Instability of De Sitter Spacetime and Eternal Inflation Hiroki - PowerPoint PPT Presentation

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary Instability of De Sitter Spacetime and Eternal Inflation Hiroki Matsui Department of Physics, Tohoku University, Sendai, Miyagi 980-8578, Japan, hiroki.matsui.c6@tohoku.ac.jp Based on: H, Mastui and F, Takahashi ,....... arXiv:1806.10339, arXiv:1807.1193....... August 11, 2018

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary Cosmological Inflation Inflation solves several problems in big bang cosmology: Horizon, flatness, magnetic-monopole problem. And, it precisely matches cosmological observations of CMB, etc. But, we do not know the origin of the inflation and the shape of the inflaton potential. Additionally, most inflation models are thought to be eternal. Broadly speaking, there are three types for eternal inflation: old, new and chaotic inflation. [Guth, J. Phys. A40, 6811 (2007), hep-th/0702178]

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary Eternal Inflation

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary Eternal Inflation The vacuum decay rate in de Sitter spacetime is given by the Hawking-Moss instanton B ≈ 8 π 2 V ( ϕ max ) Γ decay = A exp (− B ) , 3 H 4 However, the inflation increase the number of the Hubble-horizon patches exponentially N patch ∼ exp ( 3 Ht ) Thus, the number of patches continuing the inflation grows exponentially with Hubble time N inflation ∼ N patch · ( 1 − Γ decay ) Ht ∼ e Ht · { 3 + ln ( 1 − Γ decay ) } ≫ O ( 1 )

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary Eternal Inflation and Multiverse Multiverse from Andrei Linde, Stanford University

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary String Landscape

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary Eternal Inflation and Multiverse

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary Anthropic Principle String Landscape + Eternal Inflation = ⇒ Finetuning Problem

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary Swampland Conjectures Swampland De Sitter Conjecture | ∇ V | M P > c V [G. Obied, H. Ooguri, L. Spodyneiko, and C. Vafa, 1806.08362] where c is a numerical constants of order unity, but its precise value depends on details of the compactification. Swampland De Sitter Conjecture = ⇒ No dS vacuua or minima √ The slow-roll inflation requires c < 2 and the CMB measurements show ϵ < 0 . 0045 which leads to c < 0 . 094. ) 2 ϵ ≃ M 2 ( ∇ V c ϵ 1 / 2 > P √ ⇒ 2 V 2

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary Eternal Inflation vs Swampland Conjectures • Swampland De Sitter Conjecture forbids de Sitter vacua or minima. • The eternal old/hilltop inflation is impossible for this criteria. • The chaotic eternal inflation is only possible for c ∼ O ( 0 . 01 ) and 1 / D ∼ O ( 0 . 01 ) , and that the Hubble parameter H inf during the eternal inflation is parametrically close to the Planck scale, and we √ get a new constraint 2 π c ≲ H inf / M P < 1 / 3. [ H, Mastui and F, Takahashi , arXiv:1807.1193]

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary Anthropic Principle ? String Landscape + Eternal Inflation = ⇒ Finetuning Problem

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary De Sitter Spacetime (FLRW) Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetimes ds 2 = − dt 2 + a 2 ( t ) δ ij dx i dx j Einstein’s field equation G µν + Λ g µν = 8 π G N T µν . with no matter and lead to de Sitter solution ⇒ H 2 = Λ ⇒ ˙ ⇒ a ( t ) = e H · t G µν + Λ g µν = 0 ⇐ H = 0 ⇐ 3 ⇐ The most famous examples of the de Sitter spacetime are cosmic inflation and dark energy, Λ ∼ V ( ϕ ) .

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary De Sitter Spacetime Instability The quantum fluctuations on de Sitter spacetime δϕ ≈ H / 2 π. The Einstein’s equation G µν + Λ g µν = 8 π G N ⟨ T µν ⟩ . The de Sitter instability from quantum backreaction G µν + Λ g µν ≃ H 4 = ⇒ dS spacetime may be destabilized M 2 P [Mottola ’85 ’86, Tsamis, Woodard ’93, Abramo, Brandenberger, Mukhanov ’97, Goheer, Kleban, Sussking ’03, Polyakov ’07, Anderson, Mottola ’14, Dvali, Gomez, Zell ’17, Markkanen ’16 ’17]

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary Evaporation of DS Spacetime The de Sitter entropy S dS = A A = 4 π H − 2 4 G , The de Sitter thermodynamics like black hole P = ˙ ρ vac 2 ˙ HM 2 dU = TdS − PdV = 3 H , ⇒ ( 4 π ( 4 π ( 4 πρ vac ) ) ) TdS = HM 2 dU = − d , P d , PdV = − p vac d 3 H 3 H 2 3 H 3 The de Sitter thermodynamics P = ˙ ρ vac 2 ˙ HM 2 ρ vac ≃ O ( H 5 ) 2 ˙ HM 2 P = O ( H 4 ) 3 H , ˙ = ⇒ which shows the time-dependent cosmological constant. [Spradlin, Strominger, Volovich ’01, Padmanabhan ’03, Markkanen ’17]

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary De Sitter Instability from Quantum Conformal Anomaly We focus on quantum backreaction from conformal massless fields and trace of EMT is classically zero. T µ µ = 0 . But, the vacuum expectation values of EMT is non-zero T µ ⟨ ⟩ ̸ = 0 = ⇒ conformal anomaly µ We persist in the semiclassical approach of the gravity 1 G µν + ρ Λ g µν + a 1 H ( 1 ) µν + a 2 H ( 2 ) µν + a 3 H ( 3 ) µν = ⟨ T µν ⟩ 8 π G N The semiclassical gravity has no unitary problem about the gravitational S-matrix since the gravity is not quantized.

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary Quantum Conformal Anomaly Quantum trace of energy momentum tensor [ 1 ∫ m 2 + Cm 2 − 5 C 2 m 4 D 2 D ′ + D 2 ) ⟨ T µ ⟩ dkk 2 ( = µ 8 ω 5 32 ω 7 4 π 2 C ( η ) ω k k k − Cm 2 D ′′′ + 4 D ′′ D + 3 D ′ 2 + 6 D ′ D 2 + D 4 ) ( 32 ω 7 k + C 2 m 4 28 D ′′ D + 21 D ′ 2 + 126 D ′ D 2 + 49 D 4 ) ( 128 ω 9 k − 231 C 3 m 6 + 1155 C 4 m 8 D 4 ] D ′ D 2 + D 4 ) ( 256 ω 11 2048 ω 13 k k = − m 4 ϵ + 1 − γ + ln 4 π + ln µ 2 [ 1 ] 32 π 2 m 2 + m 2 D 2 1 2 D ′ + D 2 ) D ′′′ − D ′ D 2 ) ( ( − 192 π 2 C 960 π 2 C 2

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary Quantum Conformal Anomaly Quantum trace of energy momentum tensor ⟨ T µ ⟩ ⟨ T µ ⟩ anomaly = lim µ µ ren m → 0 1 D ′′′ − D ′ D 2 ) ( = − 960 π 2 C 2 1 [( R µν R µν − 1 ) ] 3 R 2 = − + □ R 2880 π 2 1 ( E − 2 ) − 1 = 3 □ R + 270 ( 4 π ) 2 □ R 360 ( 4 π ) 2 1 1 = 360 ( 4 π ) 2 E − 180 ( 4 π ) 2 □ R where m → 0 for conformal massless fields

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary Quantum Conformal Anomaly The general form of conformal anomaly for four dimensions ( E − 2 ) T µ ⟨ ⟩ = bF + b ′ + b ′′ □ R 3 □ R µ = bF + b ′ E + c □ R E is Gauss-Bonnet invariant term and F is the square of the Weyl tensor. ∗ R µνκλ = R µνκλ R µνκλ − 4 R µν R µν + R 2 E ≡ ∗ R µνκλ F ≡ C µνκλ C µνκλ = R µνκλ R µνκλ − 2 R µν R µν + R 2 / 3 , [Capper, Duff ’74, Deser, Duff, Isham ’76, Duff ’77]

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary Quantum Conformal Anomaly The dimensionless parameters b , b ′ and c are given by: 1 b = − 120 ( 4 π ) 2 ( N S + 6 N F + 12 N G ) 1 ( N S + 11 ) b ′ = 2 N F + 62 N G 360 ( 4 π ) 2 1 c = − 180 ( 4 π ) 2 ( N S + 6 N F − 18 N G ) , where we consider N S scalars (spin-0), N F Dirac fermions (spin-1/2) and N G abelian gauge (spin-1) fields. MSSM : N S = 104 , N F = 32 , N G = 12 SM : N S = 4 , N F = 24 , N G = 12 Curent Universe : N S = 0 , N F = 0 , N G = 1

Introduction Eternal Inflation and Multiverse De Sitter Spacetime Instability Conclusion and Summary Quantum Backreaction The differential equation derived from Einstein equations . ... .. .. . ) . . 2 2 4 2 2 a a a 2 + 2 a a a 3 + 2 b ′ a 1 1 a Λ ( − − a 4 − 3 + a 2 = 0 . a 2 a 3 c 8 π c G 8 π c G The differential equation with respect to Hubble parameter, H 2 − 2 b ′ 1 Λ 1 6 H 2 ˙ c H 4 − 8 π c G H 2 = 0 H + 2 H ¨ H − ˙ 3 + 8 π c G For the relatively small cosmological constant 8 b ′ Λ/ 3 ≪ M P , √ √ Λ 1 H C ≃ H Q ≃ 3 , 16 π b ′ G