Holographic self-tuning of the cosmological constant

Holographic self-tuning of the cosmological constant Francesco Nitti Laboratoire APC, U. Paris Diderot IX Aegean Summer School Sifnos, 19-09-2017 work with Elias Kiritsis and Christos Charmousis, 1704.05075 Holographic self-tuning of the cosmological constant p.1

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 µν = Λ 0 g µν +8πG N T µν QFT: the source of semiclassical gravity becomes T µν : T µν = E vac g µν in the vacuum G µν = Λ eff g µν, Λ eff = Λ 0 +8πG N E vac. Curvatures of order Λ eff Holographic 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 µν =Λ 0 g µν +8πG N T µν QFT: the source of semiclassical gravity becomes T µν : T µν = E vac g µν in the vacuum G µν = Λ eff g µν, Λ eff =Λ 0 +8πG N E vac. Curvatures of order Λ eff Self-tuning: any mechanism which allow flat spcacetime solutions for generic values of E vac. Holographic self-tuning of the cosmological constant p.2

Content of this talk Self-tuning possible in the a general framework of a dilatonic, asymmetric braneworld with general 2-derivative induced terms. Model based on holographic model building: dual of 4-dimensional, strongly coupled, non-gravitational fundamental theory. Previously explored around 2000: Arkani-Hamed et al. 00; Kachru,Schulz,Silverstein 00; Csaki et al, 00; All presented problems due to singularities or absence of localized 4d gravity on the brane Holographic self-tuning of the cosmological constant p.3

Content of this talk Self-tuning possible in the a general framework of a dilatonic, asymmetric braneworld with general 2-derivative induced terms. Model based on holographic model building: dual of 4-dimensional, strongly coupled, non-gravitational fundamental theory. Previously explored around 2000: Arkani-Hamed et al. 00; Kachru,Schulz,Silverstein 00; Csaki et al, 00; All presented problems due to singularities or absence of localized 4d gravity on the brane Outline AdS/CFT micro-review Setup Flat vacua: self-tuning Tensor perturbations: emergent braneworld gravity Scalar perturbations: stability Perspectives Holographic self-tuning of the cosmological constant p.3

AdS/CFT detour The AdS/CFT duality: conjecture that certain quantum field theories are equivalent to theories of gravity in higher dimensions Maldacena 98. Holographic self-tuning of the cosmological constant p.8

AdS/CFT detour Conformal field theory in d dimension Anti de Sitter spacetime AdS d+1 ds 2 = du 2 + e 2u/l η µν dx µ dx ν x µ : QFT coordinates; u dual to energy scale E e u/l. Holographic self-tuning of the cosmological constant p.7

AdS/CFT detour Conformal field theory in d dimension Anti de Sitter spacetime AdS d+1 ds 2 = du 2 + e 2u/l η µν dx µ dx ν x µ : QFT coordinates; u dual to energy scale E e u/l. bulk scalar field ϕ(u) running coupling g(e). The corresponding holographic RG-flow geometry breaks conformal invariance (except at fixed points where ϕ =0). ds 2 = du 2 + e A(u) η µν dx µ dx ν, ϕ = ϕ(u). E e A(u) Holographic self-tuning of the cosmological constant p.7

Setup Consider a 4d QFT with a UV conformal fixed point, made out of: 1. A strongly coupled large-n CFT, deformed by a relevant operator; 2. The weakly coupled Standard Model fields; 3. Some heavy messangers with mass scale Λ, coupling the first two. Integrating out the messangers leaves as an EFT the (broken) CFT, coupled to the SM, with some effective couplings set by Λ. semi-holographic description: Describe the strongly coupled large-n theory by a 5d gravity dual with the metric g ab and some bulk scalar fields ϕ i, dual to the operators that drive the CFT to the IR. The weakly coupled SM fields have a standard field-theoretical description, and they sit on a 4d defect in th 5d dual geometry. Holographic self-tuning of the cosmological constant p.5

Semi-holographic setup S = M 3 d 4 x du g [ R 1 2 gab a ϕ b ϕ V (ϕ) ] + d 4 σ γl(ψ i,h,w a,...,ϕ, γ µν ). Σ 0 Σ 0 Holographic tuning of the cosmological constant p.7

Effective brane-world action S = M 3 d 4 x du g [ R 1 ] 2 gab a ϕ b ϕ V (ϕ) + d 4 σ γl(ψ i,h,w a,...,ϕ, γ µν ) Σ 0 Holographic tuning of the cosmological constant p.10

Effective brane-world action S = M 3 d 4 x du g [ R 1 ] 2 gab a ϕ b ϕ V (ϕ) + d 4 σ γl(ψ i,h,w a,...,ϕ, γ µν ) Σ 0 Quantum effects from the localized fields generically induce localized effective potentials for ϕ and γ µν on the brane Holographic tuning of the cosmological constant p.10

Effective brane-world action S = M 3 d 4 x du g [ R 1 ] 2 gab a ϕ b ϕ V (ϕ) +M 3 Σ 0 d 4 σ γ [ W B (ϕ) 1 2 Z(ϕ)γµν µ ϕ ν ϕ + U(ϕ)R (γ) +... ] Quantum effects from the localized fields generically induce localized effective potentials for ϕ and γ µν on the brane. Generically expect: W B Λ 4 U Z Λ 2 W B (ϕ) includes the brane fields vacuum energy Holographic tuning of the cosmological constant p.11

Effective brane-world action S = M 3 d 4 x du g [ R 1 ] 2 gab a ϕ b ϕ V (ϕ) +M 3 Σ 0 d 4 σ γ [ W B (ϕ) 1 2 Z(ϕ)γµν µ ϕ ν ϕ + U(ϕ)R (γ) +... ] Quantum effects from the localized fields generically induce localized effective potentials for ϕ and γ µν on the brane. Generically expect: W B Λ 4 U Z Λ 2 W B (ϕ) includes the brane fields vacuum energy Action is the most general up to two derivates preserving 4d diffeos. Holographic tuning of the cosmological constant p.11

Effective brane-world action S = M 3 d 4 x du [ g R 1 ] 2 gab a ϕ b ϕ V (ϕ) +M 3 Σ 0 d 4 σ γ [ W B (ϕ) 12 Z(ϕ)γµν µ ϕ ν ϕ + U(ϕ)R (γ) ] Holographic self-tuning of the cosmological constant p.12

Field equations and matching conditions S = M 3 d 4 x du g [ R 1 2 gab a ϕ b ϕ V (ϕ) ] +M 3 Σ 0 d 4 σ γ [ W B (ϕ) 12 Z(ϕ)γµν µ ϕ ν ϕ + U(ϕ)R (γ) ] Einstein equations + Israel junction conditions ([ ] jump across Σ 0 ): G ab = 1 2 aϕ b ϕ 1 ( ) 1 2 g ab 2 gcd c ϕ d ϕ + V (ϕ), [γ µν ] = [ ] ϕ = 0; [ ] K µν γ µν K = 1 δs [ ] Σ0 γ δγ µν ; n a a ϕ = 1 γ δs Σ0 δϕ Holographic self-tuning of the cosmological constant p.12

Field equations and matching conditions S = M 3 d 4 x du g [ R 1 2 gab a ϕ b ϕ V (ϕ) ] +M 3 Σ 0 d 4 σ γ [ W B (ϕ) 12 Z(ϕ)γµν µ ϕ ν ϕ + U(ϕ)R (γ) ] Einstein equations + Israel junction conditions ([ ] jump across Σ 0 ): G ab = 1 2 aϕ b ϕ 1 ( ) 1 2 g ab 2 gcd c ϕ d ϕ + V (ϕ), [γ µν ] = [ ] ϕ = 0; [ ] K µν γ µν K = 1 δs [ ] Σ0 γ δγ µν ; n a a ϕ = 1 γ δs Σ0 δϕ Self tuning if solutions with flat defect for generic W B Λ 4. Holographic self-tuning of the cosmological constant p.12

Bulk equations S 5 = M 3 d 4 x du g [ R 1 2 gab a ϕ b ϕ V (ϕ) ] Vacuum (Poincaré invariant) solutions: ds 2 = du 2 + e 2A(u) η µν dx µ dx ν, ϕ = ϕ(u) 6Ä + ϕ2 =0, 12 A 2 1 2 ϕ2 + V (ϕ) =0. Holographic tuning of the cosmological constant p.15

Bulk equations S 5 = M 3 d 4 x du g [ R 1 2 gab a ϕ b ϕ V (ϕ) ] Vacuum (Poincaré invariant) solutions: ds 2 = du 2 + e 2A(u) η µν dx µ dx ν, ϕ = ϕ(u) 6Ä + ϕ2 =0, 12 A 2 1 2 ϕ2 + V (ϕ) =0. One has to solve independently on each side of the defect (at u = u 0 ), and glue the solutions using Israel junction conditions: [ ] A = [ ] ϕ = 0; [ ] A = 1 6 W B(ϕ(u 0 )); [ ] ϕ = dw B dϕ (ϕ(u 0)) Holographic tuning of the cosmological constant p.15

Vacuum Geometry A UV (u), ϕ UV (u) A IR (u), ϕ IR (u) e A UV +, ϕ UV 0 UV-AdS boundary e A IR 0, ϕ IR ϕ 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 auxiliary scalar function W (ϕ) ( = d/dϕ): A = 1 6 W (ϕ) Φ = W (ϕ), d 4(d 1) W 2 + 1 2 ( W ) 2 = V Holographic self-tuning of the cosmological constant p.17

Superpotential Write Einstein s equations as first order flow equations, with an auxiliary scalar function W (ϕ) ( = d/dϕ): A = 1 6 W (ϕ) Φ = W (ϕ), d 4(d 1) W 2 + 1 ( ) W 2 = V 2 Up to a rescaling of the scale factor, W completely determines the geometry. W (ϕ) = { W UV (ϕ) ϕ < ϕ 0 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: W IR (ϕ 0 ) W UV (ϕ 0 )=W B (ϕ 0 ), dw UV dϕ (ϕ dw IR 0) dϕ (ϕ 0)= dw B dϕ (ϕ 0) 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 W UV (ϕ 0 )=W IR (ϕ 0 ) W B (ϕ 0 ), dw UV dϕ (ϕ 0)= dw IR dϕ (ϕ 0) dw B dϕ (ϕ 0) 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 W UV (ϕ 0 )=W IR (ϕ 0 ) W B (ϕ 0 ), dw UV dϕ (ϕ 0)= dw IR dϕ (ϕ 0) dw B dϕ (ϕ 0) For generic brane vacuum energy Λ 4, geometry (VEVs and brane position) adjusts so that the brane is flat and the UV glues to the regular IR through the junction (self-tuning). Holographic self-tuning of the cosmological constant p.22

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 tuning of the cosmological constant p.23

Emergent gravity on the brane In the model considered, solutions with flat 4d brane are generic. Do gravitational interactions between brane sources look 4d? The transverse volume of holographic dimension is infinite in the UV no (normalizable) zero-mode gravitons exist. The induced Einstein term on the defect allows for the existence of a 4d-like graviton resonance (Dvali,Gabadadze,Porrati, 00) S = M 3 du d 4 x gr 5 +...+ M 3 u=u 0 d 4 x γu(ϕ 0 )R 4 Holographic tuning of the cosmological constant p.23

Emergent gravity on the brane In the model considered, solutions with flat 4d brane are generic. Do gravitational interactions between brane sources look 4d? The transverse volume of holographic dimension is infinite in the UV no (normalizable) zero-mode gravitons exist. The induced Einstein term on the defect allows for the existence of a 4d-like graviton resonance (Dvali,Gabadadze,Porrati, 00) S = M 3 du d 4 x gr 5 +...+ M 3 u=u 0 d 4 x γu(ϕ 0 )R 4 Localized Ricci term graviton exchange is effectively 4d at short distances. Holographic tuning of the cosmological constant p.23

Emergent gravity on the brane In the model considered, solutions with flat 4d brane are generic. Do gravitational interactions between brane sources look 4d? The transverse volume of holographic dimension is infinite in the UV no (normalizable) zero-mode gravitons exist. The induced Einstein term on the defect allows for the existence of a 4d-like graviton resonance (Dvali,Gabadadze,Porrati, 00) S = M 3 du d 4 x gr 5 +...+ M 3 u=u 0 d 4 x γu(ϕ 0 )R 4 Localized Ricci term graviton exchange is effectively 4d at short distances. Bulk curvature 4d massive graviton at very large distances. Holographic tuning of the cosmological constant p.23

Scales of braneworld gravity Two competing scales: 1. DGP transition length: r c U(ϕ 0 ) 2. Bulk curvature length r t =(e A 0 R 0 ) 1, R 0 W UV (ϕ 0 ) Holographic tuning of the cosmological constant p.24

Scales of braneworld gravity Two competing scales: 1. DGP transition length: r c U(ϕ 0 ) 2. Bulk curvature length r t =(e A 0 R 0 ) 1, R 0 W UV (ϕ 0 ) r t >r c Holographic tuning of the cosmological constant p.24

Scales of braneworld gravity Two competing scales: 1. DGP transition length: r c U(ϕ 0 ) 2. Bulk curvature length r t =(e A 0 R 0 ) 1, R 0 W UV (ϕ 0 ) r t >r c r t <r c Holographic tuning of the cosmological constant p.24

Scales of braneworld gravity Two competing scales: 1. DGP transition length: r c U(ϕ 0 ) 2. Bulk curvature length r t =(e A 0 R 0 ) 1, R 0 W UV (ϕ 0 ) r t >r c r t <r c M 2 p M 3 U 0, m 2 g R 0 U 0 Holographic tuning of the cosmological constant p.24

Scalar perturbations Determine whether vacuum solution (flat brane at r = r 0 )is stable. Possible light scalar mediated interactions (fifth force, violations of equivalence principle) pheno constraints. Analysis of linear flucutations show that there exist conditions on the background solution which guarantee stability. Holographic tuning of the cosmological constant p.28

Scalar perturbations Determine whether vacuum solution (flat brane at r = r 0 )is stable. Possible light scalar mediated interactions (fifth force, violations of equivalence principle) pheno constraints. Analysis of linear flucutations show that there exist conditions on the background solution which guarantee stability. 1. τ 0 > 0, Z 0 > 0, Z 0 τ 0 > 36 ( W B τ 0 6 6 W IR W UV No ghost instabilities ( dub dϕ ) U, ϕ 0 Z 0 Z(ϕ 0 ) ϕ0 ) 2 Holographic tuning of the cosmological constant p.28

Scalar perturbations Determine whether vacuum solution (flat brane at r = r 0 )is stable. Possible light scalar mediated interactions (fifth force, violations of equivalence principle) pheno constraints. Analysis of linear flucutations show that there exist conditions on the background solution which guarantee stability. 2. M 2 ( d 2 W B dϕ 2 (ϕ 0) [ d 2 W dϕ 2 ] IR UV ) 0 No tachyonic instabilities. Holographic tuning of the cosmological constant p.28

Conclusion and outlook We constructed a framework where Self-tuning of the CC is generically realized. Challenge now is model-building: construct phenomenologically viable model. Acceptable values of M p, r c, m g given large UV cutoff; Compliance with stability requirements; Deal with vdvz discontinuity (Role of non-linearities, Veinshtein mechanism); Avoidance of fifth force constraints; If this all goes through, one can do more phenomenology: Add SM and Higgs field (see Lukas Witkowski s talk) Study the space of solutions: non-flat brane, time-dependent solutions (cosmology) (ongoing work with Lukas Witkowski and Jewek Ghosh) The framework can potentially addess EW hierarchy problem (via stabilized warped extra dimensions) and late-time acceleration (cosmology close to the equilibrium Holographicposition). tuning of the cosmological constant p.40

Example V (ϕ) = 12 ( (4 ) 2 b2 4 ) ϕ 2 V 1 sinh 2 bϕ 2, supports an AdS fixed point at ϕ =0(l UV =1) good IR solution: W IR (ϕ) 2 (32/3) b 2 exp bϕ 2, ϕ +. Holographic tuning of the cosmological constant p.44

How large can Λ be? [ W B (ϕ) =Λ 4 1 ϕ ( ϕ s + s ) 2 ] b = 1 6, =3,V 1 =1 ϕ 0 ϕ 1.6 s Holographic tuning of the cosmological constant p.45

Effective 4d Green s function Introuce tensor perturbations: δg µν = e 2A(r) h µν (r, x α ), h µ µ = µ h µν =0 Holographic tuning of the cosmological constant p.30

Effective 4d Green s function Introuce tensor perturbations: δg µν = e 2A(r) h µν (r, x α ), h µ µ = µ h µν =0 Solve classical linearized equation for tensor fluctuationswith localized source: h µν (x, r) = d 4 xgµν ρσ (x x ; r, r 0 )T ρσ (x,r 0 ), Tree-level interaction described in purely 4d terms by an effective Green s function: d 4 p S int (T )= (2π) G 4 4 (p) [T µν (p)t µν ( p) 13 ] T (p)t ( p) G 4 (x) G(x, r 0,r 0 ). Holographic tuning of the cosmological constant p.30

4d-5d transition r c <r t : DGP-like transition, at intermediate distances. r c = U 0,r t = e A 0 R 0, M 2 p M 3 U 0, m 2 0 R 0 U 0, Holographic tuning of the cosmological constant p.31

Massless/Massive gravity transition r c >r t massive graviton propagator all the way. r c = U 0,r t = e A 0 R 0, M 2 p M 3 U 0, m 2 0 R 0 U 0, Holographic tuning of the cosmological constant p.32

Looking for solutions Junction conditions can be rewritten as a non-linear equation for ϕ 0 : Q2 2 ( ) 2 W IR (ϕ 0 ) W B 1 (ϕ 0 ) + 2 ( dw IR dϕ dw B dϕ ) 2 ϕ 0 = V (ϕ 0 ), Q d 2(d 1) V, W B and W IR are fixed functions of ϕ. 1. Solve for ϕ 0 2. Solve superpotential equation for W UV (ϕ) with initial condition: W UV (ϕ 0 )=W IR (ϕ 0 ) W B (ϕ 0 ) Holographic tuning of the cosmological constant p.45

Consistent self-tuining Two possibilities: W UV > 0 W UV < 0 Holographic tuning of the cosmological constant p.46

Genericity As we will see, it is desirable (but not strictly necessary) that W B (ϕ 0 ) > 0, i.e. 0 <W UV (ϕ 0 ) <W IR (ϕ 0 ) (in this case, the solution is manifestly ghost-free). It turns out that that for such solutions to exist, it is enough that W ( ϕ) =0, W ( ϕ) > 0 for some ϕ. Then the equations are solved, with W B (ϕ 0 ) > 0, for: ϕ 0 ϕ + ϕ(w 2 IR ) 4 V ϕ= ϕ provided: W B (ϕ 0 ) W IR (ϕ 0 ) 1 Holographic tuning of the cosmological constant p.47

Relating scales We can relate bulk parameters M,e A 0, l UV to those of the dual field theory N,g 0, : e A 0 (l UV g 0 ) 1/(d ), (Ml UV ) 3 N 2 Bulk superpotentials set the scale of the bulk curvature scale: W (ϕ(u)) R(u) M R 0 N 2/3 l UV W UV (ϕ 0 ) The scale of brane potentials is set by the UV cut-off Λ: W B Λ4 M 3, U B Λ2 M 3 Holographic tuning of the cosmological constant p.50

DGP scenario Requires r t >r c Gravity must be modified at cosmological distances: M p r c = ( ) 3/2 MU0 4 ( Λ M ) 3 u 3 (ϕ 0 ) 10 60 The assumption r t >r c translates into: e A 0 U 0 R 0 1 ( Λ M ) 2 u(ϕ 0 ) l UV W UV (ϕ 0 ) N 2/3 (l UV g 0 ) 1 (d ) 1 Holographic tuning of the cosmological constant p.51

Massive gravity scenario 1 Requires r t >r c Large distance modification (graviton mass) must be at cosmological scales m g M p 10 60 ( M Λ ) 2 (luv W UV (ϕ 0 )) 1/2 u(ϕ 0 ) 1 N 1/3 < 10 60 Short distance modification must be below (tenths of) mm: ( ) M r t M p 10 30 l UV W UV (ϕ 0 ) Λ u 1/2 1 > 10 30 (ϕ 0 )(l UV g 0 ) (d ) N 2/3 Holographic tuning of the cosmological constant p.52

Massive gravity scenario 2 Alternatively, r t <r c (no DGP regime) Same large scale condition: m g M p 10 60 ( M Λ ) 2 (luv W UV (ϕ 0 )) 1/2 u(ϕ 0 ) 1 N 1/3 < 10 60 No short distance modification until the UV cut-off. Holographic tuning of the cosmological constant p.53

Scalar-mediated interaction Define metric and dilaton sources: T µν = 2 γ δs m γ µν, O = δs m δϕ. Interaction between brane-localized sources: S int = 1 d 4 q 2 (2π) 4 T (q)g s (q)t ( q), T ( T µ,o µ ) G s (q) 1 2M 3 P [ Σ ( Γ 1 + q 2 Γ 2 ) + D 1 (r 0 ; q) ] 1 P P z IRz UV [z] 1 z IR 1 z UV 1 1 Modes coupling to O can be parametrically heavy, m M. Modes coupling to T remain light.. Holographic tuning of the cosmological constant p.54

DGP regime (short distance) Overall interaction for light modes in the 4d regime: V(q) 1 q 2 [ 1 2M 3 U 0 ( T µν (q)t µν ( q) 1 ) 3 T µ µ (q)tν ν ( q) + ] 1 2M 3 T µ µ (q)tν ν ( q) τ 0 Holographic tuning of the cosmological constant p.55

DGP regime (short distance) Overall interaction for light modes in the 4d regime: V(q) 1 q 2 [ 1 2M 3 U 0 ( T µν (q)t µν ( q) 1 ) 3 T µ µ (q)tν ν ( q) + ] 1 2M 3 T µ µ (q)tν ν ( q) τ 0 Something interesting happens if W B W IR W UV ϕ0 U 0, τ 0 6U 0. V(q) 1 q 2 [ 1 2M 2 p ( T µν (q)t µν ( q) 1 )] 2 T µ µ (q)tν ν ( q),mp 2 = M 3 U 0 Tensor Structure becomes that of a 4d massless graviton! Leftover interaction is light scalar with ultra-weak coupling Warning: need to check explcitly about ghosts Holographic tuning of the cosmological constant p.55