Holographic self-tuning of the cosmological constant Francesco - PowerPoint PPT Presentation

Holographic self-tuning of the cosmological constant Francesco Nitti Laboratoire APC, U. Paris Diderot Gravity and Cosmology 2018 YITP Kyoto, 16-02-2018 work with Elias Kiritsis and Christos Charmousis, 1704.05075 Holographic self-tuning of the cosmological constant – p.1

Outline • Introduction • AdS/CFT mini-review • Holographic model building • The self-tuning model • Emergent 4d gravity • Graviton propagator • Scales of brane-world gravity • Linear perturbations and stability • Conclusion and outlook Holographic self-tuning of the cosmological constant – p.2

Outline • Introduction • AdS/CFT mini-review • Holographic model building • The self-tuning model • Emergent 4d gravity • Graviton propagator ( ) • Scales of brane-world gravity • Linear perturbations and stability • Conclusion and outlook Holographic self-tuning of the cosmological constant – p.2

Introduction: the CC and QFT The Cosmological Constant (CC) problem arises as a clash between classical GR and QFT (in the modern effective FT sense). In classical GR: G µ ν = Λ 0 g µ ν + 8 π G N T µ ν Holographic self-tuning of the cosmological constant – p.3

Introduction: the CC and QFT The Cosmological Constant (CC) problem arises as a clash between classical GR and QFT (in the modern effective FT sense). In classical GR: G µ ν = Λ 0 g µ ν + 8 π G N T µ ν QFT: the source of semiclassical gravity becomes ⟨ T µ ν ⟩ . In flat space QFT with unbroken Lorentz symmetry: E vac ≈ M 4 ⟨ T µ ν ⟩ = E vac η µ ν , Holographic self-tuning of the cosmological constant – p.3

Introduction: the CC and QFT The Cosmological Constant (CC) problem arises as a clash between classical GR and QFT (in the modern effective FT sense). In classical GR: G µ ν = Λ 0 g µ ν + 8 π G N T µ ν QFT: the source of semiclassical gravity becomes ⟨ T µ ν ⟩ . In flat space QFT with unbroken Lorentz symmetry: E vac ≈ M 4 ⟨ T µ ν ⟩ = E vac η µ ν , For curvatures R � M the flat result gets small corrections: ( R/M ) 2 � � ⟨ T µ ν ⟩ = E vac g µ ν + O ⇒ Λ eff = Λ 0 +8 π G N E vac ⇒ Solution to Einstein eq. has curvature of order Λ eff Holographic self-tuning of the cosmological constant – p.3

Possible way out Modify gravity to disconnect vacuum energy from curvature: allow large E vac but make it so it does not gravitate. • Self-tuning: any mechanism which allows flat spcacetime solutions for generic values of E vac . Holographic self-tuning of the cosmological constant – p.4

Possible way out Modify gravity to disconnect vacuum energy from curvature: allow large E vac but make it so it does not gravitate. • Self-tuning: any mechanism which allows flat spcacetime solutions for generic values of E vac . • Braneworld in extra dimension: E vac curves the bulk, but not the brane. Previous attempts: Arkani-Hamed et al. ’00; Kachru,Schulz,Silverstein ’00. They all either lead to bad singularities, or failure to reproduce 4d gravity, or need for fine-tuning. See also Charmousis, Gregory, Padilla ’07 Holographic self-tuning of the cosmological constant – p.4

Content of this talk • Self-tuning possible in the a general framework of a dilatonic, asymmetric braneworld with general 2-derivative induced terms. Brane (we live here) Infinite Finite volume volume Bulk (only gravity and a scalar live here) Previously explored around 2000: Arkani-Hamed et al. ’00; Kachru,Schulz,Silverstein ’00; Csaki et al , ’00. See also Charmousis, Gregory, Padilla ’07 Holographic self-tuning of the cosmological constant – p.4

Effective brane-world action � � � � R − 1 du √− g S = M 3 d 4 x 2 g ab ∂ a ϕ∂ b ϕ − V ( ϕ ) 5d gravity dual of 4d CFT � � � − W B ( ϕ ) − 1 d 4 σ √− γ + M 3 2 Z ( ϕ ) γ µ ν ∂ µ ϕ∂ ν ϕ + U ( ϕ ) R ( γ ) Σ 0 The QFT vacuum Localized Effective action induced by quantum energy is in here! effects of weakly coupled QFT (up to two derivative in the bulk fields) We take this class of actions as the starting point and the definition of our model The unknown functions appearing in the localized action can be taken as a phenomenological input or motivated by weakly coupled calculation. work in progress with E. Kiritsis and L. Witkowski Holographic self-tuning of the cosmological constant – p.12

Bulk equations � � � � R − 1 du √− g S 5 = M 3 d 4 x 2 g ab ∂ a ϕ∂ b ϕ − V ( ϕ ) Vacuum (Poincaré invariant) solutions: ds 2 = du 2 + e 2 A ( u ) η µ ν dx µ dx ν , ϕ = ϕ ( u ) A 2 − 1 ϕ 2 = 0 , ϕ 2 + V ( ϕ ) = 0 . 6 ¨ 12 ˙ A + ˙ 2 ˙ One has to solve independently on each side of the defect (at u = u 0 ), and glue the solutions using Israel junction conditions: = − 1 = dW B � � � � � � � � ˙ A = ϕ = 0; A 6 W B ( ϕ ( u 0 )); ϕ ˙ d ϕ ( ϕ ( u 0 )) Holographic self-tuning of the cosmological constant – p.19

Vacuum Geometry Regular Asymptotically AdS interior Boundary A UV ( u ) , ϕ UV ( u ) A IR ( u ) , ϕ IR ( u ) e A UV → + ∞ , ϕ UV → 0 e A IR → 0 , ϕ IR → ϕ ∗ UV- AdS boundary Interior of IR- AdS space Holographic self-tuning of the cosmological constant – p.16

Superpotential Write Einstein’s equations as first order flow equations, with an ( ′ = d/d ϕ ): auxiliary scalar function W ( ϕ ) A = − 1 ˙ ˙ Φ = W ′ ( ϕ ) , 6 W ( ϕ ) 4( d − 1) W 2 + 1 d W ′ � 2 = V � − 2 • Up to a rescaling of the scale factor, W completely determines the geometry. Holographic self-tuning of the cosmological constant – p.17

Superpotential Write Einstein’s equations as first order flow equations, with an ( ′ = d/d ϕ ): auxiliary scalar function W ( ϕ ) A = − 1 ˙ ˙ Φ = W ′ ( ϕ ) , 6 W ( ϕ ) 4( d − 1) W 2 + 1 d W ′ � 2 = V � − 2 • Up to a rescaling of the scale factor, W completely determines the geometry. � W UV ( ϕ ) ϕ < ϕ 0 W ( ϕ ) = W IR ( ϕ ) ϕ > ϕ 0 • On each side of the interface ( ϕ = ϕ 0 ), W is determined by one integration consntant C . Holographic self-tuning of the cosmological constant – p.17

Junction conditions for the superpotential Junction conditions take a simple form: dW UV ( ϕ 0 ) − dW IR ( ϕ 0 ) = dW B W IR ( ϕ 0 ) − W UV ( ϕ 0 ) = W B ( ϕ 0 ) , d ϕ ( ϕ 0 ) d ϕ d ϕ Holographic self-tuning of the cosmological constant – p.18

Junction conditions for the superpotential UV side: Solutions arrive at the AdS fixed point for all values of the integration constant C UV : UV fixed point is an attractor. Holographic self-tuning of the cosmological constant – p.19

Junction conditions for the superpotential UV side: Solutions arrive at the AdS fixed point for all values of the integration constant C UV : UV fixed point is an attractor. IR side: Only certain IRs are acceptable (e.g. IR AdS fixed point) This picks out a single solution W IR and fixes C IR = C ∗ ∗ Holographic self-tuning of the cosmological constant – p.19

IR Selection UV side: Solutions arrive at the AdS fixed point for all values of the integration constant C UV : UV fixed point is an attractor. IR side: Only certain IRs are acceptable (e.g. IR AdS fixed point) This picks out a single solution W IR and fixes C IR = C ∗ ∗ Holographic self-tuning of the cosmological constant – p.20

Equilibrium solution dW UV ( ϕ 0 ) = dW IR ( ϕ 0 ) − dW B W UV ( ϕ 0 ) = W IR ∗ ∗ ( ϕ 0 ) − W B ( ϕ 0 ) , d ϕ ( ϕ 0 ) d ϕ d ϕ Two equations for two unknowns C UV , ϕ 0 . Generically there exist a unique (or a discrete set of) solutions with C UV , ϕ 0 determined. Holographic self-tuning of the cosmological constant – p.21

Equilibrium solution dW UV ( ϕ 0 ) = dW IR ( ϕ 0 ) − dW B W UV ( ϕ 0 ) = W IR ∗ ∗ ( ϕ 0 ) − W B ( ϕ 0 ) , d ϕ ( ϕ 0 ) d ϕ d ϕ Two equations for two unknowns C UV , ϕ 0 . Generically there exist a unique (or a discrete set of) solutions with C UV , ϕ 0 determined. Holographic self-tuning of the cosmological constant – p.22

Equilibrium solution dW UV ( ϕ 0 ) = dW IR ( ϕ 0 ) − dW B W UV ( ϕ 0 ) = W IR ∗ ∗ ( ϕ 0 ) − W B ( ϕ 0 ) , d ϕ ( ϕ 0 ) d ϕ d ϕ For generic brane vacuum energy ∼ Λ 4 , geometry and brane position adjust so that the brane is flat and the UV glues to the regular IR ( self-tuning ). Holographic self-tuning of the cosmological constant – p.26

Emergent gravity on the brane In the model considered, solutions with flat 4d brane are generic. Do gravitational interactions between brane sources look 4d? Holographic self-tuning of the cosmological constant – p.27

Emergent gravity on the brane In the model considered, solutions with flat 4d brane are generic. Do gravitational interactions between brane sources look 4d? Recall Randall-Sundrum type braneworld e A � u � u u 0 • Volume is finite on both sides ⇒ Normalizable 4d graviton zero mode mediates 4d gravity at large distances Holographic self-tuning of the cosmological constant – p.27