Mini workshop “Cosmology and Strings 2016” (2-5 November 2016, Nis, Serbia)

1. Tachyon inflation in
DBI and RSII context
Based 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 AdS
Braneworld With Back-reaction, (2016) 19. http://arxiv.org/abs/1607.04524
Mini Workshop „Cosmology and String 2016“
Nis, November 2-5, 2016

2. 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.

3. 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

4. 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

5. Tachyon inflation
 Consider the tachyonic field T minimally coupled to Einstein's
gravity
 Where R is Ricci scalar, g – determinant of the metric tensor and
tachyon action
 Friedman equation:
 The energy-momentum conservation equation:
41
16 T
S gRd x S
G
4
( , ) , ( ) 1T
S g T T d x V T g T T
2
2
2 2 1/2
1
3 (1 )Pl
a V
H
a M T
2
3 ( )
3 0
1
H P
T V
HT
VT
2
2
( )
1
( ) 1
V T
T
P V T T
Energy density and pressure:

6. 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
x
T 0
1 ( )
( ) .
T V x
U x V
T 0
,
t
t
3 2
0
'( ) '( )
3 3 0.
( ) ( )o
U x U x
x HT x x HT x
U x U x
0
.H T H
2
'( )
3 0.
( )1
x U x
Hx
U xx
22
2 0
2 2 2 1/22
0
1 ( ) ( )
33 (1 )1pl
XH U x U x
H
T M xx

7. Tachyon inflation
 Parameters:
 The system of dimensionless equations
 In addition, the Friedman acceleration equation
2 4
0
0 2 3
wher,
(2 )
e s
Pl s
T M
X
M g
2
2 0
2
2
2 3/2
0
( )
3 1
(1 ) ( )
3 ( )(1 ) 0
( )
X U x
H
x
x dU x
x X U x x x
U x dx
2
0
( )
2
X
H P 2
2
( )
1
( ) 1
U T
x
P U x x

8. 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 H
i
dN H
1 2 12
2
1 2
1
, 2
3
, 2
2
H H
H HH
x
H
x
x
1
(16 )i
r x
1 2
1 2 ( ) ( )is i
x xn
( ) ( )
e
i
t
t
N t H t dt
2
2
0 1
( )
where ( ) 1( ) ,
| ( ) |
e
ix
e
x U x
N x X dx x
U x
2
2 0
2
2
2 3/2
0
2
20
2
( )
3 1
(1 ) ( )
3 ( )(1 ) 0
( )
( )
( ( ) 1 )
2 1
X U x
H
x
x dU x
x X U x x x
U x dx
X U T
H U x x
x

9. Tachyon inflation
 Numerical results
0
4
60 120, 1 12
1
( ) (left)
1
( ) (right)
cosh( )
N X
U x
x
U x
x

10. 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

11. 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
5
x z
0z  z l
5( )d x
z  
N. Bilic, “Space and Time in Modern Cosmology”

12. 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”

13. 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 1
1 ( )
1 ( )
a b
abds G dX dX k z x g dx dx dz
k z k z x
  


 
    
  
The bulk space-time metric
The spatially flat FRW metric

14. Randal-Sundrum model and
tachyon-like inflation

, ,4 4 2 2 2
, , 4 4 2 2 3
1
(1 ) 1
16 2 (1 )
gR
S d x g g d x g k
G k k

 
 


 
  
           
    
1/k   2
sinh 4 / 3 G  
br
S
(0) 4
br , ,4
1S d x g g
2
, ,
, , 4 3
1
1
2
g
g

 
 


 
    
 4
k

  2 2
1 k   
the brane tension
2
8 2
1
3 3
a G G
H
a k
  
   
 

15. Hamilton’s Equations
 The Hamiltonian density
 Hamilton’s equations
 Nondimensionalization
2
2 2 8 2
4
1
1 / ( )
2
H
3
3
H
H
H
H
H
H
2
4
/ ,
/ ( ), / ( )),
, / ( )
h H k
k k
k k

16. 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
1
3 12
Gk
a
h
a
 
 
 

 
   
 
2
2
2
2 2
2
2 2 3
4
2
2
4 2 3
1 ,
sinh ,
6
2
sinh ,
6 3
1
1 / ,
2
1 1
2 1 /
d
d
p
  

 
  
 


  


 
  

 
 
   
 
 
    
 
  
 

A combined dimensionless coupling
The Hubble
expansion rate
preassure
energy density
2 2
( ) 1
2 6
h p
N h
 
 
 
    
 

Additional equations,
solved in parallel

17. 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
d
i
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
1
16 ( ) 1 ( ) ( )
6
8
1 2 ( ) ( ) 2 ( ) 2 ( ) ( ) ( ) ( )
3
r C
n C C
  
      
 
    
  
        
  

18. 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 i
N N N
6
1 4
1 0
2 1
1, 192
3 ( )
N

19. Numerical results

20. Numerical results
0
60 120
1 12
0.05 0.5
N

21. Numerical results

22. Numerical results

23. Numerical results
0
0
60 120, 1 12 and 0 0.5 (left)
115 120, 0.05, 1.25 (right)
N
N

24. 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

25. 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 AdS
braneworld 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);