Accelerated Expansion and AdS/CFT Thomas Hertog Institute for Theoretical Physics University of Leuven, Belgium with JB Hartle, SW Hawking arXiv:1205...... arXiv:1111.6090 arXiv:0803.1663 Can AdS/CFT be applied to cosmology? • Time-dependent Lorentzian AdS/CFT ! "cosmological" backgrounds Can AdS/CFT be applied to cosmology? • Time-dependent Lorentzian AdS/CFT ! "cosmological" backgrounds • Analytic continuation from Euclidean AdS ! wave function of perturbations Wave function of the universe: • Euclidean AdS/CFT [Horowitz & Maldacena '04] R exp(−IADS[h; χ]=~) = ZQF T [h; χ] • No-boundary State [Hartle & Hawking '83] Ψ[h; χ] = exp(−IdS[h; φ]=~) This talk: connect both ideas. Motivation • precise formulation of no-boundary state • singularity resolution • theoretical foundations of inflation • probabilities in eternal inflation No-Boundary State 3 R Ψ[ g; χ] = C δgδφ exp(−IdS[g; φ]=~) "The amplitude of configurations (3g; χ) on a three- surface Σ is given by the integral over all regular metrics g and matter fields φ that match (3g; χ) on their only boundary." [Hartle & Hawking '83] Saddle Point Limit 3 3 Ψ[ g; χ] ≈ expf[−IdS( g; χ)]=~g REAL COMPLEX REGULARITY 3 3 3 IdS( g; χ) = IR( g; χ) − iS( g; χ) WKB Interpretation 3 3 3 Ψ[ g; χ] ≈ expf[−IR( g; χ) + iS( g; χ)]=~g Lorentzian space-time evolution emerges if jrAIRj jrASj The predicted classical histories are pA = rAS and have conserved probabilities Phistory / exp[−2IR=~] No-boundary state ! prior on multiverse Complex Saddle points 2 2 i j ds = dτ + gij(τ; x)dx dx ; φ(τ; x) Hy CLASSICAL HISTORY TUNING AND REGULARITY SP Hx iγ Tuning at SP: φ(0) = φ0e Example Homogeneous/isotropic ensemble: Ψ[b; χ] 2 2 2 ds = dτ + a (τ)dΩ3; φ(τ) 1 2 2 V (φ) = Λ + 2m φ Classical evolution requires tuning: ! multiverse of FLRW backgrounds Probability measure Hy SP Hxr Hx 3π R 2 2 I(υ) = 2 C dτa[a (H + 2V (φ)) − 1] π IR(χ) ≈ − 4V (φ0) Probability measure Hy SP Hxr Hx 3π R 2 2 I(υ) = 2 C dτa[a (H + 2V (φ)) − 1] π IR(χ) ≈ − 4V (φ0) Including perturbations: 2 3 2 IR(δζn) ≈ +(/H )n (δζn) Inflation pA = rAS h^ ≈ mφ^ No-boundary state predicts inflation Was there a Beginning? large φ0 small φ0 Was there a Beginning? REAL COMPLEX Saddle points everywhereREGULARITY regular ! singularities in classical extrapolation no obstacle to asymptotic predictions No-Boundary State: ADS form Complex Saddle points Hy SP Hxr Hx Complex Saddle points Hy SP Hxr Hx 1 e.g. a(τ) = H sin(Hτ); φ(τ) = 0 horizontal part: 2 2 1 2 2 ds = dτ + H2 sin (Hτ)dΩ3 vertical part: 2 2 1 2 2 ds = −dy + H2 cosh (Hy)dΩ3 Complex Saddle points Hy SP Hxr Hx Representations Different representation of same solution: Hy 2 Hxa SP Hxr Hx Representations Hy 2 vertical part: Euclidean ADS Hxa SP Hxr Hx 2 2 1 2 2 ds = −dy − H2 sinh (Hy)dΩ3 Representations Hy 2 With matter: Euclidean ADS domain wall Hxa SP Hxr Hx Hy • V acts as effective −V in AdS regime because the signature of the complex saddle point metric varies in the τ-plane, Veff = −V • Somewhat reminiscent of Domain Wall/Cosmology correspondence in SUGRA. [Cvetic; Skenderis,Townsend, Van Proeyen] • Realized here at the level of the wave function of the universe which involves a given complexified theory. Saddle Point Action IdS(b; χ) = Iv + Ih Hy • Contribution from vertical part: 2 R R 3 Iv = IdS[g; φ] = −IAdS( g;~ χ~) + Sct(a; χ~) Hxv a SP Hxr Hx R where IAdS is finite when y ! 1. φ = αe−λ−y + βe−λ+y Saddle Point Action IdS(b; χ) = Iv + Ih Hy • Contribution from horizontal part: 2 R Ih = I[g; φ] = −Sct(a; χ~) + iSct(b; χ) hHxa SP Hxr Hx and no finite contribution. Asymptotic Structure Expanded in small u ≡ eiτ = e−y+ix, −1 (2) 2 gij(u; Ω) = 4u2 [hij(Ω) + hij (Ω)u (−) λ− (3) 3 +hij (Ω)u + hij (Ω)u + ···] λ− φ(u; Ω) = u (α(Ω) + α1(Ω)u + ···) λ+ +u (β(Ω) + β1(Ω)u + ···) 3 p 2 with λ± ≡ 2[1 ± 1 − (2m=3) ] and arbitrary `boundary values' (hij; α). R 3 3 Ih = h I[g; φ] = −Sct(a; g;~ χ~) + iSct(b; g;~ χ) A universal AdS/dS connection follows directly from an asymptotic analysis Saddle Point Action Hy 3 R 3 3 IdS( g; χ) = −IAdS( g;~ χ~) + iSct( g; χ) with 2 3 R p3 R p3 (3) Sct( g; φ) = a0 g + a1 gR + ··· Hxa SP Hxr Hx No-Boundary State: ~ R ~ ~ Ψ[b; h; χ] = expf[+IAdS(h; χ~) − iSct(b; h; χ)]=~g Two Sets of Saddle points φ(υ) = φ(υ2) = χ, g~ij(υ) =g ~ij(υ2) Hy Hy 2 SP Hxr Hx 2 COMPLEX φ0 REAL φ0 Hxa SP Hxr Hx Accelerated Expansion and AdS/CFT Holographic Cosmology • No-boundary State: ~ R ~ ~ Ψ[b; h; χ] = expf[+IAdS(h; χ~) + iSct(b; h; χ)]=~g • Euclidean AdS/CFT: R ~ ~ exp(−IAdS[h; χ~]=~) = ZQF T [h; χ~] ~ 1 ~ Ψ[b; h; χ] = ~ expf[iSct(b; h; χ)]=~g ZQF T [h;χ,~ ] l with complex source χ~ and UV cutoff ∼ b Cosmology with AdS Gravity Using AdS/CFT we evaluate the holographic no-boundary state, R ~ ~ exp(−IADS[h; χ~]=~) = ZQF T [h; χ~] There are two sets of saddle points Cosmology with AdS Gravity Using AdS/CFT we can evaluate the holographic no-boundary state, R ~ ~ exp(−IADS[h; χ~]=~) = ZQF T [h; χ~] There are two sets of saddle points A: Real χ~ ! real Euclidean AdS domain walls Hy SP Hx ~ R ~ ~ Ψ[b; h; χ] ≈ expf[+IAdS(h; χ~) − Sct(b; h; χ)]=~g Cosmology with AdS Gravity Using AdS/CFT we can evaluate the holographic no-boundary state, R ~ ~ exp(−IADS[h; χ~]=~) = ZQF T [h; χ~] There are two sets of saddle points B: Complex χ~ ! complex Euclidean domain walls Hy 2 SP Hx ~ R ~ ~ Ψ[b; h; χ] ≈ expf[+IAdS(h; χ~) − iSct(b; h; χ)]=~g Saddle points correspond to inflationary universes Remarks ~ R 3 p~ ZQF T [h; χ~] = hexp d x hχ~Oi • The dependence of Z on the external sources provides a cosmological measure on the space of configurations (b; h;~ χ). • AdS/CFT implements no-boundary condition of regularity in saddle point limit • Scale factor evolution arises as inverse RG flow • Physical interpretation of counterterms in AdS What models? What models? ~ R 3 p~ ZQF T [h; χ~] = hexp d x hχ~Oi • Lorentzian stability criteria too stringent • Unitary physics in given classical background • Ensemble of inflationary backgrounds 1 2 2 VAdS(φ) = −Λ − 2m φ Classical evolution constrains V Singularity Resolution (for Gary) Singularity Resolution (for Gary) ~ 1 ~ Ψ[b; h; χ] = ~ expf[iSct(b; h; χ)]=~g ZQF T [h;χ,~ ] Given an asymptotic structure one can probe the deep interior by taking the RG flow all the way down to the IR. Singularity described by IR fixed point. Conclusion The no-boundary proposal and Euclidean AdS/CFT are intimately connected. • A wave function defined in terms of a gravitational theory with a negative cosmological constant Λ can predict expanding universes with an ‘effective' positive cosmological constant −Λ. • The Euclidean AdS/CFT dual provides a more precise, `holographic' formulation of the semiclassical no-boundary state. Applications: • Holographic calculation of CMB correlators. • Get a better handle on eternal inflation? Euclidean Eternal Inflation [Hartle, Hawking & TH, in progress] Proposal: replace the inner region of eternal inflation by a dual CFT on the threshold surface at υi: Hy 2 ei i Hxa Hxr Hx r • IR CFT with a deformation set by threshold φEI. (similar to [Maldacena '10]) • < O > on inner boundary replaces regularity condition at origin..