Stefano Ansoldi and Lorenzo Sindoni- Shell–mediated tunnelling between (anti–)de Sitter vacua

8/3/2019 Stefano Ansoldi and Lorenzo Sindoni- Shellmediated tunnelling between (anti)de Sitter vacua

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arXiv:0704.1073v

3

[gr-qc]10Sep2007

MIT-CTP-3732

Shellmediated tunnelling between (anti)de Sitter vacua

Stefano Ansoldi

Center for Theoretical Physics - Laboratory for Nuclear Scienceand Department of Physics, Massachusetts Institute of Technology

and International Center for Relativistic Astrophysics (ICRA) - Pescara - ITALY

Lorenzo Sindoni

International School for Advanced Studies, SISSA/ISASvia Beirut, 2-4 I-34014 Miramare, Trieste (TS), ITALY

and INFN, Sezione di Trieste

We give an extensive study of the tunnelling between arbitrary (anti-)de Sitter spacetimes sepa-rated by an infinitesimally thin relativistic shell in arbitrary spacetime dimensions. In particular, wefind analytically an exact expression for the tunnelling amplitude. The detailed spacetime structuresthat can arise are discussed, together with an effective regularization scheme for before tunnellingconfigurations.

PACS numbers: 04.60.Kz, 04.60.Ds, 98.80.Cq, 98.80.Qc

I. INTRODUCTION

The study of vacuum decay initiated more than 30years ago with the work of Callan and Coleman [ 1, 2]; inthe following years the interest in the subject increasedand the possible interplay of true vacuum bubbles withgravitation was also studied [3, 4], together with bub-bles collisions and their importance in the early universe[5, 6]. At the same time, and as opposed with the truevacuum bubbles of Coleman et al., false vacuum bubbleswere also considered. In connection with gravity, the be-havior of regions of false vacuum, first studied by Satoet al. [7, 8, 9, 10, 11, 12], was for example analyzedin [13, 14, 15, 16, 17, 18, 19, 20]. For additional pa-

pers analyzing it in the context of inflation, we refer thereader to [21, 22, 23]; interesting links with more phe-nomenologically oriented approaches can also be drawn,as witnessed, for instance, by the recent [24].

In these last works, a minisuperspace approximationwas adopted to quantize the system. In more detail, sincegeneral relativistic shells1 can be used as a convenientmodel and since in spherical symmetry the system onlyhas one degree of freedom, standard semiclassical meth-ods might be suitable to analyze the decay process (see[25] for an early, in principle, discussion of this point).In connection with cosmology, spaces equipped with acosmological constant (i.e. de Sitter space and general-izations) have been naturally considered. In this context

Electronic address: [email protected];URL: http://www-dft.ts.infn.it/ansoldi(Mailing address) Dipartimento di Matematica e Informatica -Universita degli Studi di Udine, via delle Scienze 206, I-33100 Udine(UD), Italy; INFN, Sezione di Trieste, Trieste, ItalyElectronic address: [email protected] In this paper we will use interchangeably the terms shell, bub-

ble and brane, which in different epochs have appeared in theliterature to designate the system under consideration.

it also worth to remark the important role that they play

in connection with the problem of gravitational entropy,causal structure and the presence of horizons (see, for in-stance [26, 27, 28, 29, 30] as well as the suggestive [31]).

Notwithstanding many interesting results, after 30years, and with different flavors, the problem of the sta-bility of (the de Sitter) vacuum in connection with thedynamics of false vacuum bubbles, is still a debated one[19, 32, 33].

The still open issues are highly non-trivial and goback to the, also long lasting, problem of formulatinga consistent framework for a quantum theory of gravity[34, 35, 36, 37, 38, 39, 40], but we will not take explicitlythis point of view here. We will instead analyze a specific

situation, described below, in which the nucleation ratecan be computed exactly in arbitrary spacetime dimen-sions. We will, then, explicitly compute in closed formthe nucleation rate in the semiclassical approximation,compare it with existing results, and discuss in detail theassociated spacetime structures: an analysis of how quan-tum effects may be relevant in the context of tunnelling

from nothing configurations [41] will also be given. Theseresults, extend some results present in the literature (forinstance [16, 42, 43, 44]) and can also provide a use-ful limiting case of more general situations, for instancethose discussed in [33]. At the same time, to considerspacetimes of higher dimensionality and negative cosmo-

logical constant is especially important in view of recentresults in the context of the AdS/CFT correspondence[45] and of the braneworld cosmological scenario [46, 47](see also [48] and references therein for a study in thecontext of noncommutative branes).

Apart from the papers already cited above, the in-stanton approach has also been discussed by other au-thors (see for instance [49, 50, 51]) and although it willnot be directly related to the present paper, we cannotavoid mentioning the suggestive relationship between thedecay of the cosmological constant, membranes gener-ated by higher rank gauge potentials and black holes,
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which have appeared in many papers in the literature[52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62].

The structure of the paper is the following: in sec-tion II we recall the formalism by which a general rela-tivistic thin spherical brane/shell can be described; thisalso gives us the opportunity to fix conventions and no-tations and briefly describe the canonical approach toits semiclassical quantization (subsection II C); a conve-

nient dimensionless formalism is also introduced (subsec-tion II D). We then directly come (section III) to themain result of this paper, which is the calculation ofthe tunnelling rate between the classical configurationsof the system, in arbitrary spacetime dimensions; the re-sults for the cases of 3, 4 and 5 spacetime dimensionsare explicitly presented with dedicated plots of the val-ues of the action as a function of the two dimensionlessparameters of the model (subsection IIIC). The exact re-sults for the tunnelling amplitude/probability calculatedin section III correspond to specific transitions which takeplace in spacetime and that will be discussed later on, ina dedicated appendix. We discuss, instead, in the main

text (section IV) a proposal to regularize some space-time configurations which appear to be singular, relatingthis issue to the description of the brane energy-mattercontent. In the concluding section V, the results of thepaper are summarized; two appendices follows with thedetailed description of the parameter space of the system(appendix A) and of all the Penrose diagrams associatedwith different values of the parameters of the problem(appendix B): they are crucial to fully grasp the physi-cal system under consideration.

II. SHELL IN N + 1 DIMENSIONS

In this section we are going to briefly review someresults about the dynamics of co-dimension one branesin an (N + 1)-dimensional spacetime, where, under thewords co-dimension one brane, we understand an N-dimensional hypersurface separating the (N + 1)-dimensional manifold in two domains, M and M+ hav-ing as a common part of their boundary: in briefM M+ = . In what follows we are going touse the clear notation of [63]: the formulation developedthere can be readily extended to higher dimensions, justby letting the indices run on an extended set of values.

We will thus quickly report this paraphrase with the pur-pose of recalling some notations and conventions. In par-ticular let us choose two arbitrary systems of N + 1 in-dependent vector fields E(a) in M, respectively, withdual forms

(b) . Denoting with g() the four dimen-

sional metric tensors in the two manifolds we can write2,

2 In a few equations below, we will use the notation , todenote the scalar product. We also anticipate the notation YXfor the covariant derivative of the vector fiend X in the direction

in general,

g() = g()ab(a)()

(b)();

in our notation Latin indices a, b (as well as all other latinindices) will vary in the set {0, 1, 2, . . . , N }.

Let us first concentrate our attention on : it is alsoa manifold, as

M, and we will denote by e() an N-

dimensional system of (commuting) independent vectorfields on , with dual system (); the indices and (as well as all other Greek indices) will vary in the set{0, 1, . . . , N 1}. Because of the physical interpretationwe will be interested in the case in which is embeddedas a timelike surface in M. Since is timelike thenormal to in M is a spacelike vector n, which wechoose normalized so that n,n = +1. Moreover, ourconvention is that the normal points from the to the+ domain of spacetime.

The components ofn (an unambiguously defined non-null vector) will be different, in general, when measuredby an observer in

M or by one in

M+, according to

the following definition:

na| = n,E(a)| = n,E(a).In terms ofn the extrinsic curvature of the surface canbe expressed as3

K = n,e()e()and, in general, it is different on the and + side. More-over, by the Gauss-Codazzi formalism, the geometry ofspacetime around can be described in terms of the in-trinsic geometry of the hypersurface and of its extrinsiccurvature. In the spirit of this formalism, let us denoteby h the intrinsic metric of the hypersurface , i.e.

h = h() ().

When we will work with quantities defined in the bulk wewill need to distinguish the ones defined in M+ from theones defined in M, and to this end we will use, as wedid above, superscripts or subscripts. In many caseswe will also be interested in the jump of these quanti-ties across : for instance, if we consider the extrinsiccurvature, we may need the to consider the differenceK+ K : following [63, 64] we are going to rewritethis difference as [K ]. Throughout this paper this will

be the only meaning that we will give to the square brack-ets, i.e.

[A]def.= A+ A.

To avoid confusion no other use of the square bracketswill be done.

of the vector field Y.3 We stress the normalization condition on n.


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A. Junction conditions

The brane can be more than just a mathematicalsurface, i.e. we can (and, in most of the cases, we wantto) equip it with a matter-energy content: it is then an in-finitesimally4 thin distribution of matter-energy. Thanksto the above mentioned Gauss-Codazzi formalism, we canrewrite Einstein equations to make explicit the contribu-tion from the localized matter. Then, the dynamics of as a surface separating M+ from M is obtained solvingthe following system of equations,

K+ K [K ] = 8GN+1 (S hS/2) ; (1)

these are Israels junction conditions [63, 64] which re-late the jump in the extrinsic curvature [K ], i.e. thejump in the way the surface is embedded in each ge-ometry, to the stress-energy tensor S of the mattercontained on . The tensor Smust also satisfy a conser-vation equation,

(N),S = [T(n,e())],

where T is the stress energy tensor describing the con-tent of the complete spacetime manifold. Once we havespecified the matter content of the bulk (and hence thegeometry according to Einstein equations) the descrip-tion of the dynamics of the system is obtained by solvingthe system of equations (1); we refer the reader to [63]and [66] for additional material and related considera-tions.

B. Spherical symmetry

The set-up that is of interest for us is a simplified one,in which all the system is spherically symmetric and thesurface stress-energy tensor is that of the, so called, ten-sion model, with

S= h,

where > 0 is a constant called the tension of the brane.In what follows, it will be convenient to also define

= 4(N

2)GN+1.

Thanks to the spherical symmetry, the system of equa-tions (1) can then be reduced to a single equation [67],

R

R2 + f(R) +

R2 + f+(R)

= R2; (2)

4 Strictly speaking is a source defined in a distributional way:for additional material on this point, the reader is referred, forinstance, to [65].

in the above, f(r) are the metric functions of the staticline element adapted to the spherical symmetry, i.e., tak-ing the four dimensional case as a convenient example,we choose in both M the coordinate system xa =(t, , , r) corresponding to the basis vectors et ,e , e and er , such that the metric is reduced to the

form g()ab = diag(f(r), r2, r2 sin2 , 1/f(r)).Generalizations to spacetimes with higher dimensionalityare just more cumbersome to write but trivial in theirsubstance. In these coordinate systems, we denote theradius of the brane by R(), where is the proper timeof an observer comoving with the brane and an over-dot denotes the derivative with respect to . Moreover,as anticipated in the introduction, we consider that thisN-dimensional brane separates two (N + 1)-dimensionalspacetimes of the de Sitter/antide Sitter type, in gen-eral with different cosmological constants . We thushave

f(r) = 1 2N(N

1)

r2.

Finally are signs, defined as

(R) = sign(n, er)|r=Rand are crucial quantities to obtain the Penrose diagramsassociated with the considered brane configuration.

C. The effective action/the momentum

Before proceeding with the analysis of the physicalsystem that we introduced in the previous subsection,

we believe it is useful to recall some important pointsabout the structure of the junction conditions in spheri-cal symmetry. For a generic junction between sphericallysymmetric spacetimes across a (spherical) brane carry-ing matter described by a given, but otherwise arbitrary,equation of state, it would prove very useful to extractan effective action for the dynamics of the brane start-ing directly from the Einstein-Hilbert action and from asuitable action for the matter fields and the brane [67].This is the content of a consistent literature on the sub-ject [68, 69, 70, 71, 72, 73, 74, 75, 76] among whichwe would like to single out the more general, and recent,[77]: the problem is not trivial at all, because of the same

subtleties that appear, for instance, in the Hamiltonianformulation of General Relativity. Among the variouspossible approaches we will closely follow the quantiza-tion procedure originally proposed by Farhi, Guth andGuven [16], which has, lately, been followed also in [67]and [78]. For the Lagrangian approach to the classicaldynamics we will instead follow the clear and more gen-eral exposition of [77]. We summarize here the relevantelements of these approaches referring the reader to theliterature for additional details. In particular, the effec-tive action for the shell can be obtained starting fromthe Einstein-Hilbert action with the Gibbons-Hawking


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boundary terms [16, 77, 79]. Using the Gauss-Codazziformalism [80], it is readily seen that the Einstein-Hilbertaction for a spherical co-dimension one brane (which inour case will separate two domains of (anti)de Sitterspacetime) can be decomposed in non-dynamical bulkcontributions (indeed, we consider the geometry of thebulk spacetimes M fixed) and a dynamical boundarycontribution described in terms of the extrinsic curva-

ture of the brane . It is enlightening (and, perhaps,necessary) not to use all the freedom in fixing the coor-dinate systems. As an exemplification we do not use the

freedom given by the reparametrization invariance withrespect to the proper time of the brane and we, thus,introduce a lapse function N(s), so that the brane in-duced metric can be written as h(

N) = diag( N(s)2,

R2(s), . . .) in the coordinates (s , R , . . .) (the ... standfor the trivial spherically symmetric part). Then, follow-ing [67, 77], the effective action for the degrees of freedomassociated with the radial and time coordinates takes theform

Seff

RN2R2 +N2f(R) R arctanh

R

R2 +N2f(R)

sign(f) NRN1

d. (3)

Thus, the effective Lagrangian of the system is given by

L = P(N)R H,where, following [67], we have defined the effective Hamiltonian as

H = RN2

R2 +N2f(R)

+ NRN1

and the effective momentum as

P(N)(R, R) = RN2

arctanh

R

R2 +N2f(R)

sign(f)

. (4)

Then, the equations of motion for the R and N degreesof freedom are

dHd

=N

NH and H(R, R,N) = 0;

the second equation is a first integral of the former one,and is the Hamiltonian constraint. We see that it en-codes all the information about the dynamics of the sys-tem, being identical to the only remaining junction con-dition. The first equation is the second order equation of

motion of the system, given by the total derivative withrespect to the proper time of the Hamiltonian constraint.In what follows, we will be mostly interested in the ex-pression of the momentum evaluated on a solution of theequation of motion. In order to build a canonical struc-ture (as it is done, for example, in [46]) we first have toprove the existence of a well defined symplectic structureon the phase space of the Lagrangian system; in partic-ular the Legendre transform has to be invertible, whichin our case means nothing but the invertibility of theconjugate momentum as a function of the velocity [81].As appreciated already in [16], for generic co-dimension

one branes this cannot be proved to be always satisfied5.However, for the system under consideration the canon-ical structure always exists and P(N) can rigorously beconsidered the canonical momentum conjugate to the co-ordinate R; thus, in the specific case of interest here,the canonical construction of the corresponding quantumtheory is a well posed problem. To prove this statement,let us consider

P(N)

R

=

RN2

1

R2 +N2f(R).

This quantity can be zero if and only if two conditionsare simultaneously satisfied:

1. + = ;

5 In these cases the quantum theory cannot be built with thecanonical formalism. Alternative formulations using the pathintegral approach have been considered in the seminal work ofFarhi, Guth and Guven [16].


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2. there is a point R in the configuration space suchthat f+(R) = f(R).

For our particular choice of branes in (anti)de Sitterspacetime, we see that in order for the second conditionto hold we must have = +; but this implies

6 + =1, = +1 so that the first condition then fails. Thus,for the class of junctions that we are considering one can

always build a canonical structure.

D. Dimensionless formalism

Following, for instance, [82], it is possible to deter-mine the classical dynamics of the system by studyingan equivalent one dimensional problem for the radial de-gree of freedom R, taking also into account the values ofthe signs (R).

Before proceeding we will set up a different systemof dimensionless quantities, in order to remove the ar-bitrariness in the definition of the normalization of the

cosmological constants and of the bubbles tension. If wedefine the quantities

x = R,

= ,

=2

2N(N 1) ,

= + +,

= +,equation (2) then becomes7

x2 + 1 x2

x = x2. (5)

For completeness we also report the dimensionless formof the effective momentum (4) when we impose the gaugechoice N 1:

P(x, x) = xN2 arctanh

x

x2 + f(x)

f|f|

. (6)

This quantity will be central in what follows, but for themoment we keep our attention on equation (5). Despite

its unusual look, it is easily proved that it is equivalentto a system of equations, which, in the notation that weare using, takes the form

x2 + V(x) = 0(x) = sgn( 1) , (7)

6 This can be easily seen from the junction condition (2) remem-bering that > 0.

7 An overdot will denote, from now on, a derivative with respectto .

where the potential V(x) has a simple parabolic form

V(x) = 1 x2

x20(8)

and x0, the turning point, is given by

x20 =4

1 + 2 + 2

, (9)

when the right-hand side is positive; otherwise there areno nontrivial classical solutions. All the solutions of theproblem can be easily classified in according to the valuesof and : we refer the reader to appendix A for details.The expressions for the signs relate, instead, the globalgeometrical structure of spacetime for this brane configu-ration only to the difference between the inner and outercosmological constants, not to the details of the trajec-tory itself. This is a portion of the content of the junc-tion condition, that is necessary for the description ofthe global spacetime structure: a careful discussion ofthis point can be found in appendix B.

The form of equation (5) for a brane separating two(anti-)de Sitter spacetimes, which, in turn, is responsiblefor the simple quadratic form of the potential (8), allowsexact solution of this particular case. It is, in fact, notdifficult to see that (5) has:

1. the trivial solution x() 0, which always exists;2. the solution

x() = x0 cosh

(0)

x0

, (10)

which satisfies the initial condition x((0)) = x0;

this solution is known as the bounce solution andexists only if the condition

1 + 2 + 2 > 0

is satisfied.

We also, incidentally, note that

= 1 1x20

=

i.e. when || = 1 the turning point radius coincides withone of the two cosmological horizons, if they exist. We

also anticipate that a more careful analysis of the trivialsolution x() 0 will be required, since, although it isgiven to us by the mathematics of the problem, we aremostly interested in its physical role, especially when wewill turn on (semiclassical) quantum effects (see sectionIV).

III. TUNNELLING

Using the notation introduced in the last section wecan now describe how the semiclassical regime of the


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brane dynamics looks like. In particular we can considerthe tunnelling between the zero-radius solution towardsthe bouncing solution (10), and vice versa.

For example in one direction the semiclassical pictureis as follows: we have a brane, of very small radius; ofcourse, when it is very small, the quantum properties ofits matter content will be non-negligible (we will furtherdiscuss this issue later on in section IV) and their in-

terplay with gravity will be non-trivial; although we donot know the full quantum gravity description of the sys-tem, we will consider that, thanks to quantum effects, thebrane will have a certain probability to tunnel under thepotential barrier given by the effective potential (8) intothe bounce solution. When emerging after the tunnellingquantum effects will likely become less and less impor-tant as compared with gravitational ones (and as far asthe interaction with the bulk spacetime is concerned),so that the evolution of the brane will closely resemblethat of the classical junction. This is not the case in thefirst stage of the evolution, involving the tunnelling pro-cess, and we will try to understand, at least at an effec-

tive level, the responsibilities of both the quantum andgravitational realms in this process. In particular, quan-tum effects will be considered in a modification at smallscales of the stress-energy tensor (see section IV). Grav-itational effects, at small scales, are also present, anddescribed by the geometric character of Israel junctionconditions. We think that our description might be ableto take into account both these effect, at least within thelimits represented by the semiclassical approximation. Inthis sense, although our treatment will be effective, it willgive us the possibility to consistently solve the ambiguityarising at the mathematical level and represented by thex()

0 solution. We will suggest that, at the semiclassi-

cal level, a more detailed treatment of both, the quantumand gravitational aspects, is unlikely to be necessary.

A. Tunnelling trajectories

For the system under consideration, tunnelling trajec-tory can be described by constructing the instanton inthe Euclidean sector. The construction of the instantonin the general case is still an unsolved problem: we willfollow the approach of [16]; moreover, the problems leftopen in [16] do not affect the present case. In fact it is

easy to see that in our case the instanton describing thetunnelling can always be constructed without the neces-sity to introduce the pseudo-manifold that in [16] wasnecessary to deal with multiple covering of points in theEuclidean sector. Moreover the different descriptions ofthe tunnelling process that were obtained in [16] usingthe canonical or the path-integral approaches in our casealso coincide, because they are consequences of proper-ties of the dynamical variables during the tunnelling tra-jectory which are not present in the system that we areconsidering (please, see [83, 84, 85] for a more detaileddescription of these issues).

In this way we know that the tunnelling can be de-scribed directly at the effective level, where, the relevantaspects of the Euclidean junction can be determined bythe Wick rotated classical effective system. We thus de-fine e = i; then, denoting with a prime the derivativewith respect to e, we have that the Euclidean system isobtained with the formal substitutions x ix for thevelocity and

P iPefor the momentum (it is not difficult to prove that theabove substitution rules are rigorous results and that theycan be obtained performing an Euclidean junction, asdone, for instance, in [16]). In particular

Pe(x, x) = xN2

arctan

x

f(x) (x)2

. (11)

In this way, the tunnelling process can be modelled us-ing the effective tunnelling trajectory, which solves theEuclidean equation:

(x)2 + 1 (x/x0)2 = 0. (12)The analytic expression of the tunnelling trajectory ofa brane expanding from zero radius at Euclidean timee = 0 to the bouncing solution is:

x(e) = x0 sin(e/x0).

The opposite process is then described by

x(e) = x0 cos(e/x0),

if we assume that at Euclidean time e = 0 the branestarts contracting from x

0. These two processes occur

with certain probabilities, whose amplitudes A can beexpressed in the usual semiclassical approximation as

A(0 x0) exp(Ie[x]),where Ie[x] is the Euclidean and can be obtained as theintegral of the Euclidean momentum evaluated on a so-lution of the Euclidean equation of motion (12). If, forsimplicity, we define

Ie[x] =N1

16GN+1

1

N1Ie[x],

where N1 is the volume of SN1, then Ie[x] can be

obtained as

Ie[x] =

x00

Pe(x)dx. (13)

We stress again that Pe(x) is the Euclidean momentumevaluated on a solution of the Euclidean equation of mo-

tion and can be obtained by substituting x =

V(x) in(11). We thus have

Pe(x) = xN2

arctan

2

V(x)

(+ )x

,


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where, to make writing more compact8, we have intro-duced the quantities defined as = 1.

Note that in the Euclidean case we have some freedomin appropriately choosing one of the possible branchesof the inverse tangent function. Different choices willaffect, in general, the result that we obtain for the action.As in [16] we observe that non-careful choices will makethe action a discontinuous function of the parameters;

this can be seen without difficulties. Preliminarily, let usanticipate that with the symbol arctan, we will indicatethe branch of the inverse tangent function with rangein [/2, /2]. Let us then consider = and let usintegrate by parts the integral (13). We obtain

Ie[x] =

xN1

N 1 arctan

2

V(x)

(+ )x

x0

0

+

+

x00

xN1

N 1 d

arctan

2

V(x)

(+ )x

.

Let us now consider the first term above. Clearly, if wechose the branch of the arctan function to be the onewith range in [/2, /2], then we have

lim

xN1

N 1 arctan

2

V(x)

(+ )x

x0

0

= xN10

2(N 1)

but

lim+

xN1

N 1 arctan

2

V(x)

(+ )x

x0

0

= +xN102(N 1) ,

so that the action develops a discontinuity at = .This discontinuity can be eliminated if we choose dif-ferent branches of the inverse tangent functions (pleaseremember that in the expression that we are consideringwe are using a compact notation to indicate the differenceof two inverse tangent functions); in particular the twodiscontinuities at = 1 can be eliminated by choosing:

1. the branch with range [, 0] for the arctan func-tion containing quantities of the + spacetime;

2. the branch with range [, 0] for the arctan func-tion containing quantities of the

spacetime.

Since in our notation arctan has range [/2, /2], thismeans that in the equations above must be rewritten withthe substitutions ( is the step function)

arctan( + ) arctan( + ) (+ 1)

8 In view of the definition of the s, and of the meaning of thesquare brackets, the equation above is a shorthand for Pe(x) =

xN2

+ arctan

2V(x)

(+1)x

arctan

2V(x)

(1)x

.

and

arctan( ) arctan( ) ( 1);

for the jump of the quantity which appears inside theexpression of the Euclidean momentum, this implies thefollowing substitution:

[arctan] [arctan] (1 2).In this way the action integral is continuous also at thepoints = and is given by the integral

Ie[x] = x00

dxxN2

arctan

2

V(x)

(+ )x

(1 2)

.(14)

After these preliminary considerations, we can proceed toevaluate the tunnelling amplitude in the WKB approxi-

mation; as we anticipated and as we will see, in arbitraryspacetime dimensions this result can be expressed ana-lytically in terms of known functions.

B. General result for the tunnelling amplitude

To calculate the first contribution to the integral (14)we proceed as follows. As a preliminary step, we observethat it can be written as a difference of two integrals withthe same general structure. Let us then consider the twointegrals containing the inverse tangent functions: smalldifferences, which do not substantially affect the calcu-lation, can be taken into account by properly using thes introduced above, as we already did in the expres-sions for the momentum. This said, we can perform9

an integration by parts in (14). The terms evaluated atthe limits of integration, which appear in this process, dovanish and by changing the integration variable from xto = (x/x0)

2 (which also transforms the integration do-main into the unit interval (0, 1)) we obtain the followingexpression,

Ie[x] = xN0

4(N 1)

(+ )10

(N2)/21

11 zd

+ (15)

+xN10

(N 1) (1 2),

9 Strictly speaking this can be done only when = . The casesin which = are trivial and can be dealt separately; or,more simply, since we already know from the discussion in theprevious subsection that the action is continuous, we can justextend it to = by continuity.


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8

where

z =

1 x0 +

2

1 + x0

+

2

. (16)

When z < 1 one of the above integrals diverges sincethere is a pole of the integrand on the domain of inte-gration. On the other hand, it is easy to see that thecondition z

< 1 implies

x204

(+ )2 < 0

and cannot be realized for any choice of the parametersif a tunnelling trajectory has to exist. The value z = 1can instead be obtained if = and we know thatthe action can be extended by continuity to these values,although the above procedure to calculate the integral isnot valid. Thus, under the conditions = , we havethat z > 1 is satisfied and equation (15) gives

Ie[x] =

xN0

4(N 1)(N/2)(1/2)

((N + 1)/2)

(+ )2F1

1,

N

2,

N + 1

2, z

+ (17)

+xN10

(N 1) (1 2),

where is the Eulers gamma function, 2F1 the hyper-geometric function. Note that x0 depends on , and sothe first factor of the formula, depending on the numberof spacetime dimensions, cannot be ignored even for aqualitative description of the tunnelling amplitude.

C. Some cases of interest

It is useful to specialize the result (17) to particularsituations. We are going to do this by considering thecases in which spacetime is three, four and five dimen-sional. Below we are going to explicitly discuss thesethree cases. A comparative presentation of the resultscan be found in the contour plots of figure 1.

We start then with the three dimensional case, which islower-dimensional gravity; in three spacetime dimensionsit seems more clear how to build a quantum theory outof the classical junction conditions [46]. Moreover three

dimensional gravity is an interesting system by itself, itallows an easier visualization of some results and can beused for interesting specific toy models. Anyway, in thiscase we have N = 2 and we can express 2F1 (1, 1; 3/2; x)in terms of elementary functions as

2F1

1, 1;

3

2; y

=

arcsin(

y)1 yy . (18)

Correspondingly the action becomes

Ie = x20

4

(1)(1/2)

(3/2)

(+ )arcsin(

z)

1 zz

+ (19)

+ x0(1 2).

The corresponding probability is plotted as a function of in figure 2 for some non-negative values of and infigure 3 for some negative values of .

Not many comments are necessary, of course, for thefour dimensional case, the original arena on which thiscalculation was performed. Again we can take advantageof a simple expression for the corresponding hypergeo-metric function

2F1 (1, 3/2;2; y) =2

y

1 y 2

y, (20)

which brings the final result in the form

Ie = x20

4 (21)

sgn(+ ) 1 (1 z)1/2z

(1 2) .The plots of the probability as a function of can be

found in figure 4 for some non-negative values of andin figure 5 for some negative values of . This result isthe same as the one obtained in [86] and it also reducesto the one calculated in [3]: it, thus, represents a usefulconsistency check.

Finally, the five dimensional case is interesting in thecontext of the Randall-Sundrum scenario. In this case,using

2F1

1, 2;5

2 ; y

=3

2y arcsin(y)1 yy1/2 1 , (22)

the result for the action integral can be put in the fol-lowing form:

Ie = x40

12

(2)(1/2)

(5/2)

(+ )

arcsin(

z)

1 z(z)3/2 1

z

+ (23)

+x30

3(1 2).

Again we present two plots of the corresponding proba-bility as a function of ; figure 6 shows the behavior forsome non-negative values of , whereas plots for somenegative values of can be found in figure 7.

IV. DISCUSSION

Up to this point we have discussed the physics of thesystem rather quickly, focusing mainly on the mathe-matics necessary to describe the classical and the semi-classical phases. There is still a point which deserves a


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9

a b c+10

+5

0

5

10

+10

+5

0

5

104 2 0 +2 +4 4 2 0 +2 +4 4 2 0 +2 +4

4 2 0 +2 +4 4 2 0 +2 +4 4 2 0 +2 +4

no tunnel zone no tunnel zone no tunnel zone

FIG. 1: Contour plots of the values of the tunnelling probability in 3 (case a), 4 (case b) and 5 (case c) spacetime dimensions inthe plane. The area inside the crosshatched parabola in the bottom of each picture is the region in parameter space wheretunnelling cannot occur since no bounce solutions exists (no tunnel zone). A better understanding of what happens close tothe limiting parabola can be understood from the following figures 3, 5 and 7. Next to next contours indicate a jump of theprobability of 0.1, with the lightest gray tone indicating the range (0.9, 1.0) and the darkest the range (0.0, 0.1). The samecolor indicates the same range of values of the probability in all the three sub-diagrams.

detailed discussion, since it involves non-trivial aspects ofthe dynamics of the brane. Indeed, the physical processwe have considered is the tunnelling of spacetime, froma classical situation representing a spacetime containinga small brane to another classical configuration describ-ing a bouncing brane. In our picture, the pre-tunnelling

state is represented by the x 0 solution of the junctioncondition. Although this case might appear rather simpleat first sight and one could be tempted to just state thatthe initial state is, for example, the full M+ spacetime (apoint of view which has been taken for example in [67]),much more care has to be taken. The main reason is thatwhen x = 0, the branes world-volume degenerates to acurve. Thus, the formalism that we used to describe thebrane breaks down and the analysis that we have madecannot be considered as rigorous as it is in the case of thebouncing solutions. As a manifestation of this fundamen-tal problem, we observe that the equations determiningthe signs do not hold for the particular solution x

0.

In particular, for this solution the junction condition aswritten in (5) does not provide any mean to solve thisproblem.

A way out of this situation appears when we reflect onthe fact that this process, although described semiclassi-cally, is quantum in its true nature. Thus an approxima-tion of the system which considers it completely classicalbefore the tunnelling, i.e. in the degenerate configuration,might be too rough and might require a more careful con-sideration. During the tunnelling trajectory, and evenmore for the x 0 classical solution, quantum effects are

supposed to be, if not dominant, at least relevant enoughto modify the classical picture of the junction.

We will propose here an attempt to address the prob-lem. Our proposal should be considered a first step fur-ther, but far from a first principle solution of the com-

plex quantum problem; it just aims to show that themathematical and physical aspects of the problem statedabove can be dealt with by giving an effective formulationfor the phenomena that might arise at small scales. Atthe same time, although we will just build an effectivemodel, we will not be very demanding about its mainproperties: in this way, hopefully, the model, althougheffective, could mimic well enough effects produced byquantum gravity whatever will be their, still undiscov-ered, true nature.

In this respect, we would also like to point out the fol-lowing: i) the main effect of our proposal is to slightlyperturb the effective potential (8) in the tunnelling re-

gion; then, we will ii) show that the perturbed problemis free from ambiguities and iii) use for the unperturbedcase the results obtained in the perturbed one, when theperturbation becomes smaller and smaller. To have adefinite model, we will regard the small scale behaviorof the matter composing the brane/shell as the physicalorigin for the perturbations (see below). On the otherhand, the consequence of these perturbations in the ef-fective formulation is completely generic: it is, in fact,possible to show that other physical motivations, as forinstance the quantum properties of spacetime at smallscales, could be reflected in a similar way in the effective


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10

-4 -2 0 2 4

0

0.2

0.4

0.6

0.8

4.0 2.0 0.0

+2.0 +4.00.0

0.2

0.4P

0.6

0.8

1.0

-0.4 -0.2 0 0.2 0.4

0.005

0.01

0.015

0.02

0.025

0.030.03

0.02

0.01

0

0.04 0.02 0 +0.02 +0.04

= 10.0 = 7.0 = 4.0 = 3.0 = 2.5 = 2.0 = 1.5 = 1.0 = 0.5

= 0.0

FIG. 2: Plots of the values of the tunnelling probability in a spacetime of dimension 3, as a function of for fixed non-negativevalues of as listed above. The detailed behavior for = 0 around = 0, is also shown (to better see that the probability issmall but non-zero).

formulation. For these reasons, we can regard our conclu-sions model independent to a high degree, at least withinthe context defined by the semiclassical approximation.

To introduce our model we can naturally think thebrane as sourced by some matter fields whose nature is,ultimately, quantum. We thus argue that, although atlarge scales the approximation for the brane stress en-ergy tensor that we made in subsection II B might berough but still appropriate, at small scales it will insteadbreak down due to quantum effects. We will model theseadditional quantum effects by adding a term to the branestress-energy tensor as follows:

S S+ Sq, (24)

where the quantum contribution Sq will be parametri-zed as

Sq = qu u + qh. (25)

The conservation equation for Sq under the assumptionof spherical symmetry gives (we are going to use, fromnow on, the dimensionless versions of the parameters qand q, which following the notation above, are called q

and q)

dqdx

+ Nqx

=dqdx

. (26)

We will choose preliminarily

q(x) = axq (27)

so that

q(x) = aq + N

qxq . (28)

In this case the junction condition becomes10

(q)

x2 + 1 x2

x = x2(x), (29)

10 We remember that we are following the convention in whichsquare brackets indicate the jump of the enclosed quantity, i.e.the difference of it when evaluated in the + and domains.We are now using the suffix . . . q or . . . (q) to indicate quanti-ties when the energy momentum tensor is modified, as describedin the text, to take into account quantum effects; thus the twosigns will now be (q).


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11

0 1 2 3 4 5

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

0.2

0.4P

0.6

0.8

1.0

= 0.50

= 1.00

= 1.50

= 2.00

= 2.50

= 3.00

= 4.75

= 6.50

= 8.25

= 10.00

FIG. 3: Plots of the values of the tunnelling probability in a spacetime of dimension 3, as a function of for fixed negative valuesof as listed above. For < 1/2, not all values of are allowed. We have put the mark on each curve in correspondence ofthe minimum value of for which (at the given value of ) the tunnelling process can exist. For smaller values, the infinitelyexpanding solution, in fact, is not present.

with

(x) = 1 + axq (30)

and, for short, a = a(q + N)/q. Now we are going toslightly restrict the parameters a and q to make thesegeneral settings appropriate for our model. In particular,the modification to the stress energy tensor, described bya and q was introduced to model the quantum effects atsmall scales; thus it should be negligible at large scalesand this can be achieved if q + 1 < 0, i.e. q < 1.

We can now compute the modified effective potentialVq. The potential turns out to be

Vq(x) = x2

2 + 2(1 + axq)2 + (1 + axq)4

4(1 + axq)2

(31)

x0

a2

4x2q2 (32)

a0 V(x) +

(1 2)a22xq2

. (33)

Extracting the behavior of (31) for small x we get (32),whereas for small a the leading contribution is given by(33). These are useful results. Indeed we see from (32)that, quite generally, and certainly under the above q 1, sothat is always greater than +. The classical junc-tion is seen (by an observer travelling toward increasingvalues of the radius) as a transition from a more nega-tive to a less negative value of the cosmological constant,when he/she crosses the brane. The global picture ofthe tunnelling process is then obtained according to the


18/23

18

prescription discussed in section IV. The pre-tunnellingspacetime has been discussed above and now consists oftwo small parts of antide Sitter spacetime with differ-ent cosmological constant. The tunnelling process showsthe transition between a compact spacetime composedby two anti-de Sitter regions to a spacetime similar toantide Sitter, except for the fact that at some radiusthe cosmological constant changes its value.

r

=

0

r

=

0

AdS, = +1 AdS+, + = 1

r

=

0r

=

0

r

=

0r

=

0

AdS+/AdS Tunnelling

FIG. 12: Case A2: AdS/AdS, = +1; + = 1

The junction of type A2, is instead a junction whereeffectively a tiny junction of two anti-de Sitter space-times is inflated (figure 12). In this case the signs aregiven by = 1. Thus the normal to the brane pointsin the direction of increasing radii in the antide Sitterspacetime with cosmological constant , but it pointsin the direction of decreasing radii in the antide Sitterspacetime with cosmological constant +. The net effectof the tunnelling process, which starts from the alreadydiscussed initial state, is thus to inflate the pre-tunnellingcompact spacetime into a similar, but much larger, one

(note that for graphical reasons the scales in the pre-tunnelling and post-tunnelling parts of the diagram aredifferent; we remember that the maximum radius of thepre-tunnelling state will be much smaller than any otherlength scale in the problem). The various diagrams arein figure 12.

The final type A diagram is the A3 one. This is verymuch as the A1 type, only that now = +1; in bothspacetime the normal to the shell trajectory points inthe direction of increasing radii. Thus (apparently) therole ofM is interchanged, as shown in figure 13 wherethe dark and light gray parts looks complementary to

r

=

0

r

=

0

AdS, = +1 AdS+, + = +1

r

=

0

r

=

0

AdS+/AdS Tunnelling

FIG. 13: Case A3: AdS/AdS, = + = +1

those in figure 11; on the other hand, now we have . The global spacetime structurein the bottom left corner of figure 13 implies that anobserver crossing the brane in the direction of increasingvalues of the radius, perceives again a transition from amore negative to a less negative value of the cosmologicalconstant. So this diagram describes exactly the sameprocess described by type A1, as it is possible to see fromthe diagram in figure 13.

2. Type B

The type B solutions, B1 and B2, correspond to the junction of a part of a de Sitter spacetime, M, witha part of antide Sitter spacetime, M+. Although thepre-tunnelling picture does not change very much, theconfigurations before the tunnelling are now de Sitteranti-de Sitter junctions.

Let us first discuss the case B1, shown in figure 14. Inboth spacetimes, the normal to the shell points in thedirection of decreasing radii, since we have = 1.The M part of spacetime corresponds to the regionof de Sitter complementary to the region between r = 0and the branes world-volume, whereas the antide Sitterpart M+ is the bounded region in the upper right pictureof figure 14. Thus the spacetime M shows a transitionfrom a negative to a positive value of the cosmologicalconstant for an observer crossing the shell in the direc-tion of increasing radius. The configuration before thetunnelling has been described above; the tunnelling pro-


19/23

19

r

=

0r

=

0

I

I+

r=

H

r =H

r

=

0

dS, = 1 AdS+, + = 1

r

=

0r

=

0

I

I+

r=

H

r=H

I+

r=

H r =

H

r

=

0r

=

0

AdS+/dS Tunnelling

FIG. 14: Case B1: dS/AdS, = + = 1

cess again inflates an antide Sitterde Sitter junctionwith small volume to a much larger one, as shown in thebottom right diagram of figure 14.

r

=

0r

=

0

I

I+

r=H

r=H

r

=

0

dS, = +1 AdS+, + = 1

r

=

0r

=

0

I

I+

r =H

r=

H

r

=

0r

=

0

I+

r =H

AdS+/dS Tunnelling

FIG. 15: Case B2: dS/AdS, = +1; + = 1

We then come to the case B2, shown in figure 15. Since

in this case the signs which fix the orientation of thenormals are = 1, the situation for M+ is as in theprevious case, but the M part of spacetime changesbecause of the change in the sign of +. This gives forthe junction the diagram in the bottom left part of figure15. The configuration before and after the tunnellingis shown in the bottom right corner of figure 15. Thesituation is very similar to the one in the previous case.

3. Type C

Type C solutions are the de Sitter counterpart oftype A solution. The main difference (not related to thespacetime structure) is that whereas for antide Sitterjunctions the values of the cosmological constant are re-stricted (i.e. given two arbitrary values a junction mightnot exists) de Sitter junctions exists for all values of thecosmological constants (this is clear from figure 10).Please, also remember that now the pre-tunnelling dia-gram will consist of a junction of two de Sitter spacetimewith different cosmological constants and with all theother properties described above.

r

=

0r

=

0

I

I+

r=

H

r =H

r

=

0r

=

0

I

I+

r=

H+

r =H+

dS, = 1 dS+, + = 1

r

=

0r

=

0

I

I+

r=

H+

r =H+

r =H

r=

H

r

=

0r

=

0

I+

r =H+

r=

H

dS+/dS Tunnelling

FIG. 16: Case C1: dS/dS, = + = 1

Let us start with case C1, shown in figure 16. Forthis junction, the signs are = 1, so in both de Sitterspacetimes the normals to the brane point in the directionof decreasing radii. Moreover, since in this case > 1,the cosmological constant in M will be bigger than thecosmological constant in M+, so that the cosmologicalhorizon H will be smaller than H+. The diagram for thejunction is shown in the bottom left part of figure 16. It is


20/23

20

interesting to see some effects of the brane on the space-time structure. There are observers that crossing thebrane can move behind the cosmological horizon ofMwithout going through H. They can also come out, bycrossing the brane in the opposite direction. At the sametime observers, by moving accurately and with propertiming, might end up behind the cosmological horizon of

M+ without crossing any horizon but using the shell as

a gate11. As usual no additional considerations are re-quired for the before tunnelling configuration. We note,instead, that the semiclassical transition has again theeffect of inflating a small quantum spacetime region,as clearly shown in figure 16.

r

=

0r

=

0

I

I+

r=

H

r =H

r

=

0r

=

0

I

I+

r=

H+

r =H+

dS, = +1 dS+, + = 1

r

=

0r

=

0

I

I+

I

I+

r =H

r=

H

r=

H+

r =H+

r

=

0r

=

0

I+

I+

r =H

r=

H+

r =H+

dS+/dS Tunnelling

FIG. 17: Case C2: dS/dS, = +1; + = 1

We now discuss case C2, in which the signs are =1. Then the normal to the brane in the de Sitter spacewith cosmological constant points in the direction ofincreasing radii, whereas the normal to the brane in thede Sitter space with cosmological constant + points inthe direction of decreasing radii. We have again tworegions of spacetime of the de Sitter type but with adifferent cosmological constant joined across the brane(bottom right part of figure 17). The full picture of thetunnelling process requires the consideration of the sameinitial configuration used in case C1 above. We also point

11 These considerations might be modified if we consider the influ-ence of the observer on spacetime. To neglect this influence, as afirst approximation, is fairly common in the literature to obtainsome hints about the global spacetime structure.

out that in this case, the sign of is not fixed, so either ofthe cosmological constants may be the bigger. In par-ticular, the junction in the bottom left corner of figure 17shows the situation in which + < , so that H+ > H.In the bottom right part of figure 17 a representation ofthe tunnelling process is shown.

r

=

0r

=

0

I

I+

r=

H

r =H

r

=

0r

=

0

I

I+

r=

H+

r =H+

dS, = +1 dS+, + = +1

r

=

0r

=

0

I

I+

r=

H+

r =H+

r =H

r=

H

r

=

0r

=

0

I+

r=

H+

r =H

dS+/dS Tunnelling

FIG. 18: Case C3: dS/dS, = + = +1

The last junction of the C type is C3, for which theusual set of diagrams is shown in figure 18. As all C typeones, this is a junction between two de Sitter spacetimesand since < 1 the relation between the cosmologi-cal constants is + > , so that H+ < H. At thisstage we might wonder if this junction is the same as C1(mirroring what happens between the junctions of the A1and A3 type). We will see that this is indeed the case.The normals now point in both spacetimes toward thedirection of increasing radii (since = +1). Thus the

junction is as in the bottom left part of figure 18. Whenwe consider also the configuration before tunnelling, wesee that the diagram in the bottom right part of figure16 is the switched color and reflected version of theone in the bottom right part of figure 18: we remembernow that the in the C1 case in the light gray partwas bigger than + in the dark side, but now exactly theopposite happens. Thus, despite the different coloring,case C3 represents the same physical situation of caseC1. This observation makes interesting to complete theanalysis for the remaining two cases, which is done in thefollowing subsection.


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21

4. Type D

To conclude the analysis of the global spacetime struc-ture we have to analyze what happens for the last kind ofjunctions, those of type D. These are junctions betweende Sitter and antide Sitter spacetimes, so that the pre-tunnelling configuration will also change accordingly, aswe have discussed for the previous cases.

r

=

0

r

=

0r

=

0

I

I+

r=

H+

r =H+

AdS, = +1 dS+, + = 1

r

=

0r

=

0

I

I+

r=

H+

r =H+

r

=

0r

=

0

I+

r=

H+

r =H+

dS+/AdS Tunnelling

FIG. 19: Case D2: AdS/dS, = +1;+ = 1

The case of the junction of type D2 is characterizedby the following signs: = 1. This means that in theantide Sitter side the normal points in the direction ofincreasing radii, whereas in the de Sitter part, it pointsin the direction of decreasing radii. Thus the junction,shown in the bottom left corner of figure 19, is the mir-ror image of the B2 case. Again, the considerations insection IV determine the geometry before the tunnelling.This brings, in complete analogy with the B2 case, tothe picture of the tunnelling process given in the bottomright part of figure 19; again, apart from the different col-

ors, this case is the same as B2 also in connection withthe semiclassical tunnelling process.

We have thus only one case left, which is quickly dealtwith, namely D3. It is now not difficult to anticipatethat this process will turn out to be identical to B1, thatappears in figure 14. Indeed the signs are now = +1,

so that in both spacetime the normals point in the direc-tion of increasing radii. By the usual procedure we arethus led to the spacetime diagram shown in the bottomleft part of figure 20. Part of this diagram will constitutethe final state of the tunnelling process. The initial stateis obtained as usual and the picture of the tunnelling

r

=

0

r

=

0r

=

0

I

I+

r=

H+

r =H+

AdS, = +1 dS+, + = +1

r

=

0r

=

0

I

I+

r=

H+

r =H+

I+

r=

H+ r =

H+r

=

0r

=

0

dS+/AdS Tunnelling

FIG. 20: Case D3: AdS/dS, = + = +1

process is as in the bottom right part of figure 20.

5. Comments

From the analysis developed so far we have seen that,although in principle we have ten different tunnelling pro-cesses, in fact process A1 is the same as A3, process B1is the same as D3, process B2 is the same as A2 and pro-cess C1 is the same as C3. We are thus left with onlysix distinct processes. From symmetry consideration,it is natural to notice that ||, not itself, is the rele-vant quantity (together with ) to classify the possible

tunnelling configurations. We will thus not be surprisedif the tunnelling amplitude will be a function of ||, or,which is the same, an even function of . We would alsolike to note already here that all the tunnelling processescan be interpreted as a very large expansion of a mallquantum region of spacetime.

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22

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Stefano Ansoldi and Lorenzo Sindoni- Shell–mediated tunnelling between (anti–)de Sitter vacua

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