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Observing 4d baby universes in quantum gravity

arXiv:hep-th/9303041v1 8 Mar 1993NBI-HE-93-2January 1993Observing 4d baby universes in quantum gravityJ. AmbjørnThe Niels Bohr InstituteBlegdamsvej 17, DK-2100 Copenhagen Ø, DenmarkS.

JainCentre for Theoretical Studies, Indian Institute of ScienceBangalore 560012, IndiaJ. Jurkiewicz1Inst.

of Phys., Jagellonian University.,ul. Reymonta 4, PL-30 059, Krak´ow 16, PolandC.F.

KristjansenThe Niels Bohr InstituteBlegdamsvej 17, DK-2100 Copenhagen Ø, DenmarkAbstractWe measure the fractal structure of four dimensional simplicial quantum gravityby identifying so-called baby universes. This allows an easy determination of thecritical exponent γ connected to the entropy of four-dimensional manifolds.1Supported by the KBN grant no.

2 0053 91 011

1IntroductionA simple way to characterize the fractal structure of 2d quantum gravity was recentlyintroduced in ref. [1].

Using very few assumptions it was possible to prove that thefunctional behaviour of the number of two-dimensional surfaces with area N2 andcoupled to matter with central charge c < 1N (N2) ∼Nγ(c)−32eµ0N2(1)is directly related to the fractal structure of 2d quantum gravity. In particular onecan determine γ(c) by simply counting the number of so-called “minimum bottleneckuniverses”, abbreviated “minbu’s”, living on a typical surface in the ensemble ofsurfaces of fixed total area.

This number is given byNN2(V2) ∼N2V γ(c)−22,(2)where N2 is the total area of the surface and V2 ≪N2 is the area of the minbu’sbeing counted.A ”baby universe” of area V2 is any simply connected region ofthe surface of area V2 and boundary length l such that V2 ≫l2, and a minbu is ababy universe whose l is the smallest possible length consistent with the ultravioletcutoff. This boundary, along which the minbu is connected to the remainder ofthe (parent) surface is called a minimal bottleneck.The only assumption goinginto the derivation of (2) is the distribution (1) and the approach is well suitedfor dynamically triangulated surfaces [2, 3, 4, 5].

Here the area of a region can beidentified with the number of triangles in the region and a minimal bottleneck isa simple closed path consisting of three links which partitions the surface into twoparts - the minbu containing V2 ≫1 triangles, and the parent containing N2 −V2triangles.Until now the measurement of γ by numerical simulations has been a ratherpainful procedure since one had to perform a whole sequence of simulations (or aso-called grand canonical simulation where the number of triangles changes, whichis also an unpleasant task) for different areas N2 and in the end fit the measuredquantities to a formula related to (1). Following the suggestion in [1] one can nowextract γ from a single simulation using (2).

This is neater also because one doesnot have to subtract out the leading eµ0N2 piece while making the fit; (2) containsonly the universal power law piece. This method also works in practise.

Indeed, ina recent numerical simulation it was possible to extract γ for pure 2d gravity withhigh precision (less than 1%) [9].2

2The modelA model of 4d quantum gravity which generalizes the discretized 2d approach wassuggested a year ago [7, 8]. The partition function is given byZ(κ2, κ4) =XT∈Te−κ4N4+κ2N2(3)where the sum is over triangulations T in a suitable class of triangulations T .

Thequantity N4 denotes the number of 4-simplexes in the triangulation and N2 thenumber of triangles. The coupling constant κ2 is inversely proportional to the baregravitational coupling constant, while κ4 is related to the bare cosmological constant.The most important restriction to be imposed on T is that of a fixed topology.

Ifwe allow an unrestricted summation over all topologies in (3) the partition functionis divergent [8]. In the following we will always restrict ourselves to considering onlymanifolds with the topology of S4.Z(κ2, κ4) is the grand canonical partition function.It is defined in a regionκ4 ≥κc4(κ2) in the (κ2, κ4) coupling constant plane.

The only way in which we canhope to obtain a continuum limit is by letting κ4 approach κc4(κ2) from above. Thistentative continuum limit depends only on one coupling constant κ2.

We can write(3) asZ(κ2, κ4) =XN4Z(κ2, N4)e−κ4N4. (4)Z(κ2, N4) is the canonical partition function where N4 is kept fixed.

Then we havein practise only one coupling constant, κ2, and the aspects of gravity which do notinvolve the fluctuation of the total volume of the universe can be addressed in thelimit of large N4. However it is actually possible to extract the critical exponentfor volume fluctuations from Z(κ2, N4).

Let us assume that the canonical partitionfunction has the form:Z(κ2, N4) = Nγ(κ2)−34eκc4(κ2)N4 · (1 + O(1/N4))(5)When κ4 is close to κc4(κ2) we can approximate (3) withZ(κ2, κ4) ∼analytic +C(κ2)(κ4 −κc4(κ2))γ(κ2)−2(6)where “analytic” means possible analytic terms of the form (κ4 −κc4)n, 0 ≤n <2 −γ(κ2).The entropy exponent of the number of 4d simplicial manifolds, γ,determines the volume fluctuations since we haveDN24E−⟨N4⟩2 = d2 ln Z(κ2, κ4)dκ24∼analytic +C(κ2)(κ4 −κc4(κ2))γ(κ2)(7)3

if we assume γ(κ2) < 2.We will try to extract the critical exponent γ(κ2) by numerical simulations. Atthis point it is important to note that it is by no means obvious that the exponent γexists.

Equation (5) is an ansatz, inspired from 2d quantum gravity where we knowthat a similar formula holds when the central charge of matter c < 1. In the caseof c = 1 there are logarithmic corrections, and the form is not known for c > 1.

In3d simplicial gravity it was shown [15, 17, 18] that a more likely form of Z(κ1, N3)(the 3d analogue of (5)) isZ(κ1, N3) ∼exp"κc3(κ1)N3 1 −C(κ1)Nα(κ1)3!#(8)and a possible power law correction would be subleading. However, 3d simplicialgravity seems to differ from 4d simplicial gravity in many respects ([7, 8, 16]), and itwas clear from the fine tuning process κ4 →κc4(κ2) in the Monte Carlo simulationsthat the corrections to the leading term exp(κc4N4) were less severe in 4d than in 3d.It is therefore indeed possible that the finite size corrections in 4d are power-like, asin 2d simplicial gravity, rather than of the exponential form (8) encountered in 3d.Let us now define the minimal bottleneck baby universes, which we, following [1],will denote “minbu’s”.

On a 2d triangulated manifold one can check whether wehave a minbu in the following simple way: pick a link and two neighbouring links,attached to the two vertices of the first link. If the two links have a vertex in commonthe three links form a closed loop.

In general this loop will be trivial in the sensethat it will just be the boundary of one of the two triangles in the surface whichcontains our starting link. However, it might turn out that there is no triangle in thesurface which has the closed loop as its boundary.

If we cut the surface along sucha closed loop we will have separated it into two disconnected open surfaces each onehaving the closed loop as its boundary, provided the topology of the surface is thatof a sphere (which we will assume). Both components will have spherical topologywhen closed by adding the triangle which has the loop as boundary.

The smallestof the spheres will be a minbu, the largest the “mother universe”.This construction can immediately be generalized to 4d triangulated manifolds.We check for minbu’s as follows: pick a 3-simplex (a tetrahedron) and in this thefour 2-simplexes (triangles) which constitute its boundary. Identify for each triangleall the 3-simplexes in the manifold which have this triangle as their boundary.

Wenow have four groups of 3-simplexes and all 3-simplexes have one vertex which doesnot belong to the triangle from which the 3-simplex was constructed. Pick now a3-simplex from each of the 4 groups and check whether their 4 free vertices coincide.4

If that is the case the four 3-simplexes together with the original 3-simplex can bethought of as the (closed) boundary of a 4-simplex. In general the 4-simplex foundby this procedure will just be one of the two 4-simplexes which shared the original3-simplex.

However, if that is not the case we will, if the topology of our original4-manifold is that of a sphere (which we assume as usual), separate it into twonon-trivial components if we cut along the closed piecewise linear 3d manifold builtout of the five 3-simplexes. Both components have this closed 3-manifold as theirboundary and both components will have the topology of S4 if we close them byadding to each of them a 4-simplex in such a way that their boundary is identifiedwith the boundary of the 4-simplex, just with opposite orientation.

The smallest ofthe 4-spheres will be called the minbu, the largest the “mother”, in agreement withthe 2d notation.It is almost clear that the above description can be turned into an efficient nu-merical algorithm (after, admittedly, some pain with double counting etc.) providedone is given the coincidence matrix of the triangulated 4-manifold.Let us now argue that the counting arguments in [1] extend to the 4d triangulatedmanifolds as well.

Let us by Z′(κ2, N4) denote the canonical partition function for4-manifolds consisting of N4 4-simplices where one 4-simplex is marked. Genericallywe will get a different manifold for each mark so for large N4, where accidentalsymmetries are expected to play no role, we get Z′(κ2, N4) ≈N4Z(κ2, N4).

Such amarked manifold can also be viewed as a manifold with (N4 −1) 4-simplexes anda minimal boundary of the kind associated with minbu’s (by removing the interiorof the marked 4-simplex). A moments reflection will convince the reader that theaverage number of minbu’s of volume V4 on 4-manifolds of total volume N4 will begiven byNN4(κ2, V4) ≈60Z(κ2, N4)Z′(κ2, V4)Z′(κ2, N4 −V4)(9)where 60 is the number of ways one can glue the two boundaries of the minbu andthe mother together with opposite orientation of the boundary.If we assume the canonical partition function is given by (5), we getNN4(κ2, V4) ∼C(κ2) N4 V γ(κ2)−24(10)The scene is now set for a numerical determination of the number of minbu’s since itis already by now [7, 8, 10, 11, 12, 13, 14] standard how to generate by Monte Carlosimulations the class of triangulated 4-manifolds corresponding to the partition func-tion (3).

It is also well known [8] how one can effectively stay in the neighbourhoodof a given volume, N4, even if one is using the grand canonical partition function5

(3), as one is forced to by ergodicity requirements in 4d, contrary to what is thecase in 2d.3Numerical resultsBy Monte Carlo simulations we have generated a number of 4d-manifolds accordingto the distribution dictated by the partition function (3). For details about thisprocedure we refer e.g.

to [8, 12]. We have considered κ2 = 0.0, 0.8, 1.0, 1.1 and1.4 and N4 = 4000, 9000, 160002.

For each value of κ2 and for each value of N4 wehave generated approximately 1000 configurations.Let us at this point review some of the results of [8]: For κ2 ≈1.0 −1.1 weobserved a transition in geometry from a highly connected phase with a seeminglylarge Hausdorffdimension to a phase with an elongated, almost one-dimensionalgeometry. This is why our main efforts have been concerned with this region ofcoupling constant space.The thermalization time is short in the highly connected phase, but quite longfor κ2 > 1.0.

In the case of N4 = 16000 we have for these values of the couplingconstant used 10000 sweeps3 for thermalization and performed measurements aftereach successive fifty sweeps (after each successive hundred sweeps for κ2 = 1.4). Theresult of the measurements of minbu’s is shown in fig.

1 for N4 = 16000. We haveplotted the number of minbu’s on a log-log scale since the formula (10) suggests thedependence:log NN4(κ2, V4) = Const(κ2) + (γ(κ2) −2) log V4(11)The straight lines are the result of a least χ2 fit.

In fig. 2 we have shown the resultsfor k2 = 1.1 and different values of N4 and finally in fig.

3 we have shown γ(κ2) asextracted from fig. 1.A few comments should be made.

We have only taken into account minbu’s with avolume V4 ≥9. For small values of V4 the numbers NN4(κ2, V4) fall into two classesaccording to whether V4 = 4n −1 or V4 = 4n + 1.

This is a clear finite size effect.We have only included in our analysis the last class of numbers as they seem to fit(11) all the way down to V4 = 9. For large volumes V4 of the minbu’s the results areaverage values for volumes in the neighbourhood of the plotted V4 since the statisticsfor a specific large volume is not good.

The errorbars resulting from the binningprocedure are smaller than the symbols used in the figures.2We have also made a few runs for N4 = 32000 and k2 = 1.0.3By a sweep we mean N4 accepted updatings.6

In general we see a clear deviation from (11) when N (V4) < 0.1 and whenκ2 < 1.1. In this case the number of minbu’s begins to drop faster than given by(11).

We have not shown these data, since they have bad statistics, and we do notknow whether the above mentioned behaviour reflects that (11) is not really valid forlarge V4 or rather that the statistics is just not good enough (there is typically lessthan one such large minbu per universe) for the volumes of manifolds we consider.The message we get by comparing different N4’s does not allow us at the moment toanswer this question. For κ2 larger than 1.1 the situation is different: The number ofbaby universes is much larger and even if the number of very large baby universes issmall and errorbars large their numbers seem not to drop below numbers predictedby (11).It is seen that the curve for κ2 = 1.4 bends at V4 ≈30 −40 to a different slope.We have used the part with the larger values of V4 to extract γ(κ2).

Such a bendingis not observed for κ2 ≤1.0, i.e. in the phase with a highly connected geometry.

Aclose look at the data for κ2 = 1.1 reveals a slight tendency to such a bending, andsince κ2 = 1.1 is just at the borderline of the transition to the elongated geometrythe bending is probably related to the fact that the very fractal structure observedin this phase is only unfolded for large volumes V4.4DiscussionWith the above reservation concerning the interpretation of the numerical data, westill consider the results as somewhat remarkable. It seems possible by numericalsimulations to study in detail the fractal structure of quantum gravity.

Indeed, ourdata contains a lot of information about the branching of the geometry which willbe published in a more extensive report elsewhere [19]. Let us just here mentionthat the two phases of 4d simplicial quantum gravity observed in former works ap-pear quite clearly in the minbu measurements.

In the phase with highly connectedgeometry there is one “big mother” which contains the major fraction of the volume(from 95% to 80% depending on κ2). In the other phase of very elongated geometrythis is not so and there seems genuine democracy in “mother size”.

Nevertheless theγ(κ2) extracted seems perfectly smooth when we pass the transition in geometry, asis apparent from fig. 3.

It is interesting to note that a similar smoothness has beenobserved in the numerical simulations of 2d quantum gravity [9, 20] when we passthe c = 1 barrier. It points to the fact that subleading corrections to (1) might playa very important geometrical role, and it might be possible to understand analyti-cally the interplay between the subleading terms in (1) and the fractal geometry ofquantum gravity.

The seed to such an analysis is already to be found in [1].7

We would like to stress that the results here, especially for κ2 ≤1.1, are obtainedfor quite small baby universes and we see some deviations for larger volumes, asmentioned above. It would clearly be desirable to let the babies grow, but as iswell know from biology this requires time, in our case computer time.

Work in thisdirection is in progress. It is nevertheless somewhat intriguing that the transitionbetween the two types of geometry takes place at a value of γ ≈−0.5 −0.0, i.e.the same values of γ which are of main interest in 2d gravity.

Note further thatthe results for γ seem internally consistent in the sense the value in the phase withalmost linear geometry (Hausdorffdimension not larger that two) seems close to thevalue γ = 1/2 for the so-called branched polymers. In addition, assuming that theextraction of γ for κ2 < 1.1 will survive the test of large babies, one could wonder ifthis reflects that the situation is like in 2d gravity where we have a γ(c) dependingon the central charge.

In our case κ2 would then play a role somewhat similar to thecentral charge. The range of γ’s is in agreement with such a picture.

An interestingcontinuum approach somewhat related to such an interpretation can be found in arecent paper by Antoniadis, Mazur and Mottola [21], where the authors develop aquantum theory for the conformal mode of gravity.8

References[1] S. Jain and S.D. Mathur, World-Sheet Geometry and Baby Universes in 2-DQuantum Gravity, HUTP-92/A002.

[2] F. David, Nucl. Phys.

B 257 (1985) 45. [3] J. Ambjørn, B. Durhuus and J. Fr¨ohlich, Nucl.

Phys. B 257 (1985) 433;Nucl.Phys.

B270 (1986) 457; Nucl. Phys.

B275 (1986) 161. [4] F. David, Nucl.

Phys. B257 (1985) 543.

[5] V. A. Kazakov, I. K. Kostov and A. A. Migdal, Phys.

Lett. 157B (1985) 295.

[6] D. Boulatov, V. A. Kazakov, I. K. Kostov and A. A. Migdal, Nucl.

Phys. B275(1986) 641.

[7] M.E. Agishtein and A.A. Migdal, Mod.

Phys. Lett.

A7 (1992) 1039. [8] J. Ambjørn and J. Jurkiewicz, Phys.Lett B278 (1992) 42.

[9] J. Ambjørn, S. Jain and G. Thorleifsson, Baby Universes in 2d Quantum Grav-ity, NBI-HE-93-4. [10] S.

Varsted,UCSD-PhTh-92-03. [11] M.E.

Agishtein and A.A. Migdal, Nucl. Phys.

B385 (1992) 395. [12] J. Ambjørn, J. Jurkiewicz and C.F.

Kristjansen, NBI-HE-92-53 (to appear inNucl. Phys.

B). [13] B. Br¨ugmann, Non-uniform Measure in Four-Dimensional simplicial QuamtumGravity, SU-GP-92/9-1.

[14] B. Br¨ugmann and E. Marinari, 4d Simplicial Quantum Gravity with a Non-Trivial Measure, SU-GP-92/9-2. [15] B. Boulatov and A. Krzywicki, Mod.Phyd.Lett A6 (1991) 3005.

[16] J. Ambjørn, D. Boulatov, A. Krzywicki and S. Varsted, Phys.Lett B276 (1992)432. [17] J. Ambjørn and S. Varsted, Phys.

Lett (B226), (1991) 285. [18] J. Ambjørn and S. Varsted, Nucl.Phys.

B373 (1992) 557. [19] J. Ambjørn, J. Jurkiewicz and C.F.

Kristjansen, To appear. [20] J. Ambjørn, D. Boulatov and V.A.

Kazakov, Mod. Phys.

Lett A5 (1990) 771. [21] I. Antoniadis, P.O.

Mazur and E. Mottola, Scaling behavior of quantum four-geometries, Preprint CPTH-A214.12929

Figure CaptionsFig. 1 The number of minbu’s for N4 = 16000 and k2 = 0.0(✷), 0.8(⃝), 1.0(△),1.1(+) and 1.4(×).

The straight lines represent the best fits to (10).Fig. 2 The number of mimbues for k2 = 1.0 and N4 = 4000(✷), 9000(⃝), 16000(△)and 32000(+), normalized such that NN4(V ) is multiplied by 16000/N4.

(log-log scale)Fig. 3 γ(κ2) as extracted from fig.1.10

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