1406.0504

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arXiv:1406.0504v1

[hep-th]

2Jun2014

Spacetime Average Density

(SAD)Cosmological Measures

Don N. Page

Department of Physics4-183 CCIS

University of AlbertaEdmonton, Alberta T6G 2E1

Canada

2014 June 2

Abstract

The measure problem of cosmology is how to obtain normalized proba-

bilities of observations from the quantum state of the universe. This is par-

ticularly a problem when eternal inflation leads to a universe of unbounded

size so that there are apparently infinitely many realizations or occurrences of

observations of each of many different kinds or types, making the ratios am-

biguous. There is also the danger of domination by Boltzmann Brains. Here

two new Spacetime Average Density (SAD) measures are proposed, Maximal

Average Density (MAD) and Biased Average Density (BAD), for getting a

finite number of observation occurrences by using properties of the Space-

time Average Density (SAD) of observation occurrences to restrict to finite

regions of spacetimes that have a preferred beginning or bounce hypersurface.

These measures avoid Boltzmann brain domination and appear to give results

consistent with other observations that are problematic for other widely used

measures, such as the observation of a positive cosmological constant.

Alberta-Thy-10-14, arXiv:yymm.nnnn [hep-th]Internet address: [email protected]

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1 Introduction

Theoretical cosmology is plagued with the measure problem, the problem of how to

predict the probabilities of observations in the universe from the quantum state (or

from some other such input) [1, 2,3, 4,5, 6,7, 8,9,10,11, 12,13,14, 15,16, 17,18,

19,20, 21,22,23, 24,25,26, 27,28,29,30,31, 32,33,34, 35,36,37, 38,39,40, 41,

42,43, 44,45,46, 47,48,49, 50,51,52,53,54, 55,56,57, 58,59,60, 61,62,63, 64,

65,66, 67,68,69, 70,71,72, 73,74,75,76,77, 78,79,80, 81,82,83, 84,85,86, 87,

88,89,90,91,92, 93, 94, 95,96,97,98,99,100,101,102,103,104,105,106,107,

108,109, 110, 111, 112, 113,114, 115, 116,117, 118, 119,120, 121,122, 123, 124,

125,126, 127, 128, 129, 130,131, 132, 133,134, 135, 136,137, 138,139, 140, 141,

142,143, 144, 145, 146, 147,148, 149, 150,151, 152, 153,154, 155,156, 157, 158,

159,160, 161, 162, 163, 164,165, 166, 167,168, 169, 170,171, 172,173, 174, 175,176,177, 178, 179,180,181,182, 183,184,185,186, 187, 188,189,190,191, 192, 193,

194,195, 196, 197,198,199, 200, 201,202,203,204, 205, 206,207,208,209, 210, 211,

212,213, 214,215,216,217, 218, 219,220,221,222, 223, 224,225,226,227, 228, 229,

230,231,232, 233, 234, 235, 236,237,238,239,240,241,242,243,244, 245]. (See

[204] for a fairly recent review.) This problem is particularly severe for universes

that contain infinitely many observations of each of more than one kind, for then

one has the ambiguity of taking ratios of infinite numbers. However, there is also

the challenge even for large finite universes, because of the failure of Borns rule[158,168, 173,183].

A solution to the measure problem should not only be able to give normalized

probabilities for observations (that is, normalized so that the sum over all possible

observations is unity, not just the sum over all the possible observations of one

observer at one time) but also make the normalized probabilities of our observations

not too small. One can take the normalized probability of ones observation as given

by some theory as the likelihood of that theory. (Here a theory includes not only the

quantum state but also the rules for getting the probabilities of observations from it.)Then if one weights the likelihoods of different theories by the prior probabilities

one assigns to them and normalizes the resulting product, one gets the posterior

probabilities of the theories. One would like to find theories, including their solutions

to the measure problem, that give high posterior probabilities.

A major threat to getting high posterior probabilities for theories is the possibil-

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ity that they may predict that observations are dominated by those of Boltzmann

brains that arise from thermal and/or vacuum fluctuations [246,247,248,85,249,

106,110, 112, 114, 115, 117,118, 122, 250,251, 130, 252,253, 138,139, 145, 146,

254, 255, 152, 156, 256, 189, 196, 245]. Most such Boltzmann brain observations

seem likely to be much more disordered than ours, so if the probabilities of our or-

dered observations are diluted by Boltzmann brain observations in theories in which

they dominate the probabilities, that would greatly reduce the likelihood of such

theories. Therefore, one would like to find theories with solutions to the measure

problem that suppress Boltzmann brains if this can be done without too great a

cost in complexity that would tend to suppress the prior probabilities assigned to

such theories.

Most proposed solutions to the measure problem of cosmology tend to lean to-

ward one or the other of two extremes. Some, particularly those proposed by Hartle,

Hawking, Hertog, and/or Srednicki[33,55, 57,68,130,141,143,149,151,170,171,

180,184, 192,212,229,237], which often apply the consistent histories or decoher-

ing histories formalism [257, 258, 259, 260, 261, 262,263, 264,265, 266, 267, 268,

269, 270] to the Hartle-Hawking no-boundary proposal for the quantum state of

the universe [271], tend to suggest that the measure is determined nearly uniquely

by the quantum state (at least if a fairly-unique typicality assumption is made

[130,171,184]). Others, particularly those proposed in the large fraction of papers

cited above for the measure problem that are focused on eternal inflation, tend to

suggest that the quantum state is mostly irrelevant and that the results depend

mainly on how the measure is chosen in the asymptotic future of an eternally inflat-

ing spacetime. Here I shall steer a middle course and suggest that both the quantum

state and the measure are crucial, further suggesting that the measure is dominated

by observations not too late in the spacetime.

I am sceptical of the extreme view that the measure is determined uniquely

(or nearly uniquely) by the quantum state, partly because of my demonstrations

of the inadequacy of Borns rule [158, 168, 173, 183] when it is interpreted to im-

ply that the probabilities of observations are the expectation values of projection

operators, and mainly because of the logical freedom of what the operators are

whose expectation values give the relative probabilities of observations in what I

regard as the simplest formalism for connecting observations to the quantum state

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[14,15,16, 17,52,107,200]. It also appears to me that the particular assumptions

used in the papers of Hartle, Hawking, Hertog, and/or Srednicki cited above, which

tend to give nearly equal weights to observations that almost certainly occur at least

once in the spacetime histories that dominate the probabilities for histories (which

would be a vast range of observations if the histories whose probabilities dominate

have eternal inflation and so become so large that the probability of at least one

occurrence of even very rare observations would be unity or nearly unity), would

lead to the normalized probability for our observations to be extremely low (diluted

by the vast number of other observations of nearly equal probabilities), therefore

giving extremely low likelihoods for theories making these assumptions. Thus I be-

lieve that, unless one by fiat assigns nearly all the prior probability to such theories,

their posterior probabilities will be very small, much less than better theories that

use more sophisticated measures.

I am also sceptical of the opposite extreme, the view that the probabilities of

observations are essentially independent of the quantum state and only depend upon

the dynamics of eternal inflation. It is logically possible that this is the case, say

by having the probabilities of observations given by the expectation values of the

identity operator multiplied by coefficients that will then be the probabilities of the

observations for any normalized quantum state [175]. But then our observations

of apparent quantum effects would just be delusions, since the observations would

not depend upon the quantum state. It seems to me much more plausible that

our observations appear to depend upon the quantum state because indeed they do

depend upon the quantum state.

Now a less-extreme view would be that whether eternal inflation occurs depends

upon the quantum state, but that if the quantum state is such that eternal inflation

does occur, the probabilities of observations do not depend upon further details of

the quantum state. This is more nearly plausible, but I find it also a rather implau-

sible view. If the relative probabilities of observations are given by the expectation

values of positive operators that are not just multiples of the identity operator, they

would generically be changed by generic changes in the quantum state. That is, it

seems very hard to have the relative probabilities insensitive to generic changes in

the quantum state if these relative probabilities are nontrivial linear functionals (i.e.,

not just the expectation values of multiples of the identity operator) of the quantum

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state, which to me appears to be the simplest possibility[14, 15,16, 17,52,107,200],

though I do not claim to see any logical necessity against nonlinear functionals [ 168]

for the relative probabilities.

If the gross asymptotic behavior of an eternally inflating universe is insensitive to

the details of the quantum state (say other than requiring that the state be within

some open set), then I would think it implausible that the relative probabilities of

observations would depend only on the gross asymptotic behavior of the universe.

Therefore, I do not favor measures that have temporal cutoffs that are eventually

taken to infinityandhave the property that for a very large finite cutoff they depend

mainly on the properties of the spacetime at very large times (e.g., times near the

cutoff). If there is a time-dependent weighting to the measure, I suspect that it

should not depend mainly on the asymptotic behavior. In particular, I am scepti-

cal of the specific form of Assumption 3. Typicality. of Freivogel [204] that we

are equally likely to be anywhere consistent with our data . . . [so that with] a finite

probability for eternal inflation, which results in an infinite number of observations,

. . . we can ignore any finite number of observations. In a footnote to this statement,

Freivogel admits, This conclusion relies on an assumption about how to implement

the typicality assumption when there is a probability distribution over how many ob-

servations occur [183]. In this paper I shall reject this assumption, which Freivogel

notes[204]that I have called observational averaging [183], and instead investigate

non-uniform measures over spacetimes that suppress the asymptotic behavior.

As an example of what I mean, in [157] (see also [158,168, 173,214] for further

discussion and motivation) I proposed volume averaging instead of volume weighting

to avoid divergences in the measure of Boltzmann brain observations on spatial

hypersurfaces as they expand to become infinitely large. However, summing up over

all hypersurfaces still gave a divergence if that were done by a uniform integral over

proper time t and if indeed the proper time goes to infinity. One could make this

integral finite by cutting it off at some finite upper bound to the proper time, say

t, but then as t is taken to infinity, asymptotically half of the integral would be

given by times within a factor of two of the temporal cutoff t. Therefore, as t

is taken to infinity, the relative probabilities will be determined by the asymptotic

behavior of the spacetime. For example, if it were an asymptotically de Sitter

spacetime that does not have bubble nucleation to new hot big bang regions that

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2 Spacetime Average Density (SAD) Measures

Here I wish to propose new solutions to the measure problem with a weighted

distribution over variable, rather than fixed, proper-time cutoffs depending on the

spacetime average density of observation occurrences up to the cutoff. The ad hoc

weighting function dt/(1 +t2) of Agnesi weighting will be eliminated, though at

the cost of two different rather ad hocalgorithms for constructing a weighting over

proper time to damp the late-time contribution to the measure for observations.

Let me use the index i to denote the theoryTi, which I take to include not only

the quantum state of the universe but also the rules for getting the probabilities of

the observations from the quantum state. I shall assume that the quantum state

given byi gives, as the expectation value of some positive operator depending upon

the spacetime and perhaps also upon the theory, a relative probability distribution

or measure ij for different quasiclassical inextendible spacetimes Sj, labeled by

the index j, that each has definite occurrences of the observation Ok, labeled by

the indexk, occurring within the spacetime at definite location regions that I shall

assume are much smaller than the spacetime itself and so can be idealized to be at

points within the spacetime. Because each spacetime includes definite observation

occurrences at definite locations, it is not simply a manifold with a metric (though

it needs to have that as well) but is a spacetime description of all the observation

occurrences within it.I should note that k labels the complete content of the observation Ok (which

is generically not sufficient to specify uniquely the location of an occurrence of the

observation Ok within a spacetime Sj), so that there can be multiple occurrences of

the observationOk at different locations within the spacetime. However, since I am

assuming that different observations as such are intrinsically distinguished only by

their content and not by their locations, the probability of an observation Ok given

by a theory Ti,

Pik = P(Ok|Ti), (1)is the total normalized measure for all occurrences of the observation Ok at all the

different locations at which the observation occurs in each spacetime Sj and in all

spacetimes Sj given by the quantum state specified by the theory Ti in which the

observation occurs. Besides specifying the quantum state, a theory Ti must also

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specify how to get the total measure for the observationOk from the quantum state.

In the present proposals, I am assuming that this measure is obtained by a suitably

weighted sum (depending on the spacetime Sj that specifies not only a geometry

but also the locations and types of all observation occurrences within it) of the

occurrences of the observation Ok within a spacetime Sj , further weighted by the

measureij for the spacetime Sj in the quantum state given by the theory Ti, and

then summed over all spacetimesSj. I shall also assume that the measureij is a lin-

ear functional of the quantum state, given by the expectation value in the quantum

state of a suitable positive operator for the existence of that spacetime. Then the

probabilityPik of the observation Ok given the theory Ti would be the normalized

expectation value of some positive operator (depending upon the operators for the

existence of the spacetimes Sj and upon the weighting for the observation Ok that

may occur multiple times within various ones of these spacetimes), as proposed in my

formalism for what I have called Sensible Quantum Mechanics [14,15,16,17,107]

or Mindless Sensationalism [52,200].

I shall also assume that each such spacetime Sj with positive measure ij in the

theoriesTi that I shall be considering is globally hyperbolic with compact Cauchy

surfaces and has the equivalent of an initial compact Cauchy hypersurface, either

the Terminally Indecomposable Future Set (TIFS) [272, 273] of a big bang, or the

extremal hypersurface of globally minimum volume for spacetime with a bounce.

Then I shall introduce a time function t that is the supremum of the absolute value

of the proper time of any causal curve from the point where it is evaluated to this

preferred initial hypersurface. Next, evaluate the 4-volume Vj(t) of the spacetime

region, say Rjt , in the spacetime Sj that is within a time t or less of the preferred

initial hypersurface, and the number Njk(t) of occurrences of the observation Ok

that occur within this spacetime region Rjt that are of type k. The sum of the

number of occurrences of all observation types Ok that occur within this spacetime

region of spacetime Sj up to time t is

Nj(t) =k

Njk(t). (2)

A subtlety is the fact that presumably different types of observations Ok have

different measures as well as different spacetime frequencies of occurring. Therefore,

Njk should not literally be the number of occurrences of the observation Ok, but

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some sort of weighted number, weighted by some factor that depends upon the

particular observation type Ok [14,15, 16,17,52, 274,107,200,275]. For example,

one might suppose that on earth there are far more ant observations than human

observations, but it seems plausible that most human observations have much greater

weight than most ant observations (perhaps correlated with the generally increased

complexity of human observations over ant observations, though I do not know of

any unique obvious detailed form for this correlation). Then the weighted number

Njk of human observations could be greater than that of ant observations, despite the

greater number of ant observations if they were simply counted equally weighted,

helping to make it not statistically surprising why we find ourselves experiencing

human observations rather than ant observations. Henceforth I shall assume that

the numbers of occurrences Njk in the spacetime Sj of the observation Ok, and

the total number of occurrences in that spacetime of all observations, are weighted

numbers of occurrences.

From these quantities, calculate the Spacetime Average Density of occurrences

of the observationOk of typek for the spacetime regionRjt of 4-volumeVj(t) within

timet of the preferred initial hypersurface in the spacetime Sj,

njk(t) = Njk(t)/Vj(t). (3)

The sum of these over all observation types Ok is the total Spacetime Average

Density of all observation occurrences in the spacetime region Rjt ,

nj(t) =k

njk(t) =Nj(t)/Vj(t). (4)

I shall call this the SAD of that region, or the SAD function oft for the spacetime

Sj.

Now I wish to construct spacetime analogues of spatial volume averaging [157],

weighting each observation Ok by some form of its Spacetime Average Density.

The simplest procedure would appear to be to take the full Spacetime AverageDensity of each observation Ok over all of each spacetime Sj and then weight by

the quantum measures ij that the theory Ti assigns to that spacetime. However,

if the spacetime extends to arbitrarily large t and asymptotes in some region that

locally approaches the vacuum with a positive density per 4-volume of Boltzmann

brains, the full Spacetime Average Density will be dominated by Boltzmann brain

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2.1 Maximal Average Density (MAD) Measure

First, consider theories Ti that employ what I shall call the Maximal Average Den-

sity (MAD) measure. These make use of the time tj that is the value of t that

gives the global maximum value of the SAD function of t for the spacetime Sj ,nj(t) = Nj(t)/Vj(t), the Spacetime Average Density of the total occurrences of all

observations up to proper time t in the spacetime Sj . That is, nj(t) nj(tj) forallt in the spacetime Sj. (For simplicity, I shall assume that there is zero quantum

measure ij for spacetimes with more than one value of tj at which nj(t) attains

its global maximum, so that nj(t)< nj(tj) for all t =tj in all spacetimes Sj withpositive measures.)

In particular, the Maximal Average Density or MAD measure is the one in which

fij(t) = (t tj), (10)

the Heaviside step function, being 0 for times t before the global maximum for nj(t)

and being 1 for times after this global maximum. Then dfij/dt= (t tj), a Diracdelta function centered on the global maximum for nj(t) for the spacetime Sj, so

Eq. (5) gives

nijk = njk(tj). (11)

This then leads to the normalized probability for the observation Ok given a MAD

theoryTi as being

Pik P(Ok|Ti) =

jijnjk(tj)j,kijnjk(tj)

. (12)

Of course, there is not a unique MAD theory, since for this MAD function fij(t) =

(t tj), there are many different MAD theories giving different quantum statesand hence different quantum measures ij for the spacetimes Sj .

One might suppose that a typical inextendible spacetime which gives a large

contribution to the probabilityPik in a plausible theoryTiof a typical human obser-

vationOk would have something like a big bang at small t (though perhaps actuallya bounce at t= 0 [172]), a relatively low density of observation occurrences until a

period around t t0 when planets heated by stars exist and have a relatively highdensity of observation occurrences produced by life on the warm planets (compared

with that at any greatly different time), and then a density of observation occur-

rences that drops drastically as stars burn out and planets freeze, until the density

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of observation occurrences asymptotes to some very tiny but still positive spacetime

density of Boltzmann brain observations. In this case it is plausible to expect that

nij(t) will start very small for small t when the spacetime Sj is too hot for life (and

when life has not had much time to evolve), rise to a maximum at a time tt0when

planetary life prevails, and then drops to a very small positive asymptotic constant

when planetary life dies out and Boltzmann brains dominate.

For example, a mnemonic k = 0 CDM Friedmann-Lematre-Robertson-Walker

model [276] with = 3H2 (10 Gyr)2 ten square attohertz 3/(532400)

Planck units (and so fairly accurately applicable to our universe only after the end

of radiation dominance but here used for all times) gives

Vj(t) V3(2/27) H1 x(t) V3(2/27) H1 [sinh (3Ht) 3Ht] (13)

withV3the present 3-volume (unknown and perhaps very large because the universe

appears to extend far beyond what we can see of it, though here I shall assume that

it is finite) andx(t) = sinh y(t) y(t) with y(t) = 3Ht=

3 t.

A very crude toy model for the SAD function nj(t) for the Spacetime Average

Density of all observations might be

nj(t) A

x(t)

1 +x(t)2+

, (14)

where A is an unknown constant that parametrizes the peak density of ordinary

observations that are represented by the first term that rises and then falls, andwhere A is the much, much smaller density of Boltzmann brain observations that

is crudely assumed to be constant. (For a Boltzmann brain that is the vacuum

fluctuation of a human-sized brain, one might expect 101042 [112].) The totalnumber of ordinary observation occurrences in this crude model is finite, A V3, but

the total number of Boltzmann brain observations grows linearly with the 4-volume

Vj(t) = V3x(t) and hence diverges if indeed t and x(t) go to infinity as assumed.

The SAD function rises from A at t = 0 and x = 0 (probably an overestimate,

as I would suspect that when the universe is extremely dense, Boltzmann brainproduction would be suppressed, but since 1 I shall ignore this tiny error,no doubt much smaller than the error of the crude time-dependent term for the

Spacetime Average Density of ordinary observations) monotonically to A(0.5 +)

at t= tj that gives xj x(tj) = 1 and then drops monotonically back to A att= that gives x= .

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Then the MAD measure gives

nij = nj(tj) = A(0.5 +). (15)

The first term in the sum corresponds to ordinary observations, and the secondcorresponds to Boltzmann brain observations. If all the different spacetimes Sj that

have positive quantum measure ij had SAD functions nj(t) that were proportional

to this one (with the same ratio of ordinary and Boltzmann brain observations),

then the total normalized probability for Boltzmann brain observations would be

only /(0.5 +) 2 1. Thus the MAD measure would solve the Boltzmannbrain problem, even for models in which Boltzmann brain production is faster than

the production of new bubble universes and even for models that asymptote to

Minkowski spacetime with its infinite spacetime volume and presumed positive den-

sity per 4-volume of Boltzmann brain observations that would cause Boltzmann

brain domination in most other proposed solutions to the measure problem.

2.2 Biased Average Density (BAD) Measure

Next, consider what I shall call the Biased Average Density (BAD) measure. A

motivation for going from MAD to BAD is that the MAD measure does not give

any weight to observations within a spacetime that is after the time tj at which

the SAD, nj(t), is maximized. It does not seem very plausible that any observation

occurrence within a spacetime of positive measure would contribute zero weight to

that kind of observation Ok, so the BAD measure replaces the MAD measure by

a weighting that is positive for all observation occurrences within an inextendible

spacetime (except possibly for a set of measure zero). The auxiliary function in the

BAD measure is given by

fij(t) =

t0dt|dnj(t)/dt|

0 dt|dnj(t)/dt| , (16)

which again increases monotonically from 0 at t = 0 to 1 at t =, but now con-tinuously rather than suddenly jumping from 0 to 1 as the MAD auxiliary function

fij(t) =(t tj) does.In particular, if nj(t) increases monotonically from n0 at t= 0 to a single local

maximum (the global maximum) value n at t = tj and then decreases monotoni-

cally to n at t =(where for now I am suppressing the overbar and j index on

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nj(t) at these three special times), then

fij(t) = (tj t) nj(t) n02n n0 n +(t

tj)2n n0 nj(t)2n n0 n , (17)

Then Eq. (6) gives

nij = 2n2

n2

0 n2

2(2n n0 n) . (18)

In the case in which n0 = n (as was assumed above in the crude toy model),

one gets

nij =1

2(n+n0). (19)

This is what was assumed above in the crude toy model, which has n0=n=A

andn= A(0.5 +), giving

nij =A(0.25 +). (20)

Again taking the first term in the sum to correspond to ordinary observations and

the second term to correspond to Boltzmann brain observations, and assuming that

all the different spacetimes with positive quantum measure given the same ratio of

the first term to the second term, one gets that the total normalized probability for

Boltzmann brain observations in the BAD measure would be /(0.25 + ) 4 1,roughly twice what it would be in the MAD measure but still extremely small.

Therefore, both the MAD and BAD measures would solve the Boltzmann brain

problem.

3 Conclusions

One might compare the MAD and BAD measures, which are SAD measures using

the Spacetime Average Density, with the Agnesi measure [196], which uses spatial

averaging over hypersurfaces and a weighting of hypersurfaces by the Agnesi function

of time, dt/(1 +t2) with time t in Planck units. In some ways the MAD and BAD

measures appear to have more complicated algorithms to define, but they do avoid

the use of an explicit ad hocfunction of time such as the Agnesi function, though it

is one of the simplest functions that is positive and gives a finite integral over the

entire real axis.

The weighting factor of the Agnesi measure does favor earlier times or young-

ness (as both the MAD and BAD measures do in different ways), but in a fairly

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weak or light way, without exponential damping in time. Therefore, it might be

called a Utility Giving Light Youngness (UGLY) measure. As a result, I have now

made alternative proposals for measures that are MAD, BAD, and UGLY. I am still

looking for one that is GOOD in a supreme way of giving a high posterior proba-

bility by both giving a likelihood (probability of ones observation given the theory

that includes the measure) that is not too low (which it seems that all three of my

proposed measures would do with a suitable quantum state, such as perhaps the

Symmetric Bounce state[172])and giving a prior probability (assumed to be higher

for simpler or more elegant theories) that is not too low (which my measures might

not be in comparison with a yet unknown measure that one might hope could be

much simpler and more elegant). However, if GOOD were interpreted as simply

meaning Great Ordinary Observer Dominance, then since all three of my proposed

measures suppress Boltzmann brains relative to ordinary observers, one could saythat the MAD, the BAD, and the UGLY are all GOOD.

4 Acknowledgments

I have benefited from discussions with many colleagues on the measure problem

in cosmology, of which a proper subset that presently comes to mind includes

Andy Albrecht, Tom Banks, Raphael Bousso, Adam Brown, Steven Carlip, Sean

Carroll, Brandon Carter, Willy Fischler, Ben Freivogel, Gary Gibbons, Steve Gid-

dings, Daniel Harlow, Jim Hartle, Stephen Hawking, Simeon Hellerman, Thomas

Hertog, Gary Horowitz, Ted Jacobson, Shamit Kachru, Matt Kleban, Stefan Le-

ichenauer, Juan Maldacena, Don Marolf, Yasunori Nomura, Joe Polchinski, Steve

Shenker, Eva Silverstein, Mark Srednicki, Rafael Sorkin, Douglas Stanford, Andy

Strominger, Lenny Susskind, Bill Unruh, Erik Verlinde, Herman Verlinde, Aron

Wall, Nick Warner, and Edward Witten. Face-to-face conversations with Gary Gib-

bons, Jim Hartle, Stephen Hawking, Thomas Hertog, and others were enabled by the

gracious hospitality of the Mitchell family and Texas A & M University at a work-

shop at Great Brampton House, Herefordshire, England. This work was supportedin part by the Natural Sciences and Engineering Research Council of Canada.

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