New no-go theorems and the
costs of cosmic acceleration

         Daniel Wesley
         (Cambridge DAMTP)




     0802.2...
Two data sets
Two data sets
Two data sets




          ä>0            ä > 0 (?)
      Late universe   Early universe
      (dark energy)    (inflation?)
Old no-go theorems

 Theorem                  (Gibbons ’84, Maldacena and Nunez ‘01)
 To obtain a four-dimensional de Sitt...
Old no-go theorems

 Theorem                          (Gibbons ’84, Maldacena and Nunez ‘01)
 To obtain a four-dimensional...
Old no-go theorems

 Theorem                          (Gibbons ’84, Maldacena and Nunez ‘01)
 To obtain a four-dimensional...
Some energy conditions

   Null         TMNnMnN ≥ 0       RMNtMtN ≥ 0   Strong
                          P           P    ...
Some energy conditions

   Null         TMNnMnN ≥ 0   RMNtMtN ≥ 0   Strong
                   P             P        “geod...
Some energy conditions

   Null            TMNnMnN ≥ 0                 RMNtMtN ≥ 0          Strong
                       ...
The Null Energy Condition
...keeps us safe from...   Causality violations




                           Instabilities & o...
The Null Energy Condition
...keeps us safe from...   Causality violations
                           Superluminal travel, ...
The Null Energy Condition
...keeps us safe from...   Causality violations
                           Superluminal travel, ...
Types of new theorems

 de Sitter theorems




Non-de Sitter (w>-1) theorems
Types of new theorems

 de Sitter theorems
• concerned with eternal de Sitter expansion
• must violate null energy conditi...
Types of new theorems

 de Sitter theorems
• concerned with eternal de Sitter expansion
• must violate null energy conditi...
Types of new theorems

 de Sitter theorems
• concerned with eternal de Sitter expansion
• must violate null energy conditi...
Types of new theorems

 de Sitter theorems
• concerned with eternal de Sitter expansion
• must violate null energy conditi...
Types of new theorems

 de Sitter theorems
• concerned with eternal de Sitter expansion
• must violate null energy conditi...
Types of M


       Curvature-free                           Curved

      Intrinsic Ricci scalar R
                      ...
Types of M


       Curvature-free                           Curved

      Intrinsic Ricci scalar R
                      ...
Types of M


       Curvature-free                           Curved

      Intrinsic Ricci scalar R
                      ...
Comparison with previous theorems

            curvature-free      curved

                       Strong
                 ...
Comparison with previous theorems

             curvature-free         curved

                         Strong
           ...
Comparison with previous theorems

             curvature-free            curved

                          Strong
       ...
A simple example (I)
    Ricci-flat extra dimensions with breathing-mode dynamics
                                        ...
A simple example (I)
    Ricci-flat extra dimensions with breathing-mode dynamics
                                        ...
A simple example (I)
    Ricci-flat extra dimensions with breathing-mode dynamics
                                        ...
A simple example (I)
    Ricci-flat extra dimensions with breathing-mode dynamics
                                        ...
A simple example (II)



          Reconstruct (4+k)D metric and use Einstein
           equations to compute stress-energ...
A simple example (II)



                   Reconstruct (4+k)D metric and use Einstein
                    equations to co...
A simple example (II)
          Lesson: potentials that look perfectly reasonable in 4D
           require exotic physics ...
Assumptions

     • M closed and compact, or a quotient of c.c. M/G




     • Higher-dimensional action has Einstein-Hilb...
Three tools...
Three tools...
      1. Decomposition of metric time derivative.
         (Proxy for KK scalars and/or metric moduli)
Three tools...
      1. Decomposition of metric time derivative.
         (Proxy for KK scalars and/or metric moduli)
Three tools...
      1. Decomposition of metric time derivative.
         (Proxy for KK scalars and/or metric moduli)



 ...
Three tools...
      1. Decomposition of metric time derivative.
         (Proxy for KK scalars and/or metric moduli)



 ...
Three tools...
      1. Decomposition of metric time derivative.
         (Proxy for KK scalars and/or metric moduli)



 ...
...many challenges



     •   non-uniqueness of KK inversion
     •   (equiv) 4D theory gives only nonlocal information o...
...many challenges



     •   non-uniqueness of KK inversion
     •   (equiv) 4D theory gives only nonlocal information o...
Averages of NEC probes
Averages of NEC probes
Averages of NEC probes
                         1. Metric deformations
                         enter with negative sign
Averages of NEC probes
                         1. Metric deformations
                         enter with negative sign

...
Averages of NEC probes
                         1. Metric deformations
                         enter with negative sign

...
Averages of NEC probes
                         1. Metric deformations
                         enter with negative sign

...
Averages of NEC probes
                            1. Metric deformations
                            enter with negative ...
The v-equation
          Minimax strategy -- choose A to minimise
            the maximum number of e-foldings
The v-equation
          Minimax strategy -- choose A to minimise
            the maximum number of e-foldings
The v-equation
              Minimax strategy -- choose A to minimise
                the maximum number of e-foldings



...
The v-equation
              Minimax strategy -- choose A to minimise
                the maximum number of e-foldings



...
The v-equation
              Minimax strategy -- choose A to minimise
                the maximum number of e-foldings



...
The v-equation
              Minimax strategy -- choose A to minimise
                the maximum number of e-foldings



...
Phase plots
Phase plots



                  k=2       k=6
v                       u




    w                             w
Phase plots



                  k=2       k=6
v                       u




    w                             w
Phase plots



                  k=2       k=6
v                       u




    w                             w
e-folds (I)

              k=1-4       k=5-10 and k = 11 ...

 N                    N




               w                ...
e-folds (II)



 w




 w




               number of extra dimensions
e-folds (II)



 w



                               nS ≥ 0.90    w ≤ -0.97



 w



                               nS ≥ 0...
Conclusions


      •   de Sitter case: weakened the energy condition to
          the NEC in a variety of cases

      • ...
STOP
New no-go theorems and the costs of cosmic acceleration
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New no-go theorems and the costs of cosmic acceleration

  1. 1. New no-go theorems and the costs of cosmic acceleration Daniel Wesley (Cambridge DAMTP) 0802.2106 and 0802.3214
  2. 2. Two data sets
  3. 3. Two data sets
  4. 4. Two data sets ä>0 ä > 0 (?) Late universe Early universe (dark energy) (inflation?)
  5. 5. Old no-go theorems Theorem (Gibbons ’84, Maldacena and Nunez ‘01) To obtain a four-dimensional de Sitter universe from a static warped reduction on closed compact manifold M, one must violate the Strong Energy Condition (SEC).
  6. 6. Old no-go theorems Theorem (Gibbons ’84, Maldacena and Nunez ‘01) To obtain a four-dimensional de Sitter universe from a static warped reduction on closed compact manifold M, one must violate the Strong Energy Condition (SEC). Evasive maneuvers (a) Time dependent M (b) Non-de Sitter expansion (c) Transient de Sitter (d) Accept SEC violation ** (e) Non-compact M (finite GN?) ** not always an option, depending on the theory
  7. 7. Old no-go theorems Theorem (Gibbons ’84, Maldacena and Nunez ‘01) To obtain a four-dimensional de Sitter universe from a static warped reduction on closed compact manifold M, one must violate the Strong Energy Condition (SEC). Evasive maneuvers New improvements (a) Time dependent M (1) Weaken energy condition (b) Non-de Sitter expansion (2) Include non-de Sitter (w>-1) (c) Transient de Sitter (3) Treat time-dependent M (d) Accept SEC violation ** (e) Non-compact M (finite GN?) ** not always an option, depending on the theory
  8. 8. Some energy conditions Null TMNnMnN ≥ 0 RMNtMtN ≥ 0 Strong P P “geodesics “gravity is stable” ρ ρ converge” ≈ “gravity is ρ attractive” Weak TMNtMtN ≥ 0 TMNtM not S.L. Dominant P “energy density P “subluminal positive for all ρ ρ phase velocity” observers” ρ≥0 ρ ≥ |P | ρ+P ≥0
  9. 9. Some energy conditions Null TMNnMnN ≥ 0 RMNtMtN ≥ 0 Strong P P “geodesics “gravity is stable” ρ ρ converge” ≈ “gravity is attractive”
  10. 10. Some energy conditions Null TMNnMnN ≥ 0 RMNtMtN ≥ 0 Strong P P “geodesics “gravity is stable” ρ ρ converge” ≈ “gravity is attractive” •ρ+P≥0 • ρ + P ≥ 0 and ρ + 3P ≥ 0 • Two-derivative actions with positive • Scalars with V(φ) ≤ 0 definite kinetic terms, any V(φ) • ... classical 11D SUGRA + others • D-branes & positive tension objects • anti-de Sitter Λ<0 • Implied by all other energy conditions • dust and radiation Satisfied by • Casimir energy (...unless averaged) • Scalars with V (φ) >0 anywhere • Negative tension (orientifold planes) • de Sitter Λ>0 • ghost condensates • D-branes Violated by
  11. 11. The Null Energy Condition ...keeps us safe from... Causality violations Instabilities & other pathologies
  12. 12. The Null Energy Condition ...keeps us safe from... Causality violations Superluminal travel, “warp drives,” traversable wormholes, time machines, CTCs, chronology (non-)protection ... Morris, Thorne Am. J. Phys. 56 (1988) 395 ; Visser, Kar, Dadhich Phys. Rev. Lett. 90 (2003) 201102 ; Alcubierre Class. Quant. Grav. 11 (1994) L73 ; Krasnikov Phys. Rev. D 57 (1998) 4760 ; Morris, Thorne, Yurtsever Phys. Rev. Lett. 61 (1988) 1446 ; Hawking Phys. Rev. D 46 (1992) 603 Instabilities & other pathologies
  13. 13. The Null Energy Condition ...keeps us safe from... Causality violations Superluminal travel, “warp drives,” traversable wormholes, time machines, CTCs, chronology (non-)protection ... Morris, Thorne Am. J. Phys. 56 (1988) 395 ; Visser, Kar, Dadhich Phys. Rev. Lett. 90 (2003) 201102 ; Alcubierre Class. Quant. Grav. 11 (1994) L73 ; Krasnikov Phys. Rev. D 57 (1998) 4760 ; Morris, Thorne, Yurtsever Phys. Rev. Lett. 61 (1988) 1446 ; Hawking Phys. Rev. D 46 (1992) 603 Instabilities & other pathologies Classical: Big Rips, Big Bounces, gradient instabilities ... Quantum: unitarity violation, rapid vacuum decay, perpetuum mobile ... Cline, Jeon, Moore Phys. Rev. D 70 (2004) 043543 ; Hsu, Jenkins, Wise Phys. Lett. B 597 (2004) 270 ; Dubovsky, Gregoire, Nicolis, Rattazzi JHEP 0603 (2006) 025 ; Buniy, Hsu, Murray Phys. Rev. D 74 (2006) 063518 ; Caldwell Phys. Lett. B 545 (2002) 23 ; Caldwell, Kamionkowski, Weinberg Phys. Rev. Lett. 91 (2003) 071301, Arkani-Hamed, Dubovsky, Nicolis, Trincherini, Villadoro JHEP 05 (2007) 055 ; Dubovsky, Sibiryakov Phys. Lett. B 638 (2006) 509
  14. 14. Types of new theorems de Sitter theorems Non-de Sitter (w>-1) theorems
  15. 15. Types of new theorems de Sitter theorems • concerned with eternal de Sitter expansion • must violate null energy condition under broad circumstances • previous theorems used R00 because it does not involve the intrinsic curvature R of M. To prove the new theorems, you look at the other components and engage with the complexity of dealing with R. Non-de Sitter (w>-1) theorems
  16. 16. Types of new theorems de Sitter theorems • concerned with eternal de Sitter expansion • must violate null energy condition under broad circumstances • previous theorems used R00 because it does not involve the intrinsic curvature R of M. To prove the new theorems, you look at the other components and engage with the complexity of dealing with R. Non-de Sitter (w>-1) theorems • For each number k of extra dimensions** • There exists a threshold wthresh • For w < wthresh there is a bound N(w) on the number of e-foldings • Constant w: undergoing N > N(w) e-folds violates an energy condition. • Varying w (1): for any w(t) the bound N[w(t)] obtained by quadrature. • Varying w (2): if w < w✻ then N[w(t)] < N(w✻). ** depends on other properties of M, to be described
  17. 17. Types of new theorems de Sitter theorems • concerned with eternal de Sitter expansion • must violate null energy condition under broad circumstances • previous theorems used R00 because it does not involve the intrinsic curvature R of M. To prove the new theorems, you look at the other components and engage with the complexity of dealing with R. Non-de Sitter (w>-1) theorems • For each number k of extra dimensions** • There exists a threshold wthresh • For w < wthresh there is a bound N(w) on the number of e-foldings • Constant w: undergoing N > N(w) e-folds violates an energy condition. • Varying w (1): for any w(t) the bound N[w(t)] obtained by quadrature. • Varying w (2): if w < w✻ then N[w(t)] < N(w✻). ** depends on other properties of M, to be described
  18. 18. Types of new theorems de Sitter theorems • concerned with eternal de Sitter expansion • must violate null energy condition under broad circumstances • previous theorems used R00 because it does not involve the intrinsic curvature R of M. To prove the new theorems, you look at the other components and engage with the complexity of dealing with R. Non-de Sitter (w>-1) theorems • For each number k of extra dimensions** • There exists a threshold wthresh • For w < wthresh there is a bound N(w) on the number of e-foldings • Constant w: undergoing N > N(w) e-folds violates an energy condition. • Varying w (1): for any w(t) the bound N[w(t)] obtained by quadrature. • Varying w (2): if w < w✻ then N[w(t)] < N(w✻). ** depends on other properties of M, to be described
  19. 19. Types of new theorems de Sitter theorems • concerned with eternal de Sitter expansion • must violate null energy condition under broad circumstances • previous theorems used R00 because it does not involve the intrinsic curvature R of M. To prove the new theorems, you look at the other components and engage with the complexity of dealing with R. Non-de Sitter (w>-1) theorems • For each number k of extra dimensions** • There exists a threshold wthresh • For w < wthresh there is a bound N(w) on the number of e-foldings • Constant w: undergoing N > N(w) e-folds violates an energy condition. • Varying w (1): for any w(t) the bound N[w(t)] obtained by quadrature. • Varying w (2): if w < w✻ then N[w(t)] < N(w✻). ** depends on other properties of M, to be described
  20. 20. Types of M Curvature-free Curved Intrinsic Ricci scalar R R≠0 somewhere on M vanishes everywhere on M • One-dimensional (KK & RS) • Manifolds with non-Abelian • Tori -- continuous isometries ** • realised by periodic ... includes models which obtain 4D identification of Rn gauge symmetries by KK reduction • with R ≥ 0 everywhere • Special holonomy -- • Rugby-ball SLED • Sp(n) • Spin(7) ** We only know these cannot be Ricci- • SU(n) (Calabi-Yau) flat, which is a stronger condition than • G2 (M theory) “curvature-free.”
  21. 21. Types of M Curvature-free Curved Intrinsic Ricci scalar R R≠0 somewhere on M vanishes everywhere on M • One-dimensional (KK & RS) • Manifolds with non-Abelian • Tori -- continuous isometries ** • realised by periodic ... includes models which obtain 4D identification of Rn gauge symmetries by KK reduction • with R ≥ 0 everywhere • Special holonomy -- • Rugby-ball SLED • Sp(n) • Spin(7) ** We only know these cannot be Ricci- • SU(n) (Calabi-Yau) flat, which is a stronger condition than • G2 (M theory) “curvature-free.”
  22. 22. Types of M Curvature-free Curved Intrinsic Ricci scalar R R≠0 somewhere on M vanishes everywhere on M • One-dimensional (KK & RS) • Manifolds with non-Abelian • Tori -- continuous isometries ** • realised by periodic ... includes models which obtain 4D identification of Rn gauge symmetries by KK reduction • with R ≥ 0 everywhere • Special holonomy -- • Rugby-ball SLED • Sp(n) • Spin(7) ** We only know these cannot be Ricci- • SU(n) (Calabi-Yau) flat, which is a stronger condition than • G2 (M theory) “curvature-free.”
  23. 23. Comparison with previous theorems curvature-free curved Strong (Gibbons et. al.) de Sitter w > -1
  24. 24. Comparison with previous theorems curvature-free curved Strong (Gibbons et. al.) de Sitter NEW NEW Null Null* *bounded avg. condition w > -1
  25. 25. Comparison with previous theorems curvature-free curved Strong (Gibbons et. al.) de Sitter NEW NEW Null Null* *bounded avg. condition NEW NEW w > -1 Null Strong (transient) (transient)
  26. 26. A simple example (I) Ricci-flat extra dimensions with breathing-mode dynamics 2c ds2 4+k = A(η) 2 −dη + 2 dx2 3 + exp ψ(η) ds2 k k
  27. 27. A simple example (I) Ricci-flat extra dimensions with breathing-mode dynamics 2c ds2 4+k = A(η)2 −dη + 2 dx2 3 + exp ψ(η) ds2 k k Conversion to 4D Einstein frame 2k c= a(η) = ecψ/2 A(η) k+2
  28. 28. A simple example (I) Ricci-flat extra dimensions with breathing-mode dynamics 2c ds2 4+k = A(η)2 −dη + 2 dx2 3 + exp ψ(η) ds2 k k Conversion to 4D Einstein frame 2k c= a(η) = ecψ/2 A(η) k+2 Apply 4D Friedmann equations 2 1+w dψ a(η) ∼ η 2/(1+3w) ρ+P = H = 2 3 dη
  29. 29. A simple example (I) Ricci-flat extra dimensions with breathing-mode dynamics 2c ds2 4+k = A(η)2 −dη + 2 dx2 3 + exp ψ(η) ds2 k k Conversion to 4D Einstein frame 2k c= a(η) = ecψ/2 A(η) k+2 Apply 4D Friedmann equations 2 1+w dψ a(η) ∼ η 2/(1+3w) ρ+P = H = 2 3 dη Yields η-dependence of ψ and A. 1+w ψ(η) = ± ln η 6 + ψ0 1 + 3w
  30. 30. A simple example (II) Reconstruct (4+k)D metric and use Einstein equations to compute stress-energy tensor 1+w √ ψ(η) = ± ln η 6 + ψ0 V (ψ) = V0 e 3(1+w)ψ 1 + 3w
  31. 31. A simple example (II) Reconstruct (4+k)D metric and use Einstein equations to compute stress-energy tensor w NEC OK NEC OK TMN nMnN P = -2 ρ NEC violated NEC violated w k 1+w √ ψ(η) = ± ln η 6 + ψ0 V (ψ) = V0 e 3(1+w)ψ 1 + 3w
  32. 32. A simple example (II) Lesson: potentials that look perfectly reasonable in 4D require exotic physics in higher-dimensional context. Reconstruct (4+k)D metric and use Einstein equations to compute stress-energy tensor w NEC OK NEC OK TMN nMnN P = -2 ρ NEC violated NEC violated w k 1+w √ ψ(η) = ± ln η 6 + ψ0 V (ψ) = V0 e 3(1+w)ψ 1 + 3w
  33. 33. Assumptions • M closed and compact, or a quotient of c.c. M/G • Higher-dimensional action has Einstein-Hilbert form ...includes g(φ)R and F(R) models • Arbitrary other matter fields present • All four-dimensional statements refer to the Einstein frame metric and its associated cosmology. Denote by claim proven in the “long” paper 0802.3214
  34. 34. Three tools...
  35. 35. Three tools... 1. Decomposition of metric time derivative. (Proxy for KK scalars and/or metric moduli)
  36. 36. Three tools... 1. Decomposition of metric time derivative. (Proxy for KK scalars and/or metric moduli)
  37. 37. Three tools... 1. Decomposition of metric time derivative. (Proxy for KK scalars and/or metric moduli) 2. One-parameter family of averages on the manifold
  38. 38. Three tools... 1. Decomposition of metric time derivative. (Proxy for KK scalars and/or metric moduli) 2. One-parameter family of averages on the manifold 3. “NEC probes.” If negative the NEC must be violated.
  39. 39. Three tools... 1. Decomposition of metric time derivative. (Proxy for KK scalars and/or metric moduli) 2. One-parameter family of averages on the manifold 3. “NEC probes.” If negative the NEC must be violated.
  40. 40. ...many challenges • non-uniqueness of KK inversion • (equiv) 4D theory gives only nonlocal information on M • lack of explicit moduli space description • apparent ghosts from conformal deformations of M • “arbitrary” warp factor and deformations of M • nature of the physics that causes acceleration? • ...
  41. 41. ...many challenges • non-uniqueness of KK inversion • (equiv) 4D theory gives only nonlocal information on M • lack of explicit moduli space description • apparent ghosts from conformal deformations of M • “arbitrary” warp factor and deformations of M • nature of the physics that causes acceleration? • ...
  42. 42. Averages of NEC probes
  43. 43. Averages of NEC probes
  44. 44. Averages of NEC probes 1. Metric deformations enter with negative sign
  45. 45. Averages of NEC probes 1. Metric deformations enter with negative sign 2. Negative definite
  46. 46. Averages of NEC probes 1. Metric deformations enter with negative sign 2. Negative 3. Coefficients depend on definite averaging parameter A.
  47. 47. Averages of NEC probes 1. Metric deformations enter with negative sign 2. Negative 3. Coefficients depend on definite averaging parameter A. 4. “Arbitrary” warp function enters the NEC condition
  48. 48. Averages of NEC probes 1. Metric deformations enter with negative sign 2. Negative 3. Coefficients depend on definite averaging parameter A. 4. “Arbitrary” warp function enters the NEC condition 5. kN depends on dξ/dt
  49. 49. The v-equation Minimax strategy -- choose A to minimise the maximum number of e-foldings
  50. 50. The v-equation Minimax strategy -- choose A to minimise the maximum number of e-foldings
  51. 51. The v-equation Minimax strategy -- choose A to minimise the maximum number of e-foldings “Optimising” A: value of A for which all coefficients are positive**
  52. 52. The v-equation Minimax strategy -- choose A to minimise the maximum number of e-foldings “Optimising” A: value of A for which all coefficients are positive** An optimising A always exists.
  53. 53. The v-equation Minimax strategy -- choose A to minimise the maximum number of e-foldings “Optimising” A: value of A for which all coefficients are positive** An optimising A always exists. Optimal solution: saturates the inequalities 3N and kN. Any other solution has fewer e-foldings consistent with the NEC.
  54. 54. The v-equation Minimax strategy -- choose A to minimise the maximum number of e-foldings “Optimising” A: value of A for which all coefficients are positive** An optimising A always exists. Optimal solution: saturates the inequalities 3N and kN. Any other solution has fewer e-foldings consistent with the NEC. This gives a differential equation and boundary conditions obeyed by the optimal solution
  55. 55. Phase plots
  56. 56. Phase plots k=2 k=6 v u w w
  57. 57. Phase plots k=2 k=6 v u w w
  58. 58. Phase plots k=2 k=6 v u w w
  59. 59. e-folds (I) k=1-4 k=5-10 and k = 11 ... N N w w
  60. 60. e-folds (II) w w number of extra dimensions
  61. 61. e-folds (II) w nS ≥ 0.90 w ≤ -0.97 w nS ≥ 0.90 w ≤ -0.97 number of extra dimensions
  62. 62. Conclusions • de Sitter case: weakened the energy condition to the NEC in a variety of cases • w > -1 case: constrained the number of allowed e- foldings of accelerated expansion consistent with energy conditions • Instead of constructing examples of models, we make statements about broad classes. • Challenge -- many models with acceleration must violate energy conditions consistently to be viable.
  63. 63. STOP

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