1. 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

2. 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

3. Historical remarks. Non-locality in SFT Hikedi Yukawa (1949) Yukawa’s field as a reduced SFT field Phys.Rev., 76(1949) 300

4. 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 61:137-159,1957D.I. BlokhintsevDubna, JINR 1010) Nonlocal quantum field theory. D.A. Kirzhnits, (Lebedev Inst.) Usp.Fiz.Nauk 90:129-142,1966.D.A. KirzhnitsLebedev Inst. 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.V.L. GinzburgV.I. MankoLebedev Inst.

5. 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

6. 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

7. 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 metric Level 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

8. 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

9. 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

10. Nonlocal Models in Cosmology Nonlocality in Matter (mainly string motivated)Nonlocality in Matter (mainly string motivated) Nonlocality in Gravity Nonlocality in Gravity

11. Nonlocal Models in Cosmology 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 ) 0705.3043; G. Calcagni, M.Montobbio,G.Nardelli,0705.3043; 0712.2237; 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)

12. Nonlocal Models in Cosmology II. Nonlocality in Gravity Arkani-Hamed at al hep-th/0209227; Khoury, hep-th/0612052 Arkani-Hamed at al hep-th/0209227; Khoury, hep-th/0612052 T.Biswas, A.Mazumdar, W.Siegel hep-th/0508194, G.Dvali, S. Hofmann, JKhoury, hep-th/0703027, G.Dvali, S. Hofmann, J Khoury, hep-th/0703027, S.Deser, R.Woodard, arXiv0706.2151S. S.Deser, R.Woodard, arXiv:0706.2151S. UV - completion

13. 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

14. Non-locality in level truncation for auxiliary fields

15. Non-locality in level truncation Sen’s conjecture : f=1/4, I.A, D.Belov,A.Koshelev,P.Medvedev, Nucl.Phys.(2000), Ohmori (2001); Assumption:

16. How to understand. Non-locality in level truncation in SFT ? ? How to solve.

17. 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,

18. Non-locality in level truncation. with the Fourier transform. where This definition is natural is SFT, where all expressions came from calculations in the momentum space

19. Non-locality in level truncation where Definition depends on “c” as a symbol with the Laplace transformation

20. 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

21. I.A., Joukovskaya, Koshelev; I.A., Joukovskaya, Koshelev; Vladimirov, Ya.Volovich; Prokhorenko Non-locality in level truncation. Rolling tachyon Solution: kink Later oscillations Boundary problem BUT NO Cauchy problem for K 0 is heat kernel We cannot write (on our call of functions) We can write

22. Rolling tachyon in curved background New conjecture: Effective cosmological constant I.A., astro-ph/0410443

23. Numbers. Hubble Parameter

24. Local Mode Decomposition of Nonlocal Models F - entire function of order N Weierstrass (Hadamard) product Linearization

25. Non-local Local (Linear Approximation) The non-local model is equivalent to the local model Here are the roots of the characteristic equation We assume This equivalence is background independent Ostrogradski (1859), Pais, Uhlenbeck (1950), Volovich (1969), Nakamura, Hamamoto (1996),… IA, A.Koshelev, hep-th/0605085

26. Nonlocal Stringy Models – Linear Approximation n-s branch of Lambert Quadruples of complex roots IA, L. Joukovskaya, S.Vernov, hep-th/0701184

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

28. Summary Non-local interacting theories can define global cosmological background Cosmological perturbations of nonlocal theories are the same as in local models with w<-1 without singularity at t=0 Specifying class of functions in the problem we erase singularities/instabilities