Analytical and numerical
aspects of tachyonic inflation
VII Mathematical Physics Meeting:
Summer School and Conference on Modern Mathematical Physics
Belgrade, 9 – 19 September 2012
Goran S. Djordjević
SEEMET-MTP Office & Department of Physics,
Faculty of Science and Mathematics
University of Niš, Serbia
In cooperation with:
D. Dimitrijevic, M. Milosevic, Z. Mladenovic and D.
Vulcanov
Quantum cosmology and tachyons

Introduction and motivation

Tachyons

p-Adic Inflation

Minisuperspace quantum cosmology as quantum mechanics over
minisuperspace

Tachyons-From Field Theory to Classical Analogue

Reverse Engineering and (Non)Minimal Coupling

Conclusion and perspectives
Introduction






The main task of quantum cosmology is to describe the evolution of
the universe in a very early stage.
Since quantum cosmology is related to the Planck scale phenomena
it is logical to consider various geometries (in particular
nonarchimedean, noncommutative …)
Supernova Ia observations show that the expansion of the Universe
is accelerating, contrary to FRW cosmological models.
Also, cosmic microwave background (CMB) radiation data are
suggesting that the expansion of our Universe seems to be in an
accelerated state which is referred to as the ``dark energy`` effect.
A need for understanding these new and rather surprising facts,
including (cold) ``dark matter``, has motivated numerous authors to
reconsider different inflation scenarios.
Despite some evident problems such as a non-sufficiently long
period of inflation, tachyon-driven scenarios remain highly
interesting for study.
Tachyons



A. Somerfeld - first discussed about possibility of particles to
be faster than light (100 years ago).
G. Feinberg - called them tachyons: Greek word, means fast,
swift (almost 50 years ago).
p
2
According to Special Relativity:
m  0,
v
.
p m
2

2
From a more modern perspective the idea of faster-than-light
propagation is abandoned and the term "tachyon" is recycled
2
to refer to a quantum field with
m  V ' ' 0.
Tachyons


Field Theory
Standard Lagrangian (real scalar field):
L ( ,    )  T  V 



1
2

      V ( 0 )  V ' ( 0 ) 
1
2
V ' ' ( 0 )  ...
2
Extremum (min or max of the potential): V ' ( 0 )  0
Mass term: V ' ' ( 0 )  m 2
Clearly V ' ' can be negative (about a maximum of the
potential). Fluctuations about such a point will be
unstable: tachyons are associated with the presence of
instability.
L ( ,    )  T  V 
1
2

    
1
2
m   const
2
2
Tachyons


String Theory
A. Sen – proposed (effective) tachyon field action
(for the Dp-brane in string theory):
S   d
n 1
xV (T ) 1    i T  j T
ij
 00   1
    
- tachyon field
V (T ) - tachyon potential
Non-standard Lagrangian and DBI Action!
 T ( x)


 ,  1,..., n
p-Adic inflation from p-adic strings

p-Adic string theory was defined (Volovich, Freund, Olson (1987); Witten at al
(1987,1988)) replacing integrals over R (in the expressions for various
amplitudes in ordinary bosonic open string theory) by integrals over Q p , with
appropriate measure, and standard norms by the p-adic one.

This leads to an exact action in d dimensions, , .
 t  


2
1
1
mp
4 
d
x


e
 

  2
p 1

2
S 
m
4
s
2
gp




2
p 1


,


1
2
gp

1
p
gs p 1
2
2
2
m
2
p

2m s
.
ln p
The dimensionless scalar field describes the open string tachyon.
m s is the string mass scale and
g s is the open string coupling constant
Note, that the theory has sense for any integer and make sense in the limit
p 1
Minisuperspace quantum cosmology as quantum
mechanics over minisuperspace
(ADELIC) QUANTUM COSMOLOGY









The main task of AQC is to describe the very early stage in the evolution of the
Universe.
At this stage, the Universe was in a quantum state, which should be described
by a wave function (complex valued and depends on some real parameters).
But, QC is related to Planck scale phenomena - it is natural to reconsider its
foundations.
We maintain here the standard point of view that the wave function takes complex
values, but we treat its arguments in a more complete way!
We regard space-time coordinates, gravitational and matter fields to be adelic, i.e.
they have real as well as p-adic properties simultaneously.
There is no Schroedinger and Wheeler-De Witt equation for cosmological models.
Feynman’s path integral method was exploited and minisuperspace cosmological
models are investigated as a model of adelic quantum mechanics [Dragovich (1995),
G Dj, Dragovich, Nesic and Volovich (2002), G.Dj and Nesic (2005, 2008)…].
Adelic minisuperspace quantum cosmology is an application of adelic quantum
mechanics to the cosmological models.
Path integral approach to standard quantum cosmology
p-Adic case


p-prime number
Q p - field of p-adic numbers
Q   R  Q 
||
|
||
|
Q 
p  Q p

Ostrowski: There are just two nonequivalent (and nontrivial) norms
on Q : | | p and || ||

Rich analysis on Q p
| p

a
b
|p  p

p-Adic case


p-Adic Quantum Mechanics
Feynman p-adic kernel of the evolution operator
K p ( y , T ; y , 0 ) 
 Dy
T
 p (   Ldt )
0

Aditive character –  p ( x )  exp( 2 i { x} p )
Rational part of the p-adic number – { x} p
Semi-classical expression also holds in p-adic case.

Roots of the Freund-Witten formula


  ( x )   p ( x )  1, x  Q !
   exp(  2  x )
p-Adic Inflation

Setting in the action, the resulting potential
 m  1
takes the form

1
  
V 
 

4
s
2
p
 g  2


V
p= 19
0 .4
0 .2
-1
1
p 1
2

p 1

p-Adic Inflation

The corresponding equation of motion:
2

e

 t  
2
mp
2
 
p
In the limit (the limit of local field theory) the above eq. of motion
becomes a local one
(  t   )  2 ln 
2



2
Very different limit p  1 leads to the models where nonlocal
structures is playing an important role in the dynamics
Even for extremely steep potential p-adic scalar field (tachyon) rolls
slowly! This behavior relies on the nonlocal nature of the theory: the
effect of higher derivative terms is to slow down rolling of tachyons.
Approximate solutions for the scalar field and the quasi-de-Sitter
expansion of the universe, in which starts near the unstable
maximum   1 and rolls slowly to the minimum   0 .
Solvable Tachyonic Inflatory Potential
in a Classical Friedman Framework


There are at least several
interesting tachyonic
potentials which leads to
the solvable Friedman
equations.
Which ones of them can
be quantized?

Case 1
V ( x)  e

Case 2
V0
V ( x) 

 x
cosh(  x )
Case 3
V ( x) 

x
2
2V 0

e
x
e
 x
From Field Theory to Classical Analogue
Case 1 - V ( x )  e   x
 x
2
L ( x , x )   e
1  x

If we scale
b
 


m
b
 g
m
xbx
 g
b

mg
S    dte
L  e


y
m

1
2
y
mg
y
m
1

mg
2
y
From Field Theory to Classical Analogue

Euler-Lagrange equation (equation of motion):
m y   y

2
 mg
System under gravity in the presence of quadratic
damping; can be also obtained from:
Le
2
 x
m
(
1
2
m x 
2
2

m g
2
)
Solution:
x ( t )  c 2  ln[cosh(
g
m
t )  c1 ]
From Field Theory to Classical Analogue

Classical action:
S ( x ' ' , T , x ' ,0 ) 
2 sinh(
x '  x (0)
x ' '  x (T )
2
x ''
 2 x'
m
m
e
) cosh(
(e
g
T) 
m
mg
g
m

T )  2e
m
( x '  x '')



From Field Theory to Classical Analogue

x y
Transformation:
L ( y , y )  (
1
m y 
2
2


e
x
m
m
 g
2
y )
2
 2
2
 ( y '  y ' ' ) cosh(
g
T) 
m
mg
S ( y ' ' , T , y ' ,0 ) 
2 sinh(

T )  2 y ' y ' '
m

g
Case 2

V ( x) 
V0
cosh(  x )
Euler-Lagrange equation (equation of motion):
x
1
dV
2
V ( x ) dx

1
x 
dV
V ( x ) dx
With potential:
V (t ) 
Vm
cosh(  x )
2V m

e
x
e
 x

Takes a form:
x   tanh(  x ) x   tanh(  x )
2

The solution:
x (t ) 

1
2t
t
arcsinh  A ( C 1 e
 1) e 

Where
C1  1
2
A
2 C1 C 2
Classical action
S 

ln 1 
2

C 2 ( C 1  1)

2
2
C 2 ( C 1  1) 


C1

1  4 A C2
2
S 
AC 2
2
C1  1  2 t 

e
2 C1 



C 2 
1
ln 1  2 AC 2 ( A ( C 2 p  1) 

C1  1
2
pe
2t
Quadratic one
A
2 C1 C 2
( C 1  1)( C 2 e
2
e
2t

2
2t
4 C1 C 2
p  A ( C 2 p  1) ) 

2
2
2

 1) 



A General Quantization


Quantization:
Feynman: dynamical evolution of the system is
completely described by the kernel K ( y , T ; y , 0 ) of the
evolution operator:
2 i
K ( y , T ; y , 0 ) 
  dy ( t )e
h
T
 Ldt
0
2 i

 Dye
S
h
t

Semi-classical expression for the kernel if the classical
action S ( y , T ; y ,0 ) is polynomial quadratic in y  and y  :
 i  2S 

K ( y , T ; y , 0 )  

 h  y  y  
1/ 2
2 i
e
h
S ( y  ,T ; y  , 0 )
Review of Reverse Engineering Method - “REM”
Table 1.
where we denoted with an "0" index all values at the initial
actual time.
Cosmology with non-minimally
coupled scalar field
We shall now introduce the most general scalar field
as a source for the cosmological gravitational field,
using a lagrangian as :
L=
1
1
2
 1
2 
g 
R       V (  )  ξR 
2
2
 16 π

where  is the numerical factor that describes the
type of coupling between the scalar field and the
gravity.
Cosmology with non-minimally
coupled scalar field

Although we can proceed with the reverse method
directly with the Friedmann eqs. it is rather complicated
due to the existence of nonminimal coupling. We
appealed to the numerical and graphical facilities of a
Maple platform in the Einstein frame with a minimal
coupling!.

For sake of completeness we can compute the Einstein
equations for the FRW metric.

After some manipulations we have:
Cosmology with non-minimally
coupled scalar field
1
2
2 
3 H (t )  3


(
t
)

V
(
t
)

3

H
(
t
)(

(
t
)
)
2

R (t )
2

3

2
2
2 
3 H ( t )  3 H ( t )     ( t )  V ( t )   H ( t )( ( t ) ) 
2


2
k
 (t ) 
V

k
 6
R (t )
2
 6  H ( t ) ( t )
 1 2  H ( t )  ( t )  3 H ( t ) ( t )
2
Where 8 G  1, c  1
These are the new Friedman equations !!!
Some numerical results

The exponential expansion and the
corresponding potential depends on the field
The potential in terms of the scalar field for   1 ,   0 (with green line in both panels)
and for   0 .1 (left panel) and   0 .1 (right panel) with blue line

Some numerical results

The exponential expansion and the corresponding
potential depends on the field and ``omega factor``
The potential in terms of the scalar field and  , for   0 (the green surface in both
panels) and for    0 .1 (left panel) and   0 .1 (right panel) the blue surfaces
The ekpyrotic universe

The ekpyrotic universe and the corresponding
potential depends on the field
The ekpyrotic universe: the potential in terms of the scalar field for   1 and k  1 ,
for   0 (the green lines in both panels) and for    0 .1 (left panel) and   0 .1
(right panel) the blue lines
The ekpyrotic universe and the corresponding
potential depends on the field and ``omega factor``

The ekpyrotic universe and the
corresponding potential
depends on the field and
``omega factor``
The ekpyrotic universe: the potential in
terms of the scalar field and  , and
for k  1 and   0 .0 5
Conclusion and perspectives








Sen’s proposal and similar conjectures have attracted important interests.
Our understanding of tachyon matter, especially its quantum aspects is still quite pure.
Perturbative solutions for classical particles analogous to the tachyons offer many
possibilities in quantum mechanics, quantum and string field theory and cosmology on
archimedean and nonarchimedean spaces.
It was shown [Barnaby, Biswas and Cline (2007)] that the theory of p-adic inflation can
compatible with CMB observations.
Quantum tachyons could allow us to consider even more realistic inflationary models
including quantum fluctuation.
Some connections between noncommutative and nonarchimedean geometry (q-deformed
spaces), repeat to appear, but still without clear and explicitly established form.
Reverse Engineering Method-REM remains a valuable auxiliary tool for investigation on
tachyonic–universe evolution for nontrivial models.
Eternal inflation and symmetree, as a new stage for modelin with ultrametricnonarchimedean structures,
L. Susskind at al, Phys. Rev. D 85, 063516 (2012) ``Tree-like structure of eternal inflation:
a solvable model``
Some references
G. Dj, D. Dimitrijevic, M. Milosevic, Z. Mladenovic and D. Vulcanov
CLASSICAL AND QUANTUM APPROACH TO TACHYONIC INFLATION (in press)

D.N. Vulcanov and G. S. Djordjevic
ON COSMOLOGIES WITH NON-MINIMALLY COUPLED SCALAR FIELDS,
THE “REVERSE ENGINEERING METHOD” AND THE EINSTEIN FRAME
Rom. Journ. Phys., Vol. 57, No. 5–6 (2012) 1011–1016

G.S. Djordjevic and Lj. Nesic
TACHYON-LIKE MECHANISM IN QUANTUM COSMOLOGY AND INFLATION
in Modern trends in Strings, Cosmology and Particles
Monographs Series: Publications of the Astronomical Observatory of Belgrade
No 88 (2010) 75-93

D.D. Dimitrijevic, G.S. Djordjevic and Lj. Nesic
QUANTUM COSMOLOGY AND TACHYONS
Fortschritte der Physik, 56 No. 4-5 (2008) 412-417

G.S. Djordjevic, B. Dragovich, Lj. Nesic and I.V. Volovich
p-ADIC AND ADELIC MINISUPERSPACE QUANTUM COSMOLOGY

Int. J. Mod. Phys. A17 (2002) 1413-1433
Acknowledgement
Work of G.S.Dj and M.M is supported by
the Serbian Ministry of Education and Science
Project No. 176021 “Visible and invisible matter in nearby galaxies: theory and
observations”.
The financial support under
the Abdus Salam International Centre for Theoretical Physics (ICTP) – Southeastern
European Network in Mathematical and Theoretical Physics (SEENET-MTP)
Project PRJ-09 “Cosmology and Strings” is kindly acknowledged by G.S.Dj, D.D.D, M.M
and D.N.V.
D.N.V was partially supported by the grant
POSDRU 107/1.5/G/13798, inside POSDRU Romania 2007-2013 co-financed by the
European Social Fund - Investing in People.
Z.M. was partially supported by a Serbian Ministry for Youth and Sport scholarship for
talents.
Thank you! Хвала!