PHYSICAL REVIEW D, VOLUME 58, 083503

Quantum creation of a universe with VÞ1: Singular and nonsingular instantons

Raphael Bousso and Andrei LindeDepartment of Physics, Stanford University, Stanford, California 94305

~Received 23 March 1998; published 1 September 1998!

We propose two new classes of instantons which describe the tunneling and/or quantum creation of closedand open universes. The instantons leading to an open universe can be considered as generalizations of theColeman–De Luccia solution. They are nonsingular, unlike the instantons recently studied by Hawking andTurok, whose prescription has the problem that the singularity is located on the hypersurface connecting to theLorentzian region, which makes it difficult to remove. We argue that such singularities are harmless if they arelocated purely in the Euclidean region. We thus obtain new singular instantons leading to a closed universe;unlike the usual regular instantons used for this purpose, they do not require complex initial conditions. Thesingularity gives a boundary contribution to the action which is small for the instantons leading to sufficientinflation, but changes the sign of the action for smallf corresponding to short periods of inflation.@S0556-2821~98!08016-3#

PACS number~s!: 98.80.Cq

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I. INTRODUCTION

It is well known that most of the models of inflationarcosmology predictV5161024. It is possible to have infla-tion with VÞ1, but it is rather difficult. The basic idea is tsolve the homogeneity and isotropy problem not bylong stage of inflation, but by quantum tunneling to a stdescribing an open or closed universe. Then the univewill be homogeneous if the probability of tunneling is sufciently and strongly suppressed. In this scenario an infinumber of open universes can be created in one of two wOne may consider a purely classical evolution of an intionary universe in the false vacuum and a subsequentation of inflating open universes by tunneling to the trvacuum@1–3#. Alternatively, one may consider the quantucreation from nothing of a closed inflationary universwhich later decays into an infinite number of open univerby the process described above@4,5#. In all such models, it isnecessary to assume a potential with a false vacuum.

Recently, Hawking and Turok claimed that open uverses can be obtained without an intermediate stage inving false vacua. They described a process in which an ouniverse is created from nothing in the chaotic inflation snario with a generic effective potential@6#. They used thestandard deformed-four-sphere Euclidean solution, in whthe inflaton field is constant on the lines of constant latituThis solution generically has a singularity on one of tpoles. Usually it is cut along the equator; the singular hesphere is discarded, and the regular one is analytically ctinued to yield a closed Lorentzian universe.

Instead, Hawking and Turok cut the Euclidean solutithrough the poles, thus including the singularity on the hpersurface through which the Euclidean and Lorentzian stors are joined. The hypersurfaces of constant inflaton fiwill then form infinite open spacelike sections in a part of tresulting Lorentzian universe.

This approach suffers from two problems. The first prolem is that they obtainV51022 for the ratio of the presendensity to the critical density. This contradicts observatiodata by almost two orders of magnitude, and even this re

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was obtained only after invoking the anthropic principwithout which one would getV50. The prediction comesfrom the probability measure associated with the HarHawking wave function. In@5# it was argued that, accordinto @7,8#, this wave function should not be used for the dscription of the creation of the universe and it was shothat the use of the tunneling proposal would typically leadV'1. It thus appears that with either choice of the wafunction of the universe we are not currently in a positionobtain a realistic value ofV50.3 unless customized potentials are employed.

The second problem is associated with the presencesingularity on the nucleation surface. Vilenkin@9# arguedthat instantons of the Hawking-Turok type lead to vacuuinstability and should therefore be excluded from the pintegral. In@5# it was shown that not every instanton is pemitted even if it is nonsingular. On the other hand, it wsuggested that singular instantons are not necessarily foden, but one should be extremely careful about the analytcontinuation involving singularities which was used in@6#.For other problems associated with this issue see also@10#.

In this paper, we will perform a more detailed investigtion of these issues. We will suggest two ways of avoidithe problems associated with the Hawking-Turok singularFirst, we will consider potentials with a local maximum, fowhich there are non-singular solutions. They includeColeman–De Luccia instantons as well as some new, relsolutions that we found. We will discuss the structure aapplication of these solutions in Sec. II. They describenucleation of open universes, and allow the correct predtion of V for suitable potentials. Of course, this means ththe generality claimed by Hawking and Turok is lost, butwe pointed out above, generic inflaton potentials do not severy promising in any case when one tries to predict uverses which are both non-flat and non-empty.

We will allow generic potentials in Sec. III, where we wiuse variants of the deformed-four-sphere instanton to nuate closed universes. We cut them along the equatordiscard theregular hemisphere. The Hawking-Turok singularity will be present in this case. It will not, however, lie o

© 1998 The American Physical Society03-1

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RAPHAEL BOUSSO AND ANDREI LINDE PHYSICAL REVIEW D58 083503

the nucleation hypersurface. Therefore it can be ‘‘surgicaremoved,’’ or viewed as a small region of Planckian densWe calculate the boundary contribution to the action ashow that it is small in all cases where sufficient inflatiensues. We discuss a possible interpretation of these stions as the birth of a closed inflationary universe by tunning from space-time foam. Finally, we construct instantothat are symmetric about the equator and contain two sinlarities. They allow the construction of nucleation pathswhich all variables are everywhere real.

II. NONSINGULAR INSTANTONS

Suppose we have an effective potentialV(f) with a localminimum at f1 , and a global minimum atf50, whereV50 ~see Fig. 1!. In anO(4)-invariant Euclidean spacetimwith the metric

ds25dt21a2~t!~dc21sin2cdV22!, ~1!

the scalar fieldf and the three-sphere radiusa obey theequations of motion

f913a8

af85V,f , a952

8pG

3a~f821V!, ~2!

where primes denote derivatives with respect tot.These equations have several nonsingular solutions,

simplest of which are theO(5) invariant four-spheres onobtains when fieldf sits at one of the extrema of its potetial. In this case the first of the two equations above is trially satisfied, anda(t)5H21sinHt. Here H258pV/3Mp

2.Using the solution for whichf5f1 , Hawkins and Moss@11# found the rate at which fieldf in a single Hubble vol-ume tunnels to the top of the potential, from which it can rdown towards the true vacuum. For a recent discussionthis instanton and its interpretation see@5#. The main otheruse of these trivial instantons is to find the action of the favacuum background solution, which must be subtracted frthe bounce action to obtain a tunneling rate.

We shall consider potentials for whichV,ff@H2 in theregion where the tunneling occurs. In this case, tunnelingof the false vacuum does not occur primarily on the scalean entire Hubble volume via the Hawking-Moss instantoInstead the transition will proceed via more complicated E

FIG. 1. Effective potentialV(f)5 m2/2 @f2(f2v)21Bf4# form252, B50.12, andv50.5. It has a shallow minimum atf0

;0.357 and a local maximum atf150.312. All quantities in thefigures in this section are in units ofMp /A8p.

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clidean solutions with varying fieldf. These include theColeman–De Lucia instanton, and related instantons thafound.

A. Bubble instantons

A Euclidean solution that describes the creation ofopen universe was first found by Coleman and De Luccia1980 @12#. It is given by a slightly distorted de Sitter foursphere of radiusH21(f0). Typically, fieldf is very close tofalse vacuumf0 throughout the four-sphere except insmall region~whose center we may choose to lie att50!, inwhich it lies on the ‘‘true vacuum’’ side of the maximum oV. The behavior of the field and scale factor for the potenin Fig. 1 is shown in Fig. 2. The scale factor vanishes atpoints t50 andt5t f'p/H, which we will call the northand south pole of the four-sphere. In order to getsingularity-free solution, one must havef850 anda8561on the poles.

This solution can be cut in half along the linec5p/2,which removes half of each three-sphere. Then one cantinue analytically to a Lorentzian spacetime@13,6# with thetime variables, given byc5p/21 is. This gives region IIof the Lorentzian universe~see Fig. 3!

ds252a2~t!ds21dt21a2~t!cosh2sdV22 . ~3!

FIG. 2. The upper panel shows the behavior of scalar fieldf forexamples of the Coleman–De Luccia ‘‘bubble’’ instanton~solidline! and the new ‘‘double-bubble’’ instanton that we found~dashedline!. For both instantons, the field is in the domain of the trvacuum at smallt, forming a bubble. For the bubble instanton, thfield is closest to the false vacuum at the pole opposite the bubFor the double-bubble instanton, this happens on the equator amoment of the maximal expansion. The behavior of the thrsphere radiusa(t) shown in the lower panel is very similar for botinstantons, though it is not identical.

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QUANTUM CREATION OF A UNIVERSE WITHVÞ1: . . . PHYSICAL REVIEW D 58 083503

Field f will still depend ont in the same way as before, anwill be independent ofs. This describes a shell of widthH21, which is mostly near the false vacuum and expanexponentially. The shell separates two bubbles, regions IIII, in which the universe looks open.

Region I is obtained by takings5 ip/21x and t5 i t ,giving the metric

ds252dt21a2~ t !~dx21sinh2 xdV22!, ~4!

wherea(t)52 ia@t(t)#. Its spacelike sections~defined bythe hypersurfaces of constant inflaton field! are open. Thusregion I looks from the inside like an infinite open universwhich inflates while fieldf slowly rolls down to the truevacuum. The evolution will then undergo a transition toradiation or matter-dominated open Friedmann-RobertsWalker universe.

In region III, which is obtained by choosings5 ip/21x andt5t f1 i t , field f rolls to the local minimum atf0 ,and one gets indefinite open inflation in the false vacuum

The analytic continuations we have given support theterpretation of such solutions as the spontaneous nucleaof a bubble of true vacuum on the background of de Sispace expanding in the false vacuum. For this reason wecall them ‘‘bubble instantons.’’ The nucleation rate is givby

G5e2DS, ~5!

where DS is the difference between the action of the fuEuclidean bubble solution, and the action of a Euclideanlution describing the background spacetime. Except for nePlanckian potentials, both actions will be large and nega~about22.63104 in our example!. The background solutionis given by an exact Euclidean four-sphere on which fieldfis constant and equal tof0 , the false vacuum. Its action wilbe 2(3Mp

4)/8V(f0). Subtracting this from the action of thbubble solution, one obtains a positiveDS ~'4.9 in our ex-ample!. This means that bubble formation by tunnelingsuppressed, as it should be.

One usually requires instanton solutions to interpolatetween the initial and final spacelike sections~in this case, asection of pure de Sitter space in the false vacuum ansimilar section containing a bubble of true vacuum!. Theabove description, which seems to use two disjoint inst

FIG. 3. The Lorentzian de Sitter-like spacetime obtained frthe analytic continuation of Coleman–De Luccia instantons ctains three regions. In regions I and III the hypersurfaces of consfield f form open spacelike sections. Region II is a shell separathe two bubbles.

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tons, is actually consistent with this formal requiremesince the instantons may be connected by virtual domwalls after small~Planck size! four-balls are removed. Thiswill cause the background instanton to contribute to the toaction with a negative sign. If one connects the backgrouinstanton to the region of the bubble instanton wheref isclosest to its false vacuum, the discontinuity inf will besmall, so the volume contributions of the removed regiocancel almost exactly. Requiring continuous instantotherefore, does not change the pair creation rate significa@14#.

Cosmological instantons have frequently been interpreto describe the creation of a universe from nothing, i.e. wiout a pre-existing background. This case is considerablywell-defined than the quantum nucleation of structures ogiven background solution. In particular, the sign with whithe large, negative action enters the exponent in the pintegral is subject to debate@6,5,15#. Leaving such questionsaside for now, we will take the position that isolated cosmlogical instantons are indeed related to universe creationdependently of the formalism used to assign probabilitiessuch processes.

B. Double-bubble instantons

We have found a new instanton in which there are tbubbles, one on each pole. In this solution,f is in the do-main of the true vacuum in small regions near the poles,near the false vacuum elsewhere; this can be seen fromdashed line in Fig. 2. The geometry is still approximatelyfour-sphere. As before,f8 vanishes on the poles; but nowalso vanishes on the equator, att5tmax. The Northern andSouthern hemispheres are exactly symmetric.

Not surprisingly, the action of the double-bubble solutioafter the background subtraction described above, is apprmately twice that of the bubble~Coleman–De Luccia! in-stanton. For the instanton shown in Fig. 2 one hasDS2'9.8.

The analytic continuations will be the same as befowith a different result. Region II will be mostly in the domain of the false vacuum. Regions I and III will be identicaeach containing an open inflating universe in which the firolls down to the true vacuum. Globally, therefore, we obtatwo bubbles of true vacuum separated by a shell whichflates in the false vacuum.

This solution can be interpreted as the spontaneous pcreation of bubbles of open inflation on the backgroundfalse vacuum inflation. Alternatively, one may view it as tcreation from nothing of two open inflating universes seprated by a metastable shell.

C. Anti-double-bubble instantons

In addition we have found another family of instantontwo examples of which are shown in Fig. 4. In these insttons, the field is in the domain of the false vacuum in twregions surrounding the poles. They are separated by ashell at the equator, where the field is in the true vacudomain. These instantons have a much greater action dience to the background instanton, since the true-vacuum

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RAPHAEL BOUSSO AND ANDREI LINDE PHYSICAL REVIEW D58 083503

gion is significantly larger than in the previous two casesparticular, DS593.6 for the instanton shown by the solline in Fig. 4, andDS5124.7 for the instanton shown by thdashed line.

1. Open cut

With the analytic continuation used for the previous twinstantons, regions I and III will become open inflationauniverses in which the field rolls down to the false vacuuThey are separated by the region II, which contains a shewhich f is in the domain of the true vacuum.

Therefore we may interpret this solution as the nucleatof a shell of true vacuum on a false vacuum inflationabackground, or alternatively, as the creation of such a uverse from nothing. Because of the larger action differenspontaneous shell creation will be quite suppressed cpared to bubble formation.

2. Closed cut

A more intriguing application of this instanton can bfound by choosing a different analytic continuation. Insteof cutting at c5p/2, we may choose to leave the threspheres intact, and cut across the equator. Lorentzianwill be defined byt5tmax1iT, and we obtain a metric withclosed spacelike sections:

ds252dT21a2~T!dV32 . ~6!

The inflaton field is in the domain of the true vacuum on tnucleation surface~the equator!, so it will start rolling downtowards the absolute minimum. During this time, the spalike three-spheres grow exponentially:

a~T!'H21~T!coshE H~T!dT. ~7!

Thus we obtain a closed inflationary universe in whichscalar field rolls towards the true vacuum.

One could interpret this instanton as describing the cation of such a universe from nothing. But this would juadd an alternative to the usual instantons on which the fi

FIG. 4. Two examples of ‘‘anti-double-bubble’’ instantonswhich the field is in the false vacuum domain near the polesreaches into the domain of the true vacuum on a shell nearequator. It can be cut through the poles to describe shell nucleaor across the equator, describing the tunneling to true-vacuumflation in a closed universe.

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is entirely in the domain of the true vacuum. A much mointeresting interpretation is the one associated with a pexisting background of false vacuum inflation. In this cathe anti-double-bubble instanton is seen to describe the staneous tunneling to the true vacuum in an entire Hubbvolume of de Sitter space. Unlike the Coleman–De Lucbubbles, these regions will not contain an open universethe example we consider, for which Hawking-Moss tunning is not possible, this shows that one can neverthenucleate true vacuum bubbles containing a closed unive

III. SINGULAR INSTANTONS

A. Standard instantons

We now assume a generic effective potential of chaoinflation, with a minimum,V50, at f50, and no other sta-tionary points. The standard Euclidean solution used in qutum cosmology is obtained by requiring regularity at tnorth pole, att50. This means one must takea851 andf850 there. A Euclidean solution with these initial condtions is shown in Fig. 5 for a massive scalar field.

On most of the manifold, the solution will be almostfour-sphere, withf increasing very slowly:

a~t!5H21sin Ht, f~t!5fN , ~8!

wherefN denotes the value of the inflaton field on the reglar pole. A first approximation to the Euclidean action of tstandard instanton will therefore be given by the volumterm for a four-sphere of radiusH21(fN), S;2(3Mp

4)/8V(fN). This will be a good approximation for large valueof fN leading to long periods of inflation.

Near the south pole, att5t f , the anti-damping term;f8b8/b starts to dominate the equation of motion forf.

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FIG. 5. The standard solution. The scalar field is in units ofMp .Field f as well as the curvature are singular at the south pole. Nthatf8 does not vanish at the equator, so one should complexifyscalar field in order to make the analytical continuation to a clouniverse. The analytic continuation to an open universe suggeby Hawking and Turok@6# involves the singularity on the nucleation surface, which makes it rather problematic.

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QUANTUM CREATION OF A UNIVERSE WITHVÞ1: . . . PHYSICAL REVIEW D 58 083503

The field diverges logarithmically, and the potential termcan be neglected. Approximate solutions are given by@6,9#

a~t!5A~t f2t!1/3, ~9!

f~t!521

A12pln~t f2t!1fm, ~10!

whereA andfm are constants.There are many questions associated with this singula

First of all, even though the divergence of the scalar fieldonly logarithmic, the energy density and curvature diveaccording to a power-law. If the singularity is part of thnucleation geometry, i.e. if it is included in the Euclideinstanton and its Lorentzian analytic continuation, the corsponding method can no longer be called ‘‘the no-boundproposal.’’

The boundary att f contributes a Gibbons-Hawking@16#term to the action

St f52

Mp2

8p E d3xh1/2KUt5t f

, ~11!

whereh is the determinant of the three-metrichi j , andK isthe trace of Ki j , the second fundamental form. For aO(4)-invariant metric, we findK53a8/a. This yields

St f52

p

4Mp

2 d~a3!

dt Ut5t f

. ~12!

By Eq. ~9!, a(t) goes like (t f2t)1/3 near the singularity.Therefore, the boundary term will be positive and finite1

Note that for any power other than 1/3, the boundary tewould either vanish, or diverge.

The contribution of this boundary term to the total actiis relatively small if one considers the creation of an infltionary universe. For example, numerical investigatshows that in the theorym2f2/2 this correction, as compareto the action2 (3Mp

4)/8V(fN), is suppressed by a factoO(Mp /fN) for fN@Mp :

Sstd'23Mp

4

8V~fN! S 12Mp

2fND . ~13!

There is also a correction to the volume term becausef isnot exactly constant anywhere; we will not discuss this crection here.

As is obvious from this result, at smallfN the total actionincluding the boundary term may become positive. Oneconfirm numerically that this is indeed the case. This irather unexpected conclusion as it indicates that the absovalue of the action reaches its maximum not atfN50 but atfN;Mp .

1This result was also independently found by Vilenkin@9#.

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1. Closed cut

This Euclidean solution can be cut in different waysobtain an instanton that allows an analytic continuation tLorentzian spacetime. The standard method is to cut althe equator, att5tmax and to discard the Southern hemsphere, which contains the singularity. Witht5tmax1iT,one can join the regular hemisphere across the equatoLorentzian de Sitter space with the metric given by Eqs.~6!and~7!. The surfaces of constant inflaton field will be closspacelike slices in the Lorentzian sector. Therefore thiscorresponds to closed inflation.

In general,f8 will be small but non-zero on the equatoThis means that it will acquire an imaginary part in thLorentzian sector. But at late Lorentzian times, when msurements are made, one must demand that all variableexactly real. It is therefore necessary to compensate by aing a small imaginary part to the initial value off @17,18#:

fN5fNRe2 i

Mp2

8fNRe ~14!

where the superscript Re denotes the real part of a quanFor nucleation geometries which are everywhere real,

real part of the Euclidean action comes entirely from tEuclidean sector, since the Lorentzian sector contribuonly a purely imaginary part. But in the current case, tLorentzian sector will not be purely real for a time of ordH21 after the nucleation hypersurface. It will therefore gia further correction to the real part of the Euclidean actioncan be easily checked both analytically and numerically tthe ratio of this term to the the total action is of orderfN

22 .This corrects claims in@17# that the correction is of the samorder as the total action.

2. Open cut

Hawking and Turok@6# have suggested using the samanalytic continuation for the standard solution that had btraditionally used for the Coleman–De Luccia solutiowhich we gave explicitly in Sec. II. This involves cutting thEuclidean space through the poles, thus including half ofsingularity on the nucleation surface. The resulting Lorenian solution contains regions I and II of Fig. 3, while regioIII is cut off by the singularity. In region I the hypersurfaceof constantf trace out infinite open spacelike sections.

The interpretation of this instanton and the analytical cotinuation proposed in@6# may be rather problematic. It waargued in@5# that even if instantons are nonsingular, theynot always describe tunneling. But in this case one has ational problems associated with cutting the singularity in hand performing the analytical continuation there@5#.

Vilenkin @9# has argued that the admission of such nucation geometries, in which there is a singularity on the hpersurface of the vanishing second fundamental form, leto problems of vacuum instability. Such instantons exist foMinkowski space background, where they may cause anmost unsuppressed nucleation of singular bubbles spreaout nearly at the speed of light. Clearly, this is physicaunacceptable. It poses a problem for the prescription s

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RAPHAEL BOUSSO AND ANDREI LINDE PHYSICAL REVIEW D58 083503

gested by Hawking and Turok unless one finds sound aments why such instantons should be admitted for inflatiary universes, but not for flat space.

According to@9#, singular instantons may not be alloweat all because they are not true~nonsingular! solutions ofequation of motion that would correspond to an extremumaction. However, the singularity is not really a part of tmanifold. Moreover, one can sometimes cut it out, and csider a configuration that is nonsingular but coincides witsingular instanton everywhere except for a small vicinitythe singularity. If the action of the instanton converges, thfor a sufficiently small size of the patch replacing the singlar region, the action will differ from the instanton action bless thanDS51. Such ‘‘almost solutions’’ are perfectly admissible and play the same role in the functional integrathe true solutions, see e.g.@19,14#. However, if one makes ananalytical continuation through the singularity, one caneasily remove it by the method described above, and themay pose a real problem.

A possible way out of this problem would be to use nosingular instantons of the type we described in Sec. II. Trequire special potentials with a false vacuum. This hasadvantage that one can, in principle, obtain any given vaV,1. The price we pay is some loss of generality.2

An interesting result which appears after the boundterm is taken into account is that the action at very smafchanges its sign and becomes positive. This means thamaximal absolute value of the action is reached not atpoint whenV(f)50, but at some other point, where inflation is still possible. It would be very interesting if this poinwere at a sufficiently large value off, which would providea realistic value ofV within the Hartle-Hawking approachwithout any use of the anthropic principle. Unfortunatehowever, our numerical investigation of this question shothat in all realistic models with potentials;fn, the absolutevalue of the action is maximal atf,Mp , which does notlead to long inflation, and which, consequently, yieldsexponentially small value ofV.

3. Closed cut revisited

We will now turn to a different possibility. We wish tostudy the consequences of allowing the singularity to beof the nucleation geometry but not of the hypersurface joing the Euclidean and Lorentzian section. The simplest sinstanton is obtained by cutting the standard solution, oagain, across the equator, but discarding theregular hemi-sphere, and keeping thesingular hemisphere.

The instanton thus corresponds to the intervaltmax<t<tf . The Lorentzian section is obtained by takingt5tmax1iT, so it will be the same as that in Sec. III A 1: a closinflationary universe.

2There is a second, more brute-force way of eliminating the sgularity. One may take a spherical region around the regular pand join it to its mirror image across a domain wall of positienergy density. This method, which will be described in a sepapublication@20#, does not require false vacua, but assumes the pence of fields supporting the topological defect.

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The Euclidean region looks mostly like a four-sphereradius;H21(fN). But there will be a region near the singularity, where the curvature and energy density diverge.may impose a Planck-scale cut-off here, and considersingularity and its vicinity as a small Planckian region immersed in the large four-sphere. The presence of this reon the south pole is the crucial difference between this nuation geometry and the one studied in Sec. III A 1.

How should we interpret this difference? The regularstanton, viewed in isolation, has often been interpretedrepresenting the creation of the universe from nothing. Twas motivated by its self-contained nature; one might thof ‘‘nothing’’ as the vanishing of the scale factora, whichoccurs on the regular north pole. In contrast, the interpretion of the singular hemisphere actually seems less vagWe can think of this instanton as an interpolation betweePlanckian regime, and a large closed inflating univerTherefore we propose that it describes the spontaneoustion of inflation from a bubble of spacetime foam. In facthis agrees with the interpretation of creation from ‘‘noting’’ proposed in@7#. We are speaking about a state whethe classical part of metric strongly fluctuates, so that ocannot measure distance using any measuring devices.is what may happen at the Planckian epoch. But atPlanck time one would expect all physical fields to take laand strongly fluctuating values, rather than a definite vacorresponding to the north pole of the usual nonsingularSitter instanton att50. In this sense the use of singulainstantons seems quite appropriate for the descriptionquantum creation of a closed universe from space-time fo

The main difference of this use of the singular instantto that proposed by Hawking and Turok is that the singulity in our case does not reach into the Lorentzian sector.limited to a tiny region in the Euclidean regime, where it ceasily be smoothed out, or removed. Since the neitherboundary term nor the volume term have divergences nthe singularity, the action will be finite. For largefN it willnot differ noticeably from the action of the other hemispheor the action of the Hawking-Turok instanton.

B. Gondola instantons

In the previous subsection we argued that weak localisingularities inside the Euclidean sector of a tunneling geoetry can be interpreted as interpolations to space-time foand can thus be quite useful. Once this point of viewadopted, however, it is easy to see that the standard Euean solution is only a special case in a one-parameter faof solutions. Generically, these solutions will have singulaties on both poles.

We will now focus on a particular member of this famithat is exactly symmetric about the equator, shown in FigIt can be constructed by specifying very simple boundconditions on the equator: one is free to choose the invalue of the field,f5fE ; the derivatives of all fields andmetric components are set to zero. There will thus be idtical singularities on the north and south pole. We will cthis the ‘‘gondola’’ solution.

If we cut this solution along the equator, we obtain twidentical hemispheres, each containing a small Planckian

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QUANTUM CREATION OF A UNIVERSE WITHVÞ1: . . . PHYSICAL REVIEW D 58 083503

gion at its pole. As we discussed above, we may consthis region to interpolate to spacetime foam. Performingusual analytic continuation,t5tmax1iT, we obtain, oncemore, a closed Lorentzian universe. But the gondola instons have the great advantage that the second fundamform, and all field derivatives, vanishes on the nucleathypersurface by construction. This means that all variabwill be perfectly real in the entire Euclidean and Lorentzisectors. There is no need for introducing complex initial coditions in this case.

The gondola solution has two boundaries that contribterms to the action. For comparable values offE on theequator, we found numerically that these terms add up

FIG. 6. Gondola solution. Fieldf as well as the curvature arsingular at the south and north poles. Unlike the standard soluthis one is symmetric about the equator, where all derivatives vish.

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most exactly to the contribution of the single boundary tein the standard solution. The instanton given by half of tgondola solution will contain only one Planckian boundaTherefore, compared to the singular instanton studied in SIII A 3, the boundary contribution will be only half as larghere.

For small values offE , which give barely enough inflation, this means that the absolute value of the action is laest for the regular closed instanton of Sec. III A 1, followby the gondola instanton, and the singular instanton of SIII A 3. For large values offE , which lead to a long periodof inflation and a very flat universe, the difference is copletely negligible. Then the gondola instanton will be tmost practical to use, since it requires no analysis of compvariables.

IV. SUMMARY

We have described a number of nonsingular instantleading to open inflating universes. They include tColeman–De Luccia solution, in which a bubble of truvacuum expands inside a universe inflating in the favacuum. We found new solutions that contain two bubblor a shell of true vacuum.

We also constructed instantons with a singularity. If tsingularity does not lie on the hypersurface of nucleationcauses no problems in the Lorentzian region, and caninterpreted as a small region of Planckian density. Suchstantons can be used to describe the quantum creationclosed inflationary universe from space-time foam withothe need to use complex solutions.

ACKNOWLEDGMENTS

It is a pleasure to thank A. Chamblin, N. Kaloper, andA. Kofman for useful discussions. This work was supportin part by NSF Grant No. PHY-9219345 and by NATODAAD.

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