A lattice universe from M-theory

1 April 1999

Ž .Physics Letters B 451 1999 38–45

A lattice universe from M-theory

M.J. Duff a, P. Hoxha a, H. Lu b,1, R.R. Martinez-Acosta a, C.N. Pope a,2¨a Center for Theoretical Physics, Texas A&M UniÕersity, College Station, TX 77843, USA

b Department of Physics and Astronomy UniÕersity of PennsilÕania, Philadelphia, PA 19104, USA

Received 11 January 1999Editor: M. Cvetic

Abstract

A recent paper on the large-scale structure of the Universe presented evidence for a rectangular three-dimensional latticeof galaxy superclusters and voids, with lattice spacing ;120 Mpc, and called for some ‘‘hitherto unknown process’’ toexplain it. Here we report that a rectangular three-dimensional lattice of intersecting domain walls, with arbitrary spacing,emerges naturally as a classical solution of M-theory. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction

In a recent paper on the large-scale structure ofthe Universe at the 100-million parsec scale, Einasto

w xet al. 1–3 report seeing hints of a network ofgalaxy superclusters and voids that seems to form athree-dimensional lattice with a spacing of about 120

y1 Ž y1h Mpc where h is the Hubble constant in unitsy1 y1.of 100 km s Mpc . These authors remark that

‘‘If this reflects the distribution of all matterŽ .luminous and dark , then there must exist somehitherto unknown process that produces regularstructure on large scales.’’ In this paper we point outthat a three-dimensional lattice of orthogonally inter-

1 Research supported in part by DOE Grant DE-FG02-95ER40893.

2 Research supported in part by DOE Grant DE-FG03-95ER40917.

secting domain walls, with arbitrary lattice spacing,naturally appears as a solution of the classical equa-tions of M-theory.

Until recently, the best hope for an all-embracingtheory that would reconcile gravity and quantum

w xmechanics was based on superstrings 4 : one-di-mensional objects whose vibrational modes representthe elementary particles and which live in a universewith ten spacetime dimensions, six of which arecurled up to an unobservably small size. Unfortu-nately, there seemed to be fiÕe distinct mathemati-cally consistent string theories and this was clearlyan embarrassment of riches if one is looking for aunique Theory of EÕerything. In the last three years,however, it has become clear that all five stringtheories may be subsumed by a deeper, more pro-found, new theory called M-theory, which is re-

w xviewed in Refs. 5–8 . M-theory is an eleÕen-dimen-sional theory in which the extended objects are notone-dimensional superstrings but rather two-dimen-sional objects called supermembranes and five-di-mensional objects called superfiÕebranes. In the limit

0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 00188-4

( )M.J. Duff et al.rPhysics Letters B 451 1999 38–45 39

of low energies, M-theory is approximated byw xeleven-dimensional supergravity 9 which, ironically

enough, was the favourite candidate for superunifica-tion before it was knocked off its pedestal by the1984 superstring revolution.

Like string theory before it, M-theory relies cru-cially on a supersymmetry which unifies bosons andfermions and indeed eleven is the maximum space-time dimension that supersymmetry allows. The ba-sic objects of M-theory are solutions of the super-gravity equations of motion which preserve the frac-tion ns1r2 of this supersymmetry. In addition tothe supermembrane and superfivebrane there are alsoother solutions called plane waÕes and Kaluza-Kleinmonopoles which also preserve ns1r2. Again,seven of the eleven dimensions are assumed to becompactified to a tiny size and, when wrapped aroundthese extra dimensions, these membranes, five-branes, waves and monopoles appear as zero-dimen-sional particles, one-dimensional strings or two-di-mensional membranes when viewed from the per-spective of the physical four-dimensional spacetime.Of particular interest for the present paper will be themembranes which act as domain walls separatingone region of the three-dimensional uncompactifiedspace from another. Solutions preserving a smallerfraction ns2yn of the supersymmetry may then beobtained by permitting ns2,3 of the basic objects

w xto intersect orthogonally 10 . In particular, a solu-tion describing 3 domain walls intersecting orthogo-nally in three space dimensions will, if it exists,preserve just Ns1 of the maximum Ns8 super-symmetries allowed in four spacetime dimensions.

Although astrophysicists and cosmologists havew xconsidered topological defects 13 or solitons such

as monopoles, cosmic strings and domain walls aspossible seeds for galaxy formation, they have tradi-tionally 3 eschewed the kind of solitons arising in

w xsuperstring theory 12 . This was not without goodreason. They were interested in energy scales lessthan the order of 1016 GeV typical of the energy atwhich the strengths of the strong, weak and electro-

3 w xSee, however, 11 , where the periodic-like distribution ofw xmatter in space 15 was noted in the context of stringy topologi-

cal defects.

magnetic forces are deemed to converge in GrandUnified Theories of the elementary particles. Bycontrast, the solitons arising in string theory aregravitational in origin, having as their typical energyscale the Planck energy of 1019 GeV which corre-sponds to much too early an epoch in the history ofthe universe to be relevant to galaxy formation.

w xHowever, recent work by Witten 14 indicates thatM-theory differs from traditional superstring theoryprecisely in this respect: the phenomenologicallymost favoured size of the eleventh compact dimen-sion of M-theory is such that all four forces con-verge at a common scale of 1016 GeV. Thus gravita-tional effects are much closer to home than previ-ously realised and topological defects in M-theoryare likely to be of greater cosmological significancethan those of old-fashioned string theory.

2. The lattice universe

We now show that solutions describing threeorthogonally intersecting domain walls do indeedexist and that they may be generalised to describeany number of walls so as to form a three-dimen-

Žsional lattice, with arbitrary lattice spacing. A re-view of domain walls in Ns1 Ds4 supergravity

w x .can be found in Ref. 16 . We begin by consideringa four-dimensional universe with cartesian coordi-

m Ž 0 1 2 3. Ž .nates x s x , x , x , x s t, x, y, z whose gravi-Ž .tational field is described by a metric tensor g xmn

and which in addition contains three massless scalarfields described by the three-dimensional vectorŽ . Ža . Žf x and a set of three 3-form potentials C , asmnr

.1,2,3 . The Lagrangian is given by the EinsteinLagrangian for gravity plus kinetic energy terms forŽ . Ža .f x and C ,mnr

11 m'LLs yg Ry E fPE fm222k

31 Ž . Ž .yc Pf a mnrs aay e G G , 1Ž .Ý mnrs48

as1

where k 2 s8p G and G is Newton’s constant andwhere GŽa . ' 4E C Ža . are the 4-form fieldmnrs w m nrs x

( )M.J. Duff et al.rPhysics Letters B 451 1999 38–4540

strengths associated with the 3-form potentials. Infour spacetime dimensions such 3-forms do not cor-respond to propagating degrees of freedom but maynevertheless give rise to non-trivial topological ef-fects which are akin to the presence of a cosmologi-

w xcal constant 17 . The three vectors c are constantsa

describing the interaction of the scalars with the fieldstrengths and satisfy the relations

c Pc s1q6d . 2Ž .a b a b

A convenient choice is

' 'c s 1,1q 2 ,1y 2 ,Ž .1

' 'c s 1y 2 ,1,1q 2 ,Ž .2

' 'c s 1q 2 ,1y 2 ,1 . 3Ž .Ž .3

We shall shortly indicate the M-theoretic origin ofŽ .the Lagrangian 1 but first we provide the desired

solution of the resulting field equations:

21 yc Pf Ža .aIfsy c e G ,Ž .Ý a48

a

E e eycaPf GŽa .mnrs s0 ,Ž .m

21 1 yc Pf Ža .aR s E fPE fq e GŽ . mnÝ žmn m n2 12a

23 Ža .y G g . 4Ž . Ž ./mn8

Let us make the following ansatz for the line element

ds2 sg dx mdxn4 mn

y1r2 2 2s H H H ydt qH dxŽ . Ž1 2 3 1

qH dy2 qH dz 2 , 5Ž ..2 3

for the scalars

31fsy c log H , 6Ž .Ý a a2

as1

and for the 4-form field strengths

GŽa .sH H E Hy1dtndxndyndz , 7Ž .b g a a

where a b and g are all different, and take theirvalues from the set 1,2 and 3. Here the functions

Ž . Ž .have the coordinate dependences H x , H y ,1 2Ž .H z . Substituting into the field equations resulting3

from the Lagrangian above, we find that all aresatisfied provided the functions H are harmonic:a

E 2H E 2H E 2H1 2 3s s s0 . 8Ž .2 2 2E x E y E z

These harmonic conditions can be satisfied by takingH to have the forma

asNN

< <H x s1q M xyx 9Ž . Ž .Ý1 a aas1

Ž . Ž .and similarly for H y and H z . From the form2 3

of the metric, we see that if H and H are tem-2 3

porarily taken to be unity, then the function H1

describes a stack of NN parallel domain walls lyingŽ .in the y, z plane at the locations xsx whosea

mass per unit area M is also equal to the 3-forma

charge. Similarly, the functions H and H by2 3

themselves can describe stacks of domain walls inŽ . Ž .the x, z and x, y planes respectively. When all

three functions are of the general form given above,the solution describes the triple intersection of do-main walls lying in the three planes orthogonal to x,y and z axes, respectively. Actually, the expressionsfor the H are not harmonic everywhere, since theya

have delta-function singularities at the locations ofthe domain walls:

2 asNNE H1s2 M d xyx , etc. 10Ž . Ž .Ý a a2E x as1

This may be remedied by including in the fieldequations a source term 4 for each membrane in a

w xway that preserves the supersymmetry 12 .

4 In eleven-dimensions, duality relates the elementary singular‘‘electric’’ membrane, which requires a source term in the Ds11supergravity field equations, to the solitonic non-singular ‘‘mag-netic’’ fivebrane, which solves the source-free equations. It thusappears that the process of dualisation can eliminate the need for asource! This confusion is presumably a consequence of the inade-quacy of Ds11 supergravity to capture the full essence ofM-theory.


3. Higher dimensional origins

Ž .The Lagrangian 1 we have been considering isobtained from the bosonic sector of eleven-dimen-

w xsional supergravity 9 which contains a metric gM N

and a 3-from potential A where Ms0,1, . . . ,10.M NP

For simplicity we assume that the seven compacti-fied dimensions have the topology of a seven-dimen-sional torus. The three scalars f are a subset of the7 scalars coming from the diagonal components ofthe internal metric, while the three 4-form fieldstrengths GŽa . appear after dualising a subset of themnrs

0-form field strengths which arise via a generalisedScherk-Schwarz type of dimensional reduction inwhich tensor potentials coming both from theeleven-dimensional metric g and 3-form potentialM N

A , are allowed to depend linearly on the com-M NP

pactification coordinates. By retracing these steps itis possible to re-express our four-dimensional solu-tion as an eleven-dimensional solution.

To see the origin of dilaton vectors satisfying theŽ .relations 2 , we note that the vectors c are just thea

negatives of the dilaton vectors for the 0-form fieldstrengths that we are dualising in order to obtain the4-forms GŽa .. The 0-form field strengths arise fromthe Scherk-Schwarz reduction of 1-form fields; inother words, from reductions where the axionic scalarpotentials for the 1-form field strengths are allowedlinear dependences on the compactification coordi-nates. Details of such Scherk-Schwarz reductionsand subsequent dualisations may be found in Refs.w x18,19,10 . There are two sources of 0-form fieldstrengths, namely those coming from the Scherk-Schwarz reduction of the antisymmetric tensor inDs11, and those coming from the Scherk-Schwarzreduction of the Kaluza-Klein vectors coming fromthe metric in Ds11. These 0-forms, denoted by

Ž i jk ll . Ž i jk . w xF and FF respectively in Ref. 10 , have0 0

associated dilaton vectors a and b , given byi jk ll i jk

a s f q f q f q f yg , b syf q f q f ,i jk ll i j k ll i jk i j k

11Ž .

where

gPgs7 , gP f s3 , f P f s1q2d 12Ž .i i j i j

in Ds4. Note that the indices i, j, . . . range overthe 7 compactification coordinates z here, startingi

with is1 for the reduction step from Ds11 toDs10. It is now easy to see that one can indeedfind three dilaton vectors that satisfy the conditionsŽ .2 . There are many different combinations that willwork, but they are all equivalent, up to index rela-bellings, to the two choices

c sa , c sa , c sb , 13Ž .1 1234 2 1567 3 125

or

c sa , c sb , c sb . 14Ž .1 1234 2 235 3 345

Ž .Note that for the first choice 13 we must necessar-ily choose two 0-forms that originate from F sdA4 3

in Ds11, and one coming from the metric. Inparticular we must necessarily use terms which origi-nate, in ten-dimensional type IIA language, fromboth the NS-NS and R-R sectors of the theory. Thismeans that our intersecting membranes are solutionsof the type IIA theory or M-theory, but not of theheterotic or type I strings. However, it is also possi-

Ž .ble, as in choice 14 to choose the dilaton vectors ca

such that the Ds4 solution originates only fromDs10 NS-NS sector. As such it could equally wellbe regarded as a solution of the Ds10 heterotic ortype I string.

The reason why we chose to work with the dualformulation with 4-form field strengths, rather thanthe original 0-form field strengths coming from theScherk-Schwarz reduction, is that the former descrip-tion is necessary, in the framework of a four-dimen-sional theory itself, if one wants to have the possibil-ity of multiple membranes in a stack along eachcoordinate axis. The reason for this is that a 0-form

1 2 cPffield strength term of the form y M e in the2

four-dimensional Lagrangian would lead to an equa-tion of motion that required the associated harmonicfunction H to have slope "M, and hence we could

< <only have a solution of the form Hs1qM xyx .0

In order to have straight-line segments of differentŽ .slopes, as is required in 9 in order to describe

multiple membranes, it is necessary that the slopecan take different values in different regions, and soit must arise as an arbitrary constant of integrationŽ .related to the the electric charge of F rather than4

Ž .as a fixed, given parameter i.e. M in the La-grangian.


From the higher-dimensional point of view, therestriction of needing to work in the dualised formu-lation with 4-form field strengths in Ds4 is actu-ally an artificial one. The reason for this is that the0-form field strengths in the four-dimensional theorythemselves arise from the generalised Scherk-Schwarz reductions of standard degree 1 or higherfield strengths in higher dimensions. In these reduc-tions, unlike in standard Kaluza-Klein reductions, theassociated potentials are allowed linear dependenceson certain compactifying coordinates. In our exampleabove, for instance, the 0-form field strength F Ž1234.

0

arises from a generalised Scherk-Schwarz reductionstep from Ds8 to Ds7, where the axion AŽ123. is0

Ž123.Ž . Ž123.Ž .reduced according to A x, z ™ A x q0 4 0

M z . This gives the 0-form field strength term41 2 a Pf1234y M e in Ds7, and lower, dimensions.2

However the constant M is itself an integrationconstant from the viewpoint of the theory in Ds8,and so provided we trace our four-dimensional solu-tion back to Ds8 or a higher dimension, the pa-rameter M of the four-dimensional Lagrangian in the0-form formulation can be understood as nothing buta free integration constant. Thus again, it can takedifferent values in different regions, and so the possi-bility of having multiply-charged solutions is re-gained.

In view of the above, it is therefore useful tore-interpret our intersecting membrane solutions backin higher dimensions. This is easily done by retrac-ing the steps of the Kaluza-Klein and Scherk-Schwarzreductions. Taking our first choice of field strengths

Ž .corresponding to 13 , we find that in Ds10 thesolution becomes

y1r42 y1r8 2ds s H H H ydtŽ . Ž10 1 2 3

qH dx 2 qdz 2 qdz 2Ž .1 3 4

qH dy2 qdz 2 qdz 2Ž .2 6 7

qH dz 2 qH H dz 2 qH H dz 2 ,.3 1 3 2 2 3 5

y1r2f 3r41e s H H H ,Ž .1 2 3

F Ž1.syE H dz ndz ndz3 x 1 2 3 4

yE H dz ndz ndz ,y 2 5 6 7

FF Ž1.syE H dz ndz , 15Ž .2 z 3 2 5

where f , the first component of the fields f in1

Ds4, is the dilaton of the type IIA theory.The supersymmetry transformation parameter of

this solution satisfies the following relations

1yG es0 , 1yG es0 ,Ž . Ž .ˆ ˆˆ ˆ ˆ ˆ023567 013234

1yG es0 , 16Ž .Ž .ˆ ˆ ˆˆ0123467

where e is given in terms of a constant spinor e :0

y1r16 y1r32es H H H e , 17Ž . Ž .1 2 3 0

where G are the Ds10 Dirac matrices, andMŽG sG PPP G . Note that the explicit nu-M PPP M w M M x1 n 1 n

merical indices 0, 1, 2 and 3 refer to the four-dimen-ˆ ˆsional spacetime, while 1, 2, etc. refer to the reduc-

.tion steps is1,2 etc. from Ds11 to Ds10,9 etc.Therefore, the above solution preserves the fraction

Ž .ns1r8 of the supersymmetry. The solution 15can be viewed as a non-standard intersection of twoNS-NS 5-branes and one D6-brane. The harmonicfunctions here depend on the relative transverse co-ordinates, which differ from the situation in standardintersections, where the harmonic functions dependonly on the overall transverse space. The two 5-branecharges are both carried by the NS-NS 3-form fieldstrength F Ž1., and the D6-brane charge is carried by3

the R-R 2-form field strength FF Ž1.. The solution2

with vanishing D6-brane charge was also obtained inw xRefs. 20–22 .

Going back one step further, to Ds11, the solu-tion becomes

y1r32 2 2 2 2ds s H H ydt qH dx qdz qdzŽ . Ž .ž11 1 2 1 3 4

qH dy2 qdz 2 qdz 2 qH dz 2Ž .2 6 7 3

qH H dz 2 qH H dz 21 3 2 2 3 5

2y1qH H H dz qE H z dz ,Ž . /1 2 3 1 z 3 5 2

F syE H dz ndz ndz ndz4 x 1 1 2 3 4

yE H dz ndz ndz ndz . 18Ž .y 2 1 5 6 7

This solution can be viewed as a non-standard inter-Žsection of two M5-branes and one NUT again with

the harmonic functions depending on the relativetransverse coordinates rather than those of the over-

.all transverse space .It is also possible to choose the dilaton vectors ca

such that the Ds4 solution originates from Ds10


Ž .NS-NS sector as with 14 . By retracing the steps ofthe Kaluza-Klein and Scherk-Schwarz reductions wefind that in Ds8 the solution becomes

ds2 sHy1r6 ydt 2 qH dx 2 qH dy2 qH dz 2Ž8 3 1 2 3

qH H dz 2 qH H dz 2 qdz 2 qdz 2 ,.1 3 4 2 3 5 6 7

1 1 1fsy a log H y b log H y b log H ,123 1 23 2 3 32 2 2

F Ž123.sE H dz , FF Ž23.sE H dz ,1 x 1 4 1 y 2 5

FF Ž3.sE H dz ndz , 19Ž .2 z 3 4 5

where

3 33 4a s 1, , 2 , b s 0,y , 2 ,( (123 237 7ž / ž /' '7 7

7b s 0, 0 , y . 20( Ž .ž /3 3

Ds8 is the highest dimension where the metricremains diagonal in the oxidation process.

The solution in Ds10 takes the form

ds2 sHy1r4 ydt 2 qH dx 2 qH dy2 qH dz 2ž10 1 1 2 3

qH H dz 2 qH H dz 21 3 4 2 3 5

2y1qH H H dz qz E H dzŽ .1 2 3 3 4 z 3 5

22 2 y1qdz qdz qH H dz yz E H dz ,Ž . /6 7 1 2 2 3 y 2 5

ef1 sHy1r2 ,1

F Ž1.sE H dz n dz yz E H dzŽ .3 x 1 4 2 3 y 2 5

n dz qz E H dz . 21Ž . Ž .3 4 z 3 5

The Ds10 supersymmetry transformation parame-ter satisfies the conditions

1qG es0 , 1qG es0 ,Ž . Ž .ˆ ˆ ˆ ˆ ˆ ˆ1234 3345

1yG es0 , 22Ž .Ž .ˆ ˆ ˆ2235

where

esHy1r16 e . 23Ž .1 0

Thus, the fraction of the supersymmetry preservedby this solution is also ns1r8. This can regardedas a solution of the type 1, heterotic or type IIA

string. In the latter case, the solution can be furtheroxidised to Ds11, where it becomes becomes

ds2 sHy1r3 ydt 2 qH dx 2 qH dy2 qH dz 2ž11 1 1 2 3

qH H dz 2 qH H dz 2 qqdz 2 qdz 21 3 4 2 3 5 6 7

2y1qH H dz yz E H dzŽ .1 2 2 3 y 2 5

2y1qH H H dz qz E H dzŽ .1 2 3 3 4 z 3 5

qH dz 2 ,/1 1

F sE H dz n dz yz E H dzŽ .4 x 1 4 2 3 y 2 5

n dz qz E H dz ndz . 24Ž . Ž .3 4 z 3 5 1

In both Ds10 and Ds11 dimensions, the solu-tions can be viewed as non-standard intersections ofa 5-brane with two NUTs. In Ds10, the 5-branecarries the NS-NS charge.

In this section, we have considered two examplesof non-standard intersections in Ds10 string theoryor Ds11 M-theory that can give rise to a four-di-mensional lattice universe. The first example is in-trinsic to M-theory, while the second example canalternatively be embedded in the heterotic string.More solutions can be obtained by invoking theT-duality of the type IIA and type IIB theories. Weshall not enumerate such examples here.

4. Conclusions

The worldwide web solution we have presented isadmittedly an idealisation, and by itself would notyet satisfy hard-nosed astrophysicists. First, it repre-sents a static universe with an unbroken supersym-metry, whereas in the real world the universe is

Ž .expanding and supersymmetry if it exists at all is abroken symmetry. In fact these two features areintimately related. The reason we were able to find astable static three-dimensional lattice, rather than onewhich is collapsing under its own gravity or onewhose tendency to collapse is overwhelmed by ex-pansion, is precisely because of the famous ‘‘no-

w xstatic-force’’ phenomenon 12 of supersymmetricvacua which saturate a Bogomol’nyi-Prasad-Sommerfield bound between the mass and the charge.The mutual gravitational attraction due to gravityg and the massless scalar fields f is exactlymn

cancelled by a repulsion due the the 3-forms C Ža . ,mnr

which act in many ways like a cosmological con-


Žstant. In writing this, of course, we are acutelyaware that the introduction of a cosmological con-stant in order to obtain a static rather than expandinguniverse was, on his own admission, Einstein’s

.‘‘greatest blunder’’. A more realistic descriptionmust therefore await a satisfactory explanation ofhow M-theory breaks supersymmetry and, unfortu-nately, this remains M-theory’s biggest unsolvedproblem. Of course, our solution is but one of manysolutions of M-theory. We make no apology for this.The lattice structure is no more a prediction ofM-theory than the Friedman-Robertson-Walker cos-mology is a prediction of General Relativity.

The intersecting domain wall configuration is onewhere the regions of high density are concentratedon the faces of the lattice cubes. One might askwhether there are also solutions where the regions ofhigh density are concentrated on the edges of the

Žcube intersecting strings associated with 3-form field. Žstrengths or on the vertices of the cube point

.singularities associated with 2-form field strengths .The experimental data do not sharply distinguishbetween these possibilities, since they depend on theoverdensity drrr chosen in making the statisticalanalysis. We have found no such intersecting stringsolutions and consider their existence unlikely. Thereare solutions describing any number of isolated pointswhich may, in particular, be chosen to lie on a cubiclattice. These are M-theoretic generalisations of thewell-known Papapetrou-Majumber solutions of gen-eral relativity. Once again the mutual attraction dueto gravity and scalars is exactly cancelled by arepulsion due to the 1-form potentials.

Another reason why astrophysicists might objectis that domain walls whose mass per unit area is toogreat are ruled out experimentally. In the supersym-metric idealisation presented here, the mass per unitarea is also a free parameter depending on the vac-uum expectation values of the scalar fields which,for simplicity of presentation, we have arbitrarily setequal to zero. Once again, the actual value of thesescalar expectation values must await a resolution ofthe supersymmetry-breaking problem.

Finally, we are aware that with the success ofinflationary models of the universe, topological de-fects have fallen out of favour as the mechanism for

w xgalaxy formation 23 , although the issue remainsw xcontroversial 24 . In any event, it is difficult to see

how inflation alone could account for the three-di-w xmensional cubic lattice reported by Einasto et al. 1 .

To the best of our knowledge, all attempts to fit thislattice structure data with data on the cosmic mi-crowave background have been based on entirely adhoc assumptions on the initial spectrum of density

w xperturbations. See, for example, 25 . Moreover, thephenomenology of the kinds of defect appearing inM-theory has yet to be scrutinized. Consequently, inspite of the idealised nature of our solution we hopeto have shown that M-theory is indeed a rich sourceof possible explanations for ‘‘hitherto unexplainedphenomena’’ and in particular allows for three-di-mensional lattice cosmologies.

Acknowledgements

We are grateful to John Barrow and AlexeiStarobinsky for useful correspondence.

References

w x1 J. Einasto, M. Einasto, S. Gottlober, V. Muller, V. Saar, A.A.Starobinsky, E. Tago, D. Tucker, H. Andernach, P. Frisch,

Ž .Nature 385 1997 139.w x Ž .2 R. Kirshner, Nature 385 1997 .w x3 J. Einasto, Has the universe a honeycomb structure?, astro-

phr9711320.w x4 M.B. Green, J.H. Schwarz, E. Witten, Superstring theory,

Cambridge University Press, 1987.w x Ž .5 J.H. Schwarz, Phys. Lett. B 367 1996 97, hep-thr9510086.w x Ž .6 M.J. Duff, I.J.M. P A 11 1996 5623, hep-thr9608117.w x7 P.K. Townsend, Four lectures on M-theory, hep-thr9612121.w x8 T. Banks, Matrix Theory, hep-thr9710231.w x Ž .9 E. Cremmer, B. Julia, J. Scherk, Phys. Lett. B 76 1978 409.

w x10 H. Lu, C.N. Pope, T.A. Tran, K.W. Xu, Classification of¨p-branes, NUTs, waves and intersections, to appear in Nucl.Phys. B, hep-thr9708055.

w x Ž .11 S. Elitzur, A. Forge, E. Rabinovici, Nucl. Phys. B 359 1991581.

w x Ž .12 M.J. Duff, R.R. Khuri, J.X. Lu, Phys. Rep. 259 1995 213,hep-thr9412184.

w x13 A. Vilenkin, E.P.S. Shellard, Cosmic strings and other topo-logical defects, Cambridge University Press, 1994.

w x Ž .14 E. Witten, Nucl. Phys. B 471 1996 135, hep-thr9602070.w x15 T.J. Broadhurst, R.S. Ellis, D.C. Koo, A.S. Szalay, Nature

Ž .343 1990 726.w x Ž .16 M. Cvetic, H. H Soleng, Phys. Rep. 282 1997 159, hep-

thr9604090.


w x Ž .17 M.J. Duff, P. van Nieuwenhuizen, Phys. Lett. B 94 1980179.

w x18 E. Bergshoeff, M. de Roll, M.B. Green, G. Papadopoulos,Ž .P.K. Townsend, Nucl. Phys. B 470 1996 113, hep-

thr9601150.w x19 P.M. Cowdall, H. Lu, C.N. Pope, K.S. Stelle, P.K. Townsend,

Ž .Nucl. Phys. B 486 1997 49, hep-thr9608173.w x Ž .20 R.R. Khuri, Phys. Rev. D 48 1993 2947, hep-thr9305143.w x21 K. Behrndt, E. Bergshoeff, B. Janssen, Phys. Rev. D 55

Ž .1997 3785, hep-thr9604168.

w x22 J.P. Gauntlett, D.A. Kastor, J. Traschen, Nucl. Phys. B 478Ž .1996 544, hep-thr9604179.

w x Ž .23 U. Seljak, U-L. Pen, N. Turok, Phys. Rev. Lett. 79 19971615.

w x24 J. Pando, D. Valls-Gabaud, Li-Zhi-Fang, Evidence for scale-scale correlations in the cosmic microwave background radi-ation, astro-phr9810165.

w x25 F. Atrio-Barandela, J. Einasto, S. Gottlober, V. Muller, A.A.Ž .Starobinsky, JETP Lett. 66 1997 397, astro-phr9708128.

Documents

A lattice universe from M-theory