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6D Brane Cosmological Solutions

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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

~ Introduction

~ 6D braneworld

~ 6D brane cosmological solutions

~ Tensor perturbations

~ Stability

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 , …

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

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 ?

The property of a codimension 2 brane is quite different from that of the codimension 1 brane .

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 ?

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,…

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

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)

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

Northern pole (+-brane)

Southern pole (--brane)

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

D-dimensional Einstein space

has two positive root at

We compactify (D-4) dimensions in

Magnetic charge

Upper bound

(D+2)-dimensional seed solutions

Northern pole (+-brane)

Southern pole (--brane)

Warped

generalization

Power-law inflationary solutions since

From the (D+2)-dimensional de Sitter brane solutions

D-dimensional de Sitter spacetime

6D cosmological solutions

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)

KK decomposition

TT polarization tensor

Tensor perturbations in 6-dim dS solutions

= Tensor perturbations in (D+2)-dim dS solutions

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

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

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

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)

Scalar perturbations

KK decomposition

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.

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

Thermodynamical relations

Area of de Sitter horizon

Magnetic flux

Deficit angles (=brane tensions)

D-dimensional de Sitter

has two positive root at

Upper bound

Intensive variables

The (+)-brane point of view

“Thermodynamics”

Somewhat similar to the BH therodynamics

“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 .

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.

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

Two possibilities: toward a stable solution with a smaller radius

decompactification

effective potential

Flux conservation relates the initial vacuum to final one.

Two possibilities: toward a stable solution with a smaller radius

decompactification

effective potential

Flux conservation relates the initial vacuum to final one.

Two possibilities: toward a stable solution with a smaller radius

decompactification

effective potential

Flux conservation relates the initial vacuum to final one.

Two possibilities: toward a stable solution with a smaller radius

decompactification

effective potential

Flux conservation relates the initial vacuum to final one.

Two possibilities: toward a stable solution with a smaller radius

decompactification

effective potential

Flux conservation relates the initial vacuum to final one.

Two possibilities: toward a stable solution with a smaller radius

decompactification

effective potential

Flux conservation relates the initial vacuum to final one.

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.

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.

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

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 !