Branes with finite thickness

K – Fields and Brane Worlds

arXiv:0805.3278 [hep-th] C. Adam, P. Klimas, J. Sanchez-Guillen (Santiago de Compostela U) , N.E.G. (La Plata U.), A. Wereszczynski (Jagiellonian U.)

arXiv:0711.3550 [hep-th] C. Adam, J. Sanchez-Guillen (Santiago de Compostela U), N.E.G. (La Plata U.) , A. Wereszczynski (Jagiellonian U.)

arXiv:0705.3554 [hep-th] C. Adam, J. Sanchez-Guillen (Santiago de Compostela U), N.E.G. (La Plata U.) , A. Wereszczynski (Jagiellonian U.)

Kaluza-Klein dimensional reduction Background solution Perturbations Gravity localization Matter localization

Randall-Sundrum dimensional reduction Background solution Perturbations Gravity localization Matter localization

Randall-Sundrum on thick branes Background solution Perturbations Gravity localization Matter localization

Randall-Sundrum on finite thicknes branes Matter localization on V-compactons Matter localization on K-compactons

Conclusions and outlook

Plan of the talk

No materialsupport

No extra matter

Anzatz2 2 2

Minkds ds dz

Product space(3,1) (1)ISO U

Kaluza-Klein

Solution of Einstein(3,1)ISO 2AB ABG T

0ABT

Kaluza-Klein dimensional reduction

DelocalizedsolutionSolution

Equations2

4( ) 0m h 2( ) ( ) 0z m z

Decomposition( ) ( )g z h x , ,zz zg g g

No materialsupport

Kaluza-Klein

Perturbations

AB AB ABg g 2

AB ABG T

( ) imzz e

Kaluza-Klein dimensional reduction

Delocalizedsolution

A priori gravity localization

Compactification

Kaluza-Klein

z z l 4d Graviton

Quantization2 /m n l

0m

No materialsupport

Kaluza-Klein dimensional reduction

A priori matter localization

Solution2 /( ) i n z lf z e

No materialsupport

Kaluza-Klein

Matter

vac

Equations

Decomposition

24( ) ( ) 0m x

( ) ( )f z x

2''( ) ( ) 0f z m f z

( ) imzf z e

A priori gravity localization

Kaluza-Klein dimensional reduction

With material support

With extra matter

Solution of Einstein

3( ) | |A z z

Anzatz2 ( ) 2 2A z

Minkds e ds dz

Randall-Sundrum

Warped space(3,1)ISO 2

AB ABG T

2 2( )T

AB A B ABT z g

1 2 212T

Randall-Sundrum dimensional reduction

Localized solution

Dynamical gravity

localization

Potential

Perturbations

2( ) (| | ) ( )V z z b c z

Decomposition( )/2 ( ) ( )A zg e z h x

With materialsupport

Randall-Sundrum

AB AB ABg g g

Equations2

4( ) 0m h 2AB ABG T

, ,zz zg g g

2 2( ) ( ) ( )z V z z m z

Randall-Sundrum dimensional reduction

A priori matter localization

A priori matter localization

With materialsupport

Randall-Sundrum

Matter

5 4( ) ( ) ( )z L LSolution

Equation

4( ) 0z

( ) ( ) ( )z z x

Dynamical gravity

localization

Randall-Sundrum dimensional reduction

With thickmaterial supportWith extra matter

( ) ( )

2 2

T z zAB A B ABT g

2 2( ) 3 '( )z A z

2 ( ) 3 ''( )T z A r

Randall-Sundrum on thick branes

Anzatz2 ( ) 2 2A z

Minkds e ds dz

Thick branes

Warped space(3,1)ISO

Solution of Einstein2AB ABG T( )A z

Localized solution

Dynamical gravity

localization

Potential

Decomposition

With thickmaterial support

PerturbationsEquations

2( ) z zzV z aA bA

( )/2 ( ) ( )A zg e z h x

Thick branes

AB AB ABg g g 2

4( ) 0m h 2AB ABG T

, ,zz zg g g

2 2( ) ( ) ( )z V z z m z

Randall-Sundrum on thick branes

Equations12 ( )!

(0) ... 0n nz n

V

With thick material support

With infinitematerial support

2 ( ) 0z V

Infinite support

Thick branes

Matter2 2

5 ( ) ( ) ( )V L

Anzatz0 ( )vac z

Dynamical gravity

localization

Close to the vaccum2/(2 )( ) na z b

Randall-Sundrum on thick branes

Dynamical matter localizationClose to the

vacum

With infinite material support

PerturbationsEquations

Anzatz

Dynamical gravity

localization

Thick branes

0

( ) ( )f z x

2 20( ''( )) ( ) ( )z V f z m f z

24( ) ( ) 0m x 5 0''( )V

2 ''( )( ) vacm V zf z e

Randall-Sundrum on thick branes

With finite material support

With infinite material support

“V” shaped potentialsEquations12 ( )!

(0) ... 0n nz n

V

Dynamical matter localization

Compactons

Randall-Sundrum on V-compactons

Dynamical gravity

localization

Matter2 2

5 ( ) ( ) ( )V L

Anzatz0 ( )vac z

2 ( ) 0z V ( ) ( )nvacV

Close to the vaccum2/(2 )( ) na z b

With infinitematerial support

With finitematerial support

Fields with “K” termsMatter

Compactons

2 25 ( ) ( ) ( )V L 4 25 ( ) ( ) ( )V L

Equations14 ( )!

(0) ... 0n nz n

V

Dynamical matter localization

Randall-Sundrum on K-compactons

Dynamical gravity

localization

Anzatz

0 ( )vac z

4 ( ) 0z V

Close to the vaccum4/(4 )( ) na z b

Randall-Sundrum on K-compactons

Dynamical matter localization

With finite material support

Perturbations0

Equations

Anzatz( ) ( )f z x

1 120 0 02 2' '' 3 '' ' ''( ) 0f m f f V f

Dynamical gravity

localization

24( ) ( ) 0m x 2 2

0 0 0 0' 2 6 '' ') ''( ) 0z z V

Compactons

Matter is confinedMatter is confined0''( ) 0V f

Can gravity be also confined?

K-essence like theoriesStability

Matter is confinedSelf graviting solution

Exact compacton solution

Fine tunning problem

Applications to brane cosmology

Other geometries

Conclusions and Outlook

Dynamical matter localization

Dynamical gravity

localization

With finitematerial support

Compactons

¡Thanks!