Intersecting membraneIntersecting membraneandand

an anisotropic models of dark energyan anisotropic models of dark energy

Dmitry G. Orlov (NCU, Taiwan; VNIIMS, Dmitry G. Orlov (NCU, Taiwan; VNIIMS, Russia)Russia)

1st June, 2008 NDHU, Taiwan

• IntroductionIntroduction - effective cosmological constant- effective cosmological constant- world on brane- world on brane- space like brane- space like brane

• S - brane intersectionS - brane intersection

• Anisotropic cosmologyAnisotropic cosmology

• ConclusionsConclusions

Effective cosmological constantEffective cosmological constant

Equation of state:Equation of state:

1

3

pw

In particularly, this condition must be peformed for an inflation stage.In particularly, this condition must be peformed for an inflation stage.The matter (energy) which satisfied this equation is named The matter (energy) which satisfied this equation is named a dark matter. It’s easy check, that an equation of state a dark matter. It’s easy check, that an equation of state for a cosmological constant is for a cosmological constant is , what provide an inflationary , what provide an inflationary stage, but until now it’s still unclear what is physical meaning stage, but until now it’s still unclear what is physical meaning of such quantity and a mechanism which turn it on and off. of such quantity and a mechanism which turn it on and off.

- accelerated expansion of universe- accelerated expansion of universe

1w

From another hand, it’s possible to consider evolution of field, which From another hand, it’s possible to consider evolution of field, which on some particular stage of evolution generate an effective on some particular stage of evolution generate an effective cosmological constant.cosmological constant. The simplest sample for such field is a scalar field.The simplest sample for such field is a scalar field. If for some interval of time we obtain that a potential energy of field If for some interval of time we obtain that a potential energy of field is positive and greater than kinetic, that produce a negative pressure is positive and greater than kinetic, that produce a negative pressure and an accelerated stage in an evolution of system.and an accelerated stage in an evolution of system.

The contribution of scalar field in action for this interval of time The contribution of scalar field in action for this interval of time may be consider like effective cosmological constant. may be consider like effective cosmological constant.

Exterior dimension:Exterior dimension:- a long range (Randall-Sundrum models)- a long range (Randall-Sundrum models)- a compactified (periodical)- a compactified (periodical)

World on braneWorld on brane

From string theory D-brane is known like hypersurface which From string theory D-brane is known like hypersurface which is is described by an end of open string satisfied of Dirichlet described by an end of open string satisfied of Dirichlet boundary condition. Such object is supported by form field with boundary condition. Such object is supported by form field with RR-charge.RR-charge. If we consider Dirichlet condition for time-like direction we If we consider Dirichlet condition for time-like direction we obtain space-like hyperbrane or simple s-brane. Exist also obtain space-like hyperbrane or simple s-brane. Exist also another description of such object like unstable tachyon another description of such object like unstable tachyon condensate, which exist only one moment and then decay.condensate, which exist only one moment and then decay. From begining intesions of people about s-brane was to From begining intesions of people about s-brane was to construct dS/CFT correspondence, but when it was found only construct dS/CFT correspondence, but when it was found only avaible for theory II*, s-brane was gotten another application avaible for theory II*, s-brane was gotten another application in construction of cosmological model. in construction of cosmological model.

Space like braneSpace like braneChen, Gal'tsov, Gutperle, Phys.Rev. D66 (2002) 024043Chen, Gal'tsov, Gutperle, Phys.Rev. D66 (2002) 024043Kruczenski, Myers, Peet, JHEP Kruczenski, Myers, Peet, JHEP 02050205, 039 (2002), 039 (2002)

S-brane intersectionS-brane intersection

We consider system consists of gravity and form field We consider system consists of gravity and form field coupled with dilaton (scalar field):coupled with dilaton (scalar field):

2 2[ ]

1 1( )

2 2D a

qS d x g R e Fq

We choose follow ansatz for metric:We choose follow ansatz for metric:

22 2 2 2 2 2 2 2 21

0

( )i

pBA C E

i k q ki

ds e dt e dx e d e dz dz

for k=-1,0,1 - for k=-1,0,1 - for cases of hyperbolic, flat and spherical exterior for cases of hyperbolic, flat and spherical exterior space.space. The equations of motion for this model are invariant The equations of motion for this model are invariant under the discrete S-duality:under the discrete S-duality:

which transform electrical charged soluion to magnetic one and v.v., which transform electrical charged soluion to magnetic one and v.v., so we restrict our further investigation only to purely magnetic case.so we restrict our further investigation only to purely magnetic case.

ag g F F

In previous papers it was consider s-brane solution for and In previous papers it was consider s-brane solution for and without a flat part of exterior space. For every type solution was without a flat part of exterior space. For every type solution was obtained an inflationary stage, but without large enough e-folding (<70),obtained an inflationary stage, but without large enough e-folding (<70),so we interesting of result so we interesting of result anisotropizationanisotropization and effect of cylindrical and effect of cylindrical exterior space. exterior space.

i jB B

1 0 1 0

1 0 1 0

1 0

1 0

1( )

( )

1

( )

1

2( 1) ( )

1 1.

2( 1) ( )

i ii

qB G g g d d

p q

a pG f f E G

p q

k pA H G c c

k p q

pC H G c c

k p q

The solution is:The solution is:

[ ] vol( )k kF b

The metric of s-brane for p=2 can be represented in the follow form:The metric of s-brane for p=2 can be represented in the follow form:

Anisotropic cosmologyAnisotropic cosmology

2 2 2 2 2 2 24 ,k q kds a ds b ds c ds

now we compatify an axterior dimensions on q-torus and withnow we compatify an axterior dimensions on q-torus and with2 ( ) 2 2 2

4 ( ) ( )k q ka b c ds ds ln b ln c

obtain the reducted action in Einstein frame:obtain the reducted action in Einstein frame:

2 2 244

4

1 1 1( ) ( 2) ( ) ( 2)( )( )

2 2 2

( )

R k k q k q kS dx g

k q k V

with a potential of scalar with a potential of scalar fields :fields :

and four dimensions metric:and four dimensions metric:

23 3( ) ( 2) ( )( 1) ,

2a k q k k q kb

V e k k e

2 222 ( ) 2 2

4 20( 1)

iBkC q k E Ai

i

dds e e e dx

k

The completely anisotropic modelThe completely anisotropic model is decribed by metric:is decribed by metric:

2 2 2 2 2 2 220 1 3ˆds d b dx c dx d dxt

wherewhere

andand

Scale factors and shears:Scale factors and shears:

Hubble constant for every direction:Hubble constant for every direction:

2

0

3 3( )ˆ1 2 2

Ki

i

d k q kdt e K C E B

k

0 1 22 ( ) 2 2 ( ) 2 2 ( ) 2kC q k E B kC q k E B kC q k E Bb e c e d e

1 1 13 3 33

1 2( ) ( ) ( )Ka bcd e b c b d

0 1где ( 1) ( ) ( 1)2 2

K i Ki i

k EH k e C q k B d k e M

Further investigation of obtained model was done by numerical method. We has few which solution is dependent from, vary this parameters we try to extend maximum amount e-folding.

First we consider the question of flat component of compactified exterior space, if we have such part, it’s only decreasing e-folding, so we need set q=k, exclude flat part.

Second question was a changing in behaviour of system after introducing anisotropy.For the cases of flat and spheric exterior space we have change the property of universe from a tube to a pancake (or vice versa depending from a sign of initial parameters) during evolution through inflation stage. Another property we obtain for hyperbolic space, in this case initial anisotropy almost neglect and finally disappear on a time infinity.

1

thick curve - a condition for an expansion,thick curve - a condition for an expansion,thin curve - a condition for an acceleration phasethin curve - a condition for an acceleration phase(left picture - anisotropic case, right - tuned anisotropic) (left picture - anisotropic case, right - tuned anisotropic)

Consideration an anisotropic improve e-folding of solution, Consideration an anisotropic improve e-folding of solution, but it still not enough for standart model (60-70 e-folding), but it still not enough for standart model (60-70 e-folding), so we guess that resources of this model was over and so we guess that resources of this model was over and if we like better e-folding we have to consider hybryd models.if we like better e-folding we have to consider hybryd models.

1

Dependence w of equation of state during inflation stage for Dependence w of equation of state during inflation stage for hyperbolic exterior hyperbolic exterior space (isotropic and tuned anisotropic cases).space (isotropic and tuned anisotropic cases).

1. Considering anisotropic space like brane allow constract anisotropic 1. Considering anisotropic space like brane allow constract anisotropic cosmological solution with inflation stage.cosmological solution with inflation stage.

2. Flat part of exterior space only make worse inflation stage and 2. Flat part of exterior space only make worse inflation stage and must be exluded.must be exluded.

3. In the cases of a pure flat and spherical exterior space the solution 3. In the cases of a pure flat and spherical exterior space the solution has diverge (or tends to zero) of metric functions on time infinity, has diverge (or tends to zero) of metric functions on time infinity, so they don’t make agree with modern theories.so they don’t make agree with modern theories.

4. The most interesting result was obtained for case of hyperbolic space.4. The most interesting result was obtained for case of hyperbolic space.We have greatest value of e-folding and also an initial We have greatest value of e-folding and also an initial anisotrophy tend to zero after inflation stage.anisotrophy tend to zero after inflation stage.

5. Although amount of e-folding in anisotropic model was increase, 5. Although amount of e-folding in anisotropic model was increase, it’s still not enough.it’s still not enough.

ConclusionsConclusions