Is There Eternal Inflation in the Cosmic Landscape ?

Is there eternal inﬂation in the cosmic landscape ?

hep-th/0611148 ArXiv:0708.4374 [hep-th] with Qing-Guo Huang, ArXiv:0803.0663 [hep-th] with Dan Wohns and Yang Zhang, ArXiv:0811.3753 [hep-th]

IPMU, 04/01/09 Expansion of the universe k ρm ρr H2 =Λ+ + + a2 a3 a4 • a(t) = the cosmic scale factor ~ size of universe • H = Hubble constant =(da/dt)/a • Λ = dark energy ~ cosmological constant ~ effective potential • k = curvature • ρm = matter density at initial time • ρr = radiation density at initial time • H = constant ⇒ a = eHt Inﬂation BRIEF ARTICLE

THE AUTHOR

(1) d = s(Λc)/ξ +1

∼ ∼ 1 ∼ −1/4 (2) ξ s(Λs) Λs ms (3) d>60 BRIEF ARTICLE

(4) T (n) " T0/n THE AUTHOR 4 (5) Γ = msT

(6) n ∼ 1/Hs 1 Eternal Inﬂation Ht 3 3Ht (7) (1) a(t) " e → V = a(t) " e τ > 1/H Suppose the universe is sitting at a local minimum, with a ln L (2)lifetime longer than the Hubble lntime:g ∼− L = −e Then the number of Hubble patches will increase −Si (3) exponentially. Even after some Hubble patchesαi have∼ e decayed, there would be many remaining Hubble patches (4) that continue to inflate. |βij| < αi

(5) Eternal inﬂation impliesφi = thatϕi/f i somewhere in the universe (outside our horizon),d −1 (6) Γt ∼ n Γ0 inﬂation is still happening today. T (A → B)T (B → C) (7) T (A → C)= ∼ T0/2 T (A → B)+T (B → C)

−S (8) ΓA→B = ΓB→C = Γ0 ∼ e 2 (9) ΓA→C ∼ ΓA→BΓB→C = Γ0 1 1 2 (10) tA→C = tA→B + tB→C = + ∼ → → 1 ΓA B ΓB C Γ0

Γ0 (11) Γ → = A C 2 −S (12) Γ(1) = Γ0 ∼ e − (13) Γ(2) ∼ e 2S − (14) Γ(2) ∼ e S − (15) Γ(n) ∼ e S

(16) +H2φ2

2 2 (17) H = U/3MP 1 Flux compactiﬁcation in Type II string theory where all moduli of the 6-dim. “Calabi-Yau” manifold are stabilized • There are many meta-stable manifolds/vacua, 10500 or more, probably inﬁnite, with positive, zero, as well as negative cosmological constants. • There are many (tens to hundreds or more) moduli (scalar modes) that are dynamically stabilized. Giddings, Kachru, Polchinski, KKLT vacua Kachru, Kallosh, Linde, Trivedi and many others, 2001.... A cartoon of the stringy landscape

many local minima Pros : It seems that eternal inflation can be quite generic (unavoidable) in the cosmic landscape. Its presence leads to a non-zero probability for every single site in the landscape, including our very own universe.

Cons : Since there are many (10500 or more) vacua, and some parts of the universe are still inflating, it is difficult to see why we end up where we are without invoking some strong version of the anthropic principle.

It is very difficult to make any testable prediction.

There is also a measure problem. Scenarios with or without eternal inflation are very different. BRIEF ARTICLE BRIEF ARTICLE

THE AUTHOR THE AUTHOR

A simple question11 : Consider tunneling from A to C : −−S (1) assume for simplicity : Γ → = Γ → = Γ0 ∼ e S (1) ΓAA→BB= ΓBB→CC = Γ0 ∼ e 2 (2) ΓA→C ∼ ΓA→BΓB→C = Γ02 (2) ΓA→C ∼ ΓA→BΓB→C = Γ0 1 1 2 (3) tA→C = tA→B + tB→C = 1 + 1 ∼ 2 (3) tA→C = tA→B + tB→C = ΓA→B +ΓB→C ∼Γ0 ΓA→B ΓB→C Γ0 Γ (4) Γ → = A C 2Γ ? (4) Γ → = A C 2 (5) +H2φ2 2 2 (5) 2+H φ 2 (6) H = U/3MP 2 2 (6) H = U/− 3MH (7) Γ ∼ e E/T P ∼ −E/TH (7) (8) ΓTH =e H/2π (8) T = H/82ππ2τ (9) E/TH → H H3 8π2τ (9) E/T →∼ −B (10) Γ(CDLH ) He 3 − − (10)(11) $Γ=(CDLU(φ+) ∼Ue(φB−) − − (12) τ = 10 12, $ = 10 22 (11) $ = U(φ+) − U(φ−)

(13) d−= s(Λc)/ξ +1 − (12) τ = 10 12, $ = 10 22 ∼ ∼ 1 ∼ −1/4 (14) ξ s(Λs) Λs (13) d = s(Λcm)/sξ +1 (15) d>60 ∼ ∼ 1 ∼ −1/4 (14) ξ s(Λs) Λs (16) T (n) $mTs0/n 1 (15) d>60

(16) T (n) $ T0/n 1 BRIEF ARTICLE BRIEF ARTICLE

THE AUTHOR BRIEF ARTICLETHE AUTHOR BRIEF ARTICLE

THE AUTHOR THE AUTHOR A simple question11 : Consider tunneling from A to C : 1 ∼∼−S−S (1)(1) ΓΓAA→→1B == ΓΓBB→→CC==ΓΓ0 0 e e ∼ 2 2 (2)(2) ΓΓA→→C ∼ ΓΓA→→BΓΓB→→C ==Γ0Γ A C ∼ −AS B B C 0 (1) ΓA→B = ΓB→C = Γ0 e∼ −S (1) ΓA→B = ΓB→C = Γ0 e 1 1 2 (3) tA→C = tA→B + tB→C = 1 + ? 1 ∼ 2 (3) t → = t → + t → 2 = → + → ∼ (2) ΓA→AC ∼C ΓA∼→ABΓBB→C B= ΓC0 Γ2A B ΓB C Γ0 (2) ΓA→C ΓA→BΓB→C = Γ0ΓA→B ΓB→C Γ0 Time : 1 1 Γ 2 (4) 1ΓA→C = 1∼Γ 2 (3) (3) tA→Ct =→tA=→Bt +→tB+→Ct =→ = + + ∼ (4) A C A B B CΓ → ΓAΓ→C→=2 Γ0 A ΓBA→B B ΓCB→2C Γ0 (5) Γ0 Γ +H2φ2 (4) (4) ΓA→CΓ =→ = 2 2 (5) A C2 +H φ ? 22 2 (6) H = U/3MP 2 2 (6) 2 2 2 2H = U/3M (5) (5) +H +φH φ − H P (7) Γ ∼ e E/T 2 2 2 2∼ −E/TH (6) (6) (7) H =HU/=3MU/P3MΓP e (8) TH = H/2π −E/T−HE/TH (7) (7) (8) Γ ∼ Γe ∼ e TH = H/22π → 8π τ (9) E/TH 3 (8) T = H/2π H 2 (8) TH =HH/2π → 8π τ (9) E/TH −B3 (10) Γ(CDL2 ) ∼ eH 8π2τ8π τ (9) E/T→H → (9) E/TH 3 3 −B (10)(11) H$ =ΓH(UCDL(φ+))−∼Ue(φ−) − (10) Γ(CDL∼ )−∼Be−12B −22 (10) (11)(12) Γ(CDL) τ =e$ = 10 U(φ, +) −$ U=( 10φ−) − (11) $ = U(φ−+) U−(φ−) − (11) (12)(13) $ = U(φ+) τ =U( 10dφ−=)12s(,Λc)/ξ$+1= 10 22 −12 −22 (12) τ =−12 10 , $ =−22 10 (12) τ = 10 , $∼= 10 ∼ 1 ∼ −1/4 (13)(14) ξ sd(Λ=s)s(Λc)/ξ +1Λs ms (13) d = s(Λc)/ξ +1 (13) d = s(Λc)/ξ +1 1 − (14)(15) ξ ∼ s(Λ )d>∼ 60 ∼ Λ 1/4 ∼ ∼ 1 ∼s −1/4 s (14) ∼ ξ s(∼Λs)1 ∼ −1Λ/4s ms (14) (16) ξ s(Λs) msΛTs(n) $ T0/n ms (15) d>1 60 (15) d>60 (15) d>60 (16) $ T (n) $ T0/n (16) T$(n) T0/n (16) T (n) T01/n 1 1 BRIEF ARTICLE

THE AUTHOR

(1) v =< σ˙ >

(2) τ > 1/H

(3)BRIEF ARTICLE ln g ∼−L = −eln L to the approximate geometry for a warped− throat, or as a variation of the simplest warped (4) α ∼ e Si deformed throat in the KKLMMTTHE AUTHOR scenario.i

(5)Sharp features and/or non-Gaussianity|βij| < α ini the cosmic microwave background radiation due to steps in the inflaton potential have been studied [7, 8]. The duality cascade feature (6) φ = ϕ /f shows up in theBRIEF warped ARTICLE geometry1 BRIEF (andi in ARTICLE thei i running of the coupling (the dilaton value)) d−1 in the(7) D3-brane potential as steps. SuchΓt ∼ n stepsΓ0 in the inflaton potential can introduce sharp THE AUTHOR features in the cosmicTHE AUTHOR microwave background radiation, which may have been observed al- (1) τ > 1/HT (A → B)T (B → C) ready(8) [9, 10]. So the stepT (A → functionC)= behavior of the throat,∼ T0/ th2ough small, can have distinct T (Aln→L B)+T (B → C) (2)observable stringy signatures ln g ∼− inL the= −e cosmic microwave background radiation. The possibil- 1 −S (9) 1 ΓA→−BSi= ΓB→C = Γ0 ∼ e (3) ity of detecting and measuringαi ∼ thee duality cascade is a strong enough motivation to study the throat more carefully. Although both Seiberg2 duality and gauge/gravity duality are (10) ΓA→C ∼ ΓA→BΓB→C = −ΓS0 (4) (1) |βijΓ|A<→αBi= ΓB→C = Γ0 ∼ e strongly believed to be true, neither−S has been mathematically proven; so a cosmological (1) Γ → = Γ → = Γ0 ∼ e A B B C 1 2 1 2 (5) test(11)(2) is highly desirable.tA→ ThisC =φitA= willΓ→AϕB→i/f+ alsoC it∼B→Γ provideAC→=BΓB→C strong=+Γ0 evid∼ence for string theory. 2 Γ → Γ → Γ0 → ∼ → → A B B C (2) The analysisΓA C ofΓ theA B stepΓB featureC =d−Γ10 in inflation in comparison with WMAP data has been (6) Γt ∼ n Γ0 1 1 ∼ 2 carried(3) out for larget fieldA→C = models,tA→B + tB specifically→C = Γ0 + the quadratic chaotic inflationary model (12) 1 1ΓA→∼C =2ΓA→B ΓB→C Γ0 (3) wheretA→C the= tA inflaton→B + tB→ fieldC = startsT (A →+ atB)T values(B → C much) 2 bigger than the Planck mass (φ > 15M ) (7) T (A → C)= ΓA→B ΓB→C Γ0Γ∼0 T0/2 Pl (4) T (A → B)+TΓ(B→→ =C)−S and(13) then decreases to zero. In braneΓ(1) = inflation,AΓ0C∼ e 2 ∼ theT0 inflaton being the position of the brane, Γ0 − → S (4) (8) is bounded by theΓ sizeAΓAC→ of=B the= ΓB flux→C = compactificationΓ0 ∼ e −2S−S volume, and so typically φ MPl. This (14)(5) 2 Γ(1)Γ(2) = Γ∼0 e∼ e ! requires a new analysis, which− is carried out2 here. (9) ΓA→C∼∼ ΓAS→BΓB→C = Γ0 −−2SS (5) (15)This(6) paper isΓ(1) organized =WhichΓ0 e as follows.is correctΓΓ(2)(2) Sec∼∼ee 2 reviews? the setup of the geometry and SEc. 3 1 1 2 −2S −−S negligible (6) (10) reviews(16)(7) D3-branetA→C =Γ potentialt(2)A→B∼+e tB→ whereC = theΓΓ((2)n step+) ∼∼ ee appearsnS ∼ as a small correction. Sec. 4 compares ΓA→B ΓB→C Γ0 the model to the WMAP data and gives1 the predictions. Sec. 5 starts with a qualitative ? ∼ −S ∼ −S (7) analysis(8) of the non-GaussianityΓ(2) e dueΓ0 Γ( ton) thee steps and then presents a more quantitative (11) ΓA→C = − 2 2 2 (8) analysis.(9) Sec. 6 containsΓ(n) ∼ e aS summary. +H φ −S Transmission (12) Γ(1) = Γ0 ∼ e ∼ T0 (10) H2 = U/3coefficientM 2 (9) +H2φ2 P −2S (13) 3 SetupConsider of the the geometryΓ(2) ∼ e −E/TH (11) 2 2 Γ ∼ e (10) secondH case= : U/3M − (14) ΓP(2) ∼ e S (12) A→C = 0 2 TH = H/2π T −E/TH T / T (n) T0/n (1) (11) Γ ∼ e − " (15) Γ(n) ∼ e S 2 First it is useful to define a basis of coordinates→ 8π τ and a metric. The ten dimensional metric (13) E/TH 3 (12) TH = H/2π 2 2 H 1,1 (16) which describes the throat is+H thatφ of AdS5 X5, where X5 has the T geometry in the − (14) 2 Γ(CDL) ∼×e B (17) UV region. Including the8H expansionπ2 τ= U/3M 2 of the universe, the metric has the form (13) E/TH → P H3 1 − (15) 2 2 $ 2= U(φ+) 2 U(2φ−) −2 2 2 2 ds = h (−r)( dt + a(t) dx )+h (r)(dr + r ds ), (2) ∼ B − − X5 (14) (16) Γ(CDL) e −τ = 10 12, $ = 10 22 2 (15) Far away(17) from$ the= U bottom(φ+) − U of(φ− the) throat,d = s(Λ ds)/ξX+15 = dsT 1,1 is the metric that describes a base 1,1 3 c 2 of the conifold T which is an S fibered1 over S . Here h(r) is the warp factor, that is, a −12 −22 (16) generic massτ m= 10 mh, (r)$ in= the 10 presence of h(r). → (17) Let Φ be thed dilaton= s(Λc)/ andξ +1gs be the coupling, 1 d −Φ e = Sl (3) d log (r/r0) −

2 to the approximate geometry for a warped throat, or as a variation of the simplest warped BRIEF ARTICLE deformed throat in the KKLMMT scenario. Sharp featuresBRIEF and/or ARTICLE non-GaussianityBRIEF in ARTICLE the cosmic microwave background radiation due to steps in the inflatonBRIEF potential ARTICLE haveTHE been AUTHOR studied [7, 8]. The duality cascade feature shows up in the warped geometry (and in the running of the coupling (the dilaton value)) THE AUTHOR THE AUTHOR in the D3-brane potential asTHE steps. AUTHOR Such steps in the inflaton potential can introduce sharp features in the cosmic microwave background1 radiation, which may have been observed already [9, 10]. So the step function behavior of the throat, though small, can have distinct 1 observable stringy signatures1 in the1 cosmic microwave back−Sground radiation. The possibil- (1) Γ → = Γ → = Γ0 ∼ e ity of detecting and measuring theA dualityB cascadeB C is a strong enough motivation to study ∼ −S2 the(1)(2) throat more carefully. AlthoughΓΓ→A→C both= ΓΓ→A Seiberg→B=ΓBΓ→0 duality∼C =e Γ0 and gauge/gravity duality are A −BS ∼B C−S (1) (1) strongly believedΓA→B = toΓB beΓ→AC→ true,B==Γ neither0ΓB∼→eC = hasΓ0 beene mathematically proven; so a cosmological 1 2 1 2 (2) Γ → ∼ Γ → Γ → = Γ ∼ test(3) is highly desirable.t ThisA→C∼ will= tAA also→BC2+ providetB→AC B=2 strongB C evid+ 0ence for string theory. (2) ∼ ΓA→C ΓA→BΓB→C = Γ0 Γ → Γ → Γ0 (2) The analysisΓA→C of theΓA→ stepBΓB feature→C = Γ in0 inflation inA comparisonB B C with WMAP data has been 1 1 ∼ 2 carried(3) out for large fieldtA→C models,=1 tA→B + specifically1t1B→C = 21 theΓ0 ∼ quadrati+2 c chaotic inflationary model (3) (4) tA→C = tA→B + tB→C = ΓA∼+→C Γ=A→B ΓB→C Γ0 (3) twhereA→C = thetA→ inflatonB + tB→ fieldC = starts at+ valuesΓA→B muchΓB bigger→C2 thanΓ0 the Planck mass (φ > 15M ) Γ → Γ → Γ0 Pl A B B C Γ0 and then decreases to zero. In brane inflation, the inflaton−S being the position of the brane, (4)(5) ΓΓ0Γ(1)A→ =C Γ=0 ∼ e (4) Γ0ΓA→C = 2 is bounded by the size of the flux compactification2 volume, and so typically φ MPl. This (4) ΓA→C = −2 ! requires(6) a new analysis, which2 is carriedΓ out(2) here.∼∼e −SS (5) Γ(1)−S = Γ0 e (5) Γ(1) = Γ0 ∼ e This paper is organized as− follows. Sec 2 reviews− the setup of the geometry and SEc. 3 ∼ S ∼−2SS (5) (6)(7) Γ(1) = Γ0 e − Γ(2)Γ(2)∼ e e (6) reviews D3-brane potential whereΓ(2) ∼ thee 2 stepS appears as a small correction. Sec. 4 compares − the model to the WMAP∼ data−2S and gives the∼ predictions.−S S Sec. 5 starts with a qualitative (6) (7)(8) Γ(2) eX − ΓΓ(2)(n)∼ enegligiblee (7) analysis of the non-GaussianityΓ(2) due∼ e toS the steps and then presents a more quantitative −S 2 −2S (7) analysis.(8)(9) Sec. 6 containsΓ(2) ∼ ae summary. − Γ(n+) H∼ eφ (8) Γ(n) ∼ e S − 2 2 2 2 (9)(10) ∼ S H+=H U/φ 3MP (8) (9) Γcorrect(n) e +H2φ2 3 Setup of the geometry − (11) 2Γ ∼ e E/T2H (9) (10) +H2φ2 2 H2 = U/3MP (10) H = U/3MP TA→C = T0/2 T (n) − T0/n (1) (12) T∼H =E/TH/H2π (11) 2 2 −E/THΓ e" (10) (11) H = U/3MΓP ∼ e First it is useful to define a basis of coordinates and8π2τ a metric. The ten dimensional metric (12) − TH = H/→ 2π 1,1 (12) which(13) describes the∼ throatE/T isHT that= H/ of AdS2πE/T5 H X5, where3 X5 has the T geometry in the (11) Γ e H × H UV region. Including the expansion of the universe,2 the metric has the form 2 8π τ− (13) 8E/Tπ τ H → ∼ B (12) (14) TH = H/2π → Γ(CDL) e3 (13) 2 2 E/TH 2 3 2 2 H −2 2 2 2 ds = h (r)( dt +Ha(t) dx )+h (r)(dr + r dsX5 ), (2) 2 − −B − (14)(15) 8π τ− $Γ=(−CDLU(φ+))∼ eU(φ ) (14) → Γ(CDL) ∼ e B (13) E/TH 3 − 2 − Far away from the bottom ofH the throat, ds12X5−= dsT 1,1 is22 the metric that describes a base (15)(16) 1,1 3τ $== 10U(φ+, ) U$2(=φ− 10) (15) of the conifold T which$ is= anU(φS+)fibered− U(φ− over) S . Here h(r) is the warp factor, that is, a ∼ −B (14) (17) Γ(CDL) e d−=12 s(Λ )/ξ +1−22 generic(16) mass m mh(r) in the−12 presenceτ = 10 of, −h22(cr).$ = 10 (16) → τ = 10 , $ = 10 1 Let Φ be the dilaton and gs be the coupling, (15) $ = U(φ+) − U(φ−) (17) d = s(Λc)/ξ +1 (17) d = s(Λc)/ξ +1 1 −12 −22 d −Φ (16) τ = 10 , $ = 10 1 e = Sl (3) d log (r/r0) −

(17) d = s(Λc)/ξ +1 1 2 To understand why : resonance tunneling

Y. Zhota, PRB 41 (1990) 7879. Resonance tunneling in semi-conductors Wikipedia A resonant tunnel diode (RTD) is a device which uses quantum effects to produce negative differential resistance (NDR). As an RTD is capable of generating a terahertz wave at room temperature, it can be used in ultra high-speed circuitry. Therefore the RTD is extensively studied. Chang, Esaki, Tsu, 1974 to the approximate geometry for a warped throat, or as a variation of the simplest warped deformed throat in the KKLMMT scenario. Sharp features and/or non-Gaussianity in the cosmic microwave background radiation due to steps in the inflaton potential have been studied [7, 8]. The duality cascade feature shows up in the warped geometry (and in the running of the coupling (the dilaton value)) in the D3-brane potential as steps. Such steps in the inflaton potential can introduce sharp features in the cosmic microwave background radiation, which may have been observed already [9, 10]. So the step function behavior of the throat, though small, can have distinct observable stringy signatures in the cosmic microwave background radiation. The possibil- ity of detecting and measuring the duality cascade is a strong enough motivation to study the throat more carefully. Although both Seiberg duality and gauge/gravity duality are strongly believed to be true, neither has been mathematically proven; so a cosmological test is highly desirable. This will also provide strong evidence for string theory. The analysis of the step feature in inflation in comparison with WMAP data has been carried out for large field models, specifically the quadratic chaotic inflationary model where the inflaton field starts at values much bigger than the Planck mass (φ > 15MPl) and then decreases to zero. In brane inflation, the inflaton being the position of the brane, is bounded by the size of the flux compactification volume, and so typically φ M . This ! Pl requires a new analysis, which is carried out here. BRIEF ARTICLE This paper is organized as follows. Sec 2 reviews the setup of the geometry and SEc. 3 reviews D3-brane potentialTHE AUTHOR where the step appears as a small correction. Sec. 4 compares the model to the WMAP data and gives the predictions. Sec. 5 starts with a qualitative analysis ofGeneric the non-Gaussianity wavefunction due to the steps and then presents a more quantitative 1 analysis. Sec.spread 6 contains in energy a summary. : T (A → B)T (B → C) (1) T (A → C)= ∼ T0/2 3 Setup of theT (A geometry→ B)+T (B → C) naive −S (2) ΓA→B = ΓB→C = Γ0 ∼ e

2 T (n) T0/n (1) (3) ΓA→C ∼ ΓA→BΓB→C = Γ0 " First it is useful to define a basisIn of any coordinates meta-stable and site a in metric . The ten dimensional metric 1 1 ∼ 2 (4) tA→C = tA→B + tB→C = the+ cosmic landscape, we 1,1 which describes the throat isΓ thatA→B ofΓAdSB→C 5 Γ0 X5, where X5 has the T geometry in the cannot focus× only at its UV region. Including the expansionΓ0 of the universe, the metric has the form (5) Γ → = nearest neighbors. A C 2 2 2 2 2 2 −2 2 2 2 ds = h (r)( dt−S + a(t) dx )+h (r)(dr + r ds ), (2) (6) ResonanceΓ(1) = Γ 0tunneling∼ e in QFT : X5 Saffin, Padilla and Copeland,− arXiv:0804.3801 [hep-th] − (7) Sarangi, Shiu,Γ Shlaer,(2) ∼ earXiv:0708.43752S [hep-th] 2 Far away from the bottom of the throat, dsX5 = dsT 1,1 is the metric that describes a base 1,1 − 3 2 (8) of the conifold T whichΓ(2) is∼ ane SS fibered over S . Here h(r) is the warp factor, that is, a generic mass m mh(r) in the−S presence of h(r). (9) → Γ(n) ∼ e Let Φ be the dilaton and gs be the coupling, (10) +H2φ2

(11) H2 = U/3M 2 d −Φ P e = Sl (3) − d log (r/r0) − (12) Γ ∼ e E/TH

(13) T = H/2π H 2 8π2τ (14) E/T → H H3 − (15) Γ(CDL) ∼ e B

(16) $ = U(φ+) − U(φ−) 1 landscape is described by a set of scalar modes, namely, light closed string modes known as moduli and positions of branes and fluxes. (We shall refer to them collectively as moduli.) Depending on the location of any specific site, the number d of moduli in the stringy landscape describing its neighborhood may vary from dozens to hundreds. So a large d parameterizes the vastness of the landscape. In Ref.[13], it is argued that resonance tunneling effect can enable a fast decay of a site with a string scale vacuum energy Λ in the stringy cosmic landscape. The key point may be illustrated with the following simplistic example. Let the tunneling rate from one site to any of its neighboring site be Γ0, which is dictated by the minimum (not typical) barrier tunneling path. Tunneling from a positive Λ site to a larger Λ site (with smaller entropy) takes a very long time and is ignored here. Note that tunneling from a positive Λ site to a negative Λ site is also ignored for at least one of the following reasons : (1) it takes an exceedingly long time, (2) it leads to a big crunch [14], (3) it pops right back uBRIEFp [15, 16]. ARTICLEIn this sense, Λ = 0 is special. Consider tunneling in a d-dimensional hyper-cubic lattice. Ignoring resonance tunneling for the momeBRIEFnt, then ARTICLEthe total tunneling rate is given by In a d-dimensionalTHE AUTHORpotential : THE AUTHORnaive : Γnr 2d Γ (1.1) t ∼ 0 The time for 1 e-fold of inflation is Hubble since there are 2d nearest neighbors. Here, Γnr grows linearly with d. Since Γ is expo- time 1/H, so the lifetime oft a typical site 0 nr 1 1 nentially small, Γt is alsseemso exp toon been muchtially longersmal thanl. No thew Hubblelet us include the resonance/efficient tunneling effect (see the apscale,pend andice eternals for ainflationbrief review is unavoidable.and further discussions). For a generic (1) T (n) ! T0/n wavefunction at a site, its tunneling rate to its next-to-d−1 neighboring sites will have compa- (1) Very4 roughly : Γt ∼ n Γ0 (2) 2 Γ = msT rable Γ0, not Γ0, as naively expected. Since the number of paths for resonance tunneling increase like the volume, it is arguedFor largethatT enough(Ath→e e dffB (andec)Tti v(maybeBe total→ nC) tun) neling rate goes like (2) T∼(A → C)= ∼ T0/2 (3) n 1/Hs theT tunneling(A → B can)+ beT fast.(B → C) Γ nd Γ (1.2) t ∼ 0 −S → → ∼ (3)where n signifies the effective nΓumA beBr =ofΓsBtepsC w=heΓre0 reseonance/efficient tunneling acts with tunneling rate not much suppressed with respect to Γ0.2 That is, as a function of d, (4) ΓA→C ∼ ΓA→BΓB→C = Γ0 Γt actually grows exponentially, not linearly. Now it becomes possible that Γt can be of order unity for large enough d and n. 1 1 2 (5) tA→C = tA→B + tB→C = + ∼ The scenario then goes as follows. Tunneling willΓA→bBe fastΓwhenB→C theΓsite0 has a relatively large Λ, since it has many paths and nearby sites to tunnel to. This fast tunneling process Γ0 (6)can happen repeatedly, until it reaches aΓsit→e with= Λ smaller than some critical value Λ . A C 2 c Most if not all nearby sites of this low Λ site have larger Λs, so it will not tunnel to. That is, −S (7)the effective n 1 in most directions sΓo(1)its total = Γ0dec∼aey width becomes exponentially small. → This low Λ site may describe today’s universe. For this scenario to work, such a shut off of ∼ −2S (8)fast tunneling must be sharp, so that the unΓ(2)iverseenever enters eternal inflation. Otherwise, the universe would arrive at a meta-stable site with−some intermediate Λ, and have enough (9) Γ(2) ∼ e S time to expand away the radiation/matter present and enter into eternal inflation 1. If this − (10)happens, some (most) parts of our universΓe(wnou) ∼ldestillS be in an eternally inflationary phase

1 In brane inflation, eternal inflation of the random w2alk2 type generically does not take place [17]. So (11)here we are concerned only with eternal inflation of+theHtunnelφ ing type in the landscape.

1 2 2 (12) H = U/3MP − (13) Γ ∼ e E/TH – 2 – (14) TH = H/2π

8π2τ (15) E/T → H H3 − (16) Γ(CDL) ∼ e B 1 Outline • Tunneling in the cosmic landscape is quite different from tunneling between 2 sites. • It is harder to trap a wavefunction in higher dimensions. Also, the barriers in some directions in the landscape are known to be exponentially low. • The vacuum energy in the landscape plays the role of a ﬁnite (Gibbons-Hawking) temperature effect, enabling faster tunneling. • Treating the landscape as a random potential, we can borrow the knowledge developed in condensed matter physics to estimate the mobility of the wavefunction in the landscape. The QM wavefunction is harder to be trapped in higher dimensions:

Repulsive

A 1-dim. attractive delta-function potential always has a bound state but not a 3-dim one. Harder to trap in higher dimensions

-V 0 R Spherical square well :

V (r) = −V0 r < R = 0 r > R k0R

k02 = 2mV0

ψ(r) ∼ e−ar a2 = 2m|E| BRIEF ARTICLE BRIEF ARTICLE THE AUTHOR THE AUTHOR Axionic potentials 1 1

−Si (1) (1) φi = ϕi/fi αi ∼ e d−1 (2) (2) Γt ∼ n Γ0 |βij| < αi

(3) T (ForA →lightB axions,)T (B the→ potentialC) φ heightsi = ϕi /farei (3) T (A → C)= very low. In some directions, ∼theT barriers0/2 → → d−1 (4) T (A canB )+easilyT be(B exponentiallyC)Γt ∼ low.n Γ0

Svrcek and Witten, hep-th/0605206−S → → ∼ T (A → B)T (B → C) (4) (5) ΓA B = ΓB McAllisterTC(A=→Γ and0C )=Easther,e etc ∼ T0/2 T (A → B)+T (B → C) ∼ 2 (5) ΓA→C ΓA→BΓB→C = Γ0 −S (6) ΓA→B = ΓB→C = Γ0 ∼ e 1 1 2 2 (6) tA→(7)C = tA→B + tB→C = Γ+A→C ∼ ΓA∼→BΓB→C = Γ0 ΓA→B ΓB→C Γ0 1 1 2 (8) tA→C = tA→B + tB→C = + ∼ Γ0 Γ → Γ → Γ0 (7) Γ → = A B B C A C 2 Γ0 (9) Γ → = −S A C (8) Γ(1) = Γ0 ∼ e 2 −S (10) − Γ(1) = Γ0 ∼ e (9) Γ(2) ∼ e 2S − (11) Γ(2) ∼ e 2S ∼ −S (10) Γ(2) e − (12) Γ(2) ∼ e S − ∼ S − (11) (13) Γ(n) e Γ(n) ∼ e S 2 2 (12) (14) +H φ +H2φ2

(15) 2 2 H2 = U/3M 2 (13) H = U/3MP P − − E/TH (14) (16) Γ ∼ e E/TH Γ ∼ e

(17) TH = H/2π (15) TH = H/2π 1

8π2τ (16) E/T → H H3 1 Outline • Tunneling in the cosmic landscape is quite different from tunneling between 2 sites. • It is harder to trap a wavefunction in higher dimensions. Also, the barriers in some directions in the landscape are known to be exponentially low. • The vacuum energy in the landscape plays the role of a finite (Gibbons-Hawking) temperature effect, enabling faster tunneling. • Treating the landscape as a random potential, we can borrow the knowledge developed in condensed matter physics to estimate the mobility of the wavefunction in the landscape. so 2 2 2 2 1 1 (H+ + H−) κ τ 2 = 2 + + , (2.18) R R0 2 16 2 where R0 =3τ/#. Note that the last term is proportional to κ and so is very small most so BRIEF ARTICLE so 2 2 of2 the2 time. According to Eq.(2.16), we can easily check that the bubble radius at the 1 1 (H+ + H−) κ τ 1 1 (H2 + H2 ) κ2τ 2 2 = 2 + + , (2.18)+ − R R0 2 moment16 of materialization2 = 2 + is not+ larger, than the event horiz(2.18)on R+ of the de Sitter space THE AUTHOR R R0 2 16 so 2 where R0 =3τ/#. Note that the last term is proportionalin false to κ and vacuum so is very . small This most is reasonable; otherwise the bubble cannot be generated causally. 1 1 (H2 + H2 ) κ2τ 2 2 where R0 =3+ τ/−#. Note that the last term is proportional to κ and so is very small most of the time. According to Eq.(2.16),2 we= can2 + easily check+ that the, bubble radius at the(2.18) R R0 2 A16 comment on the thin wall approximation : to be concrete, consider the potential moment of materialization is not largerof thanthe time. the event According horizon R+ toof Eq.(2.16), the de Sitter we space can easily check that the bubble radius at the 2 where R0 =3τ/#. Note that the last term is proportional to κ and so is very small most in false vacuum . This is reasonable; otherwisemoment the of materialization bubble cannot be is generated not larger causally. than the event horizon R+ of the de Sitter space of the time. According to Eq.(2.16), we can easily check that the bubble radius at the A comment on the thin wall approximation1 in false vacuum : to be concrete, . This is con reasonable;sider the potential otherwise the bubbleU(φ)= cannotU0( beφ)+ generatedU+ + f causally.(φ) (2.19) moment of materialization is not larger than the event horizon R+ of the de Sitter space Coleman-dein false vacuum . This Luccia is reasonable; (CDL)A otherwise comment theTunneling on bubble the ca thinnnot wall be generated approximation causally. : to be concrete, consider the potential U(φ)=U0(φ)+U+ + f(φ) (2.19) 2 2 A comment on the thin wall approximation− : to be concrete, consider the potential λ 2 µ ∼ B 2 U (φ)= φ (2.20) (1) Γ(CDL) e λ µ2 U(φ)=U0(φ)+U+ + f(φ0) (2.19) U (φ)= φ2 (2.20) 8 − λ 0 U(φ)=8 U0(BRIEF−φ)+λ U+ + ARTICLEf(φ) (2.19) ! " ! " 2 2 3 (2) d = s(Λc)/ξ +1 2 2 3 λ µ and f(φ) is a smooth function where f(φ+)=# λand fand(φ−µ) =f( 0.φ Then) is aτ = smoothµ /3λ and function the 2 where f(φ+)=# and f(φ−) = 0. Then τ = µ /3λ and the U (φ)= φ2 U0(φ)= (2.20)φ (2.20) 4 0 THE AUTHOR 8 − λ 4 thin wall approximation is good if µ λ#, or equivalently,8 thin− λ if wall the topFalse approximation of the potential Utop! is good if" µ λ#, or equivalently, if the top of the potential U 1"Domain−1 /wall!4 " top between φ+ and φ−∼satisfies ∼ ∼ 3 " 3 (3) and f(φ)ξ is a smooths(Λs function) and wherefΛf(φ(sφ)+ is)= a# smoothand f(φ− function) = 0. Then whereτ = µf/(3φλ+and)= the# and f(φ−) = 0. Then τ = µ /3λ and the between φ+ and φ− satisfies thin wall approximation is goodUmtops if µU4 (φ+λ#) , or#/ equivalently,8 if the top of the4 potential U thin− wall" approximation" is good if µ λ#, ortop equivalently, if the top of the potential Utop Forbetween the specialφ+ caseand φ with− satisfiesr R , the bubble size is much1 smaller than the curvature" Utop U(φ+) #/8 (4) d>60+between φ+ and φ− satisfies # Utop U(φ+) #/8 − " radius of the background and gravity does not− play a big" role. In this limit BUbecomesU(φ ) #/8 For the specialtop case− with+ "r R+, the bubble size is much smaller than the curvature For the special case with" r 0 R+, the bubble2 size is much−B smaller than the curvature (5) (1) T (n) T#/n2 3 π Γ4(CDL) ∼ e # radius of the background andB =2 gravityπ Forr doesτ the not specialr playradius#. a big case role. of with theIn thisr background limitR+B, thebecomes(2.21) bubble and size gravity is much does smaller not than play the a big curvature role. In this limit B becomes − 2 # 4 − (6) (2) Γ = mradiusT of the! = backgroundUπ(2φ+) U(φ and−) gravity does not play a big role. In this limit B becomes The parameter B is stationary at s B =2π2r3τ r4#. (2.21) 2 3τ −−122 −22 2 2 3 π 4 (3) r = R0 =τ =, 10 , ! = 10 (2.22) π B =2π r τ r #. (2.21) # B =2π2r3τ r4#. (2.21) The parameter B is stationary at − 2 − 2 which is consistent with Eq.(2.16).∼ This result is nothing3τ but the energy conservation. (7) (4) n 1/Hs r = R0 d== s,(Λc)/ξ +1 (2.22) Since the curvature of the backgroundThe is small, parameter the energyThe#B is co parameter stationarynservation reads atB is stationary at which is consistent with Eq.(2.16). This∼ result is∼ nothing1 ∼ but− the1/4 energy conservation.3τ (5) ξ s(Λs) Λs 3τ Since the curvature of the4π background3 is2 small,4π the3 energyms conservation readsr = R0 = , (2.22) Ht R U+ =4πR3 τ + 3HtR U−, (2.23) # r = R0 = , (2.22) (8) a(t) " e →3 V = a(t) " e3 # (6) 4π d>4π 60 whichR3U is=4 consistentπR2τ + R with3U , Eq.(2.16). This result(2.23) is nothing but the energy conservation. and thus we recover Eq.(2.22). Requiring R+ R± yieldswhich is− consistent with Eq.(2.16). This result is nothing but the energy conservation. 3Since the# curvature3 of the background is small, the energy conservation reads (7) T (n) # T0/n 2 and thus we recover Eq.(2.22). Requiring3τ 1R RSince± yields the curvature of the background is small, the energy conservation reads ,# (2.24) (8) #2 # κU Γ = m4T 4π 3 2 4π 3 3τ 2 1 s R U+ =4πR τ + R U−, (2.23) , 3 (2.24) 43π 4π and the tunneling rate per unit volume is given#2 by κU 3 2 3 # R U+ =4πR τ + R U−, (2.23) and thus we recover∼ Eq.(2.22). Requiring R R± yields3 3 and(9) the tunneling rate per unit volume27π is2 givenτ 4 byn 1/Hs # B 3 . (2.25) ∼ 2 # 2 4 2 27π andτ thus we recover3 Eq.(2.22).τ 1 Requiring R R± yields B Ht 3 . 3 3Ht (2.25) , (2.24) It is also(10) interesting for us to ask how toa modify(t)∼# e the2 tunnelin#→ V =ga rate(t) if# theree are radiations#2 # κU # in the false vacuumIt is also and/or interesting in the for true us to vacuum. ask how to In modify the case the with tunnelintheg neglected rate if there background are radiations 3τ 2 1 curvature, the energy conservation impliesand the that tunneling the bubble ratesize should per unit be volume is given by in the false vacuum and/or in the true vacuum. In the case with the neglected background 2 , (2.24) curvature, the energy conservation implies that the bubble size should be # # κU 3τ 27π2 τ 4 R 4 4 , B(2.26) . (2.25) % # + g+T+ g−3Tτand− the tunneling rate∼ per2 unit#3 volume is given by R − 4 4 , (2.26) % # + g+T+ g−T− It is also interesting− for us to ask how to modify the tunneling rate if there are radiations 27π2 τ 4 in the false vacuum and/or in the true vacuum. In the case with the neglected background B 3 . (2.25) curvature,– 4 – the energy conservation implies that the bubble size should∼ 2 be# – 4 – It is also interesting for3τ us to ask how to modify the tunneling rate if there are radiations R 4 4 , (2.26) % # + g T g−T 1 in the false vacuum and/or+ + in− the− true vacuum. In the case with the neglected background curvature,1 the energy conservation implies that the bubble size should be 3τ – 4 – R 4 4 , (2.26) % # + g T g−T + + − −

– 4 – BRIEF ARTICLE

THE AUTHOR

BRIEF ARTICLE 1 In Coleman-de Luccia Tunneling THE AUTHOR − (1) Γ(CDL) ∼ e B In the thin wall approximation : BRIEF ARTICLE (2) 1 d = s(Λc)/ξ +1 log10 B 20 1 BRIEF−1 4 ARTICLE −B∼ ∼ ∼ THE/ AUTHOR (1) (3) Γ(CDL) ∼ eξ s(Λs) Λs 15 ms −12 −22 (2) τ = 10 , " = 10 THE10 AUTHOR (4) d>60 5 (3) d = s(Λc)/ξ +1 1 log10 U! !25 !20 !15 " !100 !5 (5) 1 −1 4T (n) T /n ∼ ∼ ∼ / ! 1 (4) ξ s(Λs) Λs 5 ms 4 −B (1) Γ(CDL!) ∼ e (6) Γ = msT 10 (5) d>60 − (1) −12 Γ(CDL) ∼ e −B22 −B (2)forFigure fixed 1: domainTaken from [4]. wall Huge tension variation of the τ CDL= and 10 tunneling , rate Γ "e =from 10 U+(= U− + !) (6) T (n) " T0/n ∼ to U− via a fixed barrier between them. This point is illustrated by a specific example− here.− Keeping (false-true)(2) energy density−12 difference ! = U(φ−+22 ) U(φ ) (7) fixed the domain wall tension τ 10n ∼and1 the/Hs energy difference ! 10 (both in Planck units), (3) 4 ∼ d = s(Λ∼c)/ξ +1 (7) B logΓ Γ=ism givenT as a function of the potential U−. In this case,− we12 see that B varies 30−22 orders ∼− s of magnitude.(3) Tunneling is exponentially enhanced whenτ the=wavefunction 10 , is higher! = up 10 (i.e., larger 1 −1 4 U−) in the landscape. ∼ ∼ ∼ / (4) (4) Ht ξ s(Λ3s) d =3Hts(Λ )/Λξ s+1 (8) (8) n ∼ 1/Hsa(t) " e → V = a(t) " e ms c here the metric is multiplied by an overall minus sign and dΩH3 is the element of length ∼ ∼ 1 ∼ −1/4 (5) for a unit(5) hyperboloid with timelike normal. The metricξ d>s( withΛs60in) the bubbleΛ describess a 3 3 m (9) a(t) "spatiallyeHt → openV = Friedmann-Robertson-Walkera(t) " e Ht Universe. s (6) In [1](6) the tunneling rate for the potential in Fig.T 3(n in) paper" Td> is0/n computed60 in the thin wall limit. In the semiclassical limit, the tunneling rate per unit volume takes the form 4 # (7) (7) Γ = Ae−B Γ =Tm(ns)T T0/n (2.3) 4 When we(8) include gravitation the effective 4-dimensional actionΓ for= am singlesT scalar field is given by ∼ (8) 1 n 1R/Hs S = d4x√ g gµν ∂ φ∂ φ U(φ) , (2.4) − 2 µ ν − − 2κ∼ (9) ! " n # 1/Hs where κ =8πG. The tunneling rate is determined by the minimum value of the Euclidean Ht 3 3Ht (9) action. The Euclidean action is defineda( ast) minus" e the formal→ V an=alytica(t continuation) " e of the action in(10) Lorentzian frame to imaginary time. Sucha(t a) tunneli# eHtng→ processV = is describeda(t)3 # bye3 theHt CDL instanton which exhibits an O(4) symmetry in Euclidean space. The metric of the instanton in Euclidean space is given in Eq.(2.1) and the Euclidean action is

φ"2 3r S =2π2 dχ r3( + U) (r"2 + 1) (2.5) E 2 − κ ! " # where the prime denotes d/dχ. The scalar ﬁeld equation of motion is 1 3r" dU φ"" + φ" = (2.6) r dφ

– 2 –

1 4 where g± measures the eﬀective light degrees of freedom and g±T± is the energy density of radiations in the false/true vacuum. The parameter B is now modiﬁed to be

2 2 3 π 4 4 4 B =2π r τ r (# + g T g−T ). (2.27) − 2 + + − −

The bubble size at the stationary point is just that given in Eq.(2.26), as expected.

27π2 τ 4 B 4 4 3 (2.28) ∼ 2 (# + g T g−T ) + + − −

Suppose we want to interpret the temperature to be the Hawking temp. Let g− = g+ = g, 4 4 so T T #(U + U−). + − − ∝ + It is worth discussing another limit of r R+ which corresponds to U+ U− = U 2!2 3κτ 2 $ $ % Us = 3κτ 2 + 8 . It happens at the high energy scale in the landscape. In this case the 4 bubblewhere radiusg± measures is given the by effective light degrees of freedom and g±T± is the energy density of radiations in the false/true vacuum. The parameter B is now modified to be 3 2 R . (2.29) 2 3 π 4 $ 4κU 4 B =2π r τ r (# + g+!T g−T ). (2.27) − 2 + − − Now B is dominated by the first term in Eq.(2.15), namely The bubble size at the stationary point is just that given in Eq.(2.26), as expected. 2 2 4 2π τ B27π 6√3π2τ(τκU)−3/2 = (2.30) B 4 4 3 3 (2.28) ∼ $2 (# + g+T g−T ) H + − − Suppose we want to interpret the temperature to be the Hawking temp. Let g = g = g, For TdS = H/2π, we have − + 4 4 log B so T+ T− #(U+ + U−). τ 10 − ∝ B 20 (2.31) It is worth discussing another limit of r R$+ which4πT 3 corresponds to U+ U− = U 2!2 3κτ 2 $ dS $ % Us = 3κτ 2 + 8 . It happens at the high energy scale in the landscape.15 In this case the In thebubble unit radius of M isp given= 1, by or equivalently κ = 1, the tension of the bubble satisfies τ 1. In 3 10 ' Planck region (U 1), B 1 and RΓ 1. At. low energy scale, the background(2.29) curvature ∼ ' $∼ κU radius is quite large and the bubble size! is relatively small5 , and then the tunneling rate is Now B is dominated by the first term in Eq.(2.15), namely insensitive to the vacuum energy of the false vacuum. In Fig.log10 U!in the paper, we see that B !25 !20 !15 !10 !5 2π2τ can easily change by 30 orders of√ magnitude.2 −3/2 !5 B 6 3π τ(κU) = 3 (2.30) A reasonable interpolation$ formula looks likeH !10 For TdS = H/2π, we have τ B 2 2 3/2 (2.31) 3 R R −B Figure 1: Taken from [4]. Huge variation of$ the24π CDLTdS tunneling0 + rate Γ e from U+(= U− + !) B 2π τ ∼ (2.32) to U− via a fixed barrier between them. This point is illustrated2 by 2a specific example here. Keeping In the unit of M = 1, or equivalently$κ = 1, the tensionR0 + γ ofR the+ bubble satisfies τ 1. In fixed the domainp wall tension τ 10−12 and the energy" difference ! #10−22 (both in Planck' units), Planck region (U 1), B 1∼ and Γ 1. At low energy scale,∼ the background curvature B log Γ is given∼ as a function' of the∼ potential U−. In this case, we see that B varies 30 orders whereradius one∼− is may quite take large andγ =2 the4/ bubble3. Recall size is that relativelyR =3 smallτ,/ and#. then the tunneling rate is of magnitude. Tunneling is exponentially enhancedBRIEF when0 the ARTICLEwavefunction is higher up (i.e., larger insensitiveU−) in the to landscape. the vacuum energy of the false vacuum. In Fig. in the paper, we see that B can easily changeTunneling by 30 orders is much of magnitude. faster at higher C.C. A reasonable interpolation formula looks like THE AUTHOR 3. Hawking-Mosshere the metric is multiplied by an overall minus sign and dΩH3 is the element of length for a unit hyperboloid with timelikeHawking-Moss normal. The : metric within the bubble describes a The Einstein equation yields one non-trivial equation, top 2 2 3/2 spatially open Friedmann-Robertson-Walker2 R Universe.0R+ 4 4 , , B 2π τ 2 MP MP (2.32) − 2 −224π [ − ] 1 φ!2In [1] the tunneling rate for$ the potentialBHMR0 + inγR Fig.+ 3 inU( paperφ+) U is(φ computedt) in the thin r!2 = 1 + κr2( U). , Γ e (2.7)" = e # 1 (3.1) 3 wall2 limit.− In the semiclassical limit,$ the tunneling rate per unit volume takes the form + 4/3 , where one may take γ =2 . Recall2 that R0 =3τ/#. and the equation of motion for r(χ) follows from Eqs.(2.6,2.7). Using Eq.(2.7)8 toπ simplifyδU −B BHM = Γ = Ae, 2 δU = U2(φt) U(φ+)(2.3) (3.2) the Euclidean action, one obtains (1) top 3 H2 H2 H = U/3MP − 3. Hawking-Moss + t When we include gravitation the effective 4-dimensionalE act/TionH for a single scalar field is 2 3(2) 3r Γ ∼ e SE =4π dχ r U + (2.8) given by− κ 4 4 " # 2 MP MP ! −B 1 −24π [ U φ − U φ ] R (3) S = Γ d4xe√ HMg =geµν ∂ φ∂ Tφ( H+ =U(H/φt)) 2π , (2.4)(3.1) The coefficient B in the tunneling rate (2.3) is $ − 2 µ ν − − 2κ " # ! 2 −B where(4)κ =8πG. The tunneling8π rateδU is determinedΓ( byCDL the) minimum∼ e value of the Euclidean B = SE(φ) SE(φ+). B = (2.9), δU = U(φ ) U(φ ) (3.2) − HM 2 2 – 5 – t + action. The Euclidean action3 is definedH+Ht as minus the formal− analytic continuation of the − − In the thin wall approximation, we divideaction the(5) integration in Lorentzian for B frameinto three to imaginary parts. Out- time." Such= U a(φ tunneli+) ngU( processφ ) is described by the side the bubble, φ = φ and thus − − + CDL(6) instanton which exhibits an O(4) symmetryτ = 10 in12 Euclidean, " = space. 10 22 The metric of the Bout =0instanton. in Euclidean space is given in(2.10) Eq.(2.1) and the Euclidean action is In the wall, we have (7) – 5 – d = s(Λc)/ξ +1 φ"2 3r B =2π2r3τ, S =2π2 dχ(2.11)r3( + U) (r"2 + 1) (2.5) wall E 2 − κ 1 ! " ∼ ∼ ∼ #−1/4 where r is the bubble size and τ is the tension of the(8) wall which is decided by the barrierξ s(Λs) Λs ms between the false and true vacua, where the prime denotes d/dχ. The scalar field equation of motion is (9) " d>60 φ+ 3r dU φ"" + φ" = (2.6) τ dφ 2[U(φ) U(φ+)] (2.12)r dφ # " φ− (10)− T (n) T0/n ! $ Here we take the thin wall approximation and Bwall is the energy stored in the thin wall. 4 (11) Γ = msT Inside the bubble, φ is a constant and Eq.(2.7) becomes – 2 – dχ = dr(1 κr2U/3)−1/2. (2.13) − (12) n ∼ 1/Hs Hence 12π2 r S (φ)= rd˜ r˜(1 κU(φ)˜r2/3)1/2. (2.14) Ht 3 3Ht E,in − κ (13)− a(t) # e → V = a(t) # e !0 Summing the 3 parts of B, we obtain

2 2 3 12π 1 2 3/2 1 2 3/2 B =2π r τ + 1 κr U−/3 1 1 κr U /3 1 . (2.15) κ2 U − − − U − + − " − + # %& ' ( %& ' ( The decay coeﬃcient B is stationary at r = R which satisﬁes 1 &2 κ(U + U ) κ2τ 2 = + + − + , (2.16) R2 9τ 2 6 16 1 where & = U U−. So + − 2 4π 3/2 3/2 B =2π2R3τ + R2 1 H2 R2 1 R2 1 H2 R2 1 (2.17) κ − − − − − + − + − ) % ( % (* where & ' & ' −1 −1/2 H± = R± =(κU±/3)

– 3 – 4 where g± measures the eﬀective light degrees of freedom and g±T± is the energy density of radiations in the false/true vacuum. The parameter B is now modiﬁed to be

2 2 3 π 4 4 4 B =2π r τ r (# + g T g−T ). (2.27) − 2 + + − −

The bubble size at the stationary point is just that given in Eq.(2.26), as expected.

27π2 τ 4 B 4 4 3 (2.28) ∼ 2 (# + g T g−T ) + + − −

2π2τ B 6√3π2τ(κU)−3/2 = (2.30) $ H3

For TdS = H/2π, we have τ B 3 (2.31) $ 4πTdS In the unit of M = 1, or equivalently κ = 1, the tension of the bubble satisﬁes τ 1. In p ' Planck region (U 1), B 1 and Γ 1. At low energy scale, the background curvature ∼ ' ∼ radius is quite large and the bubble size is relatively small, and then the tunneling rate is insensitive to the vacuum energy of the false vacuum. In Fig. in the paper, we see that B can easily change by 30 orders of magnitude. A reasonable interpolation formula looks like

R2R2 3/2 B 2π2τ 0 + (2.32) $ R2 + γR2 " 0 + # BRIEF ARTICLE where one may take γ =24/3. Recall that R =3τ/#. 2 Formulae : 0 THE AUTHOR 3. Hawking-Moss

CDL thin wall : 4 4 2 MP MP −B 1 −24π [ U φ − U φ ] Γ e HM = e ( +) ( t) (3.1) $ 8π2 2 δU 2 (1) Hawking-Moss : B = H = U/3, MP δU = U(φ ) U(φ ) (3.2) HM 3 H2 H2 t − + + −t (2) Γ ∼ e E/TH

(3) Gibbons-Hawking temperature TH = H/2π − (4) Γ(CDL) ∼ e B – 5 – (5) " = U(φ+) − U(φ−) − − (6) τ = 10 12, " = 10 22

(7) d = s(Λc)/ξ +1

∼ ∼ 1 ∼ −1/4 (8) ξ s(Λs) Λs ms (9) d>60

(10) T (n) # T0/n

4 (11) Γ = msT

(12) n ∼ 1/Hs

(13) a(t) # eHt → V = a(t)3 # e3Ht

1 Coleman-de Luccia (CDL) tunneling Bounce B as the vacuum energy is increased while the shape of the barrier is ﬁxed

CDL thin wall

close to HM Thermal Quantum

HM CDL

Afﬂeck :

Brown and E. Weinberg arXiv: 0706.1573 where φit is the value of φi at the top of the barrier. However, here U(φi) should be replaced by U(φi,T). Instead of calculating this, we where φcanit is make the value another of φ estimate.i at the top of the barrier. However,In hereQM, theU(φ tunnelingi) should through be replaced a barrier by U becomes(φi,T). less Instead suppress of calculatinged when the this, energy we of can makethe another particle estimate. increases. This happens when the particle is in a thermal bath with rising whereIn QM,temperature.φit is the the tunneling value In of QFT,φi at through the the top thermal a of barrier the barrier. effect becomes typically less lifts suppress the potentialed when in a the way energy such that of the particleHowever,the tunneling increases. here U becomes(φi This) should happens faster, be replaced until when the by the barrierU(φi particle,T). disappears Instead is in of(a calculating i.e., thermal the tunneling this, bath we with probability rising temperature.can makeapproaches another In QFT, unity). estimate. the thermal effect typically lifts the potential in a way such that In QM, theFinite tunneling4 Hawking through a barrier temperature becomes less suppress effected when the energy of wherethe tunnelingφ isFor the becomesλφ valuetheory, of faster,φ withat until the the top barrier of the disappears barrier. ( i.e., the tunneling probability theit particle increases. Thisi happens when the particlem is2 in a thermalλ bath with rising approaches unity). U(φ,T = 0) = φ2 + φ4 However,temperature. here In QFT,U(φ thei) thermal should eff beect replacedtypically lifts− by the2 U p(otentialφi,T4! ). in a Instead way such of that calculating this, we For λφ4 theory, with can makethe tunnelingwe another have, becomes for estimate. high faster, temperature until theT barrier, disappears ( i.e., the tunneling probability approaches unity). m2 λ Expanding about theU(φ top,T =of 0)a symmetric = φ2 +barrierφ4 : In QM,For λφ the4 theory, tunneling with through a barrier2 becomes less suppressed when the energy of 1 λ−TH2 2 4!2 λ 4 U(φ,TH )= m( 2 λm )φ + φ + ... (5.7) thewe particle have, for increases. high temperature ThisU(Tφ happens,T, = 0) = when2 24φ2 + the− φ4 particle4! is in a thermal bath with rising − 2 4! + . . . temperature. In QFT, the thermal effect typically lifts the potential in a way such that we have,so the for e highffective temperature mass termT , is no longer2 tachyonic for T > Tc, where the critical temperature 2 2 1 λTH 2 2 λ 4 the tunnelingis given becomes by Tc 24 faster,Um(φ/,Tλ.H until)= ( the barrierm )φ disappears+ φ + ...( i.e., the tunneling(5.7) probability " 1 λT 2 2 24 − λ λ 4! For any directionU(φ,T )= in the( moduliH m2 space,)φ2 + if φ2 + φ4 + ... (5.7) approaches unity). H 2 24 − 48 4! so the effective4 mass term is no longer tachyonic for T > Tc, where the critical temperature For λφ theory,No trapping with if for 24m2 so the effective2 mass term2 is no longerT tachyonic2 > fori ,iT > Tc,=1 where, 2, ...., the dcritical temperature (5.8) is given by Tc 2 24m 2/λ. H 2 is given by T "any24 mmodulus/λ. : λi m λ Expandingc the" above potentialV (φ about,T = a 0) maximum = φ (top2 + of aφ barri4 er), we have Expanding the above potential about a maximum (top of a barrider), ~100 we have Expanding the above potential about a maximum− 2 (top4! of a barrier), we have α α V (φ)=α 2 φ2α+ 4 φ4 + ... we have, for high temperatureV (φ)=T , 2 φ 2+ α 4 φ +4...α V−(2−φf)=2f 4!f φ4!2 f+ φ4 + ... −2f 2 4!f 4 we see that, if in any direction, 2 we see that, if in any direction, 1 λT 2 2 λ 2 λ 4 we see that, if inV any(φ,T direction,)= (24f 2 T THc,neff whereect is coherent will the enable critical along that temperature di- 2 2 isrection. givento wavefunction byrection. EvenT if Even this24 tom be if condition coherent this/λ. condition along is not some is satisfied, not direction satisfied, we quickly see we, that see approaching that the t finitehe afinite wavefunctionTHTeffecteffect will will enable enable c " H toExpanding wavefunctionsimilarto to wavefunction a Block the to wave. be above coherent to be potential coherent along along some about some direction a maximum direction quickly quickly, (top approaching, approaching of a barri a wavefunctioner), a wavefunction we have similar tosimilar a Block to a wave. Block wave. α α V (φ)= φ2 + φ4 + ... References −2f 2 4!f 4 References[1] S.References R. Coleman and F. De Luccia, “Gravitational Effects On And Of Vacuum Decay,” Phys. we see that,Rev. D if21 in, 3305 any (1980). direction, [1] S. R. Coleman and F. De Luccia, “Gravitational2 2 Effects On And Of Vacuum Decay,” Phys. [1] S.[2] R. S. W. Coleman Hawking and and F. I. G. De Moss, Luccia, “Supercooled “Gravitational Phase24fi Transiti

[2] S. W. Hawking and I. G. Moss, “Supercooled Phase Transitions In The Very Early 110 Universe,” Phys. Lett. B , 35 (1982).– 8 – [3] A. R. Brown and E. Weinberg ArXiv:0706.1573. [4] Huang and Tye ArXiv:0803.0663.

[5] M. J. Duncan and L. G. Jensen, Phys. Lett. B 291, 109 (1992).

[6] C. G. Callan and S. R. Coleman, Phys. Rev. D 16, 1762 (1977).

[7] P. M. Saﬃn, A. Padilla and E. J. Copeland, “Decay of an inhomogeneous state via resonant tunnelling,” arXiv:0804.3801 [hep-th].

– 8 – A comment :

• Tunneling of a D3-brane can be significantly enhanced by its Dirac-Born-Infeld action.

CDL : Brown, Sarangi, Shlaer and Weltman, ArXiv 0706.0485 [hep-th] Hawking-Moss : Tolley and Wyman, ArXiv 0801.1854 [hep-th] Wohns, ArXiv 0802.0623 [hep-th] Outline • Tunneling in the cosmic landscape is quite different from tunneling between 2 sites. • It is harder to trap a wavefunction in higher dimensions. Also, the barriers in some directions in the landscape are known to be exponentially low. • The vacuum energy in the landscape plays the role of a ﬁnite (Gibbons-Hawking) temperature effect, enabling faster tunneling. • Treating the landscape as a random potential, we can borrow the knowledge developed in condensed matter physics to estimate the mobility of the wavefunction in the landscape. The vastness of cosmic landscape • At a typical meta-stable site, count the number of parameters or the number of light scalar fields. This gives the number of moduli, or directions in the field space. • This number at any neighborhood in the landscape may be taken as the dimension d of the landscape around that neighborhood. • The landscape potential is not periodic (though it may be close to periodic along some axion directions). It is very complicated. • Let us treat it as a random potential. a cartoon :

Landscape : looks like a random potential in multi-dimensions The wavefunction of the universe may be crudely approximated by that of a D3-brane

moduli open string modes

cosmic scale factor

The behavior of the wavefunction in the landscape is a quantum diffusion and percolation problem. Anderson localization transition

• random potential/disorder medium • insulation-superconductivity transition • quantum mesoscopic systems • conductivity-insulation in disordered systems • percolation • strongly interacting electronic systems • doped systems, alloys, ...... Some references : • P. W. Anderson, Absence of Diffusion in Certain Random Lattices, Phys. Rev. Lett. 109, 1492 (1958). • E. Abrahams, P. W. Anderson, D. C. Licciardello and T. V. Ramakrishnan, Scaling Theory of Localization : Absence of Quantum Diffusion in Two Dimensions, Phys. Rev. Lett. 42, 673 (1979). • B. Shapiro, Renormalization-Group Transformation for Anderson Transition, Phys. Rev. Lett. 48, 823 (1982). • P. A. Lee and T. V. Ramakrishnan, Disordered electronic systems, Rev. Mod. Phys. 57, 287 (1985). • M. V. Sadovskii, Superconductivity and Localization,'' (World Scientific, 2000). • Les Houches 1994, Mesoscopic Quantum Physics. is an unstable fixed point, implying that the transition between mobility and localization is a sharpthone ep.article is scattered randomly and its wavefunction usually changes at the scale of the order l. However, the wavefunction remains extendr /ξed plane-wave-like (Bloch wave- Consider a specific (enveloping) wavefunction ψ(r) e−| | at site r = 0 with a typical like) through the medium. If the wavefunction is∼initially localized at some site, it would Λ in the cosmic landscape (see Figure 1(b)). The wavefunction seems to be completely quickly spread and move towards low potential sites. This mobility implies that the system localized at the site if the localization length ξ a, where ξ measures the size of the is a “conductor”. " site and a is the typical spacing between sites. They are determined by local properties Anderson [19] showed that the wave function of a quantum particle in a random poten- and ξ is known as the localization length. With resonance tunneling, ξ can be somewhat tial can qualitatively change its nature if the randomness becomes large enough (strongly bigger, althdisoughordereψd).(r)Inmathisy casstille, btheeewxapvonefunctionentiallybsecuomesppressloedcaliatzed,dissotanthatce aits. amplAt ditudistancee (enve- scales around a, the conductance goes like lope) drops exponentially with distance from the center of localization r0: a/ξ g(a) ψ(a) e− (1.3) ∼ψ|(r) | ∼exp( r r0 /ξ) (2.1) | | ∼ −| − | 2 2a/ξ Note that, for a ξ, Γ0 ψ(a) e− . where ξ#is the loc∼ali|zation| length∼ . This situation is shown qualitatively in Figure 1. When Given g = g(a) (1.3) at scale a, what happens to g at distance scales L a ? That is, the particle wavefunction is completely localized at a single site, ξ app# roaches a value how does g scale ? The scaling theory of g is well studied in condensed matter physics. It comparable to the typical spacing a between sites. In d 3, it will take an exponentially turns out that there is a critical conductance g which is a function≤ of d. If g(a) < g , then long time to tunnel to a neighboring site.c The lack of mobility implies that thec system is g(L) e L/ξ 0 as L and the landscape is an insulating medium. The wavefunction ∼ −an “in→sulator”.→ ∞ is truly localizeThed.phTyunnsicalelingmeanwingill tofakAndersone exponenlotialcalilyzatilongon isanredlativeternalely siminpflationle: coherecomntestuninnelingto play. If g(a) > g , the conductivity is finite (g(L) (L/a)(d 2)) as L and the of particlecs is possible only between energy levels∼with the −same energy→. Ho∞wever, in case landscapeofisstrongin a condrandomnesucting/mobis, l aleandphasethe. stateThe swwitavehfunthectisamoneisenfreeergytoarmoe tovoe.farArouaparndt inthspacee → transitionforpointunt,nelinthereg toisbae ceorfferelationctive. Sincelengthwe thatare inbtereslowstedupinatlarthgee dp,hasewhiletranconsdenseition.d Wmatte er shall see thphat,ysicats slengthystems stcaleypicallya, thhaisvecorrd elat3, ionwe slengthhall introcanducebe aidselnightifiedtly rwithefinedξdefin. Thatitionis,hereξ : ≤ plays the(1)roleanofextethencdedorrelationstate if lξength.a, asThshoiswnis hoinw,Figugivreen1(a);g(a) (1.3) at scale a, g can grow % large for lar(2)gea wLeakly. localized state if ξ ! a, as shown in Figure 1(b); In this(3)papa loer,calizweedargustatee thatif ξ, g. c; given g(a) (1.3), we see that fast tunneling happens 2(d 1) when Γ0 >Fore−d −3, ,aorweakly localized state will have lost mobility already (although it is believed ≤ that, under the right conditions, supaerconductivity can take place in the localization re- d > + 1 (1.4) gion). That a strongly localized stateξmay still lead to mobility for large d is not surprising. For large d, the particle has many directions for coherent or resonant tunneling. In a ran- so the wavefunction of the universe can move freely in the quantum landscape even for dom potential environment, it is much more likely that some direction allows easy motion. an exponentially small g(a). Since the β-function has a positive slope at the localization The goal is to quantify this intuition. We shall continue to use the language of condensed transition point (implying an unstable fixed point), the transition between the insulating matter physics : a is the microscopic scale while L is the macroscopic scale. (localization) phase and the conducting (mobile) phase is sharp. We see that it is precisely For fixed d, we expect a transition from a “conductor” to an “insulator” as the potential the vastnbaressriersof therisceosmicand fastlandsctunnapeleing(asispsupparameresterizesed. dTboyleaarlarn mgeorde) abthouattallothiswstrmobilansitionity, we even if thshoule wadvefunfocusctonionthise bloehacalizeviordof. thThie csonmductivobilitityy,alloor,wsmorthee apwpraoprivefunatecltiy,onthetocondsamuctanceple . the landsItcapturne vseryoutquthaticklythe. Itbehaalsoviormeansof suachsemi-a transitionclassicalcandesbcripe dtionescribofedthbey landa scalscapingetheoryis inadequate.similar to that used in the theory of critical phenomena [20]. In the scaling theory of the In thetranmobsitionilitybetphweeasne,mtobheilitwyaandvefulnctionocalizatanion,d/orone theconsDiders3-branthe ebseha(andvior moof thdue li)condmuctanceove down theaslanda fuscapnctione toofathsitee samplewith asizsme Lal.leDimensionalr Λ. At thatly, swite,e shalsoml cehoofosietssomneighe micborinroscopig sitesc units have larger Λs, so the effective a increases, leading to a larger effective critical conductance gc. This happens until the condition (1.4) is no longer satisfied and the D3-brane is stuck

– 6 –

– 4 – is an unstable fixed point, implying that the transition between mobility and localization is a sharp one. Consider a specific (enveloping) wavefunction ψ(r) e r /ξ at site r = 0 with a typical ∼ −| | Λ in the cosmic landscape (see Figure 1(b)). The wavefunction seems to be completely localized at the site if the localization length ξ a, where ξ measures the size of the " site andofatishethore dertypicofal ξspacing. Thenbeonetweenexspiteecs.ts Ththatey arefor dLeterminξ, ethd ebyeffloecctival peropcondertieuctances becomes and ξ is known as the localization length. With resonance tun!neling, ξ can be somewhat of the order of ξ. Thenexponeonexepntiallyects thsmat alforl: L ξ, the effective conductance becomes bigger, although ψ(r) may still !be exponentially suppressedL/atξ distance a. At distance exponentially small: gd(L) e− (2.2) scales around a, the conductanceL/goξ es like ∼ gd(L) e− (2.2) For a small disorder,∼ the medium is in a mea/ξtallic state and the conductivity σ is independent g(a) ψ(a) e− (1.3) For a small disorder, the ofmethdiume samis inplea mesiztalle ificthstatee sizDefineande∼ist|hem ucconda |hdimensionless∼largeructivitythanσ is ithndepe conductancemeenandenfreet path, g L l. Conductance of the sample size if the size is much larger than th2 e mean2a/ξfree path, L l. Conductance ! Note thisat,determinedfor a ξ, Γi0n thψis(ac)aseinjust ea− d-dim.by. the usualhypercubicOhm’s laregionw and offor sizea d-dimensional L hypercube, # ∼ | | ∼ ! is determined in this cGivaseenjustg =byg(thea) (usual1.3) atOhm’sscale ala, wwhatandhforappaends-dimensionalto g at distancehypescrcalesube,L a ? That is, we have: # we have: how does g scale ? The scaling theory of g is well studied in condensed 2 d matter physics. It d 2 gd(L) = σL − (2.3) g (L) = σL − (2.3)conductivity turns out that there isda critical conductance gc which is a function of d. If g(a) < gc, then L/ξ based on a simpleg(dLime) nbasedes−ional onargu0 aasmsLienmt.ple(Recdandimeallthnthatseionlanalindscargudap=e is3,maneng t.=insulatiσ(Rec(Area)ngallme/Lthatdium.σinL.)Thed =w3,avefug =nctionσ(Area)/L σL.) ∼ → → ∞ ∼ ∼ The conductanceisg tr=ulyg(Thealo) calizeatclengthonddu. Tcunntanscaleecleingagis=waillgm(tiaakc)roscopiceatexplengthonenmeasuretialscalely longofa thisaneaddmisorder.eternalicroscopicinWflatione measurecomesofintothe disorder. We (d 2) play. Ifsege(ath) at> gg(c,Lthe) macony duendctiuvitpThoulessywithis finonitee(gof(Lth) e 2(L/avery) di− ff)erasentL asymptotandictheforms for L a. see that g(L) may end up with one of the 2 very different asymptotic∼forms for L a. → ∞ landscape is in a conducting/mobile phase. The wavefunction is free !to move. Around the ! Here σ is rescaled so thatHereg is σdimeis rnesscionlesaleds.so thatConducting/mobileg is dimensionles (metalic)s. with finite conductivity Elementary strancalinsgitiontheorypoinElofet,melotherecalizationntaryis sacalincasorsumesrelationg theorythatlenofgthg loofcalizationthata d-dimensblows asiuponalsumesaththypeetrhatpchaseubeg oftranasdition.-dimensWeional hypercube shall see that, at length scale a, this correlation lengthConductancecan b =e id Mobilenessentified with ξ. That is, ξ of size L satisfies the simplof esizstedLiffesatisrentifialesequtheationsimplof eastrenordiffmeralienConductivityztiatialonequgroupation = , Mobilitywhereof a renor malization group, where plays the role of the correlation length. This is how, given g(a) (1.3) at scale a, g can grow large for large L. d ln gd(L) d ln g (L) βd(gd(L)) = β (g (L)) = d (2.4) (2.4) In this paper, we argue that,dforln Llarge d, thed crditical conducdtanln cLe is exponentially small, that is, the β-function β (g) depends only on the dimensionless conductance g (L). Then d (d 1) d that is, the β-function βd(g)gdepc ende− s−only on the dimensionless conductance gd(L). Then the qualitative behavior of βd(g) can be analyzed in a simple&way by interpolating it between the 2 limiting forms giventhebyquEq.alitativ(2.2) eanbdehaEqvi.(or2.3of). βFdor(gthe) caninsulbeatinganalypzhaseed in, (ag simple0), itway by interpolating it between So it is not difficult for g(a) > gc; given g(a) (1.3), we see that→fast tunneling happens follows from Eq.(2.2) andthEeq.(2 2.4li2(md)it1)thating :forms given by Eq.(2.2) and Eq.(2.3). For the insulating phase, (g 0), it when Γ0 > e− − , or → follows from Eq.(2.2) and Eq.(2.4) that : g a lim βd(g) ln d > + 1 (2.5) (1.4) g 0 → gc ξ g → lim βd(g) ln (2.5) g 0 → g which is negativesofortheg wa0.vefunForctitheon ofconthductine univgeprshasee can(larmgeoveg),fre→itelyfollinowsthefromquanEctuq.(m2.3lan)dscape even for → and Eq.(2.4) thatan: exponwhicentihallyis nsmallegativg(ea).forSingce th0.e βF-fuorncthetionconhasductina posgitivpehaseslop(lare atgetheg),locitalizfollationows from Eq.(2.3) → transition point (implylim ingβd(gan) undstab2le fixed point), the transition be(2.6)tween the insulating and Eq.(2.4g ) that : → − (localization) phase→∞and the conducting (mobile) phase is sharp. We see that it is precisely which is positive for d > 2. Assuming the existence of two pertlimurbationβd(g)expand sion2s over (2.6) the vastness of the cosmic landscape (as parameg terized by→a lar−ge d) that allows mobility the “coupling” g in the limits of weak and strong “couplings”, one→∞can write corrections to even if whicthe whaisvefunposcititionve isforlocalized > 2.d. AssThius mmiobngilitthyealloexwsistethencewofavtewfuno cptieonrturbto ationsampleexpansions over Eq.(2.5) and Eq.(2.6) in the following form : the landthscape “coupe verylinqug”ickglyin. Itthealsolimmitseansof awesemi-ak andclasssicaltrongdes“coupcriptionlinofgsth”,eonelandcsancapwritee is corrections to inadequate. g Eq.(2.5)βandd(g Eq.(0) =2.6ln) in(1th+e bfoldglo+wing) form : (2.7) In the mobility →phase, thegwc avefunction· ·an· d/or the D3-branes (and moduli) move αd g down βthe(glandscap) =e dto a2site with+ a smalleαr Λ>. 0At that site, some of(i2.8)ts neighboring sites d → ∞ − − g · · · dβd(g 0) = ln (1 + bdg + ) (2.7) have larger Λs, so the effective a increases, leading→to a largergecffective critical· · c·onductance αd Assuming these andgc. Tha smois hapothpmonotonens untilousthe βcdon(gditi),βitond(isg(1.4eas)yisto)no=plotlond thege2r βsat-fuisnctionfie+d anqdualitthe a-Dαd3-bran> 0 e is stuck (2.8) tively for all g and d, as shown in Figure 2. → ∞ − − g · · · For d > 2, β (g) must have a zero: β (g ) = 0, where g is the critical conductance. d Assuming these andd c a smooth monotonc ous βd(g), it is easy to plot the β-function qualita- The slope at the zero is ptivositivelye,forso althlisgzeandro ofdβ, das(g)shocorrwnespinondsFigutorean2u.nstable fixed point of Eq.(2.4). This means that a small positive or negative depar– 4 –ture from the zero will lead For d > 2, βd(g) must have a zero: βd(gc) = 0, where gc is the critical conductance. asymptotically to very different behaviors of the conductance. As g moves from g > gc to The slope at the zero is positive, so this zero of βd(g) corresponds to an unstable fixed point g < gc, mobility is lost and the wavefunction is truly localized. This sharp transition is the of Eq.(2.4). This means that a small positive or negative departure from the zero will lead mobility edge. asymptotically to very different behaviors of the conductance. As g moves from g > gc to g < gc, mobility is lost and the wavefunction is truly localized. This sharp transition is the mobility edge. – 8 –

– 8 – is an unstable fixed point, implying that the transition between mobility and localization is a sharp one. Consider a specific (enveloping) wavefunction ψ(r) e r /ξ at site r = 0 with a typical ∼ −| | Λ in the cosmic landscape (see Figure 1(b)). The wavefunction seems to be completely localized at the site if the localization length ξ a, where ξ measures the size of the " site and a is the typical spacing between sites. They are determined by local properties of the order of ξ. Then one expects that for L ξ, the effective conductance becomes ! and ξ is knoexpwnonasenthetiallylosmcalizationall: length. With resonance tunneling, ξ can be somewhat bigger, although ψ(r) may still be eHowxpon doesent gially scale ?suppreL/ssξed at distance a. At distance gd(L) e− (2.2) scales around a, the conductancGivene go ge ats scalelike a, what is g at ∼scale L For a small disorder, the medasium L becomesis in a largeme tall? ic state and the conductivity σ is independent of the sample size if the size is much larger thana/ξ the mean free path, L l. Conductance g(a) ψ(a) e− ! (1.3) of the order of ξ. Thisendeterminedone expecitns ththisatcasefor jLust∼b|yξthe, th|usuale∼effectivOhm’se condlaw uctanceand for abdec-dimensionalomes hypercube, 2 !2a/ξ exponentiallyNotesmthalat,l: forwe ahave:ξ, Γ0 ψ(a) e− . # ∼ | | ∼ L/ξ d 2 Given g = g(a) (1.3) at gscale(L) a, ewhat− happgedn(Ls )to=gσatL −distance scales L(2.2) a ? That is,(2.3) d ∼ # how does g scale ? The scaling theory of g is well studied in condensed matter physics. It For a small disorder, thbasede medoniuma sisiminplea medimetallnicsionstateal arguandmtenhet.cond(Recuctivitall thaty σinis dindep= 3,engden= tσ(Area)/L σL.) turns out that there is a criInsulating,tical c localized,onductance g Conducting,which ismobilea function of d. If g(a) < g , th∼en of the sample size if thThee sizcone isdumctanuchtrapped,celargerg = eternalgthan(a )inflationatthlengthe meansccalefreea pisath,a mLicroscopicl. Condumeasurectancofe the dcisorder. We g(L) e L/seξ e th0atasgL(L) mazeroyand endconductivitythupe withlandsconape eofisthane 2insulativery ding!ffermeentdium.asymptotTheicwforavmefus nctionfor L a. is determined i∼n th−is case→just by →the∞usual Ohm’s law and for a d-dimensional hypercube, ! Here σ is rescaled so that g is dimensionless. we have: is truly localized. Tunneling will take exponentially long and eternal inflation comes into (d 2) play. If g(a) >Elgecme, thentaryconscalinducgtithviteoryy disof2finloitecalization(g(L) assumes(L/at)hat−g )ofasa d-dimensL ionalandhyptheercube gd(L) = σL − ∼ (2.3)→ ∞ landscape isofinsizaecondL satisuctfiing/mobies the simplle epsthasediff.erTheentialwaequvefationunctiofonaisrenorfreemtoalizmoativone. groupArou,ndwherethe based on a simple dimensional argument. (Recall that in d = 3, g = σ(Area)/L σL.) transition point, there is a correlation length that blodwlns gupd(Lat) the phase∼ transition. We The conductance g = g(a) at length scale a is a microscopicβd(gd(L))m=easure of the disorder. We (2.4) shall see that, at length scale a, this correlation length cand lnbLe identified with ξ. That is, ξ see that g(L) may end up with one of the 2 very different asymptotic forms for L a. plays the rolethatof iths, ethceorβrelation-functionleβngth.(g) depThendis iss onholyw,ongivtheendimensionlesg(a) (1.3) ats csoncale!ducatan, gccane g (groL).wThen Here σ is rescaled so that g is dimensionless.d d large for largetheLqu. alitative behavior of β (g) can be analyzed in a simple way by interpolating it between Elementary scaling theory of localization assumesd that g of a d-dimensional hypercube In this paptheer,2 liwmeitarguing formse thatgiv, foren larbygeEq.d(,2.2th)e ancriticald Eq.(c2.3on).duFcortanthece isinsulexpatingonenptiallyhase, sma(g ll,0), it of size L satisfies the simplest differential equation of a renormalization group, where → follows from Eq.(2.2) and Eq.(2.4) that : (d 1) g e− − d ln gcd&(L) g βd(gd(L)) = lim βd(g) ln (2.4) (2.5) d ln L g 0 → gc So it is not difficult for g(a) > gc; given g(a→) (1.3), we see that fast tunneling happens that is, the β-function β2(d(g)1)depends only on the dimensionless conductance g (L). Then when Γ0 > ewhic− d h−is,noregative for g 0. For the conducting phase (larged g), it follows from Eq.(2.3) the qualitative behavior of β (g) can be analy→zed in a simple way by interpolating it between and Eq.(d 2.4) that : a the 2 limiting forms given by Eq.(2.2) and Eq.(2.3d).>Forlim+the1βdinsul(g) atingd 2phase, (g 0), it (1.4)(2.6) ξg → − → follows from Eq.(2.2) and Eq.(2.4) that : →∞ so the wavewhicfunchtionis pofositithvee forunivde>rse2.canAssummoivnge fthreeelyexisintenthece ofquantwotupmerturblandscationapexpane evensionfors over the “coupling” g in the limits of gweak and strong “couplings”, one can write corrections to an exponentially small g(alim). Sinβd(cge) thelnβ-function has a positive slope at(2.5)the localization Eq.(2.5) and Eq.(g 2.60 ) in th→e follogcwing form : transition point (implying →an unstable fixed point), the transition between the insulating which is negative for g 0. For the conducting phase (large g), itgfollows from Eq.(2.3) (localization) ph→ ase and the conducting (mβobd(gile) 0)ph=aselnis s(1harp+ b.dgW+e see) that it is precisely(2.7) and Eq.(2.4) that : → gc · · · the vastness of the cosmic landscape (as parameterizeαdd by a large d) that allows mobility lim βd(βgd)(g d )2= d 2 + αd > 0 (2.6) (2.8) even if the wavefunction gis localized. This mobility allows the wavefunction to sample →∞ →→ ∞− − − g · · · which is pthoseitilandve forscapde>ve2.ryAssquicumklying. Itthalsoe exismteeansnce ofa stemi-wo pclasserturbicalationdescripexpantionsionofs thoveerlandscape is Assuming these and a smooth monotonous βd(g), it is easy to plot the β-function qualita- the “coupinlinadequate.g” g in thetivlielymitsforofalwl geakandandd, asstrshoongwn“coupin Figulingsre”,2.one can write corrections to In the mobility phase, the wavefunction and/or the D3-branes (and moduli) move Eq.(2.5) and Eq.(2.6) in thFore fold lo>w2,inβgdform(g) m:ust have a zero: βd(gc) = 0, where gc is the critical conductance. down the landThescapslopeetato athesitezerwo isithpositiva smg e,alsolerthΛis. zeArot tofhatβd(sgite,) corrsomespeondsof itstoneighan unstableboringfixedsitespoint βd(g 0) = ln (1 + bdg + ) (2.7) have larger Λofs,Esq.(o the2.4).effTehctivis →mee ana ins thcrateasegac sms, lealadl pingositiv·to·e· aorlargernegativeffeecdepartiveturcriticale fromcthoneduzeroctanwillce lead gc. This hapasympensptoticallyuntil thetoconverditiy doniαffedr(e1.4nt )beishanoviorslonofgetrhesatcondisfieuctanced and .theAsDg 3-branmoves efromis stucg >kgc to βd(g ) = d 2 + αd > 0 (2.8) g < gc,→mob∞ility is −lost−andgthe ·w·a·vefunction is truly localized. This sharp transition is the mobility edge. Assuming these and a smooth monotonous βd(g), it is easy to plot the β-function qualitatively for all g and d, as shown in Figure 2.

For d > 2, βd(g) must have a zero: βd(gc) = 0, where– 4 – gc is the critical conductance. The slope at the zero is positive, so this zero of βd(g) corresponds– 8to– an unstable ﬁxed point of Eq.(2.4). This means that a small positive or negative departure from the zero will lead asymptotically to very diﬀerent behaviors of the conductance. As g moves from g > gc to g < gc, mobility is lost and the wavefunction is truly localized. This sharp transition is the mobility edge.

– 8 – of the order of ξ. Then one expects that for L ξ, the effective conductance becomes of the order of ξ. Then one expects that for L ξ, th!e effective conductance becomes exponentially small: ! L/ξ exponentially small: gd(L) e− (2.2) L/ξ ∼ g (L) e− (2.2) For a small disorder, the meddium is∼in a metallic state and the conductivity σ is independent of the order of ξ. Theofn thonee samexppecletssiztheatif forthe Lsize isξ,mthucehelargerffectivethancondthuctancee mean bfreeecomespath, L l. Conductance For a small disorder, the medium is in a!metallic state and the conductivity σ!is independent exponentially small: is determined in this case just by the usual Ohm’s law and for a d-dimensional hypercube, L/ξ of the sample size if the size gis(Lm)uche−larger than the mean free path,(2.2)L l. Conductance we have: d ∼ ! is determined in this case just by the usual Ohm’s law andd 2for a d-dimensional hypercube, For a small disorder, the medium is in a metallic state and thegcondd(L)uctivit= σLy−σ is independent (2.3) we have: of the sample size if thbasede size isonmaucshimlargerple dimethannsionthealmearguan mfreeent.path,(RecLall thatl. Conduin dc=tan3,ceg = σ(Area)/L σL.) d 2 ! ∼ is determined in this caseThejustconbduy cthetanusualce g =Ohm’sg(a)gatdla(wLlength)and= forσscaleLa−da-dimensionalis a microscopichypermcueasurebe, of the disorder.(2.3)We we have: see that g(L) may end up with one of the 2 very different asymptotic forms for L a. based on a simple dimensional argumden2t. (Recall that in d = 3, g = σ(Area)/L σ!L.) Here σ is rescalgedd(Lso) =thatσLg−is dimensionless. (2.3) ∼ The conductance g = g(a) at length scale a is a microscopic measure of the disorder. We based on a simple dimensionElealmearguntarymenscalint. (Recg thaleoryl thatof loincalizationd = 3, g as=sumesσ(Area)that/Lg ofσaLd.)-dimensional hypercube of the order of ξ. Then one expects that for L ξ, th∼e effective conductance becomes see that g(L) maofysizende L satisup fiwithes theonsimple ofestthdeiffe2revnetiraly ediquffationerenoft asymptota renormaliiczatiforonmgroups for, whereL a. The conductance g = g(a) at length scale a is a microscopic measure of th!e disorder. We ! seeHerethat gσ(Lexpis) maroneysceendalntiallyedupsowithtsmhatonall:ge ofis thdimee 2 vnesriyonlesdiffers.ent asymptotic forms for L a. d ln gL/d(ξL) ! Here σ is rescaled so that g is dimensionless. βd(ggd((LL)))= e− (2.4) (2.2) Elementary scaling theory of localization asdsumes∼thatd lngL of a d-dimensional hypercube ofElsizemeentaryL satisscalinfiegsththeeorysimplof localizationest diffeasresumesntial tehatquationg of a dof-dimensa renorionalmhaliypzeatircuonbe group, where of size L satisForfiesathesmsimplalthlatdeisorder,ists, thdiffeeβre-funthtincaletionemequationβddium(g) ofdepisaendinrenoras onmemalilytallzonatiictheonstategroupdimensionles,andwherethes concondductanuctivitce gd(yLσ). isThenindependent the qualitative behavior of βd(g) can be analyzed in a simple way by interpolating it between of the sample sizBRIEFe if th ARTICLEe size is much largerd ln g than(L) the mean free path, L l. Conductance the 2 limiting forms givβ end(lngbgy(dL(Eq.L)))(=2.2) and dEq.(2.3). For the insulating phase, (!g (2.4)0), it is determined in thβdis(gdc(aseL)) =justd dby the usual Ohm’s law and(2.4)for a d-dimensional→ hypercube, of the order of ξ. ThenfolonelowsexfrompectTHEEq.s th AUTHOR(2.2at)forandLdElnq.(L2.4ξ,)ththate ed:fflnectivL e conductance becomes ! expontheatntiallyis, thesmwβ-fuealhancl: tvione: β (g) depends only on the dimensionless conductance g (L). Then that is, the β-funcd tion βd(g) depends only on the dimensionlesg ds conductance gd(L). Then L/ξ lim βd(g) lnd 2 (2.5) the qualitative behavior of βd(g) can beganaly1 (L) zedein− a simple gwgda0y(Lby) in=terp→σLolating− g it between (2.2) (2.3) the qualitative behavior of β (dg) can be analyzed→in a simple wc ay by interpolating it between the 2 limiting forms given by Eq.(2.2d) and Eq∼.(2.3). For the insulating phase, (g 0), it ln L → For a smthalel d2(1)isorder,libasedmitingthoneformswhicmea dshiumimgivispnisenlee lngativing d∼−baimeyLemeEq.=for−ntalles(gion2.2ic stateal)0.anarguFdorandEqthemt.(enhec2.3ont.condductin).(RecFuctivitorg alpthehasel ythatσinsul(laris igeinndepatinggd),e=itnpdenfoll3,haseotwsg ,=f(romgσ(Area)Eq.(0),2.3it/L) σL.) follows from Eq.(2.2) and Eq.(2.4) that : → → and Eq.(2.4) that −:Si ∼ of the samfollopwle(2)ssizfrome if thEq.e siz(2.2e is) andmuchEαlargerq.(i ∼ e2.4)thanthatth:e mean free path, L l. Conductance The conductance g = g(a) at lengthg scale a is a m!icroscopic measure of the disorder. We (3) |limβ | <βαd(g) ln lim βd(g) d 2 (2.5) (2.6) is determined in this case just by theg usualij 0 i Ohm’s→ glaw andg for a d-dimensional→ − hypercube, see that g(L) may end→ up with cone of→∞theg2 very different asymptotic forms for L a. we have: (4) which is positiφiv=eϕfori/fi d >lim2. Assβd(ugm)ing thlne existence of two perturbation expansion(2.5)s over ! which is negativHeree fσorisg resc0. alFeordtheso ctonhatd−1ducting isgg pd0haseime(larnsi→geonlesg), gs.itc follows from Eq.(2.3) (5) th→e “coupling”Γt ∼gnin Γthe0 lim→itsd of2 weak and strong “couplings”, one can write corrections to and Eq.(2.4) that : gd(L) = σL − (2.3) Elementary scalinT (A → gB)Tth(Be→oryC) of localization assumes that g of a d-dimensional hypercube which(6) is negativEqeT (.(fAor2.5→ Cg))=and 0.Eq.(F2.6or) thein thce∼onfolT0/ductinlo2 wing formg phase: (large g), it follows from Eq.(2.3) T (Alim→ B)+βdT((gB)→ C)d 2 (2.6) based on a simple dimensional argu→ gment. (Rec→ al−l that in d = 3, g = σ(Area)/L σL.) of size L satisfies the→∞simple−stS differential equation of a renor∼malization group, where and Eq.((7) 2.4) that : ΓA→B = ΓB→C = Γ0 ∼ e g The whicconduh isctanposcitievge =forgd(a>) at2. lengthAssumingscaletheaexisistaenmceicofroscopicβtdw(go pert0)murb=easureationln (1ofexpan+thbdesgiond+isorder.s ov)er We (2.7) 2 → gc · · · (8) ΓA→C ∼ ΓA→BΓB→C = Γlim0 βd(g) d 2 (2.6) see ththate “coupg(L)linmag”yg endin theuplimwithits ofonweeakofandthestr2onggver“coupy difflinergsen”,t oneasymptotcanαddwritelnicgforcorrectionsd(Lm)s for Lto a. 1 β (1g 2β) (=g→d(L2))−= + α > 0 ! (2.8) (2.4) (9) tA→C = tA→B + tB→C = + d→∞∼ d d d HereEqσ.(is2.5r)escandaleEdq.(so2.6that) in gthise foldimelowinnsgionlesformΓ →s.: Γ → → ∞Γ0 − − g · · · which is positive for d > 2. AssAuBmingB C the existence of twdolnpLerturbation expansions over Γ0 Elementary scaling theory of localization→ assumesg that g of a d-dimensional hypercube (10) Assuming thesΓAe Cand= a smooth monotonous βd(g), it is easy to plot the β-function qualita- the “coupthatlinig”s, thg ine βthe-funcliβmdt(iiongts of0)β2dw=(eglnak) depand(1 end+sbtrdgsong+only“coup) on thelingsdimensionles”, one c(2.7)an writes concorrectionsductance gtod(L). Then of size L satisfies the simplest differen→tial e−quS ationgc of a renor· · ·malization group, where (11) tively for alΓl(1)g =andΓ0 ∼de, as shown in Figure 2. Eq.(2.5) and Eq.(2.6) in the followinαdg form : the qualitative behavior−of βd(g) can be analyzed in a simple way by interpolating it between (12) βd(gFor d >) =Γ2,(2)dβ∼de(2g2S) must+have a zeαro:d > β0d(gc) = 0, where gc (is2.8)the critical conductance. → ∞ − − g d ln·g· ·d(L) the 2 limiting forms giv−en by Eq.(2.2) and Eq.(2.3). For the insulating phase, (g 0), it (13) The slope atβdtΓ(he(2)gd∼(zeLer))oS is=positive, so this zeg ro of βd(g) corresponds to an(2.4)unstable fixed point βd(g d ln0)L= ln (1 + bdg + ) (2.7) → Assuming these and a smooth monotonous− β (g), it is easy to plot the β-function qualita- (14)follows fromof Eq.(Eq.2.4().2.2TΓh()nis) and∼mee SanEds q.(th→at2.4a sm) thatall positivg:c e or negativ·e·depar· ture from the zero will lead thattivis,elythfore βal-ful gncandtiond,βas(gsho) depwn inendFigus onre2 ly22.on the dimensionless conductance g (L). Then (15) asymd ptotically+toH φvery different behαaviorsd of the conductanced. As g moves from g > gc to βd(g ) = d 2 + αdg> 0 (2.8) For d > 2, βd(g) mgust< gha, mveobailize2tyro:is βlosd2(tgcand) =the0, wherewavefugnctionc is theiscrititrulycalloccalizeonductand. Thisce. sharp transition is the the qualitativ(16)e behavior of βd(cg) canHbe=→analyU/3∞M zed in−a simple−limg wβady·(·gb·y) interpln olating it between (2.5) P g 0 The slope at the zero ismobilpositivitye,esodge.this −zero of βd(g) corresponds to an→unstablegcfixed point the 2 limiting(17)forms given by Eq.(2.2Γ ∼) eanE/TdH Eq.(2.3). For→the insulating phase, (g 0), it of EAssq.(2.4um).inTghisthesmeane sandthataa ssmmoalothl positiv1 monotone or negatousiveβdepard(g),turiteisfromeasthyetozeroplotwillthelead→β-function qualita- follows from Eq.whic(2.2h) andis neEgativq.(2.4e) fthator g: 0. For the conducting phase (large g), it follows from Eq.(2.3) asymtivptoticallyely for altolvgeryanddiffedr,enast bshoehaviorswn inof Figuthe condre 2uctance. . As g moves from g > gc to → g g < gc, mobandility isEq.(lost2.4and)thethatwav:efunction is truly localized. This sharp transition is the For d > 2, βd(g) mustlimhavβed(ag)zero:ln βd(gc) = 0, where gc is the criti(2.5)cal conductance. mobility edge. g 0 → gc → lim βd–(g8)– d 2 (2.6) The slope at the zero is positive, so this zero gof βd(g) corr→esponds− to an unstable fixed point which is negative for g 0. For the conducting phase →∞(large g), it follows from Eq.(2.3) of Eq.(2.4). This→means that a small positive or negative departure from the zero will lead and Eq.(2.4) thatwhic: h is positive for d > 2. Assuming the existence of two perturbation expansions over asymptotically to very different behaviors of the conductance. As g moves from g > gc to the “coupling” g inlimtheβlidm(gi)ts ofd we2ak and strong “couplings”, one(2.6)can write corrections to g < g , mobility is lost andg the w–av8efu–→nction− is truly localized. This sharp transition is the c Eq.(2.5) and Eq.(2.6→∞) in the following form : which ismobilpositiitvyeefordge.d > 2. Assuming the existence of two perturbation expansions over the “coupling” g in the limits of weak and strong “couplings”, one cangwrite corrections to βd(g 0) = ln (1 + bdg + ) (2.7) Eq.(2.5) and Eq.(2.6) in the following form : → gc · · · αd βd(g g ) = d 2 + αd > 0 (2.8) βd(g 0) = ln→ ∞(1 + bdg−+ −) g · · · (2.7) → gc – 8 – · · · α Assuminβg(gthese and) = da smo2 othd +monotonαous>β0 (g), it is easy to plot(2.8) the β-function qualita- d → ∞ − − g · · · d d tively for all g and d, as shown in Figure 2. Assuming these and a smooth monotonous βd(g), it is easy to plot the β-function qualita- For d > 2, βd(g) must have a zero: βd(gc) = 0, where gc is the critical conductance. tively for all g and d, as shown in Figure 2. The slope at the zero is positive, so this zero of βd(g) corresponds to an unstable fixed point For d > 2, βd(g) must have a zero: βd(gc) = 0, where gc is the critical conductance. of Eq.(2.4). This means that a small positive or negative departure from the zero will lead The slope at the zero is positive, so this zero of βd(g) corresponds to an unstable fixed point of Eq.(2.4). Thasymis mepantoticallys that a smtoalvleprositivy diffeeorrenntegatbehivaeviorsdeparofturtehefromcondtheuctancezero will. leadAs g moves from g > gc to asymptoticallygto. gThisc to sharp transition is the g < gc, mobilitmobily is losittyandedge.the wavefunction is truly localized. This sharp transition is the mobility edge.

– 8 – – 8 – of the order of ξ. Then one expects that for L ξ, the effective conductance becomes ! exponentially small: L/ξ g (L) e− (2.2) d ∼ For a small disorder, the medium is in a metallic state and the conductivity σ is independent of the sample size if the size is much larger than the mean free path, L l. Conductance ! is determined in this case just by the usual Ohm’s law and for a d-dimensional hypercube, we have: d 2 gd(L) = σL − (2.3) based on a simple dimensional argument. (Recall that in d = 3, g = σ(Area)/L σL.) ∼ The conductance g = g(a) at length scale a is a microscopic measure of the disorder. We see that g(L) may end up with one of the 2 very different asymptotic forms for L a. ! Here σ is rescaled so that g is dimensionless. Elementary scaling theory of localization assumes that g of a d-dimensional hypercube of size L satisfies the simplest differential equation of a renormalization group, where d ln g (L) β (g (L)) = d (2.4) d d d ln L that is, the β-function βd(g) depends only on the dimensionless conductance gd(L). Then the qualitative behavior of βd(g) can be analyzed in a simple way by interpolating it between the 2 limiting forms given by Eq.(2.2) and Eq.(2.3). For the insulating phase, (g 0), it → follows from Eq.(2.2) and Eq.(2.4) that : g O lim βd(g) ln (2.5) g 0 → g → c which is negative for g 0. For the conducting phase (large g), it follows from Eq.(2.3) → and Eq.(2.4) that :

lim βd(g) d 2 (2.6) g →∞ → − which is positive for d > 2. Assuming the existence of two perturbation expansions over the “coupling” g in the limits of weak and strong “couplings”, one can write corrections to Eq.(2.5) and Eq.(2.6) in the following form : g βd(g 0) = ln (1 + bdg + ) (2.7) → gc · · · α β (g ) = d 2 d + α > 0 (2.8) d → ∞ − − g · · · d

Assuming these and a smooth monotonous βd(g), it is easy to plot the β-function qualitatively for all g and d, as shown in Figure 2.

For d > 2, βd(g) must have a zero: βd(gc) = 0, where gc is the critical conductance. The slope at the zero is positive, so this zero of βd(g) corresponds to an unstable ﬁxed point of Eq.(2.4). This means that a small positive or negative departure from the zero will lead asymptotically to very diﬀerent behaviors of the conductance. As g moves from g > gc to g < gc, mobility is lost and the wavefunction is truly localized. This sharp transition is the mobility edge.

– 8 – of the order of ξ. Then one expects that for L ξ, the effective conductance becomes ! exponentially small: L/ξ g (L) e− (2.2) d ∼ For a small disorder, the medium is in a metallic state and the conductivity σ is independent of the sample size if the size is much larger than the mean free path, L l. Conductance ! is determined in this case just by the usual Ohm’s law and for a d-dimensional hypercube, we have: d 2 gd(L) = σL − (2.3) based on a simple dimensional argument. (Recall that in d = 3, g = σ(Area)/L σL.) ∼ The conductance g = g(a) at length scale a is a microscopic measure of the disorder. We see that g(L) may end up with one of the 2 very different asymptotic forms for L a. ! Here σ is rescaled so that g is dimensionless. Elementary scaling theory of localization assumes that g of a d-dimensional hypercube of size L satisfies the simplest differential equation of a renormalization group, where d ln g (L) β (g (L)) = d (2.4) d d d ln L that is, theofβthe-fuorncdertofionξ. Tβhen(oneg) edepxpectsendthatsforonL ly ξon, the theeffectivdimensionlese conductance becomess conductance g (L). Then d ! d exponentially small: the qualitative behavior of β (g) can be analyL/ξ zed in a simple way by interpolating it between d g (L) e− (2.2) d ∼ the 2 limitingFor a smformsall disorder,givthene mebdyiumEq.is in a(2.2metall)icanstatedandEqthe.(cond2.3uctivit). Fyorσ is ithendepeninsuldent ating phase, (g 0), it of the sample size if the size is much larger than the mean free path, L l. Conductance → ! follows fromis determinedEq.(2.2in)thandis caseEjustq.(by2.4the )usualthatOhm’s: law and for a d-dimensional hypercube, we have: g (L) = σLd 2 g (2.3) d lim β−d(g) ln (2.5) based on a simple dimensional argument.g(Rec0all that in→d = 3, gg=c σ(Area)/L σL.) → ∼ The conductance g = g(a) at length scale a is a microscopic measure of the disorder. We see that g(L) may end up with one of the 2 very different asymptotic forms for L a. which is negative for g 0. For the conducting phase (large!g), it follows from Eq.(2.3) Here σ is rescaled so →that g is dimensionless. and Eq.(2.4) thatElementary: scaling theory of localization assumes that g of a d-dimensional hypercube of size L satisfies the simplest differential equation of a renormalization group, where lim βd(g) d 2 (2.6) g d ln gd(L) → − β (g (L))→∞= (2.4) d d d ln L which is positiveFigurfored2: >The2.β-fuAssnctionuofmtheingdimthensione leessxiscondteuctannceceofg, βtdw(g)oversuspertlnurbg, forationdifferentexpanvalues sions over that is, thofe βg-fuandnctdion. Tβhe(gcas) depes endforsdon 2, there is always the “coupling” g in the lim→itsd −of weak→ ∞and strong “couplings”, one can write corrections to the 2 limitaingfixedformspoingivt (enthebyzeEq.ro (of2.2th) eanβd-fuEqnction.(2.3).atForg =thegcinsul(d))atingand thphasee slop, (ge at 0),theitfixed point is positive. → Eq.(2.5) andfolloEwsq.(fromFor2.6Eq.large)(2.2ind), andththe Esepq.(acfol2.4ings)lothatofwthein: figxedformpoints:for adjacent d values are approximately equal in ln g. g lim βd(g) ln g (2.5) g 0 → gc Figurβ→de(2:g The0)β-fu=nctionln of(1the+ dibdmgension+ le)ss conductance g, βd(g) versus(2.7)ln g, for different values which is negativThee forstateg of0. aFormethediumconiductins s→uppg osphaseedly(lardegetermingg),c itedfollboywsdisorderfrom Eq.(·at·2.3·microsc) opic distances → of g and d. The cases for d < 2, d = 2 and d = 3, 4, 5, 6 are shown. For g 0, βd(g) is linear in and Eq.(2.4of )ththate ord: er of typical spacing a between neighboring sites. Using g(a) as an initial value → ln g, and βd(g) d αFigurd2 as ge 2: The. Thereβ-functionis no fiofxethed pdioimntensionfor d le c2,e thg,eβrde(gis)alwversusays ln g, for different values and integratinβ (gg Eq.(2.4),lim)it=βisd(gdeas) y dto2 fi2nd its prop+ erties for theα 2 >c(2.6)ases0 : (2.8) d g → − → ∞ d a fixed→∞ point→(the− zeroofofgthande βd-fu. nctionThe casatesg for= gd( g , →the ∞conductivity−σ =−g(L)gL2 d tend· · ·s for L to a concstant non-zero which is positive for d > 2.c Assuming the existence of two p−erturbation expansions over → • For large d, the spacingsln gof, anthed βfidxed(g)→poi∞ndts for2 asadjgacent d. vTherealues arise noappfiroxeximatelyd point forequald 2, there is always the “coupvlinalue.g” g in the limits of weak and strong “couplings”, one can write corrections→ −to → ∞ a fixed point (the zero of the β-function at g = g (d)) and the slope at the fixed point is positive. Assuming thesEq.(2.5e) andForEq.(ga(2.6as))mo 2, β (g) must have Thea zestatero:αd βof(ag me) =dium0, whereis supposgedlisy thedetermincritiecdalbycondisorderductanatce.microscopic distances expdonFeorntiallyg βgc(,gsmβd(gal))isl:=lineard 2 in ln +g (sede Fcigurα >e 20), so we have thecfol(2.8)lowing approximation: ∼ d → ∞ − − g · · · d of the order of typical spacing a beL/twξeen neighboring sites. Using g(a) as an initial value The slope at the zero is positive, so this zero of1 βgd(g)1 gcorrg esponds to an unstable fixed point Assuming these and a smooth monotonous β (g), it is easy to plotgThedthe(Lβ)state-fucnctione−ofqualita mea- dium is supposedly determined(2.2)by disorder at microscopic distances and integratindβd(g)g Eq.(ln2.4), it is− ∼easy to find its properties(2.9)for the 2 cases : of Eq.(2.4).tivTelyhforis almel g andands, asthshoatwnainsmFigualre l2.positiv≈eν orgcn≈egatν givc e departure from the zero will lead For g(a) > g , theofcondtheuctivitorderyofσt=ypigc(alL)Lspacing2 d tenda sbfeortweLen neightoboria congnsstiantes.t nonUsing-zerog(a) as an initial value FForord >a 2,smβdal(g)lmdustisorder,have a zethro: eβdme(gc)d=ium0,c whereis ingc isathemecrititallcalicconstateductancande. the− conductivity σ is independent where 1/ν is positi•ve and is equal to the bracket part of Eq.(2.7) evaluated at gc. First we → ∞ asymptoticallyThe sloptoe atvtheerzeyrodisipffositiveree,nsot thbisezehroaviorsof β (g) coforretspheandondscondtoinantegratinunstableuctancefigxedE.pq.(oiAsnt2.4g),moit isveseasfyromto fignd>itsgcpropto erties for the 2 cases : of thconsidere samthpelecassize wherevealue.if thg(ae) >sizged. isSincemucβ (hg)larger> 0, thethanflow isthtoelarmege gan, sofreewe inptegrateath, L l. Conductance g < g , mobofiliEq.(ty2.4is). Tloshis tmeandans thatthea smwallapvositivefuenctionorc negativeideparsd trturulye fromlocalizethe zero willd. leadThis sharp transition! is2thde of the orderc of ξ. TheEqn.(2.4one) toefixndpecg(Lt).sForWthegatc(ana)forappctivfirstgc,inethtecondwgrateecondgeinguctancet Einuctivitq.(su2.4latin) usinybgσecgb=ehaomesg(viorL)L; t−hattendis, σs =for0.L to a constant non-zero asymisptoticallydeterminedto very diffeirnen•tthbeishaviorscaseof tjheustcondbuctancey the• . Asusualg movesOhm’sfrom→g∞>lagcwto and for a d-dimensional hypercube, → ∞ g < g , mEqobili.(t2.9y is)losunttiandl βdthe(g)wreacavefuhesnctiond is2tratulyg∗lo, calize!thenvdwalue.. eThisusesharpβd(g)tran= sdition2 i(Eq.(s the 2.6)) to reach large exponmobilentiallyitysmedge.all:wc e have: We see th−at the behavior of βd(g) −close to its zero determines the critical behavior at the mobility edge. transition. ThereL/isξ a Fshorarpg(atran) gc. Since βd(g) > 0, the flow is to large g, so we integrate Elementary scaling theory ofd lo2 calization assumes that g of a d-dimensional hypercube Eq.(g2.4d(L) )to=findσLg(−L). We can approximate the integral by(2.3)first integrating Eq.(2.4) using of size L satisfies the simplest diffewhererential1e/quν isationpositiofveaanrenord ismequalalizatitoonthegroupbrack,ewheret part of Eq.(2.7) evaluated at gc. First we Eq.(2.9) until βd(g) considerreaches dthe2 casat ge∗where, then wg(eause) >βgd(.g)Si=nced β2((Eq.(g) > 2.60, ))thetoflroewacish largeto large g, so we integrate based on a simple dimensional argument. (Recall that in d−= 3, g = σ(Area)/Lc σL−.) d Eq.(2.4) to fidndlngg(dL(L).)We can appro∼ximate the integral by first integrating Eq.(2.4) using The conductance g = g(a) at length scale a is a miβcdroscopic(gd(L)) =measure of the disorder. We (2.4) Eq.(2.9) until βddln(g)Lreaches d 2 at g∗, then we use βd(g) = d 2 (Eq.(2.6)) to reach large see that g(L) may end up with one of the 2 very different asymptotic forms −for L a. − that is, the β-function β (g) depends only on the dimensionless condu!ctance g (L). Then Here σ is rescaled so that g is dimensionlesds. – 9 – d the qualitative behavior of β (g) can be analyzed in a simple way by interpolating it between Elementary scaling theory of localization dassumes that g of a d-dimensional hypercube the 2 limiting forms given by Eq.(2.2) and Eq.(2.3). For the insulating phase, (g 0), it of size L satisfies the simplest differential equation of a renormalization group, where – 9 – → follows from Eq.(2.2) and Eq.(2.4) that :

d ln gd(L) g βd(gd(L)) = lim βd(g) ln (2.4) (2.5) g 0 → g d ln L → c that is, the β-functionwhicβhd(isg)ndepegativende sforongly on0.theFordimensionlesthe conductins gconphaseductan(larcgee ggd),(Lit).follThenows from Eq.(2.3) → the qualitative behaviandor Eq.(of βd2.4(g)) thatcan b:e analyzed in a simple way by interpolating it between the 2 limiting forms given by Eq.(2.2) and Eq.(2.3). Flimor βthed(g)insuldating2 phase, (g 0), it (2.6) g → − → follows from Eq.(2.2) and Eq.(2.4) that : →∞ which is positive for d > 2. Assuming the existence of two perturbation expansions over the “coupling” g in the limits of wegak and strong “couplings”, one can write corrections to lim βd(g) ln (2.5) g 0 → g Eq.(2.5) and Eq.(2.6→) in the followinc g form : which is negative for g 0. For the conducting phase (large g),g it follows from Eq.(2.3) → βd(g 0) = ln (1 + bdg + ) (2.7) and Eq.(2.4) that : → gc · · · αd lim ββdd((gg) d) =2d 2 + αd > 0 (2.6) (2.8) g → ∞ − − g · · · →∞ → − which is positive forAssd u>min2.gAssthesuemandingathsmoe eothxistmonotonence of touswo βpde(rtg),urbit ationis easyexpanto plotsionthes oβv-fuernction qualita- the “coupling” g in tivtheelylimforitsalofl gwandeak dand, as sshotrongwn in“coupFigulinregs2.”, one can write corrections to Eq.(2.5) and Eq.(2.6) inForthedfol>lo2,wβind(gg)formmust: have a zero: βd(gc) = 0, where gc is the critical conductance. The slope at the zero is positive,gso this zero of βd(g) corresponds to an unstable ﬁxed point of Eq.(2.4). Thβisd(meg an0)s th=atlna sm(1all+pbositivdg +e or )negative departure from(2.7)the zero will lead → gc · · · asymptotically to very diﬀerαent behaviors of the conductance. As g moves from g > gc to β (g ) = d 2 d + α > 0 (2.8) g < gdc, m→obili∞ty is los−t and− theg wa·v·efu· nctiond is truly localized. This sharp transition is the mobility edge. Assuming these and a smooth monotonous βd(g), it is easy to plot the β-function qualitatively for all g and d, as shown in Figure 2.

For d > 2, βd(g) must have a zero: βd(gc) = 0, where gc is the critical conductance. The slope at the zero is positive, so this zero of βd(g) corresp–onds8 – to an unstable ﬁxed point of Eq.(2.4). This means that a small positive or negative departure from the zero will lead asymptotically to very diﬀerent behaviors of the conductance. As g moves from g > gc to g < gc, mobility is lost and the wavefunction is truly localized. This sharp transition is the mobility edge.

– 8 – Now we have: g(a) g β (g g ) (d 2) − c (2.18) d ∼ c ≈ − g ! c " and for the critical exponent of localization length in Eq.(2.16), we get : 1 ν (2.19) ≈ d 2 − which may be considered as the ﬁrst term of the #-expansion from d = 2 (where # = d 2), − i.e., near “lower critical dimension” for localization.

We see from Eq.(2.17) that gc depends on αd, which is clearly model-dependent. For a Fermi gas with spin-1/2 particles in 3-dimensions (d = 3), α = π 2, so g 1/π2 [23]. Now we have: 3 − c ≈ Here we are interested in g ’s dependence on d for large d. c g(a) gc For tunneling in theβldand(g scapgce)to b(edfast 2)when abov−e the transition, it would mean that(2.18) Now we have: ∼ ≈ − gc the critical conductance g for large d 102 shoul! d be exp"onentially small. That is, the c ≈ g(a) gc and for thelandsccriticalape shoulexpdonenbe intβthdof(egloconcalizationgduc)cting(dphaselen2)gtheveninforE−q.(an 2.16exponen), wtialle geytsm: all microscop(2.18)ic ∼ ≈ − gc conductance g(a). Noting that gc 1 for d = !3., we see that" gc as a function of d should and for the critical exponent of localization∼ len1gth in Eq.(2.16), we get : decrease exponentially. A glance of Fiνgure 2 suggests that the zeros of βd (as a function (2.19)of ≈ d 2 d) are approximately equally spaced in ln g, im1−plying that gc should decrease exponentially which maasyabfue nctionconsidereof d.dIfaswthee letfirst termν of the #-expansion from d = 2 (where # = (2.19)d 2), ≈ d 2 − i.e., near “lower critical dimension” for localiz−ation. g 0, β (g) ln(g/g ), reproducing∆ ln gc =Eq.(ln gc(2.5d) ).ln gFc(ord + 1)g = k > 0, β (g) d (2.20)2, reproducing d which may be conc sidered as the first term of t−he #-expansion from d d= 2 (where # = d 2), → W→e see from Eq.(2.17) that gc depends on αd, which→is clearly∞ mode→l-depen−dent.− For Eq.(2.6). Thei.e., nearcritiw“loeccalanwerexpcriticalreonenss gc(didt)mincension”antermsbeofforexprgc(3)loc(forealssizdated=ion.3)in, terms of gc, 2 2 a Fermi gas with spin-1/2 particles in 3-dimensions (d = 3), α3 = π− , so gc 1/π [23]. We see from Eq.(2.17) that gc depends on αd, which is clearly model-depe≈ndent. For Here we are intergested0,inβdg(g)’s delnpendence(g/gc), reonprodd(ducingfor3)klargeEq.(d2.5. ). For g , βd(g) d 2, reproducing → c → gc(d)1 +e− g−c gc(3) →2 ∞ →(2.21)2 − a Fermi gas with spin-1/2 particνles=in 3-dimensions$ (d = 3), α3 = π− , so gc 1/π [23]. (3.2) For tunnelingEqin.(2.6the).landThescapcritiecalto expb1e fastonen(dwhent can2)babge oexprve theessedtransitionin terms,ofit gwcould, ≈ mean that Here we Aaresiminptelereglstedance inat gFiguc’s dreep2endencesuggests thaton2dkfor l1.argecTo db.e more specific, we like to know the critical conductance gc for large d −10 −shoul∼ d be exp1 +onengc tially small. That is, the For tuqunnantitatielingvinelyththee ldepandendscapenceetoof bgce≈onfastd. whenIt is liνkabe=lyovtheatthethtris ansitionspecific qu, ialitatit wouldve propmeanertythat (3.2) In thelan#-edscxpapansione should, fbore insmallthe con#du=ctidng phase2 > ev0,enonefor1solvan(dexpesonen2)forgc tiallthey smzeroall microscof theopβic-function the criticalis insecondnsitivuctancee to thegdefortailslarofgeanydparticular102 shoulmoddel.bIne−expthe onen−next tiallsectiony small., we shalThl exatamis,inethe conductance g(a). Notingcthat g 1−for d = 3., we see that g as a function of d should (3.1) to obtain g a=simp1/le2mo#. delTInhtotheeobntain#E-eq.(xpthansione3.2c quan),titatr≈feorcoivsmallve edeprs#ent=dencehed valueof2 >gc 0,onofonedc.thsolve cesriticalfor theexpzerooneof thnetβν-fu=nction1/# landscapce should be in the conduc∼ting phase even for−an exponentially small microscopic decrease exponen(ti3.1ally) t.o Aobtaiglancen gc =of1Fi/2gure#. Th2ensuggestsEq.(3.2)trhatecovtheres tzheerosvalueof βofd th(ase carifticalunctionexponeof nt ν = 1/# in Eq.(2.19condu). Wcteancseee gth(a).atNotinthisg isthata gscimpl1 eformdo=del3., thweatseerethatprogdc uceas asfuthnctione appofroprd shouliated features d) are app3.roEximatelyxpoinneEnq.(tequallia2.19lly Smal).y sWpeaclsedCee∼riithtinatcallnthgC,isimonductispalyisngimplancesthate mogdelc shoulthatdredecreprodaseucesethxpeonenapproprtiallyiate features decrease exponentially. A glance of Figure 2 suggests that the zeros of β (as a function of expected. as a function of dexp. Ifectewed.let d d) are appConsiderroximatelya latticeequallmodyelspofacd-dimeed innlnsionalg, imdisorderedplying thatsystegmc shoulstudiedd decreby Shapiraseoe[xp22].onenSintialce ly Comparing to Eq.(2.17Com), pitaringivWhatg toesEq.(α is22.17 the),1 criticalit/2.givesForα 2gcd ?1=/2.3,Fora dn=umerical3, a numericalanalyanalysissisofofthithiss we are interested in the asymptotic behavior of gc as a ≈function of d, where this qualitative as a function of dmo. Ifdelweyi∆leeldtlns ggc == l0n.255gc(d≈an) d lnν g=c(1d.68.+ 1)Thi= ks v>alu0e for the critical expon(2.20)ent ν is higher model yields g =featu0re.255shouldanbde unνivers=cal1, .the68.detThiails−ofs thevalumoedelfdooresthnote concritcernicuals. eWxpe monay eviewnt ν is higher c than that in Eq.(2.19) for d = 3 from the #-expansion. However, this value is compatible this model as a simpl∆ lne ingterp= olatiln g on(d)thatlnsatgis(fides+all1)the= generalk > 0 properties expected. (2.20) than that wine cEanq.(exp2.19ress)gfwithcor(d) dinthe=termsestimate3 frcofomgofc(3)1the.c25(for<#−-eνd 2, just as before. For (3.4) →c is insensitive to the details of any particulard=1mo : del. In the next section, we shall examine that is, gc aissimpexpleonenmodeltialtosolyobw etainAnderson,sobtainmallthe Thouless,fortquheandleargetitatsired Abraham,ivbdeeha,dep Fisher,andvieorn Phys.dencefor Rev.theof Bcri 22,gticc 3519onal condd(1980). uctance gc. Comparing (3.3) to the 3. ExponentiamicllyrosSmalcopiclpropCreitirtycal(1.3C) leonductads to thancese conductng/mobile condition (1.4) for large d. Recall 2a/ξ that the typical tunnelinνg probab–111ilit– y between 2 neighboring sites is Γ0 e− , w(3.4)e see 3. Exponentially Small Critical Conductances ∼ Consider a latticethmoat thdeel mobilof d-dimeity connsiditionalon→disisordered, for largesysted, m studied by Shapiro [22]. Since we are interested in the asymptotic behavior of gc as a function of d, where this qualitative so we obtainConsiderthe daelatticesiredmobehadel ofvidor-dimefornstheionalcridisorderedtical condsysteuctancem2(studd 1) iedgbcy. ShComparapiro [22ing]. Sin(c3.3e ) to the feature should be universal, the details of the modΓe0l >doees− not− concern us. We may view (3.5) microscopicweparrope ineterrteysted(1.3in)thleeadasymptotis to thcebcehaonviorducoftngg/mc as aobfuilenctioncondof itiond, whe(re1.4this) fquoralitativlargeed. Recall this model as a sthimplat is,e inanterpexpolationenontiallythatsmallsatistunnfiesealllingtheprobgeneralability canpropstillertieleads exptoectheted.conducting phase, feature should be universal, the details of the model does not concern us. We may view2a/ξ that the typical tunneling probability between 2 neighboring sites is Γ0 e− , we see this model as a sthimplat eis,inmobiliterpolatity foronthethatwasvatefunisficetsion.all the general properties expec∼ted. that the mobil3.1 AitySimplconeditiMoondelis, for large d, 3.2 Disorder with Percolation 3.1The AcorrSimplesponedinMog βdel-function in the Shapiro model is given by This above model correspond2(s tod a1)microscopically (i.e., at scales ! a) random medium The corresponding β-function in thΓe0Sh>apiero− mo−del is given by (3.5) but homogeβdn(egou) =s at(da larger1) s(cale.g + 1)Theln(1cos+mic1/gland) scape probably looks (3.1)random even at scales beyond a. To mimic− the− cosmic landscape, one may introduce additional randomness that is, an exponentially small βtdunn(g) =eling(d p1)rob(abilg + 1)itlyn(1can+ 1still/g) lead to the condu(3.1)cting phase, Recall that thatescalecritics laralgeconr thduanctan−a bucet−gstillc is smgivallenerbthany thethzee mroacroscof thope icβ-fuscalenction.L. ThisIt isscale can be that is, mobilieasy ttoysefore thatthethemacwasysterosvefuncopic,m hcorastion.iantenonrm-editrivialate, whicfixedh pisoinsomt eontimely fors redfe>rred2, tojustasasthebmeforeesoscopic. For scale. Let Recall that the critical conductance gc is given by the zero of the β-function. It is us introduce disorder at this scale. This may be mimicked by the more general Shapiro easy to see that the system has a non-trivial fixed point only for d > 2, just as before. For 3.2 Disorder with Percolatmodel thationactually considers localization in a macroscopically inhomogeneous medium, with percolation disorder. Percolation can be incorporated into the above model by intro- This above model corresdupciondng a sprobabto ailitmicroscy p (1 popic0)althatly a(it.eypical., atsitescisaloccesupi!edab)y arandrandom scmatterer,ediumor ≥– 11≥ – but homogeneous at a largerif a pathsremcale.ainsTheopen cosfor emicfficienlandt tunnelinscapg.eNoprobw, βabld(g)y loβgoks(g, p)randbecomomes aevfuenctionn at – 11 – → scales beyond a. To mimicof p asthewecll,osmic landscape, one may introduce additional randomness 1 p at scales larger than a but still smβg(allg, pe)r=than(d 1)th1e+m−acroscln(1 oppic) scale(g + 1)Ll.n(1This+ 1/g)scale can(3.6)be − p − − macroscopic, or intermediate, which is some time! s referred to "as the mesoscopic scale. Let us introduce disorder at this scale. This may be mimicked by the more general Shapiro model that actually considers localization in a macrosc– 12op– ically inhomogeneous medium, with percolation disorder. Percolation can be incorporated into the above model by intro- ducing a probability p (1 p 0) that a typical site is occupied by a random scatterer, or ≥ ≥ if a path remains open for efficient tunneling. Now, β (g) β (g, p) becomes a function d → g of p as well, 1 p β (g, p) = (d 1) 1 + − ln(1 p) (g + 1) ln(1 + 1/g) (3.6) g − p − − ! "

– 12 – is an unstable fixed point, implying that the transition between mobility and localization is a sharp one. r /ξ Consider a specific (enveloping) wavefunction ψ(r) e−| | at site r = 0 with a typical g 0, β (g) ln(g/g ), reproducing Eq.(2.5). ∼For g , β (g) d 2, reproducing → Λ ind the c→osmic landsccape (see Figure 1(b)). The wavefunction→see∞ms tod be c→omple−tely localized at the site if the localization length ξ a, where ξ measures the size of the Eq.(2.6). The critical exponent can be express"ed in terms of gc, site and a is the typical spacing between sites. They are determined by local properties 1 + g and ξ is known as the localization lengtν =h. With resonanc ce tunneling, ξ can be somewhat (3.2) bigger, although ψ(r) may still be expone1ntially(dsuppr2)egssced at distance a. At distance scales around a, the conductance goes like − − In the #-expansion, for small # = d 2 > 0, one solves for the zero of the β-function a/ξ g(a) ψ(−a) e− (1.3) (3.1) to obtain gc = 1/2#. Then Eq.(∼3.2| ) re|co∼ vers the value of the critical exponent ν = 1/# g 0, βd(g) ln(g/gc), reproducing Eq.(2.5). For g , βd(g) d 2, reproducing → in Eq.(→Note2.19th).at,Wfore asee ξth, Γat thψis(a)is2 a esimpl2a/ξ.e →mo∞del that→repro−duces the appropriate features Eq.(2.6). The critical exponen#t can0 b∼e|expr|ess∼ed−in terms of gc, Given g = g(a) (1.3) at scale a, what happens to g at distance scales L a ? That is, expected. 1 + g # how does g scale ? The νscalin= g theory cof g is well studied in condensed matter ph(3.2)ysics. It Comparing to Eq.(2.17),1 it (givd es2)gα2 1/2. For d = 3, a numerical analysis of this turns out that there is a critical−conductan− cce g≈c which is a function of d. If g(a) < gc, then modelgyi(Le)ldse gL/cξ = 00as.255L andandν th=e lan1.68.dscapThie is ans ivnsulatialue ngformethdium.e critTheicwalaveefuxpnctiononent ν is higher In the #-expansion∼ −, for→small #→=∞d 2 > 0, one solves for the zero of the β-function than thisattrulyinloEcalizeq.(2.19d. T)unnforelingd =−will3takfrome exptheonential#-elyxplongansionand.eternalHoweinvflationer, thcomis vesalueinto is compatible (3.1) to obtain gc = 1/2#. Then Eq.(3.2) recovers the value of the critical exponent ν = 1/# play. If g(a) > g , the conductivity is finite (g(L) (L/a)(d 2)) as L and the in Eq.(2.19with). thWee esstimateee that thofis c1is.25a s gc; given g(a) (1.3), we see that fast tunneling happens → a/ξ bigger, alth2(dough1) ψ(r) may still(dbe1)exponegn(taially) sψu(ppra) esseed− at distance a. At distance (1.3) when Γ0 > e− − , or gc = e− − ∼ | | ∼ (3.3) scales around a, the conductance goes like so we obtain the desired behavior for the 2critic2a/alξ conductance gc. Comparing (3.3) to the Note that, for a ξ, Γ0 ψ(aa) e− . that is, gc is exponentially small for large#d, and∼d|> |+∼1 a/ξ (1.4) microscopic propertyGiv(1.3en)gle=adg(as) to(1.3thg)(atae)scaleconξψ(adua,)whatctne−g/mhappeobns toilegcondat distitionance sc(ales1.4L) f(1.3)ora ?largeThat dis,. Recall ∼ | | ∼ # how does g scale ? The scaling theory of g is well studied in condensed matter phys2ics.a/ξIt that the typical tunneling probabν ilit12 y b2a/etξween 2 neighboring sites is(3.4)Γ0 e− , we see so theNotewavthefat,uncfortiona of ξth, Γe0univψe(ras)e cane−move. freely in the quantum landscape even for turns out# that there∼ | →is |a cri∼ tical conductance gc which is a function of d. If g(∼a) < gc, then that thane emobilxponeGivnittiyenallycgon=smallditig(a)gonGeneric(L/(1.3aξ).)isatSin, decayscaleforce ratethlargeae, whattoβ -funearestdnch,apption enhass to ag patosdd itis~iv t100ancee slopscealesat Lthe loa c?alThatizationis, so we obtain the desired behag(Lvi) oref−or the0crias ticL al condanductancethe landscgap.e isComparan insulatiingng(me3.3d#)ium.to thThee wavefunction transihotionw dopoines gt sc(implyale∼? ingTheneighboran→scalinun insgt athetheoryb→le landscape∞fixedof gp :isoinwt),ell thestudiedctraninsitcondenseion betwdeenmattether insulphysics.atingIt microscopic property (1.3) leisadtrsulytolothcalizee cond.duTunnctneg/mling wobilliletakconde2(expditiononen1) tial(1.4ly )longforanlarged eternald. Recinflationall comes into (localituzationrns out) phthasatethereand theis a ccrionticdualctincongductan(mΓ ob>ile)ceeg−phc whicas−ehisisshaarpfunction. We sofeedthat.(dIf 2)git(a)is gc, the condu0 ctivity is finite (g(L) (L/a) − ) as L and the L/ξ 2a/ξ that the typicalthetunvasg(nelintnL)essgeof−ptherobabcosmic0ilitas Lylanbedsctwandeapeneth(as2e nlanpearameighdscbaporeterizeingis andsiinsulatitebysaislarngΓge0me∼dd)ium.eth−atTheallo, wwwsaevefumobilseenction→ it∞y ∼ landsc→ ape is inMobile→a∞conducting/mobile phase. The wavefun∼ction is free to move. Around the that ththe mobilat is,evitanyencisonifextrthditipulyeononwloeacalizevntranisefunt,iallyfsoridction.tionTlargesunnpmalloinisellot,ding,calizetheretunnwilldist.eakalingThiecorexpsrelationponenmrobobtialilitabilleylyngthalloitlongythatwscananthedbloestillternalwwsavupeleadfinunatflationcthtitoone phasethecomto samestranconinptolesduition.ctiWnge phase, (d 2) the landplays.capIfegv(eary) >qugicc,klythe. Itconalsoductimviteansy isafinsemi-ite (classg(L)ical (desL/acrip) −tion) asof Lthe landsandcapetheis that is, mobility forshallthesee thwat,avefunat lengthc2(tion.dscale1) a, this correlation∼ length can be iden→tified∞ with ξ. That is, ξ inadequate.landscape plisainysathconde roleucΓ0ofting/mobi>the−correlationle− phasele.nThegth. wThavisefunis hoctiw,ongivis fenreeg(toa)mo(1.3ve.) atArou(3.5)scalendath, gecan grow transition larpoinget,fortherelargeisLa. correlation length that blows up at the phase transition. We that is,3.2an exDisordponenIntiallytheer witmobsmallilith tyPunnphercolatasele,ingtheprobionwaabilvefuitnctiony cananstilld/orleadthetoD3-branthe conesdu(andctingmopduhaseli) ,move downshallthe landsee thscapat, eInatttholengthisa papsiteser,calewiwthea,arguathsmisealcorrthatlere,latΛfor.ionAlartlengthgethatd, thscite,eancriticalbsome ideecnonoftifiedduitsctanneighwithce isξb.orinexpThatong esitesis,ntiallyξ small, that is, mobility forpltheays thwearolevefunof cthtion.e correlation length. This is how, given g(a) (1.3) at scale a, g can grow This abhaovevelargermodΛes,lsocorthereseffpectivonde sa intocreaasemicroscs, leadingoptoicaallargerly(d (i1)e.effec.,tivate cscriticalalesconduac)tanrandce om medium large for large L. gc e− − ! g . This happens until the condition (1.4) is no longer&satisfied and the D3-brane is stuck 3.2 Disordbut homogeer cwith nPeInercolatouthsisatpapionaer,largerwe arguesthatcale., forThelarge cosd, thmice criticallandconsducapctane cprobe is expabloneyntiallylookssmarlandl, om even at So it is not difficult for g(a) > gc; given g(a) (1.3), we see that fast tunneling happens scales beyond a. Twheno mimicΓ > ethe2(d c1)osmic, or landscap(d e,1) one may introduce additional randomness This above model corresponds to0 a microsc− − opicallygc(i.ee.−, at− scales ! a) random medium & but homogeat scaleneouss larat gea largerr thanscale.a buThet stillcosmicsmalllanderscapthane probtheablamyacrosclooksoprandic omscaleeveLn.atThis scale can be So it is not difficult for g(a) > g ; given g(a) (1.3d >), w+e 1see that fast tunneling happens (1.4) c – 4 – ξ scales bmaceyondrosacopic,. To mimicor intheterc2(mosmicd edi1) ateland, whicscape,h oneis sommaye intimetrodus creefadderreditiontoalasrandomnthe meesssoscopic scale. Let when Γ0 > e− − , or so the wavefunction of the universe can move freely in the quantum landscape even for at scaleuss laringetror thducean adisordbut stiller smat alltheisr thanscale.theThismacroscmaa opy icbescalemimicL. kThised bscyalethecanmorebe general Shapiro an exponentially small g(a). dSin>ce th+e1β-function has a positive slope at th(1.4)e localization macroscopic, or intermediate, which is some times referredξ to as the mesoscopic scale. Let model that actuallytransconsidition poinerts(implylocaliingzanationunstainble fixeda macroscpoint), theoptranicalslyitioninbhomogeetween theninsuleouatings medium, us introduce disorder at this scale. This may be mimicked by the more general Shapiro with percolatiso theonwadisordv(loefuncalictizationeonr.of)Pphthereascuneolatanivderionthese cancconandumocbvtineegfrein(melycoborinile)ptheoratphasquanedistusinhmarptolan. thWdsceeapseeabethateovveneitformois prdeciselelyby intro- model that actuallyanconsidexponeethnrtiesallyvloasctnsalimallesszationofg(thea). incSinosmicace macroscthlane dscβ-fuapncopet(asionicalphaslyarameainphomogeterizeositivde bsynlopeaouelaratsgemethde)dlothiucatalmizallo,ationws mobility ducing a probability p (1 p 0) that a typical site is occupied by a random scatterer, or with percolation disordtransietionr. Pevpeoinerncolattif(implythione≥wingacvanefun≥anbcunetioninstacborisleplofixedoratcalizeepddoin. inThit),tothesthmeobtranabilitsoyitvionalloe mobwsetdwtheeelenbwytheavinefinsultruno-ctiatingon to sample ducing ifa parobabpathiliremty(lopcaliai(1znsationthpop)e phenland0)asforesthatcapandeethevaffierytcienycpicalonquduictkclytuntins.itgItenelin(misalsooboccile)mg.ueanspiphNoeasdaew,sbiemi-sysahβarpclassdrand(g. ical)Wome desseescβcripthatgatterer,(gtion,itpis)ofprbortheecisecomlandely esscapae isfunction the vastn≥einssadequate.≥of the cosmic landscape (as parameterized by a large →d) that allows mobility if a pathof remp asainsweopll,en for efficient tunneling. Now, β (g) β (g, p) becomes a function even if the waInvefunthectmobion ilitis ylocalizephase,d.theThiwdsavmefuob→nctionilitygalloanwsd/orthethewaDve3-branfunctieons (andto sammopdule li) move of p as well, the landscapdowne vethery qulandickslycap. Ite talsoo a smite1eanswithpa asemi-smalclassler Λical. Atdesthatcripstionite, somof the eoflandits neighscapebisoring sites inadequate.β (gha, pv)e l=arger(dΛs, s1)o the1eff+ective−a inclnrease(1s, lepad)ing to a(glarger+ 1)efflecn(1tive+critical1/g)conductance (3.6) g 1− p p − − βg(g, p) =In(dthegcmob1). Thilitis1 y+happhpasens−e, utlnnhetil(1wthaevefupcon)nctiondition(gan(1.4+d/or1)) isln(1nothelon+Dge3-bran1r/gsat) isesfie(andd andmothedu(3.6)Dli)3-branmovee is stuck − p ! − − " down the lands!cape to a site with a sm"aller Λ. At that site, some of its neighboring sites have larger Λs, so the effective a increases, leading to a larger effective critical conductance gc. This happens until the condition (1.4) is no longer satisfied and the D3-brane is stuck – 4 – – 12 – – 12 –

– 4 – The picture from the scaling theory High CC sites

mobile

Sharp transition Low CC sites trapped, exponentially long lifetimes is an unstable ﬁxed point, implying that the transition between mobility and localization is a sharp one. Consider aTosp eciﬁcestimate(enve loptheing) transitionwavefuncti CCon ψ (inr) e r /ξ at site r = 0 with a typical ∼ −| | Λ in the cosmic landthescap cosmice (see F landscape:igure 1(b)). The wavefunction seems to be completely localized at the site if the localization length ξ a, where ξ measures the size of the • Tunneling from a positive CC site to a negative" CC site and a is sitethe ist ypignoredical spacing(CDL crunch).between sites. They are determined by local properties and ξ is kno•wnTunnelingas the fromlocalization a dS site tole anotherngth. WdSith site rwithesonan a ce tunneling, ξ can be somewhat bigger, althoughlargerψ CC(r) isma ignored.y still be exponentially suppressed at distance a. At distance scales around a, the conductance goes like

a/ξ g(a) ψ(a) e− (1.3) ∼ | | ∼ Note that, for a ξ, Γ ψ(a) 2 e 2a/ξ. # 0 ∼ | | ∼ − Given g = g(a) (1.3) at scale a, what happens to g at distance scales L a ? That is, # how does g scale ? The scaling theory of g is well studied in condensed matter physics. It turns out that there is a critical conductance gc which is a function of d. If g(a) < gc, then g(L) e L/ξ 0 as L and the landscape is an insulating medium. The wavefunction ∼ − → → ∞ is truly localized. Tunneling will take exponentially long and eternal inflation comes into play. If g(a) > g , the conductivity is finite (g(L) (L/a)(d 2)) as L and the c ∼ − → ∞ landscape is in a conducting/mobile phase. The wavefunction is free to move. Around the transition point, there is a correlation length that blows up at the phase transition. We shall see that, at length scale a, this correlation length can be identified with ξ. That is, ξ plays the role of the correlation length. This is how, given g(a) (1.3) at scale a, g can grow large for large L. In this paper, we argue that, for large d, the critical conductance is exponentially small,

(d 1) g e− − c &

So it is not difficult for g(a) > gc; given g(a) (1.3), we see that fast tunneling happens 2(d 1) when Γ0 > e− − , or a d > + 1 (1.4) ξ so the wavefunction of the universe can move freely in the quantum landscape even for an exponentially small g(a). Since the β-function has a positive slope at the localization transition point (implying an unstable fixed point), the transition between the insulating (localization) phase and the conducting (mobile) phase is sharp. We see that it is precisely the vastness of the cosmic landscape (as parameterized by a large d) that allows mobility even if the wavefunction is localized. This mobility allows the wavefunction to sample the landscape very quickly. It also means a semi-classical description of the landscape is inadequate. In the mobility phase, the wavefunction and/or the D3-branes (and moduli) move down the landscape to a site with a smaller Λ. At that site, some of its neighboring sites have larger Λs, so the effective a increases, leading to a larger effective critical conductance gc. This happens until the condition (1.4) is no longer satisfied and the D3-brane is stuck

– 4 – Since f < 1 for Λˆ < Ms, we see that a(Λˆ) > a(Ms), and a increases as Λˆ decreases. ˆ 4 Following Eq.(4.1), we see that there is a critical Λc = Λc , below which mobility stops and the wavefunction is truly localized. The universe lives exponentially long at any low Λ site, where Λ < Λc. BRIEF ARTICLE ˆ ˆ s ˆ ˆ s/d As an illustratiEstimateBRIEFon, one ARTICLEm aofy takcriticale f(Λ) C.C.(Λ/Ms) , so a(Λ)/a(Ms) (Λ/Ms)− . THE AUTHOR ≈ ≈ Taking a(Ms) assume1/Ms a andrandoms imilardistributionly :ξ 1/Ms, we have ∼ THE AUTHOR ∼ d/s fractionBRIEF of sites ARTICLE : Λˆ c d− Ms (4.5) 1 ≈ 1 4 Since the cosmological constanTHE AUTHORt Λ in 4-dimensional spacetime goes like Λˆ , s = 4 cor-

(1) (1) responds to a ﬂat Λd d=d=istribss(Λ(Λcc))//ξuti+1+1on. For this distribution, the critical 4-dim. cosmological

11 −1 4 (2) constant goes like∼ξ ∼ s(Λ )∼∼ ∼∼Λ −/1/4 (2) ξ s(Λs)s m Λss1 mss (3) For ﬂatd> distribution60 : d 4 (3) d>60 Λc d− Ms (4.6) (4)(1) T (n) " T0/nd>60 ∼ (4) T (n) " T0/n 30 20 so d 60 looks quite reasonab4 le. (Recall that Λˆ /M 10 to 10 .) In fact, a smaller (5) Γ = msT ! 0 c s − − (2) 4T (n) T /n (5) ∼ Γ = m T ∼ s or a larger d will lead to sa much4 too small Λc. We see that the vastness of the cosmic (3) ∼ Γ = msT (6) landscape, as parameterizen 1/Hsd by the large d, is crucial to this particular approach to the (6) n ∼ 1/Hs (4)cosmological constant problem.n ∼ 1/Hs

5. Discussions

As we have discussed, we like to avoid eternal inflation totally in the history of our universe. If the universe goes through inflation for more than one e-fold in a Λ meta-stable site, then eternal inflation is unavoidable, since tunneling of some patches of the inflating universe will still leave other patches to continue inflating. At a given site, the time to go through one e-fold of inflation is a Hubble time, i.e., ∆t = H 1 M /Λˆ 2 (recall that we have − ∼ P lanck defined Λˆ to be the scale of the cosmological constant). (If the universe at that site starts with some radiation, then it will take additional time to reach the inflationary stage. So the time the universe can1 stay in a Λ meta-stable site without eternal inflation may be longer than the Hubble time. Let us ignore the presence of radiation for the moment.) In the absence of resonance tunneling, the time δt that the universe stays at the Λ site is

δt 1/HΓ 1/H, so ete1 rnal inflation is unavoidable. ∼ 0 # To avoid eternal inflation in the quantum landscape, fast tunneling has to be faster for larger Λ. In the mobility phase, we can crudely estimate the time δt it takes for the universe to move from one site to another, using the simple formula : potential difference V equals the product of the current1 and the resistance 1/g. To get an order of magnitude ∼ estimate, we let V Λˆ, the current (a/δt)/L and g(L) = σLd 2 B(L/a)d 2 where σ ∼ ∼ − ∼ − is the rescaled conductivity and B is a dimensionless finite constant. This yields the time it takes the state to move from a site to a neighboring site δt (a/L)d 1(1/Λˆ), or ∼ − δt < 1/Λˆ < ∆t

So we see that the avoidance of eternal inﬂation in the early universe is probably automatic when the universe is in the conducting component of the landscape, i.e., when Λ > Λc. So it is reasonable to assume that no eternal inﬂation happens during the mobile phase.

– 15 – Is there eternal inflation ? • Gibbons-Hawking temperature makes it harder to trap the wave function of the universe at a classical local vacuum site in the landscape. • The presence of many moduli (tens to hundreds) makes it harder to trap the wavefunction. Furthermore, the moduli-axion potential has many low barrier directions. • The vastness of the landscape (both in the number of moduli- axions and in the number of metastable sites) enhances the decay of any metastable site. • Treating the cosmic landscape as a random potential, one can borrow the RG techniques developed in condensed matter physics to argue that there is no eternal inflation in the landscape, until the CC is exponentially small. Inﬂation in the landscape • Inflation takes place while the brane is moving in the landscape (and in the bulk). • It rolls, scatters, percolates (bouncing around), hops and tunnels, happens at a rate of 20 to 10,000 times per e-fold. So it is like “rapid old inflation”, but may look like slow roll. • It is like inflation with a random multi-dimensional potential. This will lead to random fluctuations in the CMBR Power spectrum.

Freese, Spolyar, Liu Huang . . . . . Davoudiasal, Sarangi, Shiu Sarangi, Shlaer, Shiu, Podolsky, Majumder, Jokela Chialva, Danielsson Watson, Perry, Kane, Adams H.T., Zhang, Xu Treating the landscape as a multi-dimensional random potential, inflation in such a potential will lead to random oscillations in the CMBR power spectrum.

WMAP

HT, J. Xu, Y Zhang, ArXiv 0812.1944 Summary

• The universe is freely moving in the stringy landscape when the vacuum energy density is above a critical value. Because of mobility, there is probably no eternal inflation. • When the universe drops below the critical C.C. value, it stays there. Its lifetime there is exponentially long. • The critical C.C. value is exponentially small compared to the string/Planck scale. • This scenario suggests an inflationary scenario in the landscape, which hopefully leads to observable fluctuations in the CMBR power spectrum.