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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!