A New Model of Nonlocal Modified Gravity

A NEW MODEL OF NONLOCAL MODIFIED GRAVITY arXiv:1411.5034v1 [hep-th] 18 Nov 2014 Ivan Dimitrijevic, Branko Dragovich, Jelena Grujic, and Zoran Rakic Abstract. We consider a new modified gravity model with nonlocal term of the form R−1 F ()R. This kind of nonlocality is motivated by investigation of applicability of a few unusual ansätze to obtain some exact cosmological solutions. In particular, we find attractive and useful quadratic ansatz R = qR2 . 1. Introduction In spite of the great successes of General Relativity (GR) it has not got status of a complete theory of gravity. To modify GR there are motivations coming from its quantum aspects, string theory, astrophysics and cosmology. For example, cosmological solutions of GR contain Big Bang singularity, and Dark Energy as a cause for accelerated expansion of the Universe. This initial cosmological singularity is an evident signature that GR is not appropriate theory of the Universe at cosmic time t = 0. Also, GR has not been verified at the very large cosmic scale and dark energy has not been discovered in the laboratory experiments. This situation gives rise to research for an adequate modification of GR among numerous possibilities (for a recent review, see [1]). Recently it has been shown that nonlocal modified gravity with action Z  √  R − 2Λ (1.1) S = d4 x −g + CRF ()R , 16πG ∞ X where R is scalar curvature, Λ – cosmological constant, F () = fn n is an √n=0 1 ∂µ −gg µν ∂ν , g = analytic function of the d’Alembert-Beltrami operator  = √−g det(gµν ) and C is a constant, has nonsingular bounce cosmological solutions, see [2, 3, 4, 5]. To solve equations of motion it was used ansatz R = rR+ s. In [6] we 2010 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 Technological Development, contract No. 174012. 1 2 IVAN DIMITRIJEVIC, BRANKO DRAGOVICH, JELENA GRUJIC, AND ZORAN RAKIC introduced some new ansätze, which gave trivial solutions for the above nonlocal model (1.1). In this paper we consider some modification of the above action in the nonlocal sector, i.e. Z  √  R + R−1 F ()R (1.2) S = d4 x −g 16πG and look for nontrivial cosmological solutions for the new ansätze (see [6]). Note that the cosmological constant Λ in (1.2) is hidden in the term f0 , i.e. Λ = −8πGf0 . 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. [7] and references therein), but it is without nonlocality. 2. Equations of motion By variation of action (1.2) with respect to metric g µν one obtains the equations of motion for gµν 1 Rµν V − (∇µ ∇ν − gµν )V − gµν R−1 F ()R 2 ∞ n−1 X
fn X + gµν ∂α l (R−1 )∂ α n−1−l R + l (R−1 )n−l R 2 (2.1) n=1 l=0
Gµν , − 2∂µ  (R−1 )∂ν n−1−l R = − 16πG V = F ()R−1 − R−2 F ()R. l The trace of the equation (2.1) is RV + 3V + (2.2) ∞ X n=1 fn n−1 X ∂α l (R−1 )∂ α n−1−l R + 2l (R−1 )n−l R l=0
R . − 2R−1 F ()R = 16πG The 00 component of (2.1) is (2.3) 1 R00 V − (∇0 ∇0 − g00 )V − g00 R−1 F ()R 2 ∞ n−1 X
fn X + g00 ∂α l (R−1 )∂ α n−1−l R + l (R−1 )n−l R 2 n=1 l=0
G00 . − 2∂0  (R−1 )∂0 n−1−l R = − 16πG We use Friedmann-Lemaı̂tre-Robertson-Walker (FLRW) metric ds2 = −dt2 +
2 dr 2 2 2 2 2 2 and investigate all three possibilities for curvature a (t) 1−kr 2 +r dθ +r sin θdφ   2 parameter k (0, ±1). In the FLRW metric scalar curvature is R = 6 äa + ȧa2 + ak2 l and h(t) = −∂t2 h(t) − 3H∂t h(t), where H = ȧa is the Hubble parameter. In the sequel we shall use three kinds of ansätze (two of them introduced in [6]) and A 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) = a0 |t − t0 |α . 3. Quadratic ansatz: R = qR2 Looking for solutions in the form a(t) = a0 |t − t0 |α this ansatz becomes (3.1) α(2α − 1)(qα(2α − 1) − (α − 1))(t − t0 )−4 + qk 2 αk −2α−2 (1 − α + 6q(2α − 1))(t − t ) + (t − t0 )−4α = 0. 0 3a20 a40 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 = 21 , q ∈ R, (3) k = 0, α 6= 0 and α 6= α−1 q = α(2α−1) , (4) k = −1, α = 1, q 6= 0, a0 = 1, (5) k = 6 0, α = 0, q = 0, (6) k 6= 0, α = 1, q = 0. 1 2, In the cases (1), (2) and (4) we have R = 0 and therefore R−1 is not defined. The case (5) yields a solution which does not satisfy equations of motion. Hence there remain two cases for further consideration. α−1 . For this case, we have the following expressions 3.1. Case k = 0, q = α(2α−1) depending on the parameter α : α−1 q= , R = 6α(2α − 1)(t − t0 )−2 , α(2α − 1) (3.2) a = a |t − t |α , H = α(t − t )−1 , 0 0 0 −2 R00 = 3α(1 − α)(t − t0 ) 2 G00 = 3α (t − t0 )−2 . , We now express n R and n R−1 in the following way: n R = B(n, 1)(t − t0 )−2n−2 , B(n, 1) = 6α(2α − 1)(−2)n n! (3.3) n R−1 = B(n, −1)(t − t0 )2−2n , n Y (1 − 3α + 2l), n > 1, l=1 n Y B(n, −1) = (6α(2α − 1))−1 2n (2 − l)(−3 − 3α + 2l), n > 1, l=1 B(0, 1) = 6α(2α − 1), B(0, −1) = B(0, 1)−1 . 3α+1 = −2(3α + 1)B(0, 1)−1 and B(n, −1) = 0 if Note that B(1, −1) = − 3α(2α−1) n > 2. Also, we obtain ∞ X F ()R = fn B(n, 1)(t − t0 )−2n−2 , (3.4) n=0 F ()R−1 = f0 B(0, −1)(t − t0 )2 + f1 B(1, −1). 4 IVAN DIMITRIJEVIC, BRANKO DRAGOVICH, JELENA GRUJIC, AND ZORAN RAKIC Substituting these equations into trace and 00 component of the EOM one has ∞ X r−1 fn B(n, 1) (−3r + 6(1 − n)(1 − 2n + 3α)) (t − t0 )−2n n=0 +r 1 X fn (rB(n, −1) + 3B(n + 1, −1)) (t − t0 )−2n n=0 ∞ X + 2r (3.5) n=1 ∞ X fn γn (t − t0 )−2n = fn r−1 B(n, 1) n=0 + 1 X n=0 r 2 r2 (t − t0 )−2 , 16πG  − An (t − t0 )−2n fn rB(n, −1) An (t − t0 )−2n + ∞ rX fn δn (t − t0 )−2n 2 n=1 α −r2 (t − t0 )−2 , 32πG 2α − 1 where r = B(0, 1) and = γn = n−1 X B(l, −1)(B(n − l, 1) + 2(1 − l)(n − l)B(n − l − 1, 1)), n−1 X B(l, −1)(−B(n − l, 1) + 4(1 − l)(n − l)B(n − l − 1, 1)), l=0 (3.6) δn = l=0 An = 6α(1 − n) − r α−1 r 3 − 2n − α = . 2(2α − 1) 2 2α − 1 Equations (3.5) can be split into system of pairs of equations with respect to each coefficient fn . In the case n > 1, there are the following pairs:
fn B(n, 1) (−3r + 6(1 − n)(1 − 2n + 3α)) + 2r2 γn = 0,  r  r2  (3.7) fn B(n, 1) − An + δn = 0. 2 2 to be a natural number one obtains: Taking 3α−1 2 (3.8) B(n, 1) = 6α(2α − 1)4n n! (3.9) B(n, 1) = 0, (3.10) (3.11) (3.12) n> ( 32 (α − 1))! , 3 ( 2 (α − 1) − n)! n< 3α − 1 , 2 3α − 1 , 2 3α − 1 , 2 3α − 1 δn = 2B(0, −1)B(n − 1, 1)(2n2 + 3n + 3α − 3αn + 1), n 6 , 2 3α − 1 γn = δn = 0, n > . 2 γn = 2B(0, −1)B(n − 1, 1)(3nα − 2n2 − 3α − 1), n6 A NEW MODEL OF NONLOCAL MODIFIED GRAVITY 5 If n > 3α−1 then B(n, 1) = γn = δn = 0 and hence the system is trivially 2 satisfied for arbitrary value of coefficients fn . On the other hand for 2 6 n 6 3α−1 2 the system has only trivial solution fn = 0. When n = 0 the pair becomes
(3.13) f0 − 2r + 6(1 + 3α) + 3rB(1, −1) = 0, f0 = 0 and its solution is f0 = 0. The remaining case n = 1 reads
r f1 − 3r−1 B(1, 1) + rB(1, −1) + 2γ1 = , 16πG (3.14)   α −r2 1 , f1 A1 (rB(1, −1) − r−1 B(1, 1)) + (B(1, 1) + rδ1 ) = 2 32πG 2α − 1 3α(2α−1) . and it gives f1 = − 32πG(3α−2) 3.2. Case k 6= 0, α = 1, q = 0. In this case a = a0 |t − t0 |, s = 6(1 + (3.15) k ), a20 H = (t − t0 )−1 , R = 0, R00 = 0, n R−1 = D(n, −1)(t − t0 )2−2n , D(0, −1) = s−1 , R = s(t − t0 )−2 , D(1, −1) = −8s−1 , D(n, −1) = 0, n > 2. Substitution of the above expressions in trace and 00 component of the EOM yields 3f0 + 1 X n=0 (3.16) fn sD(n, −1)(t − t0 )−2n + 4f1 (t − t0 )−2 = 1 X 1 − 6f0 s + f0 + 6 fn D(n, −1)(1 − n)(t − t0 )−2n 2 n=0 s (t − t0 )−2 . + 2f1 (t − t0 )−2 = − 32πG −1 This system leads to conditions for f0 and f1 : (3.17) s (t − t0 )−2 , 16πG s =− (t − t0 )−2 . 32πG −2f0 − 4f1 (t − t0 )−2 = 1 f0 + 2f1 (t − t0 )−2 2 The corresponding solution is (3.18) s (t − t0 )−2 , 16πG f0 = 0, f1 = −s , 64πG fn ∈ R, n > 2. 6 IVAN DIMITRIJEVIC, BRANKO DRAGOVICH, JELENA GRUJIC, AND ZORAN RAKIC 4. Ansatz n R = cn Rn+1 , n > 1 Presenting n+1 R in two ways: n+1 R = cn Rn+1 = cn ((n + 1)Rn R − n(n + 1)Rn−1 Ṙ2 ) = cn (n + 1)(c1 Rn+2 − nRn−1 Ṙ2 ) = cn+1 Rn+2 it follows (4.1) Ṙ2 = R3 , (4.2) cn+1 = cn (n + 1)(c1 − n), where Ṙ2 means (Ṙ)2 . General solution of equation (4.1) is (4.3) R= 4 , t0 ∈ R. (t − t0 )2 Taking n = 1 in the ansatz yields R = c1 R2 . (4.4) Substitution of (4.3) in (4.4) gives H = (4.5) 2c1 +3 3(t−t0 ) . a(t) = a0 |t − t0 | 2c1 +3 3 , This implies a0 > 0. Using (4.3) in equation (4.6) R=6  ä ȧ2 k + 2 + a a2 a  gives (4.7) 4 (t − t0 )2 ÿ − y = −2k(t − t0 )2 , where y = a2 (t). 3 It can be shown that general solution of the last equation is (4.8) √ √ 3− 57 3+ 57 a2 (t) = C˜1 |t − d1 | 6 + C˜2 |t − d1 | 6 − 3k|t − d1 |2 , C˜1 , C˜2 ∈ R. By comparison of the last equation with (4.5) one can conclude: (1) If c1 = 0 then k must be equal to −1. In this case n R = √0, n > 1. (2) If c1 6= 0 then k must be equal to 0. In this case c1 = −9±8 57 . A NEW MODEL OF NONLOCAL MODIFIED GRAVITY 7 4.1. Case n R = cn Rn+1 , c1 = 0. From the previous analysis, it follows: k = −1, (4.9) R= a(t) = 4 , (t − t0 )2 √ 3|t − t0 |, n R = 0, H(t) = n > 1, 1 , t − t0 F ()R = f0 R. It can be shown that n R−1 = (−1)n 4n−1 (4.10) n−1 Y l=0 n From (4.10) follows  R −1 (1 − l)(2 − l)(t − t0 )2−2n . = 0, n > 1. Then F ()(R−1 ) = f0 R−1 + f1 R−1 . (4.11) Substituting (4.9) and (4.11) in the 00 component of the EOM one obtains 1 f0 (t − t0 )2 + 2f1 + =0 2 8πG (4.12) and it follows −1 , fn ∈ R, n > 2. 16πG Substituting (4.9) and (4.11) in the trace equation one has f0 = 0, (4.13) f1 = −2f0 (t − t0 )2 − 4f1 − (4.14) 1 =0 4πG and it gives the same result (4.13). 4.2. Case n R = cn Rn+1 , c1 = (4.15) k = 0, R= 4 , (t − t0 )2 √ −9± 57 . 8 H= In this case: 2c1 + 3 , 3(t − t0 ) a = a0 |t − t0 | 2c1 +3 3 , a0 > 0, R00 = 3α(1 − α)(t − t0 )−2 , G00 = (3α(1 − α) + 2)(t − t0 )−2 , α = n R = 4n+1 cn (t − t0 )−2n−2 , 2c1 + 3 , 3 c0 = 1. One can show that n R−1 = M (n, −1)(t − t0 )2−2n , (4.16) where (4.17) M (0, −1) = 1 , 4 M (1, −1) = −(c1 + 2), M (n, −1) = 0, Also one obtains (4.18) F ()R = ∞ X n=0 4n+1 fn cn (t − t0 )−2n−2 , F ()R−1 = f0 M (0, −1)(t − t0 )2 + f1 M (1, −1). n > 1. 8 IVAN DIMITRIJEVIC, BRANKO DRAGOVICH, JELENA GRUJIC, AND ZORAN RAKIC Substituting (4.18) in the trace equation it becomes (4.19) − ∞ X 1 (t − t0 )−2 − 2f0 − 3 4n fn cn (t − t0 )−2n 4πG n=1 + ∞ X fn n=1 n−1 X l=0
M (l, −1)4n−l+1 ((1 − l)(n − l)cn−1−l + 2cn−l ) (t − t0 )−2n + f1 (4M (1, −1) + 3M (2, −1))(t − t0 )−2 = 0. To satisfy equation (4.19) for all values of time t one obtains: f0 = 0, (4.20) (4.21) fn − 3cn + 1 X l=0 f1 (2c1 + 1) = − 1 , 16πG
M (l, −1)41−l((1 − l)(n − l)cn−1−l + 2cn−l ) = 0, n > 2. Suppose that fn 6= 0 for n > 2, then from the last equation follows (4.22) −3cn + 1 X l=0 M (l, −1)41−l((1 − l)(n − l)cn−1−l + 2cn−l ) = 0 and it becomes cn−1 (n2 − c1 n − 2c1 − 4) = 0. (4.23) Since cn−1 6= 0, condition (4.23) is satisfied for n = −2 or n = c1 + 2. Hence, we conclude that fn = 0 for n > 2. Since fn = 0 for n > 2, the 00 component of the EOM becomes (4.24) 1 (−3α2 + 3α + 2)(t − t0 )−2 16πG 3 9 1 + f0 ( α2 − α + 1) + f1 c1 (3α2 − 3α + 2)(t − t0 )−2 2 2 2 + 8f1 M (0, −1)(1 − c1 )(t − t0 )−2 + 3α(3 − α)M (0, −1)f0 − 3α(α − 1)M (1, −1)f1(t − t0 )−2 = 0. In order to satisfy equation (4.24) for all values of time t it has to be 4 10 1 2 f1 ( c31 + c21 + 2c1 + 1) = ( c21 + c1 − 1). 3 3 16πG 3 The necessary and sufficient condition for the EOM to have a solution is (4.25) f0 = 0, c1 (8c21 + 18c1 + 3) = 0. (4.26) Since c1 = √ −9± 57 , 8 the last condition is satisfied. A NEW MODEL OF NONLOCAL MODIFIED GRAVITY 9 5. Cubic ansatz: R = qR3 Recall that we are looking for solutions in the form a(t) = a0 |t − t0 |α . In the explicit form it reads   k α(α − 1) 3(2α − 1)(t − t0 )−4 + 2 (t − t0 )−2α−2 a0  3 k = 18q α(2α − 1)(t − t0 )−2 + 2 (t − t0 )−2α . a0 (5.1) It yields the following seven possibilities: (1) (2) (3) (4) k k k k = 0, α = 0, q ∈ R, = 0, α = 12 , q ∈ R, = −1, α = 1, q 6= 0, a0 = 1, = 0, α = 1, q = 0, (5) k 6= 0, α = 0, q = 0, (6) k = 6 0, α = 1, q = 0, a4 (7) k 6= 0, α = 21 , q = − 720 . Cases (1), (2) and (3) contain scalar curvature R = 0, and therefore we will not discuss them. Cases (4), (5) and 6 are also obtained from the quadratic ansatz and have been discussed earlier. The last case contains: a(t) = a0 (5.2) R(t) = p |t − t0 |, 1 , 2(t − t0 ) 3 = . 4(t − t0 )2 H(t) = 6k |t − t0 |−1 , a20 R00 One can derive the following expressions: n R = N (n, 1)|t − t0 |−2n−1 , N (0, 1) = 6k , a20 N (0, −1) = N (0, 1)−1 , n N (n, 1) = N (0, 1)(−1) (5.3) n R−1 = N (n, −1)|t − t0 |1−2n , n−1 Y 1 (2l + 1)(2l + ), 2 l=0 n−1 Y n N (n, −1) = N (0, 1)−1 (−1) F ()R = F ()R−1 = ∞ X n=0 ∞ X n=0 l=0 3 (2l − 1)(2l − ), 2 fn N (n, 1)|t − t0 |−2n−1 , fn N (n, −1)|t − t0 |1−2n . n > 1, n > 1, 10 IVAN DIMITRIJEVIC, BRANKO DRAGOVICH, JELENA GRUJIC, AND ZORAN RAKIC Substituting (5.3) in the trace equation we obtain − 2N (0, 1)−1 ∞ X n=0 fn N (n, 1)|t − t0 |−2n + N (0, 1) ∞ X n=0 fn (N (n, −1) − N (0, 1)−2 N (n, 1))|t − t0 |−2n (5.4) +3 ∞ X n=0 + ∞ X n=1 fn (N (n, −1) − N (0, 1)−2 N (n, 1))|t − t0 |−1−2n fn n−1 X l=0 N (l, −1)((1 − 2l)(−2n + 2l + 1)N (n − l − 1, 1) N (0, 1) |t − t0 |−1 . 16πG This equation implies the following conditions on coefficient f0 : + 2N (n − l, 1))|t − t0 |−2n = N (0, 1) = 0. 16πG Since N (0, 1) 6= 0, the last equation never holds and therefore there is no solution in this case. (5.5) f0 = 0, 6. Concluding remarks Using a few new ansätze we have shown that equations of motion for nonlocal gravity model given by action (1.2) yield some bounce cosmological solutions of the form a(t) = a0 |t − t0 |α . These solutions lead to f0 = 0 and hence Λ = 0, and when t → ∞ then R → 0. In particular, quadratic ansatz R = qR2 is very promising. Note that ansatz n R = cn Rn+1 , n > 1, can be viewed as a special case of ansatz R = qR2 . It is worth noting that equations of motion (2.2) and (2.3) have the de Sitter 2 solutions a(t) = a0 cosh(λt), k = +1 and a(t) = a0 eλt , k = 0, when f0 = −3λ 8πG = −Λ 8πG , fn ∈ R, n > 1. This investigation can be generalized to some cases with R−p F ()Rq nonlocal term, where p and q are some natural numbers satisfying q − p > 0. It will be presented elsewhere with discussion of various properties. References 1. T. Clifton, P. G. Ferreira, A. Padilla, C. 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Woodard, “Avoiding dark energy with 1/R modifications of gravity”, Lect. Notes Phys. 720, 403–433 (2007) [arXiv:astro-ph/0601672]. Faculty of Mathematics, University of Belgrade, Belgrade, Serbia Institute of Physics, University of Belgrade, Belgrade, Serbia Teachers Training Faculty, University of Belgrade, Belgrade, Serbia Faculty of Mathematics, University of Belgrade, Belgrade, Serbia