Cosmological models with nonlocal scalar fields Sergey Yu. Vernov - PowerPoint PPT Presentation

Cosmological models with nonlocal scalar fields Sergey Yu. Vernov SINP MSU based on the following papers I.Ya. Aref’eva, L.V. Joukovskaya, S.V., J. Phys A 41 (2008) 304003, arXiv:0711.1364 S.V., Class. Quant. Grav. 27 (2010) 035006, arXiv:0907.0468 S.V., arXiv:1005.0372 S.V., arXiv:1005.5007 1

Papers about cosmological models with nonlocal fields: I.Ya. Aref’eva, Nonlocal String Tachyon as a Model for Cosmological Dark Energy , astro-ph/0410443, 2004. I.Ya. Aref’eva and L.V. Joukovskaya, 2005; I.Ya. Aref’eva and A.S. Koshelev, 2006; 2008; I.Ya. Aref’eva and I.V. Volovich, 2006; 2007; I.Ya. Aref’eva, 2007; A.S. Koshelev, 2007; L.V. Joukovskaya, 2007; 2008; 2009 I.Ya. Aref’eva, L.V. Joukovskaya, S.Yu.V., 2007 J.E. Lidsey, 2007; G. Calcagni, 2006; G. Calcagni, M. Montobbio and G. Nardelli, 2007; G. Calcagni and G. Nardelli, 2007; 2009; 2010 N. Barnaby, T. Biswas and J.M. Cline, 2006; N. Barnaby and J.M. Cline, 2007; N. Barnaby and N. Kamran, 2007; 2008; N. Barnaby, 2008; 2010; D.J. Mulryne, N.J. Nunes, 2008; B. Dragovich, 2008; A.S. Koshelev, S.Yu.V., 2009; 2010 2

The SFT inspired nonlocal cosmological models From the Witten action of bosonic cubic string field theory, considering only tachyon scalar field φ ( x ) one obtains: � α ′ � S = 1 � 2 φ ( x ) � φ ( x ) + 1 2 φ 2 ( x ) − 1 3 γ 3 Φ 3 ( x ) − ˜ d 26 x Λ , (1) g 2 o where 4 k = α ′ ln( γ ) , Φ = e k � φ, γ = √ 3 . (2) 3 g o is the open string coupling constant, α ′ is the string length 6 γ − 6 is added to the potential to set the local squared and ˜ Λ = 1 minimum of the potential to zero. The action (1) leads to equation of motion ( α ′ � + 1) e − 2 k � Φ = γ 3 Φ 2 . (3) 3

In the majority of the SFT inspired nonlocal gravitation mod- els the action is introduced by hand as a sum of the SFT action of tachyon field and gravity part of the action: d 4 x √− g � M 2 � S = 1 � 2 R + 1 2 φ � g φ + 1 2 φ 2 − 1 3 γ 3 Φ 3 − Λ P , (4) g 2 o Action (4) includes a nonlocal potential. Using a suitable redefinition of the fields, one can made the potential local, at that the kinetic term becomes nonlocal. This nonstandard kinetic term leads to a nonlocal field be- havior similar to the behavior of a phantom field, and it can be approximated with a phantom kinetic term. The behavior of an open string tachyon can be effectively simulated by a scalar field with a phantom kinetic term. Another type of the SFT inspired models includes nonlocal modification of gravity. Recently G. Calcagni and G. Nardelli have considered non- local gravity with nonlocal scalar field (arXiv: 1004.5144). 4

Nonlocal action in the general form We consider a general class of gravitational models with a non- local scalar field, which are described by the following action: d 4 x √− gα ′ � � 1 � � � R + 1 S = 2 φ F ( � g ) φ − V ( φ ) − Λ , (5) 16 πG N g 2 o G N is the Newtonian constant: 8 πG N = 1 /M 2 P , M P is the Planck mass. We use the signature ( − , + , + , +) , g µν is the metric tensor, R is the scalar curvature, Λ is the cosmological constant. Hereafter the d’Alembertian � g is applied to scalar functions and can be written as follows √− gg µν ∂ ν . 1 √− g∂ µ � g = (6) 5

The function F ( � g ) is assumed to be an analytic function: ∞ � f n � n F ( � g ) = g . (7) n =0 Note that the term φ F ( � g ) φ include not only terms with derivatives, but also f 0 φ 2 . In an arbitrary metric the energy-momentum tensor 2 δg µν = 1 δS � � E µν + E νµ − g µν ( g ρσ E ρσ + W ) √− g T µν = − , (8) g 2 o ∞ n − 1 E µν ≡ 1 � � ∂ µ � l g φ∂ ν � n − 1 − l f n φ, (9) g 2 n =1 l =0 ∞ n − 1 W ≡ 1 φ − f 0 2 φ 2 + V ( φ ) . � � � l g φ � n − l f n (10) g 2 n =2 l =1 6

From action (5) we obtain the following equations G µν = 8 πG N ( T µν − Λ g µν ) , (11) F ( � g ) φ = dV dφ , (12) where G µν is the Einstein tensor. 7

From action (5) we obtain the following equations G µν = 8 πG N ( T µν − Λ g µν ) , (13) F ( � g ) φ = dV dφ , (14) where G µν is the Einstein tensor. It is a system of nonlocal nonlinear equations !!! HOW CAN WE FIND A SOLUTION? 8

The Ostragradski representation . • M. Ostrogradski , M´ emoire sur les ´ equations differentielles relatives aux probl` emes des isoperim´ etres , Mem. St. Pe- tersbourg VI Series, V. 4 (1850) 385–517 • A. Pais and G.E. Uhlenbeck , On Field Theories with Nonlo- calized Action , Phys. Rev. 79 (1950) 145–165 Let F is a polynomial: N � � 1 + � � F ( � ) = F 1 ( � ) ≡ , (15) ω 2 j j =1 all roots, which are equal to − ω 2 j , are simple. We want to get the Ostrogradski representation for L F = φ F 1 ( � ) φ. (16) We should find such numbers c j , that the Lagrangian L F can 9

be written in the following form N � c j φ j ( � + ω 2 L l = j ) φ j . (17) j =1 N � 1 + 1 � � � � + ω 2 � φ j = φ, ⇒ φ j = 0 . (18) � j ω 2 k k =1 ,k � = j Substituting φ j in L l , we get N c k ω 4 1 L l ∼ � k = L F ⇔ k + � = F 1 ( � ) . (19) ω 2 k =1 All roots of F 1 ( � ) are simple, hence, we can perform a partial fraction decomposition of 1 / F 1 ( � ) . c k = F ′ 1 ( − ω 2 k ) k ) ′ ≡ d F 1 F 1 ( − ω 2 , where d � | � = − ω 2 k . (20) ω 4 k Let F 1 ( � ) has two real simple roots. F ′ 1 > 0 in one and only one root. We get model with one phantom and one real root. 10

An algorithm of localization in the case of an arbi- trary quadratic potential V ( φ ) = C 2 φ 2 + C 1 φ + C 0 . � � C 2 − f 0 φ 2 + C 1 φ + C 0 + Λ . V eff = (21) 2 We can change values of f 0 and Λ such that the potential takes the form V ( φ ) = C 1 φ . In other words, we put C 2 = 0 and C 0 = 0 . There exist 3 cases: • C 1 = 0 • C 1 � = 0 and f 0 � = 0 • C 1 � = 0 and f 0 = 0 I will speak about the case C 1 = 0 . Cases C 1 � = 0 have been considered in S.V., arXiv:1005.0372 . 11

Let us consider the case C 1 = 0 and the equation F ( � g ) φ = 0 . (22) We seek a particular solution of (14) in the following form N 1 N 2 ˜ � � φ 0 = φ i + φ k . (23) i =1 k =1 ( � g − J i ) φ i = 0 , (24) J i are simple roots of the characteristic equation F ( J ) = 0 . ˜ J k are double roots. The fourth order differential equation J k ) 2 ˜ ( � − ˜ φ k = 0 (25) is equivalent to the following system of equations: ( � − ˜ J k )˜ ( � − ˜ φ k = ϕ k , J k ) ϕ k = 0 . (26) 12

Energy–momentum tensor for special solutions If we have one simple root φ 1 such that � g φ 1 = J 1 φ 1 , then ∞ n − 1 ∂ µ φ 1 ∂ ν φ 1 = F ′ ( J 1 ) E µν ( φ 1 ) = 1 � � J n − 1 f n ∂ µ φ 1 ∂ ν φ 1 . 1 2 2 n =1 l =0 ∞ n − 1 ∞ 1 = J 1 F ′ ( J 1 ) W ( φ 1 ) = 1 1 = J 1 � � � J n f n nJ n − 1 1 φ 2 φ 2 φ 2 f n 1 . 1 2 2 2 n =1 l =0 n =1 In the case of two simple roots φ 1 and φ 2 we have E µν ( φ 1 + φ 2 ) = E µν ( φ 1 ) + E µν ( φ 2 ) + E cr µν ( φ 1 , φ 2 ) , (27) where the cross term E cr µν ( φ 1 , φ 2 ) = A 1 ∂ µ φ 1 ∂ ν φ 2 + A 2 ∂ µ φ 2 ∂ ν φ 1 . (28) ∞ n − 1 � l � J 2 A 1 = 1 = F ( J 1 ) − F ( J 2 ) � � f n J n − 1 = 0 , (29) 1 2 J 1 2( J 1 − J 2 ) n =1 l =0 A 2 = 0 . (30) 13

So, the cross term E cr µν ( φ 1 , φ 2 ) = 0 and E µν ( φ 1 + φ 2 ) = E µν ( φ 1 ) + E µν ( φ 2 ) (31) Similar calculations shows W ( φ 1 + φ 2 ) = W ( φ 1 ) + W ( φ 2 ) . (32) In the case of N simple roots the following formula has been obtained: N � ∂ µ φ k ∂ ν φ k − 1 �� � F ′ ( J k ) g ρσ ∂ ρ φ k ∂ σ φ k + J k φ 2 � T µν = 2 g µν . (33) k k =1 Note that the last formula is exactly the energy-momentum tensor of many free massive scalar fields. If F ( J ) has simple real roots, then positive and negative values of F ′ ( J i ) alternate, so we can obtain phantom fields. 14

Let ˜ J 1 is a double root. The fourth order differential equation J 1 ) 2 ˜ ( � − ˜ φ 1 = 0 is equivalent to the following system of equa- tions: ( � − ˜ J 1 )˜ ( � − ˜ φ 1 = ϕ 1 , J 1 ) ϕ 1 = 0 . (34) It is convenient to write � l ˜ φ 1 in terms of the ˜ φ 1 and ϕ 1 : � l ˜ 1 ˜ φ 1 = ˜ φ 1 + l ˜ J l J l − 1 ϕ 1 . (35) 1 E µν (˜ φ 1 ) = B 1 ∂ µ ˜ φ 1 ∂ ν ˜ φ 1 + B 2 ∂ µ ˜ φ 1 ∂ ν ϕ 1 + B 3 ∂ µ φ 1 ∂ ν ˜ ϕ 1 + B 4 ∂ µ ϕ 1 ∂ ν ϕ 1 , (36) where B 1 = F ′ ( ˜ B 2 = B 3 = F ′′ ( ˜ B 4 = F ′′′ ( ˜ J 1 ) J 1 ) J 1 ) = 0 , , . 2 4 12 Thus, for one double root we obtain the following result: φ 1 ) = F ′′ ( ˜ ϕ 1 ) + F ′′′ ( ˜ J 1 ) J 1 ) E µν (˜ ( ∂ µ ˜ φ 1 ∂ ν ϕ 1 + ∂ µ φ 1 ∂ ν ˜ ∂ µ ϕ 1 ∂ ν ϕ 1 . 4 12 Similar calculations gives � ˜ � J 1 F ′′ ( ˜ ˜ J 1 F ′′′ ( ˜ + F ′′ ( ˜ J 1 ) J 1 ) J 1 ) W ( ˜ ˜ ϕ 2 φ 1 ) = φ 1 ϕ 1 + 1 . (37) 2 12 4 15