Dynamics of p-brane system

Kunihito Uzawa

References:(1) G.W. Gibbons, H. Lu, C.N. Pope; Phys.Rev.Lett.94:131602,2005(2) D.Kastor & J. Traschen soltuion; Phys.Rev. D 47 (1993) 5370

(3) H. Kodama & K. Uzawa; JHEP 0507:061,2005(4) H. Kodama & K. Uzawa; JHEP 0603:053,2006(5) P. Binetruy, M. Sasaki, K. Uzawa, arXiv:0712.3615

[1] Introduction

・Analysis of the early universe
Construction of the cosmological model that explains theobservational facts on the basis of fundamental theory (String orSUGRA theory)

It is necessary to obtain the dynamical solution of the field equations(including the Einstein equations) , and to extract information of
the cosmological behavior.
dynamics of 4d or internal space, symmetry breaking (SUSY, … )Cosmological solution in the higher dimensional theory
String theory

Higher dimensional gravity(e.g. Brane world model)
・ Inflation model・ Stabilization of the scaleof internal spacecomplementary・Low energy SUGRA
・Dynamical solution ofEinstein equation

cosmological model

Construction of the cosmological scenario

Overview☆ 4 dimensional Gravity・ Charged BH solution (RN)
(Reissner & Nordstrøm )・ Dynamical solution in Einstein-Maxwell
theory (Kastor & Traschen)★ Higher dimensional Gravity
・ p-brane solution of SUGRA(Horowitz & Strominger)

・ Intersecting brane(Guven, Papadopoulos & Townsend, Ohta)

・ Dynamical p-brane solution(Gibbons, Binetruy, Kodama,Sasaki, Uzawa)・ Dynamical solution ofintersecting brane(Binetruy, Sasaki, Uzawa)

[2] Dynamical solution of 4-dim gravity(D.Kastor & J. Traschen ; Phys.Rev. D 47 (1993) 5370 )・ 4-dim Einstein-Maxwell system + cosmological constant◇ 4-dimensoinal action ：☆ Solution of field equations ：⇒Time dependent solution☆ Cosmological dynamics ⇒ linear function of time◆ Dynamics of the spatial background

Near the, the metric denotes the geometry for cylinder.

Spatial surface are asymptotically flat.

Spatial infinity asymptotically cylindrical.

： typical coordinate distance from the center

Singularity moves in from spatial infinity.

Singularity splits and eventually surrounds
each of the throats individually.

[3] Dynamical solutoin of p-brane system(G.W. Gibbons, H. Lu, C.N. Pope Phys.Rev.Lett.94:131602,2005)(P. Binetruy, M. Sasaki, K. Uzawa, arXiv:0712.3615)

Let us consider the case of an arbitrary p-brane backgroundThe dynamical background of the p-brane can be written by

In thecase, the field equations are reduced to・ Internal and external space are Ricci flat.

From the assumption, the Einstein equations are

Last equation leads

The gauge field equation yields

Under the assumptions, the scalar field equation reads

Thus, unless the parameter c is zero, the warp factor should satisfy the equations

In thecase, the field equations are reduced to◆ For example, in the case ofthe solution is(G.W. Gibbons, H. Lu, C.N. Pope; Phys.Rev.Lett.94:131602,2005）: constant parameters

For example,

★ Kaluza-Klein compactification
・ "product type" ansatz

・ Kastor & Traschen
・ Dynamical p-brane

★ Dynamics of 4-dimensional spacetime (scalar field: constant)◇ Let us consider the casein more detail.

In this case, the solution for the warp factor
explicitly as

can be obtained

In the following, we consider the simple case

If we introduce a new time coordinateby■ metric of ten-dimensional spacetime :; 4d scale factor is proportional to☆ Constants

are nonzero ；★metric of 3-brane in ten dimension

For, the metric exists inside a domain； bounded by the level set

Small positive,

is large.

Large positive,

is small.

increases, domain

shrinksSmall positive,

Is large.

Large positive,

is small.
Universe splits into disconnect regions.

increases

D3-brane

Individual D3-brane

★(p+1)-dimensional effective theory (No flux case)★ Ansatz for background★ Field equations are reduced to□ lower-dimensional effective action

・No flux and internal space is Ricci flat space
・Scalar field satisfies the equation of motion.
★ Ansatz for background□ (p+1)-dimensional field equations

★(p+1)-dimensional effective theory with Flux

◇ D-dimensional model● Ansatz for background・Internal space is Einstein space・ Gauge fields satisfy field equations.
◎ D-dimensional action□ (p+1)-dimensional effective action

・No flux and internal space is Ricci flat space

Conformal transformation :★ Field equations□ (p+1)-dimensional effective action

・No flux and internal space is Einstein space

Conformal transformation :★ Field equations

☆ p=3, D=10 case, ⇒ 4-dimensional moduli potential (Einstein frame)[4] Summary :★ The solutions we found have the property that they are genuinely higherdimensional in the sense that one can never neglect the dependence on
say of.
・ The same results hold for other intersecting brane model(D2-D6 brane, D2-D4-D8 brane, D3-D7,brane, M2-M5 brane system).

☆ Warped structure :linear combination of theand→ 10-dimensional IIA, IIB and 11-dimensional supergravity→ metric ansatz or supersymmetry★ Further calculations of the cosmological dynamics :・ Construction of the special solution of Einstein equation→ other black intersecting brane solution→ stabilization and application to cosmology