Puzzles with tachyon in SSFT and Cosmological Applications

I. Aref'eva Steklov Mathematical Institute, Moscow

String Field Theory 2010

Kyoto, Japan 18-22 October, 2010

Outlook

• Historical remarks (about non-locality)• Non-locality in SFT or SFT inspired models• Schemes of calculations • Applications of SFT non-locality to cosmology

Time dependent solutions in cubic (super)string field theories

Key word ----- non-locality

Historical remarks. Non-locality in SFT

• Hikedi Yukawa (1949)

2

2

( ) ( , ) 0

( ) ( , ) 0

m U x r

r r U x r

...)()()]([

...)()()(),(

1

xAxx

rrxBrxAxrxU

Yukawa’s field as a reduced SFT field

Phys.Rev., 76(1949) 300

Historical remarks. H.Yukawa (1949)

Refs.:9) Non-local and non-linear field theories. D.I. Blokhintsev, (Dubna, JINR) Fortsch.Phys.6:246-269,1958, Usp.Fiz.Nauk 61:137-159,195710) Nonlocal quantum field theory. D.A. Kirzhnits, (Lebedev Inst.) Usp.Fiz.Nauk 90:129-142,1966.154) Relativistic Wave Equations with Inner Degrees of Freedom and Partons. V.L. Ginzburg, V.I. Manko, (Lebedev Inst.) .Sov.J.Part.Nucl.7:1,1976.

Historical remarks. Non-locality in QFT

A.Pais, G.Uhlenbeck, 1950

Field theories with non-localized action.

Motivation: eliminate divergences

Strings

One of the main goals: eliminate divergences

• Veneziano amplitudes?• Nambu-Goto string --- extended object

(we expected “strings” from QCQ,

QCD string -- Wilson criteria – lattice calculations)• Polyakov’s approach? • Light-cone Kaku-Kikkawa SFT?• Covariant midpoint SFT : cubic/nonpolynomial,

Witten, AMZ-PTY, Zwiebach, Berkovich

• Covariant light-cone-like Hata-Itoh-Kugo-Kunitomo-Ogawa SFT

Historical remarks. Non-locality in strings

Question: can we see non-locality in

Non-locality in String Field Theory

Time–dependent solutions:Rolling solutions vs wild oscillations Moeller and Zwiebach, hep-th/0207710; Fujita and Hata, hep-th/0304163

Non-flat metricLevel truncation method vs analytical solutions

• In fundamental setting locality and causality non-locality and noncommutative geometry (Witten)

• In practical setting (cosmological applications)

Hata and Oda, Causality in Covariant SFT, hep-th/9608128 Erler and Gross, Locality, Causality, and an Initial Value Formulation for Open SFT, hep-th/0406199

• Cosmological constant

why it is now so small• Dynamical DE

w < - 1

periodic crossing the w=-1 barrier

Non-local Cosmology from String Field Theory

Questions we want to address:

• Physics after Big Bang • Inflation; Non- Gausianity • Primordial Black Holes

Non-local Cosmology from String Field Theory

• String Field Theory is the UV-complete theory

• Non-locality is the key point of UV completion

Motivation

Nonlocal Models in Cosmology

• Nonlocality in Matter (mainly string motivated)

• Nonlocality in Gravity

Nonlocal Models in Cosmology

24

422

( )1 1( ( ))2 2

g

s

pFM

d x g V Rg M

I. Nonlocality in Matter (mainly string motivated)

• Later cosmology w<-1 • Inflation steep potential, non-gaussianity

• Bouncing

solutions

• I.A., astro-ph/0410443 I.A., L.Joukovkaya, JHEP,05109 (2005) 087 I.A., A.Koshelev, JHEP, 07022 (2007) 041 L.Joukovskaya, PR D76(2007) 105007; JHEP (2009 )  G. Calcagni, M.Montobbio,G.Nardelli,0705.3043; 0712.2237; Calcagni, Nardelli, 0904.4245;

• I.A., L.Joukovskaya, S.Vernov, JHEP 0707 (2007) 087

• Nunes, Mulryne, 0810.5471; N. Barnaby, T. Biswas, J.M. Cline, hep-th/0612230, J.Lidsey, hep-th/0703007; IA, N.Bulatov, L.Joukovskaya, S.Vernov, 0911.5105; PR(2009)

Nonlocal Models in Cosmology

II. Nonlocality in Gravity

24

2 24

1 1( ( ))2

( ,2

...)

p

s

Md x g V G R

g MFM

Arkani-Hamed at al hep-th/0209227; Khoury, hep-th/0612052T.Biswas, A.Mazumdar, W.Siegel hep-th/0508194 ,

G.Dvali, S. Hofmann, J Khoury, hep-th/0703027,

S.Deser, R.Woodard, arXiv:0706.2151S. UV - completion

Non-local Cosmology from String Field Theory

• Rolling tachyon in flat background• Rolling tachyon in curved background• Mathematical aspects

(rolling vs wide oscillations )• Non-local vs local models

Non-locality as a key point

Non-locality in level truncation

4 2 31 1[ ]

2 2 3open bosontachyon

gS d x

( ) exp[ ] ( )x r x

r is a number

2 20 i

24

( ) ( ) ( ) ( ) ( ) ( ) ( )

1[ ]

2 2 3ijki

open i i i ij i j i j kscalars

S d x M

2 2( ) ( ) ( )( ) 0r

i ik ik k ijk j kM e 2 0i for auxiliary fields

Non-locality in level truncation

Sen’s conjecture : f=1/4, I.A, D.Belov,A.Koshelev,P.Medvedev, Nucl.Phys.(2000), Ohmori (2001);

42 4

24

1 1[ ]

2 2 4NS GSOtachyon

d xS f

g

646

2 24

1

s s

co

v M MMg g

u u

Assumption:

How to understand .

2 2( ) ( ) ( )( ) 0r

i ik ik k ijk j kM e

2 2 2 2( ) 0m e c

Non-locality in level truncation in SFT

( )F J

?

?How to solve .

Non-locality in level truncation.

There are several options:

with the Fourier transform; the Laplace transform.

F is an analytical function in a neighbourhood of z=0

In particular,

( ) ?F

Non-locality in level truncation.

with the Fourier transform .

( )F

where

This definition is natural is SFT, where all expressions came from calculations in the momentum space

2t t

( )F

Non-locality in level truncation

where

Definition depends on “c”

as a symbol with the Laplace transformation

( ) ?F

2 0t t

Non-locality in level truncation

With the Laplace transformation with a closed contour

where

N.Barnaby, N.Kamran to study cosmological perturbations, arXiv:0712.2237;arXiv:0809.4513 (previous works by R.Woodard and coauthors)

Suitable for the Cauchy problem

0 t

2 2 32( 1) e 3( ') ( ') ' ( )K t t t dt t

I.A., Joukovskaya, Koshelev; Vladimirov, Ya.Volovich; Prokhorenko

Non-locality in level truncation. Rolling tachyon

Solution: kink2 02 1 Later oscillations

Boundary problem 1)(

30 0

2e 30 0 0K

0( ) 1 2Je

BUT NO Cauchy problem for

K0 is heat kernel

12( ) Je We cannot write (on our call of functions)

32Je We can write

14

f

Rolling tachyon in curved background

New conjecture:

( )t

t Effective cosmological constant

4 42 24

21 1 1( ( ) )2 24

g

s

pMg fFg

Rd xM

( ) ( ) zF z z e 2 1

I.A., astro-ph/0410443

4 42 24

21 1 1( ( ) )2 24

g

s

pMg fFg

Rd xM

Numbers. Hubble Parameter

62 22

26

p o c

pss

M g Mm

v MM

1/ pM

1/ sM

,s PM stri ng scal e M Pl anck scal e

2 pmH

pc MM 9

p

sp M

MMH

10sM

ps MM 6.610 pMH 6010

Local Mode Decomposition of Nonlocal Models

( ) gS d x F

F - entire function of order N

{ . . . }( )( ) (1 )

N

Nn n

z z

Nm P z

n n

zF z z e e

Weierstrass (Hadamard) product

2( ) ( ) . .g

n n g n ns

F c cM

Linearization

Non-local <--> Local (Linear Approximation)

The non-local model

42 24

21

( )2

)2

g

s

pMg RS d x Fg M

4 ' 22

,24

1( )( )

2 2M i i i ii

piS d

Mg Rx F M M

g

is equivalent to the local model

Here are the roots of the characteristic equation ( ) 0iF M iM'( ) 0iF M We assume

This equivalence is background independent

Ostrogradski (1859), Pais, Uhlenbeck (1950), Volovich (1969), Nakamura, Hamamoto (1996),…

IA, A.Koshelev, hep-th/0605085

Nonlocal Stringy Models – Linear Approximation

22 2( )F a ce

( )( ) WzWz e z

n-s branch of Lambert nW

Quadruples of complex roots

IA, L. Joukovskaya, S.Vernov, hep-th/0701184

214 2 ( 2 ) 0, 1, . . .

2a

n na W ce n

Nonlocal Stringy Models. Removing complex masses -----> zeta-function

1

1( ) , , 1

zn

z z in

1( ) ( )

2F z iz

1( )2

S i

/ 2( 1)( ) ( ) ( )

2 2zz z z

z z

The zeros of the Riemann zeta-function become the masses of particles

I.A., I.Volovich, hep-th/0701284;B.Dragovich, hep-th/0703008

zeta-function ksi-function

1½

tachyons

1( ) (1 ) z

p

z p

Summary

Non-local interacting theories can define globalcosmological background

Cosmological perturbations of nonlocal theoriesare the same as in local models

with w<-1without singularity at t=0

Specifying class of functions in the problem we erase singularities/instabilities