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B. Dragovich: On Modified Gravity and Cosmology

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B. Dragovich: On Modified Gravity and Cosmology

  1. 1. ON MODIFIED GRAVITY AND COSMOLOGY Branko Dragovich http://www.ipb.ac.rs/˜ dragovich dragovich@ipb.ac.rs Institute of Physics Belgrade, Serbia Balkan Workshop BW 2013 25 – 29. 04. 2013 Vrnjaˇcka Banja – SERBIA BW - 2013 B. Dragovich Balkan Workshop BW2013 1/15
  2. 2. Contents 1 Introduction 2 Nonlocal modified gravity 3 Nonsingular bounce cosmological solutions 4 Concluding remarks BW - 2013 B. Dragovich Balkan Workshop BW2013 2/15
  3. 3. 1. Introduction: Einstein theory of gravity (ETG)    Einstein Theory of Gravity (1915) 4 8 2 R G R g T g c μν μν μν μν π − = − Λ 2 2 0 d x dx dx d d d μ α β μ αβ τ τ τ + Γ =   BW - 2013 B. Dragovich Balkan Workshop BW2013 3/15
  4. 4. 1. Introduction: Einstein theory of gravity (ETG) ETG is the simplest self-consistent theory of gravity It is General Relativity (GR) and contains Newton theory of gravity Its predictions are confirmed mainly in Solar System It gives possibility to understand gravitational phenomena from laboratory scale to cosmological scales ETG predicts existence of Dark Energy (DE) and Dark Matter (DM) BW - 2013 B. Dragovich Balkan Workshop BW2013 4/15
  5. 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 different reasons. To get nonsingular bounce cosmological solutions we consider a Nonlocal Modified Gravity. BW - 2013 B. Dragovich Balkan Workshop BW2013 5/15
  6. 6. 2. Nonlocal Modified Gravity: Relevant references T. Clifton, P. G. Ferreira, A. Padilla, C. Skordis, “Modified gravity and cosmology”, Phys. Rep. 513 (1), 1–189 (2012). [arXiv:1106.2476v2 [astro-ph.CO]]. 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:1005.0590v2 [hep-th]]. A. S. Koshelev, S. Yu. Vernov, “On bouncing solutions in non-local gravity”, Phys. Part. Nuclei 43, 666–668 (2012) [arXiv:1202.1289v1 [hep-th]]. I. Dimitrijevic, B. Dragovich, J. Grujic, Z. Rakic, “On modified gravity”, to appear in Springer Proc. in Math. and Statistics 36 (2013) [arXiv:1202.2352v2 [hep-th]]. I. Dimitrijevic, B. Dragovich, J. Grujic, Z. Rakic, “New cosmological solutions in nonlocal modified gravity”, to appear in Romanian J. Physics (2013) [arXiv:1302.2794 [gr-qc]]. BW - 2013 B. Dragovich Balkan Workshop BW2013 6/15
  7. 7. 2. Nonlocal Modified Gravity Nonlocal gravity action without matter (Biswas et al.) S = d4 x √ −g R − 2Λ 16πG + C 2 RF(2)R F(2) = ∞ n=0 fn2n , 2 = µ µ = 1 √ −g ∂µ √ −ggµν ∂ν Equations of motion C 2RµνF(2)R − 2( µ ν − gµν2)(F(2)R) − 1 2 gµνRF(2)R + ∞ n=1 fn 2 n−1 l=0 gµν gαβ ∂α2l R∂β2n−1−l R + 2l R2n−l R − 2∂µ2l R∂ν2n−1−l R = −1 8πG (Gµν + Λgµν). BW - 2013 B. Dragovich Balkan Workshop BW2013 7/15
  8. 8. 2. Nonlocal Modified Gravity Trace and 00-component 62(F(2)R) + ∞ n=1 fn n−1 l=0 ∂µ2l R∂µ 2n−1−l R + 22l R2n−l R = 1 8πGC R − Λ 2πGC C 2R00F(2)R − 2( 0 0 − g002)(F(2)R) − 1 2 g00RF(2)R + ∞ n=1 fn 2 n−1 l=0 g00 gαβ ∂α2l R∂β2n−1−l R + 2l R2n−l R − 2∂02l R∂02n−1−l R = −1 8πG (G00 + Λg00) BW - 2013 B. Dragovich Balkan Workshop BW2013 8/15
  9. 9. 3. Nonsingular bounce cosmological solutions We use FLRW metric ds2 = −dt2 +a2 (t) dr2 1 − kr2 +r2 dθ2 +r2 sin2 θdφ2 , k = 0, ±1 We use ansatz 2R = rR + s 2n R = rn (R + s r ), n ≥ 1, F(2)R = F(r)R + s r (F(r) − f0) We look for a solution of the form (Dimitrijevic, B.D., Grujic, Rakic) a(t) = a0(σeλt + τe−λt ), 0 < a0, λ, σ, τ ∈ R BW - 2013 B. Dragovich Balkan Workshop BW2013 9/15
  10. 10. 3. Nonsingular bounce cosmological solutions H(t) = ˙a a = λ(σeλt − τe−λt ) σeλt + τe−λt R(t) = 6 a2 (a¨a + ˙a2 + k) = 6 2a2 0λ2 σ2e4tλ + τ2 + ke2tλ a2 0 (σe2tλ + τ) 2 2R = − 12λ2e2tλ 4a2 0λ2στ − k a2 0 (σe2tλ + τ) 2 2R = 2λ2 R − 24λ4 , r = 2λ2 , s = −24λ4 BW - 2013 B. Dragovich Balkan Workshop BW2013 10/15
  11. 11. 3. Nonsingular bounce cosmological solutions From trace and 00-component two equations follow as polynomials in e2λt a4 0τ6 4πG 3λ2 − Λ + 3a2 0τ4 Q1e2λt + 6a2 0στ3 Q2e4λt − 2στQ3e6λt + 6a2 0σ3 τQ2e8λt + 3a2 0σ4 Q1e10λt + a4 0σ6 4πG 3λ2 − Λ e12λt = 0 τ6a4 0 8πG 3λ2 − Λ + 3τ4 a2 0R1e2λt + 3τ2 R2e4λt + 2στR3e6λt + 3σ2 R2e8λt + 3σ4 a2 0R1e10λt + σ6a4 0 8πG 3λ2 − Λ e12λt = 0 BW - 2013 B. Dragovich Balkan Workshop BW2013 11/15
  12. 12. 3. Nonsingular bounce cosmological solutions where Q1 = 36Cλ2 KF(2λ2 ) + a2 0(−96Cf0λ4 + λ2 πG − Λ 2πG )στ + 24Cf0kλ2 + k 8πG , Q2 = 72Cλ2 KF(2λ2 ) + a2 0(−192Cf0λ4 + 7λ2 8πG − 5Λ 8πG )στ + 48Cf0kλ2 + k 4πG , Q3 = −324Ca2 0λ2 στKF(2λ2 ) + 144Cλ2 K2 F (2λ2 ) − a2 0k(216Cf0λ2 + 9 8πG )στ + a4 0(864Cf0λ4 − 3λ2 πG + 5Λ 2πG )σ2 τ2 BW - 2013 B. Dragovich Balkan Workshop BW2013 12/15
  13. 13. 3. Nonsingular bounce cosmological solutions and R1 = Q1 − 3λ2 − Λ 4πG στa2 0 R2 = −6C k − 12a2 0λ2 στ KF(2λ2 ) − 36Cλ2 K2 F (2λ2 )+ a2 0k 2πG 192πGCf0λ2 + 1 στ − a4 0 8πG 3072πGCf0λ4 + λ2 + 5Λ σ2 τ2 R3 = −18C k − 6a2 0λ2 στ KF(2λ2 ) + 36Cλ2 K2 F (2λ2 )+ 9a2 0k 8πG 192πGCf0λ2 + 1 στ − a4 0 4πG 3456πGCf0λ4 + 3λ2 + 5Λ σ2 τ2 and K = 4a2 0λ2στ − k. BW - 2013 B. Dragovich Balkan Workshop BW2013 13/15
  14. 14. 3. Nonsingular bounce cosmological solutions Equations of motion are satisfied when λ = ± Λ 3 , as well as Q1 = Q2 = Q3 = 0 and R1 = R2 = R3 = 0. There are three cases of solutions. Case 1. F 2λ2 = 0, F 2λ2 = 0, f0 = − 1 64πGCΛ Case 2. 3k = 4a2 0Λστ Case 3. F 2λ2 = 1 96πGCΛ + 2 3 f0, F 2λ2 = 0, k = −4a2 0Λστ BW - 2013 B. Dragovich Balkan Workshop BW2013 14/15
  15. 15. 4. Concluding Remarks We have considered a nonlocal gravity model with cosmological constant Λ and without matter. Using ansatz 2R = rR + s we found three types of nonsingular bouncing solutions for cosmological scale factor in the form a(t) = a0(σeλt + τe−λt ). Solutions exist for all three values of spatial curvature constant k = 0, ±1. All these solutions depend on cosmological constant Λ, which is here an arbitrary positive parameter. Nonsingular bounce solution a(t) = a0e 1 2 Λ 3 t2 for k = 0 was found by A. Koshelev and S. Vernov. Nonsingular bounce solution a(t) = a0 cosh Λ 3 t for k = 0 and some radiation was found by T. Biswas, T. Koivisto, A. Mazumdar and W. Siegel. There are also some other ans¨atze which can be also useful in finding cosmological solutions. BW - 2013 B. Dragovich Balkan Workshop BW2013 15/15

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