Anthony Aguirre, Steven Gratton and Matthew C Johnson- Hurdles for Recent Measures in Eternal Inflation

8/3/2019 Anthony Aguirre, Steven Gratton and Matthew C Johnson- Hurdles for Recent Measures in Eternal Inflation

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arXiv:hep-th/0611221v22

3Mar2007

Hurdles for Recent Measures in Eternal Inflation

Anthony Aguirre,1, Steven Gratton,2, and Matthew C Johnson1,

1SCIPP, University of California, Santa Cruz, CA 95064, USA2Institute of Astronomy, Madingley Road, Cambridge, CB3 0HA, UK

(Dated: February 2, 2008)

In recent literature on eternal inflation, a number of measures have been introduced which at-tempt to assign probabilities to different pocket universes by counting the number of each type ofpocket according to a specific procedure. We give an overview of the existing measures, pointing outsome interesting connections and generic predictions. For example, pairs of vacua that undergo fasttransitions between themselves will be strongly favored. The resultant implications for making pre-dictions in a generic potential landscape are discussed. We also raise a number of issues concerningthe types of transitions that observers in eternal inflation are able to experience.

I. INTRODUCTION

In eternal inflation, different post-inflationary regionsmay have different properties. How even in princi-ple to statistically describe these properties so as tomake probabilistic cosmological predictions is a major

outstanding problem in current cosmology. Recently,a number of proposals have been advanced for gauge-independent measures that do not depend on the choiceof a time coordinate [1, 2, 3, 4]. In this note we compare,contrast, and assess the existing proposals, and point outsome predictions that they seem to share. We focus hereon eternal inflation as driven by a potential with multi-ple minima; transitions between these correspond to thenucleation of bubbles or pocket universes containinga new phase of different vacuum energy [5, 6]. If tran-sitions are sufficiently slow, the growing bubbles neverpercolate, and inflation is eternal.

A form of predictions in a multiverse is a set of state-ments such as The probability that a randomly chosenX is in a region with properties is PX(), where X issome conditionalization object such as a point in space,a baryon, a galaxy, or an observer that arguably makesPX relevant to what we will actually observe in some fu-ture experiment (see, e.g., [7, 8]). This probability isgenerally split into two components:

PX() Pp()nX,p(). (1)

Here, Pp is a prior probability distribution defined interms of some type of object p regardless of the condi-tionalization object X, and is a vector of propertieswe might hope to compare to locally observed properties

of our universe. For example, if p=pocket universethen Pp() describes the probability that a randomlychosen bubble has low-energy observable properties .The factor nX,p conditions these probabilities by the re-quirement that some X-object exists; for example with

Electronic address: [email protected] address: [email protected] address: [email protected]

X=galaxy, nX,p() might count the (-dependent)number of galaxies in a pocket with properties .

The measures discussed here are proposals for calculat-ing Pp (though we will also discuss some relevant issuesconcerning nX,p). We shall see that many of the mea-

sures share some properties for example, they all accordvery high probability to regions in a potential landscapewhich allow for very rapid transitions between nearbyminima. Unfortunately, these regions of the landscapelook nothing like our universe: the resulting spacetimeswould almost certainly be dominated by a very high vac-uum energy and be devoid of structure. This, of courseis nothing new the whole idea of the anthropic ap-proach to explaining our observed universe is that nX,p,where X=observer, will unweight such states. Butwe shall see that employing the measures under consid-eration makes the problem very acute.

More generally, while the measures we discuss are allstated and formulated in rather different ways, many ofthem are, in fact, either fully or partially equivalent (asacknowledged by the authors in some cases); we will at-tempt to sort out these relations comprehensively. Differ-ences do exist however, and we will also see that certaindesirable properties hold in some measures and notothers.

In Section II, we present a scorecard of features thatmight be desirable in a measure, and we summarize anumber of recent measures and the connections betweenthem. We then compute the prior distribution for a num-ber of sample landscapes in Sec. III and use the resultsto highlight important predictions and connections. Theimplications of these predictions are discussed in Sec. IV.Sec. V describes some problems associated with the as-sumptions usually made about the global picture of aneternally inflating spacetime, and we conclude in Sec. VI.In appendix A, a quick matrix method for calculatingbubble abundances is introduced and a number of themore technical results of the paper are derived.
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II. MEASURE DESIDERATA AND PROPOSALS

A. Desirable measure properties: a scorecard

To test a theory of eternal inflation yielding diversepost-inflationary predictions, we would like to knowwhat physical properties are most likely, and comparethem to our local observations. This question, however, issimply ambiguous any answerable version of this ques-tion will entail a tacit choice of a conditionalization Xand calculation ofPX as described above. The measureswe will discuss correspond to different attempts to (atleast implicitly) propose a plausible candidate for X, andto calculate the prior distribution Pp that might be usedin calculating PX for that X.

A fundamental property that a well-defined measureshould have is that its answer should be gauge-invariant,by which we simply mean that its answer can be calcu-lated in any coordinate system we choose. This is distinctfrom gauge-independence as we shall discuss shortly.

Beyond this, it is important to consider what proper-

ties we might want a sensible measure to have. Some suchdesiderata, either stressed previously in the literature orfirst mentioned here, are given below. We note, however,that it is quite possible that the correct measure (if itexists) does not satisfy every item.

Physicality The p to which the measure applies,and the choice ofPp, should be such that (a) theprobabilities do not appear to have been pickedout of a hat, and (b) nX,p is plausibly calcula-ble. For example, we might choose p =vacuumand set Pp proportional to the tenth power of thehyperbolic tangent of the energy of the vacuum in

Planck units. However, (a) this measure is obvi-ously rather arbitrary, and (b) since there is nophysical process behind the creation of regions de-scribed by the different vacua, the measure seemsuseless in calculating nX,p for, say X=baryon.Note, however, that different physically reasonableconditionalization objects may require different Pp for example were X=vacuum, then the measurewould still violate condition (a), but would satisfycondition (b) by definition.

Gauge-independence The relative probabilitiesshould not depend on an arbitrary decompositionof spacetime into space and time. For instance, ithas been shown [9, 10, 11, 12] that measures thatweight based on the physical volume in a given stateat late times give a result that depends sensitivelyon the assumed foliation of spacetime into equal-time hypersurfaces. In the absence of a strongphysical reason for choosing a particular decompo-sition, such measures thus seem ambiguous.

Ability to cope with varieties of transitions andvacua The measure should be general enough to

treat all of the types of vacua (e.g. positive, nega-tive, or zero energy), and the various types of tran-sitions between them.

Independence of initial conditions It is often ar-gued that eternal inflation approaches a steady-state, and that essentially all observers exist atlate times, so a physically reasonable measure

should become independent of initial conditions.This criterion is not obviously necessary; althoughit may be appropriate for a particular condition-alization object (e.g. X=a randomly chosen ob-server), it may not be appropriate for others. Forexample, if one were interested in knowing whata given observer (or worldline) will experience inthe future, then a dependence on initial conditionsseems quite reasonable.

Ability to cope with various and/or varying topo-logical structures The measure should potentiallybe applicable to spacetimes with non-trivial topo-logical structures as may arise in eternal inflation(as discussed at length in Sec. V).

Accurate and robust treatment of states andtransitions this entails several sub-criteria:

General principles the basic ideas behind themeasure should allow it to be used (in the-ory) for the complicated spacetimes of land-scapes that cannot simply be encapsulated bytransition rates between vacua.

Physical description of transitions transi-tion rates must be clearly linked to the phys-

ical process that describes the transition (e.g.Coleman-De Luccia bubble nucleation).

Reasonable treatment of split states themeasure should deal properly with very sim-ilar states and/or very large transition rates.(For example, a vacuum split by the insertionof a small potential barrier should, in the limitof an infinitesimal barrier, act just as a singlevacuum.)

Continuity in transition rates When tran-sition rates are used, the measure should becontinuous in these rates. For example, thereshould be no discontinuity in the probabili-

ties between a stable vacuum and a metastablevacuum with a lifetime , in the limit .

We would argue that all of these potentially pleasingfeatures are absent in at least one measure proposal inthe literature, and that no extant proposal clearly fulfillsthem all. But the good news is that the bubble-countingprocedures discussed here satisfy many of them, so letus summarize these measures and provide a listing ofconnections between them.


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B. The Measures and their Properties

We now examine the various measures under consid-eration. All of these have subtleties, so we refer thereader to the original papers, and also to the review byVilenkin [13] and to the lectures of Shenker [14]. Here,we will mainly provide brief summaries, but will also addextended comments on some measures.

Restricting the discussion to eternal inflation as drivenby a potential with multiple minima, it is useful to clas-sify vacua as terminal or recycling: terminal vacuacan be reached, but never exited; recycling vacua canexit to the state from which they originated, and mayalso transition to other states. Following [1], we can alsolabel entire landscapes as terminal or recycling; the for-mer contain at least one terminal vacuum whereas thelatter do not.

As a first step in this analysis, we can divide the mea-sures into three categories: first, those that calculate vol-umes in different vacua on some equal-time surface; sec-ond, those that count individual bubbles; third, those

that focus on the vacua experienced by an observer fol-lowing a single worldline.There are two basic volume-counting methods, count-

ing either physical volume (i.e. p=unit of physical vol-ume) or comoving volume (p=unit of comoving vol-ume). See, e.g., [9, 10, 11, 12, 15] for the former; herewe focus on:

The Comoving Volume (CV) method: Put forwardby Garriga and Vilenkin [16], this method might beconsidered the counterpart for bubble nucleations(in comoving volume) to the work of Linde, Lindeand Mezhlumian [9] in stochastic inflation. Onestarts with some region on an initial spacelike sur-

face, and considers a congruence of hypersurface-orthogonal geodesics (the comoving observers)emanating from that region. As a function of someglobal time coordinate t, the number of worldlines(to which the comoving volume fraction is definedto be proportional) in different vacua is calculated.The probability, Pcv, to be in a given vacuum isthen defined to be proportional to the fraction ofcomoving volume (or number of worldlines), fi(t),in that pocket, in the t limit. Note that ifthere are terminal vacua, then as t all of thecomoving volume will be distributed among the ter-minal vacua, except for a set of measure zero (albeit

one that corresponds to infinite physical volume!).Metastable vacua are thus accorded zero weight.This measure depends heavily on initial conditions,because the fraction of comoving volume in a giventerminal vacuum can only increase with time [34].

The next two methods, rather than counting total rela-tive volume in different bubble types, count relative totalnumbers of bubbles, i.e. p=bubble.

The Comoving Horizon Cutoff (CHC) method: In

the proposal of Garriga et al. [3], the measure is de-fined by directly counting bubbles of a given phase.One has in mind performing the count at late times,or future infinity. We follow the most recent de-scription of this procedure as given by Vilenkin [13].First, just as in the CV method, a spacelike hyper-surface in the spacetime is chosen, and a congru-ence of geodesics is extended from this hypersur-

face. The geodesics are followed arbitrarily far intothe future, passing into any bubbles they may en-counter. These lines are used to project bubbles inthe spacetime back onto the initial hypersurface ascolored shadows. The relative frequency of bub-bles of different colors is defined to be the ratios ofthe numbers of their shadows on the initial hyper-surface. The shadows are very clumped, gatheringaround the rare regions where inflation continueslongest, with an arbitrarily large number of arbi-trarily small overlaid shadows surrounding the set(of measure zero) of points on the surface whereinflation continues forever. Thus, all counts areinfinite numbers and require regularization to bewell-defined. The authors propose only countingshadows larger than a size and then taking thelimit 0. This measure is argued to be indepen-dent of initial conditions on the surface and appliesto terminal and recycling vacua. It also has theimportant feature of giving metastable states non-zero weight. While the idea of counting bubbles atfuture infinity is intuitively clear, it is somewhatunclear that the shadow counting used to actu-ally implement the cutoff is particularly physical.

Moreover, converting this idea into an actual calcu-lation is a subtle matter. To date, such calculationshave been performed in a rate-equation frameworkin which one follows the fractions of comoving vol-ume in the various vacua and then effectively di-vides through by the bubble volume in order toobtain the bubble count. The shadow-size cutoffis then implemented by imposing a set of late-timecutoffs, one for each bubble type out of which thecounted bubbles are nucleated (on the assumptionthat this determines the comoving size of the nu-cleated bubbles, and thus the size of the shadow,

to which the cutoff applies). This cutoff, t()ij , for

transitions out of vacuum j into vacuum i, is givenby [3]

t()ij = ln (Hj) , (2)and is designed so that when bubbles intersectingthe cutoff surface are projected back onto the initialsurface, only bubbles of size exceeding will beobtained.

There are, however, some features of this calcu-lation that warrant a closer look. For example,the formalism allows situations in which a bub-ble formed soon after its parent can be assigned


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a larger asymptotic comoving size than the par-ent (we thank Alex Vilenkin for discussions of thispoint) and may therefore be included in the count-ing while its parent is not. It is, however, unclearif or how the nucleation of bubbles larger thantheir parent actually occurs, or what asymptoticsize should really be assigned to them. One mighthope that such events lead to a small error, but this

is not clear because the ratio of 4-volume betweenthe cutoff surfaces to the full 4-volume before thecutoffs may be large. Thus rather than the timeperiod between the cutoffs being unimportant, nu-cleations during this period may actually dominatethe bubble statistics. Details of this sort shouldserve to encourage the development of calculationaltechniques in which spacetime dependence is moreexplicitly taken into account.

The Worldline (W) method: Easther et al. [2],whose measure we denote the Worldline (W)method, assume that at some initial time (defined

by a spacelike hypersurface), the universe is in someplaces in a non-terminal vacuum. They then sug-gest considering a finite number of randomly cho-sen points on this initial data surface and follow-ing forward worldlines with randomly chosen veloc-ities [35] from these initial data points. Only bub-bles that are encountered by at least one of theseworldlines are counted in determining the relativebubble abundance (no bubble is counted more thanonce, even if multiple worldlines enter it). Onethen takes the total number of worldlines to in-finity. Like CHC, this measure is claimed to beessentially independent of initial conditions as longas inflation is eternal. It was argued in [3] that

the CHC and W methods of bubble counting yieldidentical answers for terminal landscapes (the Wmethod is ill-defined for fully recycling landscapesas discussed in [4]).

The remaining two measures focus on the transitionsbetween vacua experienced by a single eternal world-line, and accord a probability to a vacuum that is pro-portional to the relative frequency with which it is en-tered (p=segment of a worldline between vacuum tran-sitions).

The Recycling Transition (RT) method: The pro-posal of Vanchurin and Vilenkin [4], which we willrefer to as the Recycling Transition (RT) method,is to follow the evolution of a given geodesic ob-server and set the probability to be in a given vac-uum proportional to the frequency with which thisvacuum is entered, in the limit where the propertime elapsed goes to infinity. As presented, themethod only applies to landscapes with no termi-nal vacua, and was argued to be equivalent to theCHC method in that case [4].

The Recycling and Terminal Transition (RTT)method: The Bousso proposal [1], which we de-note the Recycling and Terminal Transition (RTT)method, covers the cases of terminal and recyclingvacua. Here, one chooses an initial condition forthe worldline (the predictions of this measure aredependent on initial conditions), and considers therelative probabilities of the worldline entering var-

ious other vacua, averaging over possible realiza-tions. This is equivalent to the RT measure in thecase where there are no terminal vacua.

The focus in RTT on the worldline of an observeris presented as being motivated by holography andthe desire to only consider regions of spacetimethat an observer can signal to and receive sig-nals from (the causal diamond). However, thisviewpoint makes essentially no difference to themathematics and as mentioned below the timeaverage over histories for Boussos observer couldequally well be thought of as spatial averages overwidely-separated worldlines in any of the above ap-

proaches. A similar observation is made in [15].Of course, a holographic point of view might leadone to strongly disfavor further possible weightingfactors to apply such as volume weighting.

Although we will not treat them further, let us alsomention some other approaches to asking about predic-tions in eternal inflation. In [12], Tegmark advances asimple and direct possible answer to the question of therelative numbers of different vacuum regions: becauseeternal inflation should produce a countably infinite num-ber of each type of vacuum region, and because all count-able infinities are equal in the sense of being relatable bya one-to-one mapping, each vacuum should be assignedequal weight. In [17], the authors put a measure onthe space of classical FRW solutions to the Einstein plusscalar field equations. If this could be extended to allowfor quantum jumps analogous to bubble nucleations, itmight help address the distribution of vacua within andamongst solutions. In [18], the authors focus on historiesthat might be/might have been observed, in the contextof single-field inflation with a monotonic potential.

C. Relations between the measures

Although the methods, both in their motivation and intheir presentation here, have been categorized into vol-ume counting, bubble counting and worldline follow-ing, there are relations between them that cross thesedivisions, so that in fact there are actually very few es-sentially different measures under consideration.

Some of the relations between measures (as presentedby their authors) have been mentioned above (e.g. theequality of CHC and W for terminal landscapes, and theequality of CHC and RT for fully recycling landscapeswith no terminal vacua). More, however, exist.


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W

RTT

RT

CHC

CV

FIG. 1: A summary of the connections between the variousmeasures. Solid green lines indicate equivalence between themeasures for a terminal landscape. Dashed blue lines indicateequivalence in the case of a fully recycling landscape. Dashed-dotted red lines indicate that the measures assign the samerelative weights to terminal vacua.

In particular, the RTT method accords the same rel-ative probabilities to terminal vacua as does the CVmethod (though the methods differ for non-terminalvacua, which have zero probability in CV and nonzero

probability in RTT). To see this, consider a congruenceof comoving worldlines starting in some vacuum. Now,as t , every worldline that will eventually end up ina terminal vacuum will do so (by definition); moreover,each terminal vacuum will only be entered once (also bydefinition). Since RTT accords relative probability totwo terminal vacua A and B equal to the relative prob-ability of a worldline entering them, this will be equalto the relative numbers of worldlines terminating in Aversus B, which is in turn equal to the relative t comoving volume fractions as defined in the CV method.In appendix A, we show this correspondence by directlycomparing the results of the RTT and CV methods inthe context of a specific model. More generally, the re-sults of the RTT method, for terminal as well as recyclinglandscapes, can be obtained by integrating the incomingprobability current into the various vacua [15, 19].

These relations between the measures (as formulatedin the original papers) are summarized in Fig. 1. It alsoappears possible to use what is understood about theseconnections to devise some hybrid or generalized versionsof the methods.

For example, take the CV procedure, where only asingle late-time hypersurface is considered, and attemptto count the number of bubbles intersecting this surfacefrom the volume distribution and some appropriately de-fined cutoff. This is not quite the CHC method since,as described above, the CHC calculation requires a dif-ferent time cutoff for bubbles formed in different parentvacua. But this CV-CHC hybrid prescription does notseem any less reasonable to us. One could also generalizethe CHC prescription to obtain an infinite number of re-lated measures by altering the limiting procedure: ratherthan only counting shadows larger than a size indepen-dent of the bubble type, one could instead only countshadows larger than a given size relative to, say, somefunction of their Hubble radius. That is, for bubbles of

B

V1

V

V

V

2

34

AB

ABB

BA

B Z

ZZ

Z

A

B

FIG. 2: Some sample landscapes. Potential V1 depicts theABZ example discussed by Bousso [1]. V2 splits the B vacuumby introducing a small barrier. Potential V3 lowers the A vac-uum to zero or negative energy, so that it becomes terminal.The potential V4 has a low energy minimum with high-energyneighbors that have short lifetimes (relative to other vacua inthe landscape).

type P, rather than only counting those that have shad-ows larger than on the initial surface, count those that

have shadows larger than

HP or

/HP say. This wouldcorrespond to replacing the time cutoff of ln(HM) forbubbles of type P forming out of bubbles of type Mwith ln(HMHP) or ln(HM/HP). It would be interest-ing to investigate how (in)sensitive the probabilities areto the choice of a particular cutoff procedure.

Having described the various bubble counting mea-sures and their connections, we now use a set of samplelandscapes to illustrate some of their predictions.

III. SOME SAMPLE LANDSCAPES

Consider the related one-dimensional landscapes pic-tured in Fig. 2. They all contain both terminal and re-cycling vacua (where we assume here that a vacuum isterminal if and only if its energy is zero or negative),and we now discuss the predictions made by the RTTmethod for each. In light of the close connections be-tween the measures, many of the conclusions drawn fromthese calculations will hold more generally.

Following Bousso, we define the relative probabilityNM to transition from vacuum M to vacuum N as

NM NMP PM

(3)

where P is summed over all decay channels out ofM, andNM is the probability per unit time of tunneling fromvacuum M to vacuum N. Note that all summations inthis paper are expressly indicated. NM typically takesthe form of a three-volume times a nucleation rate perunit four-volume, the latter being calculated using semi-classical instanton techniques. Note that

P PM = 1

ifM is metastable and PM = 0 ifM is terminal, andalso that MN = NM in general. Bousso introducesthe concepts of trees and pruned trees in order to calcu-late the prior distribution in the RTT method. He also


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presents a matrix formulation, which we develop furtherin appendix A.

It will be important for what follows to obtain anindication of the magnitudes of tunneling rates in atypical landscape. We model this landscape by a sin-gle scalar field with a potential V() expressed asV() = 4v(/m). We further assume that v is a smoothfunction that varies over a range of order unity as its ar-

gument changes by order unity, and sets the energyscale. For the semi-classical approximation that we areworking in to make sense, we must have 4 M4Pl, whereMPl is the Planck Mass. For ColemanDe Luccia instan-tons to exist, m must be less than some O(1) multiple ofMPl. See [20] for more on the motivation for this form ofthe potential.

As mentioned above, we will estimate tunneling ratesbetween the potential minima using semiclassical instan-ton techniques, notwithstanding thorny issues of inter-pretation, particularly for upward transitions. ThenNM e(S(NM )SM ), the bracketed exponential fac-tor being the difference between the action S(NM) ofthe Coleman-De Luccia or Hawking-Moss instanton link-ing the two vacua and the action SM of the Euclideanfour-sphere corresponding to the tunneled-from space-time. Note that the same instanton applies to uphilland downhill transitions (hence the use of symmetrisingbrackets in its label). Using the Euclidean equations ofmotion, S(NM) can be written as

S(NM) =

g V() d4x (4)

where the integral is performed over the Euclidean man-ifold of the instanton. The background subtraction term(which is negative and larger in magnitude than the in-stanton action) is given by the same expression and eval-uates to

SM = 3M4Pl

8V(M), (5)

where V(M) is the value of the potential of the pre-tunneling vacuum M at = M.

From these formulae we can immediately deduce twoimportant facts. First, we can compare uphill and down-hill rates between two vacua. In the ratio of the ratesthe instanton part cancels out, and only the backgroundparts are left. IfV(M) = V(N) + V, then

MN

NM exp 3M4Pl

8

V

V2(M) = exp 3

8

v

v2MMPl

4

.(6)

So, unless v is tuned to be much smaller than v, the up-hill rate is exponentially smaller than the downhill rate.

Second, we can compare the rates to two vacua N andP from the same parent vacuum M. This time the back-ground parts cancel and we are left with the exponentialof the difference of the instanton actions:

PMNM

exp (S(PM) S(NM)). (7)

Both instanton actions will be of order (MPl/)4, so we

typically expect the tunneling rates to differ exponen-tially. In particular, if VN and VM are somewhat atyp-ically similar and there is only a small barrier betweenthe two, then, as long as VP is not atypically close to VMalso, tunneling from M to P will be exponentially disfa-vored relative to tunneling to N. This holds even if thetunneling from M to N is uphill and that from M to P

is downhill. This difference in tunneling rates can be ex-treme: for a typical inflationary energy scale of 1016GeV, PM/NM e1012 .

A. Coupled pairs dominate in terminal landscapes

We begin by considering the potential V2 depicted inFig. 2. We assume that the barrier separating B and B

is very small, so that rapid transitions occur between thetwo wells. Thus we take BB AB and BB ZB .Using the results of appendix A, in the limit we obtain:

PA,B,B

A

PA,B,B

B

PA,B,B

B

PA,B,B

Z

BBABBBBBBBBBBBZB

(8)

where PMN is the prior probability of Eq. 1 (with sub-script p dropped) to be in the vacuum N, given an initialstate in vacuumM. A multiple superscript indicates thatthe same distribution applies to the listed initial statesfor the transition rates under consideration.

There are a number of interesting points to note here.First

PA,B,B

B

PA,B,B

A

=BBAB

1 (9)

PA,B,B

B

PA,B,B

Z

=BB

ZB 1. (10)

These ratios hold independent of initial conditions. Vac-uum B is similarly weighted relative to A and Z. Wetherefore see that (as might be expected in a measurethat counts transitions) metastable vacua participatingin fast transitions with their neighbors are weighted very

heavily. Such regions certainly exist in a landscape withsufficient complexity, and it is these regions that the priordistribution in the RTT method will favor. From ourabove estimates of typical transition rates in regimes withenergies somewhat below the Planck scale, factors of or-

der e1012

should be commonplace.Of course, arbitrarily fast transitions between B and

B (which give arbitrarily high weighting to both vacua)are unrealistic. In reality, bubble collisions will becomeimportant, and at high enough nucleation rates there will


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be percolation. In this limit, there should then be a tran-sition to a treatment in terms of field-rolling and diffu-sion. In this regard, it would be desirable to treat fielddiffusion as described by the stochastic formalism andbubble nucleation (with collisions taken into account) ina unified way (see [19] for work in this direction).

Although the CHC measure is inequivalent to the RTTmeasure in landscapes with terminal vacua, it (and hencethe W method) nevertheless gives similar qualitative pre-dictions. We can see this by analyzing the FABImodel of [3], which, in the limit where BB AB andBB ZB , gives the same ratios as Eqs. 9 and 10.Thus the CHC and W proposals weight fast-transitioningstates exponentially more than others in exactly the sameway the RTT method does. The weighting can easilybe large enough to dominate any volume factors, whichappear in the full probability defined using the CHCmethod [3], unless the number of e-folds during the slow-roll period after a transition is extreme.

We have seen that pairs of vacua undergoing fast tran-

sitions in both directions are weighted very heavily, butwhat about transitions that are fast in one direction only?For example, consider V4 in Fig. 2, where there are quicktransitions into B, but transitions out ofB are stronglysuppressed. Requiring only BB ZB in the proba-bility tables from appendix A yields:

PA,B,B

A

PA,B,B

B

PA,B,B

B

PA,B,B

Z

BBABBB (AB + BB)

BBBBBBZB

. (11)

It is apparent that vacuum B will be the most probablevacuum in this sample landscape. The relative weight ofA to B is very sensitive to the details of the potentialsince, as shown above, there is an exponential dependenceon the difference in instanton actions (which itself tendsto be quite large). In the absence of extremely fine-tunedcancellation in this difference (which would be requiredto make AB BB), one of the two will be vastly moreprobable than the other. We have already considered thecase where vacuum B is much more likely than vacuumA with landscape V2 above. So the other generic alterna-tive is for vacua A and B to have probabilities very close

to one-half, vacuum B

to be exponentially suppressedand vacuum Z to be even more suppressed.

These two examples together make it clear that in or-der to obtain the large weighting observed for potentialsV2 and V3, there must be pairs of vacua which undergofast transitions in both directions. This allows for closedloops that produce large numbers of bubbles of each ofthe vacua in the pair; in such cases the probabilities ofboth vacua scale with the product of the transition ratesbetween them.

B. Coupled pairs dominate in cyclic landscapes

As one might expect, the extreme weighting of cou-pled pairs persists if we raise the height of the Z well ofV2 in Fig. 2 so that it is no longer terminal. From thecalculations in appendix A, we find:

PA,B,B,Z

B

PA,B,B

,ZA

BB

AB (12)

PA,B,B,Z

B

PA,B,B,Z

Z

BBZB

(13)

with the same results for the ratios ofPB in place ofPBto PA and PZ . This is of special interest because for cycliclandscapes the predictions of the RTT method agree withthose of the CHC and RT methods (see Fig. 1). Thus allof these measures will weight rapidly transitioning vacuaheavily.

C. Splitting vacua

A closely related test to which we can put the RTTmethod to is to consider the situation where potentialV2 is obtained from potential V1 (The ABZ exampleof[1]) by inserting a small potential barrier in the middle(B) well. The ratio of weights in the A and Z wells inpotential V1 is given by:

PA,BAPA,BZ

=ABZB

, (14)

which can be found from the result of [1] by substituting = AB/ (AB + ZB) and 1 = ZB/ (AB + ZB ).Now let us insert the barrier in such a way that the tran-sition rates into and out of the A and Z wells remainunaffected. After the insertion, the relative weights ofvacuum A and Z (in potential V2) are then found fromEq. 8 to be

PA,B,B

A

PA,B,B

Z

=BB

BB

ABZB

. (15)

Now we can consider two cases. First, if there is no sym-metry as B is interchanged with B, then we see thatinserting the barrier has changed boththe absolute prob-abilities (which are now strongly weighted toward B andB), and also the relative weights of the other vacua. Sec-ond, if the problem is symmetric under interchange ofBand B (so that BB = BB and ZB = AB), thenthe relative weights ofA and Z are unaffected; however,the absolute weights of both are still altered drasticallyby this decomposition ofB into two identical vacua withfast transitions between them. This is somewhat disturb-ing, and again points to the need for a smooth connectionbetween vacuum transitions and field evolution.


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D. Continuity of predictions

The next landscape we wish to consider is one ofthe simplest imaginable just a double well potential.In this example, the predicted ratio of weights in vac-uum A to that in Z (in the case of full recycling) isidentical for the CHC, RT, and RTT methods, withPA/PZ = 1, independent of the relative lifetimes of the

states. The ratio of weights predicted by the CV methodis [4] PA/PZ = (HA/HZ)4eSASZ , where HA,Z is theHubble constant and SA,Z the entropy of vacuum A andZ respectively. The difference is due to the fact thatthe CHC, RT, and RTT methods count the frequencyof transitions while the CV method weights according tothe time spent in a given vacuum [4].

Now consider shifting the entire potential down, suchthat the lower well becomes a terminal vacuum. Thepredictions of the CHC, RT, and RTT methods will re-main identical until the lower well is exactly terminal, atwhich point the CHC and RTT methods (the RT methodbreaks down when the lower well becomes terminal) pre-

dict PA = 0, PZ = 1 [36]. Were this a correct descriptionof relevant probabilities, it would be very important inmaking predictions to know if the energy of a minimum

were zero or different from zero by one part in 1010100

.The CV method will predict this distribution as well, butwill approach it in a continuous manner (SZ , send-ing the ratio PA/PZ to zero). The predictions of the CVmethod are for this reason much more robust under smallchanges of the potential.

One possible way to avoid this discontinuity might beto reverse the order of limits t and 1AZ .All of the measures discussed in this paper take the t limit first, but one could perhaps define a measure

where the duration in time is held finite while

1

AZ .Applying this to the two-well example, as the lifetime ofthe lower well goes to infinity, the expectation value ofthe number of transitions observed would smoothly goto zero. Alternatively, it may be the case that there areno truly terminal vacua (with strictly zero probabilityof being tunneled from) [37]. Finally, it may be thatthere is simply something conceptually flawed in the waybubble-counting measures treat the borderline between avacuum being terminal and non-terminal.

IV. CONSEQUENCES FOR PREDICTIONS IN

A LANDSCAPE

The previous section pointed out some interesting fea-tures of bubble-counting measures (all the measures heresave CV) as somewhat abstract procedures applied tosmall toy landscapes. What might these features im-ply for predictions (in the form ofPp or PX) in a morerealistic landscape with many, many vacua and transi-tions connecting them?

Without a well-specified model of such a landscapethis is a difficult question to answer; however the strong

preference for pairs of fast-transitioning vacua does sug-gest some general and possibly troubling predic-tions. Within a landscape, imagine the set of all pairsof neighboring vacua (M,N) with similar pairs of ener-gies (VM, VN), and suppose that for each pair, the barrierbetween M and N is independent of the barriers separat-ing M and N from other nearby vacua. Then we mightexpect that members of different pairs will be accorded

exponentially differing probabilities depending on the de-tails of the barrier. In Sec. III we found in our samplelandscapes that the probabilities for the vacua in a fast-transitioning pair (N,M) are approximately proportionalto the product NMMN of the transition rates betweenthem. What determines this product? We fix VM andVN, and imagine the possible potentials v in-between (i.e.consider we consider many pairs in the landscape). Wehave

MNNM e2S(MN )(v)eSM+SN , (16)where S(MN)(v) is the instanton action of Eq. 4 and SM,Nare the background subtractions for vacua M and N,

given by Eq. 5. With SM and SN fixed, the product thendepends just on S(MN). As argued above, this action

will be of order (MPl/)4, and vary by order unity as theparameters governing the potential v are varied. Thusthe weightings of the members of each pair do appear tobe exponentially sensitive to the shape of the potentialin-between.

Now imagine that our vacuum is one tunnel away fromone of the vacua with energy VN. All other things beingequal, we should be likely to come from any given oneaccording to its weight. The evolution towards our vac-uum depends on the shape of the potential, and becausev is smooth this will not be independent of the shape of

the potential between the endpoints of the instanton. Ifan observable depends on the shape of the potential asour vacuum is approached, then this raises the possibilityof it having an exponentially varying prior over an obser-vationally relevant range. A good example might be thenumber of post-tunneling e-folds, which might possess aprior exponentially favoring a particular number.

One might hope to compensate the prior probabilitiesPp favoring cosmologies unlike ours using a conditional-ization factor nX,p that disfavors them (e.g. conditional-izing on the existence of a galaxy). In some cases, thisseems plausible. For example, if we consider the cos-mological constant and (unrealistically) assume thatall other cosmological parameters stay fixed to our ob-served values, then nX,p() decreases as an exponentialin /4Q3, where Q 105 is the fluctuation amplitudeand 1028 is the matter mass per photon in Planckmasses (e.g., [21]). Because this scale is so much smallerthan the scale over which the parameters of the poten-tial vary (i.e. 4Q3 M), the exponential variationsof Pp() are likely to be nearly constant over a rangeof order 4Q3, so nX,p() would be effective in forcingPX to give most weight to a region of parameter spacenear to what we observe [22, 23]. But in other cases this


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is far from clear; for example, the number of inflation-ary e-folds is determined by the high energy structure ofthe potential at and near tunneling, and the number ofe-folds is linked to the field value to which tunneling oc-curs, which is in turn linked to the instanton solution andhence the tunneling rate. Thus nX,p and Pp might easilyvary over the same scale in the parameters governing thelandscape potential, and the conditionalization may be

ineffective at forcing PX to peak in the observed range.

V. OBSERVERS IN ETERNAL INFLATION

Measures relying on properties experienced by a localobserver (generally equated with a causal worldline) re-quire that observers can actually transition between thedifferent vacua. It is not, however, clear that this is al-ways the case. In [24], two of the authors found that insemi-classical Hamiltonian descriptions of thin-wall tun-neling, there are always two qualitatively different typesof transitions described by the same formalism.

One, called the R tunneling geometry, is a gen-eralization of Coleman-De Luccia [5]/Lee-Weinberg [6](CDL/LW) true and false vacuum bubbles. It corre-sponds to the fluctuation of a bubble of the new phasewhich is always in causal contact with the backgroundregion, in the sense that worldlines in the old phase canboth tunnel with the bubble, and also enter the bubbleof new phase soon after it forms.

In the other, which was called the L tunneling ge-ometry (a generalization of the Farhi-Guth-Guven mech-anism [25]), the bubble of new phase lies behind a worm-hole separating it from the original background space-time. In this case, no causal curve from the original

phase can enter the new phase after the tunneling event(in marked contrast to the usual picture of an expandingbubble of new phase, or to the R mechanism). Some rareworldlines might tunnel with the bubble, but the phys-ical connection between pre-and post-tunneling phasesrepresented by such worldlines is obscure at best; more-over such worldlines do not exist in the (highest proba-bility) limit in which the bubble has zero mass.

If both L and R processes occur, then the L mechanismis the most probable path by which regions ofhigher vac-uum energy emerge, while the R geometry dominates de-cay to a lower vacuum [24]; both processes are dominatedby the lowest-mass bubbles.

At the semi-classical level of these calculations, the au-thors of [24] found no convincing reason that one butnot the other of these two tunneling processes would oc-cur. Holographic considerations would seem to conflictwith the L geometries (at least for transitions to highervacuum energy), and [26] argued using AdS/CFT thatsuch events tunneling from AdS to dS would correspondto non-unitary processes; however the question has notbeen settled with any clarity. (See [27] for another treat-ment of L tunneling geometries using AdS/CFT.) In thissection we will therefore consider how the L-tunneling

process would impact eternal inflation, and the measuresas applied to it.

Let us consider an initial parcel of comoving volume ina metastable state residing in an arbitrary potential land-scape. This is shown at the bottom of Fig. 3. As timegoes on, bubbles of either higher or lower vacuum en-ergy will nucleate by either the L or R tunneling geome-tries. Since low-mass bubbles are most probable, most

downward transitions will be CDL bubbles (the R geom-etry in the zero mass limit), and most upward transitionswill be L-geometry tunneling events corresponding to avery small mass black hole forming in the backgroundspacetime. Such small black holes affect the backgroundspacetime in a completely negligible way as long as thenucleation rate is rather small [38]. In particular, theseupward nucleations remove zero comoving volume fromthe old phase.

The pre-and post-tunneling spacetimes in an L-tunneling event are described comprehensively in,e.g., [24]; the portion of the post-tunneling spacetime ex-isting behind the wormhole consists of regions with both

new and old vacuum energy separated by a thin wall, andin the zero-mass limit is just the Lorentzian CDL bouncegeometry. Both vacuum regions are larger than theircorresponding Hubble radii and so will unavoidably con-tinue to inflate, independent of the precise details of theinitial nucleated space (i.e. how the instanton is slicedto be continued into Lorentzian space; see [19] for thecorresponding issue concerning the CDL instanton).

The result is that an entirely new branch of eternalinflation is created, with some initial physical volume,having essentially no effect on the original spacetime. Ifa comoving volume is assigned to this physical volumeusing the scale factor time of the background geometrynear the nucleation event, then the effect will be to createnew comoving volume [39]. The new branch will in turnspawn more branches and more comoving volume via L-events, so that the comoving volume appears toactually grow exponentially (though in what time thisoccurs is unclear since there is no foliation of the entirespacetime). This process is shown in Fig. 3.

How do the measures we have been discussing con-nect with this new picture? Consider first the measuresRTT, RT and W that explicitly follow causal worldlines.As formulated, these measures would essentially ignoreL transitions. This seems quite artificial, however, asregions with high vacuum energy (reached by upwardtransitions) would almost all arise from this process; put

another way, choosing a random point in the entire space-time (including the tree of new universes formed by theL tunneling geometry) and projecting any geodesic back,it would almost certainly hit an L-geometry nucleationsurface in the past rather than the assumed initial slice.

Now consider the CV and CHC prescriptions. Asstated, the idea is to count the relative comoving volumeor number of bubbles of different types on future nullinfinity. But as described in Sec. II B and in [3, 13, 16],these measures are actually calculated with very strong


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FIG. 3: A picture of an eternally inflating universe which takes into account both L and R tunneling geometries. At thebottom, there is an original parcel of comoving volume (defined by the horizontal spacelike slice at the bottom of the figure),which evolves in time (vertically). True and false vacuum bubble nucleation events occur via the R geometry in this volume,

denoted by the shaded regions which in the case of true vacuum bubbles grow to a comoving Hubble volume and in the caseof false vacuum bubbles shrink to a comoving Hubble volume. The vertical black lines denote the black holes formed duringL geometry tunneling events. On the other side of a wormhole (inside the captions), the initial distribution, which is fixed bythe tunneling geometry, undergoes L and R tunneling events as well, spawning more disconnected parcels of volume in whichthis process repeats. The original parcel of comoving volume will spawn an infinite amount of new comoving volume via Lgeometry tunneling events. Shown on the bottom of each parcel is the set of bubble shadows that might be used in the CHCmethod to calculate probabilities PVi for each region Vi.

reliance on a congruence of geodesics emanating from aninitial surface; thus as calculated in this formulation theywould be as unaffected by L-geometry events as RTT,RT, and W. It is interesting, however, to speculate abouttaking these prescriptions seriously as counting bubbles

on future infinity, as this would actually include the bub-bles in the other branches created by L-events.

Consider, then, a volume Vi nucleated by an L-event(with the subscript i labeling the particular region underconsideration), and imagine a congruence of geodesicsemanating from it, denoting by J+(Vi) the part ofthe spacetimes future null infinity reachable by thesegeodesics. Then we might count bubbles of comovingsize exceeding (for CHC) or count comoving volume(for CV) on J+(Vi), to define a set of relative probabil-ities PVi .

Now, it is very unclear how precisely to combine thePVi in all of the branches i formed from L-tunnelings

out of both the original spacetime, and out of the futureof Vi, and from the descendants of these branches, etc.Nonetheless, some general statements might be madeeven in the absence of such precision.

Consider first CHC. Since its probabilities are essen-tially independent of Vi, it seems that PVi will be thesame in all branches, so it is hard to see how anythingelse could result from combining them.

Now consider CV, which is dependent on the initialconditions for Vi. Here, the initial conditions for a

branch are not provided by the original spacetime, butrather by the dynamics of the L-tunneling process, with adifferent set corresponding to each pair of vacua betweenwhich the nucleations can occur. Whatever way we calcu-late all of the PVi , it seems likely that the original space-

times initial conditions will be completely overwhelmedby those of all of the branches in the infinite self-similartree depicted in Fig. 3. One might then imagine that thetotal prior distribution P is given by a weighted sum ofthese separate distributions, and is independent of theinitial conditions of the original spacetime.

We also point out that these questions may apply tostochastic eternal inflation as well. It is generally im-plicitly assumed in these models that the global space-time is causally connected, but this is far from proven.Indeed, large fluctuations generically cause a large backreaction, and it is not obvious that the large stochas-tic fluctuations driving eternal inflation do not cause theproduction of universes behind a wormhole (this is sug-gested by singularity theorems [28, 29, 30]). This dis-cussion is also relevant for hypothetical transitions outof negative energy minima. While no instanton has beenconstructed for such a transition (see [31] for a proposalconcerning the probability of such a process), if one existsthen (considering thin-wall constructions [26]) it wouldhave to be an L geometry. Based on the considerationsabove, it is unclear how or if including such transitionswould change the predictions of extant measures.


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VI. DISCUSSION AND CONCLUSIONS

We have analyzed a number of existing measures foreternal inflation, exploring connections that exist be-tween them, and highlighting some generic predictionsthat they make. With this perspective, let us return tothe list of desiderata presented in Sec. IIA. Shown in Ta-ble I is a scorecard detailing which of the measures, in

at least a majority of the authors humble and irresoluteopinions, satisfy the properties listed in Sec. II A.

First, which measures are physical, in the senseof providing a non-arbitrary prior probability Pp, forsome counting object p, useful for calculating PX?Physical volume weighting (discussed little here) wouldseem quite physical but appears to lead to gauge depen-dence [9, 10], and incorrect predictions in at least somegauges (see [11, 12]). The related CV (p=unit of co-moving volume) method may avoid some of this dif-ficulty, but at some cost to physicality: comoving vol-umes are generally meaningful only insofar as they arere-converted to physical ones, or if there are conserved

objects (baryons, galaxies, etc.) with fixed density perunit comoving volume. The latter may be true after re-heating, but it is unclear to us that comoving volumeis as meaningful during a complex, inhomogenous infla-tionary period. Another option is to weight accordingto the integrated incoming probability current [15, 32]across reheating surfaces, which can be found directlyfrom volume distributions. This proposal, which is tiedmore closely to the conditionalization, avoids the gaugedependence and spurious predictions of standard volumeweighting (as discussed above, this prescription can re-produce the results of the RTT method [15, 19]).

The CHC and W methods have p =bubbles, which

might be tied to conditionalization objects associatedwith the various reheating surfaces (though this involvesconsiderable uncertainty since those reheating surfacesare generically infinite). However, the objects (world-lines and shadows) actually used to arrive at a bubblecount seem rather less physical, particularly as they de-mand a cutoff prescription that while natural alsoseems as if it could easily be different. The RT and RTTmethods use p =segment of a worldline between vac-uum transitions, and has been suggested as an appropri-ate measure if we identify X=unit of entropy produc-tion [1, 33]. This connection is not entirely compelling,however, as the results of these holographic measurescan be found by considering an ensemble of observers (as

noted in Sec. II B and by [15]). These connections sug-gest that CV, RT, and RTT are very closely related, butwith a consistent and appropriate physical interpretationsomewhat lacking.

Consider now gauge independence. Physical volumeweighting is gauge dependent, but the other measuresappear gauge-independent, albeit with some caveats.For RT, RTT, W, and CHC, gauge-independence stemsfrom their counting of objects (bubbles) or events (tran-sitions); in CV it occurs via use of a congruence of

geodesics, which are also then counted to obtain co-moving volume. The caveats stem from subtleties con-nected with a time variable choice in defining cutoffs,transitions rates, and initial conditions, and we hope toelucidate some of these further in future work. (We singleout CV as partially gauge-dependent because the resultswill depend on the time slicing used to characterize theinitial value surface.)

Drawing on the description of the various measurespresented in Sec. IIB, we can see that not all of the mea-sures under discussion have the ability to cope with alltypes of transitions and vacua. For instance, the CVmethod accords zero weight to metastable minima (par-ticularly disturbing as we may live in one), and the RTmethod in its current formulation is not able to describea landscape with terminal vacua. We also note that theCV and RTT methods are dependent on initial condi-tions.

In Sec. V, we argued that it is possible if certain typesof L bubble nucleation events occur for different re-gions of the eternally inflating multiverse to be separated

by wormholes, and therefore causally disconnected. Noneof the evaluated measures are, as formulated, equippedto deal with such spacetimes in a reasonable way. Thephilosophy behind CV and CHC of counting bubblesor volume on future infinity might reasonably apply tosuch spacetimes, and if this could be implemented tech-nically we argued that in this case CV would probablybecome independent of initial conditions. The philosophybehind RTT and RT would suggest simply ignoring theseevents (as indeed those measures effectively do) but it israther unclear to us that this is appropriate. Accountingfor such tunneling events in measure prescriptions is verydifficult but this merely highlights the possible impor-tance of such transitions, and of determining whether ornot they occur.

Even thornier problems might arise from consideringtransitions in greater generality. All of the measures con-sidered rely on a congruence of worldlines and a fairlystraightforward spacetime structure. Were we to includetransitions between different string/M theory flux vacua,including even different numbers of large spacetime di-mensions, it is unclear whether the principles of extantmeasures would apply. Without having a welldefineddescription of such transitions this is difficult to asses,hence we do not consider this in our table.

But even confining our attention to (relatively) well-understood spacetime evolution in a general scalar poten-tial landscape, the measures differ somewhat in how gen-erally and robustly they treat vacua and transitions.All of the measures under discussion have been applied tothe brand of eternal inflation driven by metastable min-ima. However, it would be desirable to include the effectsof all the dynamics of an eternally inflating universe, andthe effective scalar fields that are imagined to drive it.This includes a description of the diffusion and classicalrolling of the field that will occur. There has been workextending CV and CHC methods to these cases, but little


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mology grant from the John Templeton Foundation dur-ing the preparation of this work. SG is supported byPPARC.

APPENDIX A: MATRIX CALCULATIONS AND

SNOWMAN DIAGRAMS

In this appendix we present a quick way of calculat-ing normalized probabilities for terminal and cyclic land-scapes in a unified manner, which also sheds light on thenature of the regularizing limit taken in the cyclic case.

First, assemble the relative transition probabilitiesNM into a matrix (equivalent to Boussos matrix).Starting in an initial state represented by a vector qwith components qN (NqN = 1), after one transitionthe mean number of entries (or raw probability) foreach vacuum will be given by q. At the second tran-sition an additional 2q entries will occur and so on.After n transitions the raw probability will be given by(+2 + . . .n)q. If we set Sn

+2 + . . .n, then

(1 )Sn = (1 n

). In the terminal case we caninvert (1 ) and take the n limit to obtain Sdirectly (n 0 since asymptotically all the probabilitygoes into the terminal vacua and so fewer and fewer vac-uum entries occur). In the cyclic case det(1) = 0 andn does not tend to zero, and things are not so simple.It is convenient to proceed by replacing by (1 )(where is an auxiliary parameter to be taken to zeroafter the calculation), which can be inverted. Neglect-ing the troublesome determinant factor (since we shall belater normalizing to obtain probabilities from numbers ofvacuum entries anyway), we take the limits n and 0 in that order, and for both terminal and recyclinglandscapes obtain the simple expression:

S T (adj(1 )) (A1)

where adj denotes the adjoint matrix operation (i.e. thetranspose of the matrix of cofactors of the matrix in ques-tion). Multiplying T into q and normalizing yields theprobabilities for the vacua given the initial state in ques-tion.

This procedure yields exactly the same results as thepruned tree method. We thus see that the latter proce-dure is equivalent to considering sequences of transitionsup to some length n and then taking the limit n .

The NMs in question can conveniently be depictedin snowman-like diagrams such as those shown in Fig. 4,which apply to the calculations in Sec. III. These dia-grams emphasize that the path between any two vacuacan involve an arbitrary number of circulations in closedloops between recycling vacua. In fact we treat bothcases at once by leaving ZB arbitrary and only set itto 1 or 0 as appropriate after having calculated T. Wealso allow for the possibility of vacuum A being terminalin the same manner.

Suppressing the normalizing factor for clarity, we ob-

A

B

B

Z

AB

BA

BB BB

BZ

ZB

A

B

AB

BA

BB BB

B

BZ

Z

FIG. 4: Examples of snowman diagrams summarizing rela-tive transition probabilities NM. The one on the left is for arecycling landscape and the one on the right is for a terminallandscape.

tain

PA,B,B

,ZA

PA,B,B,Z

B

PA,B,B,Z

B

PA,B,B,Z

Z

AB(1 BZZB)

1 BZZB

B

B

BBZB

(A2)

in the recycling case with the full set of superscripts in-dicating that the results are independent of initial condi-tions.

In the terminal case we can only start in states A, Bor B and we obtain:

PAAPABPAB

PAZ

AB1

BBZBBB

, (A3)

PBAPBBPBB

PBZ

ABABBA + BBBB

BBBBZB

(A4)

and PB

A

PB

B

PB

B

PB

Z

ABBB

BB

BBBBZB(1 ABBA)

. (A5)

The relative transition probabilities are related to the

transition rates byBA = 0 or 1 (A6)

AB =AB

AB + BB(A7)

BB =BB

AB + BB(A8)

BB =BB

ZB + BB(A9)

ZB =ZB

ZB + BB(A10)


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14

where BA = 0 ifA is terminal and BA = 1 if it isnt.Substituting these expressions into equations A3, A4,and A5, we can then take the limits discussed in Sec. IIIto produce the appropriate probability tables.

In the case where vacuum A is terminal (AB = 0),there are a number of ratios of interest. The probabilitiesassigned by the CV method to this sample landscape werecalculated in [3] (the FABI model), and using these

results, we can directly compare the results of the CVand RTT methods. For initial conditions in B or B, wefind:

PBAPBZ

=AB (BB + ZB)

BBZB(A11)

PB

A

PB

Z

=ABBB

ZB (AB + BB)(A12)

As expected given the argument of Sec. IIC, these resultsagree with the predictions of the CV method.

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(2000), gr-qc/9811037.[30] A. Aguirre and M. C. Johnson, Phys. Rev. D72, 103525

(2005), gr-qc/0508093.[31] T. Banks and M. Johnson (2005), hep-th/0512141.[32] J. Garcia-Bellido, A. D. Linde, and D. A. Linde, Phys.

Rev. D50, 730 (1994), astro-ph/9312039.[33] R. Bousso and B. Freivogel (2006), hep-th/0610132.[34] Note that this is worse than it may sound, because the

same spacetime might be sliced with different initial sur-faces so as to lead to completely different probability dis-tributions.

[35] It is unclear the extent to which the velocities of indi-vidual points can be chosen at random, as discussed byVilenkin in [13]

[36] It is worth noting that that this is completely indepen-dent of the ratio of the lifetimes of the states, which mightbe arbitrarily large [20].

[37] For example, if the L process describe below in Sec. Voccurs, it might mediate transitions away from negativeor zero-energy vacua. A heuristic argument in favor oftunneling from negative big crunch vacua was givenin [31]. Finally, we note that after tunneling to a nega-tive vacuum, the spacetime is an open FRW model withenergy density. Thus there may conceivably be tunnelingbefore the crunch even if such tunneling is impossible

from pure AdS or Minkowski space.[38] In fact even more probable is the zero-mass limit in which

there is no black hole at all, which also clearly does notaffect the background spacetime.

[39] How to actually define comoving volume in the newphase is very unclear; comoving volume is related to aparticular coordinatization of a spacetime, and its defi-nition is tied to a congruence of geodesics; here no suchcongruence continues through the nucleation to fill theinitial slice.
http://online.itp.ucsb.edu/online/strings06/shenker/http://online.itp.ucsb.edu/online/strings06/shenker/http://online.itp.ucsb.edu/online/strings06/shenker/

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