A Model for the Expansion of the Universe

Transcription

1 Issue 2 April PROGRESS IN PHYSICS Volume A Model for he Expansion of he Universe Nilon Penha Silva Deparmeno de Física Reired Professor, Universidade Federal de Minas Gerais, Belo Horizone, MG, Brazil nilon.penha@gmail.com One inroduces an ansaz for he expansion facor a = e H H 0 /β for our Universe in he spiri of he FLRW model; β is a consan o be deermined. Considering ha he ingrediens acing on he Universe expansion > s Gyr are mainly maer baryons plus dar maer and dar energy, one uses he curren measured values of Hubble consan H 0, he Universe curren age, maer densiy parameer Ω m and dar energy parameer Ω Λ ogeher wih he Friedmann equaions o find β = and ha our Universe may have had a negaive expansion acceleraion up o he age T = 3.24 Gyr maer era and posiive afer ha dar energy era, leading o an eernal expansion. An ineracion beween maer and dar energy is found o exis. The deceleraion q has been found o be qt = 0 and q = Inroducion The Cosmological Principle saes ha he Universe is spaially homogeneous and isoropic on sufficienly large scale [ 4] and [7]. This is expressed by he Friedmann spaceime meric: ds 2 = R 2 dψ 2 +R 2 f 2 ψ dθ 2 + sin 2 θ dϕ 2 c 2 d 2, ψ, θ and ϕ are comoving space coordinaes 0 ψ π, f or closed Universe, 0 ψ, f or open and f la Universe, 0 θ π, 0 ϕ 2π, is he proper ime shown by any observer cloc in he comoving sysem. R is he scale facor in unis of disance; acually R is he radius of curvaure of he Universe. The proper ime may be idenified wih he cosmic ime. In erms of he usual expansion facor a = R R, 2 being he curren age of he Universe, equaion becomes ds 2 = R 2 a 2 dψ 2 + f 2 ψ dθ 2 + sin 2 θ dϕ 2 c 2 d 2, f 2 ψ assumes he following expressions: f 2 ψ f 2 ψ = sin2 ψ closed Universe ψ = ψ2 fla Universe f 2 ψ = sinh2 ψ open Universe f 2 0 The expansion process one will be considering here is he one sared by he ime of s Gyr when he so called maer era began. Righ before ha, he Universe wen hrough he so called radiaion era. In his paper one considers only he role of he maer baryonic and non-baryonic and he dar energy Einsein s field equaions Le one uses Einsein s Field Equaions [5], wih he inclusion of he Λ cosmological consan erm. G µν = R µν 2 g µν R = 8πG Tµν + T Λ c 4 µν g µν is he meric ensor, R µν is he Ricci ensor, R is he Ricci scalar curvaure, T µν is he energy-momenum ensor, and, T Λ µν he dar-energy-momenum ensor, 5 T Λ µν = ρ Λ c 2 g µν, 6 ρ Λ = Λc2 8πG ; 7 Λ is he cosmological consan, which will be here allowed o vary wih ime. The meric ensor for he meric above, equaion 3, is g µν = R R 2 f 2 ψ R 2 f 2 ψsin2 θ c 2 The Ricci ensor is given by 8 R = R a. 9 R µν = λ Γ λ µν ν Γ λ µλ + Γη µνγ λ ηλ Γη µλ Γλ ην 0 he Chrisoffel symbols Γ λ µν are Γ λ µν = 2 gλσ µ g σν + ν g σµ σ g µν. The Ricci scalar curvaure is given by R = g µν R µν, 2 Nilon Penha Silva. A Model for he Expansion of he Universe 93

2 Volume PROGRESS IN PHYSICS Issue 2 April and he energy-momenum ensor is T µν = ρ m + c p 2 m u µ u ν + p m g µν, 3 ρ m is he maer densiy and p m is he maer pressure, boh only ime dependen. By maing sraighforward calculaions, one ges R = 6 R 2 a 2 + ȧ 2 c 2 + ä a a = 6 K + ȧ 2 c + ä 4 2 a a. Here K is Gaussian curvaure a cosmic ime : K = The Einsein s field equaions are and G ii = 8πG Tii + Tii Λ c 4 R 2 = R 2 a 2. 5 ȧ 2 6 c2 K + + a 2ä a = 8πG c p 2 m ρ Λ G = 8πG T + T Λ c 4 ȧ 2 3 c2 K + a = 8πG ρ m + ρ Λ 7 i = ψ, θ, ϕ; all off-diagonal erms are null. The equaion of sae for dar energy is p Λ = ρ Λ c 2. 8 Simple manipulaion of equaions above leads o ä a = 4πG 3 ȧ a ρ m + 3 c 2 p m 2ρ Λ, c 2 K = 8πG 3 ρ m + ρ Λ. 20 Equaions 9-20 are nown as Friedmann equaions. Having in accoun ha ȧ = H, 2 a ä a = Ḣ + H2, 22 H is ime dependen Hubble parameer, and ha pressure p m = 0 maer is reaed as dus, one has Ḣ + H 2 = 8πG 3 2 ρ m + ρ Λ, 23 or c 2 K + H 2 = 8πG 3 ρ m + ρ Λ, 24 Ḣ H 2 + = 2 ρ ρ m + ρ Λ, 25 cri c 2 K H 2 + = ρ cri ρ m + ρ Λ, 26 ρ cri = 3H2 8πG 27 is he so called criical densiy. From equaions one obains, afer simple algebra, or, ρ m = c 2 K Ḣ, 4πG 28 ρ Λ = 4πG 2 c2 K H2 + Ḣ, 29 2 c 2 K Ω m = 3 H 2 2 Ḣ, 3 H 2 30 c 2 K Ω Λ = 3 H Ḣ 3 H 2 +, 3 Ω m = ρ m /ρ cri and Ω Λ = ρ Λ /ρ cri are, respecively, he cosmological maer and dar energy densiy parameers. The Ricci scalar curvaure sands as R = 6 K + 2H 2 + Ḣ. 32 c 2 3 The ansaz Now le one inroduces he following ansaz for he expansion facor: a = e H H 0 /β 33 is he curren age of he Universe, H 0 = H is he Hubble consan, and β is a consan parameer o be deermined. From equaions 2-23 one obains β H = H 0 34 Ḣ = H β. 35 By insering equaions ino equaion 25 one has: β H 0 β + = ρ cri 2 ρ m + ρ Λ 36 β β = H 0 2 Ω m + Ω Λ Nilon Penha Silva. A Model for he Expansion of he Universe

3 Issue 2 April PROGRESS IN PHYSICS Volume Fig. : a = e β β T H 0 0 Fig. 3: ȧ = a H 0 β Fig. 2: H = H 0 β Fig. 4: ä = a H β 0 β H β 0 Since β is assumed o be a consan, and, ha Ω m, Ω Λ and H = H 0 are measured quaniies, one has for =, β H 0 = 2 Ω m + Ω Λ 38 which solved for β gives β = + H 0 2 Ω m + Ω Λ = H 0 = ms Mpc = Gyr, = Gyr, Ω m = and Ω Λ = [6]. The plo of he expansion acceleraion ä = Ḣ + H 2 a 40 as funcion of = age of he Universe reveals ha for < T, he acceleraion is negaive and for > T, he acceleraion is posiive. See Figure 4. This means ha when he Universe is younger han T, he regular graviaion overcomes dar energy, and afer T, dar energy overcomes graviaion. The resul is an eernal posiive acceleraed expansion afer T = 3.24 Gyr. See ahead. Fig. 5: q = + H 0 β β In fac, by seing equaion 40 o zero and jus solving i for, one ges H β + H2 = 0, 4 β β = T = = 3.24 Gyr. 42 H 0 From equaion 26, one wries c 2 R 2 H 2 = Ω m + Ω Λ. 43 The nown recenly measured values of Ω m and Ω Λ [6] do no allow one o say, from above equaion, ha he Nilon Penha Silva. A Model for he Expansion of he Universe 95

4 Volume PROGRESS IN PHYSICS Issue 2 April Fig. 6: Lef hand side of equaion 43 is ploed for some values of R. A he curren Universe age = Gyr, he righ side of he referred equaion has he margin of error equal o ±0.09. Fig. 8: Maer and dar energy densiy parameers for 2 =, 0, : Ω m = 3H 2 c 2 K β H ; ΩΛ = 3H 2 c 2 K + 2 β H + 3H 2. The radius of curvaure is aen as R = 02 Gly. Fig. 7: Gaussian curvaure K = and Ricci scalar curvaure R = 6 K + H 2H + β. R a 2 c 2 Universe is clearly fla = 0. The referred measured values have a margin of error: { Ω Λ = { Ω m = Considering also he margin of errors of he oher measured parameers [6], one canno disinguish beween =, or 0. The mach beween boh sides of equaions 43 requires ha, he oday s curvaure radius of he Universe be R > 00 Gly, in he conex of his paper. See Figure 6. The so called deceleraion parameer is q = äa ȧ 2 = Ḣ H 2 + = + β β H 0 44 which, a curren Universe age is q = See Figure 5. The expansion scalar facor a, Hubble parameer H, expansion speed ȧ, expansion acceleraion ä, and he deceleraion parameer q are ploed in Figures -5. Fig. 9: Maer and dar energy densiies for =, 0, : 6sssmmm ρ m = 2 8πG c 2 K β H ; ρ Λ = 8πG c 2 K + 2 β H + 3H 2. The radius of curvaure is aen as R = 02 Gly. The sequence of Figures 7-0 shows he Gaussian K and R curvaures, maer densiy parameer Ω m, dar energy densiy parameer Ω Λ, maer densiy ρ m, dar energy densiy ρ Λ and cosmological dar energy Λ. The ime derivaives of ρ Λ and ρ m reveal ineresing deail of he model in quesion: ρ m + 3H ρ m + c p 2 m = ρ m + 3Hρ m = Q 45 ρ Λ + 3H ρ Λ + c p 2 Λ = ρ Λ = Q 46 Q = 2H β 2β + 3Ḣ 2 c2 K p m = 0 and p Λ = ρ Λ c 2. This implies ha 47 ρ m + ρ Λ = 3Hρ m. 48 The wo perfec fluids inerac wih each oher. In Figure one shows he plos for ρ m, ρ Λ and ρ m + ρ Λ as funcions of cosmic ime. 96 Nilon Penha Silva. A Model for he Expansion of he Universe

5 Issue 2 April PROGRESS IN PHYSICS Volume Islam J.N. An Inroducion o Mahemaical Cosmology. Cambridge Universiy Press Baryshev, Y.V. Expanding space: The roo of concepual problems of he cosmological physics. arxiv: gr-qc/ Benne, C.L. e al. Nine-Year Wilinson Microwave Anisoropy Probe WMAP Observaions: Final Maps and Resuls. arxiv: asro-ph.co Ellis, G.F.R. e al. Relaivisic Cosmology. Cambridge Universiy Press, 202. Fig. 0: Dar energy Λ, in unis of cm 2 for =, 0,. Λ = 8πG ρ c 2 Λ. The radius of curvaure is aen as R = 02 Gly. The resul for Λ saisfies he following inequaliy: Λ < 0 42 cm 2 [4]. Fig. : Time derivaives of ρ Λ, ρ m and of he sum ρ Λ + ρ m for =, 0,. The radius of curvaure is aen as R = 02 Gly. 4 Conclusion H 0 T β 0 β, The expression for he expansion facor a = e β = , consiues a model for he expansion of he Universe for > s Gyr in which graviy dominaes up o he Universe age of T = 3.24 Gyr and afer ha dar energy overcomes graviy and ha persiss forever. The acceleraion of expansion is negaive in he firs par maer era and posiive afer ha dar energy era. The mahemaical expressions for dar energy and maer densiies indicae a clear ineracion beween he wo perfec fluids dar energy and maer. The classical deceleraion parameer q is found o have he value q = for he curren Universe age and he curren radius of curvaure should be R > 00 Gly. References Submied on March 2, 204 / Acceped on March 4, 204. Raine, D. An Inroducion o he Science Of Cosmology. Insiue of Physics Publishing Ld, Peacoc, J.A. Cosmological Physics. Cambridge Universiy Press, Harrison, E.R. Cosmology: The Science of he Universe. Cambridge Universiy Press, 2 nd ed Nilon Penha Silva. A Model for he Expansion of he Universe 97