Friedman(n) Models of the Universe. Max Camenzind Modern Cosmology Oct-13-D9

Transcription

1 Friedman(n) Models of the Universe Max Camenzind Modern Cosmology Oct-13-D9

2 Topics Matter in the Universe? DM, B, DE, phot The 2 Friedmann equations Equation of state EoS for matter parameter w Density evolution as a function of w. Expansion of the Universe as a function of w. Age of the Universe, Age as a function of redshift, Look-back Time. Luminosity distances for particular models how to measure the Expansion with SN Ia. Apparent angular diameters in the expanding Universe. The Universe of Dark Energy

3 Matter of the Universe Universe is expanding. Components of the Universe are: Universe is 13.7 Billion years old. Expansion is currently accelerating.

4 Dynamics: Friedmann Equs Einstein s field equations: 2 Friedmann equations. couples expansion rate with matter content. Solution will depend on the EoS: Matter dominance. Vacuum dominance. Radiation dominance.

5 Matter in Equilibrium Fermions Bosons

6 Equilibrium Distributions

7 Relativistic Equilibrium Distributions Bosons Fermions Riemann zeta function

8 From Einstein sequations to Friedmann-Equations Gµν = Rµν 1/2 δµν R = 8π G Tµν G00 = - 3/a2 (å2 + k) Gki = - (2aä + å2 + k)/a² δki Tµν = diag[-ρ(t), P(t), P(t), P(t)] = Energy-Momentum-Tensor 3 (å2 + k) /a2 = 8 π G ρ(t) + c²λ ä /a = - 4πG(ρ + 3P/c²)/3 + c²λ/3 Einstein s Field Eqs GR Friedmann-Equs Look for P(ρ) to solve a(t)

9 Spin Connection and Curvature Exterior derivative for 1-forms: d² = 0 β = f(x) α dβ = df α + f dα Cartan s equations: 0 = dθa + ωab θb : 1st struct equ Ωab = dωab + ωac ωcb : 2nd struct equ Ωab = ½ Rabcd θc θd : 2-form

10 Friedmann Derivation

11 Friedmann Derivation

12 Friedmann Derivation

13 Friedmann Derivation

14 Einstein Tensor for FLRW Metric

15 Friedmann Derivation

16 Raychouduri Equation

17 Energy Conservation 1st Law of Thermodynamics Use Friedmann equations (c=1): to show that d/dt(ρa3) = - P (d/dt)a3 du = -P dv, ds = 0 for EoS of the form P = w ρ we then find a3 dρ = -(w+1) 3 ρ a2 da provided w=constant ρ ~ a-3(w+1)

18 The Cosmological Equation of State EoS Defined via the EoS: P=wρ is known Cosmic Equation of State (EoS) Particular EoS: w=0 P=0 : Dark Matter (DM), Baryons (B) well satisfied for non-relativistic Baryons! w=1/3 Radiation, masseless Neutrinos w=-1 Vacuum Energy, looks like a cosmological constant.

19 General Density Evolution ρ ~ a-3(w+1) Matter dominated (w=0): Radiation dominated (w=1/3): Cosmological Constant (w=-1): Dark Energy with w<-1, eg w=-2: ρ ~ a-3 ρ ~ a-4 ρ = const ρ ~ a3 Energy density would increase with expansion! would dominate even matter, which consists of normal Elements! ( socalled Big Rip ). w < -1 is therefore quite improbable. -1 < w < -1/3 is however possible.

20 Density Evolution; 1+z=1/a Dark Energy aeq

21 Density Evolution as Func of a ρtot = ρr + ρm + ρde ρr = ρr 0 a-4, since ρ ~ a-4 and a = 1 today ρr0 present radiation energy density Ωr0 ρr0 / ρcrit via Definition Index 0 is usually just omitted Ωr0 Therefore ρr = ρcrit Ωr a-4 and similarly for ρm, ρde For this, Density Evolution in F-Equ ρtot = ρcrit [ Ωr a-4 + Ωm a-3 + ΩDE a-3(1+w)] if w for DE constant

22 1st Friedmann Equation (å/a)2 = 8 π G ρ(t) /3 kc²/a² ρ(t) = ρcrit [ Ωr a-4 + Ωm a-3 + ΩDE a-3(1+w)] Ωk = 0 w = const Replace in Friedmann-equ., H0 (å/a)0: (å/a)2 = H02 ( Ωr a-4 + Ωm a-3 + ΩDE a-3(1+w) )

23 Cosmo Density Parameters Hubble-Radius RH = c/h0 = 4200 Mpc The Universe appears flat, whenever R0 > 10 RH Ωk ~ 0 Fundamental Plane of Cosmology

24 Density Evolution; 1+z=1/a DE Dominated Future Matter dominated Radiation dominated

25 Fundamental Plane of Cosmology Each Point is a Cosmological Model

26 Parameters Friedmann Universe (i) Hubble Constant H0; (ii) Density parameter of non-relativistic Matter : Ωm = ΩDM + ΩB. (iii) Curvature parameter: Ωk = -k RH²/R0². Inflation R0 >> RH LCDM-Model. (iv) Parameter of Dark Energy: ΩDE. (v) Equation of state of Dark Energy: w ~ -1 ( Vacuum Energy ).

27 Models of Expanding Universe a ~ t2/(3(w+1)) Matter-dominated (w=0): a ~ t2/3 Einstein-deSitter Is decelerating Radiation-dominated (w=1/3): a ~ t1/2 Is decelerating Cosmological Vacuum (w=-1): a ~ eλ t (desitter) Is accelerating When is the transition? w > -1/3 Deceleration w < -1/3 Acceleration

28 Classical Models without DE Ωm < 1: OCDM Expands forever k = -1 Ωm = 1: SCDM k=0 Ωm > 1 k = +1 Big Bang Collapsing later Big Crunch

29 Einstein-deSitter Universe Flat Model (k=0) Einstein-deSitter Universum R t 2/3 R 2 H0 = = R 3t0 R 2 t0 = = 9.3 Gyrs 3H 0 t

30 desitter Universe - DE Universe without matter, only with Λ De Sitter 1917: the first expanding Universe Λ a (t ) = a (t ) 3 a (t ) e H = H t Λ / 3 = const Exponentially accelerated Expansion

31 desitter Solutions

32 Summary Only Relativistic Cosmos is correct FLRW 2 Friedmann-Equations determine the Expansion of the Universe for given EoS. Matter in the present Universe consists of various components: Baryons, Photons, Neutrinos, Dark Matter and Dark Energy. Expressed in terms of Omega-parameters. Since 1997, Supernovae observations give evidence for an accelerated expansion Dark Energy is therefore required and Ωk ~ 0. EoS w of Dark Energy is one of crucial problem.