Tachyon inflation in

DBI and RSII contextBased on:

M. Milosevic, D.D. Dimitrijevic, G.S. Djordjevic, M.D. Stojanovic, Dynamics of tachyon fields and inflation - comparison of analytical and numerical results with observation, Serbian Astron. J. (2016). doi:10.2298/SAJ160312003M.

N. Bilic, D. Dimitrijevic, G. Djordjevic, M. Milosevic, Tachyon Inflation in an AdSBraneworld With Back-reaction, (2016) 19. http://arxiv.org/abs/1607.04524

Mini Workshop „Cosmology and String 2016“

Nis, November 2-5, 2016

Introduction and Motivation

The inflationary universe scenario in which the early

universe undergoes a rapid expansion has been

generally accepted as a solution to the horizon

problem and some other related problems of the

standard big-bang cosmology

Recent years - a lot of evidence from WMAP and

Planck observations of the CMB

Quantum cosmology: probably the best way to

describe the evolution of the early universe.

Tachyons and Non-standard

Lagrangians

Traditionally, the word tachyon was used to describe a

hypothetical particle which propagates faster than

light.

In modern physics this meaning has been changed

The field theory of tachyon matter proposed by A. Sen

String theory: states of quantum fields with imaginary mass (i.e. negative mass squared)

It was believed: such fields permitted propagation faster

than light

However it was realized that the imaginary mass creates

an instability and tachyons spontaneously decay through

the process known as tachyon condensation

Tachyion Fields

No classical interpretation of the ”imaginary mass”

The instability: The potential of the

tachyonic field is initially at a local

maximum rather than a local

minimum (like a ball at the top of

a hill)

A small perturbation - forces the

field to roll down towards the

local minimum.

Quanta are not tachyon any more, but rather an

”ordinary” particle with a positive mass.

Tachyon potential:

Dirack-Born-Infeld (DBI) Lagrangian

(0) , '( 0) 0, ( ) 0V V T V T

( ) 1V T g T T

Tachyon inflation

Consider the tachyonic field T minimally coupled to Einstein's gravity

Where R is Ricci scalar, g – determinant of the metric tensor andtachyon action

Friedman equation:

The energy-momentum conservation equation:

41

16 TS gRd x SG

4( , ) , ( ) 1TS g T T d x V T g T T

2

2

2 2 1/2

1

3 (1 )Pl

a VH

a M T

2

3 ( )

3 01

H P

T VHT

VT

2

2

( )

1( ) 1

V T

TP V T T

Energy density and pressure:

Nondimensionalization

Rescaling field, potential and time

The equation is transformed to

The nondimensional Hubble parameter

The nondimensional energy-momentum conservation equation

The nondimensional Friedman equation

0

,T

xT

0

1 ( )( ) .

T V xU x V

T 0

,t

t

3 20

'( ) '( )3 3 0.

( ) ( )o

U x U xx HT x x HT x

U x U x

0 .H T H

2

'( )3 0.

( )1

x U xHx

U xx

222 0

2 2 2 1/220

1 ( ) ( )

33 (1 )1pl

XH U x U xH

T M xx

Tachyon inflation

Parameters:

The system of dimensionless equations

In addition, the Friedman acceleration equation

2 40

0 2 3wher,

(2 )e s

Pl s

T MX

M g

22 0

2

22 3/2

0

( )

3 1(1 ) ( )

3 ( )(1 ) 0( )

X U xH

xx dU x

x X U x x xU x dx

20 ( )

2

XH P 2

2

( )

1( ) 1

U T

xP U x x

Tachyon inflation

The slow-roll parameters

Number of e-folds

In the slow-roll aproximation

Observational parameters

The scalar spectral index

The tensor-to-scalar ratio

*1 0

ln | |, 0,i

i

d Hi

dN H

1 2 12

21 2

1, 2

3, 2

2

H H

H HH

xH

x

x

1(16 )ir x

1 21 2 ( ) ( )is ix xn

( ) ( )e

i

t

tN t H t dt

220 1

( )where ( ) 1( ) ,

| ( ) |

e

ixe

x U xN x X dx x

U x

22 0

2

22 3/2

0

220

2

( )

3 1(1 ) ( )

3 ( )(1 ) 0( )

( )( ( ) 1 )

2 1

X U xH

xx dU x

x X U x x xU x dx

X U TH U x x

x

Tachyon inflation

Numerical results

0

4

60 120, 1 12

1( ) (left)

1( ) (right)

cosh( )

N X

U xx

U xx

Tachyon inflation in an AdS

braneworld

Randall–Sundrum models 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 or branes.

Proposed in 1999 by Lisa Randal and Raman Sundrum

A simple cosmological model of this kind is based on

the second Randall-Sundrum (RSII) model

Inflation is driven by the tachyon field originating in

string theory

The RSI Model The model was originally proposed

as a possible mechanism for

localizing gravity on the 3+1

universe embedded in a 4+1

dimensional space-time without

compactification of the extra

dimension.

Observer – negative tension brane;

separation - such that the strength

of gravity on observer’s brane is

equal to the observed four-

dimensional Newtonian gravity.

x

5x z

0z z l

5( )d xz

N. Bilic, “Space and Time in Modern Cosmology”

The RS II Model Observers reside on the

positive tension brane and the negative tension brane

is pushed off to infinity.

The Planck mass scale is

determined by the

curvature of the AdS space-

time rather than by the size

of the fifth dimension.

Radion – massless scalar

field; fluctuation of

interbrane distance along

extra dimension

z z 0z N. Bilic, “Space and Time in Modern Cosmology”

Randal-Sundrum model and

tachyon-like inflation

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 along the

extra dimension implies the existence of the radion.

Radion - a massless scalar field that causes a distortion

of the bulk geometry.

The bulk spacetime of the extended RSII model in

Fefferman-Graham coordinates is described by the

metric

2 2 2 2

(5) 22 2 2 2

1 11 ( )

1 ( )

a b

abds G dX dX k z x g dx dx dzk z k z x

The bulk space-time metric

The spatially flat FRW metric

Randal-Sundrum model and

tachyon-like inflation

, ,4 4 2 2 2

, , 4 4 2 2 3

1(1 ) 1

16 2 (1 )

gRS d x g g d x g k

G k k

1/k 2sinh 4 / 3 G

brS

(0) 4br , ,4

1S d x g g

2, ,

, , 4 3

11

2

gg

4k

2 21 k

the brane tension

2

8 21

3 3

a G GH

a k

Hamilton’s Equations

The Hamiltonian density

Hamilton’s equations

Nondimensionalization

22 2 8 2

4

11 / ( )

2H

3

3

H

H

HH

HH

2

4

/ ,

/ ( ), / ( )),

, / ( )

h H k

k k

k k

Randal-Sundrum model

The dimensionless Hamiltonian’s equations are

obtained

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

2 2

8

13 12

Gk

ah

a

2

22

2 2

22 2 3

4

22

4 2 3

1 ,

sinh ,6

2sinh ,

6 3

11 / ,

2

1 1

2 1 /

d

d

p

A combined dimensionless coupling

The Hubble

expansion rate

preassure

energy density

2 2

( ) 12 6

h p

N h

Additional equations,

solved in parallel

Randal-Sundrum model

Slow-roll parameters are

Observational parameters: the tensor-to-scalar

ration (r) and scalar spectral index (ns)

*0

1ln | |, 1i

i

H

H

di

Hdt

*-Hubble rate at an arbitrarily chosen timeH

1 i 1 i 2 i

2

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

116 ( ) 1 ( ) ( )

6

81 2 ( ) ( ) 2 ( ) 2 ( ) ( ) ( ) ( )

3

r C

n C C

Numerical solution

Initial values:

Tachyion field: inverse quartic tachyon potential

No a priori reason to restrict possible initial values of the

radion field; a range of initial values based on the natural

scale dictated by observations.

Conjugate momenta

After the system of equation is solved, the slow-roll

parameters are calculated

Conditions:

0 0 0

1( ) 1

( ) ( )f

f iN N N

6

1 41 0

2 11, 192

3 ( )N

Numerical results

Numerical results0

60 120

1 12

0.05 0.5

N

Numerical results

Numerical results

Numerical results

0

0

60 120, 1 12 and 0 0.5 (left)

115 120, 0.05, 1.25 (right)

N

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 slow-roll equations of the tachyon inflation are quite distinct to those of the standard tachyon inflation with the same potential.

The ns-r relation in our model is substantially different from the standard one and is closer to the best observational value.

The agreement with observations is not ideal and it is fair to say that the present model is disfavoured but not excluded.

However, the model is based on the brane dynamics which results in a definite potential with one free parameter only.

We have analysed 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

D. Steer, F. Vernizzi, Tachyon inflation: Tests and comparison with single scalar field inflation, Phys. Rev. D. 70 (2004) 43527.

N. Bilic, G.B. Tupper, AdS braneworld with backreaction, Cent. Eur. J. Phys. 12 (2014) 147–159.

P.A.R. Ade, N. Aghanim, M. Arnaud, F. Arroja, M. Ashdown, J. Aumont, et al., Planck 2015 results: XX. Constraints on inflation, Astron. Astrophys. 594 (2016) A20.

L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999)

N. Bilic, D. Dimitrijevic, G. Djordjevic, M. Milosevic, Tachyon inflation in an AdSbraneworld with back-reaction, (2016) 19. http://arxiv.org/abs/1607.04524.

M. Milosevic, D.D. Dimitrijevic, G.S. Djordjevic, M.D. Sto- janovic, 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);