Analytical and numerical aspects of tachyonic inflation

Goran S. DjordjevićSEEMET-MTP Office & Department of Physics,

Faculty of Science and MathematicsUniversity of Niš, Serbia

In cooperation with:D. Dimitrijevic, M. Milosevic, Z. Mladenovic and D. Vulcanov

VII Mathematical Physics Meeting: Summer School and Conference on Modern Mathematical Physics

Belgrade, 9 – 19 September 2012

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).

According to Special Relativity:

From a more modern perspective the idea of faster-than-light propagation is abandoned and the term "tachyon" is recycled to refer to a quantum field with

.22 mp

pv

,02 m

.0''2 Vm

Tachyons

Field Theory Standard Lagrangian (real scalar field):

Extremum (min or max of the potential): Mass term: Clearly 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.

...)(''2

1)(')(

2

1),( 2

000 VVVVTL

0)(' 0 V2

0 )('' mV

''V

constmVTL 22

2

1

2

1),(

Tachyons

String Theory A. Sen – proposed (effective) tachyon field action

(for the Dp-brane in string theory):

- tachyon field - tachyon potential Non-standard Lagrangian and DBI Action!

TTTxVdS jiijn 1)(1

100

n,...,1,

)(xT

)(TV

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 , with appropriate measure, and standard norms by the p-adic one.

This leads to an exact action in d dimensions, , .

The dimensionless scalar field describes the open string tachyon. is the string mass scale and is the open string coupling constant Note, that the theory has sense for any integer and make sense in the limit

1p

,1

1

2

1 142

42

22

pm

p

s

pexd

g

mS p

t

smsg

1

11 2

22

p

p

gg sp

.ln

2 22

p

mm s

p

pQ

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 - field of p-adic numbers

Ostrowski: There are just two nonequivalent (and nontrivial) norms on : and

Rich analysis on

QRQ ||||

pQQ p ||

pQ

Qp|| ||||

pQ

pb

ap p||

p-Adic case

p-Adic Quantum Mechanics Feynman p-adic kernel of the evolution operator

Aditive character – Rational part of the p-adic number – Semi-classical expression also holds in p-adic case.

Roots of the Freund-Witten formula

!,1)()( Qxxx p

)()0,;,(0

T

pp LdtDyyTyK

)}{2exp()( pp xix

px}{

)2exp( x

p-Adic Inflation

Setting in the action, the resulting potential takes the form

12

2

4

1

1

2

1 p

p

s

pg

mV

V

-1 1

0.2

0.4

p =19

p-Adic Inflation

The corresponding equation of motion:

In the limit (the limit of local field theory) the above eq. of motion becomes a local one

Very different limit 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 and rolls slowly to the minimum .01

1p

2 2( ) 2 lnt

2 2

2t

pm pe

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

Case 2

Case 3

( ) xV x e

0 02( )

cosh( ) x x

V VV x

x e e

2( )V x

x

From Field Theory to Classical AnalogueCase 1 -

If we scale

21),( xexxL x

xbx

mb

gb

m

g gm

b

21 ymg

eL m

y

21 ymg

dteS m

y

( ) xV x e

From Field Theory to Classical Analogue

Euler-Lagrange equation (equation of motion):

System under gravity in the presence of quadratic damping; can be also obtained from:

Solution:

mgyym 2

)22

1(

22

2

gmxmeL m

x

])ln[cosh()( 12 ctm

gctx

From Field Theory to Classical Analogue

Classical action:

)'''(''2

'2

2)cosh()(

)sinh(2

)0,',,''(xx

mx

mx

m eTm

gee

Tmg

gmxTxS

)0(' xx

)('' Txx

From Field Theory to Classical Analogue

Transformation:

'''2)cosh()'''(

)sinh(2

)0,',,''( 22 yyTm

gyy

Tmg

gmyTyS

xme

myx

)22

1(),( 22 y

gymyyL

Case 2

Euler-Lagrange equation (equation of motion):

With potential:

21 1

( ) ( )

dV dVx x

V x dx V x dx

2( )

cosh( )m m

x x

V VV t

x e e

0( )cosh( )

VV x

x

Takes a form:

The solution:

Where

2tanh( ) tanh( )x x x x

21

1( ) arcsinh ( 1)t tx t A C e e

21

1 2

1

2

CA

C C

Classical action2 2 2

2 12 2

2 1 1 2

21 1 2

22

1 22 1

( 1) 1 1 12 1 ( )( e )ln 1

2( 1)e

4

ttt C CC C C

S eC C CC C CC

22 2 2

2 2 22

1 4ln 1 2 ( ( 1) ( 1) )

A CS AC A C p p A C p

AC

2tp e

21

1 2

1

2

CA

C C

Quadratic one

A General Quantization

Quantization: Feynman: dynamical evolution of the system is

completely described by the kernel of the evolution operator:

Semi-classical expression for the kernel if the classical action is polynomial quadratic in and :)0,;,( yTyS y y

)0,;,(22/12

)0,;,(yTyS

h

i

eyy

S

h

iyTyK

)0,;,( yTyK

S

h

iLdth

i

t

DyeetdyyTyK

T

22

0)()0,;,(

where we denoted with an "0" index all values at the initial actual time.

Review of Reverse Engineering Method - “REM”

Table 1.

We shall now introduce the most general scalar field as a source for the cosmological gravitational field, using a lagrangian as :

where is the numerical factor that describes thetype of coupling between the scalar field and thegravity.

2 21 1 1

16 2 2L= g R V( ) ξR

π

Cosmology with non-minimallycoupled scalar field

Cosmology with non-minimallycoupled 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:

2 2 22

13 ( ) 3 ( ) ( ) 3 ( )( ( ) )

( ) 2

kH t t V t H t t

R t

2 2 233 ( ) 3 ( ) ( ) ( ) ( )( ( ) )

2H t H t t V t H t t

Where These are the new Friedman equations !!!

Cosmology with non-minimallycoupled scalar field

2

2

( ) 6 6 ( ) ( )( )

12 ( ) ( ) 3 ( ) ( )

V kt H t t

R t

H t t H t t

8 1, 1G c

The potential in terms of the scalar field for , (with green line in both panels) and for (left panel) and (right panel) with blue line

Some numerical results

The exponential expansion and the corresponding potential depends on the field

1 0 0.1 0.1

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 (the green surface in both panels) and for (left panel) and (right panel) the blue surfaces

0 0.1 0.1

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 and , for (the green lines in both panels) and for (left panel) and (right panel) the blue lines

1 0 0.1 0.1

1k

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 and

1k 0.05

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 ultrametric-nonarchimedean 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. VulcanovCLASSICAL AND QUANTUM APPROACH TO TACHYONIC INFLATION (in press) D.N. Vulcanov and G. S. DjordjevicON COSMOLOGIES WITH NON-MINIMALLY COUPLED SCALAR FIELDS,THE “REVERSE ENGINEERING METHOD” AND THE EINSTEIN FRAMERom. Journ. Phys., Vol. 57, No. 5–6 (2012) 1011–1016 G.S. Djordjevic and Lj. NesicTACHYON-LIKE MECHANISM IN QUANTUM COSMOLOGY AND INFLATIONin Modern trends in Strings, Cosmology and ParticlesMonographs Series: Publications of the Astronomical Observatory of BelgradeNo 88 (2010) 75-93 D.D. Dimitrijevic, G.S. Djordjevic and Lj. NesicQUANTUM COSMOLOGY AND TACHYONSFortschritte der Physik, 56 No. 4-5 (2008) 412-417 G.S. Djordjevic, B. Dragovich, Lj. Nesic and I.V. Volovichp-ADIC AND ADELIC MINISUPERSPACE QUANTUM COSMOLOGYInt. 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!