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

3 Galactic and intergalactic magnetic fields as source of structures? problems in Standard Model of Cosmology related to initial conditions required: horizon and flatness problems PART 1: PROBLEMS IN STANDARD COSMOLOGICAL MODEL Structures in early Universe 3

4 Galactic and intergalactic B fields cosmology model assumes uniform & homogeous medium Non-uniformity observed in CMB and galaxy distribution, Is clustering of matter in the universe due to galactic and intergalactic magnetic fields? At small = stellar scales : intense magnetic fields are only important in later stages of star evolution At galacticscale: l magnetic field in Milky Way 3µG At intergalactic scale : average fields of G intergalacticmagnetic ti ti fields probably bl have very small role in development of large scale structures Development of fl large scalestructures structures t mainly due to gravity Structures in early Universe 4

5 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 STATIC DH = ct t < 14 Gyr Would give as horizon distance today the Hubble radius STATIC H D ( t ) 0 = ct0 = Gpc Structures in early Universe 5

6 Horizon in expanding universe In expanding universe, particle horizon for observer at t 0 t R( t ) ( ) 0 0 DH t0 = cdt 0 Rt () Expanding, flat, radiation dominated d universe () t ( 0 ) t0 R t = R t 1 2 ( ) D t = 2ct H 0 0 Expanding,flat,matter dominated universe () t ( 0 ) t0 R t = R t 2 3 ( ) D t = 3ct H 0 0 expanding, flat, with W m =0.24, W Λ =0.76 D t = 3.3ct H ( ) 0 0 ~ 14GpcG Expanding Universe 6

7 Summary lecture 1 ( 1 ) 4 2 radiation ρ c + z ( 1+ ) 3 2 matter ρc z 2 vacuum ρc cst ( 1 ) 2 curvature ρ c + z Structures in early Universe 7

8 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 8

9 Horizon at time of decoupling Age of universe at matter-radiation decoupling t dec 5 = 4 10 y z dec 1100 Optical horizon at decoupling static D t ct ( ) γγ dec = This distance measured today ( ) ( ) dec dec angle subtended today in flat matter dominated universe dec D t ct z γγ 0 = 1+ θ dec ( 0 ) dec ( 1+ dec ) ( ) 3 ( ) Dγγ t ct z 1 D t c t t H 0 0 dec Structures in early Universe 9

10 Flatness problem 1 universe is flat today, but has been much flatter in early universe How come that curvature was so finely tuned to Ω=1 in very early universe? 2 kc Friedman equation rewritten Ω() t 1 = ( ) H( t) R t 2 2 R t ( ) n t radiation domination matter domination Ω() t 1~ Ω () t 1~ t t 2 3 At time of decoupling At GUT time (10-34 s) Ω 1~ Ω 1 ~ Structures in early Universe 10

11 Flatness problem 2 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 11

12 Horizon problem Flatness problem Non-homogenous universe The solution is INFLATION Structures in early Universe 12

13 The need for an exponential expansion in the very early universe PART 2 EXPONENTIAL EXPANSION Structures in early Universe 13

14 Accelerated expansion domination of Dark Energy Steady state expansion Radiation and matter dominate Described by SM of Cosmo Exponential expansion inflation phase Rubakov Structures in early Universe 14

15 Constant energy density = exponential expansion Assume that at Planck scale the energy density is given by a cosmological constant Λ ρtot = ρλ = 8π G Aconstantenergy density causes an exponential expansion For instance between t 1 and t 2 in flat universe H πgρ Λ Λ = = = R R 8 H = constant H = H ρ 0 ρ c Λ 2 RR1 1=( )2eHt2-t Structures in early Universe 15

16 When does inflation happen? Inflation period Structures in early Universe 16

17 When does inflation happen? Grand Unification at GUT energy scale GUT time couplings of Electromagnetic Weak Strong interactions are the same kt ~10 ( ) 16 GUT t GUT GeV 34 ~10 sec At end of GUT era : spontaneous symmetry breaking into electroweak and strong interactions Reheating and production of particles and radiation Big Bang expansion with radiation domination Structures in early Universe 17

18 How much inflation is needed? Calculate backwards size of universe at GUT scale after GUT time universe is dominated by radiation If today ( ) R t t 1 2 Then we expect ( ) ( ) 0 0 ( GUT ) ( ) R t R t t GUT 0 t0 R t ~ ct = c s ~ m ( ) ~ R( t ) R t GUT s 4 10 s EXPECTED R ( t )~1m GUT On the other hand horizon distance at GUT time was D t ~ ct ~10 m H ( ) 26 GUT GUT Structures in early Universe 18

19 How large is exponential expansion? It is postulated that before inflation the universe radius was smaller than the horizon distance 26 Size of universe before inflation R1 10 m Inflation needs to increase size of universe by factor R ( ) 2 GUT 60e1m 26 = R start inflation 10 m H ( t2 t1 ) 1 ( ) 2 1 > 60 R2 R = 1 e Ht ( -t ) 2 1 At least 60 e-folds are needed Structures in early Universe 19

20 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 20

21 Scientific American Feb Structures in early Universe 21

22 Flatness problem is solved If curvature term at start of inflation is roughly 1 R t 2 kc H t ( ) ( ) = O ( 1 ) Then at the end of inflation the curvature is even closer to 1 kc R t ( ) H 2=e12( ) 2 kc R t 1 ) H 12 R = 1 1R Ht-t 10( -52 Ω= 1± 10 at t = 10 s Structures in early Universe 22

23 Conclusion so far Constant energy density yields exponential expansion This solves horizon and flatness problems What causes the exponential inflation? Structures in early Universe 23

24 Scalar inflaton field PART 2 INFLATION SCENARIOS Structures in early Universe 24

25 Introduce a scalar inflaton field Physical mechanism underlying inflation is unknown Postulate by Guth (1981): inflation is caused by an unstable scalar field in the very early universe dominates during GUT era (kt GeV) spatially homogeneous depends only on time : f(t) inflaton field f describes a particle with mass m Scalar field causes negative pressure needed for exponential growth Potential V(f ) due to inflaton field Structures in early Universe 25

26 Chaotic inflation scenario (Linde, 1982) Inflation starts at different field values in different locations Inflaton field is displaced from true minimum by some arbitrary mechanism, probably quantum fluctuations Potential V(f) is slowly varying with f slow roll approximation Inflation ends when field f reaches minimum of potential Transformation of potential energy to kineticenergy produces particles = reheating Different models can fulfill this inflation scenario Lineweaver Structures in early Universe 26

27 Start from quantum fluctuations 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 27

28 Equation of motion of scalar field Lagrangian energy of inflaton field 2 L T V R φ V 2 3 ( φ ) = = ( φ ) φ = Mechanics (Euler-Lagrange) : equation of motion of finflaton field dv φ + 3 H ( t ) φ + = 0 dφ dφφ dt Frictional force due to expansion = oscillator/pendulum motion with damping field oscilllates around minimum stops oscillating = reheating Structures in early Universe 28

29 Slow roll exponential expansion At beginning of inflation kinetic energy T is small compared to potential energy V field varies slowly φ = 0 φ small T V << φ φ0 = cst Total energy in universe = potential energy of field ρ c φ 2 2 T + V φ = = + V 3 R 2 ( φ ) ρφ c2 ~ V Friedman equation for flat universe becomes H 2 ( ) 2 V 8πGρφ 8π φ H 2 ~ ( φ ) 0 cst R π ρ π φ = = = 2 R 3 3M PL Exponential expansion Structures in early Universe

30 Summary Planck era When the field reaches the minimum, potential energy is transformed in kinetic energy This isreheating phase Relativistic particles are created Expansion is now radiation dominated Hot Big Bang evolution starts R GUT era Structures in early Universe 30 t kt

31 Quantum fluctuations in early universe as seed for presentday structures PART 3 PRIMORDIAL FLUCTUATIONS Structures in early Universe 31

32 quantum fluctuations yield anisotropies 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 32

33 Quantum fluctuations as seed quantum fluctuations in inflaton field fluctuations in inflation end time Δt Energy fluctuations ΔE are possible within Δ t = ΔE Δφ φ Δt Δρ φ Exponential inflation: particles and anti-particles are inflated apart outside horizon Quantum fluctuations are frozen-in can be treated as classical perturbations Superposition of waves aes Structures in early Universe 33

34 Wavelength is much larger than horizon at end of inflation Structures in early Universe 34

35 inflaton field as cosmic fluid density fluctuations modify the energy-momentum tensor and therefore also the curvature of space The modification is seen over scale λ Scalecanbe anysize within R(t), even larger than horizon Change in curvature of space-time yields change in gravitation potential Particles created during reheating will fall in gravitational potential wells Φ ρ x Structures in early Universe 35

36 Gravitational potential fluctuations gravitational potential Φ due to energy density ρ spherical symmetric case 2 cases ( r ) Φ = 2π G ρr 3 Global background potential from average field f Potential change due to fluctuation Δρ over arbitrary scale λ 2π Gρ Φ= 2 3H ΔΦ = 2πGπ G 2 ( Δρ ) Fractional perturbation in gravitational potential at certain location ΔΦ 2 2 Δρ = H λ Φ ρ 2 λ Structures in early Universe 36

37 Spectrum of fluctuations During inflation Universe is effectively in stationary state Fluctuation amplitudes will not depend on space and time ΔΦ ~ cst 2 Φ H ~ cst Define rms amplitude of fluctuation ΔΦ Φ = H λ 2 2 Δρ ρ δ 2 λ Δρ = ρ rms 2 1 δ λ 2 λ Amplitude is independent of scale R(t) at which perturbation enters horizon Structures in early Universe 37

38 Clustering of particles at end of inflation role of CDM Evolution of fluctuations during radiation dominated era Evolution of fluctuations during matter dominated era Damping effects PART 4 GROWTH OF STRUCTURE Structures in early Universe 38

39 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 39

40 Jeans length = critical dimension After reheating matter condensates around primordial density fluctuations universe = gas of relativistic particles Consider a cloud of gas around fluctuation Δρ When will this lead to condensation? Compare cloud size L to Jeans length λ J Depends on density r and sound speed v s λ J π = v s G ρ x 1 2 Sound wave in gas = pressure wave p 5 Sound velocity in an ideal gas v s C p = γ γ = = ρ C Structures in early Universe 40 C V 3

41 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 coudcondenses around aou dthe density perturbations Start of structure formation Structures in early Universe 41

42 Need cold dark matter Start from upward density fluctuation Δρ Density contrast evolution after decoupling in flat matter dominated universe δ = Δ δ () ( t0 ) ~ R t R 10 GM δ ( z) 2 Δρ 3Δ R 3 c ρ = ( 1 z) + Assume photons and baryons have same fluctuations ti at decoupling (adiabatic fluctuations) z(dec) =1100 and density contrast at decoupling = Expect density contrast in matter today δ ( t 0 ) ~10 Observe from galaxy distributions : much larger value Need CDM in addition to baryons Structures in early Universe 42

43 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 = ρc 3 2 v s = 5 3 p ~ ρ p ρ v horizon distance ~ Jeans length : no condensation of matter 1 1 D 2 2 = λh ( t) ct π 32 π = = ct Gρ J 9 Density fluctuations inside horizon on survive fluctuations continue to grow as R(t) But: photon and neutrino components have damping effect Structures in early Universe 43 v s s c 3

44 Evolution of fluctuations in matter era After decoupling: dominated by non-relativistic matter Neutral atoms (H) are formed sound velocity drops ideal gas 5 p 5 kt v s = = 3 ρ 3 m v 5kT s c = mat 3 M H c Jeans length becomes smaller therefore faster gravitational collapse Matter structures can grow λ J π = v s G ρ 1 2 Cold dark matter plays vital role in growth of structures Structures in early Universe 44

45 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 t diffusion i damping (Silk damping) Structures in early Universe 45

46 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 46

47 Observations py lump Δρ ρ rms CMB smooth λ (Mpc) Structures in early Universe 47

48 From density contrast to temperature anisotropy Angular power spectrum of anisotropies Experimental observations PART 5 OBSERVATIONS OF STRUCTURES Structures in early Universe 48

49 CMB temperature map dipole ΔT ΔT T T 10 3 O ( mk ) 10 5 O ( µk ) Contrast enhanced to make 10-5 variations visible! Structures in early Universe 49

50 Small angle anisotropies Adiabatic fluctuations at end of radiation domination : photons & matter fluctuate together Photons keep imprint of density contrast Δ T 1 = T 3 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 c peaks in CMB angular a power distribution Peaks should all have same amplitude but: Silk damping by photons Δ ρ ρ Structures in early Universe 50

51 δ Δ 2 ρ 1 λ = ρ rms Scientific American Feb λ x Structures in early Universe 51

52 Angular spectrum of anisotropies 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 27K 2.7K Δ T n = T n T ( ) ( ) 0 C(θ) = correlation between temperature fluctuations in 2 directions (n,m) separated by angle θ average over all pairs with given θ Structures in early Universe 52

53 Angular spectrum of anisotropies Expand in Legendre polynomials, integrate over f C l describes amplitude of correlations Temperature fluctuations ti ΔT are superposition of fluctuations at different scales (λ) Decompose in spherical harmonics Structures in early Universe 53

54 Angular spectrum of CMB anisotropies Coefficients C l = angular power spectrum - measure level of structure found at angular separation of π θ 200 degrees Large l means small angles l = 200 corresponds to 1 = horizon at decoupling as seen today Amplitudes seen today depend on: primordial density fluctuations Fit of C l as function of l Baryon/photon ratio yields cosmological parameters Amount of dark matter Structures in early Universe 54

55 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 55

56 Physics of Young universe Large wavelengths Physics of primordial fluctuations Short wavelengths Sachs-Wolfe effect Silk damping 1 = horizon at decoupling Structures in early Universe 56

57 overview problems in Standard Model of Cosmology: horizon and flatness problems presence of structures Need for an exponential expansion in the early universe Inflation scenarios scalar inflaton field Primordial fluctuations in inflaton field Growth of density fluctuations during radiation and matter dominated eras Observations related to primordial fluctuations: CMB and LSS Structures in early Universe 57

58 Examen Book Perkins 2 nd edition chapters 5, 6, 7, 8, + slides Few examples worked out : ex5.1, ex5.3, ex6.1, ex7.2 Open book examination Structures in early Universe 58