thus, we
conclude that for all three cases, the ratio of Boltzmann
brains to normal observers is totally negligible.
Finally, let us also mention the possibility that Boltz-
mann brains might form as quantum fluctuations in sta-
ble Minkowski vacua. String theory implies at least the
existence of a 10D decompactified Minkowski vacuum;
Minkowski vacua of lower dimension are not excluded,
but they require precise fine tunings for which motiva-
tion is lacking. While quantum fluctuations in Minkowski
space are certainly less classical than in de Sitter space,
they still might be relevant. The possibility of Boltz-
mann brains in Minkowski space has been suggested by
Page [5, 7, 41]. If ΓBB is nonzero in such vacua, re-
gardless of how small it might be, Boltzmann brains will
always dominate in the scale-factor cutoff measure as we
have defined it. Even if Minkowski vacua cannot support
Boltzmann brains, there might still be a serious problem
with what might be called “Boltzmann islands.” That is,
it is conceivable that a fluctuation in a Minkowski vac-
uum can produce a small region of an anthropic vacuum
with a Boltzmann brain inside it. The anthropic vacuum
could perhaps even have a different dimension than its
23
Minkowski parent. If such a process has a nonvanish-
ing probability to occur, it will also give rise to Boltz-
mann brain domination in the scale-factor cutoff mea-
sure. These problems would be shared by all measures
that assign an infinite weight to stable Minkowski vacua.
There is, however, one further complication which might
allow Boltzmann brains to form in Minkowski space with-
out dominating the multiverse. If one speculates about
Boltzmann brain production in Minkowski space, one
may equally well speculate about spontaneous creation
of inflationary universes there, each of which could con-
tain infinitely many normal observers [63]. These issues
become complicated, and we will make no attempt to
resolve them here. Fortunately, the estimates of ther-
mal Boltzmann brain nucleation rates in de Sitter space
approach zero in the Minkowski space limit Λ 0, so
the issue of Boltzmann brains formed by quantum fluc-
tuations in Minkowski space can be set aside for later
study. Hopefully the vague idea that these fluctuations
are less classical than de Sitter space fluctuations can be
promoted into a persuasive argument that they are not
relevant.
B. Vacuum Decay Rates
One of the most developed approaches to the string
landscape scenario is based on the KKLT construc-
tion [64]. In this construction, one begins by finding a set
of stabilized supersymmetric AdS and Minkowski vacua.
After that, an uplifting is performed, e.g. by adding a D3
brane at the tip of a conifold [64]. This uplifting makes
the vacuum energy density of some of these vacua positive
(AdS dS), but in general many vacua remain AdS, and
the Minkowski vacuum corresponding to the uncompact-
ified 10d space does not become uplifted. The enormous
number of the vacua in the landscape appears because of
the large number of different topologies of the compact-
ified space, and the large number of different fluxes and
branes associated with it.
There are many ways in which our low-energy dS vac-
uum may decay. First of all, it can always decay into the
Minkowski vacuum corresponding to the uncompactified
10d space [64]. It can also decay to one of the AdS vacua
corresponding to the same set of branes and fluxes [65].
More generally, decays occur due to the jumps between
vacua with different fluxes, or due to the brane-flux anni-
hilation [55, 66–72], and may be accompanied by a change
in the number of compact dimensions [73–75]. If one does
not take into account vacuum stabilization, these transi-
tions are relatively easy to analyze [66–68]. However, in
the realistic situations where the moduli fields are deter-
mined by fluxes, branes, etc., these transitions involve a
simultaneous change of fluxes and various moduli fields,
which makes a detailed analysis of the tunneling quite
complicated.
Therefore, we begin with an investigation of the sim-
plest decay modes due to the scalar field tunneling. The
transition to the 10d Minkowski vacuum was analyzed in
Ref. [64], where it was shown that the decay rate κis
always greater than
κ&eSD= exp24π2
VdS .(121)
Here SDis the entropy of dS space. For our vacuum,
SD10120, which yields
κ&eSDexp 10120.(122)
Because of the inequality in Eq. (121), we expect the
slowest-decaying vacua to typically be those with very
small vacuum energies, with the dominant vacuum en-
ergy density possibly being much smaller than the value
in our universe.
The decay to AdS space (or, more accurately, a decay
to a collapsing open universe with a negative cosmolog-
ical constant) was studied in Ref. [65]. The results of
Ref. [65] are based on investigation of BPS and near-
BPS domain walls in string theory, generalizing the re-
sults previously obtained in N= 1 supergravity [76–79].
Here we briefly summarize the main results obtained in
Ref. [65].
Consider, for simplicity, the situation where the tun-
neling occurs between two vacua with very small vacuum
energies. For the sake of argument, let us first ignore the
gravitational effects. Then the tunneling always takes
place, as long as one vacuum has higher vacuum energy
than the other. In the limit when the difference between
the vacuum energies goes to zero, the radius of the bubble
of the new vacuum becomes infinitely large, R→ ∞ (the
thin-wall limit). In this limit, the bubble wall becomes
flat, and its initial acceleration, at the moment when the
bubble forms, vanishes. Therefore to find the tension
of the domain wall in the thin wall approximation one
should solve an equation for the scalar field describing a
static domain wall separating the two vacua.
If the difference between the values of the scalar po-
tential in the two minima is too small, and at least one
of them is AdS, then the tunneling between them may
be forbidden because of the gravitational effects [80]. In
particular, all supersymmetric vacua, including all KKLT
vacua prior to the uplifting, are absolutely stable even if
other vacua with lower energy density are available [81–
84].
It is tempting to make a closely related but opposite
statement: non-supersymmetric vacua are always unsta-
ble. However, this is not always the case. In order to
study tunneling while taking account of supersymmetry
(SUSY), one may start with two different supersymmet-
ric vacua in two different parts of the universe and find
a BPS domain wall separating them. One can show that
if the superpotential does not change its sign on the way
24
from one vacuum to the other, then this domain wall
plays the same role as the flat domain wall in the no-
gravity case discussed above: it corresponds to the wall
of the bubble that can be formed once the supersymme-
try is broken in either of the two minima. However, if
the superpotential does change its sign, then only a suf-
ficiently large supersymmetry breaking will lead to the
tunneling [65, 76].
One should keep this fact in mind, but since we are
discussing a landscape with an extremely large number
of vacua, in what follows we assume that there is at least
one direction in which the superpotential does not change
its sign on the way from one minimum to another. In
what follows we describe tunneling in one such direction.
Furthermore, we assume that at least some of the AdS
vacua to which our dS vacuum may decay are uplifted
much less than our vacuum. This is a generic situation,
since the uplifting depends on the value of the volume
modulus, which takes different values in each vacuum.
In this case the decay rate of a dS vacuum with low
energy density and broken supersymmetry can be esti-
mated as follows [65, 85]:
κexp 8π2α
3m2
3/2!,(123)
where m3/2is the gravitino mass in that vacuum and
αis a quantity that depends on the parameters of the
potential. Generically one can expect α=O(1), but it
can also be much greater or much smaller than O(1). The
mass m3/2is set by the scale of SUSY breaking,
3m2
3/2= Λ4
SUSY ,(124)
where we recall that we use reduced Planck units, 8πG =
1. Therefore the decay rate can be also represented in
terms of the SUSY-breaking scale ΛSUSY :
κexp24π2α
Λ4
SUSY ,(125)
Note that in the KKLT theory, Λ4
SUSY corresponds to the
depth of the AdS vacuum before the uplifting, so that
κexp24π2α
|VAdS|.(126)
In this form, the result for the tunneling looks very simi-
lar to the lower bound on the decay rate of a dS vacuum,
Eq. (121), with the obvious replacements α1 and
|VAdS| → VdS .
Let us apply this result to the question of vacuum de-
cay in our universe. Clearly, the implications of Eq. (125)
depend on the details of SUSY phenomenology. The
standard requirement that the gaugino mass and the
scalar masses are O(1) TeV leads to the lower bound
ΛSUSY &104–105GeV ,(127)
which can be reached, e.g., in the models of conformal
gauge mediation [86]. This implies that for our vacuum
κour &exp(1056)– exp(1060 ).(128)
Using Eq. (99), the Boltzmann brain nucleation rate in
our universe exceeds the lower bound of the above in-
equality only if M.109kg.
On the other hand, one can imagine universes very
similar to ours except with much larger vacuum energy
densities. The vacuum decay rate of Eq. (123) exceeds
the Boltzmann brain nucleation rate of Eq. (99) when
m3/2
102eV 2M
1 kg H1
Λ
108m&109α . (129)
Note that H1
Λ108m corresponds to the smallest de
Sitter radius for which the tidal force on a 10 cm brain
does not exceed the gravitational force on the surface of
the earth, while m3/2102eV corresponds to ΛSUSY
104GeV. Thus, it appears the decay rate of Eq. (123)
allows for Boltzmann brain domination.
However, we do not really know whether the mod-
els with low ΛSUSY can successfully describe our world.
To mention one potential problem: in models of string
inflation there is a generic constraint that during the
last stage of inflation one has H.m3/2[87]. If we
assume the second and third factors of Eq. (129) can-
not be made much less than unity, then we only require
m3/2&O(102) eV to avoid Boltzmann brain domination.
While models of string inflation with H.100 eV are not
entirely impossible in the string landscape, they are ex-
tremely difficult to construct [88]. If instead of ΛSUSY
104GeV one uses ΛSUSY 1011 GeV, as in models
with gravity mediation, one finds m3/2103GeV and
Eq. (129) is easily satisfied.
These arguments apply when supersymmetry violation
is as large or larger than in our universe. If supersym-
metry violation is too small, atomic systems are unsta-
ble [89], the masses of some of the particles will change
dramatically, etc. However, the Boltzmann computers
described in the previous subsection do not necessarily
rely on laws of physics similar to those in our universe (in
fact, they seem to require very different laws of physics).
The present authors are unaware of an argument that
supersymmetry breaking must be so strong that vacuum
decay is always faster than the Boltzmann brain produc-
tion rate of Eq. (118).
On the other hand, up to this point we have used the
estimates of the vacuum decay rate that were obtained
in Refs. [65, 85] by investigation of the transition where
only moduli fields changed. As we have already men-
tioned, the description of a more general class of tran-
sitions involving the change of branes or fluxes is much
more complicated. Investigation of such processes, per-
formed in Refs. [55, 69, 70], indicates that the process
of vacuum decay for any vacuum in the KKLT scenario
25
should be rather fast,
κ&exp(1022).(130)
The results of Refs. [55, 69, 70], like the results of
Refs. [65, 85], are not completely generic. In particu-
lar, the investigations of Refs. [55, 69, 70] apply to the
original version of the KKLT scenario, where the uplift-
ing of the AdS vacuum occurs due to D3 branes, but
not to its generalization proposed in Ref. [90], where the
uplifting is achieved due to D7 branes. Neither does it
apply to the recent version of dS stabilization proposed
in Ref. [91]. Nevertheless, the results of Refs. [55, 69, 70]
show that the decay rate of dS vacua in the landscape can
be quite large. The rate κ&exp(1022) is much greater
than the expected rate of Boltzmann brain production
given by Eq. (105). However, it is just a bit smaller
than the bosonic gas Boltzmann brain production rate
of Eq. (119) and much smaller than our most dangerous
upper bound on the Boltzmann brain production rate,
given by Eq. (118).
VI. CONCLUSIONS
If the observed accelerating expansion of the universe
is driven by constant vacuum energy density and if our
universe does not decay in the next 20 billion years or
so, then it seems cosmology must explain why we are
“normal observers” — who evolve from non-equilibrium
processes in the wake of the big bang — as opposed to
“Boltzmann brains” — freak observers that arise as a
result of rare quantum fluctuations [2–4, 8, 9]. Put in
experimental terms, cosmology must explain why we ob-
serve structure formation in a residual cosmic microwave
background, as opposed to the empty, vacuum-energy
dominated environment in which almost all Boltzmann
brains nucleate. As vacuum-energy expansion is eter-
nal to the future, the number of Boltzmann brains in an
initially-finite comoving volume is infinite. However, if
there exists a landscape of vacua, then rare transitions
to other vacua populate a diverging number of universes
in this comoving volume, creating an infinite number of
normal observers. To weigh the relative number of Boltz-
mann brains to normal observers requires a spacetime
measure to regulate the infinities.
Recently, the scale-factor cutoff measure was shown
to possess a number of desirable attributes, including
avoiding the youngness paradox [29] and the Q(and
G) catastrophe [31–33], while predicting the cosmolog-
ical constant to be measured in a range including the
observed value, and excluding values more than about a
factor of ten larger and smaller than this [40]. The scale-
factor cutoff does not itself select for a longer duration
of slow-roll inflation, raising the possibility that a signif-
icant fraction of observers like us measure cosmic curva-
ture significantly above the value expected from cosmic
variance [49]. In this paper, we have calculated the ratio
of the total number of Boltzmann brains to the number
of normal observers, using the scale-factor cutoff.
The general conditions under which Boltzmann brain
domination is avoided were discussed in Subsection IV F,
where we described several alternative criteria that can
be used to ensure safety from Boltzmann brains. We also
explored a set of assumptions that allow one to state con-
ditions that are both necessary and sufficient to avoid
Boltzmann brain domination. One relatively simple way
to ensure safety from Boltzmann brains is to require two
conditions: (1) in any vacuum, the Boltzmann brain nu-
cleation rate must be less than the decay rate of that
vacuum, and (2) for any anthropic vacuum jwith a de-
cay rate κjq, and for any non-anthropic vacuum j,
one must construct a sequence of transitions from jto
an anthropic vacuum; if the sequence includes suppressed
upward jumps, then the Boltzmann brain nucleation rate
in vacuum jmust be less than the decay rate of vacuum
jtimes the product of all the suppressed branching ra-
tios Bup that appear in the sequence. The condition (2)
might not be too difficult to satisfy, since it will gener-
ically involve only states with very low vacuum energy
densities, which are likely to be nearly supersymmetric
and therefore unlikely to support the complex structures
needed for Boltzmann brains or normal observers. Con-
dition (2) can also be satisfied if there is no unique dom-
inant vacuum, but instead a dominant vacuum system
that consists of a set of nearly degenerate states, some
of which are anthropic, which undergo rapid transitions
to each other, but only slow transitions to other states.
The condition (1) is perhaps more difficult to satisfy. Al-
though nearly-supersymmetric string vacua can in princi-
ple be long-lived [64, 65, 76–79], with decay rates possibly
much smaller than the Boltzmann brain nucleation rate,
recent investigations suggest that other decay channels
may evade this problem [55, 69, 70]. However, the decay
processes studied in [55, 64, 65, 69, 70, 76–79] do not
describe some of the situations which are possible in the
string theory landscape, and the strongest constraints on
the decay rate obtained in [55] are still insufficient to
guarantee that the vacuum decay rate is always smaller
than the fastest estimate of the Boltzmann brain produc-
tion rate, Eq. (118).
One must emphasize that we are discussing a rapidly
developing field of knowledge. Our estimates of the
Boltzmann brain production rate are exponentially sensi-
tive to our understanding of what exactly the Boltzmann
brain is. Similarly, the estimates of the decay rate in the
landscape became possible only five years ago, and this
subject certainly is going to evolve. Therefore we will
mention here two logical possibilities which may emerge
as a result of the further investigation of these issues.
If further investigation will demonstrate that the
Boltzmann brain production rate is always smaller than
the vacuum decay rate in the landscape, the probabil-
ity measure that we are investigating in this paper will
26
be shown not to suffer from the Boltzmann brain prob-
lem. Conversely, if one believes that this measure is cor-
rect, the fastest Boltzmann brain production rate will
give us a rather strong lower bound on the decay rate
of the metastable vacua in the landscape. We expect
that similar conclusions with respect to the Boltzmann
brain problem should be valid for the causal-patch mea-
sures [24, 25].
On the other hand, if we do not find a sufficiently con-
vincing theoretical reason to believe that the vacuum de-
cay rate in all vacua in the landscape is always greater
than the fastest Boltzmann brain production rate, this
would motivate the consideration of other probability
measures where the Boltzmann brain problem can be
solved even if the probability of their production is not
strongly suppressed.
In any case, our present understanding of the Boltz-
mann brain problem does not rule out the scale-factor
cutoff measure, but the situation remains uncertain.
Acknowledgments
We thank Raphael Bousso, Ben Freivogel, I-Sheng
Yang, Shamit Kachru, Renata Kallosh, Delia Schwartz-
Perlov, and Lenny Susskind for useful discussion. The
work of ADS is supported in part by the INFN “Bruno
Rossi” Fellowship, and in part by the U.S. Department of
Energy (DoE) under contract No. DE-FG02-05ER41360.
AHG is supported in part by the DoE under contract
No. DE-FG02-05ER41360. AL and MN are supported
by the NSF grant 0756174. MPS and AV are supported
in part by the U.S. National Science Foundation under
grant NSF 322, and AV is also supported in part by a
grant from the Foundational Questions Institute (FQXi).
Appendix A: Boltzmann Brains in Schwarzschild–de
Sitter Space
As explained in Subsection V A, Eq. (100) for the pro-
duction rate of Boltzmann brains must be reexamined
when the Boltzmann brain radius becomes comparable
to the de Sitter radius. In this case we need to describe
the Boltzmann brain nucleation as a transition from an
initial state of empty de Sitter space with horizon radius
H1
Λto a final state in which the dS space is altered by the
presence of an object with mass M. Assuming that the
object can be treated as spherically symmetric, the space
outside the object is described by the Schwarzschild–de
Sitter (SdS) metric [92]:16
ds2=12GM
rH2
Λr2dt2
+12GM
rH2
Λr21
dr2+r2d2.(A1)
The SdS metric has two horizons, determined by the pos-
itive zeros of gtt, where the smaller and larger are called
RSch and RdS, respectively. We assume the Boltzmann
brain is stable but not a black hole, so its radius satisfies
RSch < R < RdS. The radii of the two horizons are given
by
RSch =2
3HΛ
cos π+ξ
3,
RdS =2
3HΛ
cos πξ
3,
(A2)
where
cos ξ= 33GMHΛ.(A3)
This last equation implies that for a given value of HΛ,
there is an upper limit on how much mass can be con-
tained within the de Sitter horizon:
MMmax = (33GHΛ)1.(A4)
Eqs. (A2) and (A3) can be inverted to express Mand
HΛin terms of the horizon radii:
1
H2
Λ
=R2
Sch +R2
dS +RSchRdS (A5)
M=RdS
2G1H2
ΛR2
dS(A6)
=RSch
2G1H2
ΛR2
Sch.(A7)
We relate the Boltzmann brain nucleation rate to the
decrease in total entropy ∆Scaused by the the nucleation
process,
ΓBB eS,(A8)
where the final entropy is the sum of the entropies of the
Boltzmann brain and the de Sitter horizon. For a Boltz-
mann brain with entropy SBB, the change in entropy is
given by
S=π
GH2
Λπ
GR2
dS +SBB.(A9)
Note that for small Mone can expand ∆Sto find
S=2πM
HΛSBB +O(GM2),(A10)
16 We restore G= 1/8πin this Appendix for clarity.
27
giving a nucleation rate in agreement with Eq. (100).17
To find a bound on the nucleation rate, we need an up-
per bound on the entropy that can be attained for a given
size and mass. In flat space the entropy is believed to be
bounded by Bekenstein’s formula, Eq. (107), a bound
which should also be applicable whenever RRdS .
More general bounds in de Sitter space have been dis-
cussed by Bousso [54], who considers bounds for systems
that are allowed to fill the de Sitter space out to the hori-
zon R=RdS of an observer located at the origin. For
small mass M, Bousso argues that the tightest known
bound on Sis the D-bound, which states that
SSDπ
G1
H2
ΛR2
dS=π
GR2
Sch +RSchRdS,
(A11)
where the equality of the two expressions follows from
Eq. (A5). This bound can be obtained from the princi-
ple that the total entropy cannot increase when an ob-
ject disappears through the de Sitter horizon. For larger
values of M, the tightest bound (for R=RdS ) is the
holographic bound, which states that
SSHπ
GR2
dS .(A12)
Bousso suggests the possibility that these bounds have a
common origin, in which case one would expect that there
exists a valid bound that interpolates smoothly between
the two. Specifically, he points out that the function
Smπ
GRSchRdS (A13)
is a candidate for such a function. Fig. (1) shows a graph
of the holographic bound, the D-bound, and the m-bound
(Eq. (A13)) as a function of M/Mmax. While there is
no reason to assume that Smis a rigorous bound, it is
known to be valid in the extreme cases where it reduces
to the D– and holographic bounds. In between it might
be valid, but in any case it can be expected to be valid up
to a correction of order one. In fact, Fig. (1) and the as-
sociated equations show that the worst possible violation
of the m-bound is at the point where the holographic and
D– bounds cross, at M/Mmax = 36/8 = 0.9186, where
the entropy can be no more than (1 + 5)/2 = 1.6180
times as large as Sm.
Here we wish to carry the notion of interpolation one
step further, because we would like to discuss in the same
formalism systems for which RRdS , where the Beken-
stein bound should apply. Hence we will explore the con-
sequences of the bound
SSIπ
GRSchR , (A14)
17 We thank Lenny Susskind for explaining this method to us.
FIG. 1: Graph shows the holographic bound, the D-bound,
and the m-bound for the entropy of an ob ject that fills de
Sitter space out to the horizon. The holographic and D–
bounds are each shown as broken lines in the region where
they are superseded by the other. Although the m-bound
looks very much like a straight line, it is not.
which we will call the interpolating bound. This bound
agrees exactly with the m-bound when the object is al-
lowed to fill de Sitter space, with R=RdS . Again we
have no grounds to assume that the bound is rigorously
true, but we do know that it is true in the three limiting
cases where it reduces to the Bekenstein bound, the D-
bound, and the holographic bound. The limiting cases
are generally the most interesting for us in any case, since
we wish to explore the limiting cases for Boltzmann brain
nucleation. For parameters in between the limiting cases,
it again seems reasonable to assume that the bound is at
least a valid estimate, presumably accurate up to a factor
of order one. We know of no rigorous entropy bounds for
de Sitter space with Rcomparable to RdS but not equal
to it, so we don’t see any way at this time to do better
than the interpolating bound.
Proceeding with the I-bound of Eq. (A14), we can use
Eq. (106) to rewrite Eq. (A9) as
S=π
GH2
ΛR2
dSSBB,max +IBB ,(A15)
which can be combined with SBB,max SIto give
Sπ
GH2
ΛR2
dS RSch R+IBB ,(A16)
which can then be simplified using Eq. (A5) to give
Sπ
GRSch (RSch +RdS R) + IBB .(A17)
To continue, we have to decide what possibilities to
consider for the radius Rof the Boltzmann brain, which
28
is related to the question of Boltzmann brain stabilization
discussed after Eq. (111). If we assume that stabilization
is not a problem, because it can be achieved by a domain
wall or by some other particle physics mechanism, then
Sis minimized by taking Rat its maximum value, R=
RdS, so
Sπ
GR2
Sch +IBB .(A18)
Sis then minimized by taking the minimum possible
value of RSch, which is the value that is just large enough
to allow the required entropy, SBB,max IBB. Using
again the I-bound, one finds that saturation of the bound
occurs at
ξsat = 3 sin1 p13˜
I
2!,(A19)
where
˜
IIBB
SdS
=GH2
Λ
πIBB (A20)
is the ratio of the Boltzmann brain information to the
entropy of the unperturbed de Sitter space. Note that
˜
Ivaries from zero to a maximum value of 1/3, which
occurs in the limiting case for which RSch =RdS . The
saturating value of the mass and the corresponding values
of the Schwarzschild radius and de Sitter radius are given
by
Msat =˜
Ip1 + ˜
I
2GHΛ
,(A21)
RSch,sat =p1 + ˜
Ip13˜
I
2HΛ
,(A22)
RdS,sat =p13˜
I+p1 + ˜
I
2HΛ
.(A23)
Combining these results with Eq. (A18), one has for this
case (R=RdS) the bound
S
IBB 1 + ˜
Ip1 + ˜
Ip13˜
I
2˜
I.(A24)
As can be seen in Figure 2, the bound on ∆S/IBB for
this case varies from 1, in the limit of vanishing ˜
I(or
equivalently, the limit HΛ0), to 2, in the limit RSch
RdS.
The limiting case of ˜
IBB 0, with a nucleation rate
of order eIBB , has some peculiar features that are worth
mentioning. The nucleation rate describes the nucleation
of a Boltzmann brain with some particular memory state,
so there would be an extra factor of eIBB in the sum over
all memory states. Thus, a single-state nucleation rate of
eIBB indicates that the total nucleation rate, including
all memory states, is not suppressed at all. It may seem
strange that the nucleation rate could be unsuppressed,
but one must keep in mind that the system will func-
tion as a Boltzmann brain only for very special values of
the memory state. In the limiting case discussed here,
the “Boltzmann brain” takes the form of a minor pertur-
bation of the degrees of freedom associated with the de
Sitter entropy SdS =π/(GH2
Λ).
As a second possibility for the radius R, we can con-
sider the case of strong gravitational binding, RRSch,
as discussed following Eq. (111). For this case the bound
(A17) becomes
Sπ
GRSchRdS +IBB .(A25)
(Interestingly, if we take I= 0 (SBB =Smax ) this for-
mula agrees with the result found in Ref. [93] for black
hole nucleation in de Sitter space.) With R=RSch the
saturation of the I-bound occurs at
ξsat =π
23 sin1 p3˜
I
2!.(A26)
The saturating value of the mass and the corresponding
values of the Schwarzschild radius and de Sitter radius
are given by
Msat =p˜
I1˜
I
2GHΛ
,(A27)
RSch,sat =p˜
I
HΛ
,(A28)
RdS,sat =p43˜
Ip˜
I
2HΛ
.(A29)
Using these relations to evaluate Sfrom Eq. (A25), one
finds
S
IBB
=p43˜
I+p˜
I
2p˜
I
,(A30)
which is also plotted in Figure 2. In this case (R=RSch)
the smallest ratio ∆S/IBB is 2, occurring at ˜
I= 1/3,
where RSch =RdS. For smaller values of ˜
Ithe ratio be-
comes larger, blowing up as 1/p˜
Ifor small ˜
I. Thus, the
nucleation rates for this choice of Rwill be considerably
smaller than those for Boltzmann brains with RRdS ,
but this case would still be relevant in cases where Boltz-
mann brains with RRdS cannot be stabilized.
Another interesting case, which we will consider, is to
allow the Boltzmann brain to extend to R=Requil, the
point of equilibrium between the gravitational attraction
of the Boltzmann brain and the outward gravitational
pull of the de Sitter expansion. This equilibrium occurs
at the stationary point of gtt, which gives
Requil =GM
H2
Λ1/3
.(A31)
29
FIG. 2: Graph shows the ratio of ∆Sto IBB, where the nu-
cleation rate for Boltzmann brains is proportional to eS.
All curves are based on the I-bound, as discussed in the text,
but they differ by their assumptions about the size Rof the
Boltzmann brain.
Boltzmann brains within this radius bound would not
be pulled by the de Sitter expansion, so relatively small
mechanical forces will be sufficient to hold them together.
Again ∆Swill be minimized when the I-bound is sat-
urated, which in this case occurs when
ξsat =π
23 sin1
q12A(˜
I)
2
,(A32)
where
A(˜
I)sin
sin1127 ˜
I3
3
.(A33)
The saturating value of the mass and the Schwarzschild
and de Sitter radii are given by
Msat =
3[1 + A(˜
I)]q12A(˜
I)
9GHΛ
,(A34)
RSch,sat =q12A(˜
I)
3HΛ
,(A35)
RdS,sat =
3r33 + 2A(˜
I)q12A(˜
I)
6HΛ
.
(A36)
The equilibrium radius itself is given by
Requil,sat =h12A(˜
I)i1/6h1 + A(˜
I)i1/3
3HΛ
.(A37)
Using these results with Eq. (A17), ∆Sis found to be
bounded by
S
IBB
=r312A(˜
I)3 + 2A(˜
I)2A(˜
I) + 1
6˜
I,
(A38)
which is also plotted in Figure 2. As one might expect
it is intermediate between the two other cases. Like the
R=RSch case, however, the ratio ∆S/IBB blows up for
small ˜
I, in this case behaving as (2/˜
I)1/4.
In summary, we have found that our study of tunnel-
ing in Schwarzschild–de Sitter space confirms the quali-
tative conclusions that were described in Subsection V A.
In particular, we have found that if the entropy bound
can be saturated, then the nucleation rate of a Boltz-
mann brain requiring information content IBB is given
approximately by eaIBB , where ais of order a few, as in
Eq. (111). The coefficient ais always greater than 2 for
Boltzmann brains that are small enough to be gravita-
tionally bound. This conclusion applies whether one in-
sists that they be near-black holes, or whether one merely
requires that they be small enough so that their self-
gravity overcomes the de Sitter expansion. If, however,
one considers Boltzmann brains whose radius is allowed
to extend to the de Sitter horizon, then Figure 2 shows
that acan come arbitrarily close to 1. However, one must
remember that the R=RdS curve on Figure 2 can be
reached only if several barriers can be overcome. First,
these objects are large and diffuse, becoming more and
more diffuse as ˜
Iapproaches zero and aapproaches 1.
There is no known way to saturate the entropy bound
for such diffuse systems, and Eq. (117) shows that an
ideal gas model leads to aI1/3
BB 1. Furthermore,
Boltzmann brains of this size can function only if some
particle physics mechanism is available to stabilize them
against the de Sitter expansion. A domain wall provides
a simple example of such a mechanism, but Eq. (114) in-
dicates that the domain wall solution is an option only if
a domain wall exists with tension σIBB H3
Λ. Thus, it is
not clear how close acan come to its limiting value of 1.
Finally, we should keep in mind that it is not clear if any
of the examples discussed in this appendix can actually
be attained, since black holes might be the only ob jects
that saturate the entropy bound for S1.
30
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    • "This is particularly pertinent when T 2 is a theory for which the principle of mediocrity is not a viable partner. For example, there exist theories in which 'Boltzmann brains' exist and outnumber ordinary observers [see Albrecht and Sorbo (2004), and De Simone et al. (2010)]. Boltzmann brains can observe our data D 0 , but generally, their experimental record is disordered and uncorrelated with the past. "
    [Show abstract] [Hide abstract] ABSTRACT: Extracting predictions from cosmological theories that describe a multiverse, for what we are likely to observe in our domain, is crucial to establishing the validity of these theories. One way to extract such predictions is from theory-generated probability distributions that allow for selection effects---generally expressed in terms of assumptions about anthropic conditionalization and how typical we are. In this paper, I urge three lessons about typicality in multiverse settings. (i) Because it is difficult to characterize our observational situation in the multiverse, we cannot assume that we are typical (as in the 'principle of mediocrity'): nor can we ignore the issue of typicality, for it has a measurable impact on predictions for our observations. (ii) There are spectra of assumptions about both conditionalization and typicality, which lead to coincident predictions for our observations, leading to problems of confirmation in multiverse cosmology. And moreover, (iii) when one has the freedom to consider competing theories of the multiverse, the assumption of typicality may not lead to the highest likelihoods for our observations. These three entwined aspects of typicality imply that positive assertions about our typicality, such as the 'principle of mediocrity', are more questionable than has been recently claimed.
    Article · Sep 2016
    • "In particular, it was argued that certain cut-off measures (e.g. scale factor cut-off) lead to cosmological predictions which are in a good agreement with observations [21, 22]. In this section we will instead discuss the computability properties of a CS whose symmetry is common to all (global and local) cut-off measures. "
    Full-text · Dataset · May 2016
    • "This can lead to a cyclic behaviour, eternal inflation, and the spawning of a number of 'bubble universes' in which the initial conditions take different values. Since this process is endless, one encounters classical counting problems when attempting to place a measure on the possible outcomes [22, 36, 37]. To give a sketch of the behaviour of eternal inflation, we consider the same massive scalar field as above. "
    [Show abstract] [Hide abstract] ABSTRACT: It has recently been shown, in flat Robertson-Walker geometries, that the dynamics of gravitational actions which are minimally coupled to matter fields leads to the appearance of "attractors" - sets of physical observables on which phase space measures become peaked. These attractors will be examined in the context of inhomogeneous perturbations about the FRW background and in the context of anisotropic Bianchi I systems. We show that maximally expanding solutions are generically attractors, i.e. any measure based on phase-space observables becomes sharply peaked about those solutions which have $P=-\rho$.
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