Don N. Page- Agnesi Weighting for the Measure Problem of Cosmology

8/3/2019 Don N. Page- Agnesi Weighting for the Measure Problem of Cosmology

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arXiv:1011

.4932v3

[hep-th]

24Feb2011

Agnesi Weighting for the

Measure Problem of Cosmology

Don N. Page

Theoretical Physics InstituteDepartment of Physics, University of Alberta

Room 238 CEB, 11322 89 AvenueEdmonton, Alberta, Canada T6G 2G7

2011 February 24

Abstract

The measure problem of cosmology is how to assign normalized probabili-

ties to observations in a universe so large that it may have many observations

occurring at many different spacetime locations. I have previously shown

how the Boltzmann brain problem (that observations arising from thermal

or quantum fluctuations may dominate over ordinary observations if the uni-

verse expands sufficiently and/or lasts long enough) may be ameliorated by

volume averaging, but that still leaves problems if the universe lasts too long.

Here a solution is proposed for that residual problem by a simple weighting

factor 1/(1 + t2) to make the time integral convergent. The resulting Agnesi

measure appears to avoid problems other measures may have with vacua of

zero or negative cosmological constant.

Alberta-Thy-13-10, arXiv:1011.4932 [hep-th]Internet address: [email protected]

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Introduction

The high degree of spatial flatness observed for the constant-time hypersurfaces of

our universe leads to the idea that our universe is much larger than what we can

presently observe. The leading explanation for this flatness, cosmological inflation inthe early universe, suggests that in fact the universe is enormously larger than what

we can see, perhaps arbitrarily large if an indefinitely long period of eternal inflation

has occurred in the past. Furthermore, the recent observations of the acceleration

of the universe suggest that our universe may expand exponentially yet more into a

very distant future. As a result, spacetime may already be, or may become, so large

that a vast number of different observations (by which I mean observational results,

what it is that is actually observed) will recur a huge number of times throughout

the universe.

If we restrict to theories that only predict whether or not a particular observa-

tion (e.g., ours) occurs, there are likely to be many such theories predicting that

our observation almost certainly occurs, so that we would have very little observa-

tional evidence to distinguish between such theories. This would seem to imply that

observational science would come to an end for such theories.

However, even for a very large universe there can be theories that are much more

testable by predicting not just whether a particular observation occurs, but also the

probability that this particular observation is made rather than any of the other

possible observations. Then one can use the probability the theory assigns to our

actual observation as the likelihood of the theory, given the observation (actually

the conditional probability of the observation, given the theory). One can then draw

statistical conclusions about alternative theories from their likelihoods. For example,

in a Bayesian analysis in which one assigns prior probabilities to the theories, one

can multiply these priors by the likelihoods and divide by the sum of the products

over all theories to get the normalized posterior probabilities of the theories.

Therefore, it would be desirable to have theories that each predict normalized

probabilities for all possible observations. (These probabilities can be normalized

measures for a set of observations that in a global sense all actually occur, as in Ev-erettian versions of quantum theory in which quantum probabilities are not propen-

sities for a wide class of potentialities to be converted to a narrower class of ac-

tualities. All observations with positive measure could actually occur in such a

completely deterministic version of quantum theory, but with different measures,

which, if normalized, can be used as likelihoods in a statistical analysis.)

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However, in a very large universe in which many observations recur many times,

it can become problematic what rule to use to calculate the normalized measure

(or probability) for each one. If one had a definite classical universe in which each

observation occurs a fixed finite number of times, and if the total number of all

observations is also a finite number, one simple rule would be to take the normalized

measure for each observation to be the fraction of its occurrence, the number of times

it occurs divided by the total number of all observations. But in a quantum universe

in which there are amplitudes for various situations, it is less obvious what to do.

I have shown that Borns rule, taking the normalized measure of each observation

to be the expectation value of a corresponding projection operator, does not work

in a sufficiently large universe [1, 2, 3, 4, 5]. The simplest class of replacement rules

would seem to be to use instead the expectation values of other positive operators,

but then the question arises as to what these operators are.For a universe that is a quantum superposition of eigenstates that each have

definite finite numbers of each observation, one simple choice for the normalized

measures would be to take the expectation values of the frequencies of each obser-

vation (say frequency averaging), and a different simple choice would be to take the

expected numbers of each observation divided by the expected total number of all

observations (say number averaging). For example, suppose that the quantum state

giving only two possible observations (say of a loon or of a bear, to use the animals

on the one- and two-dollar Canadian coins) is

| = cos |mn + sin |MN, (1)where the first eigenstate |mn corresponds to m loon observations and n bear ob-servations and the second eigenstate |MN corresponds to M loon observations andN bear observations. (For simplicity I am assuming all of the loon observations are

precisely identical but different from all of the bear observations that are themselves

precisely identical.) Then the first choice above, frequency averaging, would give

Pf(loon) =m

m + ncos2 +

M

M + Nsin2 ,

Pf(bear) =n

m + n cos2

+N

M + N sin2

, (2)

whereas the second choice above, number averaging, would give

Pn(loon) =m cos2 + Msin2

(m + n)cos2 + (M + N)sin2 ,

Pn(bear) =n cos2 + Nsin2

(m + n)cos2 + (M + N)sin2 . (3)

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Therefore, even in this very simple case, there is no uniquely-preferred rule for con-

verting from the quantum state to the observational probabilities. One would want

P(loon) to be between the two loon-observation frequencies for the two eigenstates,

between m/(m + n) and M/(M + N), as indeed both rules above give, but un-

less one believes in the collapse of the wavefunction (which would tend to favor

frequency averaging), there does not seem to be any clear choice between the two.

(One can easily see that in this example, there is no state-independent projection

operator whose expectation value is always between m/(m + n) and M/(M + N)

for arbitrary m, n, M, N and , so Borns rule fails [1, 2, 3, 4, 5].)

The problem becomes even more difficult when each quantum component may

have an infinite number of observations. Then it may not be clear how to get

definite values for the frequencies of the observations in each eigenstate, or how to

get definite values for the ratios of the infinite expectation values for the numbers ofeach different observation. Most of the work on the measure problem in cosmology

has focused on regularizing these infinite numbers of observations [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, 1, 55,

56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77].

However, I have emphasized [1, 2, 3, 4, 5] that there is also the ambiguity described

above even for finite numbers of occurrence of identical observations.

One challenge is that many simple ways to extract observational probabilities

from the quantum state appear to make them dominated by Boltzmann brain ob-servations, observations produced by thermal or vacuum fluctuations rather than by

a long-lived observer [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]. But if Boltzmann brains

dominate, we would expect that our observations would have much less order than

they are observed to have, so we have strong statistical evidence against our obser-

vations being made by Boltzmann brains. We would therefore like theories that

does not have the measures for observations dominated by Boltzmann brains.

The main way in which theories tend to predict domination by Boltzmann brains

is by having the universe last so long that after ordinary observers die out, a much

larger number (say per comoving volume) of Boltzmann brains eventually appear.

A big part of the problem is that the volume of space seems to be beginning to grow

exponentially as the universe enters into an era in which a cosmological constant

(or something very much like it) dominates the dynamics. Therefore, the expected

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number of Boltzmann brain observations per unit time would grow exponentially

with the expansion of the universe and would eventually become larger than the

current number of observations per time by ordinary observers, leading to Boltz-

mann brain domination in number averaging (which I have previously called volume

weighting [1, 2, 3, 4, 5] because the number per time is proportional to the spatial

volume for observations at a fixed rate per four-volume).

To avoid this part of the problem that occurs for what I have called volume

weighting (or what I now prefer to call number averaging, setting the measure for a

observation proportional to the expectation value or quantum average of the number

of occurrences of the observation), I have proposed using instead volume averaging

[1, 2, 3, 4, 5] (or what I now prefer to call spatial density averaging), setting the

measure for each observation on a particular hypersurface to be proportional to the

expectation value of the spatial density of the occurrences of that observation, theexpected number divided by the spatial volume. This would lead to the contribution

per time for hypersurfaces at late times being the very low asymptotically constant

spacetime density of Boltzmann brains. This density is presumably enormously

lower than the spacetime density of ordinary observers today, so per time, obser-

vations today dominate greatly over Boltzmann brains at any time in the distant

future.

However, if the universe lasts for a sufficiently long time (exponentially longer

than the time it would have to last for Boltzmann brains to dominate in number av-

eraging), then integrating over time would cause even the contribution from the verytiny spatial density of Boltzmann brains eventually to dominate over the contribu-

tions of ordinary observers that presumably exist only during a finite time. (For the

moment I am ignoring the contributions from tunnelings to new vacua, which will be

discussed below.) Therefore, it appears that we need not only a shift from number

averaging (volume weighting) to spatial density averaging (volume averaging), but

that we also need something else to suppress the divergence in the Boltzmann brain

contributions at infinite times.

In the scale-factor measure [53, 55, 56, 68], one puts a cutoff where the volume

of cross sections of certain sets of timelike geodesics reaches some upper limit. This

avoids the divergence one would get from number averaging (volume weighting)

without a cutoff, in which the volume and the number of observations would go

to infinity as the time is taken to infinity. By putting a cutoff on the volume, the

scale-factor measure gives results rather similar to spatial density averaging (volume

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averaging), because for exponentially expanding universes, most of the observations

occur near the cutoff where the volume is fixed, so that the total number of observa-

tions is essentially proportional to the spatial density of observations (up to a factor

for the amount of time over which most of the observations occur before the cutoff,

which is inversely proportional to the Hubble expansion rate if it is constant).

If the universe is dominated by a positive cosmological constant at late times,

geodesics other than those that stay within bound matter configurations expand in-

definitely at an asymptotically exponential rate, so that all such geodesics eventually

reach the cutoff. Then if the contributions within the bound matter configurations,

where the geodesics are not cut off, do not dominate, then one only gets a finite

set of observations of each type, and one can apply either frequency averaging or

number averaging. (One might expect the matter in bound matter configurations

eventually to decay away, so that one does not have to worry about the timelikegeodesics there that would never expand to the cutoff if the matter configuration

persisted.)

The usual answer that one gets is that if the universe tends to a quasi-stationary

eternal inflation picture in which new bubbles are forming and decaying at an asymp-

totically fixed rate, and if the cutoff is applied at a sufficiently great volume, then

the precise value at which it is applied does not matter [53, 55, 56, 68]. Furthermore,

if Boltzmann brains do not form at all in the longest-lived de Sitter vacuum, and if

the tunneling rate to new bubbles that lead to more ordinary observers is greater

than the rate for Boltzmann brains to form in all anthropic vacua, then ordinary ob-servers dominate over Boltzmann brains [55, 56]. (See analogous restrictions [86, 84]

for other measures that unfortunately I do not have time to discuss in detail in this

paper.) Although it is not implausible that this latter requirement may be satisfied

in a large fraction of anthropic vacua [104], it does seem quite stringent for it to be

satisfied in all of them if there are an exponentially large number. Therefore, the

scale-factor measure still has a potential Boltzmann brain problem.

Another somewhat undesirable feature of the scale-factor measure (at least from

my viewpoint, though others like Bousso disagree [107]) is that it is important in

this approach that there be a cutoff, which is crucial for defining the ensemble of

observations. It has even been noted [74] that using this cutoff, Eternal inflation

predicts that time will end.

In the scale-factor measure, it is not specified precisely where the cutoff is to

be imposed (at what volume, relative to some initial volume of each set of timelike

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geodesics), but it is just pointed out that the resulting observational probabilities

appear to be insensitive to the value of the cutoff so long as it is sufficiently late (or

large). However, for a precise theory with a cutoff, one might like a precise cutoff

(though the procedure of taking the cutoff to infinity does appear to give precise

statistical predictions). It then seems to me a bit ad hoc to have the cutoff at some

particular very late (or large) value, as seems to be necessary with the scale-factor

cutoff, unless one can really understand what it means for the cutoff to be taken

to infinity. (What simple explanation could be given for the very large value of the

cutoff time if it is finite, and on the other hand, what can be meant by a cutoff if it

is at infinite time?)

Here I am proposing to replace the scale-factor cutoff that has an unspecified

or infinite late value with a particular simple explicit weighting factor to suppress

the measures for late-time observations, such as Boltzmann brains, in a preciselyspecified way. The idea is to supplement the spatial density averaging (volume av-

eraging), which greatly ameliorates the Boltzmann brain problem, with a measure

over time that integrates to a finite value over infinite time. The measure over time

is chosen to be dt/(1 + t2), where t is the proper time in Planck units, which is the

simplest analytic weighting I could think of that gives a convergent integral over

time. Since the curve y = 1/(1 + x2) is named the witch of Agnesi, I shall call

this Agnesi weighting. (The witch of Agnesi was named after the Italian linguist,

mathematician, and philosopher Maria Gaetana Agnesi, 1718-1799, after a misiden-

tification of the Italian word for curve with the word for woman contrary toGod, so that it was mistranslated witch.)

With an appropriate quantum state, this Agnesi weighting appears to be con-

sistent with all observations and in particular avoids the potential Boltzmann brain

problem remaining with the scale-factor cutoff if not all anthropic vacua give tunnel-

ing rates (eventually leading to new anthropic bubbles and new ordinary observers)

greater than the rate of Boltzmann brain formation [86, 84, 55, 56, 104]. It also ap-

pears to avoid problems that other measures have with zero or negative cosmological

constant [77], as shall be discussed below.

Probabilities of observations with Agnesi weighting

In this paper I shall use a semiclassical approximation for gravity, since I do not

know how to do Agnesi weighting in full quantum gravity. Assume that the quantum

state corresponds to a superposition of semiclassical spacetime geometries. Further

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assume that the postulated operators whose expectation values give the measures

for observations commute approximately with the projection operators to the semi-

classical geometries, so that for the measures one can regard the quantum state as

if it were an ensemble of 4-geometries with probabilities p(4g) given by the absolute

squares of the amplitudes for each geometry. There is no guarantee that this ap-

proximation is good, but here I shall make it for simplicity. Perhaps later one can

go back and look at refinements, though it may be hard to do that without knowing

more about the postulated operators.

I shall assume that each semiclassical 4-geometry has a preferred fiducial Cauchy

hypersurface. In a standard big-bang model, this could be the singular surface at

the big bang. In my Symmetric-Bounce model for the universe [108], which I have

argued is more predictive, the fiducial hypersurface would be the hypersurface in

which the semiclassical geometry has zero trace of the extrinsic curvature. (In this

model, to semiclassical accuracy, the entire extrinsic curvature would vanish on this

hypersurface of time symmetry.) If one had a different semiclassical model in which

there is a bounce rather than a singular big bang, the fiducial hypersurface could

be the hypersurface of zero trace of the extrinsic curvature, the one that minimizes

the spatial volume if there are more than one such extremal hypersurfaces. It shall

be left to the future to extend Agnesi weighting to semiclassical spacetimes that do

not have such preferred fiducial Cauchy hypersurfaces, but one might assume that

the quantum state of the universe, if indeed it leads to semiclassical spacetimes at

all, would lead to semiclassical spacetimes having such preferred fiducial Cauchy

hypersurfaces. Or, one could just restrict attention to such quantum states.

Then for each point of the spacetime, I shall choose the simplest choice of a

time function t, the proper time of the longest timelike curve from that point to the

fiducial hypersurface. This will be a timelike geodesic intersecting the hypersurface

perpendicularly. If there are two sides to the hypersurface, as in my Symmetric-

Bounce model, arbitrarily take t positive on one side of it (its future) and negative

on the other side of it (its past). Take the preferred foliation of the spacetime given

by the hypersurfaces of constant t.

These foliation hypersurfaces may have kinks where one goes from one region of

the spacetime with one smooth congruence of timelike geodesics that maximize the

proper time to the fiducial hypersurface to another region with a discontinuously

different smooth congruence of geodesics, but they are spatial hypersurfaces, not

only locally but also globally in the sense that they are acausal, with no points

on them being null or timelike separated. It is easy to see that they are semi-

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spacelike or achronal (no points timelike separated [109]), since if any point p on

a foliation hypersurface (say to the future of the fiducial hypersurface; replacing

future with past everywhere gives the same argument for the opposite case)

were to the future of any other point q on the foliation hypersurface by a positive

proper time , a timelike curve from p back to the fiducial hypersurface could go

through q and hence be longer by than the longest timelike curve from q, in

contradiction to the assumption that each point on the hypersurface has the same

maximal proper time t back to the fiducial hypersurface. With a bit more work, one

can also get a contradiction if any point p on the foliation hypersurface were to the

null future of any other point q on the foliation hypersurface (so that there exists

a future-directed null curve from q to p, but no timelike curve), since one could

perturb the null curve going back from p to q to a timelike curve going back from p

to a point r on the longest timelike curve from q back to the fiducial hypersurface,

and this timelike curve can be chosen to have proper time from p back to r longer

than the longest timelike curve from q back to r.

Let V(t) be the spatial 3-volume of each such foliation hypersurface, at a maximal

proper time t to the future or to the past of the fiducial Cauchy surface. I shall

assume that V(0) at t = 0 is a local minimum of the spatial volume, the fiducial

hypersurface which can have V(0) > 0 in a bounce model or V(0) = 0 in a big bang

model. The proper 4-volume between infinitesimally nearby hypersurfaces of the

foliation is dV4 = V(t)dt.

In a WKB approximation to the Wheeler-DeWitt equation for canonical quan-

tum gravity, the absolute square of the wavefunctional for the hypersurfaces inte-

grated over an infinitesimal sequence of hypersurfaces in a foliation is proportional

to the conserved WKB flux multiplied by the infinitesimal proper time between hy-

persurfaces [110], so here I shall take the quantum probability for the hypersurface

to be one of the foliation hypersurfaces between t and t + dt to be p(4g)dt. Note that

for semiclassical 4-geometries that have t running to infinity, the integral of the ab-

solute square of the wavefunctional diverges when integrated over the hypersurfaces

corresponding to all t. This fact also suggests the need to put in a weighting factor

or do something else to get finite observational probabilities out from a quantum

state of canonical quantum gravity.

Let us assume that the semiclassical spacetime 4g gives a spacetime density

expectation value nj(t, xi) for the observation Oj to occur at the time t and spatial

location xi. Let nj(t) be the spatial average ofnj(t, xi) over the spatial hypersurface.

Then the expected number of occurrences of the observation Oj between t and t + dt

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is dNj = nj(t)V(t)dt. If we were doing number averaging (volume weighting),

we would seek to integrate dNj over t to get a measure for the observation Oj

contributed by the semiclassical geometry 4g if it were the only 4-geometry. However,

if t can go to infinity, this integral would diverge. If V(t) grows exponentially with

t, it would still diverge even if we included the weighting factor 1/(1 + t2). One

would need an exponentially decreasing weight factor (with a coefficient of t in

the exponential that is greater than the Hubble constant of the fastest expanding

vacuum in the landscape) to give a convergent integral if one just used a function of

t with number averaging. Such a rapidly decaying weight factor would lead to the

youngness problem [38].

Things are much better if we use spatial density averaging (volume averaging),

which divides dNj by V(t) to get nj(t)dt, the spatial average of the density of the

observation Oj multiplied by the proper time dt. If we then combine this spatial

density averaging over the spatial hypersurfaces with Agnesi weighting for the time,

we get that the semiclassical 4-geometry 4g contributes

nj(t)dt/(1 + t2) to the

measure for Oj. Next, we sum this over the quantum probabilities of the 4-geometries4g to get the relative probability of the observation Oj as

pj =4g

p(4g)

nj(t)dt

1 + t2=4g

p(4g) dNj

dt

1

V(t)

dt

1 + t2. (4)

Here, of course, the expectation value of the spatially averaged density nj(t) of the

observations Oj, and thus also the expectation value of the rate of observations

per time dNj/dt, depend implicitly on the 4-geometry4g, and by a semiclassical 4-

geometry I am including the quantum state of the matter fields on that 4-geometry,

on which the expectation value nj(t, xi) and hence nj(t) and dNj/dt are likely to

depend, as well as on the 4-geometry itself.

Finally, we get the normalized probabilities for the observations Oj by dividing

by the sum of the unnormalized relative probabilities pj:

Pj =pjk pk

. (5)

It may be noted that the weighting factor 1/[V(t)(1 + t2)] in the last expres-

sion of Eq. (4) from spatial density averaging and Agnesi weighting is essentially a

nonuniform xerographic distribution in the language of Srednicki and Hartle [63, 64].

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Consequences of Agnesi weighting for our universe

The combination of spatial density averaging (volume averaging) and Agnesi weight-

ing (so that the expectation value dNj/dt of the number of observations Oj per

proper time is divided by both the 3-volume V(t) of the hypersurface and by theAgnesi factor 1 + t2) avoids all divergences (assuming that there is a finite upper

bound on the spatial density of observations, as seems highly plausible). The re-

sults appear to be better than several other recent measures, such as the scale-factor

measure [53, 55, 56, 68], the closely related fat-geodesic measure [55], the causal-

patch measure [29, 60, 71], the stationary measure [43, 59], and the apparent-horizon

measure [77].

In particular, Agnesi weighting solves the Boltzmann brain problem without

having to assume that all anthropic vacua give tunneling rates to new vacua greater

than the rate of Boltzmann brain formation [86, 84, 55, 56, 104]. The spatial densityaveraging avoids the domination of Boltzmann brains on individual hypersurfaces, no

matter how large, and the Agnesi factor suppresses the the cumulative contributions

of the arbitrarily many hypersurfaces that occur at very late times.

Agnesi weighting also avoids the potential problem with possible Boltzmann

brains in a supersymmetric 11-demensional Minkowski vacuum. Initially, Boltz-

mann brains were postulated to arise by thermal fluctuations in asymptotically de

Sitter spacetime [78, 79, 80], but then I pointed out that if observations are given by

the expectation values of localized operators (e.g., a weighted sum of localized pro-

jection operators to have a particular brain configuration in a finite region of space),Boltzmann brain observations should also occur with positive probability even in

the vacuum [81, 85, 87, 89]. This would particularly appear to pose a problem if the

states in the landscape can tunnel to a supersymmetric 11-demensional Minkowski

vacuum, since it would have an infinite volume in which Boltzmann brains might

form. (Even if one thought that the fields corresponding to the normal excitations of

this vacuum were unable to support observations, surely there would be some posi-

tive expectation values for a finite region to have the right fluctuations of whatever

fields are necessary to give observations [111].)

The other measures mentioned above do not seem to suppress the contributions

of such Boltzmann brains. For example, the scale-factor measure could have the

congruence of geodesics entering into the Minkowski vacuum with arbitrarily small

divergence, in which case without the repulsive effects of a cosmological constant,

they can go arbitrarily far into the Minkowski spacetime, and sample an arbitrarily

large 11-volume, before they reach the scale-factor cutoff. Similarly, the causal-patch

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measure could be dominated by Boltzmann brains in an arbitrarily large causal patch

corresponding to a geodesic that lasts infinitely long in the Minkowski vacuum.

Of course, it is not absolutely certain that Boltzmann brains do form in the

vacuum [106], and recently Edward Witten told me [112] he did not believe that

Boltzmann brains form in a vacuum, where information processing and dissipation

do not occur. Despite this expert opinion, it is still hard for me to be convinced that

localized observations would not occur by purely vacuum fluctuations. If they can,

it is encouraging that Agnesi weighting would explain why they do not dominate,

even if other measures do not.

The other measures mentioned above also appear to have problems with vacua

having negative cosmological constant [77], which tend to dominate the probabilities

of observations and hence would make our observation of a positive cosmological

constant highly improbable. For the scale-factor and fat-geodesic measures, this is

because the geodesics can go for a very long proper time in a vacuum with a very

tiny negative cosmological constant before reaching the scale-factor cutoff or the big

crunch. For the causal-patch measure, it is because the causal patch can be very

large in a region with a very small negative cosmological constant. This is essentially

the same problem that arises with those measures for the Minkowski vacuum, except

that here one gets the domination by ordinary observers in excitations of vacua with

negative cosmological constants rather than by Boltzmann brains in the Minkowski

vacuum.

On the other hand, since the Agnesi weighting damps late times whether or not

geodesics are exponentially diverging or whether or not a causal patch has a bounded

spatial size, it suppresses the late-time contributions of not only the Minkowski

vacuum but also all vacua with positive or negative cosmological constant. So long

as the quantum state does not strongly favor negative values of the cosmological

constant, there is nothing in Agnesi weighting that would favor them either, so there

is no statistical conflict with our observations of a positive cosmological constant.

The youngness effect

The combination of spatial density averaging (volume averaging) and Agnesi weight-

ing does lead to a very mild youngness effect, because the expectation value dNj/dt

of the number of observations Oj per proper time is divided by both the 3-volume

V(t) of the hypersurface and by the Agnesi factor 1 + t2. This tends to favor obser-

vations early in the universe, so let us see how great a youngness effect it gives, say

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between the origin of the solar system and a time equally far in the future, near its

expected demise.

Let us use what I call the Mnemonic Universe Model (MUM, which itself might

be considered a British term of endearment for Mother Nature) for the universe, a

spatially flat universe dominated by dust and a cosmological constant, with present

age t0 = H1

0 = 108 years/, where 1/137036000 [113] is the electromagnetic

fine structure constant, and with the solar age t0/3. The present observations give

a universe age of 13.69 0.13 Gyr [113] that is 0.999 0.009 times the MUMvalue of 13.7036 Gyr, a Hubble constant of 72 3 km s1 Mpc1 [113] that is1.009 0.042 times the MUM value of 71.3517 km s1 Mpc1, and a solar systemage of 4.5681 0.0003 Gyr [114] that is 1.00005 0.00007 times the MUM valueof 4.56787 Gyr. Thus the MUM values are all within the present observational

uncertainties for the universe age, Hubble constant, and solar system age.

The metric for the MUM model is

ds2 = dt2 + sinh4/3(1.5Ht)(dx2 + dy2 + dz2), (6)

where H =

/3 is the asymptotic value of the Hubble expansion rate

H =a

a= H coth (1.5Ht). (7)

For t0 = H1

0 , we need Ht0 = tanh(1.5Ht0) or Ht0 0.858560, and thent0 = 10

8years/ gives H

(15.96115Gyr)1

61.2597 km s1 Mpc1. One can

also calculate that the MUM predicts that at present the dark energy corresponding

to the cosmological constant gives a fraction of the total (closure) energy density

that is = tanh2 (1.5Ht0) = (Ht0)

2 0.737125, in good agreement with theobservational value of 0.74 0.03 [113] that is 1.004 0.041 times the MUM value.

Some features of the MUM are that with the conformal time that is given by

=t0

dt/ sinh2/3(1.5Ht), the total conformal time is 44.76088 Gyr, and the

present value of the conformal time is 0 33.8825 Gyr 0.756967. (This isusing the normalization above that a(t) = sinh2/3(1.5Ht), which gives a0 a(t0) 1.41014; if one had instead set a0 = 1 so a(t) = sinh

2/3(1.5Ht)/ sinh2/3(1.5Ht0),

one would have =t0 dt

/a(t) giving 63.1193 Gyr and 0 47.7792 Gyr.)Thus we see that although there is only a finite proper time in the past and an

infinite proper time in the future, over three-quarters of the total finite conformal

time of the MUM has already passed.

The cosmological event horizon for the comoving observer at r =

x2 + y2 + z2 =

0 (which we shall take to be our worldline) is at r = , so on the constant-time

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hypersurface t = t0 (and hence = 0), it is at r = r1 = 0 10.8784 Gyr, ata distance along this hypersurface of a0r1 15.3401 Gyr (times the speed of lightc, which I am setting to unity; e.g., this distance is 15.3401 billion light years). The

actual spacetime geodesic distance from us to the point on the comoving worldline

at r = r1 that is crossing our cosmological event horizon when its proper time from

the big bang is the same as ours is 16.2282 Gyr, greater than the distance along a

geodesic of the constant-time hypersurface, because geodesics of that hypersurface

are not geodesics of spacetime but instead bend in the timelike direction, shortening

their length. The actual geodesic of spacetime joining the two events goes forward

in the time t from t0 to t 1.17686t0 16.1272 Gyr, to a point with a 1.18725a0,before bending back in t to get back to t0 at the cosmological event horizon.

Like de Sitter spacetime with the same value of the cosmological constant, the

MUM has a maximal separation of two events connected by a spatial geodesic, which

is /H 50.1434 Gyr. All events with r 2 0 cannot be reached by anygeodesics from our location in spacetime. The events on this boundary at t = t0are at r = r2 = 2( 0) 21.7567 Gyr, which is at a distance of 30.6802 Gyralong the t = t0 hypersurface, though the geodesic distance is the maximal value

of 50.1434 Gyr. (Actually, there is no geodesic to this boundary itself, but this

maximal value is the limit of the geodesic distance as r approaches the boundary.)

A third preferred distance on the t = t0 hypersurface of homogeneity is at r =

r3 = 0 33.8825 Gyr, which is where a comoving worldline that started at thebig bang on our past light cone reaches after the same proper time t0 from the big

bang as we are. That is, this is the present location of a worldline which started atour particle horizon. This value of r corresponds to a physical distance along this

hypersurface of 47.7792 Gyr. There are no geodesics from us to that point, so even

if we had a tachyon gun, we could not hit that worldline at a point on it after its

proper time passed our value of t0.

The MUM also allows on to calculate the geodesic distance from us to each

of these three worldlines along a geodesic that is orthogonal to our worldline at

its intersection here and now. This distance to r = r1 (the comoving worldline

that crosses our cosmological event horizon at a proper time of t0) is 11.3244 Gyr,

to r = r2 (the worldline that after proper time t0 reaches the boundary of wheregeodesics from us can reach) is 14.3274 Gyr, and to r = r3 (the worldline that starts

at the big bang on our past light cone) is 14.6863 Gyr. This spacelike geodesic never

reaches our cosmological event horizon but instead ends at the big bang at a distance

of 14.6889 Gyr from us, where r = r4 41.0459 Gyr (or a0r4 57.8806 Gyr for thedistance along the t = t0 hypersurface to the comoving worldline with r = r4), which

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is less than the value r = r5 44.7609 Gyr where our cosmological event horizonintersects the big bang, whose comoving worldline is at a distance a0r5 63.1193Gyr from us along the t = t0 hypersurface. That is, if we define simultaneity by

spacelike geodesics orthogonal to our worldline, the big bang is still going on right

now [115], at a distance of 14.7 billion light years from us in the Mnemonic Universe

Model.

Yet another comoving worldline that one may define is the one that emerges

from the big bang from the boundary of the region that can be reached from us

by spacetime geodesics. This is at r = r6 = 2 0 55.6393 Gyr, which asmeasured along the t = t0 hypersurface is at the distance a0r6 78.4594 billionlight years from us. This is the upper limit to the current distance (over a constant-

time hypersurface, not along a spatial geodesic of spacetime that has a maximum

length of 50.1434 billion light years in the MUM) of any comoving worldline that

can be reached by any geodesics from our current location in spacetime. The limitof the spatial geodesics that reach from us to comoving worldlines as r r6 is anull geodesic that goes from us to the spacelike future boundary at = and

then returns to the big bang along another null geodesic; the spacelike geodesics

approaching this limit approach the maximum spacelike geodesic length of 50.1434

billion light years, this length occurring in the de Sitter region in the arbitrarily

distant future where the spatial geodesic turns around from going toward the future

in t to going back toward the past in t.

Now let us use the MUM to calculate the youngness effect from the formation

of the solar system, at a time t0/3 before the present, or at t = 2t0/3 after the bigbang, to a time equally equally far in the future, at t = 4t0/3, which we shall use as a

very crude approximation for the mnemonic demise of the solar system. Since both

of these times are enormously longer than the Planck time (with t0 = 8.021 1060in Planck units), we can drop the 1 that is included in the Agnesi weighting to avoid

a divergence at t = 0. Then we see that on a per-time basis, the Agnesi weighting

factor of 1/(1 + t2) is four times smaller at the demise of the solar system than

at its formation. However, the spatial volume of the universe also goes up by a

factor of 7.75 during this lifetime of the solar system, so if we had a fixed comoving

density of observers, the combination of the Agnesi and spatial density averaging(volume averaging) factors would give about 31 times the weight for observations at

the formation of the solar system than at its end.

This would imply that if the same number of observations occurred per proper

time and per comoving volume throughout the lifetime of the solar system, the ones

at the demise would have only about 3% of the measure of the ones at the formation.

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Half of the measure would occur within the first 18% of the solar system lifetime.

This effect would tend to favor observations early in the history of the solar system.

However, it seems highly plausible that a factor of only about 31 would be neg-

ligible in comparison with the factors that determine the numbers of observations.

Presumably if one sampled a huge number of solar systems, only a very tiny frac-

tion of the observations would occur very close to the formation, because of the

time needed for evolution. If the probability for evolution to intelligent life to have

occurred rises sufficiently rapidly with the time after the formation time (e.g., signif-

icantly faster than the linear rise one would expect if evolution were a single event

that occurred statistically at a constant rate per time per solar system), then it

would not be at all surprising that we exist at a time when 85% of the measure

would have passed if the number of observations were instead uniform in time.

The shift of the measure (say calibrated for a fixed comoving density of observers

making a constant number of observations per time) from being uniform in thetime to having the weighting factors of the inverse three-volume (from the spatial

density averaging) and of very nearly the inverse square of the time (from the Agnesi

weighting) would have an effect on the number of hard steps n Brandon Carter

estimated for the evolution of intelligent life on earth [116, 117]. A hard step (or

critical step in the first of these papers) is one whose corresponding timescale is at

least a significant fraction of the available time for it to occur (e.g., the lifetime of

the sun). Carter emphasized [116] that unless there is an unexplained (and therefore

a priori improbable) coincidence, the timescale of a step is not likely to be close to

the available time, so generically a hard step has a timescale much longer than theavailable time. Therefore, a hard step is unlikely to occur within the available time

on a random suitable planet in which the previous steps have occurred.

In the first of these papers [116], Carter assumed that since we are about halfway

through the predicted lifetime of the sun, we arose about halfway through the life-

permitting period on earth and about halfway through the measure if the measure

were uniform in time. He then concluded that the number of hard steps n would

likely be 1 or 2. In the second paper [117], Carter used more recent information

[118] that the sun may become too luminous for life to continue on earth just one

billion years in the future rather than five. Then we would be a fraction f 5/6 ofthe way through the available period for life, and this would lead to an estimate for

the number of hard steps to be n f /(1 f) 5. (Carter suggested 4


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assumptions of Carters original paper, that the available time is the entire solar

lifetime and that we are halfway through it, without any measure factors f would

be 0.5, but with my measure factors this fraction would be changed to f = 0.85,

which would then give n

6 even without the natural global warming effects of

rising solar flux. On the alternative assumption that there is only one gigayear left

for life on earth, my measure factors change Carters f = 5/6 to f = 0.94 and hence

give the number of hard steps as n f /(1 f) 16.Therefore, if we could really learn what the number of hard steps were for the

evolution of intelligent life here on earth, we could in principle test between different

proposals for the measure, such as between the scale-factor measure and my proposed

spatial density averaging with Agnesi weighting. However, this currently seems like

a very hard problem. (Would it be another hard step for intelligent life to solve

it?) All I can say at present is that it does not seem obviously in contradiction with

observations that the number of hard steps might be higher than Carters estimates,so our present knowledge does not appear to provide strong evidence against the

proposed spatial density averaging and Agnesi weighting.

Conclusions

Agnesi weighting gives a precise weighting factor that may be an improvement over

the indefinite cutoff proposed by other proposals, such as the scale-factor measure

[53, 55, 56, 68]. Unlike what occurs in the latter, in which time comes to a sharp end

at an unspecified time [74], in Agnesi weighting old universes never die, they just

fade away. This fading away is purely in the measure for the various observations

and not in any property of the contents of the observations themselves (e.g., of the

observed spectrum of the CMB, or of how painful a toothache feels), so it cannot be

directly observed. However, if one did have an observation that a theory with this

fading said would have excessively low measure, that would be statistical evidence

against that theory. One can make similar statistical interpretations of the idea that

time ends abruptly at an unspecified time, so that difference by itself is a matter of

the assumed ontology rather than of different testable statistical predictions.

When combined with number density averaging (which I previously called volume

averaging [1, 2, 3, 4, 5]) and with a suitable quantum state for the universe (such

as the Symmetric-Bounce state [108]), Agnesi weighting gives a finite measure for

observations in the universe and appears to avoid the Boltzmann brain problem

17


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and other potential problems of cosmological measures, even without restrictions on

the decay rates of anthropic vacua used to solve the Boltzmann brain problem in

other measures [86, 84, 55, 56], even allowing for Boltzmann brains to form in the

11-dimensional Minkowski vacuum that states in the string landscape may lead to

[81, 85, 87, 89, 111], and even allowing vacua with negative cosmological constants

that tend to dominate the probabilities in other measures [77]. Agnesi weighting

leads to a very mild youngness effect, but one which is well within the current

uncertainties of how rapidly intelligent life is likely to have evolved on earth.

Phenomenologically, Agnesi weighting appears to work well. However, it is surely

not the last word on the subject. For one thing, although it is quite simple, it is

rather ad hoc (like all other solutions to the measure problem proposed so far, at

least in my mind), so one would like to learn some principle that would justify it or

an improvement to it. Second, it is presently formulated only in the semiclassical

approximation to quantum cosmology, so one would want a fully quantum version.

These challenges will be left for future work.

Acknowledgments

The idea for this paper came while I was visiting James Hartle, Donald Marolf,

Mark Srednicki, and others at the University of California at Santa Barbara in

February 2010. I am especially grateful for the hospitality of the Mitchell family

and of the George P. and Cynthia W. Mitchell Institute for Fundamental Physics

and Astronomy of Texas A&M University at a workshop in April 2010 at Cooks

Branch Conservancy, where I had many more valuable discussions on this subject,

especially with James Hartle, Stephen Hawking, and Thomas Hertog. I was further

stimulated by discussions with many colleagues at Peyresq Physics 15 in June 2010

under the hospitality of Edgard Gunzig and OLAM, Association pour la Recherche

Fondamentale, Bruxelles. Comments by Raphael Bousso, both by email and while

I was enjoying his hospitality at the University of California at Berkeley in Febru-

ary 2011, were instrumental for various revisions of the paper, as well as further

discussions with James Hartle, Stefan Leichenauer, Vladimir Rosenhaus, and Ed-

ward Witten. This research was supported in part by the Natural Sciences and

Engineering Research Council of Canada.

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