Is there eternal inflation in the cosmic landscape ? Henry Tye Cornell University 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 (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 (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.