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B. Dragovich: On Modified Gravity and Cosmology

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  • 1. ON MODIFIED GRAVITYAND COSMOLOGYBranko Dragovichhttp://www.ipb.ac.rs/˜ dragovichdragovich@ipb.ac.rsInstitute of PhysicsBelgrade, SerbiaBalkan Workshop BW 201325 – 29. 04. 2013Vrnjaˇcka Banja – SERBIABW - 2013 B. Dragovich Balkan Workshop BW2013 1/15
  • 2. Contents1 Introduction2 Nonlocal modified gravity3 Nonsingular bounce cosmological solutions4 Concluding remarksBW - 2013 B. Dragovich Balkan Workshop BW2013 2/15
  • 3. 1. Introduction: Einstein theory of gravity (ETG)  Einstein Theory of Gravity (1915)482R GR g T gcμν μν μν μνπ− = − Λ220d x dx dxd d dμ α βμαβτ τ τ+ Γ = BW - 2013 B. Dragovich Balkan Workshop BW2013 3/15
  • 4. 1. Introduction: Einstein theory of gravity (ETG)ETG is the simplest self-consistent theory of gravityIt is General Relativity (GR) and contains Newton theory ofgravityIts predictions are confirmed mainly in Solar SystemIt gives possibility to understand gravitational phenomenafrom laboratory scale to cosmological scalesETG predicts existence of Dark Energy (DE) and DarkMatter (DM)BW - 2013 B. Dragovich Balkan Workshop BW2013 4/15
  • 5. 1. Introduction: Some problems of Einstein theoryof gravityGeneral Relativity is non-renormalizable quantum fieldtheory.It predicts Dark Energy and Dark Matter which aremysterious and without other evidence.General Relativity has not been tested and confirmed atlarge cosmic scales, hence its application for the Universeas a whole is questionable.Cosmological solutions of GR contain Big Bang singularity.All these problems serve as motivation to investigate aModified Gravity, which is a generalization of ETG.There are many modifications motivated by differentreasons.To get nonsingular bounce cosmological solutions weconsider a Nonlocal Modified Gravity.BW - 2013 B. Dragovich Balkan Workshop BW2013 5/15
  • 6. 2. Nonlocal Modified Gravity: Relevant referencesT. Clifton, P. G. Ferreira, A. Padilla, C. Skordis, “Modifiedgravity and cosmology”, Phys. Rep. 513 (1), 1–189 (2012).[arXiv:1106.2476v2 [astro-ph.CO]].T. Biswas, T. Koivisto, A. Mazumdar, “Towards a resolutionof the cosmological singularity in non-local higherderivative theories of gravity”, JCAP 1011 (2010) 008[arXiv:1005.0590v2 [hep-th]].A. S. Koshelev, S. Yu. Vernov, “On bouncing solutions innon-local gravity”, Phys. Part. Nuclei 43, 666–668 (2012)[arXiv:1202.1289v1 [hep-th]].I. Dimitrijevic, B. Dragovich, J. Grujic, Z. Rakic, “Onmodified gravity”, to appear in Springer Proc. in Math. andStatistics 36 (2013) [arXiv:1202.2352v2 [hep-th]].I. Dimitrijevic, B. Dragovich, J. Grujic, Z. Rakic, “Newcosmological solutions in nonlocal modified gravity”, toappear in Romanian J. Physics (2013) [arXiv:1302.2794[gr-qc]].BW - 2013 B. Dragovich Balkan Workshop BW2013 6/15
  • 7. 2. Nonlocal Modified GravityNonlocal gravity action without matter (Biswas et al.)S = d4x√−gR − 2Λ16πG+C2RF(2)RF(2) =∞n=0fn2n, 2 = µµ =1√−g∂µ√−ggµν∂νEquations of motionC 2RµνF(2)R − 2( µ ν − gµν2)(F(2)R) −12gµνRF(2)R+∞n=1fn2n−1l=0gµν gαβ∂α2lR∂β2n−1−lR + 2lR2n−lR− 2∂µ2lR∂ν2n−1−lR =−18πG(Gµν + Λgµν).BW - 2013 B. Dragovich Balkan Workshop BW2013 7/15
  • 8. 2. Nonlocal Modified GravityTrace and 00-component62(F(2)R) +∞n=1fnn−1l=0∂µ2lR∂µ2n−1−lR + 22lR2n−lR=18πGCR −Λ2πGCC 2R00F(2)R − 2( 0 0 − g002)(F(2)R) −12g00RF(2)R+∞n=1fn2n−1l=0g00 gαβ∂α2lR∂β2n−1−lR + 2lR2n−lR− 2∂02lR∂02n−1−lR =−18πG(G00 + Λg00)BW - 2013 B. Dragovich Balkan Workshop BW2013 8/15
  • 9. 3. Nonsingular bounce cosmological solutionsWe use FLRW metricds2= −dt2+a2(t)dr21 − kr2+r2dθ2+r2sin2θdφ2, k = 0, ±1We use ansatz2R = rR + s2nR = rn(R +sr), n ≥ 1, F(2)R = F(r)R +sr(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 ∈ RBW - 2013 B. Dragovich Balkan Workshop BW2013 9/15
  • 10. 3. Nonsingular bounce cosmological solutionsH(t) =˙aa=λ(σeλt − τe−λt )σeλt + τe−λtR(t) =6a2(a¨a + ˙a2+ k) =6 2a20λ2 σ2e4tλ + τ2 + ke2tλa20 (σe2tλ + τ)22R = −12λ2e2tλ 4a20λ2στ − ka20 (σe2tλ + τ)22R = 2λ2R − 24λ4, r = 2λ2, s = −24λ4BW - 2013 B. Dragovich Balkan Workshop BW2013 10/15
  • 11. 3. Nonsingular bounce cosmological solutionsFrom trace and 00-component two equations follow aspolynomials in e2λta40τ64πG3λ2− Λ + 3a20τ4Q1e2λt+ 6a20στ3Q2e4λt− 2στQ3e6λt+ 6a20σ3τQ2e8λt+ 3a20σ4Q1e10λt+a40σ64πG3λ2− Λ e12λt= 0τ6a408πG3λ2− Λ + 3τ4a20R1e2λt+ 3τ2R2e4λt+ 2στR3e6λt+ 3σ2R2e8λt+ 3σ4a20R1e10λt+σ6a408πG3λ2− Λ e12λt= 0BW - 2013 B. Dragovich Balkan Workshop BW2013 11/15
  • 12. 3. Nonsingular bounce cosmological solutionswhereQ1 = 36Cλ2KF(2λ2) + a20(−96Cf0λ4+λ2πG−Λ2πG)στ+ 24Cf0kλ2+k8πG,Q2 = 72Cλ2KF(2λ2) + a20(−192Cf0λ4+7λ28πG−5Λ8πG)στ+ 48Cf0kλ2+k4πG,Q3 = −324Ca20λ2στKF(2λ2) + 144Cλ2K2F (2λ2)− a20k(216Cf0λ2+98πG)στ + a40(864Cf0λ4−3λ2πG+5Λ2πG)σ2τ2BW - 2013 B. Dragovich Balkan Workshop BW2013 12/15
  • 13. 3. Nonsingular bounce cosmological solutionsandR1 = Q1 −3λ2 − Λ4πGστa20R2 = −6C k − 12a20λ2στ KF(2λ2) − 36Cλ2K2F (2λ2)+a20k2πG192πGCf0λ2+ 1 στ −a408πG3072πGCf0λ4+ λ2+ 5Λ σ2τ2R3 = −18C k − 6a20λ2στ KF(2λ2) + 36Cλ2K2F (2λ2)+9a20k8πG192πGCf0λ2+ 1 στ −a404πG3456πGCf0λ4+ 3λ2+ 5Λ σ2τ2and K = 4a20λ2στ − k.BW - 2013 B. Dragovich Balkan Workshop BW2013 13/15
  • 14. 3. Nonsingular bounce cosmological solutionsEquations of motion are satisfied when λ = ± Λ3 , as wellas Q1 = Q2 = Q3 = 0 and R1 = R2 = R3 = 0. There arethree cases of solutions.Case 1.F 2λ2= 0, F 2λ2= 0, f0 = −164πGCΛCase 2.3k = 4a20ΛστCase 3.F 2λ2=196πGCΛ+23f0, F 2λ2= 0, k = −4a20ΛστBW - 2013 B. Dragovich Balkan Workshop BW2013 14/15
  • 15. 4. Concluding RemarksWe have considered a nonlocal gravity model withcosmological constant Λ and without matter.Using ansatz 2R = rR + s we found three types ofnonsingular bouncing solutions for cosmological scalefactor in the form a(t) = a0(σeλt + τe−λt ).Solutions exist for all three values of spatial curvatureconstant k = 0, ±1.All these solutions depend on cosmological constant Λ,which is here an arbitrary positive parameter.Nonsingular bounce solution a(t) = a0e12Λ3t2for k = 0was found by A. Koshelev and S. Vernov.Nonsingular bounce solution a(t) = a0 cosh Λ3 t fork = 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 alsouseful in finding cosmological solutions.BW - 2013 B. Dragovich Balkan Workshop BW2013 15/15

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