Oxidised cosmic acceleration

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Talk given at Cambridge DAMTP on Friday, 20 June 2008. Describes recent work on understanding what is necessary to embed accelerating cosmology in higher-dimensional theory.

Talk given at Cambridge DAMTP on Friday, 20 June 2008. Describes recent work on understanding what is necessary to embed accelerating cosmology in higher-dimensional theory.

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  • 1. Oxidised cosmic acceleration Daniel Wesley (DAMTP)
  • 2. Outline 1. Review: a. energy conditions b. past “no-go” theorems 2. New results 3. Observational connections, work-in-progress, speculations.
  • 3. Null Strong TMNnMnN ≥ 0 RMNtMtN ≥ 0 “geodesics P P “gravity converge” ≈ is stable” ρ ρ “gravity is ρ attractive” TMNtMtN ≥ 0 TMNtM not S.L. Dominant Weak P P “subluminal “energy density phase velocity” positive for all ρ ρ observers” ρ ≥ |P | ρ≥0 ρ+P ≥0
  • 4. Null Strong TMNnMnN ≥ 0 RMNtMtN ≥ 0 ρ+P ≥ 0 ρ+3P ≥ 0, ρ+P ≥ 0 •ρ+P≥0 • ρ + P ≥ 0 and ρ + 3P ≥ 0 • Scalars with pos KE, any V(φ) • Scalars with V(φ) ≤ 0 • D-branes & positive tension objs. • ... classical 11D SUGRA + others • Implied by all other E.C.’s • anti-de Sitter Λ<0 • dust and radiation • Casimir energy (...unless averaged) • Scalars with V (φ) >0 anywhere • Negative tension (orientifold planes) • de Sitter Λ>0 • ghost condensates • D-branes
  • 5. Why do we need the NEC? Causality violations Quantum instabilities “warp drives,” traversable non-unitarity, ghosts, wormholes, time machines, CTCs vacuum decay Morris, Thorne Am. J. Phys. 56 (1988) 395 ; Visser, Kar, (many references...) 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 Classical instabilities Breakdown of gravitational thermodynamics Big Rips, Big Bounces, gradient instabilities ... perpetuum mobile, apparent second law violations Cline, Jeon, Moore Phys. Rev. D 70 (2004) 043543 ; Hsu, Dubovsky, Gregoire, Nicolis, Rattazzi JHEP 0603 (2006) Jenkins, Wise Phys. Lett. B 597 (2004) 270 ; Buniy, Hsu, 025 ; Arkani-Hamed, Dubovsky, Nicolis, Trincherini, Murray Phys. Rev. D 74 (2006) 063518 ; Caldwell Phys. Villadoro JHEP 05 (2007) 055 ; Dubovsky, Sibiryakov Lett. B 545 (2002) 23 ; Caldwell, Kamionkowski, Weinberg Phys. Lett. B 638 (2006) 509 Phys. Rev. Lett. 91 (2003) 071301 (many others...)
  • 6. Previous results: (no-go theorems) 1. Gibbons 2. Maldancena, Nunez 3. Warping
  • 7. Theorem (Gibbons ’84) To obtain a four-dimensional de Sitter universe from a static warped reduction on closed compact manifold M, if the Einstein equations hold, one must violate the Strong Energy Condition (SEC). SEC requires
  • 8. Maldacena and Nunez ‘01 One cannot obtain four-dimensional de Sitter space from static compactifications with all the following properties: 1. Einstein-Hilbert gravity in higher-D 2. Potential energies all non-positive: V ≤ 0 3. Fields in the theory: a. ... have no mass terms b. ... have positive definite kinetic terms 4. Extra dimensions M: a. .... give finite four-dimensional GN (M has finite warped volume) b. ... compact, no boundary
  • 9. Maldacena and Nunez ‘01 One cannot obtain four-dimensional de Sitter space from static compactifications with all the following properties: 1. Einstein-Hilbert gravity in higher-D 2. Potential energies all non-positive: V ≤ 0 imply 3. Fields in the theory: SEC a. ... have no mass terms b. ... have positive definite kinetic terms 4. Extra dimensions M: a. .... give finite four-dimensional GN (M has finite warped volume) b. ... compact, no boundary
  • 10. A warping no-go theorem positive definite 1 Warping requires NEC p ar W SE NEC violated if... 0 C SEC violated if... -1 -> Obtaining a warp factor requires neither SEC nor NEC violation. -2 -1 0 1
  • 11. New results
  • 12. Improvements on previous results • Time-dependent compactifications • Four-dimensional cosmology is not exactly de Sitter • Weakening of energy conditions <--> strengthening of theorems. • Claim: transient acceleration is allowed, and explicit bounds on the number of allowed e-foldings can be constructed. Statement • 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 • 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✻). • Exceeding the bound violates an energy condition • Can be the strong energy condition (SEC) • Can also be the null energy condition (NEC)
  • 13. Curved Curvature-free CRF R≠0 on M R=0 on M Non-Abelian Special holonomy Klebanov-Strassler continuous warped throat Sp(n) isometries** Spin(7) SU(n) (Calabi-Yau) Giddings-Kachru- ... includes models which G2 (M theory) Polchinski flux obtain 4D gauge symmetries by KK solutions One-dimensional reduction Original Kaluza-Klein Randall-Sundrum Rugby-ball SLED Tori Rn / Zn ** We only know these cannot be with R ≥ 0 Ricci-flat, which is a stronger condition than “curvature-free.”
  • 14. curved curvature-free CRF Strong (Gibbons et. al.) dS Strong, Null* Null Null *bounded avg. condition w>-1 Strong Null Null (transient) (transient) (transient)
  • 15. A simple example
  • 16. 4D universe with constant w embedded with k extra dimensions, which are Ricci-flat and evolve by breathing mode only. a(η) ∼ η 2/(1+3w) 2 1+w dψ ρ+P = H= 2 3 dη 1+w √ ψ(η) = ± ln η 6 + ψ0 V (ψ) = V0 e 3(1+w)ψ 1 + 3w 2c ds2 = A(η) −dη + dx2 + exp ψ(η) ds2 2 2 4+k 3 k k 2k a(η) = ecψ/2 A(η) c= k+2
  • 17. 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 √ V (ψ) = V0 e 3(1+w)ψ This unusual (even pathological) behavior is completely invisible from 4D
  • 18. For a more general case... • Extra dimensions could distort •Anisotropically •Inhomogeneously • Warped extra dimensions • Non-Ricci-flat extra dimensions • Scalar field might not be breathing mode • Metric moduli may not act like quintessence • Completely different scalar could drive accel. • ... etc
  • 19. Assumptions • Higher-dimensional action has Einstein-Hilbert form ...includes g(φ)R and F(R) models • All four-dimensional statements refer to the Einstein frame metric and its associated cosmology. • M closed and compact, or a quotient of such, as M =M’/G • Arbitrary other matter fields present in the action
  • 20. Three tools: 1. Gauge choice 2. Averaging 3. NEC lemma
  • 21. A purely scalar metric transformation which leaves the total volume of M invariant is gauge-equivalent to zero. Local: Global: The last term integrates to zero, and can only cancel out the piece of the first term which integrates to zero -- eg, the non-zero mode
  • 22. A-Averaging Space of functions Q(t,y) on M
  • 23. If either then NEC violated 1. Diagonalise τab obtain real eigenvalues λ1, λ2,... λk. 2. To each λj, associate null vector na = (1,0,0,0,εnλj ) 3. But PDk is the average of the λj. Choose j* with smallest λj*.
  • 24. Einstein equations, RF example
  • 25. This action describes a system of scalars coupled to other matter in 4D Dimensional reduction commutes with action variation
  • 26. Must be non-negative to satisfy NEC
  • 27. 1. Metric deformations enter with negative sign Must be 2. Negative 3. Coefficients depend on non-negative definite when averaging parameter A. to satisfy NEC accelerating 4. “Arbitrary” warp function enters the NEC condition 5. kN depends on dξ/dt
  • 28. Non-negative for all A Positive or negative depending on A “Optimising” A: choice of A for which all coefficients are non-negative An optimising A always exists.
  • 29. “Optimal solution”: saturates the inequalities ρD+PD3 ≥ 0 and ρD+PDk ≥ 0 & any other solution has fewer e-foldings consistent with the NEC. Further assume that w = constant These equalities define the differential equation obeyed by the optimal solution for constant w.
  • 30. “Optimal solution”: saturates the inequalities ρD+PD3 ≥ 0 and ρD+PDk ≥ 0 & any other solution has fewer e-foldings consistent with the NEC. Further assume that w = constant These equalities define the differential equation obeyed by the optimal solution for constant w.
  • 31. k=2 k=6 v u w w
  • 32. k=2 k=6 v u w w
  • 33. k=2 k=6 v u w w
  • 34. k=1-4 k=5-10 and k = 11 ... Number of e-foldings w w
  • 35. (Ricci-flat only) w w (Curved and not CRF only) number of extra dimensions
  • 36. Varying w(t): if w(t) < w✻ then N[w(t)] < N(w✻). Therefore ζ goes faster, and has less to go, for the w(t).
  • 37. “Loopholes” 1. Curvature of M could go either way. what metric should we use? 2. Quantum effects If we are unable to impose the Einstein equations, how literally should we take the extra dimensions? 3. Higher-derivative corrections to GR if effectively NEC-violating, do we avoid usual problems? 4. Negative-tension objects (not really a loophole) braneworld boundaries, O-planes
  • 38. Bubbles of nothing Accelerating to decelerating transition after inflation Kate Marvel’s talk & work-in-progress Are additional sources of NEC violation required? M negative inflating tension boundary nothing M radiation dominated
  • 39. Observational prospects (work-in-progress with Paul Steinhardt) More precise measurements of w(a) and bounds on variation of fundamental constants can prove surprisingly powerful probes of extra-dimensional physics. Simple families of models give promising results. Study of more complicated models requires a little more computing power. Stay tuned.... [Preliminary]
  • 40. Conclusions • We can make very general statements about extra- dimensional physics from the observation that the Universe is/was accelerating • For a broad class of theories, this implies violation of energy conditions, more severe than already known • Raises interesting questions about the role of corrections to GR, explicit higher-D solutions, and exotic negative-tension objects • Can make refined measurements of w and variation of fundamental constants into powerful probes of extra-dimensional physics.