Holographic self-tuning of the cosmological constant

Transcription

1 Holographic self-tuning of the cosmological constant Francesco Nitti Laboratoire APC, U. Paris Diderot IX Crete Regional Meeting in String Theory Kolymbari, work with Elias Kiritsis and Christos Charmousis, Holographic self-tuning of the cosmological constant p.1

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 Holographic tuning of the cosmological constant p.2

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 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

4 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 The problem: In a low energy EFT it is possible to compute the generic contributions to E vac by (almost) flat space physics. The result is grossly incompatible with the observed small curvature of space-time. Holographic tuning of the cosmological constant p.2

5 Possible way out Disconnect vacuum energy from curvature: allow large E vac but make it so it does not gravitate. G µν = Λ eff g µν something different Holographic tuning of the cosmological constant p.5

6 Possible way out Disconnect vacuum energy from curvature: allow large E vac but make it so it does not gravitate. G µν = Λ eff g µν something different braneworld in extra dimension: Λ eff curves the bulk, but not the brane. self-tuning: For a generic value of E vac flat 4d space is a solution to the field equations. Early attempts: Arkani-Hamed et al. 00; Kachru,Schulz,Silverstein 00. Holographic tuning of the cosmological constant p.5

7 Possible way out Disconnect vacuum energy from curvature: allow large E vac but make it so it does not gravitate. G µν = Λ eff g µν something different braneworld in extra dimension: Λ eff curves the bulk, but not the brane. self-tuning: For a generic value of E vac flat 4d space is a solution to the field equations. Early attempts: Arkani-Hamed et al. 00; Kachru,Schulz,Silverstein 00. Tighlty connected to holography: fundamental theory is a (UV complete) strongly coupled QFT in flat space and 4d gravity is an emergent low-energy phenomenon. idea explored by E. Kiritsis and myself in 06 Holographic tuning of the cosmological constant p.5

8 Outline Setup Flat vacua: self-tuning Tensor perturbations: emergent braneworld gravity Scalar perturbations: stability Conclusions and perspectives Holographic tuning of the cosmological constant p.6

9 Setup and motivation Consider a model 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. Holographic tuning of the cosmological constant p.7

10 Setup and motivation Consider a model 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. Holographic tuning of the cosmological constant p.7

11 Setup and motivation Consider a model 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 tuning of the cosmological constant p.7

12 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

13 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

14 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

15 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. Holographic tuning of the cosmological constant p.11

16 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

17 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

18 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) ] Holographic tuning of the cosmological constant p.11

19 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) ] Geometry is determined by the bulk Einstein equations: G ab = 1 2 aϕ b ϕ 1 ( ) 1 2 g ab 2 gcd c ϕ d ϕ + V (ϕ), plus Israel junction conditons at the defect ([ ] jump across Σ 0 ): [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 tuning of the cosmological constant p.11

20 Self-tuning vacua Are there solutions with flat space as the 4d geometry, for generic potentials W B of order (UV cut-off) 4? if yes large 4d vacuum energy does not imply large 4d curvature! Holographic tuning of the cosmological constant p.12

21 Self-tuning vacua Are there solutions with flat space as the 4d geometry, for generic potentials W B of order (UV cut-off) 4? if yes large 4d vacuum energy does not imply large 4d curvature! Dual QFT language: the UV CFT lives in flat space. Are there vacua with unbroken Poincaré symmetry? Holographic tuning of the cosmological constant p.12

22 Self-tuning vacua Are there solutions with flat space as the 4d geometry, for generic potentials W B of order (UV cut-off) 4? if yes large 4d vacuum energy does not imply large 4d curvature! Dual QFT language: the UV CFT lives in flat space. Are there vacua with unbroken Poincaré symmetry? Self-tuning of the CC: the geometry ( the QFT vevs and the brane position) adjust so that a static flat solution arises. Holographic tuning of the cosmological constant p.12

23 Self-tuning vacua Are there solutions with flat space as the 4d geometry, for generic potentials W B of order (UV cut-off) 4? if yes large 4d vacuum energy does not imply large 4d curvature! Dual QFT language: the UV CFT lives in flat space. Are there vacua with unbroken Poincaré symmetry? Self-tuning of the CC: the geometry ( the QFT vevs and the brane position) adjust so that a static flat solution arises. Explored before ( ) but: IR geometry not fully understood (in particular meaning of the singularity) 4d gravity regime seemed incompatible with self-tuning (partly because models were not general enough) Holographic tuning of the cosmological constant p.12

24 Self-tuning vacua Are there solutions with flat space as the 4d geometry, for generic potentials W B of order (UV cut-off) 4? if yes large 4d vacuum energy does not imply large 4d curvature! Dual QFT language: the UV CFT lives in flat space. Are there vacua with unbroken Poincaré symmetry? Self-tuning of the CC: the geometry ( the QFT vevs and the brane position) adjust so that a static flat solution arises. Holographic picture clarifies how to organize the space of solutions: what integrations constants are fixed, which are dinamically determied, and what IR geometries are acceptable. Holographic tuning of the cosmological constant p.13

25 Self-tuning vacua Are there solutions with flat space as the 4d geometry, for generic potentials W B of order (UV cut-off) 4? if yes large 4d vacuum energy does not imply large 4d curvature! Dual QFT language: the UV CFT lives in flat space. Are there vacua with unbroken Poincaré symmetry? Self-tuning of the CC: the geometry ( the QFT vevs and the brane position) adjust so that a static flat solution arises. Holographic picture clarifies how to organize the space of solutions: what integrations constants are fixed, which are dinamically determied, and what IR geometries are acceptable. Framework is general enough to allow for emergent 4d gravity. Holographic tuning of the cosmological constant p.13

26 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 ϕ2 + V (ϕ) =0. Holographic tuning of the cosmological constant p.15

27 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 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

28 Vacuum Geometry A UV (u), ϕ UV (u) e A UV +, ϕ UV 0 AdS boundary as u ds 2 du 2 + e 2u/l η µν dx µ dx ν ϕ ϕ e (4 )u/l +... V 12 l 2 + m2 2 φ ( 4) = m 2. A IR (u), ϕ IR (u) e A IR 0, ϕ IR ϕ Regular AdS interiorornakedsingularityas u +. Holographic tuning of the cosmological constant p.16

29 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 ( W ) 2 = V Holographic tuning of the cosmological constant p.16

30 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 ( ) W 2 = V 2 This system is equivalent to usual Einstein equations. On each side, W (Φ) contains an integration consntant C: Holographic tuning of the cosmological constant p.16

31 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 ( ) W 2 = V 2 This system is equivalent to usual Einstein equations. On each side, W (Φ) contains an integration consntant C: UV side: C UV controls the vev of the dual operator and it is not fixed by UV boundary conditions. ( ) ϕ(u) =ϕ e (1+... )+ϕ (4 )u/l /(d ) C UV e u/l Holographic tuning of the cosmological constant p.16

32 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 ( ) W 2 = V 2 This system is equivalent to usual Einstein equations. On each side, W (Φ) contains an integration consntant C: UV side: C UV controls the vev of the dual operator and it is not fixed by UV boundary conditions. ( ) ϕ(u) =ϕ e (1+... )+ϕ (4 )u/l /(d ) C UV e u/l IR side: C IR fixed by regularity of the IR geometry. Holographic tuning of the cosmological constant p.17

33 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 tuning of the cosmological constant p.17

34 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 tuning of the cosmological constant p.18

35 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 (IR AdS fixed point, good singularities). This picks out a single solution W IR and fixes C IR = C (or at most a finite number) Holographic tuning of the cosmological constant p.18

36 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 (IR AdS fixed point, good singularities). This picks out a single solution W IR and fixes C IR = C (or at most a finite number) Holographic tuning of the cosmological constant p.19

37 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 tuning of the cosmological constant p.20

38 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 tuning of the cosmological constant p.22

39 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 tuning of the cosmological constant p.21

40 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

41 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. Holographic tuning of the cosmological constant p.23

42 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 M 3 u=u 0 d 4 x γu(ϕ 0 )R 4 Holographic tuning of the cosmological constant p.23

43 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 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

44 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 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

45 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

46 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

47 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

48 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

49 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. Holographic tuning of the cosmological constant p.28

50 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

51 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 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

52 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

53 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; Holographic tuning of the cosmological constant p.40

54 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

55 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

56 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

57 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

58 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 ), Holographic tuning of the cosmological constant p.30

59 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

60 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

61 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

62 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

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

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

65 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

66 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

67 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 ) 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

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

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

70 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

71 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

72 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

73 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

74 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. Holographic tuning of the cosmological constant p.55

75 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