cosmological context Neven Bilic, Goran S. Djordjevic, Milan - - PowerPoint PPT Presentation

Neven Bilic, Goran S. Djordjevic, Milan Milosevic and Dragoljub D. Dimitrijevic

RBI, Zagreb, Croatia and FSM, University of Nis, Serbia

Tachyon scalar field in DBI and RSII cosmological context

9th MATHEMATICAL PHYSICS MEETING: School and Conference on Modern Mathematical Physics 18 - 23 September 2017, Belgrade, Serbia

1 Introduction 2 Tachyon field cosmology 3 Tachyon inflation in an AdS braneworld 4 Conclusion

Introduction

• The inflationary universe scenario in which the early universe

undergoes a rapid expansion has been generally accepted as a solution to the horizon, flatness, etc. problems of the standard big-bang cosmology.

• Recent years - a lot of evidence from WMAP, Planck, etc.
• bservations of the CMB.

Introduction

• We study (real) scalar field in cosmological context.
• General Lagrangian action:
• Lagrangian (Lagrangian density) of the standard form:
• Non-standard Lagrangian:

( , ) ( ) 1 2 ( )

tach T X

V T X T    

( , ) ( ) ( ) X V        

4

( ( ), ) S d x g X     



1 2 X g T T

  

  

Introduction

• The action:
• In cosmology, scalar fields can be connected with a perfect

fluid which describes (dominant) matter in the Universe.

• Components of the energy-momentum tensor:

4

( , ) S d x g X T  



2 S T g g

 

    

( ) T P u u Pg

   

   

Introduction

• Pressure, matter density and velocity 4-vector, respectively:

(

, ) ( , ) P X T X T 

( , ) 2 ( , ) X T X X T X      2 T u X

 

 

( ) T P u u Pg

   

   

Introduction

• Total action: term which describes gravity (Ricci scalar,

Einstein-Hilbert action) plus term that describes cosmological fluid (scalar field Lagrangian):

• Einstein equations:

 

4

( , ) S d x g R X T   



1 2 R Rg T

  

 

Tachyon field cosmology

• Tachyon lagrangian:
• EoM:

( , ) ( ) 1

tach T X

V T g T T

  

    

2 2

1 (1 ( ) ) 1 ( ) ( ) T T dV g T T T T V T dT

    

                

Tachyon field cosmology

• Tachyon lagrangian:
• Friedmann equations for spatially homogenous scalar field:

2 2 2 2 1/2

1 3 (1 )

Pl

a V H a M T ( , ) ( ) 1

tach T X

V T g T T

  

    

2

3 1 T V HT V T

Tachyon field cosmology

• Rescaling:
• EoM:
• Hubble parameter rescaling:

T x T 1 ( ) ( ) T V x U x V T t t

3 2

'( ) '( ) 3 3 ( ) ( )

• U

x U x x HT x x HT x U x U x

H T H

Tachyon field cosmology

• Dimensionless equations:

2 2 2 2 2 3/2

( ) 3 1 (1 ) ( ) 3 ( )(1 ) ( ) X U x H x x dU x x X U x x x U x dx

2 4 2 3

, (2 )

s Pl s

T M X M g

The Inflation

• Slow-roll regime, slow-roll parameters:
• Number of e-folds:

* 1

ln | |, 0,

i i

d H i dN H

1 2 1 2 2 1 2

1 , 2 3 , 2 2 H H H H H x x Hx ( ) ( )

e i

t t

N t H t dt

The Inflation

• Number of e-folds:
• The scalar spectral index:
• The tensor-to-scalar ratio:

2 2 1

( ) ( ) , where ( ) 1 | ( ) |

e i

x e x

U x N x X dx x U x

1

16 ( )

i

r x

1 2

1 2 ( ) ( )

s i i

n x x

The Inflation

• Numerical results:

4

60 120, 1 12 1 ( ) N X U x x

The Inflation

• Numerical results:

60 120, 1 12 1 ( ) cosh( ) N X U x x

Tachyon inflation in an AdS braneworld

• Randall–Sundrum models (1999) imagine that the real world

is a higher-dimensional universe described by warped

• geometry. More concretely, our universe is a five-dimensional

anti-de Sitter space and the elementary particles except for the graviton are localized on a (3+1)-dimensional brane(s).

• A simple cosmological model of this kind is based on the RSII

model .

Tachyon inflation in an AdS braneworld

• Cosmology on the brane is obtained by allowing the brane to

move in the bulk. Equivalently, the brane is kept fixed at z=0 while making the metric in the bulk time dependent.

• The fluctuation of the interbrane distance implies the existence
• f the radion.
• Radion – a massless scalar field that causes a distortion of the

bulk geometry.

Tachyon inflation in an AdS braneworld

• The bulk spacetime of the extended RSII model in Fefferman-

Graham coordinates is described by the metric

• Inverse of the AdS curvature radius – k
• Radion field –
• Fifth coordinate – z

   

2 2 2 2 (5) 2 2 2 2 2

1 1 1 ( ) 1 ( )

a b ab

ds G dX dX k z x g dx dx dz k z k z x

  

              

( ) x 

Tachyon inflation in an AdS braneworld

• Add dynamical 3-brane, i.e. tachyon field (in terms of induced

metric).

• The action, after integrating out fifth coordinate z:
• Radion field (canonical) –
• Tachyon field –

, , 4 4 2 2 2 , , 4 4 2 2 3

1 (1 ) 1 16 2 (1 ) g R S d x g g d x g k G k k

     

                         

 





 

2

sinh 4 / 3 G    

Tachyon inflation in an AdS braneworld

• In the absence of radion – tachyon condenzate:
• Going back, lagrangian we are playing with:

(0) 4 br , , 4

1 S d x g g

2 2

1 k     

2 , , , , 4 3

1 1 2 g g

     

         

4

k   

Tachyon inflation in an AdS braneworld

• Hubble expansion rate H in standard cosmology (without

brane):

• Hubble expansion rate H in RS cosmology:

8 3 a G H a   

2

8 2 1 3 3 a G G H a k           

Tachyon inflation in an AdS braneworld

• Hamilton’s equation:

3 3 H H

Tachyon inflation in an AdS braneworld

• Dimensionless:

2 4

/ , / ( ), / ( )), , / ( ) h H k k k k k

4 8 2 8 2 2 8 2 10 2 5 8 2

1 / 4 3 / 3 2 1 / 4 3 / 3 1 / h h

          

                                          

2 2

( ) 1 2 6 h p N h               

2 2

8 Gk   

Tachyon inflation in an AdS braneworld

• Slow-roll parameters:
• Observational parameters:

1 i 1 i 2 i 2 s 1 i 2 i 1 i 1 i 2 i 2 i 3 i

1 16 ( ) 1 ( ) ( ) 6 8 1 2 ( ) ( ) 2 ( ) 2 ( ) ( ) ( ) ( ) 3 r C n C C                                      

2 2 2 2 1 2 4 4 2 1 2 2 2 2 2 2 2 4 4 4 4

8 1 1 6 12 8 1 1 1 12 4 6 6

Tachyon inflation in an AdS braneworld

• Some numerical results:

60 120 1 12 0.05 0.5 N

Tachyon inflation in an AdS braneworld

• Some numerical results:

60 120, 1 12 and 0 0.5 N

Conclusion

• We have investigated a model of inflation based on the

dynamics of a D3-brane in the AdS5 bulk of the RSII model. The bulk metric is extended to include the back reaction of the radion excitations.

• The ns/r relation here is substantially different from the

standard one and is closer to the best observational value.

• The model is based on the brane dynamics which results in a

definite potential with one free parameter only.

• We have analized the simplest tachyon model. In principle, the

same mechanism could lead to a more general tachyon potential if the AdS5 background metric is deformed by the presence of matter in the bulk.

References

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1, 1 (2016).

• M. Milosevic, D.D. Dimitrijevic, G.S. Djordjevic, M.D. Stojanovic, Serb.
• Astron. J. 192, 1-8 (2016).
• N. Bilic, D.D. Dimitrijevic, G.S. Djordjevic, M. Milosevic, M. Stojanovic, AIP
• Conf. Proc. 1722, 050002 (2016).
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12 (2014) 147–159.

• This work is supported by the SEENET-MTP

Network under the ICTP grant NT-03.

• The financial support of the Serbian Ministry for

Education and Science, Projects OI 174020 and OI 176021 is also kindly acknowledged.

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