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

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

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

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

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5 The property of a codimension 2 brane is quite different from that of the codimension 1 brane. The property of a codimension 2 brane is quite different from that of the codimension 1 brane. 6D braneworld 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 ?

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

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7 Nevertheless, as a toy braneworld model with two essential features 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 stabilized extra dimensions Higher codimensions Flux stabilization of extra dimensions would be useful 6D model (2D bulk) gives the simplest example C.f. in 5D d d is not fixed originally quantum corrections, … additional mechanism

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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

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

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

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

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

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

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

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

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

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

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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

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

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20 The 6D brane cosmological solutions are stable against the tensor perturbations. The 6D brane cosmological solutions are derived via the dimensional reduction from the higher- dimensional de Sitter brane solutions 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 Summary 1

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

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22 Scalar perturbations KK decomposition

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23 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. A tachyonic mode appears for the expansion rates

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24 Dynamical v.s. “thermodynamical” instabilities 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. See the next slide

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

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

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

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28 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. “Thermodynamical stability” conditions Some Identities

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

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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

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

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

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

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

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

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

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

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38 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. Summary

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

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

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