Structures in the early Universe. Particle Astrophysics chapter 8 Lecture 4

Transcription

1 Structures in the early Universe Particle Astrophysics chapter 8 Lecture 4

2 overview Part 1: problems in Standard Model of Cosmology: horizon and flatness problems presence of structures Part : Need for an exponential expansion in the early universe Part 3 : Inflation scenarios scalar inflaton field Part 4 : Primordial fluctuations in inflaton field Part 5 : Growth of density fluctuations during radiation and matter dominated eras Part 6 : CMB observations related to primordial fluctuations Structures in early Universe

3 problems in Standard Model of Cosmology related to initial conditions required: horizon problem flatness problem PART 1: PROBLEMS IN STANDARD COSMOLOGICAL MODEL Structures in early Universe 3

4 01-13 Structures in early Universe 4

5 The ΛCDM cosmological model Concordance model of cosmology in agreement with all observations = Standard Model of Big Bang cosmology ingredients: Universe = homogeneous and isotropic on large scales Universe is expanding with time dependent rate Started from hot Big Bang, followed by short inflation period Is essentially flat over large distances Made up of baryons, cold dark matter and a constant dark energy + small amount of photons and neutrinos Is presently accelerating Lecture Expanding Universe 5

6 The ΛCDM cosmological model Concordance model of cosmology in agreement with all observations = Standard Model of Big Bang cosmology Lecture 1 ingredients: Universe = homogeneous and isotropic on large scales Universe is expanding with time dependent rate Started from hot Big Bang, followed by short inflation period Is essentially Q: Which flat mechanism over large caused distances the exponential inflation? Made up of baryons, cold A: scalar dark matter inflaton and field a constant dark energy + small amount of photons and neutrinos Is Exponential presently accelerating inflation takes care of horizon and flatness problems Parts 1,, Expanding Universe 6

7 The ΛCDM cosmological model Concordance model of cosmology in agreement with all observations = Standard Model of Big Bang cosmology Lecture 1 ingredients: Universe = homogeneous and isotropic on large scales Universe is expanding with time dependent rate Started Structures from hot observed Big Bang, in galaxy followed surveys by short and inflation anisotropies period in CMB Is essentially flat over large distances observations: Made up of baryons, Q: What cold dark seeded matter these and structures? a constant dark energy + small amount A: quantum of photons fluctuations and neutrinos primordial universe Is presently accelerating Parts 4, 5, Expanding Universe 7

8 Horizon in static universe Particle horizon = distance over which one can observe a particle by exchange of a light signal v=c Particle and observer are causally connected In flat universe with k=0 : on very large scale light travels nearly in straight line In a static universe of age t photon emitted at time t=0 would travel t < 14 Gyr STATIC DH Would give as horizon distance today the Hubble radius STATIC H Structures in early Universe 8 ct D t0 ct0 4.Gpc

9 Horizon in expanding universe In expanding universe, particle horizon for observer at t 0 t0 Rt0 DH t0 cdt Lecture 1 0 Rt Expanding, flat, radiation dominated universe Rt Rt 0 Expanding, flat, matter dominated universe Rt 0 expanding, flat, with m =0.4, Λ =0.76 t t t t D t ct H 0 0 Rt D t0 3ct0 H D t 3.3ct H 0 0 ~ 14Gpc Structures in early Universe 9

10 Horizon problem At time of radiation-matter decoupling the particle horizon was much smaller than now Causal connection between photons was only possible within this horizon We expect that there was thermal equilibrium only within this horizon Angle subtented today by horizon size at decoupling is about 1 Why is CMB uniform (up to factor 10-5 ) over much larger angles, i.e. over full sky? Answer : short exponential inflation before decoupling Structures in early Universe 10

11 Horizon at time of decoupling Age of universe at matter-radiation decoupling tdec Optical horizon at decoupling This distance measured today angle subtended today in flat matter dominated universe y z dec 1100 D t ct dec H dec H dec dec D t0 ctdec 1 z dec dec dec H D t ct 1 z 0 dec D t 3c t t H 0 0 dec dec Structures in early Universe 11

12 01-13 Structures in early Universe 1

13 Flatness problem 1 universe is flat today, but was much closer to being flat in early universe How come that curvature was so finely tuned to Ω=1 in very early universe? Friedman equation rewritten t 1 R t kc H t R t t n radiation domination matter domination t t 1~ 1~ t t 3 At time of decoupling At GUT time (10-34 s) 1 ~ 10 1 ~ Structures in early Universe 13

14 01-13 Structures in early Universe 14

15 Flatness problem How could Ω have been so closely tuned to 1 in early universe? Standard Cosmology Model does not contain a mechanisme to explain the extreme flatness The solution is INFLATION Big Bang Planck scale? GUT scale today t s t s Structures in early Universe 15

16 Horizon problem Flatness problem Non-homogenous universe The solution is INFLATION Structures in early Universe 16

17 The need for an exponential expansion in the very early universe PART EXPONENTIAL EXPANSION Structures in early Universe 17

18 Steady state expansion Radiation dominated Described by SM of cosmology Exponential expansion inflation phase Rubakov Structures in early Universe 18

19 Constant energy density & exponential expansion Assume that at Planck scale the energy density is given by a cosmological constant 8 G A constant energy density causes an exponential expansion For instance between t 1 and t in flat universe tot H R 8 R G 3 3 R =e H t -t R Structures in early Universe 19

20 When does inflation happen? Inflation period Structures in early Universe 0

21 When does inflation happen? Grand Unification at GUT energy scale GUT time couplings of Electromagnetic Weak Strong interactions are the same At end of GUT era : spontaneous symmetry breaking into electroweak and strong interactions Next, reheating and production of particles and radiation Start of Big Bang expansion with radiation domination kt GUT t GUT 16 ~10 GeV 34 ~ 10 sec Structures in early Universe 1

22 How much inflation is needed? 1 Calculate backwards size of universe at GUT scale after GUT time universe is dominated by radiation If today R t t 1 R t R t GUT t t GUT R t0 ~ D static H t0 ct0 c 4 10 s ~ 4 10 m 1 Then we expect R t GUT ~ R t On the other hand horizon distance at GUT time was D ~ ~ 10 H s 4 10 s tgut ctgut m EXPECTED R t ~1m Structures in early Universe GUT

23 01-13 Structures in early Universe 3

24 How much inflation is needed? It is postulated that before inflation the universe radius was smaller than the horizon distance at GUT time Size of universe before inflation R1 10 Inflation needs to increase size of universe by factor 6 m R GUT 1m R start inflation 10 m e H t t1 60 R R 1 e H t -t 1 At least 60 e-folds are needed Structures in early Universe 4

25 horizon problem is solved Whole space was in causal contact and thermal equilibrium before inflation Some points got disconnected during exponential expansion entered into our horizon again at later stages horizon problem solved Structures in early Universe 5

26 Scientific American Feb Structures in early Universe 6

27 Flatness problem is solved If curvature term at start of inflation is roughly 1 Then at the end of inflation the curvature R t is even closer to 1 kc H R t kc 1 H R t kc H t 1 1 O 1 R =e 10 1 R at t 10 1 s 1 H t1 -t Structures in early Universe 7

28 Conclusion so far Constant energy density yields exponential expansion This solves horizon and flatness problems What causes the exponential inflation? Structures in early Universe 8

29 Scalar inflaton field PART 3 INFLATION SCENARIOS Structures in early Universe 9

30 Introduce a scalar inflaton field Physical mechanism underlying inflation is unknown Postulate by Guth (1981): inflation is caused by an unstable scalar field present in the very early universe dominates during GUT era (kt GeV) spatially homogeneous depends only on time : (t) inflaton field describes a scalar particle with mass m Dynamics is analogue to ball rolling from hill Scalar has kinetic and potential energy Structures in early Universe 30

31 V(Φ) Chaotic inflation scenario (Linde, 198) Inflation starts at different field values in different locations Inflaton field is displaced from true minimum by some arbitrary mechanism, probably quantum fluctuations Total energy density associated with scalar field 1 c V During inflation the potential V( ) is only slowly varying with slow roll approximation Kinetic energy can be neglected Lineweaver inflation Φ reheating Structures in early Universe 31

32 Slow roll leads to exponential expansion Total energy in universe = potential energy of field c V c ~ V ~ cst Friedman equation for flat universe becomes H H R R 8 c 3 M PL 8 G 3 V H ~ cst Exponential expansion Structures in early Universe 3

33 Reheating at the end of inflation Lagrangian energy of inflaton field 3 L T V R V Mechanics (Euler-Lagrange) : equation of motion of inflaton field dv 3H t 0 d Frictional force due to expansion = oscillator/pendulum motion with damping field oscilllates around minimum stops oscillating = reheating Structures in early Universe 33

34 V(Φ) Reheating and particle creation Inflation ends when field reaches minimum of potential Transformation of potential energy to kinetic energy = reheating Scalar field decays in particles Lineweaver inflation Φ reheating Structures in early Universe 34

35 Summary Planck era When the field reaches the minimum, potential energy is transformed in kinetic energy This is reheating phase Relativistic particles are created Expansion is now radiation dominated Hot Big Bang evolution starts R GUT era Structures in early Universe 35 t kt

36 V(Φ) Chaotic inflation scenario (Linde, 198) Inflation starts at different field values in different locations Inflaton field is displaced from true minimum by some arbitrary mechanism, probably quantum fluctuations Total energy density associated with scalar field 1 c V During inflation the potential V( ) is only slowly varying with slow roll approximation Kinetic energy can be neglected Lineweaver inflation Φ reheating Structures in early Universe 36

37 01-13 Structures in early Universe 37

38 01-13 Structures in early Universe 38

39 Quantum fluctuations in early universe as seed for presentday structures: CMB and galaxies PART 4 PRIMORDIAL FLUCTUATIONS Structures in early Universe 39

40 quantum fluctuations in inflaton field Field starting in B needs more time to reach minimum than field starting in A Potential should vary slowly Reaches minimum Δt later Quantum fluctuations set randomly starting value in A or B Structures in early Universe 40

41 Quantum fluctuations as seed 1 Introduce quantum fluctuations in inflaton field This yields fluctuations in inflation end time Δt Inflation will end at different times in different locations of universe Hence : Fluctuations in kinetic energy at end of inflation Ekin c T t Hence different density of matter at different locations And different temperatures Kinetic Energy Potential Energy Structures in early Universe 41

42 Quantum fluctuations as seed Due to rapid expansion, particles and anti-particles are inflated away from each other One of (anti-)particles moves out of horizon no causal contact with other (anti-)particle no annihilation Energy balance is restored during reheating Fluctuations can be treated as classical perturbations & superposition of waves Structures in early Universe 4

43 Perturbations in the cosmic fluid density fluctuations modify the energy-momentum tensor and therefore also the curvature of space Change in curvature of space-time yields change in gravitation potential Particles and radiation created during reheating will fall in gravitational potential wells These perturbations remain as such till matter-radiation decoupling are imprinted in CMB Structures in early Universe 43 T T ~ 10 5 x

44 Gravitational potential fluctuations gravitational potential Φ due to energy density ρ spherical symmetric case cases G r 3 Global background potential from average field within horizon Potential change due to fluctuation Δρ over arbitrary scale λ r G 3 H Fractional perturbation in gravitational potential at certain location H G Structures in early Universe 44

45 Spectrum of fluctuations During inflation Universe is effectively in stationary state Fluctuation amplitudes will not depend on space and time ~ cst H ~ cst Define rms amplitude of fluctuation rms H 1 Amplitude is independent of horizon size independent of epoch at which it enters horizon is scale-invariant Structures in early Universe 45

46 Clustering of particles at end of inflation Evolution of fluctuations during radiation dominated era Evolution of fluctuations during matter dominated era Damping effects PART 5 GROWTH OF STRUCTURE Structures in early Universe 46

47 Introduction We assume that the structures observed today in the CMB originate from tiny fluctuations in the cosmic fluid during the inflationary phase These perturbations grow in the expanding universe Will the primordial fluctuations of matter density survive the expansion? Structures in early Universe 47

48 Jeans length = critical dimension After reheating matter condensates around primordial density fluctuations universe = gas of relativistic particles x Consider a cloud of gas around fluctuation Δρ When will this lead to condensation? 1 Compare cloud size L to Jeans length λ J J v s G Depends on density and sound speed v s Sound wave in gas = pressure wave Sound velocity in an ideal gas v s P C C p V Structures in early Universe 48

49 Jeans length = critical dimension Compare cloud size L with Jeans length when L << λ J : sound wave travel time < gravitational collaps time no condensation L t v s When L >> λ J : sound cannot travel fast enough from one side to the other cloud condenses around the density perturbations Start of structure formation Structures in early Universe 49

50 Evolution of fluctuations in radiation era After inflation : photons and relativistic particles Radiation dominated till decoupling at z 1100 Velocity of sound is relativistic P c 3 horizon distance ~ Jeans length : no condensation of matter DH t ct v s λj Density fluctuations inside horizon survive fluctuations continue to grow in amplitude as R(t) 5 3 P v s ~ G P v 1 1 ct 3 9 s c Structures in early Universe 50

51 Evolution of fluctuations in matter era After decoupling: dominated by non-relativistic matter Neutral atoms (H) are formed sound velocity drops v s ideal gas 5 P 5 kt 3 3 m v s 5kT c 3M c mat H Jeans length becomes smaller 1 Horizon grows therefore faster gravitational collapse Matter structures can grow J v s G Cold dark matter plays vital role in growth of structures Structures in early Universe 51

52 Damping by photons Fluctuations are most probably adiabatic: baryon and photon densities fluctuate together Photons have nearly light velocity - have higher energy density than baryons and dark matter If time to stream away from denser region of size λ is shorter than life of universe move to regions of low density Result : reduction of density contrast diffusion damping (Silk damping) Structures in early Universe 5

53 Damping by neutrinos Neutrinos = relativistic hot dark matter In early universe: same number of neutrinos as photons Have only weak interactions no interaction with matter as soon as kt below 3 MeV Stream away from denser regions = collisionless damping Velocity close to c : stream up to horizon new perturbations coming into horizon are not able to grow iron out fluctuations Structures in early Universe 53

54 Observations lumpy smooth rms CMB λ (Mpc) Structures in early Universe 54

55 From density contrast to temperature anisotropy Angular power spectrum of anisotropies PART 6 OBSERVATIONS IN CMB Structures in early Universe 55

56 CMB temperature map dipole T 10 T 3 O mk T 5 10 O µk T Contrast enhanced to make 10-5 variations visible! Structures in early Universe 56

57 Small angle anisotropies Adiabatic fluctuations at end of radiation domination : photons & matter fluctuate together Photons keep imprint of density contrast T T Adiab horizon at decoupling is seen now within an angle of 1 temperature correlations at angles < 1 are due to primordial density fluctuations Seen as acoustic peaks in CMB angular power distribution Peaks should all have same amplitude but: Silk damping by photons Structures in early Universe 57

58 1 rms Scientific American Feb 004 x Structures in early Universe 58

59 Angular spectrum of anisotropies 1 CMB map = map of temperatures after subtraction of dipole and galactic emission extra-galactic photons Statistical analysis of the temperature variations in CMB map multipole analysis In given direction n measure difference with average of.7k T n T n T 0 C(θ) = correlation between temperature fluctuations in directions (n,m) separated by angle θ average over all pairs with given θ Structures in early Universe 59

60 Angular spectrum of anisotropies Expand in Legendre polynomials, integrate over C l describes intensity of correlations Temperature fluctuations ΔT are superposition of fluctuations at different scales λ Decompose in spherical harmonics C l T T 0, a lm 1 C l 1 C l Pl cos 4 l l 0 m l a Y lm 1 l 1 lm l m a, lm Structures in early Universe 60

61 Angular spectrum of CMB anisotropies Coefficients C l = angular power spectrum - measure level of structure found at angular separation of Large l means small angles l = 00 corresponds to 1 = horizon at decoupling as seen today Amplitudes seen today depend on: primordial density fluctuations Baryon/photon ratio Amount of dark matter. 00 degrees Fit of C l as function of l yields cosmological parameters Structures in early Universe 61

62 Physics of Young universe Large wavelengths Physics of primordial fluctuations Short wavelengths Silk damping WMAP result 1 = horizon at decoupling Structures in early Universe 6

63 Position and height of peaks Position and heigth of acoustic peaks depend on curvature, matter and energy content and other cosmological parameters Dependence on curvature Flat universe Dependence on baryon content Structures in early Universe 63

64 overview Part 1: problems in Standard Model of Cosmology: horizon and flatness problems presence of structures Part : Need for an exponential expansion in the early universe Part 3 : Inflation scenarios scalar inflaton field Part 4 : Primordial fluctuations in inflaton field Part 5 : Growth of density fluctuations during radiation and matter dominated eras Part 6 : CMB observations related to primordial fluctuations Structures in early Universe 64

65 Examen Book Perkins nd edition chapters 5, 6, 7, 8, + slides Few examples worked out : ex5.1, ex5.3, ex6.1, ex7. Open book examination Structures in early Universe 65