NONLOCAL MODIFIED GRAVITY AND ITS COSMOLOGY

Transcription

1 NONLOCAL MODIFIED GRAVITY AND ITS COSMOLOGY Branko Dragovich dragovich Institute of Physics, University of Belgrade, and Mathematical Institute SANU Belgrade, Serbia XI. International Workshop LIE THEORY AND ITS APPLICATIONS IN PHYSICS , Varna Bulgaria VARNA 2015 B. Dragovich VARNA /42

2 Contents 1 Introduction 2 Modified gravity 3 Nonlocal modified gravity 4 Cosmological solutions 5 Concluding remarks VARNA 2015 B. Dragovich VARNA /42

3 1. Introduction: Einstein theory of gravity (ETG) Einstein Theory of Gravity (1915) R 8π G R g = T Λg 2 c μν μν 4 μν μν 2 μ α β d x μ dx dx + Γ 0 2 αβ = dτ dτ dτ VARNA 2015 B. Dragovich VARNA /42

4 1. Introduction: Einstein theory of gravity (ETG) ETG is the simplest geometric and current theory of gravity It is General Relativity (GR) and contains Newton theory of gravity Its predictions are confirmed mainly in Solar System (classical tests: perihelion precession of Mercury orbit, deflection of light by the Sun, gravitational redshift of light ) It gives a possibility to understand gravitational phenomena from laboratory scale to cosmological scales ETG predicts existence black holes, gravitational waves, gravitational lensing, Dark Energy (DE) and Dark Matter (DM) VARNA 2015 B. Dragovich VARNA /42

5 1. Introduction: some problems of Einstein theory of gravity General Relativity is non-renormalizable quantum field theory. It predicts Dark Energy and Dark Matter which are mysterious and without other evidence. General Relativity has not been tested and confirmed at large cosmic scales, hence its application for the Universe as a whole is questionable. Cosmological solutions of GR contain Big Bang singularity. All these problems serve as motivation to investigate a Modified Gravity, which is a generalization of ETG. There are many modifications motivated by high energy particle physics, astrophysics and cosmology. To get (nonsingular) bounce cosmological solutions we consider a Nonlocal Modified Gravity. VARNA 2015 B. Dragovich VARNA /42

6 2. Modified Gravity: some review references T. Clifton, P. G. Ferreira, A. Padilla, C. Skordis, Modified gravity and cosmology, Phys. Rep. 513 (1), (2012). [arxiv: v2 [astro-ph.co]]. T. P. Sotiriou, V. Faraoni, f (R) theories of gravity, Rev. Mod. Phys. 82, (2010) [arxiv: v4 [gr-qc]]. S. Nojiri, S. D. Odintsov, Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models, Phys. Rept. 505, (2011) [arxiv: v4 [gr-qc]]. S. Capozziello and M. De Laurentis, Extended Theories of Gravity, [arxiv: v2 [gr-qc]]. M. Novello and S.E. Bergliaffa, Bouncing Cosmologies, Phys. Rept. 463, (2008) [arxiv: [astro-ph]]. VARNA 2015 B. Dragovich VARNA /42

7 2. Modified Gravity First modifications: Einstein 1917, Weyl 1919, Edington 1923,... Einstein-Hilbert action g S = d 4 x 16πG R + modification d 4 x g L(matter) R f (R, Λ, R µν, R α µβν,,...), = µ µ = 1 g µ gg µν ν Gauss-Bonnet invariant G = R 2 4R µν R µν + R αβµν R αβµν VARNA 2015 B. Dragovich VARNA /42

8 2. Modified Gravity: some simple actions f (R) modified gravity g S = d 4 x 16πG f (R) + d 4 x g L(matter) Gauss-Bonnet modified gravity S = g d 4 x 16πG (R + αg) + d 4 x g L(matter) nonlocal modified gravity S = g d 4 x 16πG f (R, ) + d 4 x g L(matter) VARNA 2015 B. Dragovich VARNA /42

9 3. Nonlocal Modified Gravity: some relevant references T. Biswas, T. Koivisto, A. Mazumdar, Towards a resolution of the cosmological singularity in non-local higher derivative theories of gravity, JCAP 1011 (2010) 008 [arxiv: v2 [hep-th]]. A. S. Koshelev, S. Yu. Vernov, On bouncing solutions in non-local gravity, Phys. Part. Nuclei 43, (2012) [arxiv: v1 [hep-th]]. I. Dimitrijevic, B. Dragovich, J. Grujic, Z. Rakic, New cosmological solutions in nonlocal modified gravity, Romanian J. Physics 56 (5-6), (2013) [arxiv: [gr-qc]]. I. Dimitrijevic, B. Dragovich, J. Grujic, Z. Rakic, A new model of nonlocal modified gravity, Publications de l Institute Matematique 94 (108) (2013) B. Dragovich, On nonlocal modified gravity and cosmology, Springer Proc. in Mathematics & Statistics 111 (2014) VARNA 2015 B. Dragovich VARNA /42

10 3. Nonlocal Modified Gravity Addition of d Alembert operator in Einstein-Hilbert action improves renormalizability (Modesto,...) Nonlocal modified gravity with 1 (Deser, Woodard, Odintsov,...) S = d 4 x g 16πG R f ( 1 R) The exact tree-level Lagrangian for effective scalar field ϕ which describes open p-adic string tachyon is L p = md p g 2 p p 2 [ 1 p 1 2 ϕ p 2mp 2 ϕ + 1 p + 1 ϕp+1] where p is any prime number, = 2 t + 2 is the D-dimensional d Alembertian and metric with signature ( ). Nonlocality in the matter sector of cosmological models (Arefeva,...). VARNA 2015 B. Dragovich VARNA /42

11 3. Nonlocal Modified Gravity Action for a class of models: ( R 2Λ ) g S = 16πG + CH(R)F( )G(R) d 4 x, M where F( ) = f n n, H and G are some differentiable n=0 functions of R, Λ is cosmological constant, and C is a constant. For simplicity, nonlocal modification without matter. VARNA 2015 B. Dragovich VARNA /42

12 3. Nonlocal Modified Gravity Equations of motion: G µν + Λg ( µν + C 1 16πG 2 g µνh(r)f( )G(R) + (R µν Φ K µν Φ) + 1 n 1 ( f n gµν g αβ α l H(R) β n 1 l G(R) 2 n=1 l=0 2 µ l H(R) ν n 1 l G(R) + g µν l H(R) n l G(R) )) = 0, where K µν = µ ν g µν, Φ = H (R)F( )G(R) + G (R)F( )H(R), and denotes derivative on R. VARNA 2015 B. Dragovich VARNA /42

13 3. Nonlocal Modified Gravity In homogeneous and isotropic spaces (FLRW metric) there are only two linearly independent equations of motion (Trace and 00-component): 4Λ R ( 16πG + C 2H(R)F( )G(R) + (RΦ + 3 Φ) n 1 ( + f n µ l H(R) µ n 1 l G(R) + 2 l H(R) n l G(R) )) = 0, n=1 l=0 G 00 + Λg ( 00 16πG + C 1 2 g 00H(R)F( )G(R) + (R 00 Φ K 00 Φ) + 1 n 1 ( f n g00 g αβ α l H(R) β n 1 l G(R) 2 n=1 l=0 2 0 l H(R) 0 n 1 l G(R) + g 00 l H(R) n l G(R) )) = 0. VARNA 2015 B. Dragovich VARNA /42

14 3. Nonlocal Modified Gravity: two simple models 1 Model 1: H(R) = G(R) = R S = d 4 x ( R 2Λ g 16πG + C ) 2 RF( )R 2 Model 2: H(R) = R 1 and G(R) = R S = d 4 x ( R ) g 16πG + R 1 F( )R Λ cosmological constant, C a constant and F( ) = f n n is an analytic function of = µ µ. n=0 VARNA 2015 B. Dragovich VARNA /42

15 4. Cosmological Solutions Model 1. (Biswas et al.) S = d 4 x ( R 2Λ g 16πG + C ) 2 RF( )R F( ) = f n n, = µ µ = 1 µ gg µν ν g n=0 Equations of motion ( C 2R µν F( )R 2( µ ν g µν )(F( )R) 1 2 g µνrf( )R + f n n 1 ( ( ) gµν g αβ α l R β n 1 l R + l R n l R 2 n=1 l=0 2 µ l R ν n 1 l R )) = 1 8πG (G µν + Λg µν ). VARNA 2015 B. Dragovich VARNA /42

16 4. Cosmological Solutions Model 1: Trace and 00-component 6 (F( )R) + n=1 f n n 1 l=0 ( ) µ l R µ n 1 l R + 2 l R n l R = 1 8πGC R Λ 2πGC ( C 2R 00 F( )R 2( 0 0 g 00 )(F( )R) 1 2 g 00RF( )R + f n n 1 ( ( ) g00 g αβ α l R β n 1 l R + l R n l R 2 n=1 l=0 2 0 l R 0 n 1 l R )) = 1 8πG (G 00 + Λg 00 ) VARNA 2015 B. Dragovich VARNA /42

17 4. Cosmological Solutions: Some ansätze linear ansatz: R = rr + s quadratic ansatz: R = qr 2 cubic ansatz: R = qr 3 n-degree ansatz: n R = c n R n+1 FLRW metric for homogeneous and isotropic Universe [ ds 2 = dt 2 + a 2 dr 2 (t) 1 kr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2], k = 0, ±1 VARNA 2015 B. Dragovich VARNA /42

18 4. Cosmological Solutions We use FLRW metric ds 2 = dt 2 + a 2 (t) ( dr 2 1 kr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2), k = 0, ±1 We use ansatz R = rr + s n R = r n (R + s r ), n 1, F( )R = F(r)R + s r (F(r) f 0) We look for a solution of the form (Dimitrijevic, B.D., Grujic, Rakic) a(t) = a 0 (σe λt + τe λt ), 0 < a 0, λ, σ, τ R VARNA 2015 B. Dragovich VARNA /42

19 4. Cosmological Solutions: Nonsingular bounce cosmological solutions H(t) = ȧ a = λ(σeλt τe λt ) σe λt + τe λt R(t) = 6 a 2 (aä + ȧ2 + k) = 6 ( 2a 2 0 λ2 ( σ 2 e 4tλ + τ 2) + ke 2tλ) R = 12λ2 e 2tλ ( 4a 2 0 λ2 στ k ) a 2 0 (σe2tλ + τ) 2 a 2 0 (σe2tλ + τ) 2 R = 2λ 2 R 24λ 4, r = 2λ 2, s = 24λ 4 VARNA 2015 B. Dragovich VARNA /42

20 4. Cosmological Solutions: Nonsingular bounce cosmological solutions From trace and 00-component two equations follow as polynomials in e 2λt a0 4τ 6 ( ) 3λ 2 Λ + 3a0 2 4πG τ 4 Q 1 e 2λt + 6a0 2 στ 3 Q 2 e 4λt 2στQ 3 e 6λt + 6a 2 0 σ3 τq 2 e 8λt + 3a 2 0 σ4 Q 1 e 10λt + a4 0 σ6 4πG ( ) 3λ 2 Λ e 12λt = 0 τ 6 a0 4 ( ) 3λ 2 Λ + 3τ 4 a0 2 8πG R 1e 2λt + 3τ 2 R 2 e 4λt + 2στR 3 e 6λt + 3σ 2 R 2 e 8λt + 3σ 4 a 2 0 R 1e 10λt + σ6 a 4 0 8πG ( ) 3λ 2 Λ e 12λt = 0 VARNA 2015 B. Dragovich VARNA /42

21 4. Cosmological Solutions: Nonsingular bounce cosmological solutions where Q 1 = 36Cλ 2 K F(2λ 2 ) + a0 2 ( 96Cf 0λ 4 + λ2 πg Λ 2πG )στ + 24Cf 0 kλ 2 + k 8πG, Q 2 = 72Cλ 2 K F(2λ 2 ) + a 2 0 ( 192Cf 0λ 4 + 7λ2 8πG 5Λ 8πG )στ + 48Cf 0 kλ 2 + k 4πG, Q 3 = 324Ca 2 0 λ2 στk F(2λ 2 ) + 144Cλ 2 K 2 F (2λ 2 ) a 2 0 k(216cf 0λ πG )στ + a4 0 (864Cf 0λ 4 3λ2 πg + 5Λ 2πG )σ2 τ 2 VARNA 2015 B. Dragovich VARNA /42

22 4. Cosmological Solutions: Nonsingular bounce cosmological solutions and R 1 = Q 1 3λ2 Λ 4πG στa2 0 ( ) R 2 = 6C k 12a0 2 λ2 στ K F(2λ 2 ) 36Cλ 2 K 2 F (2λ 2 )+ a0 2k ( 2πG R 3 = 18C ) 192πGCf 0 λ στ a4 0 ( 8πG k 6a0 2 λ2 στ ( ) 3072πGCf 0 λ 4 + λ 2 + 5Λ σ 2 τ 2 ) K F(2λ 2 ) + 36Cλ 2 K 2 F (2λ 2 )+ 9a0 2k ( ) 192πGCf 0 λ στ a4 ( ) πGCf 0 λ 4 + 3λ 2 + 5Λ σ 2 τ 2 8πG 4πG and K = 4a 2 0 λ2 στ k. VARNA 2015 B. Dragovich VARNA /42

23 4. Cosmological Solutions: Nonsingular bounce cosmological solutions F Equations of motion are satisfied when λ = ± Λ 3, as well as Q 1 = Q 2 = Q 3 = 0 and R 1 = R 2 = R 3 = 0. There are three cases of solutions. Case 1. ( F 2λ 2) ( = 0, F 2λ 2) 1 = 0, f 0 = 64πGCΛ Case 2. Case 3. ( 2λ 2) = 3k = 4a 2 0 Λστ 1 96πGCΛ f 0, F ( 2λ 2) = 0, k = 4a 2 0 Λστ VARNA 2015 B. Dragovich VARNA /42

24 4. Cosmological Solutions Case 1. S = d 4 x g( R 2Λ 16πG + C 2 RF( )R ) linear ansatz: R = rr + s nonsingular bounce solutions ( ) 1 Λ a(t) = a 0 cosh 3 t for k = 0. (Biswas, Koivisto, Mazumdar and Siegel) 2 a(t) = a 0 e 1 2 Λ 3 t 2 for k = 0. (Koshelev and Vernov) 3 a(t) = a 0 (σe λt + τe λt ) for k = 0, ±1. (Dimitrijevic, B.D., Grujic, Rakic) VARNA 2015 B. Dragovich VARNA /42

25 4. Cosmological Solutions Model 2. (Dimitrijevic, B.D., Grujic, Rakic) S = d 4 x ( R ) g 16πG + R 1 F( )R F( ) = f 0 + f f n n + Cosmological constant is related to f 0 by Λ = 8πGf 0 The nonlocal term R 1 F( )R is invariant under transformation R CR. It means that effect of nonlocality does not depend on the magnitude of scalar curvature R, but on its spacetime dependence, and in the FLRW case is sensitive only to dependence of R on time t. VARNA 2015 B. Dragovich VARNA /42

26 4. Cosmological Solutions Equations of motion R µν V ( µ ν g µν )V 1 2 g µνr 1 F( )R + f n n 1 ( ( ) gµν α l (R 1 ) α n 1 l R + l (R 1 ) n l R 2 n=1 l=0 2 µ l (R 1 ) ν n 1 l R ) = G µν 16πG, V = F( )R 1 R 2 F( )R. VARNA 2015 B. Dragovich VARNA /42

27 4. Cosmological Solutions Model 2. Trace and 00 component RV + 3 V + n=1 +2 l (R 1 ) n l R f n n 1 l=0 ( α l (R 1 ) α n 1 l R ) 2R 1 F( )R = R 16πG R 00 V ( 0 0 g 00 )V 1 2 g 00R 1 F( )R + f n n 1 ( ( ) g00 α l (R 1 ) α n 1 l R + l (R 1 ) n l R 2 n=1 l=0 2 0 l (R 1 ) 0 n 1 l R ) = G 00 16πG VARNA 2015 B. Dragovich VARNA /42

28 4. Cosmological Solutions Model 2. Ansätze quadratic ansatz: R = qr 2 cubic ansatz: R = qr 3 n-degree ansatz: n R = c n R n+1 cosmic scale factor of the form a(t) = a 0 t t 0 α ( ) and R = 6 ä a + ȧ2 + k a 2 a 2 = t 2 3H t, where H = ȧ a. VARNA 2015 B. Dragovich VARNA /42

29 4. Cosmological Solutions Model 2. S = d 4 x ) g( R 16πG + R 1 F( )R quadratic ansatz: R = qr 2 Case k = 0, q = α 1 α(2α 1). n R =B(n, 1)(t t 0 ) 2n 2, n R 1 = B(n, 1)(t t 0 ) 2 2n, n B(n, 1) = 6α(2α 1)( 2) n n! (1 3α + 2l), n 1, B(n, 1) = (6α(2α 1)) 1 2 n l=1 n (2 l)( 3 3α + 2l), n 1, l=1 B(0, 1) = 6α(2α 1), B(0, 1) = B(0, 1) 1. VARNA 2015 B. Dragovich VARNA /42

30 4. Cosmological Solutions Model 2. S = d 4 x ) g( R 16πG + R 1 F( )R quadratic ansatz: R = qr 2 Case k = 0, q = α 1 α(2α 1). B(n, 1) = 0 if n 2 B(n, 1) = 0 if n 3α 1 2 f 0 = 0, f 1 = 3α(2α 1) 32πG(3α 2), and f n is arbitrary for n > 3α 1 2. f n = 0 if 2 n 3α 1 2, VARNA 2015 B. Dragovich VARNA /42

31 4. Cosmological Solutions Model 2. S = d 4 x ) g( R 16πG + R 1 F( )R quadratic ansatz: R = qr 2 Case k 0, α = 1, q = 0. a(t) = a 0 t t 0 f 0 = 0, f 1 = (1 + k/a2 0 ), f n R, n 2 64πG VARNA 2015 B. Dragovich VARNA /42

32 4. Cosmological Solutions Model 2. S = d 4 x ) g( R 16πG + R 1 F( )R n-degree ansatz: n R = c n R n+1 f 0 = 0, f 1 0, f n R, n 2 cubic ansatz: R = qr 3 There is no solution. VARNA 2015 B. Dragovich VARNA /42

33 4. Cosmological Solutions Cosmological solutions with constant scalar curvature Let R = R 0 = const and we obtain 6( ä a + ( ) ) ȧ 2 a + k = R a 2 0. The change of variable b(t) = a 2 (t) yields 3 b R 0 b = 6k. Depending on the sign of R 0 we have the following solutions for b(t) R 0 > 0 : R 0 < 0 : b(t) = 6k R0 + σe 3t R 0 b(t) = 6k R 0 + σ cos + τe R0 3t, R0 R0 + τ sin. 3t 3t VARNA 2015 B. Dragovich VARNA /42

34 4. Cosmological Solutions: constant scalar curvature When we set R = R 0 = const into trace and 00-equation we obtain the following system 2f 0 = R 0 16πG, 1 2 f 0 = G 00 16πG The last system has a solution iff R 0 + 4R 00 = 0. Note that R 00 can be written in terms of function b(t) as R 00 = 3ä a = 3((ḃ)2 2b b) 4b 2. VARNA 2015 B. Dragovich VARNA /42

35 4. Cosmological Solutions: constant scalar curvature Now, from R 0 + 4R 00 = 0 we obtain the following conditions on the parameters σ and τ: R 0 > 0 : 9k 2 = R 2 0 στ, R 0 < 0 : 36k 2 = R 2 0 (σ2 + τ 2 ). VARNA 2015 B. Dragovich VARNA /42

36 4. Cosmological Solutions: constant scalar curvature (R 0 < 0) If k = 1 we can define ϕ by σ = 6 R 0 cos ϕ and τ = 6 R 0 sin ϕ and rewrite a(t) and b(t) as b(t) = 12 cos 2 1 R 0 2 ( R 0 3 t ϕ), 12 a(t) = cos 1 R 0 2 ( R 0 3 t ϕ). In the last case k = +1 we can transform b(t) to b(t) = 12 R 0 sin ( R 0 3 t ϕ), which is non positive and hence yields no solutions. VARNA 2015 B. Dragovich VARNA /42

37 4. Cosmological Solutions: constant scalar curvature (R 0 > 0) If k = 0 then a(t) = a 0 e λt. If k = +1, we obtain b(t) = 12 cosh 2 1 R 0 2 ( 12 a(t) = cosh 1 R 0 2 ( If k = 1, then R0 3 t + ϕ), R0 R0 3 t + ϕ). b(t) = 12 sinh 2 1 R 0 2 ( 3 t + ϕ), 12 a(t) = sinh 1 R 0 2 ( R0 3 t + ϕ). VARNA 2015 B. Dragovich VARNA /42

38 4. Cosmological Solutions: constant scalar curvature (R = 12λ 2 ) We consider the scale factor of the form a(t) = a 0 (σ 1 e λt + τ 1 e λt ). We have H(t) = λ(e2λt σ 1 τ 1 ) e 2λt σ 1 + τ 1, R(t) = 6(e2λt k + 2λ 2 (e 4λt σ1 2 + τ 1 2)a2 0 ) (e 2λt σ 1 + τ 1 ) 2 a0 2. In order to have R = const we have to satisfy condition k = 4λ 2 a 2 0 σ 1τ 1. From the last condition we obtain R = 12λ 2. VARNA 2015 B. Dragovich VARNA /42

39 4. Cosmological Solutions: constant scalar curvature (R = 12λ 2 ) Substituting this into equations trace and 00-component we obtain f 0 = 3λ2 8πG, f i R, i 1. In particular, if we set σ 1 = τ 1 = 1 2 the scale factor becomes a(t) = a 0 cosh(λt). In this case from condition k = 4λ 2 a 2 0 σ 1τ 1 we see that the only nontrivial case is when k is equal to 1. From this we obtain a 0 = 1 λ. If we take σ 1 = 0 or τ 1 = 0 the scale factor becomes a(t) = a 0 e λt. From condition k = 4λ 2 a 2 0 σ 1τ 1 we see that in this case k must be equal to 0. VARNA 2015 B. Dragovich VARNA /42

40 4. Cosmological Solutions: constant scalar curvature (R 0 = 0) The case R 0 = 0 can be considered as limit of R 0 0 in both cases R 0 < 0 and R 0 > 0. When R 0 < 0 there is condition 36k 2 = R0 2(σ2 + τ 2 ). From this condition, R 0 0 implies k = 0 and arbitrary values of constants σ and τ. The same conclusion obtains when R 0 > 0 with condition 9k 2 = R0 2 στ. In both these cases there is Minkowski solution with b(t) = constant > 0 and consequently a(t) = constant > 0. Note that the Minkowski space solution can be also obtained from the case R = 12λ 2. Namely, the solution a(t) = a 0 e λt satisfies H = λ. Taking the limit λ 0 in a(t) = a 0 e λt one obtains Minkowski space as a solution for f 0 = 0, f i R, i 1. VARNA 2015 B. Dragovich VARNA /42

41 5. Concluding Remarks Motivations to modify Einstein theory of gravity come from high energy particle physics, astrophysics and cosmology. We have considered two simple nonlocal gravity models with their bounce cosmological solutions. The first model is S = d 4 x ( R 2Λ ) g 16πG + RF( )R with cosmological solutions of the form: ( ) Λ a(t) = a 0 cosh 3 t, a(t) = a 0 e 1 2 a(t) = a 0 (σe λt + τe λt ) Λ 3 t2, VARNA 2015 B. Dragovich VARNA /42

42 5. Concluding Remarks The second model is given by S = d 4 x ( R ) g 16πG + R 1 F( )R. When R = R 0 < 0 there is nontrivial solution a(t) = 12 R 0 cos 1 2 ( R 0 3 t ϕ) for k = 1, is a singular cyclic solution. In the case R = R 0 > 0 there are solutions for all three values of curvature constant k = 0, ±1. The case R = R 0 = 0 was considered as limit of R 0 0 in both cases R 0 < 0 and R 0 > 0, and Minkowski space solution was obtained. Solutions for R 0 > 0 with k = 0, +1 are nonsingular bounce cosmological solutions. Research on cosmological perturbations is in progress. VARNA 2015 B. Dragovich VARNA /42