A new twist on the No-Boundary State

Transcription

1 A new twist on the No-Boundary State Stephen Hawking s 70th Birthday Conference January 7, 2012 Thomas Hertog Institute for Theoretical Physics University of Leuven, Belgium arxiv: arxiv: arxiv:

2 Stephen s recent work... No-boundary cosmology:

3 Stephen s recent work... No-boundary cosmology: 90 s: bottom-up approach the instanton

4 Stephen s recent work... No-boundary cosmology: 90 s: bottom-up approach the instanton 2002: top-down approach prior on multiverse

5 Stephen s recent work... No-boundary cosmology: 90 s: bottom-up approach the instanton 2002: top-down approach prior on multiverse The history of the universe depends on the question that is asked. local observations: conditional, coarse-grained

6 Stephen s recent work... No-boundary cosmology: 90 s: bottom-up approach the instanton 2002: top-down approach prior on multiiverse The history of the universe depends on the question that is asked. local observations: conditional, coarse-grained 2007: dominated by histories with many efolds eternal inflation 2010: local observations in eternal inflation; P (CMB 1 )/P (CMB 2 ) exp[π(1/v 1 ei 1/V 2 ei )]

7 Stephen s recent work... No-boundary cosmology: 90 s: bottom-up approach the instanton 2002: top-down approach prior on multiiverse The history of the universe depends on the question that is asked. local observations: conditional, coarse-grained 2007: dominated by histories with many efolds eternal inflation 2010: local observations in eternal inflation; P (CMB 1 )/P (CMB 2 ) exp[π(1/v 1 ei 1/V 2 ei )] Higher order corrections? 2011: dual formulation no-boundary state?

8 Outline No-Boundary state Its AdS representation dual formulation Implications for eternal inflation

9 No-Boundary Wave Function Ψ[b, h, χ] = C δgδφ exp( I[g, φ]) The amplitude of configurations (b, h, χ) on a threesurface Σ is given by the integral over all regular metrics g and matter fields φ that match (b, h, χ) on their only boundary. [Hartle & Hawking 83] Motivation: analogy w/ ground state wave function

10 Semiclassical Limit Ψ(b, h, χ) e exp{[ A e(b, h, χ)]/ } with A e = I(b, h, χ) + I (1) (b, h, χ) + Extremal geometries generally complex: I(b, h, χ) = I R (b, h, χ) + is(b, h, χ)

11 WKB Interpretation Ψ(b, h, χ) exp{[ I R (b, h, χ) + is(b, h, χ)]/ } The semiclassical wave function predicts Lorentzian, classical evolution in regions of superspace where [Hawking 84, Grischuk & Rozhansky 90] A I R A S The resulting coarse-grained classical histories of the universe are given by the integral curves of S L : p A = A S and have conserved probabilities P history exp[ 2I R / ] no-boundary measure: prior on multiverse.

12 Complex Saddle points ds 2 = dτ 2 + g ij (τ, x)dx i dx j, φ(τ, x) Hy TUNING AND REGULARITY CLASSICAL HISTORY SP Regularity at SP: g ij (0) 0, φ(0) 0 At final boundary: g ij (υ) = b 2 h ij, φ(υ) = χ Tuning at SP: φ(0) = φ 0 e iγ,...

13 Example Consider homogeneous/isotropic multiverse: ds 2 = dτ 2 + a 2 (τ)dω 3, φ(τ) V (φ) = Λ m2 φ 2 Classical evolution requires tuning of φ(0) = φ 0 e iγ at the South Pole of the saddle point: ( m 2 c = 9/4H 2 ) A I R A S at large scale factor 1-parameter set of FLRW backgrounds

14 Saddle point Action Hy SP r with r π/2 for φ 0 0 I(υ) = 3π 2 C(0,υ) dτa[a2 (H 2 + 2V (φ)) 1] I R tends to a constant along vertical part probability measure on classical histories.

15 Inflation Background histories: p A = A S no-boundary state predicts inflation

16 Origin Background histories: p A = A S large φ 0 small φ 0

17 Part II: Towards a dual description of the no-boundary state

18 Complex Saddle points Lorentzian histories always lie on asymptotically vertical curves in complex τ-plane: Hy SP r

19 Complex Saddle points Lorentzian histories always lie on asymptotically vertical curves in complex τ-plane: Hy SP r e.g. pure desitter: a(τ) = 1 H sin(hτ) horizontal part: ds 2 = dτ H 2 sin 2 (Hτ)dΩ 2 3 vertical part: ds 2 = dy H 2 cosh 2 (Hy)dΩ 2 3

20 Complex Saddle points Lorentzian histories always lie on asymptotically vertical curves in complex τ-plane: Hy SP r

21 Representations of Saddle points Hy 2 a SP r

22 Representations of Saddle points Hy 2 a SP r vertical part: Euclidean ADS ds 2 = dy 2 1 H 2 sinh 2 (Hy)dΩ 2 3

23 Representations of Saddle points Hy 2 a SP r With matter: Euclidean ADS domain wall

24 Representations of Saddle points Hy 2 a SP r With matter: Euclidean ADS domain wall signature complex saddle pt metric varies in τ-plane

25 Representations of Saddle points Hy 2 a SP r With matter: Euclidean ADS domain wall Domain Wall/Cosmology correspondence in SUGRA [Cvetic 96; Skenderis, Townsend, Van Proeyen 07] realized here through universe s quantum state

26 Representations of Saddle points Hy 2 a SP r With matter: Euclidean ADS domain wall Domain Wall/Cosmology correspondence in SUGRA [Cvetic 96; Skenderis, Townsend, Van Proeyen 07] V eff AdS = Λ V

27 Representations of Saddle points Hy 2 a SP r Vice versa: starting from Euclidean AdS SUGRA we predict Lorentzian asymptotically desitter universes.

28 Representations of Saddle points Hy 2 a SP r Does the AdS/dS connection hold independently of the saddle pt approximation? What happens along horizontal branch?

29 Asymptotic Analysis Asymptotic expansion of metric and fields in small u e iτ = e y+ix [Skenderis,...]: g ij (u, Ω) = 1 4u 2 [h ij (Ω) + h (2) ij (Ω)u2 +h ( ) ij (Ω)uλ + h (3) ij (Ω)u3 + ] φ(u, Ω) = u λ (α(ω) + α 1 (Ω)u + ) with λ ± 3 2 [1 ± 1 (2m/3) 2 ] +u λ +(β(ω) + β 1 (Ω)u + ) and (arbitrary) boundary values (h ij, α). AdS/dS connection

30 Action integral Action integral along horizontal part: I h = h I[g, φ] = +S ct(b, h, χ) is ct (b, h, χ) and no finite contribution. S ct = a 0 h + a1 hr (3) + a 2 hφ 2 Action integral along vertical part: I v = v I[g, φ] = IR AdS (h, χ) S ct(b, h, χ) where I R AdS is finite when a. starting from AdS SUGRA, horizontal branch amounts to adding the usual counterterms. I(b, h, χ) = I R AdS (h, χ) is ct(b, h, χ)

31 Towards a holographic dual Ψ(b, h, χ) exp{[+i R AdS (h, χ) + is ct(b, h, χ)]/ } AdS/CFT [Maldacena,Witten,...]: exp( I R ADS [h, χ]/ ) = Z QF T [h, χ] = exp d 3 x h χo ds/cft dual formulation of NBWF: Ψ(b, h, χ) 1 Z QF T [h, χ,ɛ] exp{[is ct(b, h, χ)]/ } where UV cutoff ɛ 1/b

32 Remarks Ψ(b, h, χ) 1 Z QF T [h, χ,ɛ] exp{[is ct(b, h, χ)]/ } Z provides measure on configurations (b, h, χ). Scale factor evolution as inverse RG flow [Strominger 02] Physical interpretation of counterterms in AdS Coarse-graining over UV modes at finite scale factor [see also Vilenkin 11] no-boundary condition of regularity implemented

33 Part III: Eternal Inflation

34 Part III: Eternal Inflation Can the dual formulation of no-boundary state be applied to regime of eternal inflation only?

35 Euclidean Eternal Inflation [Hartle, Hawking & TH, in progress] Replace inner region of eternal inflation by dual CFT on its boundary: Hy 2 i a r

36 Euclidean Eternal Inflation [Hartle, Hawking & TH, in progress] Replace inner region of eternal inflation by dual CFT on its boundary: ei r IR CFT with deformation given by φ = φ EI. (similar to [Maldacena 10]) < O > on inner boundary replaces regularity at SP. remaining saddle point with inner boundary

37 Euclidean Eternal Inflation [Hartle, Hawking & TH, in progress] Replace inner region of eternal inflation by dual CFT on its boundary: ei Ψ(b, h, χ) 2 1 Z QF T [φ EI,h e,ɛ] 2 exp{[+2ĩr AdS / } r

38 Conclusion EAdS/dS connection in no-boundary cosmology dual description in terms of relevant deformations of the CFTs that occur in AdS/CFT. Reinterpretation of eternal inflation

39 Conclusion EAdS/dS connection in no-boundary cosmology dual description in terms of relevant deformations of the CFTs that occur in AdS/CFT. Reinterpretation of eternal inflation Happy Birthday Stephen!