6D Brane Cosmological Solutions

1. 6D Brane Cosmological Solutions Masato Minamitsuji (ASC, LMU, Munich) T. Kobayashi & M. Minamitsuji, JCAP0707.016 (2007) [arXiv:0705.3500] M. Minamitsuji, CQG 075019(2008) [arXiv:0801.3080] CENTRA, Lisbon, June 2008

2. ~ Introduction ~ 6D braneworld ~ 6D brane cosmological solutions ~ Tensor perturbations ~ Stability Contents

3. Introduction bulk Brane (SM) (Gravity) Motivated from string / M-theory Braneworld One of the most popular and mostly studied higher-dimensional cosmological scenarios in the last decade Matter (SM particles) are confined on the brane while Gravity can propagate into the bulk Gauge hierarchy problem, Inflation, Dark energy , …

4. Vanishing cosmological constant cannot be obtained unless one fine-tunes the value of the brane tension. 5D braneworld Randall-Sundrum (II) model (RS 1999) Localization of gravity by strong warping Standard Cosmology 3-brane

5. Codimension 2 brane ~Conical singularity Codimension 1 Codimension 2 The tension of the brane is absorbed into the bulk deficit angle and does not curve the brane geometry Self-tuning of cosmological constant ? 6D braneworld The property of a codimension 2 brane is quite different from that of the codimension 1 brane .

6. Caroll & Guica (03), Navarro (03), Aghababaie, et.al (03) We assume that for a given After the sudden phase transition on the brane , it seems to be plausible that the brane keep the initial flat geometry. however, because of the flux conservation Vinet & Cline (04), Garriga & Poratti (03) Models with the compact bulk Rugby-ball shaped bulk The compact bulk is supported by the magnetic flux Self-tuning of the cosmological constant ?

7. Nevertheless, as a toy braneworld model with two essential features Flux stabilized extra dimensions Higher codimensions Stabilization of extra dimensions In comactifying extra dimensions, d.o.f.s associated with the shape and size appear in the 4D effective theory. Flux stabilization of extra dimensions would be useful 6D model (2D bulk) gives the simplest example C.f. in 5D d additional mechanism d is not fixed originally quantum corrections,…

8. Northern pole (+-brane) Southern pole (--brane) generalization Static warped solutions Mukohyama et.al (05) Aghababaie, et. al (03), Gibbons, Gueven and Pope (04) We derive the cosmological version of these solutions

9. Branes in higher co-dimensional bulk Brane tension develops the deficit angle Codim-2 but one cannot put ordinary matter on the brane Codim >2 = black holes or curvature singularities One cannot put any kind of matter on the brane 4D GR Scalar mode associated with the compact dimension need of regularizations of the brane Large distances scales Recovery of 4D GR Cap region 4-brane Codimension-1 Codimension-2 Peloso, Sorbo & Tasinato (06), Kobayashi & Minamitsuji (07)

10. 6D brane cosmological solutions Our purpose is to find brane cosmological solutions in the following 6D Einstein-Maxwell-dilaton theory pure Einstein-Maxwell model gauged supergravity Instead of solving coupled Einstein-Maxwell-dilaton system, we start from (D+2)-dimensional Einstein-Maxwell theory First, we consider seed solutions in higher dimensions

11. Northern pole (+-brane) Southern pole (--brane)

12. with some field identifications For a seed (D+2)-dim solution, we consider the dimensional reduction: Compactified Dimensional reduction The effective 6D theory is the same as the one we are interested in

13. D-dimensional Einstein space has two positive root at We compactify (D-4) dimensions in Magnetic charge Upper bound (D+2)-dimensional seed solutions

14. Northern pole (+-brane) Southern pole (--brane) Warped generalization

15. Power-law inflationary solutions since From the (D+2)-dimensional de Sitter brane solutions D-dimensional de Sitter spacetime 6D cosmological solutions

16. Late time cosmology Power-law solutions are always the late-time attractors From the Kasner-de Sitter solutions The early time cosmology generalizations of solutions found in KK cosmology Maeda & Nishino (85)

17. Tensor perturbations KK decomposition TT polarization tensor Tensor perturbations in 6-dim dS solutions = Tensor perturbations in (D+2)-dim dS solutions

18. 4D observers on the brane measure the KK masses The critical mass Light KK modes may decay slowly First few KK modes Dashed line= critical KK masses

19. Red= The first KK mass Dashed = The critical KK mass For the increasing brane expansion rate, the first KK mass tends to be lighter than the critical one. But one must be careful for the stability of the solutions

20. Summary 1 The 6D brane cosmological solutions are derived via the dimensional reduction from the higher-dimensional de Sitter brane solutions The 6D brane cosmological solutions are stable against the tensor perturbations. For the larger value of the brane expansion rate, the first KK mass of tensor perturbations becomes lighter than the critical one, below which the mode does not decay during inflation

21. Stability Stability of 6-dim dS solutions = Stability of (D+2)-dim dS solutions Minkowski branes stable Yoshiguchi, et. al (06), Sendouda, et.al (07) Lee & Papazoglou (06), Burgess, et.al (06) de Sitter branes unstable for relatively higher expansion rates Kinoishita, Sendouda & Mukohyama (07)

22. Scalar perturbations KK decomposition

23. A tachyonic mode appears for the expansion rates The lowest mass eigenvalue is given by An instability against the scalar perturbations appears in the de Sitter brane solutions with relatively higher expansion rates.

24. Kinoshita, et. al showed the equivalence of dynamical and “thermodynamical” instabilities in the 6D warped dS brane solutions with flux compactified bulk Dynamically unstable solutions = Thermodynamically unstable solutions The arguments can be extended to the cases of higher dimensional dS brane solutions. Dynamical v.s. “thermodynamical” instabilities See the next slide

25. Thermodynamical relations Area of de Sitter horizon Magnetic flux Deficit angles (=brane tensions)

26. D-dimensional de Sitter has two positive root at Upper bound

27. Intensive variables The (+)-brane point of view “Thermodynamics” Somewhat similar to the BH therodynamics

28. “Thermodynamical stability” conditions Some Identities The boundary between unstable and stable solutions is given by the curve, which is determined by the breakdown of one-to-one map from plane to conserved quantities .

29. Special limits 1) 6D limit : The curve is exactly boundary between dynamically stable and unstable modes Kinoshita, Sendouda & Mukohyama (07) 2) unwarped limit The same thing happens in the higher dimensional geometry.

30. Cosmological evolutions Cosmological evolutions from (D+2)-dimensional unstable de Sitter brane solutions Evolution of the radion mode The potential has one local maximum and one local minimum

31. Two possibilities: toward a stable solution with a smaller radius decompactification effective potential Flux conservation relates the initial vacuum to final one.

32. Two possibilities: toward a stable solution with a smaller radius decompactification effective potential Flux conservation relates the initial vacuum to final one.

33. Two possibilities: toward a stable solution with a smaller radius decompactification effective potential Flux conservation relates the initial vacuum to final one.

34. Two possibilities: toward a stable solution with a smaller radius decompactification effective potential Flux conservation relates the initial vacuum to final one.

35. Two possibilities: toward a stable solution with a smaller radius decompactification effective potential Flux conservation relates the initial vacuum to final one.

36. Two possibilities: toward a stable solution with a smaller radius decompactification effective potential Flux conservation relates the initial vacuum to final one.

37. a new dS brane solution The corresponding 6D solution is the stable accelerating, power-law cosmological solutions. Inflation Dark Energy Universe ? an AdS brane solution The corresponding 6D solution is the collapsing Universe.

38. Summary 6D brane cosmological solutions in a class of the Einstein-Maxwell-dilaton theories are obtained via dimensional reduction from the known solutions in higher-dimensional Einstein-Maxwell theory. Higher-dimensional dS brane solutions (and hence the equivalent 6D solutions) are unstable against scalar perturbations for higher expansion rates. This also has an analogy with the ordinary thermodynamics. The evolution from the unstable to the stable cosmological solutions might be seen as the cosmic evolution from the inflation to the current DE Universe.

39. characterizes the effective scalar potential The cosmological evolution may be seen as the evolution from the initial inflation to the current dark energy dominated Universe. Equivalent 6D point of view 4D effective theory for the final stable vacuum

40. Quantum corrections Ghilencea, et.al (05), Elizalde, Minamitsuji & Naylor (07) Stability Minkowski branes Einstein-Maxwell stable Yoshiguchi, et. al (06), Sendouda, et.al (07) Supergravity Lee & Papazoglou (06), Burgess, et.al (06) marginally stable (with one flat direction) de Sitter branes Einstein-Maxwell Kinoishita, Sendouda & Mukohyama (07) dS brane solutions are unstable for relatively higher expansion rates !