A New Model of Nonlocal Modified Gravity
Ivan Dimitrijevic, Branko Dragovich, Jelena Grujic, Zoran Rakic
aa r X i v : . [ h e p - t h ] N ov A NEW MODEL OF NONLOCAL MODIFIED GRAVITY
Ivan Dimitrijevic, Branko Dragovich, Jelena Grujic, and Zoran Rakic
Abstract.
We consider a new modified gravity model with nonlocal term ofthe form R − F ( (cid:3) ) R. This kind of nonlocality is motivated by investigationof applicability of a few unusual ans¨atze to obtain some exact cosmologicalsolutions. In particular, we find attractive and useful quadratic ansatz (cid:3) R = qR .
1. Introduction
In spite of the great successes of General Relativity (GR) it has not got statusof a complete theory of gravity. To modify GR there are motivations comingfrom its quantum aspects, string theory, astrophysics and cosmology. For example,cosmological solutions of GR contain Big Bang singularity, and Dark Energy as acause for accelerated expansion of the Universe. This initial cosmological singularityis an evident signature that GR is not appropriate theory of the Universe at cosmictime t = 0 . Also, GR has not been verified at the very large cosmic scale and darkenergy has not been discovered in the laboratory experiments. This situation givesrise to research for an adequate modification of GR among numerous possibilities(for a recent review, see [ ]).Recently it has been shown that nonlocal modified gravity with action(1.1) S = Z d x √− g (cid:16) R − πG + CR F ( (cid:3) ) R (cid:17) , where R is scalar curvature, Λ – cosmological constant, F ( (cid:3) ) = ∞ X n =0 f n (cid:3) n is ananalytic function of the d’Alembert-Beltrami operator (cid:3) = √− g ∂ µ √− gg µν ∂ ν , g = det ( g µν ) and C is a constant, has nonsingular bounce cosmological solutions, see[
2, 3, 4, 5 ]. To solve equations of motion it was used ansatz (cid:3) R = rR + s. In [ ] we Mathematics Subject Classification.
Primary 83Dxx, 83Fxx, 53C21; Secondary 83C10,83C15.
Key words and phrases. nonlocal modified gravity, cosmological solutions.Work partially supported by the Serbian Ministry of Education, Science and TechnologicalDevelopment, contract No. 174012. introduced some new ans¨atze, which gave trivial solutions for the above nonlocalmodel (1.1). In this paper we consider some modification of the above action inthe nonlocal sector, i.e.(1.2) S = Z d x √− g (cid:16) R πG + R − F ( (cid:3) ) R (cid:17) and look for nontrivial cosmological solutions for the new ans¨atze (see [ ]). Notethat the cosmological constant Λ in (1.2) is hidden in the term f , i.e. Λ = − πGf . To the best of our knowledge action (1.2) has not been considered so far. However,there are investigations of gravity modified by 1 /R term (see, e.g. [ ] and referencestherein), but it is without nonlocality.
2. Equations of motion
By variation of action (1.2) with respect to metric g µν one obtains the equationsof motion for g µν (2.1) R µν V − ( ∇ µ ∇ ν − g µν (cid:3) ) V − g µν R − F ( (cid:3) ) R + ∞ X n =1 f n n − X l =0 (cid:0) g µν (cid:0) ∂ α (cid:3) l ( R − ) ∂ α (cid:3) n − − l R + (cid:3) l ( R − ) (cid:3) n − l R (cid:1) − ∂ µ (cid:3) l ( R − ) ∂ ν (cid:3) n − − l R (cid:1) = − G µν πG ,V = F ( (cid:3) ) R − − R − F ( (cid:3) ) R. The trace of the equation (2.1) is(2.2) RV + 3 (cid:3) V + ∞ X n =1 f n n − X l =0 (cid:0) ∂ α (cid:3) l ( R − ) ∂ α (cid:3) n − − l R + 2 (cid:3) l ( R − ) (cid:3) n − l R (cid:1) − R − F ( (cid:3) ) R = R πG . The 00 component of (2.1) is(2.3) R V − ( ∇ ∇ − g (cid:3) ) V − g R − F ( (cid:3) ) R + ∞ X n =1 f n n − X l =0 (cid:0) g (cid:0) ∂ α (cid:3) l ( R − ) ∂ α (cid:3) n − − l R + (cid:3) l ( R − ) (cid:3) n − l R (cid:1) − ∂ (cid:3) l ( R − ) ∂ (cid:3) n − − l R (cid:1) = − G πG . We use Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric ds = − dt + a ( t ) (cid:0) dr − kr + r dθ + r sin θdφ (cid:1) and investigate all three possibilities for curvatureparameter k (0 , ± R = 6 (cid:16) ¨ aa + ˙ a a + ka (cid:17) and (cid:3) h ( t ) = − ∂ t h ( t ) − H∂ t h ( t ) , where H = ˙ aa is the Hubble parameter. Inthe sequel we shall use three kinds of ans¨atze (two of them introduced in [ ]) and NEW MODEL OF NONLOCAL MODIFIED GRAVITY 3 solve equations of motions (2.2) and (2.3) for cosmological scale factor in the form a ( t ) = a | t − t | α .
3. Quadratic ansatz: (cid:3) R = qR Looking for solutions in the form a ( t ) = a | t − t | α this ansatz becomes(3.1) α (2 α − qα (2 α − − ( α − t − t ) − + αk a (1 − α + 6 q (2 α − t − t ) − α − + qk a ( t − t ) − α = 0 . Equation (3.1) is satisfied for all values of time t in six cases:(1) k = 0, α = 0, q ∈ R ,(2) k = 0, α = , q ∈ R ,(3) k = 0, α = 0 and α = , q = α − α (2 α − , (4) k = − α = 1, q = 0, a = 1,(5) k = 0, α = 0, q = 0,(6) k = 0, α = 1, q = 0.In the cases (1), (2) and (4) we have R = 0 and therefore R − is not defined.The case (5) yields a solution which does not satisfy equations of motion. Hencethere remain two cases for further consideration. k = 0 , q = α − α (2 α − . For this case, we have the following expressionsdepending on the parameter α :(3.2) q = α − α (2 α − , R = 6 α (2 α − t − t ) − ,a = a | t − t | α , H = α ( t − t ) − ,R = 3 α (1 − α )( t − t ) − , G = 3 α ( t − t ) − . We now express (cid:3) n R and (cid:3) n R − in the following way:(3.3) (cid:3) n R = B ( n, t − t ) − n − , (cid:3) n R − = B ( n, − t − t ) − n ,B ( n,
1) = 6 α (2 α − − n n ! n Y l =1 (1 − α + 2 l ) , n > ,B ( n, −
1) = (6 α (2 α − − n n Y l =1 (2 − l )( − − α + 2 l ) , n > ,B (0 ,
1) = 6 α (2 α − , B (0 , −
1) = B (0 , − . Note that B (1 , −
1) = − α +13 α (2 α − = − α + 1) B (0 , − and B ( n, −
1) = 0 if n > . Also, we obtain(3.4) F ( (cid:3) ) R = ∞ X n =0 f n B ( n, t − t ) − n − , F ( (cid:3) ) R − = f B (0 , − t − t ) + f B (1 , − . IVAN DIMITRIJEVIC, BRANKO DRAGOVICH, JELENA GRUJIC, AND ZORAN RAKIC
Substituting these equations into trace and 00 component of the EOM one has(3.5) r − ∞ X n =0 f n B ( n,
1) ( − r + 6(1 − n )(1 − n + 3 α )) ( t − t ) − n + r X n =0 f n ( rB ( n, −
1) + 3 B ( n + 1 , − t − t ) − n + 2 r ∞ X n =1 f n γ n ( t − t ) − n = r πG ( t − t ) − , ∞ X n =0 f n r − B ( n, (cid:16) r − A n (cid:17) ( t − t ) − n + X n =0 f n rB ( n, − A n ( t − t ) − n + r ∞ X n =1 f n δ n ( t − t ) − n = − r πG α α − t − t ) − , where r = B (0 ,
1) and(3.6) γ n = n − X l =0 B ( l, − B ( n − l,
1) + 2(1 − l )( n − l ) B ( n − l − , ,δ n = n − X l =0 B ( l, − − B ( n − l,
1) + 4(1 − l )( n − l ) B ( n − l − , ,A n = 6 α (1 − n ) − r α − α −
1) = r − n − α α − . Equations (3.5) can be split into system of pairs of equations with respect toeach coefficient f n . In the case n > , there are the following pairs:(3.7) f n (cid:0) B ( n,
1) ( − r + 6(1 − n )(1 − n + 3 α )) + 2 r γ n (cid:1) = 0 ,f n (cid:16) B ( n, (cid:16) r − A n (cid:17) + r δ n (cid:17) = 0 . Taking α − to be a natural number one obtains: B ( n,
1) = 6 α (2 α − n n ! ( ( α − ( α − − n )! , n < α − , (3.8) B ( n,
1) = 0 , n > α − , (3.9) γ n = 2 B (0 , − B ( n − , nα − n − α − , n α − , (3.10) δ n = 2 B (0 , − B ( n − , n + 3 n + 3 α − αn + 1) , n α − , (3.11) γ n = δ n = 0 , n > α − . (3.12) NEW MODEL OF NONLOCAL MODIFIED GRAVITY 5 If n > α − then B ( n,
1) = γ n = δ n = 0 and hence the system is triviallysatisfied for arbitrary value of coefficients f n . On the other hand for 2 n α − the system has only trivial solution f n = 0.When n = 0 the pair becomes f (cid:0) − r + 6(1 + 3 α ) + 3 rB (1 , − (cid:1) = 0 , f = 0(3.13)and its solution is f = 0. The remaining case n = 1 reads(3.14) f (cid:0) − r − B (1 ,
1) + rB (1 , −
1) + 2 γ (cid:1) = r πG ,f (cid:16) A ( rB (1 , − − r − B (1 , B (1 ,
1) + rδ ) (cid:17) = − r πG α α − , and it gives f = − α (2 α − πG (3 α − . k = 0 , α = 1 , q = 0 . In this case(3.15) a = a | t − t | , H = ( t − t ) − , R = s ( t − t ) − ,s = 6(1 + ka ) , (cid:3) R = 0 , R = 0 , (cid:3) n R − = D ( n, − t − t ) − n ,D (0 , −
1) = s − , D (1 , −
1) = − s − , D ( n, −
1) = 0 , n > . Substitution of the above expressions in trace and 00 component of the EOMyields(3.16) 3 f + X n =0 f n sD ( n, − t − t ) − n + 4 f ( t − t ) − = s πG ( t − t ) − , − f s − + 12 f + 6 X n =0 f n D ( n, − − n )( t − t ) − n + 2 f ( t − t ) − = − s πG ( t − t ) − . This system leads to conditions for f and f :(3.17) − f − f ( t − t ) − = s πG ( t − t ) − , f + 2 f ( t − t ) − = − s πG ( t − t ) − . The corresponding solution is(3.18) f = 0 , f = − s πG , f n ∈ R , n > . IVAN DIMITRIJEVIC, BRANKO DRAGOVICH, JELENA GRUJIC, AND ZORAN RAKIC
4. Ansatz (cid:3) n R = c n R n +1 , n > (cid:3) n +1 R in two ways: (cid:3) n +1 R = (cid:3) c n R n +1 = c n (( n + 1) R n (cid:3) R − n ( n + 1) R n − ˙ R )= c n ( n + 1)( c R n +2 − nR n − ˙ R )= c n +1 R n +2 it follows ˙ R = R , (4.1) c n +1 = c n ( n + 1)( c − n ) , (4.2)where ˙ R means ( ˙ R ) .General solution of equation (4.1) is(4.3) R = 4( t − t ) , t ∈ R . Taking n = 1 in the ansatz yields(4.4) (cid:3) R = c R . Substitution of (4.3) in (4.4) gives H = c +33( t − t ) . This implies(4.5) a ( t ) = a | t − t | c , a > . Using (4.3) in equation(4.6) R = 6 (cid:18) ¨ aa + ˙ a a + ka (cid:19) gives(4.7) ( t − t ) ¨ y − y = − k ( t − t ) , where y = a ( t ) . It can be shown that general solution of the last equation is(4.8) a ( t ) = ˜ C | t − d | √ + ˜ C | t − d | −√ − k | t − d | , ˜ C , ˜ C ∈ R . By comparison of the last equation with (4.5) one can conclude:(1) If c = 0 then k must be equal to −
1. In this case (cid:3) n R = 0 , n > c = 0 then k must be equal to 0. In this case c = − ±√ . NEW MODEL OF NONLOCAL MODIFIED GRAVITY 7 (cid:3) n R = c n R n +1 , c = 0 . From the previous analysis, it follows:(4.9) k = − , a ( t ) = √ | t − t | , H ( t ) = 1 t − t ,R = 4( t − t ) , (cid:3) n R = 0 , n > , F ( (cid:3) ) R = f R. It can be shown that (cid:3) n R − = ( − n n − n − Y l =0 (1 − l )(2 − l )( t − t ) − n . (4.10)From (4.10) follows (cid:3) n R − = 0 , n >
1. Then F ( (cid:3) )( R − ) = f R − + f (cid:3) R − . (4.11)Substituting (4.9) and (4.11) in the 00 component of the EOM one obtains f t − t ) + 2 f + 18 πG = 0(4.12)and it follows f = 0 , f = − πG , f n ∈ R , n > . (4.13)Substituting (4.9) and (4.11) in the trace equation one has − f ( t − t ) − f − πG = 0(4.14)and it gives the same result (4.13). (cid:3) n R = c n R n +1 , c = − ±√ . In this case: k = 0 , R = 4( t − t ) , H = 2 c + 33( t − t ) , a = a | t − t | c , a > , (4.15) R = 3 α (1 − α )( t − t ) − , G = (3 α (1 − α ) + 2)( t − t ) − , α = 2 c + 33 , (cid:3) n R = 4 n +1 c n ( t − t ) − n − , c = 1 . One can show that (cid:3) n R − = M ( n, − t − t ) − n , (4.16)where M (0 , −
1) = 14 , M (1 , −
1) = − ( c + 2) , M ( n, −
1) = 0 , n > . (4.17)Also one obtains(4.18) F ( (cid:3) ) R = ∞ X n =0 n +1 f n c n ( t − t ) − n − , F ( (cid:3) ) R − = f M (0 , − t − t ) + f M (1 , − . IVAN DIMITRIJEVIC, BRANKO DRAGOVICH, JELENA GRUJIC, AND ZORAN RAKIC
Substituting (4.18) in the trace equation it becomes − πG ( t − t ) − − f − ∞ X n =1 n f n c n ( t − t ) − n + ∞ X n =1 f n (cid:0) n − X l =0 M ( l, − n − l +1 ((1 − l )( n − l ) c n − − l + 2 c n − l ) (cid:1) ( t − t ) − n (4.19) + f (4 M (1 , −
1) + 3 M (2 , − t − t ) − = 0 . To satisfy equation (4.19) for all values of time t one obtains: f = 0 , f (2 c + 1) = − πG , (4.20) f n (cid:0) − c n + X l =0 M ( l, − − l ((1 − l )( n − l ) c n − − l + 2 c n − l ) (cid:1) = 0 , n > . (4.21)Suppose that f n = 0 for n >
2, then from the last equation follows − c n + X l =0 M ( l, − − l ((1 − l )( n − l ) c n − − l + 2 c n − l ) = 0(4.22)and it becomes c n − ( n − c n − c −
4) = 0 . (4.23)Since c n − = 0 , condition (4.23) is satisfied for n = − n = c + 2.Hence, we conclude that f n = 0 for n > f n = 0 for n >
2, the 00 component of the EOM becomes(4.24) 116 πG ( − α + 3 α + 2)( t − t ) − + 12 f ( 32 α − α + 1) + f c (3 α − α + 2)( t − t ) − + 8 f M (0 , − − c )( t − t ) − + 3 α (3 − α ) M (0 , − f − α ( α − M (1 , − f ( t − t ) − = 0 . In order to satisfy equation (4.24) for all values of time t it has to be f = 0 , f ( 43 c + 103 c + 2 c + 1) = 116 πG ( 23 c + c − . (4.25)The necessary and sufficient condition for the EOM to have a solution is c (8 c + 18 c + 3) = 0 . (4.26)Since c = − ±√ , the last condition is satisfied. NEW MODEL OF NONLOCAL MODIFIED GRAVITY 9
5. Cubic ansatz: (cid:3) R = qR Recall that we are looking for solutions in the form a ( t ) = a | t − t | α . In theexplicit form it reads(5.1) α ( α − (cid:16) α − t − t ) − + ka ( t − t ) − α − (cid:17) = 18 q (cid:16) α (2 α − t − t ) − + ka ( t − t ) − α (cid:17) . It yields the following seven possibilities:(1) k = 0, α = 0, q ∈ R ,(2) k = 0, α = , q ∈ R ,(3) k = − α = 1, q = 0, a = 1,(4) k = 0, α = 1, q = 0, (5) k = 0, α = 0, q = 0,(6) k = 0, α = 1, q = 0,(7) k = 0, α = , q = − a .Cases (1), (2) and (3) contain scalar curvature R = 0 , and therefore we willnot discuss them. Cases (4), (5) and 6 are also obtained from the quadratic ansatzand have been discussed earlier. The last case contains:(5.2) a ( t ) = a p | t − t | , H ( t ) = 12( t − t ) ,R ( t ) = 6 ka | t − t | − , R = 34( t − t ) . One can derive the following expressions:(5.3) (cid:3) n R = N ( n, | t − t | − n − , (cid:3) n R − = N ( n, − | t − t | − n ,N (0 ,
1) = 6 ka , N (0 , −
1) = N (0 , − ,N ( n,
1) = N (0 , − n n − Y l =0 (2 l + 1)(2 l + 12 ) , n > ,N ( n, −
1) = N (0 , − ( − n n − Y l =0 (2 l − l −
32 ) , n > , F ( (cid:3) ) R = ∞ X n =0 f n N ( n, | t − t | − n − , F ( (cid:3) ) R − = ∞ X n =0 f n N ( n, − | t − t | − n . Substituting (5.3) in the trace equation we obtain(5.4) − N (0 , − ∞ X n =0 f n N ( n, | t − t | − n + N (0 , ∞ X n =0 f n ( N ( n, − − N (0 , − N ( n, | t − t | − n + 3 ∞ X n =0 f n ( N ( n, − − N (0 , − N ( n, | t − t | − − n + ∞ X n =1 f n n − X l =0 N ( l, − − l )( − n + 2 l + 1) N ( n − l − , N ( n − l, | t − t | − n = N (0 , πG | t − t | − . This equation implies the following conditions on coefficient f : f = 0 , N (0 , πG = 0 . (5.5)Since N (0 , = 0, the last equation never holds and therefore there is no solutionin this case.
6. Concluding remarks
Using a few new ans¨atze we have shown that equations of motion for nonlocalgravity model given by action (1.2) yield some bounce cosmological solutions of theform a ( t ) = a | t − t | α . These solutions lead to f = 0 and hence Λ = 0 , and when t → ∞ then R → . In particular, quadratic ansatz (cid:3) R = qR is very promising.Note that ansatz (cid:3) n R = c n R n +1 , n > , can be viewed as a special case of ansatz (cid:3) R = qR . It is worth noting that equations of motion (2.2) and (2.3) have the de Sittersolutions a ( t ) = a cosh( λt ) , k = +1 and a ( t ) = a e λt , k = 0 , when f = − λ πG = − Λ8 πG , f n ∈ R , n > . This investigation can be generalized to some cases with R − p F ( (cid:3) ) R q nonlocalterm, where p and q are some natural numbers satisfying q − p > . It will bepresented elsewhere with discussion of various properties.
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