tachyon inflation in dbi and rsii context
TRANSCRIPT
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);