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The Expanding Universe The expanding universe Lecture 1 The early universe chapters 5 to 8 Particle Astrophysics , D. Perkins, 2nd edition, Oxford 5. The expanding universe 6. Nucleosynthesis and baryogenesis 7. Dark matter and dark energy components 8. Development of structure in early universe Expanding universe : content • part 1 : ΛCDM model ingredients: Hubble flow, cosmological principle, geometry of universe • part 2 : ΛCDM model ingredients: dynamics of expansion, energy density components in universe • Part 3 : observation data – redshifts, SN Ia, CMB, LSS, light element abundances ‐ΛCDM parameter fits • Part 4: radiation density, CMB • Part 5: Particle physics in the early universe, neutrino density • Part 6: matter‐radiation decoupling • Part 7: Big Bang Nucleosynthesis • Part 8: Matter and antimatter 2013‐14 Expanding Universe lect1 3 The ΛCDM cosmological model • Concordance model of cosmology –in agreement with all observations = Standard Model of Big Bang cosmology • ingredients: 9 Universe = homogeneous and isotropic on large scales 9 Universe is expanding with time dependent rate 9 Started from hot Big Bang, followed by short inflation period 9 Is essentially flat over large distances 9 Made up of baryo ns, col d dark matter aadnd a con stant dark energy + small amount of photons and neutrinos 9 Dark energy is related to cosmological constant Λ 2013‐14 Expanding Universe lect1 4 2013‐14 Expanding Universe lect1 5 Lecture 3 Lecture 1 2 e rr Lectu Lecture 3 Lecture 4 © Rubakov 2013‐14 Expanding Universe lect1 6 Part 1 ΛCDM ingredients Hubble expansion – redshift Cosmological principle Geometry of the universe – Robertson‐Walker metric Some distances • Earth‐Sun = AU = 150 x 109 m = 150 x M km • Lightyear = Ly = 0.946 x 1016 m • 1 year = 31.5 106 s • parsec = pc = 3.3 Ly • Mpc = 3.3 MLy ruler in cosmology • Radius Milky Way 15 kpc ª is~ O(Mpc) • Sun to centre MW ª 8 kpc • AdAndrome dagalaxy (M31) to earth ª 800 kpc • Width Local Group of ª 30 galaxies ª 2 Mpc • Average inter‐galactic distance ª Mpc • highest redshift observed : dwarf galaxy at z ª 11 2013‐14 Expanding Universe lect1 8 Hubble law 1 • Hubble(1929): spectral lines of distant galaxies are redshifted fi galaxies move away from Earth • Receding velocity increases with distance : Amount of shift Dl depends on apparent brightness ((distance~distance D) of galaxy Original Hubble plot SuperNovae /s) mm Ia and II locity (k ee V Cepeids Distance (Mpc) 2013‐14 Expanding Universe lect1 9 Hubble law 2 • Interpret redshift z as Doppler effect • For relativistic objects Δλ z = λλobs− em 1+ β β 1 λ ==⎯⎯⎯ λλobs em→+ λ em (1 β) λβem 1− • For close‐by objects, in non‐relativistic limit λ v 11+=zzobs ≈+ββ →≈ = λem c • Hubble law becomes linear relation between velocity and Distance z c ≈ v = HD0 • Confirms theories of Friedmann and Lemaître 2013‐14 Expanding Universe lect1 10 Hubble law 3 • H0 = Hubble constant today – best direct estimate (HST, WFC3) HHt00==±( )( 73.8 2.4) (km/sec) /Mpc t0 = today HkmMpchh0 =→=±100 / sec/i 0.738 0.024 • If H(t) is same at all times → constant and uniform expansion • Observa tions show tha t expansion was not constttant and that Hubble ‘constant’ depends on time • Expansion rate evolves as function of changing energy‐ matter density of universe (see Friedman equations) ρ 2 2 8πGN t ot kc Ht()=−()t 2 3 (Rt( )) 2013‐14 Expanding Universe lect1 11 Sources of redshift • Motion of object with respect to observer – Doppler effect (red and blue shifts): e.g. rotation of stars in galaxies – see galaxy rotation curves (dark matter) • Gravitational redshift : stretching of wavelength close to heavy object ‐ local effect – negligible over long distances • Cosmological redshift : stretching of wavelength due to expansion of universe – dominant at high redshifts λobs (tRt00) ( ) 1+=z = R(t)=Scale of universe at time t λem (tRt) ( ) Also called a(t) 2013‐14 Expanding Universe lect1 12 Cosmological redshift - time • Redshift is the measured quantity – it is related to a given time during the expansion • Large z means small t , or early during the expansion t = 0 tt= Big 0 today Bang z = ∞ z = 0 • Actual (proper) distance D from Earth to distant galaxy at time t Dt( ) =⋅ rR(t ) • D can ooynly be measur ed for near by objects • r = co‐moving coordinate distance –in reference frame co‐ moving with expansion 2013‐14 Expanding Universe lect1 13 Cosmological principle - large scales • universe is isotropic and homogeneous at large scales • There is no preferred position or direction • Expansion is identical for all observers • Is true at scale of inter‐galactic distances : O(Mpc) •2dF quasar survey •Two slices in Ri declination and right g ht ascention •Plot redshift vs right asce n ascension sion •Each dot = galaxy 2013‐14 Expanding Universe lect1 14 Equatorial coordinates 2013‐14 Expanding Universe lect1 15 Cosmic Microwave Background photons WMAP 5 years data ‐ 2.75°K sky Uniform temperature up to ΔT/T ~ 10‐5 2013‐14 Expanding Universe lect1 16 Cosmic Microwave Background photons PLANCK 4 years of data – 2.75°K sky Uniform temperature up to ΔT/T ~ 10‐5 © NASA, ESA 2013‐14 Expanding Universe lect1 17 Structures - small scales • Over short distances universe is clumpy • Galaxy clusters O(10 Mpc) diameter • galaxies, eg Milky Way : 15 kpc COMA cluster: 1000 galaxies 20Mpc diameter M74‐ IR 100 Mpc(330 Mly) from Earth Optical + IR Milky Way 2013COBE‐14 ‐ radio Expanding Universe lect1 18 Olbers’ paradox • Olber (~1800): Why is the sky not bright at night? • Suppose that universe is unlimited & filled uniformly with light sources (stars) ‐ total flux expected is ~ rmax ¾ The sky does not look dark because: • Observable universe has finite age – for constant expansion: lig ht can reach us from maximum rmax = O(c t0 ) • Stars emit light during finite time Dt–flux reduced by factor Δt/t0 • Light is redshifted due to expansion ‐ eg red light Æ IR – finally undetectable flows • CMB fills universe – dark matter does not light up 2013‐14 Expanding Universe lect1 19 Robertson-Walker metric 1 ¾ assume universe = homogeneous and isotrope fluid • Distance between 2 ‘events’ in 4‐dimensional space‐time ds222=−++ c dt⎡ dx 2 dy 2 dz 2⎤ Co‐moving ⎣⎦distance dr ¾ Universe is expanding • scale factor R(t), identical at all locations, only dependent R t on time 2222 2 2 2 ds=++= c dt− ( ︵︵ ︶︶ ) ⎣⎦⎡⎤ dx++ dy dz • proper distance D from Earth to distant galaxy at time t Dt( ) =⋅ rR(t ) • r = co‐moving coordinate distance 2013‐14 Expanding Universe lect1 20 Stretching of the universe scale Example: closed universe R(t ) R(t1) 2 © L. BtöBergström 2013‐14 Expanding Universe lect1 21 Robertson-Walker metric 2 ¾ space is not necessarily flat • Introduce k = spatial curvature of universe • Distance between 2 events in 4‐dim space‐time 2 2 222R t ⎡⎤dr 2 2 2 2 ds=− c dt()⎢ 2 + r() dθθφ +sin d ⎥ ⎣⎦1− kr • k : curvature of space → geometry – obtained from observations –canbe‐1,0,1 • R(t) : dynamics of expansion → Friedman‐Lemaître equations – model dependent 2013‐14 Expanding Universe lect1 22 Curvature k in FLRW model • k = +1, 0, ‐1 depends on space geometry • Curvature is independent of time • Curvature is related to total mass‐energy density of universe kc2 Ωtotal ()t −=22 1 HtRt() () • It affects the evolution of density fluctuations in CMB radiation –pathof photon is different • Therefore it affects pattern of the CMB radiation today • For universe as a whole, over large distances, space‐time seems to be flat: k=0 from CMB observations ρρtot (tt) = crit ( ) 2013‐14 Expanding Universe lect1 23 k from CMB observations movie Curvature of universe CMB photons z = 1100 t=380.000 y model k0k<0 k>0 z = 0 observe k=0 2013‐14 Expanding Universe lect1 24 Global energy CiCosmic dtidestiny geometry density k<0 expands for ever k=0 velocity asymptotically zero k>0 big crunch 2013‐14 Expanding Universe lect1 25 Excercise • D. Perkins, chapter 5 • Oplossing meebrengen op het examen 2013‐14 Expanding Universe lect1 26 Hubble time – Hubble distance • Expansion parameter H(t) has dimension of (time)‐1 • Hubble time = expansion time today for constant expansion 1 9 t0 ==×13.6 10 year H0 • Hubble distance = Horizon distance today in flat static universe DtH ( 00) = ct= 4.2 Gpc • But! Look back time and distance in expanding universe depend on dyna mics in H(t) (see par t 2) ρ 2 2 8πGN t ot kc Ht() =−()t 2 3 (Rt( )) 2013‐14 Expanding Universe lect1 27 Example: matter dominated flat universe • Integration of Friedman equation with k=0 and only non‐ relativistic matter 1 3 2 ⎛⎞9GM 3 R(tt) = ⎜⎟ ⎝⎠2 21 1 Rt( 0 ) 3t0 9 == tyr0 ==×919.110 10 3 H HRt00() 2 0 • Dating of earth crust and old stars shows that age of uueseniverse is of ooderder14 x 109 yr • → Need other energy components (see part 2) 2013‐14 Expanding Universe lect1 28 Part 2 ΛCDM ingredients Dynamics of the expansion – Friedman Lemaître equations Energy density of universe Dynamics of the expansion • Einstein field equations of general relativity 1 RGTRg− ℜΛℜ =−8π GTgΛ μν2 μνN μν μν ⎛⎞Λgμν Cosmological =−8πGTN ⎜⎟μν constant ⎝⎠8πGN Geometry of space‐time Energy‐momentum Robertson‐Walker metric Function of energy content of universe −11 3 -1 -2 Is model dependent GN =×6.67 10 m kg s • Friedmann & Lemaître (1922) : Solutions for uniform and homogeneous universe behaving as perfect frictionless fluid 2013‐14 Expanding Universe lect1 30 Friedmann-Lemaître equation • Time dependent evolution of universe = Friedman‐Lemaître 2 equation 2 2 ⎛⎞Rt( ) 8πG ρ kc H t ≡ = N t ot t − ( ) ⎜⎟ ( ) 2 ⎝⎠Rt() 3 ()R()t kc2 R2 = curvature term ρ t t ot ()= energy density of universe globally, 'at large' Ht( ) = expansion rate = Hubble 'constant' at ti me t H =±H t = 73.8 2.4km /Mpc = Hubble constant today 0 ()(0 )( s ) • energy density ρ is model dependent - ΛCDM model : ρtot (tt) =+ρρbaryon( ) coldDM( tt) ++ ρρ rad( ) DarkEn 2013‐14 Expanding Universe lect1 31 Second Friedman equation • Consider universe = perfect fluid with energy density ρc2 • Conservation of energy in volume element dV ; P = pressure of fluid 23 3 ⎛⎞R dE= − PdV dcRρ ⋅=− PdR 2 ( ) ( ) ρρ=−3()Pc + ⎜⎟2 ⎝⎠Rc • Differentiate 1st Friedman equation3&P substitute ρ + dR ⎛⎞4πGRN ⎛⎞ R ==−⎜⎟⎜⎟c2 Equation of state dt ⎝⎠3 ⎝⎠ • ΛCDM model : Energy density consists of 3 components: matter, radiation (incl.
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