4. Two data sets
ä>0 ä > 0 (?)
Late universe Early universe
(dark energy) (inflation?)
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. 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. 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. 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. Some energy conditions
Null TMNnMnN ≥ 0 RMNtMtN ≥ 0 Strong
P P “geodesics
“gravity
is stable” ρ ρ converge” ≈
“gravity is
attractive”
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. The Null Energy Condition
...keeps us safe from... Causality violations
Instabilities & other pathologies
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. 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. Types of new theorems
de Sitter theorems
Non-de Sitter (w>-1) theorems
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. 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. 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. 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. 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. 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. 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. 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. Comparison with previous theorems
curvature-free curved
Strong
(Gibbons et. al.)
de Sitter
w > -1
24. Comparison with previous theorems
curvature-free curved
Strong
(Gibbons et. al.)
de Sitter NEW NEW
Null Null*
*bounded avg. condition
w > -1
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. 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. 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. 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. 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. 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. 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. 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. 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
35. Three tools...
1. Decomposition of metric time derivative.
(Proxy for KK scalars and/or metric moduli)
36. Three tools...
1. Decomposition of metric time derivative.
(Proxy for KK scalars and/or metric moduli)
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. 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. 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. ...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. ...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?
• ...
44. Averages of NEC probes
1. Metric deformations
enter with negative sign
45. Averages of NEC probes
1. Metric deformations
enter with negative sign
2. Negative
definite
46. Averages of NEC probes
1. Metric deformations
enter with negative sign
2. Negative 3. Coefficients depend on
definite averaging parameter A.
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. 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. The v-equation
Minimax strategy -- choose A to minimise
the maximum number of e-foldings
50. The v-equation
Minimax strategy -- choose A to minimise
the maximum number of e-foldings
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. 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. 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. 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
61. e-folds (II)
w
nS ≥ 0.90 w ≤ -0.97
w
nS ≥ 0.90 w ≤ -0.97
number of extra dimensions
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.
