JHEP12(2008)096 ) 22 lifetime of December 22, 2008 November 26, 2008 September 16, 2008 ompactifications. less than about exp(10 ult, however, is contingent Received: Accepted: Published: ll, which naively would lead estigate whether the bound is y longer than the Hubble time LT vacua. Despite the freedom y on a conjectural upper bound , U.S.A. Published by IOP Publishing for SISSA dS vacua in string theory, Flux compactifications. Recent work has suggested a surprising new upper bound on the [email protected] [email protected] Department of Physics andUniversity Center of for California, Theoretical Physics, Berkeley,Lawrence CA Berkeley 94720, National U.S.A., Laboratory, Berkeley, and CA 94720 E-mail: Department of Physics, Technion, Haifa 32000, Israel, and Department of Mathematics andUniversity Physics of Haifa atE-mail: Oranim, Tivon 36006, Israel SISSA 2008 c Ben Freivogel Matthew Lippert Abstract: de Sitter vacua in stringbut theory. parametrically The shorter bound than is thesatisfied parametricall recurrence in time. a particular We inv classto of make de the Sitter supersymmetry solutions,to breaking the scale extremely KK exponentially stable sma vacua, we find that the lifetime is always Evidence for a bound on the lifetime of de Sitter space

Hubble times, in agreement withon the several estimates proposed and bound. assumptions;on This in the res particular, Euler we number rel of the Calabi-Yau fourfolds usedKeywords: in KKLT c JHEP12(2008)096 4 6 8 1 3 9 11 12 15 17 18 10 22 19 24 (1.1) (1.2) ter space does not contain acua. Our current under- e stringent bound on the e will explain in section 2, hich violate the second law [1, 2], necessarily decaying dS S 2 2 P e 1 H M − H ∼ – 1 – ∼ dS S rec t is the entropy of the cosmological horizon, dS is the Hubble constant. S H Recently, theoretical considerations have suggested a mor 2.1 Boltzmann Brains2.2 in our causal Boltzmann patch Brains2.3 in the landscape Summary 4.1 Geometry 4.2 SUSY breaking4.3 and decay rate Gravitational corrections 4.4 Destabilization of4.5 bulk fluxes Summary where 1. Introduction String theory appears tostanding contain is a that large debefore number Sitter the of vacua Poincare de recurrence cannot Sitter time, be v completely stable 2. The Boltzmann Brain problem Contents 1. Introduction 3. False vacuum decay 4. Decay rate of the KKLT construction 5. Corrections to the tension 6. Delicacy of the KKLT construction 7. Conclusions and future directions and maximum lifetime of dethe Sitter bound vacua comes in from string demandingan that theory enormous one [3]. causal number patch of As of observers w de formed Sit from rare processes w JHEP12(2008)096 (1.3) (1.4) (1.8) (1.7) (1.6) (1.5) umbers by the owered the lifetime t the above formula ime but much shorter ing theory, focusing on e with their analysis in he point of view of low e recurrence time. Both rder the recurrence time. e minimum with negative ifetime by he least precise prediction LT) [1]. Since the KKLT other [4, 5]. 3 branes, even though the symmetry breaking scale. so the dimensional prefactor D ly stable vacua. For example, , and supersymmetry guaran- ingle scalar field whose poten-  timated in a particular context ty remains in this bound. We energy point of view there seems 3 6 ) . . 3 
. M D 5 20 s ! χ N g M ± 40 2 ( 9 s 2 / P πK 40 g 3 2 3 − 10 2 3 − e M 10 m 10 − e 1 10

−  · 1 − 3 exp – 2 – exp  exp brane is a domain
22 10 exp – 3 – < decay t 3 brane at the tip of a warped throat. The supersymmetry break D We are focusing on a tiny piece of the string theory landscape Recent work on the lifetimes of string theory vacua includes We begin, in section 2, with a discussion of Boltzmann Brains The intuitive explanation for why the lifetime is insensiti be exponentially lownonsupersymmetric due de to Sitter vacuum the isthe described exponential tip by of an redshift the NS5 throat. in br wall. In the the What 4-dimensional t happens description, is the very that same although warp the factor SUSY guaranteessmall. that breaking the We scale will tension i see of the thatthroats. doma these In two other warp factors words,the cancel the rate in decay compu is is actually a insensitive process to localized the nea length ofto the try throat to f constructmay extremely be stable highly vacua using modelsurprising other dependent. bound const demanded We byby present Boltzmann the here landscape Brain one of consideratio small string theory. piec Westphal [7], by Dine and collaboratorshowever, [8], were and concerned by Johnson with a stabilityfocus on in time on scales one corner of of ord the landscape and investigate a new t for a bound onvacuum the decay, lifetimes reminding of the de reader Sittervacua which that vacua. live for at In about section this therate 3 level recurrence using we time. it the is section brane 4 no description pre due of the to instanton, closed while in stringSitter s moduli. vacua using the In KKLTwindow section method. where 6 the In we particular, constructionthese point we is difficulties show out marginally can th the be under easily diffic control. conclude fixed in by section minor 7. modifications of 2. The Boltzmann Brain problem String theory appears towe contain normally a think vast landscape ofelectron of as mass, stable constants vary a of froma nature, one mechanism such vacuum for to as producinginflation. another. the large regions cosmo String of theor spacetime in each is of order the maximum known Euler number, we get a numerical Assuming that the Euler number of Calabi-Yau fourfolds is bo Clearly our result is highly sensitive to the maximumbreaking Euler scale n is theanti- following. Recall that KKLT break supe JHEP12(2008)096 ∼ P pocket each s that we have not ture is exponentially ch, how many Earths n infinite number of he reason is that when ular definition of what er hand, the number of rrence time. (In section 3 ervers inside it exists — a Boltzmann etical predictions for the ” [2, 12, 3]. There are two l way via inflation, reheat- volume of this causal patch d use the term “Boltzmann tions, in the multiverse, of ssume for the moment that nd law violation. t are the consequences? onstants, different electron icles can spontaneously form r ability to make predictions o observers produced in the s in sharp conflict with obser- y rare thermal fluctuations as rs form in many different re- is that the entire formulation nt of second law violation, of thermodynamics is violated. if we work in the semiclassical gulating the infinities lead to rmal fluctuations is an isolated n Brains within our horizon. As a problem of infinities. rough rare thermal fluctuations ]. The probability of a given ex- lating infinities and extracting pre- . Observers who form from rare thermal S – 4 – . measure ). So fluctuating a large, homogeneous universe full of struc . We want to ask the following question: within one causal pat S 3 − Our observations indicate that we are ordinary observers. T Fortunately, most simple prescriptions lead to prediction We restrict attention to one causally connected region; the In the eternally inflating multiverse, intelligent observe Many problems remain in making this framework precise. One i Eternal inflation produces an infinite volume of spacetime, a H vation. One test of abasic measure ways in is which the structure “Boltzmanning, can Brain and form. problem gravitational It collapse. canwhich Structure form decrease can in the also the entropy. usua form Fora example, th a planet diffuse populated gas of byusual part way intelligent as observers. “ordinary observers,”“Boltzmann and We observers Brains.” will produced refer b t structure forms by rare thermal fluctuations,The the probability of second a law rare fluctuation is supressed by the amou exp(∆ rarer than fluctuating aobservers small produced amount is of only structure. proportional to On ∆ the oth constitutes an observer, the typicalbrain observer in formed by empty the space,Brain. which We will just not need livesBrain” to to long refer refer enough to to such to any extreme observer limits realize which here forms2.1 an as Boltzmann a Brains result in of our seco To get causal used patch to this strange idea,far let as us we first discuss know, Boltzman ourwe vacuum may review have the a arguments lifetimeour of leading order vacuum to the lives this recu for conclusion.) approximately the Let recurrence us time. a Wha is fluctuations do not see stars in the sky. In fact, with a partic universe. Different seeminglydrastically different natural predictions. prescriptions A prescriptiondictions for for is regu re referred to as a gions. Differentmasses, observers and will different seeresults CMB of different multipoles. experiments cosmological areperimental In c necessarily outcome this statistical is [10, setting, proportionalthat 11 to outcome. theor the number of observa precisely defined what constitutesso an far observation. relies Another onapproximation the and semiclassical take approximation. some But, definitionis even of hindered an by observation, a ou familiar hobgoblin of theoretical physics: “pocket universes” of each type, and an infinite number of obs JHEP12(2008)096 E ... (2.5) (2.1) (2.4) (2.6) (2.7) (2.2) (2.3) of Earths produced by s is patch is roughly equal to e more ordinary observers e unimaginably large. s for about the recurrence zmann Earth is Brains. After all, it takes a er the recurrence time, the ns to ordinary observers in ry Earths, e, so we write the ratio as a rrence time, we find that our t be regulated before we can nd how many Earths form in ively, one might expect that it vation indicates that our Earth in our causal patch. pared to the number produced by 92 123 0000000000000000000000000000000000000 92 10 10 E e 10 e 1 e 1 M . 1 − − − . 123 123 H 22 H H live for the recurrence time? The answer is 10 10 e e βE e 10 years) ≈ 1 but can depend on details such as coupling = not − βe ≈ ≈ 10 – 5 – β , the time to produce a fluctuation of energy H 1 ≈ BE BE BE t OE OE decay (10 − t t ≈ N N β N N ≈ = BE t BE t BE N 100000000000000000000000000000000000000000000000000 e = In a system at finite temperature If our vacuum lives for about the recurrence time, the number Can we conclude that our vacuum must BE OE N N On the other hand, thethe number number of of ordinary stars Earths inside in our our horizon, causal where the prefactor is typically of order form from rare thermal fluctuations (“Boltzmann Earths”), a the usual way (“ordinary Earths”)? is given by Dividing, we find constants. In our case, this means that the time to form a Bolt Therefore, assuming that our vacuumcausal lives patch for contains about far the more recu Boltzmann Earths than ordina It is easy tosingle forget exponential how large double-exponential numbers ar Plugging in the values, we find Continuing to assume that thenumber lifetime of of Boltzmann our Earths vacuum produced is before of our ord vacuum decay except that it will not fit on the page. The numbers involved ar ordinary structure formation is completely negligiblerare com thermal fluctuations. Yet, aswas we formed discussed above, in obser thetime, ordinary we way. are Therefore, extraordinarily atypical if among our civilizations vacuum live that we really need adoes measure to not answer matter this if question. ourlong causal Intuit time patch for is dominated the by Boltzmannare Boltzmann Brains produced to form, elsewhere and in indefinitively the the say multiverse. meantim that The comparing infinities theone mus number causal of patch is Boltzmann a Brai meaningful thing to do. JHEP12(2008)096 (2.9) (2.8) (2.10) n that it degrees of BB S degrees of freedom given vacuum. First BB rain, we must remove S rm. in the structure is about t to compare the number horizon. On average, one e formation of interesting vide the mass of a person d in constructing ordered nimum number of degrees his structure has a size of itter vacua. Above we fo- operties which we will not e multiverse we need a more han us in general we will charac- . Finally, if the cosmological ero. If particle physics allows ent observers; the amount of ther hand, it is quite possible s for interesting structures to rticles than in a person, so we planets as observers. The earth ms most robust to characterize cal constant may be too large, so . ) σ (1 . 50 30 10 is not literally the entropy. 10 15 ∼ ± ∼ BB – 6 – ? The number of degrees of freedom in a person 35 S BB estimate BB BB S S σ S = 10 BB S is related to the entropy of the object under consideration i degrees of freedom, so BB more particles than a person, so this estimate would give . In other words, we say that any system with fewer than S BB S BB 22 S Now we can estimate the number of Boltzmann Brains formed in a In equilibrium, all of the entropy of de Sitter space is in the What is a reasonable estimate for graviton is present inentropy the from bulk. the horizonorder In and the order build Hubble to an scale, make ordered then a structure. the Boltzmann number If B of t degrees of freedom of all, the particlestructures, physics in which of case the the numberfor vacuum of the may Boltzmann formation Brains not of is allow z interestingthat for structures, there th the is cosmologi notconstant enough room is to reasonably make smallform, interesting and we structures can the estimate particle the physics number allow of Boltzmann Brains which fo Surely no more entropyintelligence than per this particle is on requiredthat the to intelligent earth form observers is intellig can miniscule. bewill produced On summarize with our the far ignorance o fewer by pa the 1 freedom is not counted as an observer; systems with greater t 2.2 Boltzmann Brains in the landscape More generally, string theory contains a large number of de S cused on the production ofof Boltzmann Boltzmann Brains Earths to in the ourgeneral number vacuum, of definition bu ordinary of observers what inobservers constitutes th by an requiring “observer.” them It toterize see have Boltzmann a Brains certain as complexity.of Th ordered freedom systems with at least a mi is about equal to the number of particles, so roughly we can di by the mass of the proton to get is the logarithm ofsystems with the number of states, but we are intereste have a chance ofexamine being here. observers if they also satisfy other pr Perhaps we only want to counthas entire about civilizations living 10 on JHEP12(2008)096 ongly (2.14) (2.13) (2.11) (2.16) (2.17) (2.15) (2.18) (2.12) uivalent to are not of ed structure. inst S The Boltzmann n. In the remainder 1 and acuum is izon. by defining BB B he two. Therefore, there mall entropy compared to BB tion t ally gives a rough bound, uild a Boltzmann Brain, we nergy must be expended in a Boltzmann Brain, because , the number of Boltzmann tures from forming efficiently. he entire system decreases its BB e of B ; since the Boltzmann Brain has BB ber of degrees of freedom removed from BB B g, then the number of degrees of freedom S − . . . , . . inst BB BB BB S BB S S B e inst BB which are not strongly gravitating, one can replace B e e BB e S t decay 1 1 1 − t e ∼ e − − − ∼ >S = H H ∼ – 7 – BB ≈ >H t ≡ decay BB BB t BB B decay N t N BB BB BB = t t t BB N , on the entropy of the brain. But for systems which are not str S . If the Boltzmann Brain is of order Hubble size then this is eq & 1 − . Therefore, the time to produce a Boltzmann Brain is given by [14]. We thank Andrei Linde for bringing this to our attentio MR MH BB 3 / S 4 is an exponentially large number. Generically, S & BB degrees of freedom from the horizon and put them into an order B . MR 3 BB / 4 BB S S The expected number of Boltzmann Brains produced in a given v by Actually, if the Boltzmann Brain is not strongly gravitatin 1 BB This process decreases the entropy of the horizon by It is helpful to make explicit the double-exponential natur in the Boltzmannthe Brain horizon will is be of significantly order smaller. The num small entropy relative to theentropy number by of about degrees of freedom, t The decay time is given by the exponential of the instanton ac the Beckenstein bound, Note that this is actually athe lower particle bound physics on the of time the toFor vacuum produce may example, prevent if ordered the struc building mass a of Boltzmann the Brain. particles Therefore is the large above then argument extra re e equal to the number of degrees of freedom removed from the hor Now the number of Boltzmann Brains is given by Recall that Our argument above gives Brain we are buildingthe is number an of ordered degrees state ofremove freedom and it therefore contains. has So, a in s order to b gravitating, Brains produced is double-exponentially small, S of this section, if one wants to restrict attention to brains the same order, soare the two exponent regimes. is dominated If by the the instanton larger action of is t smaller than JHEP12(2008)096 (2.19) (2.20) (2.21) (2.22) (2.23) class of vacua will the proposed bound is following: all proposed is either essentially zero entially large number of , so that the decay time dy ruled out [13, 15] require tzmann Brains, e estimated above that in a BB eful number to keep in mind, , it is parametrically shorter his paper. See [3, 16 – 20] for on ourselves, we found cessary before we can say that one causal patch is a problem. B e appropriate definition of the ncertainty to account for how is simply related to the number BB . t . BB 20 B . ± 15 BB 40 ± S > e BB e 10 35 1 e − 1 inst conifold with defor- H y les: the A-cycle, which is a ( B Z mn e fluxes through the cycles via 2 ˆ where the deformation becomes lly fibered ) g s 4] u 1 3 2 / . / πl − e 1 ) 1 ; this factor is often ignored. , (2 S y . 1 3 0 ( « r 2 T ) + 1 2 6 s in this region is approximately = ∼ / l ˜ r M ; further nonperturbative effects stabilize 4 6 s M 1 ds πK , the warp factor is approximately [24, 25] s 0 l s πK r 2 g r 2 r h = r/ g K 2 u mn where the fact that the CY is not simply r 6 − W g ˆ g − e + e 0 + 3 0 „ r ν p r 2 = [log( y – 11 – 4 6 dx 6 = dr 3 µ V L d is a parameter of the infrared physics and does not by a factor of = exp F S ≈ dx z Z S A S ) 2 Z ˆ s x ( d by 2 =1+ ) z µν s h , but g 1 0 ) πl and the IR cutoff ˜ r y 0 ( (2 r 2 / 1 = , and the B-cycle, which is noncompact in the conifold soluti − h S M = . The Calabi-Yau metric ˆ S 2 ds ) is the fiducial Calabi-Yau metric on the manifold, which we h y is the number of units of flux through the A-cycle and ( mn M g In the presence of fluxes, the compact manifold is a conformal Between the UV cutoff fluxes generates a tree-level superpotential Many works define the parameter depends on the UV cutoff 5 3 so that H equivalently, an F-theory compactification on an elliptica This parameter is related to our parameter the volume modulus [1]. write the string-frame metric 4.1 Geometry We start in type IIB string theory with D7 branes and O3 planes Here ˆ where This metric is valid between a UV cutoff important. The deformed conifold3-sphere has with two volume holomorphic 3-cyc a conifold becomes apparent and an IR cutoff ˜ Conifoldology [23] relates the deformation parameter to th Thus the unwarped volume of the compactification is K of flux through the B-cycle: depend on the cutoff. We assume that near somemation point the parameter Calabi-Yau looks like a def JHEP12(2008)096 3 2 3 S S H , so (4.9) other (4.14) (4.13) (4.10) (4.11) (4.12) mn g 3-brane nism of 3’s and the D D 1 units of − f the at the tip of the 5 wrapping an K 3 S , the case relevant to in the metric ˆ NS 3 flux carry a 3 D point of view, this looks 3 S N H ]); what will be important he geometry, 12 break supersymmetry by an 3s, leaving and . D 3 1 M > F − tip 3 s l 1 6 2 x h 4 s / − tip l 3 4 3 h 2 ) S d . ) 4 s V l u 4 M s 3 4 Z / M s g 3s. The e = 3 s D 4 s 3 8 g l − tip ) bg is exponentially small; equation (4.11) shows D S N s ( h 3 π 2 u g 2 S 3 3 81( D – 12 – flux they polarize into an π (2 e ∼ ) 3s sit near one pole of the 4 N 3 π / = field induced by the fluxes. 3 branes at the tip of the throat. The contibution S D = 3 tip = 2 H 5 (2 4 h D F 3 L 10 S δV = 5 is large enough to slide around the equator of the ⋆ V 3 2 anti- gives the volume of the minimal D − s NS 3 S g S D = N 5 , the 3 932. H . 5 classically sits near the original pole but can decay to the D 0 is explained in [27]; half the energy comes from the tension o 3s, conserving 3-brane charge. For N ≈ NS D δV 12 b 3 D N is exponentially large and they scale as [23] M < − , which in this process “annihilates” with the 3s at the tip of the throat are subject to decay by the KPV mecha tip is the warp factor at the infrared end of the throat. h D . If M 3s to the action is 3 tip MK S h D The Note that the deformation parameter The factor of 2 in 6 of the pole via tunneling across the equator. brane/flux annihilation [27, 28].throat, but, The due to the with other half from the potential energy in the metastable dS, the flux and We work in the stringlike frame. an Due exponentially to small the additional warp energy factor, density, from the 4d with the constant The exact metric infor the us throat is is that known the (see, proper for volume of example, the [26 minimal A-cycle is that that of the we have a useful relation between the complex structure and t where It is these exponentiallyexponentially small small parameters amount. which allow us to 4.2 SUSY breaking and decaySUSY rate is broken by adding to the other pole where it de-polarizes into charge JHEP12(2008)096 n SY is of the (4.18) (4.16) (4.15) (4.17) ffective , the instanton 3 D N 3s which provide the ll . . D mation is ius in the 4d spacetime 4 correction to the exponent 4 . 3 / / 6 cation has cancelled out, ) parating the interior true cels out of the decay rate. 7 3 ns to the decay, and in the 3 ) proximation the instanton M − tip ctually independent of the ctor appeared at all is that M π D ses localized at the tip, we 5 brane which mediates the h s 4 N then independent of the warp g 3 sensitive to how far away the s at; the / 2 ) and the difference in vacuum ( l 4 3 / 2 / NS 3 / π − tip 1 tip ous (2 6 ction to the exponent which we have 1 s b h g M s 3 l x h 2 27 3 / 2 π 2048 3 / d b 1 s 16 = g Z 2 3 3 / 3 = ) 3 D S 4 4 V / N τ M 3 δV 4 6 s 2 ( l − tip / . 2 s 3 2 h p – 13 – g b 3 π 5 2 1 3 S ) and the amount of flux V 27 π s = 6 s g l (2 = 2 s is their τ g 3 3 δV = 5 1 D ) 5 CDL π N = S NS ρ (2 S = = 5 KPV , and our has cancelled out! Although the warped geometry allows for a NS 7 2 0 B τ b . The action of the NS5-brane wrapping the 3-sphere at the tip 3 3s and SUSY restored, from the exterior false vacuum where SU tip S 3s are present. The tension of this domain wall is just the 4d e h D D is their b There is an intuitive explanation for why the warp factor can For now we will assume we can ignore gravitational correctio The radius of the domain wall (3.6) in the field theory approxi In the thin-wall limit, which is a good approximation for sma This matches the decay rate found by [27] in 2001, up to the fam 7 throat is mediating the KPV decay isand a wrapping Euclidean the NS5 bubble at a fixed rad exponentially small SUSY breakingamount of scale, warping. the Note decayso that rate that also the is the lifetime a volume depends of only the on compactifi The entire decay process is localized near the tip of the thro tension of the NS5-brane difference in vacuum energy are localized at the tip, and made in versiondiscovered; 4 our from 2006 and an additional factor of 4 corre The warp factor next section we’ll checksolution whether is we given can. by just Inenergy the the tension (4.14). field of the theory domain ap wall (4.16 From the 4d point ofvacuum, view with the no NS5-brane is just a domain wall se and the action (3.1) is decay is also localizedbulk at of the the tip. Calabi-Yau is. Sowe the In are entire fact, measuring process the quantitiescan is only relative write in reason everything to the in the warp terms fa of bulk. proper quantities For which proces are broken and the JHEP12(2008)096 (4.22) to be (4.27) (4.28) (4.20) (4.26) (4.21) (4.19) (4.24) (4.25) (4.23) S as large as B ifetime. Plugging flux is 5 F 5 n the warp factor, at least χ s = 1 to make l ormation parameter 10 2 . ion of 3 / − , D 1 s « g , a similar argument would tell us that 10 N 3 M 3 s the proper decay time is independent of 2 3 3 6 ) 2 l · / 4 s 6 ) 3 / 2 3 l D 3 3 4 / M D s 3 1 s s M N D g N M g D g s < M 3 4 ( 2 3 in any case the exponential gives the dominant N g ) 2 / N 4 ( / π . , π 3 6 π 2 3 χ 3 b 24 χ b 1 24 b M − 3 16 (2 χ 27 2 24 2048 < 10 . Combining this with the tadpole constraint, be bigger than the string scale gives the addi- = = = „ 2 M · s of the F-theory compactification. In addition, 3 M > ρ M τ – 14 – 3 g M S s s 4 4 4 3 exp s g M < / / MK < ≈ 4 − g 1 3 hδV / − 1 tip h CY 10 h h = · = K > g KPV ∼ 3 = 8 B < proper decay proper t τ proper δV KPV ρ B in equation (4.18), we find b is the Euler number of the χ Now, having computed the decay rate, we can deduce a maximum l Although we are not computing the one-loop determinant here 8 Thus the instanton action is bounded by the decay time willthe depend warping, on so the we warp get factor in such a way that which when combined with (4.25) gives a maximum for This makes it clearin that the the field instanton theory action approximation. cannot depend o in the value of factor: tional constraint It is not completely clear that this argument is correct, but behavior in the regime of interest. where Furthermore, requiring that the minimal we obtain exponentially small, which implies consistency of the warped compactification requires the def possible. The tadpole constraint coming from the conservat How big can this quantity possibly be? First, we set JHEP12(2008)096 1 up ≪ χ (4.31) (4.30) (4.29) (4.32) (4.33) x , the lifetime agrees so well , it is extremely χ 5 χ ero cosmological con- know the dimensional te. Now we must check osmological constant by lly on nn Brain considerations. ’s with Euler number e scale set by the de Sitter are negligible when 4 smological constant, because rections which can make the le is related to the amount of . 3s. The supersymmetric AdS ince the decay is a field theory CY ) because the lifetime is always . Plugging (4.16), (4.14), and cale which is much shorter than
D . 40 5 . . This is more than a technicality χ
. AdS 8 s
5 l l 2 10 6 s 2 s / χ 22 l 1 − g u ≪ 10 7 − tip 6 10 ) 10 − ρ h e π · 2 3 10 4 3 (2 · M D exp 2 s 4 = 1 N g − 3 u exp – 15 – b 6 6 10 1 4 e V exp G 2 − π − s ! (4.39) 2 . g / tip . 1 2 1 . h 2 2 0 − tip dS / . M 1 π h MK . V 2 s s ≈ 2 4 − tip g . If χ g 24 24 h 3 2 G 3 S / 4 π 1 π > 3 ≈ 3 − tip 10 – 16 – s h > g > s u u x< 4 KPV 4 πg 3 e e , therefore, brane/flux annihilation in long, large- 3 B b through the B-cycle, which controls the length of the √ M , this becomes 2 K M is too small, gravitational corrections are large. As discu s is bounded by its maximum known value of around 10 and x< which control the field theory decay rate, dialing the warp tip s χ g h M 1 the tension is supercritical and the decay rate nearly satu K > g 1, so the gravitational corrections are bounded by and x > throats, brane/flux annihilation therefore occurs extreme s ≥ g K 3 D N Although the presence of the warp factor On the other hand, if In addition, Recalling that we need rections can easily beare made big very for small, anycompactification we reasonable is want choice bigger to of than parameters. know the if Demanding volume the t in g the throat give Combining the inequalities (4.26) and (4.27),to we find be the low Assuming, as before, that Note that here the warp factor does not cancel; warping is ver itational backreaction because the energy of the process is now have For short, small- This quantity canchoices be of bigger parameters, so thanthe we one need parameters to for worry acceptable, about gravitation althoug factor controls the strength ofcorresponds the to gravitational dialing correcti the flux ever, in the regimealso where badly the broken. gravitationalimportant corrections We in will are this see regime. in the next section that other d throat as well as the deformation parameter throats occur at field-theory rates. are indeed small,bound and (4.28). we can For safely fixed use the field theory result in section 3, when the recurrence bound (3.7): JHEP12(2008)096 , < AdS V (4.43) (4.41) (4.42) (4.40) CDGKL B cles. Before su- ry approximation. fting to a dS vacuum, rocess and the decay is , the bounce action is | uum energies and not on ls of instantons mediating . x annihilation (4.18), it is ay. with vacuum energy apped on cycles which can ription of the vacuum, it is AdS pproximated by the tension V r, in this regime fied. However, all flux vacua we consider whether it is really 0 KPV he instanton action is well ap- . And, of course, with unbroken B n the size of the SUSY breaking, ≪ | 4 0 ρ | 4 AdS dS Kallosh, and Linde [6] estimated the ℓ V 2 AdS 2 ∼ π V 0 x KPV ) , | 6 4 0 2 4 B of the critical bubble for the KPV decay in ρ τ G δV 0 1 and gravitational corrections are unimpor- ∼ ρ AdS = . ∼ V – 17 – ( 0 x< ρ 0 KPV 4 0 ρ CDGKL 4 AdS B ℓ CDGKL B controls the gravitational corrections. So we can write 1 and gravity is important, the brane/flux annihilation B ∼ 2 x , so the destabilization of bulk fluxes is slower than the x> ∼ 0 KPV 4 0 CDGKL ρ 4 AdS ℓ B > B CDGKL is also approximately the size of SUSY breaking. is the action for the brane/flux annihilation in the field theo B | 1, AdS 0 KPV V | B , so the destabilization of bulk fluxes is the most important p x < Recall that in the bulk we have wrapped fluxes on a variety of cy However, if supersymmetry is broken by a small amount by upli On the other hand, when We will first consider the case when 0 KPV the field theory approximation, For to get brane/flux annihilation, and since gravity is unimportant t proximated by the field theory result 4.4 Destabilization of bulk fluxes Having computed the decay rate via brane/flux annihilation, the dominant decay mode.impossible Without a to completely be detailed sure desc have the a fastest very decay generic has decay mode truly whose been rate identi can be estimated. persymmetry is broken, there areinterpolate between BPS vacua domain with walls, different flux branessupersymmetry configurations wr there are no instabilities. some of these now near-BPS domaingenuine walls instabilities. become Ceresole, the Dall’Agata, bubble wal Giryavets, decay rate in precisely these circumstances.the bubble To first size order and i decaythe rate change depend in only tension.of on Therefore, the the the associated change bubble BPS in tension domain vac wall. can For be a a supersymmetric AdS, rate instead approaches the recurrence rate (4.39). Howeve tant. To compare thehelpful rate to (4.40) multiply to and the divide decay by rate the by radius brane/flu where where B even faster than the field theory approximation to the KPV dec approximately Recall that the quantity uplifted to slightly positive cosmological constant, JHEP12(2008)096 (4.48) (4.47) (4.46) wrapped M ormula (4.40) to te of nearly supersym- tions which are longer upersymmetry breaking gravity approximation is e KKLT vacuum. In the ge other authors to try to y to determine the dimen- nihilation and warping has n, so that the decay is just . tead by decay of bulk fluxes t for BPS domain walls, as to our proposed bound. nds sensitively on the largest y true. Finally, our estimates mply on the flux he decay rate is bounded by rrect bound. fore supersymmetry breaking , is prefactor is small compared to 22 s d surprising. g ven within our simplified context 10 e 1 (4.44) 1 1 (4.45) . −
6 x< x> M s 0 KPV

K  np 3 branes. This is essentially the same W D W 2 / + K e  flux – 19 – ∆ W explicit

= 3s, one can consider a wrapped = M D E W τ along with M indicates that this is the tension computed in the 4D Einstei E τ 3s, we expect that the tension computed from the BPS formula w 1 and D 5 brane. In fact, there are other contributions to the tensio appears to be the dominant correction. − S which controls the deformation of the conifold changes in th NS K S The superpotential is The contribution to the action from the closed string moduli To compute the full tension, including the effect of the close This correction was first computed by Frey, Lippert, and Will This supersymmetric domain wall does not constitute an inst change much, but nowvacuum the and domain a wall supersymmetricreal interpolates true instability. vacuum, between In and a the the n following correspond we compute the tension in We can compute thedomain tension wall. by Even relating in the the absence domain of wall we are i the wrapped In the previous section, we approximated the tension of the d 5. Corrections to the tension review that computation, updatingpotential it and to correcting some reflect minor an errorsliterature. which improved arose u However, due these t resultspotential remain in uncertain warped becau compactifications remains an open probl from the 4D superpotential in thewhere 4D we Einstein worked frame. in Therefo string frame, in this section we work in the wall; these contributions could haveparameter the effect of increasin taking this into account increasesthe the change bounce in action. In fac the an approximation similarinterested in to describing that the of brane/flux [6] annihilation as which described in se have fluxes domain wall which changes the flux through the B-cycle by one u are supersymmetric and we can compute the tension using the B where the notation It is unclear tothe us complications whether associated our with calculation Kahler moduli is in exact warped for c BPSsmall do number of wall. On one side of the domain wall we have fluxes JHEP12(2008)096 s g (5.9) (5.5) (5.7) (5.8) (5.6) (5.4) (5.12) (5.11) (5.10) 9 1) forms. Our explicit ,  1 = 0, we get −   1 3 0 S r tential is small and that Ω throat − olume is not. i A W 3 K 0 e throat to the superpotential, log S or the conifold throat we have Z r S 2 fluxes are (2 nventions the unwarped volume ) ∂  Ω (5.3)

G s s i S M log ≈ ∧ g πl B W  Z S πi 1 swer is correct. The conflict is resolved as w (2 G 2 W 2 i MS . w S nd on UV physics, and is equal to what one and V ∆ − S 6 s V π Z l ∆ = K Ω M D 2 M 6 8 s ) i i 6 l s Ω= − l B G π 7 + 6 ) Z 1 6 B s ) B i (2 l S π w 1) forms, but because the manifold is noncompact this w Z W Z π 2 G s , s V 6 s V i (2 g l g K = ∆ (2 (2 A – 20 –

 Z 6 = = s = 6 s l is the warped volume of the compactification, and vac  are l 2 s 2 ¯ 5 Ω = / g = 0. Once the conifold is embedded in a compact Calabi-Yau, w ) i i W flux K 1) forms. V ∧ π X e , = 0 because the ≈ ∆ vac , throat W M (2 Ω s 2 E decreases by one unit while M stays fixed. The change in W ) vac τ − s Z Ω= i/g throat K = πl ∧ W = throat S G τ W = (2 Z throat Ω = G W with A A Z Z τH − F = G Across the domain wall, The flux superpotential can be evaluated by using the formula In [28] it is claimed that 9 Evaluating this at the supersymmetric minimum does not change much in the transition, the tension is it is no longer clear that the fluxes are (2 superpotential is therefore As pointed out by, for example, [32,is 34], independent although in of these the co complex structure moduli, the warped v where We choose the following set of conventions Assuming that the change in the superpotential and Kahler po Ω is the holomorphic three-form. calculation here gives awould nonzero get answer, from which the doesfollows. field not In depe theory the noncompact analysis, conifold so the fluxes we believe this an Plugging these in, we get a formula for the contribution of th where the sum is over all symplectic pairs of three-cycles. F is not sufficient to conclude that JHEP12(2008)096 5 NS (5.13) (5.17) (5.19) (5.18) (5.16) (5.14) (5.15) 3 branes. D at the tip of the 3 explicit S all is subtle, because an do the calculation e we multiply by the is M ed from the probe ill not keep them here.

1 arped volume of the last − W . 2 K d out. To get the physical 6 s / r l 1 ifold metric and get the same 2 − tip / e domain wall, one step in the e in warped volume is just the . We can compute the warped 1 h s l − tip and ˜ M 6 s π 2 2 l s h 2 g / / 2 2 3 K 1 1 ) ) 2 ) r w e is the size of the − tip . V 3 M h M S . M = s S S to the geometrical factors appearing in s s s g r g V 3 dr S g g ( ( S / u ( 6 s c 1 3 l 6 IR π M 5 ∼ i − ℓ ∼  s 2 + ) e 1 1 g 3 branes included, but the answer can instead π 4 L − w − / M K u 1 D 3 K (2 V s = K S e ˜ S r – 21 – K s g − tip 3 branes on the metric, then one finds that the ˜ r S g S 2 h = = Z / D 6 s  1 ∆ l = 5 ∼ − tip IR W ) = ℓ 1 )], so h S w π ∆ 1 V 6 IR M − (2 ℓ K s

∆ ˜ r K g 6 s ˜ r ( ∼ l 2 s 1, as it should be in the supergravity approximation. Then w g V πK/ ≪ = ∆ 2 ) E − τ 1 units of flux through the B-cycle there are M s − g exp[ ( 3 0 K r π/ = . The relationship between the IR cutoffs ˜ S 2 / 1 is an unknown order one constant. − tip 5 computation. From (4.11), the parameter c h Gathering together the above formulas we get We would like to compare this formula to the tension we comput Computing the change in the warped volume across the domain w Since this argument will not get order one factors right, we w NS assuming that 2 conifold with the Kahlervolume modulus we and put these the back warp in: factor factore One can perform this analysis inresult. the full warped deformed con the where Recall that Performing the integral, we get on the side with volume of this step: The first part isfactor the proper volume, and to get the warped volum brane computation. To translate, we must relate The proper AdS radius in the IR is Klebanov-Strassler cascade has been eliminated.warped volume The chang of thestep eliminated of region. the Klebanov-Strassler So, cascade. we just need the w be estimated by the following intuitive argument. Across th change in the warpedcorrectly volume by has finding the a full strange metric with UV the dependence. One c If one ignores the backreaction of the JHEP12(2008)096 (5.20) (5.21) (5.22) is exponentially cisely the wrapped tip h

the throat make the 2 mpute the additional / 1 s that we are interested − tip ght of as the contribution and the volume modulus upersymmetric vacua but rbative corrections to the . Therefore it, and not the tification is about the same s ing to our formula from the not important in the regime g W h tip y zero cosmological constant. the tension becomes Therefore, we are justified in 3 s ctor is precisely the conversion h l ion mechanism can work. T dS vacua, we discover that in uppressed by additional powers 2 2 alid and the nonsupersymmetric / is challenging to find parameters 1 w M ons to the superpotential and then V s g c + . 4 2 1 / / 5 brane calculation. − tip 3 , than the first term; with some more work 1 h − tip − tip tip h h NS ∼ 3 h S ∼ – 22 – τ V ∆ 6 s l W 2 s g , and rearranging some factors, we get 5 1 ) w π V 3s is not easy to control. The basic tension is that large flux (2 . However, the warp factor at the tip ≈

s D 6 s g 2 l 9 s / l u 5 brane, could be the dominant contribution for a wide range 0 which implies 3 3 s 6 w g e V ≈ NS = δV E τ + AdS V 5 computation. The first term inside the absolute value is pre NS For us, however, this term will not be important. The reason i We have assumed in the above that the string coupling 5 brane tension, equation (4.16). The second term can be thou do not change significantly across the domain wall. One can co numbers in the throat are desirablevacuum so that is supergravity is metastable. v volume On of the the othersuperpotential compactification hand, are large. exponentially large small at flux However,for large numbers the volume. which It in the nonpertu volumesmall is enough large so enough that the to nonperturbative allow volume metastable stabilizat nons This is the tension computed in thefrom Einstein string frame. frame The to prefa Einsteinprobe frame, so we drop this in compar to the action due to changingof the closed the string volume moduli. and It is factors s of Using the approximation that the warped volume of the compac as the unwarped volume, NS large, and the second term is suppressed by fewer powers of of parameters. in a situation where the nonsupersymmetricThis vacuum requires has nearl tension of the wrapped Thus for uplifting to nearly flat space the correction term in which is now smaller, in terms of powers of one can see thatusing in the fact tension the calculated correction from is the always probe negligible. σ contribution to the action fromof these interest. terms and find that it is 6. Delicacy of the KKLT construction Upon investigating the parameter space offact controllable stabilizing KKL the volume with nonperturbativebreaking correcti supersymmetry with JHEP12(2008)096 (6.5) (6.6) (6.4) (6.1) (6.2) (6.7) (6.3) uction because s. In terms of the 3 th fluxes so that we / 3s are perturbatively 2 D  the volume modulus as mpted to make each one r the bulk where the warp 6 V is the volume of the throat throat .34) can be restated as [30] V vely, the superpotential is th the warped volume and the .  throat 3 / V , so for the supergravity solution to 2 M K s  M g aσ aσ . . . . s − −  3 5000 (6.8) 6 + e [35]. This gives roughly bg ) D V Ae throat σ √ MK N 3 M V 3 + s A< aAσ 2000 π  g 12 , and in order that the 0 ( 3 − − 3 log – 23 – W M log  − s = 3 = MK − σ > 0 M > D = 3 times a number of factors, each of which must be π N M W W 3 aσ K is given by 12 K > g 3 can be subject to solving this equation. The smallest = 3  S σ 3 σ D , or perhaps 10 1. N 3 vac π ≫ N . 3 M √ / s = 36 / 2 w g V σ 1 = 0 for the supersymmetric vacuum we get − s g W , so the requirement is actually = σ 0 σ D is about 1 is the volume of the compact manifold and W 2 6 / 3 V /σ To make use of these inequalities, we rewrite the formula for More generally, we need some room for other cycles wrapped wi Such a large volume may be difficult to obtain in the KKLT constr The compact volume has to be large enough so that the throat fit The warped solution requires | 0 W and the Kahler potential is be reliable we need so solving can tune The volume modulus is roughly 10 where the where stable against brane/flux annihilation, we need [27] larger than one by theof arguments these above. factors One large would in have order been to te obtain control. nonperturbative effects must be important. More quantitati | imaginary part of the universal Kahler modulus, equation (4 We want to know how large region. Warping is not significantunwarped in volume this of formula the because throat bo factor are approaches dominated one. by the region nea Also, the radius of the minimal JHEP12(2008)096 . 5 = a (6.9) (6.10) (6.11) 5 10 7 branes, then D 7s are wrapped. Also, be of great interest to ght it would be easy to l cosmological constant pe of bound may be an r the bound in a sector r than the Hubble time D cua should decay before s only the combination of arger volumes, but in this <σ< bilized, so we would have s has narrowly focused on are not deep in the regime mes by breaking supersym- ion can work, 7 branes is not possible, so f the analysis in this paper t that minor modifications 3 / on on 2 D n must be larger than one. In  nted out that the classic KKLT ymmetry breaking by antibranes 6 V throat V may be quite large, on which the is zero at tree level [38, 39]; this would A ) does not take an extreme value, we have 0 .  s D D ( . This latter possibility has recently been g W A 5 0 24 πχ M 2 K s 10 W e g – 24 –  ∼ to take a larger volume than our estimate of 10 ) σ < A σ M s g (  3 D N M -symmetry so that R 12  3 1. Then, if the prefactor . D 0 N 3 a> 10 ) is the Euler number of the divisor D ( . As far as we know, an extremely large number of 7 χ D Finally, one could perhaps avoid the need for such large volu Of course, the large volume scenario of [41] allows for much l There may be ways to arrange for π/N allow for a much smaller minimum value of case supersymmetry is alreadyto broken do when an the entirely moduli different estimate are of sta the decaymetry rates. in a milder wayvolume stabilization than by by nonperturbative adding effects antibranes. andwhich supers Note squeezes that us it into i the narrow window (6.10). 7. Conclusions and future directions We have investigated athey new produce bound Boltzmann stating Brains.but that much all This shorter de time thansuch Sitter scale as the va is our recurrence much own. timeof longe the for We landscape, vacua have the with foundconstruct KKLT very smal surprisingly vacua, long-lived in strong vacua. which supportconstruction one Incidentally, fo we might is have have quite poi thou difficultcan lead to to control. muchthe more However, specific controlled we de example expec Sitter ofexample vacua. KKLT of vacua, Our but a analysi we phenomenon suspect generic that to this stringy ty dS vacua. It would explored in more detailremains [40], valid and in has computing the the decay advantage rates. that all o As pointed out by Denef et al. [37], the prefactor Recall that each of thethe factors words on of the S. leftof Kachru, side calculability.” constructions of [36] in the this equatio narrow window “ one can impose a discrete If the nonperturbative effects come from gaugino condensati where 2 which leaves an extremely narrow window where the construct we assume that JHEP12(2008)096 ]. ]. , 06 , Phys. JHEP , , hep-th/0605266 arXiv:0705.1557 hey make Boltzmann Domain walls, near-BPS the decay rate is actually lightly different construc- gy to be small requires a bounds the lifetimes of de 5, the Berkeley Center for c vacua are not necessarily euzer, Andrei Linde, Liam ve valuable new information nrelated to the scale of the flation. Finally, it would be ame bound, since our results ally small, the ingredients in he early stages of this project , and Leonard Susskind. This -lived vacua. In this case, the ]. . We have also enjoyed helpful (2006) 086010 [ (2008) 012 [ Metastable domains of the A. Linde, 01 D 74 ]. De Sitter vacua in string theory JHEP , ]. hep-th/0603105 Phys. Rev. – 25 – Probabilities in the landscape: the decay of nearly , Regulating eternal inflation. II: the great divide Disturbing implications of a cosmological constant ]. ]. arXiv:0712.1397 ]. A paradox in the global description of the multiverse (2006) 046008 [ hep-th/0301240 hep-th/0208013 hep-th/0603107 D 74 (2008) 014 [ 06 Lifetime of stringy de Sitter vacua hep-th/0610132 (2003) 046005 [ JHEP Phys. Rev. (2002) 011 [ (2006) 065 [ , , 10 08 D 68 landscape JHEP Rev. (2007) 018 [ JHEP bubbles and probabilities in the landscape flat space On the other hand, it is quite possible that by considering a s The basic reason that all de Sitter vacua might decay before t [2] L. Dyson, M. Kleban and L. Susskind, [8] M. Dine, G. Festuccia, A. Morisse and K. van den Broek, [3] R. Bousso and B. Freivogel, [7] A. Westphal, [4] A. Aguirre, T. Banks and M. Johnson, [1] S. Kachru, R. Kallosh, A. Linde and S.P. Trivedi, [6] A. Ceresole, G. Dall’Agata, A. Giryavets, R. Kallosh and [5] R. Bousso, B. Freivogel and M. Lippert, tion, other authors willcurrently be viable able measures would to be construct ruledabout extremely out, the and long we correct would way ha very to interesting regulate to the findSitter infinities a vacua of model-independent without eternal invoking argument Boltzmann in which Brains. Acknowledgments We particularly thank Raphael Boussoand for Shamit collaboration Kachru in for t discussionsdiscussions and with technical Chris assistance Beem,McAllister, Yu Steve Nakayama, Giddings, Stephen Shenker, Maximilianwork Eva Kr was Silverstein supported byTheoretical Israel Physics, Science and by Foundation DOE grant grant 568/0 DE-AC0376SF00098. 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