THE COSMOLOGICAL CONSTANT AND THE BRANE WORLD SCENARIO ∗ HANSPETER NILLES Physikalische Institut, Universit¨at Bonn, Nussallee 12, D-53115 Bonn, Germany E-mail: nilles@th,physik.uni-bonn.de 1 0 Although the cosmological constant has primarilycosmological consequences, its smallness poses one of the basic problems in particle physics. Various attempts 0 have been made to explain this mystery, but no satisfactory solution has been 2 found yet. The appearance of extra dimensions in the framework of brane world n systemsseemstoprovidesomenewideastoaddressthisproblemformadifferent a pointofview. Weshalldiscusssomeofthesenew approaches andseewhether or J not they lead to an improvement of the situation. We shall conclude that we are stillfarfromasolutionoftheproblem. 2 1 v 1 Introduction 5 1 We know that the cosmological constant is much smaller than one would 0 naivelyexpect. This ledtothe beliefthata naturalapproachto this problem 1 wouldbe a mechanismthat explainsa vanishingvalue ofthis vacuum energy. 0 Whilecosmologicalobservations1,2 seemtobeconsistentwithanonzerovalue 1 0 of the cosmological constant, still the small value obtained lacks a satisfac- / tory explanation other than just being the result of a mere fine-tuning of the h parameters. p - Recentlynewtheoreticalideasinextradimensionshavebeenputforward p to attack this problem. In the present talk I shall elaborate on work done in e h collaborationwithStefanF¨orste,ZygmuntLalakandSt´ephaneLavignac3,4,5, : wheretheproblemoffine-tuninghasbeenanalyzedintheframeworkofmod- v els with extra dimensions that have attracted some attention recently. i X Oneofthemostoutstandingopenproblemsinquantumfieldtheoryisitto r find an explanationfor the stability of the observedvalue of the cosmological a constant in the presence of radiative corrections. As we will see below (and as has been discussed in several review articles6,7,8) a simple quantum field theoreticestimateprovidesnaturallyacosmologicalconstantwhichisatleast 60ordersofmagnitudetolarge. Quantumfluctuationscreateavacuumenergy which in turn curves the space much stronger than it is observed. Hence, ∗TALK GIVEN AT “INTERNATIONAL WORKSHOP ON PARTICLE PHYSICS AND THE EARLY UNIVERSE”, COSMO2000, CHEJU ISLAND, KOREA, SEPTEM- BER2000 cosmoarchive: submitted to World Scientific on February 1, 2008 1 the classical vacuum energy needs to be adjusted in a very accurate way in order to cancel the contributions from quantum effects. This would require a fine-tuning of the fundamental parameters of the theory to an accuracy of at least 60 digits. From the theoretical point of view we consider this as a rather unsatisfactory situation and would like to analyze alternatives leading to the observed cosmological constant in a more natural way. In this talk we will focus on brane world scenarios and how they might modify the above mentioned problem. In brane worlds the observed matter is confined to live on a hypersurface of some higher dimensional space, whereas gravity andpossiblyalsosomeotherfieldscanpropagateinalldimensions. Thismay givesomealternativepointofviewconcerningthecosmologicalconstantsince the vacuum energy generated by quantum fluctuations of fields living on the brane may not curve the brane itself but instead the space transverse to it. The idea of brane worlds dates back to 9,10,11. A concrete realization can be found in the context of string theory where matter is naturally confined to live on D-branes 12 or orbifold fixed planes 13. More recently there has been renewedinterestin addressingthe problemofthe cosmologicalconstant within brane worlds, for an (incomplete) list of references see [14–42] and references therein/thereof. The talk will be organized as follows. First, we will recall the cosmologi- calconstantproblemasitappearsinordinaryfourdimensionalquantumfield theory. We shall then elaborate on some of the past (four-dimensional) at- tempts tosolvethe problem. Subsequentlythe generalset-upofbraneworlds will be presented. Particular emphasis will be put on a consistency condi- tion(sometimesalsocalledasumrule)forwarpedcompactificationsthathas beenoverlookedin variousattempts toaddressthe problemofthe cosmologi- calconstantandwhichisacrucialtooltounderstandtheissueoffine-tunings in the brane world scenario. Then we will study how fine-tunings appear in order to achieve a vanishing cosmological constant in the Randall Sundrum model 14,15. We shall argue that a similar fine-tuning is needed in the set-up presented in 21,22 once the singularity is resolved. Finally, we elaborate on the issue of the existence of nearby curved solutions and we will argue that it is this questions that has to be addressed if one wants to understand the small value of the cosmologicalconstant. 2 The problem of the cosmological constant The observationalbound on the cosmologicalconstant is λM2 10 120(M )4 (1) Pl ≤ − Pl cosmoarchive: submitted to World Scientific on February 1, 2008 2 where M is the Planck mass (of about 1019 GeV) and the formula has Pl been written in such a way that the quantity appearing on the left hand side correspondsto the vacuumenergydensity. This is averysmallquantityonce one admits the possiblilty of the Planck scale as the fundamantal scale of physics. Even in the particle physics standard model of weak, strong and electromagneticinteractionsone wouldexpectatree levelcontributiontothe vacuumenergyoforderofseveralhundredGeVtakingintoaccountthescalar potentialthatleadstoelectroweaksymmetrybreaking. Moreover,inquantum fieldtheoryweexpectadditionalcontributionsfromperturbativecorrections, e.g. at one loop λM2 =λ M2 +(UV-cutoff)4Str(1) (2) Pl 0 pl inadditiontoλ thebare(treelevel)valueofthecosmologicalconstantwhich 0 can in principle be chosen by hand. The supertrace in (2) is to be taken over degreesoffreedomwhicharelightcomparedtothescalesetbytheUV-cutoff. Comparison of (1) with (2) shows that one needs to fine-tune 120 digits in λ M2 such that it cancels the one-loop contributions with the necessary 0 Pl accuracy. Supersymmetry could ease this problem of radiative corrections (for a review see43). If one believes that the world is supersymmetric above the TeV scale one would still need to adjust 60 digits. Instead of adjusting input parameters of the theory to such a high accuracy in order to achieve agreementwith observationsone would prefer to get (1) as a prediction or at leastasanaturalresultofthetheory(inwhich,forexample,onlyafewdigits need to be tuned, if at all). This is the situation within the framework of four-dimensional quantum fieldtheories. Theabovediscussionmightbe modifiedinabraneworldsetup which we will discuss in this lecture. We should however mention already at this point that “modification” does not necessarily imply an improvement of the situation. Before we get into this discussion let us first recall some attempts to solve the problem in the four-dimensional framework. 3 Some attempts to solve the problem A starting point for a natural solution would be a symmetry that forbids a cosmological constant. In fact, symmetries that could achieve this do exist: e.g. supersymmetry and conformal symmetry. Unfortunately these symme- tries are badly broken in nature at a level of at least a few hundred GeV and thereforetheproblemremains. Stillonemightthinkthatthepresenceofsuch a symmetry would be a first step in the right direction. cosmoarchive: submitted to World Scientific on February 1, 2008 3 A second possible solution could be a dynamical mechanism to relax the cosmologicalconstant. Such a mechanism could be quite similar to the axion mechanism that relaxes the value of the θ parameter in quantum chromody- namics (QCD). This mechanism needs a new ingredient, a propagating field that adjusts is vacuum expectation value dynamically. For a review of these questions see 6,44. In string theory the so-called “sliding dilaton” could play this role as has been argued in 45,46. In all these cases, however, one would thenexpecttheexistenceofanextremelylightscalardegreeoffreedomwhich would lead to new fifth force that probably should not have escaped our de- tection. Other attempts to understand the value of the vacuum energy have used the anthropic principle in one of its various forms. For a review see 6. Given the present situation it is fair to say that we do not have yet a satisfactory solution of the problem of the cosmological constant, at least in the framework of four-dimensional string and quantum field theories. Could thisbebetterinahigherdimensionalworld? Foranalternativewaytoaddress the problem in less than four space-time dimensions see 7. We should keep in mind, however, that the problem of the cosmological constant is just a problem of fine-tuning the parameters of the theory in a veryspecialway. Wenowwanttoseewhetherthiscanbeavoidedinahigher dimensional set-up. 4 Extra dimensions as a new hope In the so-called “brane world scenario” matter fields (quarks and leptons, gauge bosons, Higgs bosons) are supposed to be confined to live on a hyper surface(thebrane)inahigherdimensionalspace,whereasgravityandpossibly also some additional fields can propagate also in directions transverse to the brane. Such a picture of the universe is motivated by recent developments in (open) string theory 12 and heterotic M-theory 13,47. Since gravitational interactions are much weaker than the other known interactions, the size of the additional dimension is much less constrained by observations than in usual compactifications. In fact, the size of the additional dimensions might be directly correlatedto the strength of four-dimensionalgravitational interactions48. Looking for example on product compactifications of type I stringtheoryithasbeennotedthatitispossibletopushthestringscaledown totheTeVrangewhenoneallowsatleasttwoofthecompactifieddimensions to be “extra large” (i.e. up to a µm)53. Afirstlookatthequestionofthecosmologicalconstantdoesnotlookvery promising. The naiveexpectationwouldbe that the cosmologicalconstantin cosmoarchive: submitted to World Scientific on February 1, 2008 4 the extra (bulk) dimensions Λ and that on the brane, the brane tension T, B should vanish separately. We would then essentially have the same situation as in the four-dimensional case, with the additional problem to explain why also Λ has to vanish. The known mechanism of a sliding field 45,46 can be B carried over to this case 49,50,51,52, but does not shed any new light on the question of the cosmologicalconstant. A closer inspection of the situation reveals the novel possibility to have a flat brane even in the presence of a nonzero tension T. For a consistent picture, however, here one also has to require a non-zero bulk cosmological constant Λ that compensates the vacuum energy (tension) of the brane. In B somewaythiscorrespondstoapicturewherethevacuumenergyofthebrane does not lead to a curvature on the brane itself, but curves transverse space and leaves the brane flat. Curvature of the brane can flow off to the bulk, a mechanism that is sometimes called “self-tuning”. For such a mechanism to appear we need to consider so-called warped compactificationswherebraneandtransversespacearenotjustadirectprod- uct. Weshallseethatinthiscasewecanhaveflatbranesembeddedinhigher dimensionalantideSitterspace,providedcertainconsistencyconditionshave been fulfilled. In the following we will be considering the special case that the brane is 1+3 dimensional and we have one additional direction called y. Then the ansatz for the five dimensional metric is in general (M,N = 0,...,4 and µ,ν =0,...3) ds2 G dxMdxN =e2A(y)g˜ dxµdxν +dy2 (3) MN µν ≡ where the brane will be localized at some y. We split g˜ =g¯ +h (4) µν µν µν intoavacuumvalueg¯ andfluctuationsaroundith . Forthevacuumvalue µν µν we will be interested in maximally symmetric spaces, i.e. Minkowski space (M ), de Sitter space (dS ), or anti de Sitter space (adS ). In particular, we 4 4 4 chose coordinates such that diag( 1,1,1,1) for:M 4 − diag 1,e2√Λ¯t,e2√Λ¯t,e2√Λ¯t for:dS g¯µν =diag(cid:16)−e2√ Λ¯x3,e2√ Λ¯x3,e2(cid:17)√ Λ¯x3,1 for:adS4 (5) − − − 4  (cid:16)− (cid:17) That means we are looking for 5d spaces which are foliated with maximally symmetricfourdimensionalslices. Throughoutthistalk,the fivedimensional cosmoarchive: submitted to World Scientific on February 1, 2008 5 action will be of the form 4 S = d5x√ G R (∂φ)2 V (φ) d5x√ gf (φ)δ(y y ). 5 i i Z − (cid:20) − 3 − (cid:21)− Z − − Xi (6) We allow for situations where apart from the graviton also a scalar φ propa- gates in the bulk. The positions of the branes involved are at y . With lower i case g we denote the induced metric on the brane which for our ansatz is simply g =G δMδN. (7) µν MN µ ν The corresponding equations of motion read 1 4 2 1 √ G R G R ∂ φ∂ φ+ (∂φ)2G + V (φ)G MN MN M N MN MN − (cid:20) − 2 − 3 3 2 (cid:21) 1 + √ g f δ(y y )g δµ δν =0 (,8) 2 − i − i µν M N Xi ∂V 8 √ G+ ∂ √ GGMN∂ φ √ g δ(y y )∂ f (φ)=0. (9) M N i φ i − ∂φ − 3 (cid:16) − (cid:17)− − Xi − Afterintegratingoverthefifthcoordinatein(6)oneobtainsafourdimensional effective theory. In particular, the gravity part will be of the form S =M2 d4x g˜ R˜ λ , (10) 4,grav PlZ p− (cid:16) − (cid:17) where R˜ is the 4d scalar curvature computed from g˜. The effective Planck mass is M2 = dye2A, (note that we put the five dimensional Planck mass Pl to one). Now, fRor consistency the ansatz (5) should be a stationary point of (10). This leads to the requirement λ = 6Λ¯. Finally, the on-shell values of the 4d effective action should be equal to the 5-dimensionalone. This results in the consistency condition4 (see also24), S h 5i =6Λ¯M2 . (11) d4x Pl R IthastobefulfilledforallconsistentsolutionsoftheEinsteinequations,inde- pendentlywhetherthebranesareflatorcurved. Especiallyforfoliationswith Poincare invariant slices the vacuum energy Λ¯ should vanish. Curved solu- tions would require a correspondingnonzero value of Λ¯. It is this adjustment of the parameters that replaces the traditional four-dimensional fine-tuning in the brane world picture. cosmoarchive: submitted to World Scientific on February 1, 2008 6 5 A first example: the Randall-Sundrum model As a warm-up example for a warped compactification we want to study the model presented in 14. There is no bulk scalar in that model. Therefore, we put φ=const in (6). Moreover,we plug in V (φ)= Λ , f =T , f =T , (12) B 1 1 2 2 − where Λ , T and T are constants. There will be two branes: one at y = 0 B 1 2 and a second one at y =y . Denoting with a prime a derivative with respect 0 to y the yy-component of the Einstein equation gives Λ 2 B 6(A′) = (13) − 4 Following 14 we are looking for solutions being symmetric under y y and →− periodic under y y+2y . The solution to (13) is 0 → Λ B A= y , (14) −| |r− 24 where y denotes the familiar modulus function for y < y < y and the 0 0 | | − periodic continuationif y is outside that interval. The remaining equation to be solved corresponds to the µν components of the Einstein equation, T T 1 2 3A = δ(y) δ(y y ). (15) ′′ 0 − 4 − 4 − This equationis solvedautomatically by (14)as long as y is neither 0 nor y . 0 Integrating equation (15) from ǫ to ǫ, relates the brane tension T to the 1 − bulk cosmologicalconstant Λ , B T = 24Λ . (16) 1 B − p Integrating around y gives 0 T = 24Λ . (17) 2 B − − p These relations arise due to Λ¯ = 0 in the ansatz and can be viewed as fine- tuningconditionsfortheeffectivecosmologicalconstant(λin(10))16. Indeed, one finds that the consistency condition (11) is satisfied only when (14) to- gether with both fine-tuning conditions (16) and (17) are imposed. Since the brane tension T corresponds to the vacuum energy of matter living in the i corresponding brane, the amount of fine-tuning contained in (16), (17) is of the same order as needed in ordinary 4d quantum field theory discussed in the beginning of this talk. cosmoarchive: submitted to World Scientific on February 1, 2008 7 Next we have to address the important question: What happens if the fine-tuningsdonothold? Doesthisnecessarilyleadtodisasterordosolutions exist also in this case. Indeed it has been shown in 16 that in that case solutions exist, howeverwith Λ¯ =0. This closes the argumentof interpreting 6 conditions (16) and (17) as fine-tunings of the cosmological constant. It also emphasizesthenewproblemwiththeadjustmentofthecosmologicalconstant onthe brane: how to select the flatsolutioninsteadof these “nearby”curved solutions that are continuously connected in parameter (moduli) space. 6 Self-Tuning Thus the generic higher dimensional set-up considersnonzero values of brane tensionsandthebulkcosmologicalconstant. Afinetuningisneededtoarrive at a flat brane with vanishing cosmologicalconstant. Recently an attempt has been made to study the situation with Λ =0. B We will focus on a “ solution” discussed in 21,22 (solution II of the second reference). In this model there is a bulk scalar without a bulk potential V (φ)=0. (18) In addition we put one brane at y =0, and a bulk scalar with a very specific coupling to the brane via 4 f (φ)=T ebφ , with: b= . (19) 0 0 ∓3 Observe that this model already assumes fine-tuned values Λ and b which B would have to be explained. We now make the same warped ansatz (3) as before. Ifagainwe assumeΛ¯ =0in(5), the bulk equationsseemto be solved by A = 1φ, and ′ ±3 ′ 3log 4y+c +dfor:y <0 φ(y)= ±4 3 , (20) (cid:26) 3log(cid:12)4y c(cid:12)+dfor:y >0 ±4 (cid:12)3 − (cid:12) (cid:12) (cid:12) wheredandcareintegrationconsta(cid:12)nts(the(cid:12)ywouldcorrespondtothevacuum expectation values of moduli fields in an effective low energy description). Observe that with the logarithm appearing in (20) we are no longer dealing with an exponential warp factor as (3) would suggest. As a result of this we havetoworryaboutpossiblesingularitiesinthesolutionunderconsideration. We shall come back to this point in a moment. Finally, by integrating the equations of motion around y =0 one obtains the matching condition T0 =4e±34d. (21) cosmoarchive: submitted to World Scientific on February 1, 2008 8 Thismeansthatthematchingconditionresultsinanadjustmentofanintegra- tionconstantratherthanamodel parameter(like in the previouslydiscussed example). So, there seems to be no fine-tuning involved even though we re- quired Λ¯ = 0. As long as one can ensure that contributions to the vacuum energy on the brane couple universely to the bulk scalar as given in (19) it looks as if one can adjust the vev of a modulus such that Poincareinvariance on the brane is not broken. Infactitseemsthatamiraclehasappeared: “solution”(20)isapparently independent of the branetension T . So if one wouldaddsomething to T on 0 0 the brane,the solutiondoesnotchange. Thiswouldalsosolvethe problemof potential contributions to the brane tension in perturbation theory, as they can be absorbed in T . Is this so-called self-tuning of the vacuum energy 0 a solution to the problem of the cosmological constant? Unfortunately not, since there are some subtleties as we shall discuss now. We firstnoticethattheuniformcouplingofthe bulkscalartoanycontri- butionto the vacuumenergyonthe branemaybe problematicdue to scaling anomaliesinthetheorylivingonthebrane4. Apartfromthatonewouldhave to worry about the correct strength of gravitational interactions. In order to be in a agreement with four dimensional gravity, the five dimensional grav- itational wave equation should have normalizable zero modes in the given background. In other words this means that the effective four dimensional Planck mass should be finite. For the model considered with a single brane at y =0 and c<0 this implies that ∞ dye2A(y) should be finite. However, plugging in the solution (20) one finRd−s∞that this is not satisfied. Following 21,22 this could be solved by choosing c > 0 and simultaneously cut off the y integration at the singularities at y = 3c. This prescription then yields a | | 4 finite four dimensional Planck mass. With this choice of parameters, however, we are approaching disaster. Checking the consistency condition (11) one finds that it is not satisfied any- more. The explanation for this is simple – the equation of motions are not satisfied at the singularities, and hence for c > 0 (20) is not a solution to the equations of motion. It is the singularities that have created the miracle mentioned above. Of course, it has been often observed that singularities appear in an ef- fective low energy prescription, and that at those points new effects (such as masslessparticles)appearasa resultofanunderlyingtheoryto whichthe ef- fective description breaks down at this point. A celebratedexample is N =2 supersymmetricYang-Millstheorywheresingularitiesinthemodulispaceare duetomonopolesordyonsbecomingmasslessatthispoint54. Wemightthen hopethatasimilarmechanism(e.g. comingfromstringtheory)maysavethe cosmoarchive: submitted to World Scientific on February 1, 2008 9 solutionwith c>0 andprovide asolutionto the problemofthe cosmological constant. It should be clear by now, that this new physics at the singularity wouldbethe solutionofthecosmologicalconstantproblem,ifsuchasolution does exist at all. In the following, we will investigate such a mechanism and see whether it is connected to a potential fine-tuning of the parameters. Does the new physics at the singularity have to know about the actual value of the tension of the brane at y = 0 or does it lead to a relaxation of the cosmological constant independent of T ? 0 To start this discussion, we first modify the theory in such a way that we obtain a consistent solutionin which the equations of motion are satisfied everywhere. This can, for example, be done by adding two more branes, situated at y = 3c to the setup. We then choose the coupling of the bulk | | 4 scalar to these branes as follows, f (φ)=T eb±φ, (22) ± ± wherethe indexreferstothebraneaty = 3c. Thesetwoadditionalsource ± ±4 terms in the action lead to two more matching conditions whose solution is 4 1 b =b =b= and T =T = T . (23) + + 0 − ∓3 − −2 It is obvious that here a third fine tuning (apart from Λ = 0 and b = 4) B ∓3 hastobeperformed. Theamountoffine-tuningimpliedbytheseconditionsis againdeterminedbythedeviationofthevacuumenergyonthebranefromthe observedvalue. Hence,thesituationhasworsenedwithrespecttothequestion of fine-tuning. However, we have learned that the consistency condition (11) is a very importanttoolto analyzethe questionof the cosmologicalconstant. A short calculation shows that (23) is essential for the consistency condition (11) to be satisfied. 7 Nearby curved solutions So far, we focused on a very specific model and there remains the question whether this situation is generic. For the given set of parameters we should then scan the available moduli space of solutions parametrized by the values of the bulk cosmological constant Λ , the brane tensions T and the various B couplingslikebofthescalarstothebrane. Itisquiteeasytoseethattheabove observationappliesingeneral(forvariousexplicitexamplessee4). Thereason isthefactthattheamountofenergycarriedawayfromthebranebythebulk scalar needs to be absorbed somewhere else. In principle, it could flow off to cosmoarchive: submitted to World Scientific on February 1, 2008 10