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The Expanding Universe

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The Expanding Universe The expanding universe Lecture 1 The early universe chapters 5to85 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 2011-12 Expanding Universe 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 bar yon s, col d dark matter aadacostatnd a constant dark energy + small amount of photons and neutrinos 9 Is presently accelerating 2012-13 Expanding Universe 4 2012-13 Expanding Universe 5 © Rubakov 2012-13 Expanding Universe 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) t o 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 2012-13 Expanding Universe 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) 2012-13 Expanding Universe 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 2012-13 Expanding Universe 10 Hubble law 3 • H0 = Hubble constant today – present value (PDG 2012) H00==±Ht() (70.2 1.4) (km/sec) /M pc t0 = today HkmMpchh0 =→=±100 / sec/i 0.702 0.014 • If H(t) is same at all times → constttant an d uniform expansion • Observations show that expansion was not constant 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( )) 2012-13 Expanding Universe 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 :stretchingof: stretching of wavelength close to heavy object - local effect – negligible over long distances • Cosmological redshift :stretchingof: 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) ( ) 2012-13 Expanding Universe 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 - in which 2 objects are at rest 2012-13 Expanding Universe 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 ascention ht asce •Plot redshift vs right n ascension sion •Each dot = galaxy 2012-13 Expanding Universe 14 Cosmic Microwave Background photons WMAP 5 years data Uniform temperature up to ΔT/T ~ 10-5 2012-13 Expanding Universe 15 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 2012-13COBE - radio Expanding Universe 16 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 μ rmax ¾ The sky does not look dark because: • Observable universe has finite age – for constant expansilihtion: light can reach us from maximum rmax = O(c t0 ) • Stars emit light during finite time Dt – flux reduced by factor Dt/t0 • Light is redshifted due to expansion - eg red light Æ IR – finally undetectable flows • CMB fills universe – dark matter does not liggpht up 2012-13 Expanding Universe 17 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 2012-13 Expanding Universe 18 Stretching of the universe scale Example: closed universe R(t ) R(t1) 2 © LBL. Bergs tötröm 2012-13 Expanding Universe 19 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 R2 t ⎡⎤dr 2 ds222 c dt r 2 d 2sin 2 d 2 =−()⎢ 2 +()θθφ + ⎥ ⎣⎦1− kr • k:k : curvature of space → geometry – obtained from observations – can be -1,0,1 • R(t) : dynamics of expansion → Friedman-Lemaître equations – model dependent 2012-13 Expanding Universe 20 Curvature k in FLRW model • k = +1, 0, -1 depends on space geometry • Curvature is independent of time • Curvature is related to global mass-energy density of universe kc2 Ω−=energy− matter ()t 22 1 HtRt() () • It affects the evolution of density fluctuations in CMB radiation – path of 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 ( ) 2012-13 Expanding Universe 21 Global energy CiCosmic dtidestiny geometry density k<0 expands for ever k=0 velocity asymptotically zero k>0 big crunch 2012-13 Expanding Universe 22 Excercise • D. Perkins, chapter 5 • Oplossing meebrengen op het examen 2012-13 Expanding Universe 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 2012-13 Expanding Universe 24 Hubble time – Hubble distance • Expansion parameter H(t) has dimension of (time)-1 • Hubble time = expansion time today for constant expansion 11 9 tt= ⇒ 0 ==×13.9 10 year Ht() H0 • Horizon distance today in flat static universe (Hubble distance) DtH ()00== ct4.2 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( )) 2012-13 Expanding Universe 25 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 ooderder 14 x 109 yr • → Need other energy components (see part 2) 2012-13 Expanding Universe 26 Part 2 ΛCDM ingredients Dyypnamics 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 2012-13 Expanding Universe 28 Friedmann-Lemaître equation • Time dependent evolution of universe = Friedman-Lemaître 2 equation ⎛⎞Rt ρ 2 2 ( ) 8πGN t ot kc H (t ) ≡ ⎜⎟=−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 = 70.2 1.4km /Mpc = Hubble constant today 0 =±()(0 )( s ) • energy density ρ is model dependent - ΛCDM model : ρtot (tt) =+ρρbaryon( ) coldDM( tt) ++ ρρ rad( ) DarkEn 2012-13 Expanding Universe 29 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 3 Pc2 ()()ρρ=−() + ⎜⎟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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