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Gravitational mo dels 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

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To sp ecify different typ es of cosmic fluids one uses a relation b etween the pressure p and the energy density p = w, p = Ek - V , = Ek + V where w is the state parameter. The spatially flat Friedmann­Robertson­Walker metric: (2 ) 2 2 2 2 2 ds = - dt + a (t) dx1 + dx2 + dx3 , (1) where a(t) is the scale factor, the Hubble parameter H a/a. 2H 2Ek w(t) = - 1 - = -1+ . (2) 3 H2 Contemporary exp eriments give strong supp ort that w > 0 -- Atoms. (4%) w = 0 -- the Cold Dark Matter. (23%) w < 0 -- the Dark Energy. (73%) wDE = -1 ± 0.2. (3) We consider the case wDE < -1. Null energy condition (NEC) is violated and there are problems of instability.
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Papers ab out cosmological mo dels with nonlo cal 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
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The SFT inspired nonlo cal cosmological mo dels
From the Witten action of b osonic cubic string field theory, considering only tachyon scalar field (x) one obtains: ] [ 1 12 133 26 ~ S= 2 d x (4) (x) (x) + (x) - (x) - , go 2 2 3 4 = . (5) 33 go is the op en string coupling constant, is the string length ~1 squared and = 6 -6 is added to the p otential to set the lo cal minimum of the p otential to zero. The action (4) leads to equation of motion = ek , k = ln( ), (


where

+ 1)e-2k = 32.

(6)

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In the ma jority of the SFT inspired nonlo cal gravitation mo dels the action is intro duced by hand as a sum of the SFT action of tachyon field and gravity part of the action: (2 ) 1 MP 1 12 133 4 S = 2 d x -g R + g + - - , (7) go 2 2 2 3 Action (7) includes a nonlo cal p otential. Using a suitable redefinition of the fields, one can made the p otential lo cal, at that the kinetic term b ecomes nonlo cal. This nonstandard kinetic term leads to a nonlo cal field b ehavior similar to the b ehavior of a phantom field, and it can b e approximated with a phantom kinetic term. The b ehavior of an op en string tachyon can b e effectively simulated by a scalar field with a phantom kinetic term. Another type of the SFT inspired mo dels includes nonlo cal mo dification of gravity. Recently G. Calcagni and G. Nardelli have considered nonlo cal gravity with nonlo cal scalar field (arXiv: 1004.5144).
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Nonlo cal action in the general form
We consider a general class of gravitational mo dels with a nonlo cal scalar field, which are describ ed by the following action: ( ( ) ) R 11 +2 S = d4x -g F ( g ) - V () - , (8) 16 GN go 2
2 GN is the Newtonian constant: 8 GN = 1/MP , MP is the Planck mass. We use the signature (-, +, +, +), gµ is the metric tensor, R is the scalar curvature, is the cosmological constant. Hereafter the d'Alembertian g is applied to scalar functions and can b e written as follows 1 µ -g g µ . (9) g= -g

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The function F (

g

) is assumed to b e an analytic function: F(
g

)=

n=0

fn

n g

.

(10)

Note that the term F ( g ) include not only terms with derivatives, but also f02. In an arbitrary metric the energy-momentum tensor ) 2 S 1( Tµ = - = 2 Eµ + E µ - gµ (g E + W ) , (11) µ -g g go Eµ
n-1 1 fn µ 2 n=1 l=0 l g l g



n-1-l g

,

(12)

n-1 1 W fn 2 n=2 l=1



n-l g

-

f0 2 + V (). 2

(13)

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From action (8) we obtain the following equations = 8 GN (Tµ - gµ ) , dV F ( g ) = , d G
µ

(14) (15)

where G

µ

is the Einstein tensor.

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From action (8) we obtain the following equations = 8 GN (Tµ - gµ ) , dV , F ( g ) = d G
µ

(16) (17)

where G

µ

is the Einstein tensor.

It is a system of nonlo cal nonlinear equations !!! HOW CAN WE FIND A SOLUTION?

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The Ostrogradski representation.
· M. Ostrogradski, M´ emoire sur les ´ equations differentiel les relatives aux probl` emes des isoperim´ es, Mem. St. Peetr tersb ourg VI Series, V. 4 (1850) 385­517 · A. Pais and G.E. Uhlenbeck, On Field Theories with Nonlocalized Action, Phys. Rev. 79 (1950) 145­165 F ( ) = F1( )
N j =1

( 1+
2 j

) , (18)

2 all ro ots, which are equal to -j , are simple. We want to get for

LF = F1( ). LF Ll = =
N j =1
10

(19)

the Ostragradski representation: find such numb ers cj , that cj j (
2 + j )j .

(20)


We define j =

N (

k =1,k =j

1 1+ 2 k

) ,

(

+

2 j

)

j = 0.

(21)

Substituting j in Ll , we get ) (N 4 ck 2 k Ll = F1 ( ) . 2+ k
k =1

(22)

Ll = LF



N ck 4 1 k = . 2 k + F1( ) k =1

(23)

All ro ots of F1( ) are simple, hence, we can p erform a partial fraction decomp osition of 1/F1( ).
2 F1(-k ) ck = , 4 k

where

2 F1(-k )

dF1 | d

2 =-k

.

(24)

Let F1( ) has two real simple ro ots. F1 > 0 in one and only one ro ot. We get mo del with one phantom and one standard field.
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The Ostrogradski representation and an algorithm of lo calization in the case of gravitational mo dels with an arbitrary quadratic p otential Generalization:
· Gravitation · F ( ) is an analytic functions · F ( ) has b oth simple and double ro ots. · The p otential V () = C22 + C1 + C0. ( V
ef f

=

) f0 C2 - 2 + C1 + C0 + . 2

(25)

We can change values of f0 and such that the p otential takes the form V () = C1. So, we put C2 = 0 and C0 = 0.

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Let us start with the case C1 = 0 and the equation F(
N1 i=1 g

) = 0.
N2 k =1

(26)

We seek a particular solution of (26) in the following form 0 = (
g

i +

~ k .

(27) (28)

- Ji)i = 0,

(

~~ - Jk )2k = 0

Ji are simple ro ots of the characteristic equation F (J ) = 0. ~ Jk are double ro ots.

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Energy­momentum tensor for sp ecial solutions
If we have one simple root 1 such that
n-1 1 Eµ (1) = fn J 2 n=1 l=0 n-1 1 g 1

= J11, then

F ( J1 ) µ1 1 = µ1 1. 2
n-1 2 1 1

n-1 1 n 2 J1 W (1) = fn J1 1 = fnnJ 2 n=1 2 n=1 l=0

J1F (J1) 2 = 1. 2 (29) (30) (31) (32)

In the case of two simple roots 1 and 2 we have
cr Eµ (1 + 2) = Eµ (1) + Eµ (2) + Eµ (1, 2),

where the cross term
cr Eµ (1, 2) = A1µ1 2 + A2µ2 1. n-1 ( J2 )l F (J1) - F (J2) 1 n A1 = fnJ1 -1 = 0, = 2 n=1 J1 2(J1 - J2) l=0

A2 = 0 .
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cr So, the cross term Eµ (1, 2) = 0 and

Eµ (1 + 2) = Eµ (1) + Eµ (2) Similar calculations shows W (1 + 2) = W (1) + W (2).

(33)

(34)

In the case of N simple roots the following formula has b een obtained:
N k =1

Tµ =

1 F (Jk ) µk k - gµ g k k + Jk 2 k 2

(

(

)

) . (35)

Note that the last formula is exactly the energy-momentum tensor of many free massive scalar fields. If F (J ) has simple real ro ots, then p ositive and negative values of F (Ji) alternate, so we can obtain phantom fields.

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~ Let J1 is a double ro ot. The fourth order differential equation ~~ ( - J1)21 = 0 is equivalent to the following system of equations: ~~ ~ ( - J1)1 = 1, ( - J1 ) 1 = 0 . (36) ~ ~ It is convenient to write l 1 in terms of the 1 and 1:
l

~ ~l ~ ~l 1 = J11 + lJ1-11.

(37)

For one double ro ot we obtain the following result: ) F (J ) ~ ~1 F (J1) ( ~ ~1) = Eµ ( µ1 1 + µ1 1 + ~ µ1 1. 4 12 Similar calculations gives ( ) ~ ~ ~ ~ ~ J1F (J1) F (J1) J1F (J1) ~ ~1) = 11 + + 2 . W ( 1 2 12 4

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For any analytical function F (J ), which has simple ro ots Ji ~ and double ro ots Jk , the energy­momentum tensor (N ) N2 N1 N2 1 ~ ~ Tµ (0) = Tµ i + k = Tµ (i) + Tµ (k ), (38)
i=1 k =1 i=1 k =1

where Tµ

) 1( = 2 Eµ + E µ - gµ (g E + W ) , go JiF (Ji) 2 W (i) = i , 2 dF F dJ


(39)

F (Ji) Eµ (i) = µi i, 2

(40)

) F (J ) ~k ~ F (Jk ) ( ~ ~ ~ µk k + k µk + µk k , Eµ (k ) = 4 12 ( ) ~ ~ ~ ~ ~ ~ ~k ) = Jk F (Jk ) k k + Jk F (Jk ) + F (Jk ) 2 . W ( k 2 12 4
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(41)

(42)


Consider the following lo cal action ( ) N2 N1 R ~ - + Si + Sk , Sloc = d4x -g 16 GN i=1
k =1

(43)

where

1 Si = - 2 d 4 x go ( 4 ~k = - 1 S d x -g g µ 2 go ) ~ F ( Jk ) µk k + + 12



) F (Ji) ( µ 2 -g g µi i + Jii , 2 ( ) ~ F (Jk ) ( ~ ~ µk k + k µk + 4 ( )) ~ ~ ~ ~ ~ J k F ( Jk ) ~ Jk F (Jk ) F (Jk ) k k + + 2 . k 2 12 4

Remark 1. If F (J ) has an infinity numb er of ro ots then one nonlo cal mo del corresp onds to infinity numb er of different lo cal mo dels. In this case the initial nonlo cal action (8) generates infinity number of lo cal actions (43).
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Remark 2. We should prove that the way of lo calization is self-consistent. To construct lo cal action (43) we assume that equations (28) are satisfied. Therefore, the metho d of lo calization is correct only if these equations can b e obtained from the lo cal action Sloc. The straightforward calculations show that Sloc Sloc ~ = 0 g i = Jii; = 0 g k = Jk k . (44) ~k i Sloc ~ ~~ =0 g k = Jk k + k . k We obtain from Sloc the Einstein equations as well: G
µ

(45)

= 8 GN (Tµ (0) - gµ ) ,

(46)

where 0 is given by (27) and Tµ (0) can b e calculated by (38). Any solutions of system (44)­(46) are particular solutions of the initial nonlo cal system (14)­(15).

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~ To clarify physical interpretation of lo cal fields k and k , we diagonalize the kinetic terms of these scalar fields in Sloc. ~ Expressing k and k in terms of new fields: (( ) ( )) 1 1 ~ 1 ~ ~ ~ ~k = F (Jk ) - F (Jk ) k - F (Jk ) + F (Jk ) k , (J ) ~k 3 3 2F k = k + k , ~ we obtain the corresp onding Sk in the ( ~ F (Jk ) ( 1 4 µ ~ Sk = - 2 d x -g g µk go 4 ~k ( J 1 ~ ~ ~ + (F (Jk ) - F (Jk ))k - (F (Jk ) + 4 3 ) ) ( ~ ~ ~ Jk F ( Jk ) F ( Jk ) + (k + k )2 . + 12 4 following form: ) k - k µk + ) 1 ~ F (Jk ))k (k + k ) + 3

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Let us consider the mo del with action (8) in the case C1 = 0. If f0 = 0, then we introduce a new scalar field C1 =- (47) f0 and get the previous case in terms of new field . F ( g ) = C1 F ( g ) = 0. (48) If f0 = 0, then J = 0 is a ro ot of the characteristic equation F (J ) = 0. It is easy to show, that the function = 0 + , ~ (49) where 0 and are solutions of the following equations C1 F ( g )0 = 0, . (50) g = f1 is a solution for F ( g ) = C1. ~ (51) The function 0 is given by (27), but the sum do not include i0 , which corresp onds to the ro ot J = 0, b ecause this function can not b e separated from .
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For a quadratic potential V () = C22 + C1 + C0 there exists the fol lowing algorithm of localization :

· Change values of f0 and such that the p otential takes the form V () = C1. · Find ro ots of the function F (J ) and calculate orders of them. · Select an finite number of simple and double ro ots. · Construct the corresp onding lo cal action. In the case C1 = 0 one should use formula (43). In the case C1 = 0 and f0 = 0 one should use (43) with the replacement of the scalar field by . In the case C1 = 0 and f0 = 0 the lo cal action is (43) plus ( ) 2 1 f2C S = - 2 d4x -g f1g µ µ + 2C1 + 2 1 . 2go f1 · Vary the obtained lo cal action and get a system of the Einstein equations and equations of motion. · Seek solutions of the obtained lo cal system.
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Conclusions
For gravitational mo dels with minimally coupling SFT inspired nonlo cal scalar fields and quadratic p otentials we obtain: · The Ostrogradski representations for nonlo cal Lagrangians in an arbitrary metric. · The algorithm of lo calization. · Lo cal and nonlo cal Einstein equations have one and the same solutions. · Nonlo cality arises in the case of F ( b er of ro ots.
g

) with an infinite num-

· One system of nonlo cal Einstein equations Infinity numb er of systems of lo cal Einstein equations.

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