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 • AdAndromed agalaxy (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 1 λ ==⎯⎯⎯ obs em→+ em ( 1 ) λβem 1− λλ λ β • For close‐by objects, in non‐relativistic limit β 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 8πG kc N t ot 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
R2 2 222t ⎡⎤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 Ω ()t −=1 total HtRt22() () • 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 dyn ami cs in H(t) (see par t 2) 2 2 8πG ρ kc N t ot 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 νμ νμν μν μν μν 2 N
⎛⎞μν Λgμν Cosmological =−8πGTN ⎜⎟constant ⎝⎠8 GN Geometry of space‐time Energy‐momentumπ 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 ot ()t = 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 dcR(ρ ⋅=−) PdR( ) 2 ρρ=−3()Pc + ⎜⎟2 ⎝⎠Rc
• Differentiate 1st Friedman equation3& substitute ρ + P dR ⎛⎞4πGRN ⎛⎞ R ==−⎜⎟⎜⎟c2 Equation of state dt ⎝⎠3 ⎝⎠ • ΛCDM model : Energy density consists of 3 components: matter, radiation (incl. relativistic particles), constant vacuum energy Λ ρ =+ρρmatterradiation ρ + vacuum ρ with ρ vac ==Λ 8πGN
2013‐14 Expanding Universe lect1 32 Equations of state 1 1. Matter component: ‘dust’ of non‐relativistic particles (v< gravitational repulsion 2 Pvac =−ρccst = 2013‐14 Expanding Universe lect1 33 Equation of state 2 • Relation between pressure and energy ‐ introduce w P = w ρc2 ⎛⎞R ⎛⎞R 2 2 ρ = −+3(P ρc )⎜⎟2 ρ =−31ρcw( + )⎜⎟2 ⎝⎠Rc ⎝⎠Rc −+31( w) 31+w ρ =∝+const R()1 z ( ) 2 β R ~ t 31()+w • For small t assume that Rt()∼ t 1 radiati on w = 1 2 3 R ∼ t 2 matter w = 0 R ∼ t 3 vacuum w=−1&ρ = const R ∼ eHt 2013‐14 Expanding Universe lect1 34 31+w ρ ∝+(1+ z) ( ) wDarkEnergy =−10210 ± .2 © J. Frieman 2013‐14 Expanding Universe lect1 35 Critical density • flat universe (k=0) = equilibrium between open and close universe • Friedmann equation becomes ⎛⎞Rt 2 ( ) 8πρ G ρ 3H 2 Ht2 ()≡=c N ⇒=c ⎜⎟2 ⎝⎠Rt( ) 38π GN • density ρc is called critical density • Value today is 2 3H0 −−27 3 ρc ()tkgm0 ==×9.6 10 8πGN 23 3 ρccGeVm=≈5.4 5 protons at rest per m • Observations →universe is geometrically flat, so present average density of universe, away from galaxies, is a few protons per m3 2013‐14 Expanding Universe lect1 36 closure parameter • Define closure parameter W Ω t ρ (t) ()= ρc (t) • For general geometry with k≠0, Friedmann equation becomes 222 8πG H 2 ⎛⎞Hkc N =⇒=H ⎜⎟ − 3 ρρ1 = R2 cc⎝⎠Ω t - kc2 ρ ()t kc2 1 = −⇒( ) H2 22 t ρ (tHR) ρ ( ) R2( ) c t • Define curvature term Observations: Ω 2 flat universe k ρK kc ==− Ω(tt)()+Ωk =1 and Ω=1 at ρ H 22R c all times 2013‐14 Expanding Universe lect1 37 Observed energy densities 1 • Energy density is composed of: dust/matter (non‐rel), radiation (incl. rel particles), vacuum energy (L) Ω Ω+Ω =( m + Ω ) +Ω =1 kkr + Ω • Relative contributions unfluence dynamicsΛ of expansion ¾ Present‐day observations: 1. Radiation (from CMB energy density) ρ (t ) rad 0 −5 Ωrad (t0 )=≈×510 ρc ()t0 2. Luminous baryonic matter : p, n, nuclei in stars, gas, dust ρlum Ω=lum <0.01 ρc 2013‐14 Expanding Universe lect1 38 Observed energy densities 2 3. total baryon density – from fit of the model of primordial nucleosynthesis to measurements of light element abundances (chapter 6) −3 nmb ≈ 0.26 baryons −−28 3 ρbb≈×4.0 10kg m ⇒Ω≈ 0.04 4. TtlTotal matter ditdensity – from fit to CMB ddtata ‐ isclumpy in ΛCDM model this is composed of baryons and (mostly cold) dark matter Ωm =Ωb +ΩDM =0.31 5. Neutrino density – relics of early universe – from fit to CMB data Ω≤ν 0010.01 2013‐14 Expanding Universe lect1 39 Observed energy densities 3 5. total density: fit of ΛCDM cosmological model to large set of observation data (CMB, SN, …) – PDG 2013 Ωtotal =−Ω=1k 1.0005 ± 0.0033 k ≈ 0 6. By subtraction we obtain the vacuum energy density non clumpy – homogeneous effect – PDG 2013 ΩΛ =±0.692 0.010 ¾ Conclusions: – 95% of energy content of universe is of unknown type – Today most of it is dark vacuum energy Ω ≈ 0310.31 – See further in chapter 7 m Ω≈Λ 0.69 2013‐14 Expanding Universe lect1 40 Evolution of energy components 1 • Cosmological redshift is related to the expansion rate R(t ) 10+ zzt==0 ( ) 31( +w) Rt() 0 ρ ∝+(1 z) 1. RditiRadiation = phhtotons and reltiitilativisticpartic les Ehν 11 1 ρρcc22= ∼∼⇒ ( ) ∼ VR33λ Rrad R 4 24 ρct() Rt0 ( ) 44 ==+⇒(11zcz) (ρ 2 ) ∼ ( +) 24 rad ρct( 0 ) Rt( ) 2. Non‐relativ istic matter – clmplumpy matter Em 113 ρρccz22=⇒∼∼⇒+( ) ∼∼(1 +) VR33 Rm R 3 2013‐14 Expanding Universe lect1 41 Evolution of energy components 2 3. Vacuum energy is constant in our model ρccst2 = ( )Λ 4. Curvature term 2 −kc 1 2 ρρccz22=⇒( ) ∼∼(1 +) HR22k R 2 • Dynamics of expansion can be rewritten 8πG H 2 (ttttt)= ⎡⎤ρ ( )++ρρ( ) ( ) + ρ( ) 3 ⎣⎦mrΛ k 2 =Htt0 ⎣⎦⎡⎤ ΩΩ+Ω+Ω+Ωmr( ) +Ω( ) +ΩΛ ( tt) +Ωk ( ) =++++Hz2 ⎡⎤Ω t 113 ΩΩt z 42tt++Ω 1z 0 ⎣ mr()00() ()()Λ 0 k 0()⎦ ()() 2013‐14 Expanding Universe lect1 42 Summary 4 radiationρ c2 ∼ (1+ z) 3 matterρ c2 ∼ ()1+ z vacuumρ c2 ∼ cst 2 curvatureρ c2 ∼ ()1+ z 2013‐14 Expanding Universe lect1 43 Lookback time 11dR dR (1+ z) dR H =⇒==dt Rdt RHHR( t) dz 0 dt = − ⎡ 1+ z H ⎤ Rt()0 −1 ⎣()⎦ Rt( )= ⇒=dR Rt( 0 ) 2 dz 1+ z (1+ z) • Time elapsed between emission of a photon at expansion time tE by object at redshift zE and today t 0 0 dz tt0 −=E dt =− ∫∫⎡⎤(1+ z)Hz( ) tE zE ⎣ ⎦ 2013‐14 Expanding Universe lect1 44 age of the universe • Age of universe from integration From Big Bang time ( t=0 and z=∞) to present time ( t=t0 and z=0) t 0 0 dz tdt−=0 == =− 0 ∫∫ 0 ∞ ⎣⎡(1+ z)Ht( )⎦⎤ • Example: flat universe (Wk=0) with Wm=0.24, WΛ=0.76 (WMAP results) tGyr0 =±(13.95 0.4) • Uitwerking meebrengen op examen 2013‐14 Expanding Universe lect1 45 Age of the universe ()Ωmr+Ω +ΩΛ +Ωk =1 Ω r = 0 Ω k = 0 Ω =m 0.24 Ω Ht00 m Λ =−1Ω = 0.76 Ht00=1.026 tG0 = (13.95± 0.4) yr Only matter 2013‐14 Ωm Expanding Universe lect1 46 Particle horizon 1 • Maximum distance accessible via light signals to an observer at given time t0 ? static− flat • In static flat universe DtctGpcH ()00==4.2 • In an expanding FLRW universe, for a photon following a line with fixed θ and φ = ⎡⎤dr 2 2 cRt22dt 2 ds = 0 ( )⎢ 2 ⎥ ⎣1− kr ⎦ • Proper distance travelled by photon emitted at time tE and comoving position r E Dt( 00)=⋅ rE Rt( ) • Observed at time t0 and position r=0 2013‐14 Expanding Universe lect1 47 Particle horizon 2 • Particle horizon for observer at time t0 : tE →0 ⎡⎤rE dddr t0 cdt DH ()tRtRt000==i ( ) ( ) ⎢⎥∫∫002 ⎣⎦1− kr R t () • Expanding, flat, radiation dominated universe 1 Rt() ⎛⎞t 2 = ⎜⎟ DtH ( 00)= 2 ct Rt( 00) ⎝⎠ t • Expanding, flat, matter dominated universe 2 3 Rt() ⎛⎞t Dt( )= 3 ct = ⎜⎟ H 00 Rt( 00) ⎝⎠ t 2013‐14 Expanding Universe lect1 48 Particle horizon 3 • For a more complex cosmological model r dr cdz cI( z) Rt() =− = 0 ∫∫2 0 1− kr HH0 z dz Iz= ()∫ 1 0 ⎡⎤Ω++Ω++Ω+Ω+ttt 11z 34t z tt 1z 22 ⎣ mr( 00)( ) ( )( ) Λ ( 00) K( )( ) ⎦ • For Wk=0, Wm=0.24, WΛ=0.76 and integration to z=∞ BBexpansion c DctGpcH ∼∼3.3= 3.30 14 H0 2013‐14 Expanding Universe lect1 49 Deceleration parameter 1 • Taylor expansion of expansion parameter around t0 (present time)