SEENET-MTP, profile picture
Uploaded bySEENET-MTP
260 views

Ivan Dimitrijević "Nonlocal cosmology"

This document discusses nonlocal cosmology and modifications to Einstein's theory of gravity. It presents three cases of nonlocal modified gravity models: 1. When P(R)=R and Q(R)=R, nonsingular bounce cosmological solutions were found with scale factor a(t)=a0(σeλt+τe-λt). 2. When P(R)=R-1 and Q(R)=R, several power-law cosmological solutions were obtained, including a(t)=a0|t-t0|α. 3. For the case P(R)=Rp and Q(R)=Rq, the trace and 00 equations of motion were transformed into an equivalent

Embed presentation

Download to read offline
Nonlocal cosmology Ivan Dimitrijevi´c Nonlocal cosmology Ivan Dimitrijevi´c University of Belgrade, Faculty of Mathematics 14.06.2018.
Nonlocal cosmology Ivan Dimitrijevi´c Motivation Large cosmological observational findings: High orbital speeds of galaxies in clusters. (F.Zwicky, 1933) High orbital speeds of stars in spiral galaxies. (Vera Rubin, at the end of 1960es) Accelerated expansion of the Universe. (1998)
Nonlocal cosmology Ivan Dimitrijevi´c Problem solving approaches There are two problem solving approaches: Dark matter and energy Modification of Einstein theory of gravity Rµν − 1 2 Rgµν = 8πGTµν − Λgµν, c = 1 where Tµν is stress-energy tensor, gµν are the elements of the metric tensor, Rµν is Ricci tensor and R is scalar curvature of metric.
Nonlocal cosmology Ivan Dimitrijevi´c Dark matter and energy If Einstein theory of gravity can be applied to the whole Universe then the Universe contains about 4.9% of ordinary matter, 26.8% of dark matter and 68.3% of dark energy. It means that 95.1% of total matter, or energy, represents dark side of the Universe, which nature is unknown. Dark matter is responsible for orbital speeds in galaxies, and dark energy is responsible for accelerated expansion of the Universe.
Nonlocal cosmology Ivan Dimitrijevi´c Modification of Einstein theory of gravity Motivation for modification of Einstein theory of gravity The validity of General Relativity on cosmological scale is not confirmed. Dark matter and dark energy are not yet detected in the laboratory experiments. Another cosmological problem is related to the Big Bang singularity. Namely, under rather general conditions, general relativity yields cosmological solutions with zero size of the universe at its beginning, what means an infinite matter density. Note that when physical theory contains singularity, it is not valid in the vicinity of singularity and must be appropriately modified.
Nonlocal cosmology Ivan Dimitrijevi´c Approaches to modification of Einstein theory of gravity There are different approaches to modification of Einstein theory of gravity. Einstein General Theory of Relativity From action S = ( R 16πG − Lm − 2Λ) √ −gd4 x using variational methods we get field equations Rµν − 1 2 Rgµν = 8πGTµν − Λgµν, c = 1 where Tµν is stress-energy tensor, gµν are the elements of the metric tensor, Rµν is Ricci tensor and R is scalar curvature of metric. Currently there are mainly two approaches: f(R) Modified Gravity Nonlocal Gravity
Nonlocal cosmology Ivan Dimitrijevi´c Nonlocal Modified Gravity Nonlocal gravity is a modification of Einstein general relativity in such way that Einstein-Hilbert action contains a function f( , R). Our action is given by S = d4 x √ −g R − 2Λ 16πG + P(R)F( )Q(R) where = 1√ −g ∂µ √ −ggµν ∂ν, F( ) = ∞ n=0 fn n . We use Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric ds2 = −dt2 + a2 (t) dr2 1−kr2 + r2 dθ2 + r2 sin2 θdφ2 , k ∈ {−1, 0, 1}.
Nonlocal cosmology Ivan Dimitrijevi´c Equations of motion Equation of motion are − 1 2 gµνP(R)F( )Q(R) + RµνW − KµνW + 1 2 Ωµν = − Gµν + Λgµν 16πG , Ωµν = ∞ n=1 fn n−1 l=0 Sµν l P(R), n−1−l Q(R) , Kµν = µ ν − gµν , Sµν(A, B) = gµν α A αB − 2 µA νB + gµνA B, W = P (R)F( )Q(R) + Q (R)F( )P(R).
Nonlocal cosmology Ivan Dimitrijevi´c Trace and 00-equations In case of FRW metric there are two linearly independent equations. The most convenient choice is trace and 00 equations: − 2P(R)F( )Q(R) + RW + 3 W + 1 2 Ω = R − 4Λ 16πG , 1 2 P(R)F( )Q(R) + R00W − K00W + 1 2 Ω00 = − G00 − Λ 16πG , Ω = gµν Ωµν.
Nonlocal cosmology Ivan Dimitrijevi´c Cosmological solutions in nonlocal modified gravity We discuss the class of models given by the action S = M R − 2Λ 16πG + P(R)F( )Q(R) √ −g d4 x. In particular, we investigate the following cases: P = R, Q = R, P = R−1 , Q = R, P = Rp , Q = Rq , p ∈ N, q ∈ N, P = (R + R0)m , Q = (R + R0)m , m ∈ R, R0 ∈ R
Nonlocal cosmology Ivan Dimitrijevi´c First case P = R, Q = R In this the action becomes S = M R − 2Λ 16πG + RF( )R √ −g d4 x. This model is interesting since we obtain nonsingular cosmological solutions. To solve EOM we use the ansatz: R = rR + s, r, s ∈ R
Nonlocal cosmology Ivan Dimitrijevi´c Nonsingular bounce cosmological solutions Let the scale factor a(t) be in a form a(t) = a0(σeλt + τe−λt ), a0 > 0, λ, σ, τ ∈ R. Theorem Scale factor a(t) = a0(σeλt + τe−λt ) is a solution of EOM in a following three cases: Case 1. F 2λ2 = 0, F 2λ2 = 0, f0 = − 1 128πGCΛ . Case 2. 3k = 4a2 0Λστ. Case 3. F 2λ2 = 1 192πGCΛ + 2 3 f0, F 2λ2 = 0, k = −4a2 0Λστ. In all cases we have 3λ2 = Λ.
Nonlocal cosmology Ivan Dimitrijevi´c Summary, P = R, Q = R We studied nonlocal gravity model with cosmological constant Λ, without matter. Using anzac R = rR + s we found three types of nonsingular bounce cosmogical solutions with scale factor a(t) = a0(σeλt + τe−λt ). All solutions satisfy ¨a(t) = λ2 a(t) > 0. There are solutions for all values of parameter k = 0, ±1.
Nonlocal cosmology Ivan Dimitrijevi´c Second case P = R−1 , Q = R This model is given by the action S = M R 16πG + R−1 F( )R √ −g d4 x, where F( ) = ∞ n=0 fn n . If we set f0 = − Λ 8πG , then f0 takes the place of cosmological constant.
Nonlocal cosmology Ivan Dimitrijevi´c Second case P = R−1 , Q = R This model is given by the action S = M R 16πG + R−1 F( )R √ −g d4 x, where F( ) = ∞ n=0 fn n . If we set f0 = − Λ 8πG , then f0 takes the place of cosmological constant.
Nonlocal cosmology Ivan Dimitrijevi´c Some power-law cosmological solutions We use the following form of scale factor and scalar curvature: a(t) = a0|t − t0|α , R(t) = 6 α(2α − 1)(t − t0)−2 + k a2 0 (t − t0)−2α .
Nonlocal cosmology Ivan Dimitrijevi´c Case k = 0, α = 0 i α = 1 2 Theorem For k = 0, α = 0, α = 1 2 and 3α−1 2 ∈ N scale factor a = a0|t − t0|α is a solution of EOM if f0 = 0, f1 = − 3α(2α − 1) 32πG(3α − 2) , fn = 0 za 2 ≤ n ≤ 3α − 1 2 , fn ∈ R za n > 3α − 1 2 .
Nonlocal cosmology Ivan Dimitrijevi´c Case k = 0, α → 0 (Minkowski spacetime) Theorem For k = 0 and α → 0 EOM are satisfied if f0, f1 ∈ R, fi = 0, i ≥ 2. Since all fn = 0, n ≥ 2 this model is not a nonlocal model. Thus the power-law solutions cannot be obtained by perturbing the Minkowski solution.
Nonlocal cosmology Ivan Dimitrijevi´c Case k = 0, α → 1 2 Theorem For k = 0 and α → 1 2 EOM are satisfied if f0 ∈ R, fi = 0, i ≥ 1.
Nonlocal cosmology Ivan Dimitrijevi´c Case k = 0, α = 1 Let k = 0. To simplify the scalar curvature R(t) = 6 α(2α − 1)(t − t0)−2 + k a2 0 (t − t0)−2α we have three options: α = 0, α = 1 2 i α = 1. The first two does not yield any solutions of EOM and the last one is described in the following theorem. Theorem For k = 0 scale factor a = a0|t − t0| is a solution of EOM if f0 = 0, f1 = −s 64πG , fn ∈ R, n ≥ 2, where s = 6(1 + k a2 0 ).
Nonlocal cosmology Ivan Dimitrijevi´c Summary, P = R−1 , Q = R We the model with nonlocal term R−1 F( )R and obtained power-law cosmological solutions a(t) = a0|t − t0|α . It is worth noting that there is a solution a(t) = |t − t0| which corresponds to Milne universe for k = −1. All presented solutions a(t) = a0|t − t0|α have scalar curvature R(t) = 6 α(2α − 1)(t − t0)−2 + k a2 0 (t − t0)−2α , that satisfies R = qR2 , where parameter q depends on α.
Nonlocal cosmology Ivan Dimitrijevi´c Third case P = Rp , Q = Rq We investigate the model S = M R − 2Λ 16πG + Rp F( )Rq √ −g d4 x. Also, we assume p ≥ q.
Nonlocal cosmology Ivan Dimitrijevi´c Third case P = Rp , Q = Rq We discuss only k = 0. Scale factor is chosen in the form a(t) = a0e− γ 12 t2 , γ ∈ R. Hubble parameter and scalar curvature are given by H(t) = − 1 6 γt, R(t) = 1 3 γ(γt2 − 3), R00 = 1 4 (γ − R).
Nonlocal cosmology Ivan Dimitrijevi´c Third case P = Rp , Q = Rq Rp = pγRp − p 3 (4p − 5)γ2 Rp−1 − 4 3 p(p − 1)γ3 Rp−2 . From this relation it follows that operator is closed on the space of polynomials in R of degree at most p. In the basis vp = Rp Rp−1 . . . R 1 T the matrix of is Mp = γ       p p 3 (5 − 4p)γ 4 3 p(1 − p)γ2 0 . . . 0 0 p − 1 p−1 3 (9 − 4p)γ 4 3 (1 − p)(p − 2)γ2 . . . 0 ... ... ... ... ... 0 0 0 . . . 1 γ 3 0 0 0 . . . 0 0       .
Nonlocal cosmology Ivan Dimitrijevi´c EOM Moreover, Fp = ∞ n=0 fnMn p is a matrix of operator F( ). Let Dp be a matrix of operator ∂ ∂R and ep are the coordinates of vector Rp in basis vp. Trace and 00 equation are transformed into T = 0, Z = 0, where T = −2epvpeqFqvq + RWpq − 4γ2 (R + γ)Wpq − 2γ2 Wpq − S1 + 2S2 − R − 4Λ 16πG , Z = 1 2 epvpeqFqvq + γ 4 (γ − R)Wpq − γ(R + γ)Wpq − 1 2 (S1 + S2) + G00 − Λ 16πG ,
Nonlocal cosmology Ivan Dimitrijevi´c EOM and S1 = 4 3 γ2 (R + γ) ∞ n=1 fn n−1 l=0 epMl pDpvpeqMn−1−l q Dqvq, S2 = ∞ n=1 fn n−1 l=0 epMl pvpeqMn−l q vq.
Nonlocal cosmology Ivan Dimitrijevi´c EOM Theorem For all p ∈ N and q ∈ N we have T + 4Z = 4γZ . Trace and 00 equations are equivalent. Therefore, it is sufficient to solve only trace equation. It is of a polynomial type, degree p + q in R, and it splits into p + q + 1 equations with p + q + 1 ”variables” f0 = F(0), F(γ), . . . , F(pγ), F (γ), . . . , F (qγ).
Nonlocal cosmology Ivan Dimitrijevi´c (p, q) = (1, 1) Theorem For p = q = 1, trace equation is satisfied iff γ = −12Λ, F (γ) = 0 and f0 = 3 32γπG − 8F(γ).
Nonlocal cosmology Ivan Dimitrijevi´c (p, q) = (1, 1) Theorem Trace equation is satisfied for the following values of parameters p and q (κ = 1 16πG ): p = 2, q = 1: F(γ) = 9κ(γ+9Λ) 112γ3 , F(2γ) = 3κ(γ+9Λ) 56γ3 , f0 = −κ(4γ+15Λ) 7γ3 , F (γ) = −3κ(γ+9Λ) 8γ4 , p = 2, q = 2: F(γ) = 369κ(γ+8Λ) 9344γ4 , F(2γ) = 27κ(γ+8Λ) 4672γ4 , f0 = κ(145γ+576Λ) 876γ4 , F (γ) = −639κ(γ+8Λ) 2336γ5 , F (2γ) = −27κ(γ+8Λ) 9344γ5 , p = 3, q = 1: F(γ) = κ(107γ+408Λ) 6432γ4 , F(2γ) = −κ(173γ+840Λ) 7504γ4 , F(3γ) = 0, f0 = −κ(95γ+768Λ) 268γ4 , F (γ) = −9κ(γ+8Λ) 88γ5 . p = 3, q = 2: F(γ) = 3κ(10702γ+40497Λ) 245680γ5 , F(2γ) = −27κ(6γ+25Λ) 24568γ5 , F(3γ) = −27κ(6γ+25Λ) 49136γ5 , f0 = −3κ(7099γ+23949Λ) 15355γ5 , F (γ) = −3κ(11614γ+68865Λ) 270248γ6 , F (2γ) = 513κ(6γ+25Λ) 171976γ6 ,
Nonlocal cosmology Ivan Dimitrijevi´c Summary, P = Rp , Q = Rq , p ∈ N, q ∈ N We discuss the scale factor of the form a(t) = a0 exp(− γ 12 t2 ) Trace and 00 equations are equivalent. Trace equation is of a degree p + q in R, so it splits into p + q + 1 equations over p + q + 1 ”variables” f0 = F(0), F(γ), . . . , F(pγ), F (γ), . . . , F (qγ). For p = q = 1 the system has infinitely many solutions, and constants γ and Λ satisfy γ = −12Λ. For other values of p and q, there is unique solution, for any γ ∈ R. We obtained solutions for 1 ≤ q ≤ p ≤ 4.
Nonlocal cosmology Ivan Dimitrijevi´c Fourth case P(R) = (R + R0)m , Q(R) = (R + R0)m We take that P(R) = (R + R0)m , Q(R) = (R + R0)m S = M R − 2Λ 16πG + (R + R0)m F( )(R + R0)m √ −g d4 x. Scale factor is in the form a(t) = A tn e− γ 12 t2 .
Nonlocal cosmology Ivan Dimitrijevi´c Fourth case P(R) = (R + R0)m , Q(R) = (R + R0)m Also, we take the ansatz (R + R0)m = r(R + R0)m , (1) where R0,r, m i n realne konstante. Ansatz in the expanded for is − 648mn2 (2n − 1)2 (2m − 3n + 1) = 0, − 324n(2n − 1) −γm + 6γmn2 − 4γmn − mnR0 + mR0 + 2n2 r − nr = 0, 18n(2n − 1) 8γ2 m2 − 13γ2 m + 12γ2 mn − 3γmR0 + 24γnr + 6γr − 6rR0 = 0, − 2γ3 m − 24γ3 mn2 − 14γ3 mn + 6γ2 mnR0 + 2γ2 mR0 + 72γ2 n2 r + 12γ2 nr − 24γnrR0 + 3γ2 r − 6γrR0 + 3rR2 0 = 0, − γ2 4γ2 m2 + γ2 m + 18γ2 mn − 3γmR0 − 24γnr − 6γr + 6rR0 = 0, − γ4 (r − γm) = 0. There are five solutions of the above system 1 r = mγ, n = 0, R0 = γ, m = 1 2 2 r = mγ, n = 0, R0 = γ 3 , m = 1 3 r = mγ, n = 1 2 , R0 = 4 3 γ, m = 1 4 r = mγ, n = 1 2 , R0 = 3γ, m = −1 4 5 r = mγ, n = 2m+1 3 , R0 = 7 3 γ, m = 1 2 .
Nonlocal cosmology Ivan Dimitrijevi´c Case n = 0, m = 1 2 Trace and 00 equations are linear in R, and they split into two systems: γ 16πG + Λ 4πG − γF( γ 2 ) − γ2 3 F ( γ 2 ) = 0, − 108γ2 πG − γ2 3 F( γ 2 ) + γ3 3 F ( γ 2 ) = 0. Λ 16πG − γ 2 F( γ 2 ) + γ2 6 F ( γ 2 ) = 0, 27γ2 πG + γ2 12 F( γ 2 ) + γ3 6 F ( γ 2 ) = 0. The solution of the above system is F( γ 2 ) = 24γ + Λ 768γπG , F ( γ 2 ) = 3Λ − 3γ 16γ2πG .
Nonlocal cosmology Ivan Dimitrijevi´c Case n = 2 3, m = 1 2 Trace and 00 equations split into the following systems: F ( γ 2 ) = 0, 11γ 48πG + Λ 4πG − γF( γ 2 ) = 0, 1 16πG + F( γ 2 ) = 0, γ2 16πG + γ2 F( γ 2 ) = 0, i F ( γ 2 ) = 0, − γ 24πG − Λ 16πG + 1 2 γF( γ 2 ) = 0, 1 16πG + cF( γ 2 ) = 0, γ2 16πG + cγ2 F( γ 2 ) = 0. The solution is F( γ 2 ) = −1 16πG , F ( γ 2 ) = 0, Λ = − 7 6 γ.
Nonlocal cosmology Ivan Dimitrijevi´c Symmary, P(R) = (R + R0)m , Q(R) = (R + R0)m We take the scale factor in the form a(t) = Atn exp(− γ 12 t2 ). Ansatz is in the form (R + R0)m = r(R + R0)m . Ansatz has five solutions, from witch four satisfy EOM. In case n = 0, m = 1 2 there is unique solution in F(γ 2 ) and F (γ 2 ). In case n = 2 3 , m = 1 2 there is unique solution in F(γ 2 ) and F (γ 2 ) where Λ = −7 6 γ. In case n = 1 2 , m = −1 4 EOM have no solution.
Nonlocal cosmology Ivan Dimitrijevi´c References J. Grujic. On a new model of nonlocal modified gravity. Kragujevac Journal of Mathematics, 2015. I. Dimitrijevic Cosmological solutions in modified gravity with monomial nonlocality, Aplied Mathematics and computation 285, 195-203, 2016 Branko Dragovich. On nonlocal modified gravity and cosmology. In Lie Theory and Its Applications in Physics, volume 111 of Springer Proceedings in Mathematics and Statistics, pages 251-262, 2014. Ivan Dimitrijevic, Branko Dragovich, Jelena Grujic, and Zoran Rakic. Constant Curvature Cosmological Solutions in Nonlocal Gravity. In Bunoiu, OM and Avram, N and Popescu, A, editor, TIM 2013 PHYSICS CONFERENCE, volume 1634 of AIP Conference Proceedings, pages 18-23. W Univ Timisoara, Fac Phys, 2014. TIM 2013 Physics Conference, Timisoara, ROMANIA, NOV 21-24, 2013. Ivan Dimitrijevic, Branko Dragovich, Jelena Grujic, and Zoran Rakic On Nonlocal Modified Gravity and its Cosmological Solutions In Lie Theory and Its Applications in Physics, Springer Proceedings in Mathematics and Statistics, 2016.
Nonlocal cosmology Ivan Dimitrijevi´c References Dimitrijevic, I., Dragovich, B., Grujic J., Rakic, Z.: New cosmological solutions in nonlocal modified gravity. Rom. Journ. Phys. 58 (5-6), 550559 (2013) [arXiv:1302.2794 [gr-qc]] I. Dimitrijevic, B.Dragovich, J. Grujic, Z. Rakic, “A new model of nonlocal modified gravity”, Publications de l’Institut Mathematique 94 (108), 187–196 (2013). Ivan Dimitrijevic, Branko Dragovich, Jelena Grujic, and Zoran Rakic. On Modified Gravity. Springer Proc.Math.Stat., 36:251-259, 2013. I. Dimitrijevic, B. Dragovich, J. Grujic, and Z. Rakic. Some power-law cosmological solutions in nonlocal modified gravity. In Lie Theory and Its Applications in Physics, volume 111 of Springer Proceedings in Mathematics and Statistics, pages 241-250, 2014. I. Dimitrijevic, B. Dragovich, J. Grujic, and Z. Rakic. Some cosmological solutions of a nonlocal modified gravity. Filomat, 2014.
Nonlocal cosmology Ivan Dimitrijevi´c Thank you!

More Related Content

PDF
R. Jimenez - Fundamental Physics from Astronomical Observations
bySEENET-MTP
 
PDF
The proof complexity of matrix algebra - Newton Institute, Cambridge 2006
byMichael Soltys
 
PDF
D. Vulcanov - On Cosmologies with non-Minimally Coupled Scalar Field and the ...
bySEENET-MTP
 
PDF
Proceedings A Method For Finding Complete Observables In Classical Mechanics
byvcuesta
 
PDF
Poster Gauge systems and functions, hermitian operators and clocks as conjuga...
byvcuesta
 
PPT
Rdnd2008
byGautam Sethia
 
PPT
Adc
byGautam Sethia
 
PDF
Application of analytic function
byDr. Nirav Vyas
 
R. Jimenez - Fundamental Physics from Astronomical Observations
bySEENET-MTP
 
The proof complexity of matrix algebra - Newton Institute, Cambridge 2006
byMichael Soltys
 
D. Vulcanov - On Cosmologies with non-Minimally Coupled Scalar Field and the ...
bySEENET-MTP
 
Proceedings A Method For Finding Complete Observables In Classical Mechanics
byvcuesta
 
Poster Gauge systems and functions, hermitian operators and clocks as conjuga...
byvcuesta
 
Rdnd2008
byGautam Sethia
 
Adc
byGautam Sethia
 
Application of analytic function
byDr. Nirav Vyas
 

What's hot

PDF
Bird’s-eye view of Gaussian harmonic analysis
byRadboud University Medical Center
 
PPTX
Laplace
byDivyesh Harwani
 
PDF
Topic 8 kft 131
byNor Qurratu'aini Abdullah
 
PDF
Queueing theory
byMeenakshi Dhasmana
 
PPT
Matrices ii
bySolo Hermelin
 
PDF
Aurelian Isar - Decoherence And Transition From Quantum To Classical In Open ...
bySEENET-MTP
 
PPTX
Laplace transformation
byWasim Shah
 
PDF
Adiabatic Theorem for Discrete Time Evolution
bytanaka-atushi
 
PDF
Solution set 3
by慧环 赵
 
PDF
Geometric and viscosity solutions for the Cauchy problem of first order
byJuliho Castillo
 
PDF
2014 04 22 wits presentation oqw
byRene Kotze
 
PDF
Ps02 cmth03 unit 1
byPrakash Dabhi
 
PPTX
Laplace transformation
byamanullahkakar2
 
PDF
ENFPC 2010
byLeandro da Silva
 
PDF
Application of Convolution Theorem
byijtsrd
 
PDF
Precessing magnetic impurity on sc
byYi-Hua Lai
 
PPTX
Time series Modelling Basics
byAshutosh Kumar
 
PDF
Solucionario Mecácnica Clásica Goldstein
byFredy Mojica
 
Bird’s-eye view of Gaussian harmonic analysis
byRadboud University Medical Center
 
Laplace
byDivyesh Harwani
 
Topic 8 kft 131
byNor Qurratu'aini Abdullah
 
Queueing theory
byMeenakshi Dhasmana
 
Matrices ii
bySolo Hermelin
 
Aurelian Isar - Decoherence And Transition From Quantum To Classical In Open ...
bySEENET-MTP
 
Laplace transformation
byWasim Shah
 
Adiabatic Theorem for Discrete Time Evolution
bytanaka-atushi
 
Solution set 3
by慧环 赵
 
Geometric and viscosity solutions for the Cauchy problem of first order
byJuliho Castillo
 
2014 04 22 wits presentation oqw
byRene Kotze
 
Ps02 cmth03 unit 1
byPrakash Dabhi
 
Laplace transformation
byamanullahkakar2
 
ENFPC 2010
byLeandro da Silva
 
Application of Convolution Theorem
byijtsrd
 
Precessing magnetic impurity on sc
byYi-Hua Lai
 
Time series Modelling Basics
byAshutosh Kumar
 
Solucionario Mecácnica Clásica Goldstein
byFredy Mojica
 

Similar to Ivan Dimitrijević "Nonlocal cosmology"

PDF
B. Dragovich: On Modified Gravity and Cosmology
bySEENET-MTP
 
PDF
Dynamical Systems Methods in Early-Universe Cosmologies
byIkjyot Singh Kohli
 
PDF
First-order cosmological perturbations produced by point-like masses: all sca...
byMaxim Eingorn
 
PDF
Black hole formation by incoming electromagnetic radiation
byXequeMateShannon
 
PPT
D. Vulcanov, REM — the Shape of Potentials for f(R) Theories in Cosmology and...
bySEENET-MTP
 
PDF
Starobinsky astana 2017
byBaurzhan Alzhanov
 
PDF
NCTS--FGCPA-LeCosPA-[Kazuharu-Bamba].pdf
bysahooparbati1990
 
PDF
Alexei Starobinsky "New results on inflation and pre-inflation in modified gr...
bySEENET-MTP
 
PDF
Cosmology using numerical relativity - what hapenned before big bang?
bySérgio Sacani
 
PPT
Introduction to cosmology and numerical cosmology (with the Cactus code) (1/2)
bySEENET-MTP
 
DOCX
Outgoing ingoingkleingordon ghp
byfoxtrot jp R
 
PDF
A Fibonacci-like universe expansion on time-scale
byOctavianPostavaru
 
PDF
Observational parameters of Inflation in Holographic cosmology
byMilan Milošević
 
PDF
Numerical Calculation of the Hubble Hierarchy Parameters and the Observationa...
byMilan Milošević
 
DOCX
Outgoing ingoingkleingordon 8th_jun19sqrd
byfoxtrot jp R
 
PDF
A0440109
byinventy
 
PDF
D. Vulcanov: Symbolic Computation Methods in Cosmology and General Relativity...
bySEENET-MTP
 
DOCX
The klein gordon field in two-dimensional rindler space-time 16052020
byfoxtrot jp R
 
DOCX
The klein gordon field in two-dimensional rindler space-time -sqrdupdt41220
byfoxtrot jp R
 
DOCX
The klein gordon field in two-dimensional rindler space-time 23052020-sqrd
byfoxtrot jp R
 
B. Dragovich: On Modified Gravity and Cosmology
bySEENET-MTP
 
Dynamical Systems Methods in Early-Universe Cosmologies
byIkjyot Singh Kohli
 
First-order cosmological perturbations produced by point-like masses: all sca...
byMaxim Eingorn
 
Black hole formation by incoming electromagnetic radiation
byXequeMateShannon
 
D. Vulcanov, REM — the Shape of Potentials for f(R) Theories in Cosmology and...
bySEENET-MTP
 
Starobinsky astana 2017
byBaurzhan Alzhanov
 
NCTS--FGCPA-LeCosPA-[Kazuharu-Bamba].pdf
bysahooparbati1990
 
Alexei Starobinsky "New results on inflation and pre-inflation in modified gr...
bySEENET-MTP
 
Cosmology using numerical relativity - what hapenned before big bang?
bySérgio Sacani
 
Introduction to cosmology and numerical cosmology (with the Cactus code) (1/2)
bySEENET-MTP
 
Outgoing ingoingkleingordon ghp
byfoxtrot jp R
 
A Fibonacci-like universe expansion on time-scale
byOctavianPostavaru
 
Observational parameters of Inflation in Holographic cosmology
byMilan Milošević
 
Numerical Calculation of the Hubble Hierarchy Parameters and the Observationa...
byMilan Milošević
 
Outgoing ingoingkleingordon 8th_jun19sqrd
byfoxtrot jp R
 
A0440109
byinventy
 
D. Vulcanov: Symbolic Computation Methods in Cosmology and General Relativity...
bySEENET-MTP
 
The klein gordon field in two-dimensional rindler space-time 16052020
byfoxtrot jp R
 
The klein gordon field in two-dimensional rindler space-time -sqrdupdt41220
byfoxtrot jp R
 
The klein gordon field in two-dimensional rindler space-time 23052020-sqrd
byfoxtrot jp R
 

More from SEENET-MTP

PDF
SEENET-MTP Booklet - 15 years
bySEENET-MTP
 
PDF
Milan Milošević "The shape of Fe Kα line emitted from relativistic accretion ...
bySEENET-MTP
 
PDF
Dragoljub Dimitrijević "Tachyon Inflation in the RSII Framework"
bySEENET-MTP
 
PDF
Vesna Borka Jovanović "Constraining Scalar-Tensor gravity models by S2 star o...
bySEENET-MTP
 
PDF
Elena Mirela Babalic "Generalized alpha-attractor models for hyperbolic surfa...
bySEENET-MTP
 
PDF
Dragan Huterer "Novi pogledi na svemir"
bySEENET-MTP
 
PDF
Mihai Visinescu "Action-angle variables for geodesic motion on resolved metri...
bySEENET-MTP
 
PDF
Sabin Stoica "Double beta decay and neutrino properties"
bySEENET-MTP
 
PDF
Yurri Sitenko "Boundary effects for magnetized quantum matter in particle and...
bySEENET-MTP
 
PDF
Predrag Milenović "Physics potential of HE/HL-LHC and future circular"
bySEENET-MTP
 
PDF
Marija Dimitrijević Ćirić "Matter Fields in SO(2,3)⋆ Model of Noncommutative ...
bySEENET-MTP
 
PDF
Zvonimir Vlah "Lagrangian perturbation theory for large scale structure forma...
bySEENET-MTP
 
PDF
Vitaly Vanchurin "General relativity from non-equilibrium thermodynamics of q...
bySEENET-MTP
 
PDF
Sergey Sibiryakov "Galactic rotation curves vs. ultra-light dark matter: Impl...
bySEENET-MTP
 
PDF
Radoslav Rashkov "Integrable structures in low-dimensional holography and cos...
bySEENET-MTP
 
PDF
Nikola Godinović "The very high energy gamma ray astronomy"
bySEENET-MTP
 
PDF
Miroljub Dugić "The concept of Local Time. Quantum-mechanical and cosmologica...
bySEENET-MTP
 
PDF
Cemsinan Deliduman "Astrophysics with Weyl Gravity"
bySEENET-MTP
 
PDF
Radu Constantinescu "Scientific research: Excellence in International context"
bySEENET-MTP
 
PDF
Loriano Bonora "HS theories from effective actions"
bySEENET-MTP
 
SEENET-MTP Booklet - 15 years
bySEENET-MTP
 
Milan Milošević "The shape of Fe Kα line emitted from relativistic accretion ...
bySEENET-MTP
 
Dragoljub Dimitrijević "Tachyon Inflation in the RSII Framework"
bySEENET-MTP
 
Vesna Borka Jovanović "Constraining Scalar-Tensor gravity models by S2 star o...
bySEENET-MTP
 
Elena Mirela Babalic "Generalized alpha-attractor models for hyperbolic surfa...
bySEENET-MTP
 
Dragan Huterer "Novi pogledi na svemir"
bySEENET-MTP
 
Mihai Visinescu "Action-angle variables for geodesic motion on resolved metri...
bySEENET-MTP
 
Sabin Stoica "Double beta decay and neutrino properties"
bySEENET-MTP
 
Yurri Sitenko "Boundary effects for magnetized quantum matter in particle and...
bySEENET-MTP
 
Predrag Milenović "Physics potential of HE/HL-LHC and future circular"
bySEENET-MTP
 
Marija Dimitrijević Ćirić "Matter Fields in SO(2,3)⋆ Model of Noncommutative ...
bySEENET-MTP
 
Zvonimir Vlah "Lagrangian perturbation theory for large scale structure forma...
bySEENET-MTP
 
Vitaly Vanchurin "General relativity from non-equilibrium thermodynamics of q...
bySEENET-MTP
 
Sergey Sibiryakov "Galactic rotation curves vs. ultra-light dark matter: Impl...
bySEENET-MTP
 
Radoslav Rashkov "Integrable structures in low-dimensional holography and cos...
bySEENET-MTP
 
Nikola Godinović "The very high energy gamma ray astronomy"
bySEENET-MTP
 
Miroljub Dugić "The concept of Local Time. Quantum-mechanical and cosmologica...
bySEENET-MTP
 
Cemsinan Deliduman "Astrophysics with Weyl Gravity"
bySEENET-MTP
 
Radu Constantinescu "Scientific research: Excellence in International context"
bySEENET-MTP
 
Loriano Bonora "HS theories from effective actions"
bySEENET-MTP
 

Recently uploaded

PPTX
Control-of-Microorganisms. powerpoint presentation
bykapoyko081
 
PDF
Detection of an NH3 Absorption Band at 2.2μm on Europa
bySérgio Sacani
 
PPTX
Anaerobic respiration (Fermentation, Types, Pasteur effect, Warburg effect, R...
byRashmiMG2
 
PDF
Polymer [ बहुलक ] Chemistry Notes PDF - Irfanullah Mehar - JJ Sir Chemistry.pdf
byWorld of Wisdom
 
PDF
Control and Coordination - Short Notes (Prashant Kirad) (1).pdf
byAnu kumari
 
PPTX
Cell cycle ( mitosis and meiosis)Msc.pptx
byPallabiBehera9
 
PDF
Surface Chemistry ( पृष्ठ रसायन ) - Notes PDF - Irfanullah Mehar - JJ Sir Che...
byWorld of Wisdom
 
PDF
Removal of heavy metals in wastewater .
byrkaviyadharshini555
 
PPT
Lesson 1- Earthquake and Type of Faults.ppt
byMeldredManalon1
 
PDF
Gas-depleted planet formation occurred in the four-planetsystem around the re...
bySérgio Sacani
 
PPTX
TOOLS OF RECOMBINANT DNA TECHNOLOGY.pptx
bysivaranjini82
 
PDF
Nuclear Magnetic Resonance Hindi Medium Notes PDF Download - World of Wisdom.pdf
byWorld of Wisdom
 
PPTX
The Secret Science of MUSIC; A symphony
byneharaj1617
 
PPT
Electric Fields and capacitance AQA A-level Physics
bybensonr1993
 
PPTX
Mount and repair a fishing net for fishi
byelsavant25
 
PPTX
OECD 204 (Acute Dermal Toxicity) .pptx
byManjari36
 
PDF
The Crab Nebula Revisited Using HST/WFC3
bySérgio Sacani
 
PDF
Boyle-Turok Work: A Cosmological Inquiry
byBob Marcus
 
PPTX
VIRULENCE FACTOR OF BACTERIA: STRUCTURAL ELEMENT, ENZYMES, TOXINS...
byBIOSPHERE OF KNOWLEDGE
 
PPTX
Grade 9 - Science Fourth Quarter Physics
byModezaPonje
 
Control-of-Microorganisms. powerpoint presentation
bykapoyko081
 
Detection of an NH3 Absorption Band at 2.2μm on Europa
bySérgio Sacani
 
Anaerobic respiration (Fermentation, Types, Pasteur effect, Warburg effect, R...
byRashmiMG2
 
Polymer [ बहुलक ] Chemistry Notes PDF - Irfanullah Mehar - JJ Sir Chemistry.pdf
byWorld of Wisdom
 
Control and Coordination - Short Notes (Prashant Kirad) (1).pdf
byAnu kumari
 
Cell cycle ( mitosis and meiosis)Msc.pptx
byPallabiBehera9
 
Surface Chemistry ( पृष्ठ रसायन ) - Notes PDF - Irfanullah Mehar - JJ Sir Che...
byWorld of Wisdom
 
Removal of heavy metals in wastewater .
byrkaviyadharshini555
 
Lesson 1- Earthquake and Type of Faults.ppt
byMeldredManalon1
 
Gas-depleted planet formation occurred in the four-planetsystem around the re...
bySérgio Sacani
 
TOOLS OF RECOMBINANT DNA TECHNOLOGY.pptx
bysivaranjini82
 
Nuclear Magnetic Resonance Hindi Medium Notes PDF Download - World of Wisdom.pdf
byWorld of Wisdom
 
The Secret Science of MUSIC; A symphony
byneharaj1617
 
Electric Fields and capacitance AQA A-level Physics
bybensonr1993
 
Mount and repair a fishing net for fishi
byelsavant25
 
OECD 204 (Acute Dermal Toxicity) .pptx
byManjari36
 
The Crab Nebula Revisited Using HST/WFC3
bySérgio Sacani
 
Boyle-Turok Work: A Cosmological Inquiry
byBob Marcus
 
VIRULENCE FACTOR OF BACTERIA: STRUCTURAL ELEMENT, ENZYMES, TOXINS...
byBIOSPHERE OF KNOWLEDGE
 
Grade 9 - Science Fourth Quarter Physics
byModezaPonje
 

Ivan Dimitrijević "Nonlocal cosmology"

  • 1.
    Nonlocal cosmology Ivan Dimitrijevi´c Nonlocal cosmology IvanDimitrijevi´c University of Belgrade, Faculty of Mathematics 14.06.2018.
  • 2.
    Nonlocal cosmology Ivan Dimitrijevi´c Motivation Large cosmologicalobservational findings: High orbital speeds of galaxies in clusters. (F.Zwicky, 1933) High orbital speeds of stars in spiral galaxies. (Vera Rubin, at the end of 1960es) Accelerated expansion of the Universe. (1998)
  • 3.
    Nonlocal cosmology Ivan Dimitrijevi´c Problem solvingapproaches There are two problem solving approaches: Dark matter and energy Modification of Einstein theory of gravity Rµν − 1 2 Rgµν = 8πGTµν − Λgµν, c = 1 where Tµν is stress-energy tensor, gµν are the elements of the metric tensor, Rµν is Ricci tensor and R is scalar curvature of metric.
  • 4.
    Nonlocal cosmology Ivan Dimitrijevi´c Dark matterand energy If Einstein theory of gravity can be applied to the whole Universe then the Universe contains about 4.9% of ordinary matter, 26.8% of dark matter and 68.3% of dark energy. It means that 95.1% of total matter, or energy, represents dark side of the Universe, which nature is unknown. Dark matter is responsible for orbital speeds in galaxies, and dark energy is responsible for accelerated expansion of the Universe.
  • 5.
    Nonlocal cosmology Ivan Dimitrijevi´c Modification ofEinstein theory of gravity Motivation for modification of Einstein theory of gravity The validity of General Relativity on cosmological scale is not confirmed. Dark matter and dark energy are not yet detected in the laboratory experiments. Another cosmological problem is related to the Big Bang singularity. Namely, under rather general conditions, general relativity yields cosmological solutions with zero size of the universe at its beginning, what means an infinite matter density. Note that when physical theory contains singularity, it is not valid in the vicinity of singularity and must be appropriately modified.
  • 6.
    Nonlocal cosmology Ivan Dimitrijevi´c Approaches tomodification of Einstein theory of gravity There are different approaches to modification of Einstein theory of gravity. Einstein General Theory of Relativity From action S = ( R 16πG − Lm − 2Λ) √ −gd4 x using variational methods we get field equations Rµν − 1 2 Rgµν = 8πGTµν − Λgµν, c = 1 where Tµν is stress-energy tensor, gµν are the elements of the metric tensor, Rµν is Ricci tensor and R is scalar curvature of metric. Currently there are mainly two approaches: f(R) Modified Gravity Nonlocal Gravity
  • 7.
    Nonlocal cosmology Ivan Dimitrijevi´c Nonlocal ModifiedGravity Nonlocal gravity is a modification of Einstein general relativity in such way that Einstein-Hilbert action contains a function f( , R). Our action is given by S = d4 x √ −g R − 2Λ 16πG + P(R)F( )Q(R) where = 1√ −g ∂µ √ −ggµν ∂ν, F( ) = ∞ n=0 fn n . We use Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric ds2 = −dt2 + a2 (t) dr2 1−kr2 + r2 dθ2 + r2 sin2 θdφ2 , k ∈ {−1, 0, 1}.
  • 8.
    Nonlocal cosmology Ivan Dimitrijevi´c Equations ofmotion Equation of motion are − 1 2 gµνP(R)F( )Q(R) + RµνW − KµνW + 1 2 Ωµν = − Gµν + Λgµν 16πG , Ωµν = ∞ n=1 fn n−1 l=0 Sµν l P(R), n−1−l Q(R) , Kµν = µ ν − gµν , Sµν(A, B) = gµν α A αB − 2 µA νB + gµνA B, W = P (R)F( )Q(R) + Q (R)F( )P(R).
  • 9.
    Nonlocal cosmology Ivan Dimitrijevi´c Trace and00-equations In case of FRW metric there are two linearly independent equations. The most convenient choice is trace and 00 equations: − 2P(R)F( )Q(R) + RW + 3 W + 1 2 Ω = R − 4Λ 16πG , 1 2 P(R)F( )Q(R) + R00W − K00W + 1 2 Ω00 = − G00 − Λ 16πG , Ω = gµν Ωµν.
  • 10.
    Nonlocal cosmology Ivan Dimitrijevi´c Cosmological solutionsin nonlocal modified gravity We discuss the class of models given by the action S = M R − 2Λ 16πG + P(R)F( )Q(R) √ −g d4 x. In particular, we investigate the following cases: P = R, Q = R, P = R−1 , Q = R, P = Rp , Q = Rq , p ∈ N, q ∈ N, P = (R + R0)m , Q = (R + R0)m , m ∈ R, R0 ∈ R
  • 11.
    Nonlocal cosmology Ivan Dimitrijevi´c First caseP = R, Q = R In this the action becomes S = M R − 2Λ 16πG + RF( )R √ −g d4 x. This model is interesting since we obtain nonsingular cosmological solutions. To solve EOM we use the ansatz: R = rR + s, r, s ∈ R
  • 12.
    Nonlocal cosmology Ivan Dimitrijevi´c Nonsingular bouncecosmological solutions Let the scale factor a(t) be in a form a(t) = a0(σeλt + τe−λt ), a0 > 0, λ, σ, τ ∈ R. Theorem Scale factor a(t) = a0(σeλt + τe−λt ) is a solution of EOM in a following three cases: Case 1. F 2λ2 = 0, F 2λ2 = 0, f0 = − 1 128πGCΛ . Case 2. 3k = 4a2 0Λστ. Case 3. F 2λ2 = 1 192πGCΛ + 2 3 f0, F 2λ2 = 0, k = −4a2 0Λστ. In all cases we have 3λ2 = Λ.
  • 13.
    Nonlocal cosmology Ivan Dimitrijevi´c Summary, P= R, Q = R We studied nonlocal gravity model with cosmological constant Λ, without matter. Using anzac R = rR + s we found three types of nonsingular bounce cosmogical solutions with scale factor a(t) = a0(σeλt + τe−λt ). All solutions satisfy ¨a(t) = λ2 a(t) > 0. There are solutions for all values of parameter k = 0, ±1.
  • 14.
    Nonlocal cosmology Ivan Dimitrijevi´c Second caseP = R−1 , Q = R This model is given by the action S = M R 16πG + R−1 F( )R √ −g d4 x, where F( ) = ∞ n=0 fn n . If we set f0 = − Λ 8πG , then f0 takes the place of cosmological constant.
  • 15.
    Nonlocal cosmology Ivan Dimitrijevi´c Second caseP = R−1 , Q = R This model is given by the action S = M R 16πG + R−1 F( )R √ −g d4 x, where F( ) = ∞ n=0 fn n . If we set f0 = − Λ 8πG , then f0 takes the place of cosmological constant.
  • 16.
    Nonlocal cosmology Ivan Dimitrijevi´c Some power-lawcosmological solutions We use the following form of scale factor and scalar curvature: a(t) = a0|t − t0|α , R(t) = 6 α(2α − 1)(t − t0)−2 + k a2 0 (t − t0)−2α .
  • 17.
    Nonlocal cosmology Ivan Dimitrijevi´c Case k= 0, α = 0 i α = 1 2 Theorem For k = 0, α = 0, α = 1 2 and 3α−1 2 ∈ N scale factor a = a0|t − t0|α is a solution of EOM if f0 = 0, f1 = − 3α(2α − 1) 32πG(3α − 2) , fn = 0 za 2 ≤ n ≤ 3α − 1 2 , fn ∈ R za n > 3α − 1 2 .
  • 18.
    Nonlocal cosmology Ivan Dimitrijevi´c Case k= 0, α → 0 (Minkowski spacetime) Theorem For k = 0 and α → 0 EOM are satisfied if f0, f1 ∈ R, fi = 0, i ≥ 2. Since all fn = 0, n ≥ 2 this model is not a nonlocal model. Thus the power-law solutions cannot be obtained by perturbing the Minkowski solution.
  • 19.
    Nonlocal cosmology Ivan Dimitrijevi´c Case k= 0, α → 1 2 Theorem For k = 0 and α → 1 2 EOM are satisfied if f0 ∈ R, fi = 0, i ≥ 1.
  • 20.
    Nonlocal cosmology Ivan Dimitrijevi´c Case k= 0, α = 1 Let k = 0. To simplify the scalar curvature R(t) = 6 α(2α − 1)(t − t0)−2 + k a2 0 (t − t0)−2α we have three options: α = 0, α = 1 2 i α = 1. The first two does not yield any solutions of EOM and the last one is described in the following theorem. Theorem For k = 0 scale factor a = a0|t − t0| is a solution of EOM if f0 = 0, f1 = −s 64πG , fn ∈ R, n ≥ 2, where s = 6(1 + k a2 0 ).
  • 21.
    Nonlocal cosmology Ivan Dimitrijevi´c Summary, P= R−1 , Q = R We the model with nonlocal term R−1 F( )R and obtained power-law cosmological solutions a(t) = a0|t − t0|α . It is worth noting that there is a solution a(t) = |t − t0| which corresponds to Milne universe for k = −1. All presented solutions a(t) = a0|t − t0|α have scalar curvature R(t) = 6 α(2α − 1)(t − t0)−2 + k a2 0 (t − t0)−2α , that satisfies R = qR2 , where parameter q depends on α.
  • 22.
    Nonlocal cosmology Ivan Dimitrijevi´c Third caseP = Rp , Q = Rq We investigate the model S = M R − 2Λ 16πG + Rp F( )Rq √ −g d4 x. Also, we assume p ≥ q.
  • 23.
    Nonlocal cosmology Ivan Dimitrijevi´c Third caseP = Rp , Q = Rq We discuss only k = 0. Scale factor is chosen in the form a(t) = a0e− γ 12 t2 , γ ∈ R. Hubble parameter and scalar curvature are given by H(t) = − 1 6 γt, R(t) = 1 3 γ(γt2 − 3), R00 = 1 4 (γ − R).
  • 24.
    Nonlocal cosmology Ivan Dimitrijevi´c Third caseP = Rp , Q = Rq Rp = pγRp − p 3 (4p − 5)γ2 Rp−1 − 4 3 p(p − 1)γ3 Rp−2 . From this relation it follows that operator is closed on the space of polynomials in R of degree at most p. In the basis vp = Rp Rp−1 . . . R 1 T the matrix of is Mp = γ       p p 3 (5 − 4p)γ 4 3 p(1 − p)γ2 0 . . . 0 0 p − 1 p−1 3 (9 − 4p)γ 4 3 (1 − p)(p − 2)γ2 . . . 0 ... ... ... ... ... 0 0 0 . . . 1 γ 3 0 0 0 . . . 0 0       .
  • 25.
    Nonlocal cosmology Ivan Dimitrijevi´c EOM Moreover, Fp= ∞ n=0 fnMn p is a matrix of operator F( ). Let Dp be a matrix of operator ∂ ∂R and ep are the coordinates of vector Rp in basis vp. Trace and 00 equation are transformed into T = 0, Z = 0, where T = −2epvpeqFqvq + RWpq − 4γ2 (R + γ)Wpq − 2γ2 Wpq − S1 + 2S2 − R − 4Λ 16πG , Z = 1 2 epvpeqFqvq + γ 4 (γ − R)Wpq − γ(R + γ)Wpq − 1 2 (S1 + S2) + G00 − Λ 16πG ,
  • 26.
    Nonlocal cosmology Ivan Dimitrijevi´c EOM and S1 = 4 3 γ2 (R+ γ) ∞ n=1 fn n−1 l=0 epMl pDpvpeqMn−1−l q Dqvq, S2 = ∞ n=1 fn n−1 l=0 epMl pvpeqMn−l q vq.
  • 27.
    Nonlocal cosmology Ivan Dimitrijevi´c EOM Theorem For allp ∈ N and q ∈ N we have T + 4Z = 4γZ . Trace and 00 equations are equivalent. Therefore, it is sufficient to solve only trace equation. It is of a polynomial type, degree p + q in R, and it splits into p + q + 1 equations with p + q + 1 ”variables” f0 = F(0), F(γ), . . . , F(pγ), F (γ), . . . , F (qγ).
  • 28.
    Nonlocal cosmology Ivan Dimitrijevi´c (p, q)= (1, 1) Theorem For p = q = 1, trace equation is satisfied iff γ = −12Λ, F (γ) = 0 and f0 = 3 32γπG − 8F(γ).
  • 29.
    Nonlocal cosmology Ivan Dimitrijevi´c (p, q)= (1, 1) Theorem Trace equation is satisfied for the following values of parameters p and q (κ = 1 16πG ): p = 2, q = 1: F(γ) = 9κ(γ+9Λ) 112γ3 , F(2γ) = 3κ(γ+9Λ) 56γ3 , f0 = −κ(4γ+15Λ) 7γ3 , F (γ) = −3κ(γ+9Λ) 8γ4 , p = 2, q = 2: F(γ) = 369κ(γ+8Λ) 9344γ4 , F(2γ) = 27κ(γ+8Λ) 4672γ4 , f0 = κ(145γ+576Λ) 876γ4 , F (γ) = −639κ(γ+8Λ) 2336γ5 , F (2γ) = −27κ(γ+8Λ) 9344γ5 , p = 3, q = 1: F(γ) = κ(107γ+408Λ) 6432γ4 , F(2γ) = −κ(173γ+840Λ) 7504γ4 , F(3γ) = 0, f0 = −κ(95γ+768Λ) 268γ4 , F (γ) = −9κ(γ+8Λ) 88γ5 . p = 3, q = 2: F(γ) = 3κ(10702γ+40497Λ) 245680γ5 , F(2γ) = −27κ(6γ+25Λ) 24568γ5 , F(3γ) = −27κ(6γ+25Λ) 49136γ5 , f0 = −3κ(7099γ+23949Λ) 15355γ5 , F (γ) = −3κ(11614γ+68865Λ) 270248γ6 , F (2γ) = 513κ(6γ+25Λ) 171976γ6 ,
  • 30.
    Nonlocal cosmology Ivan Dimitrijevi´c Summary, P= Rp , Q = Rq , p ∈ N, q ∈ N We discuss the scale factor of the form a(t) = a0 exp(− γ 12 t2 ) Trace and 00 equations are equivalent. Trace equation is of a degree p + q in R, so it splits into p + q + 1 equations over p + q + 1 ”variables” f0 = F(0), F(γ), . . . , F(pγ), F (γ), . . . , F (qγ). For p = q = 1 the system has infinitely many solutions, and constants γ and Λ satisfy γ = −12Λ. For other values of p and q, there is unique solution, for any γ ∈ R. We obtained solutions for 1 ≤ q ≤ p ≤ 4.
  • 31.
    Nonlocal cosmology Ivan Dimitrijevi´c Fourth caseP(R) = (R + R0)m , Q(R) = (R + R0)m We take that P(R) = (R + R0)m , Q(R) = (R + R0)m S = M R − 2Λ 16πG + (R + R0)m F( )(R + R0)m √ −g d4 x. Scale factor is in the form a(t) = A tn e− γ 12 t2 .
  • 32.
    Nonlocal cosmology Ivan Dimitrijevi´c Fourth caseP(R) = (R + R0)m , Q(R) = (R + R0)m Also, we take the ansatz (R + R0)m = r(R + R0)m , (1) where R0,r, m i n realne konstante. Ansatz in the expanded for is − 648mn2 (2n − 1)2 (2m − 3n + 1) = 0, − 324n(2n − 1) −γm + 6γmn2 − 4γmn − mnR0 + mR0 + 2n2 r − nr = 0, 18n(2n − 1) 8γ2 m2 − 13γ2 m + 12γ2 mn − 3γmR0 + 24γnr + 6γr − 6rR0 = 0, − 2γ3 m − 24γ3 mn2 − 14γ3 mn + 6γ2 mnR0 + 2γ2 mR0 + 72γ2 n2 r + 12γ2 nr − 24γnrR0 + 3γ2 r − 6γrR0 + 3rR2 0 = 0, − γ2 4γ2 m2 + γ2 m + 18γ2 mn − 3γmR0 − 24γnr − 6γr + 6rR0 = 0, − γ4 (r − γm) = 0. There are five solutions of the above system 1 r = mγ, n = 0, R0 = γ, m = 1 2 2 r = mγ, n = 0, R0 = γ 3 , m = 1 3 r = mγ, n = 1 2 , R0 = 4 3 γ, m = 1 4 r = mγ, n = 1 2 , R0 = 3γ, m = −1 4 5 r = mγ, n = 2m+1 3 , R0 = 7 3 γ, m = 1 2 .
  • 33.
    Nonlocal cosmology Ivan Dimitrijevi´c Case n= 0, m = 1 2 Trace and 00 equations are linear in R, and they split into two systems: γ 16πG + Λ 4πG − γF( γ 2 ) − γ2 3 F ( γ 2 ) = 0, − 108γ2 πG − γ2 3 F( γ 2 ) + γ3 3 F ( γ 2 ) = 0. Λ 16πG − γ 2 F( γ 2 ) + γ2 6 F ( γ 2 ) = 0, 27γ2 πG + γ2 12 F( γ 2 ) + γ3 6 F ( γ 2 ) = 0. The solution of the above system is F( γ 2 ) = 24γ + Λ 768γπG , F ( γ 2 ) = 3Λ − 3γ 16γ2πG .
  • 34.
    Nonlocal cosmology Ivan Dimitrijevi´c Case n= 2 3, m = 1 2 Trace and 00 equations split into the following systems: F ( γ 2 ) = 0, 11γ 48πG + Λ 4πG − γF( γ 2 ) = 0, 1 16πG + F( γ 2 ) = 0, γ2 16πG + γ2 F( γ 2 ) = 0, i F ( γ 2 ) = 0, − γ 24πG − Λ 16πG + 1 2 γF( γ 2 ) = 0, 1 16πG + cF( γ 2 ) = 0, γ2 16πG + cγ2 F( γ 2 ) = 0. The solution is F( γ 2 ) = −1 16πG , F ( γ 2 ) = 0, Λ = − 7 6 γ.
  • 35.
    Nonlocal cosmology Ivan Dimitrijevi´c Symmary, P(R)= (R + R0)m , Q(R) = (R + R0)m We take the scale factor in the form a(t) = Atn exp(− γ 12 t2 ). Ansatz is in the form (R + R0)m = r(R + R0)m . Ansatz has five solutions, from witch four satisfy EOM. In case n = 0, m = 1 2 there is unique solution in F(γ 2 ) and F (γ 2 ). In case n = 2 3 , m = 1 2 there is unique solution in F(γ 2 ) and F (γ 2 ) where Λ = −7 6 γ. In case n = 1 2 , m = −1 4 EOM have no solution.
  • 36.
    Nonlocal cosmology Ivan Dimitrijevi´c References J. Grujic.On a new model of nonlocal modified gravity. Kragujevac Journal of Mathematics, 2015. I. Dimitrijevic Cosmological solutions in modified gravity with monomial nonlocality, Aplied Mathematics and computation 285, 195-203, 2016 Branko Dragovich. On nonlocal modified gravity and cosmology. In Lie Theory and Its Applications in Physics, volume 111 of Springer Proceedings in Mathematics and Statistics, pages 251-262, 2014. Ivan Dimitrijevic, Branko Dragovich, Jelena Grujic, and Zoran Rakic. Constant Curvature Cosmological Solutions in Nonlocal Gravity. In Bunoiu, OM and Avram, N and Popescu, A, editor, TIM 2013 PHYSICS CONFERENCE, volume 1634 of AIP Conference Proceedings, pages 18-23. W Univ Timisoara, Fac Phys, 2014. TIM 2013 Physics Conference, Timisoara, ROMANIA, NOV 21-24, 2013. Ivan Dimitrijevic, Branko Dragovich, Jelena Grujic, and Zoran Rakic On Nonlocal Modified Gravity and its Cosmological Solutions In Lie Theory and Its Applications in Physics, Springer Proceedings in Mathematics and Statistics, 2016.
  • 37.
    Nonlocal cosmology Ivan Dimitrijevi´c References Dimitrijevic, I.,Dragovich, B., Grujic J., Rakic, Z.: New cosmological solutions in nonlocal modified gravity. Rom. Journ. Phys. 58 (5-6), 550559 (2013) [arXiv:1302.2794 [gr-qc]] I. Dimitrijevic, B.Dragovich, J. Grujic, Z. Rakic, “A new model of nonlocal modified gravity”, Publications de l’Institut Mathematique 94 (108), 187–196 (2013). Ivan Dimitrijevic, Branko Dragovich, Jelena Grujic, and Zoran Rakic. On Modified Gravity. Springer Proc.Math.Stat., 36:251-259, 2013. I. Dimitrijevic, B. Dragovich, J. Grujic, and Z. Rakic. Some power-law cosmological solutions in nonlocal modified gravity. In Lie Theory and Its Applications in Physics, volume 111 of Springer Proceedings in Mathematics and Statistics, pages 241-250, 2014. I. Dimitrijevic, B. Dragovich, J. Grujic, and Z. Rakic. Some cosmological solutions of a nonlocal modified gravity. Filomat, 2014.
  • 38.