Notes on Real and p-Adic Inflation

Spring School on Strings, Cosmology and Particles SSSCP2009

Nis, 4 April 2009

Goran S. Djordjević

In Collaboration with

D. Dimitrijevic, M. Milosevic and Lj. Nesic,

Introduction

Tachyons

Inflation

Minisuperspace quantum cosmology as quantum mechanics over minisuperspace

Tachyons-From Field Theory to Classical Analogue

Conclusion

Plan of the talk

(Real & p-Adic)

Introduction

The main task of quantum cosmology is to describe the evolution of the universe in a very early stage (Hawking, Hartle (1983)).

Since quantum cosmology is related to the Planck scale phenomena it is logical to consider various geometries (in particular nonarchimedean, noncommutative …) and parametrizations of the space-time coordinates: real, p-adic, or even adelic.

Supernova Ia observations show that the expansion of the Universe is accelerating, contrary to the 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.

The cosmological constant as the vacuum energy can be responsible for this evolution by providing a negative pressure.

Introduction

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

Inflationary and/or Higss potential

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 [2000,02]– proposed effective tachyon field action

(for the Dp-brane in string theory):

- tachyon field - tachyon potential Non-standard Lagrangian:

TTTxVdS jiijn 1)(1

100

n,...,1,

)(xT

)(TV

Some elementary maths

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

Q p|| ||||

pQ p

b

ap p||

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 Qp, 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 .

p-Adic strings and tachyons

p-Adic tachyons and inflation

1p

,1

1

2

1 142

42

22

pm

p

s

pexd

g

mS p

t

sm

sg

1

11 2

22

p

p

gg sp

.ln

2 22

p

mm sp

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 [Arefeva, Dragovich, Frampton, Volovich(1991), Dragovich (1995), G.S. Dj, Dragovich, Nesic and Volovich (2002), G.S..Dj and Nesic (2005)…].

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-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.

)()0,;,(0

T

pp LdtDyyTyK

)}{2exp()( pp xix

px}{

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

New and old inflation (Guth(1980), Linde (1981,83)…)

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 (Barnaby, Biswas, Cline (2007), Joukovskaya (2007)…! 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 .

p-Adic Inflation

pmp

t

e

2

22

ln2)( 22 t

01

1p

1p

From Field Theory to Classical Analogue

If we scale

21),( xexxL x

xbx

mb

gb

m

g gm

b

21 ymg

eL m

y

21 ymg

dteS m

y

From Field Theory to Classical Analogue

Euler-Lagrange eq. 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

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 , generalized in G.S.Dj, Dragovich (1997), G.S.Dj, Dragovich, Nesic(2003):

)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,;,(

``Our Goal or Dream``

Conclusion and perspectives

Sen`s proposal and similar conjectures have attracted important interests among physicists.

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 for further investigations and toy models 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), as well as physical models on them, repeat to appear (Dragovich, G.S.Dj, Goshal, Volovich…), but still without clear and explicitly established form.

Acknowledgements

Many thanks to the audience for your attention and possible applause

This process continues, and eventually the universe becomes populated by inhomogeneous scalar field. Its energy takes different values in different parts of the universe. These inhomogeneities are responsible for the formation of galaxies.Sometimes these fluctuations are so large that they substantially increase the value of the scalar field in some parts of the universe. Then inflation in these parts of the universe occurs again and again. In other words, the process of inflation becomes eternal.

Simulacija 1 Simulacija 2