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# 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