Structures in the early Universe
Particle Astrophysics chapter 8 Lecture 4
overview • Part 1: problems in Standard Model of Cosmology: horizon and flatness problems – presence of structures
• Part 2 : 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
2012-13 Structures in early Universe 2 •problems in Standard Model of Cosmology related to initial conditions required: horizon problem flatness problem PART 1: PROBLEMS IN STANDARD COSMOLOGICAL MODEL
2012-13 Structures in early Universe 3 2012-13 Structures in early Universe 4 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 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
2012-13 Expanding Universe 5 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 essentiallyQ: Which flat mechanism over large causeddistances the exponential inflation? Made up of baryons, coldA: scalar dark matterinflaton and field a constant dark energy + small amount of photons and neutrinos IsExponential presently accelerating inflation takes care of horizon and flatness problems Parts 1, 2, 3
2012-13 Expanding Universe 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 StartedStructures from hot observed Big Bang, in galaxy followed surveys by short and inflationanisotropies period in CMB Is essentially flat over large distancesobservations: Made up of baryons,Q: What cold darkseeded matter these and structures? a constant dark energy + small amountA: quantumof photons fluctuations and neutrinos in primordial universe Is presently accelerating Parts 4, 5, 6
2012-13 Expanding Universe 7 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 t < 14 Gyr DH ct
• Would give as horizon distance today the Hubble radius STATIC DH t00 ct4.2 Gpc
2012 -13 Structures in early Universe 8 Horizon in expanding universe
• In expanding universe, particle horizon for observer at t0
t0 Rt0 DH t0 cdt Lecture 1 0 Rt • Expanding, flat, radiation dominated universe 1 Rt t 2 DH t002 ct Rt0 t0 • Expanding, flat, matter dominated universe 2 3 Rt t DH t003 ct
Rt0 t0 DH t003.3 ct • expanding, flat, with m=0.24, Λ=0.76 ~ 14Gpc
2012-13 Structures in early Universe 9 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
2012-13 Structures in early Universe 10 Horizon at time of decoupling • Age of universe at matter-radiation decoupling
5 tydec 4 10 zdec 1100 • Optical horizon at decoupling dec DH tdec2 ct dec
Ddec t21 ct z • This distance measured today H 0 dec dec • angle subtended today in flat matter dominated universe
dec DH t0 21 ctdec z dec dec 1 DH t003 c t t dec
2012-13 Structures in early Universe 11 2012-13 Structures in early Universe 12 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? kc2 • Friedman equation rewritten t 1 22 R t H t
tt1~ n radiation domination R t t 2 matter domination tt1~ 3
16 • At time of decoupling 1 ~ 10 • At GUT time (10-34s) 1 ~ 10 53
2012 -13 Structures in early Universe 13 2012-13 Structures in early Universe 14 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
ts 10 44 ts 10 34
2012-13 Structures in early Universe 15 Horizon Flatness problem problem
Non-homogenous universe
The solution is INFLATION
2012-13 Structures in early Universe 16
The need for an exponential expansion in the very early universe PART 2 EXPONENTIAL EXPANSION
2012-13 Structures in early Universe 17 Steady state expansion Radiation dominated Described by SM of cosmology Exponential expansion inflation phase © Rubakov 2012-13 Structures in early Universe 18 Constant energy density & exponential expansion
• Assume that at Planck scale the energy density is given by a cosmological constant tot 8 G • A constant energy density causes an exponential expansion
• For instance between t1 and t2 in flat universe
2 H t -t RG8 R 2 2 1 H 2 =e
R 33 R1
2012-13 Structures in early Universe 19 When does inflation happen?
Inflation period
2012-13 Structures in early Universe 20 When does inflation happen?
• Grand Unification at GUT energy scale kT~1016 GeV GUT • GUT time t ~ 1034 sec • couplings of GUT – 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
2012-13 Structures in early Universe 21 How much inflation is needed? 1 • Calculate backwards size of universe at GUT scale • after GUT time universe is dominated by radiation 1 1 Rt t 2 R t t 2 GUT GUT R t00 t • If today static 17 26 R t0~ DH t 0 ct 0 c 4 10 s ~ 4 10 m
• Then we expect 1 10 34 s 2 REXPECTED t~1 m R t~ R t GUT GUT 0 4 1017 s • On the other hand horizon distance at GUT time was 26 DH tGUT ~ ct GUT ~10 m 2012-13 Structures in early Universe 22 2012-13 Structures in early Universe 23 How much inflation is needed? 2 • It is postulated that before inflation the universe radius was smaller than the horizon distance at GUT time • Size of universe before inflation 26 Rm1 10
• Inflation needs to increase size of universe by factor
R2 GUT 1m 26 60 26 10 e H t21 t 60 R1 start inflation 10 m
R H t- t 2 e 21 At least R1 60 e-folds are needed
2012-13 Structures in early Universe 24 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
2012-13 Structures in early Universe 25 © Scientific American Feb 2004
2012-13 Structures in early Universe 26 Flatness problem is solved • If curvature term at start of inflation is roughly 1 kc2 O 1 R t22 H t 11 • Then at the end of inflation the curvature is even closer to 1 2 kc 2 2 1 R t H 2 R 2H t -t 2 2 =e 1 2 ∼ 10 - 52 kc2 1 2 2 R1 R t1 H
1 1052 at ts 10 34
2012-13 Structures in early Universe 27 Conclusion so far • Constant energy density yields exponential expansion • This solves horizon and flatness problems • What causes the exponential inflation?
2012-13 Structures in early Universe 28
Scalar inflaton field PART 3 INFLATION SCENARIOS
2012-13 Structures in early Universe 29 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≈ 1016 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
2012-13 Structures in early Universe 30 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
• Total energy density associated with scalar field
221
cV )
2 Φ
V( • During inflation the potential V() is only slowly varying with – slow roll approximation inflation reheating • Kinetic energy can be neglected
© Lineweaver Φ
2012-13 Structures in early Universe 31 Slow roll leads to exponential expansion • Total energy in universe = potential energy of field
2 2 cV2 c ~ V ~ cst 2 • Friedman equation for flat universe becomes 2 R 8 G H 2 R 3 H2 ~ cst 2 8 c H 2 V 3 M PL
Exponential expansion
2012-13 Structures in early Universe 32 Reheating at the end of inflation • Lagrangian energy of inflaton field 2 LTVRV3 2 • Mechanics (Euler-Lagrange) : equation of motion of inflaton field dV 3Ht 0 d Frictional force due to expansion • • = oscillator/pendulum motion with damping • field oscilllates around minimum • stops oscillating = reheating
2012-13 Structures in early Universe 33 Reheating and particle creation
• Inflation ends when field reaches minimum of potential
)
• Transformation of potential Φ
energy to kinetic energy = V( reheating
inflation reheating • Scalar field decays in particles
© Lineweaver Φ
2012-13 Structures in early Universe 34 Summary Planck era
kT • 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 R starts GUT era t 2012-13 Structures in early Universe 35 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
• Total energy density associated with scalar field
221
cV )
2 Φ
V( • During inflation the potential V() is only slowly varying with – slow roll approximation inflation reheating • Kinetic energy can be neglected
© Lineweaver Φ
2012-13 Structures in early Universe 36 2012-13 Structures in early Universe 37 2012-13 Structures in early Universe 38 Quantum fluctuations in early universe as seed for present- day structures: CMB and galaxies PART 4 PRIMORDIAL FLUCTUATIONS
2012-13 Structures in early Universe 39 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
2012-13 Structures in early Universe 40 Quantum fluctuations as seed 1 • Introduce quantum fluctuations in inflaton field t • 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 2 Ekin c T
Kinetic Potential • Hence different density of matter at Energy Energy different locations • And different temperatures
2012-13 Structures in early Universe 41 Quantum fluctuations as seed 2 • 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
2012-13 Structures in early Universe 42 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
x
• 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 T ~ 10 5 T 2012-13 Structures in early Universe 43 Gravitational potential fluctuations • gravitational potential Φ due to energy density ρ spherical symmetric case 2 Gr2 r • 2 cases 3 – Global ‘background’ potential from ‘average’ field within horizon – Potential change due to fluctuation Δρ over arbitrary scale λ
2 G 2 G 2 3 H 2 3 • Fractional perturbation in gravitational potential at certain location H 22
2012-13 Structures in early Universe 44 Spectrum of fluctuations • During inflation Universe is effectively in stationary state • Fluctuation amplitudes will not depend on space and time ~ cst H2 ~ cst
2 • Define rms amplitude of fluctuation rms
H 2 2 2 1 2
• Amplitude is independent of horizon size – independent of epoch at which it enters horizon – is ‘scale-invariant’
2012-13 Structures in early Universe 45 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
2012-13 Structures in early Universe 46 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?
2012-13 Structures in early Universe 47 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 1 • When will this lead to condensation? 2 • λ vs Compare cloud size L to Jeans length J J G Depends on density and sound speed vs
• Sound wave in gas = pressure wave P Cp 5 vs • Sound velocity in an ideal gas CV 3
2012-13 Structures in early Universe 48 Jeans length = critical dimension • Compare cloud size L with Jeans length
• when L << λJ : sound wave travel time < gravitational collaps time – no condensation t L vs
• When L >> λJ : sound cannot travel fast enough from one side to the other • cloud condenses around the density perturbations • Start of structure formation
2012-13 Structures in early Universe 49 Evolution of fluctuations in radiation era • After inflation : photons and relativistic particles • Radiation dominated till decoupling at z ≈ 1100 • Velocity of sound is relativistic c2 5 PP c P vs ~ vs 3 3 3 • horizon distance ~ Jeans length : no condensation of matter 1 1 2 32 2 DtH ct λ v ct J s G 9 • Density fluctuations inside horizon survive – fluctuations continue to grow in amplitude as R(t)
2012-13 Structures in early Universe 50 Evolution of fluctuations in matter era • After decoupling: dominated by non-relativistic matter • Neutral atoms (H) are formed – sound velocity drops 1 5P ideal gas 5 kT 2 v 5kT 5 vs s 2 10 33m c 2 mat 3McH
1 • Jeans length becomes smaller 2 – Horizon grows v J s G – therefore faster gravitational collapse – Matter structures can grow
• Cold dark matter plays vital role in growth of structures
2012-13 Structures in early Universe 51 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)
2012-13 Structures in early Universe 52 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
2012-13 Structures in early Universe 53
Observations
lumpy
CMB
rms
smooth
λ (Mpc) 2012-13 Structures in early Universe 54 From density contrast to temperature anisotropy Angular power spectrum of anisotropies PART 6 OBSERVATIONS IN CMB
2012-13 Structures in early Universe 55 CMB temperature map
dipole T 10 3 O mK T
T 10 5 O µK T
Contrast enhanced to make 10-5 variations visible! 2012-13 Structures in early Universe 56 Small angle anisotropies • Adiabatic fluctuations at end of radiation domination : photons & matter fluctuate together • Photons keep imprint of density contrast T 1 T Adiab 3 • 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
2012 -13 Structures in early Universe 57
2 1 2 rms x © Scientific American Feb 2004
2012-13 Structures in early Universe 58 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 2.7K 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
2012-13 Structures in early Universe 59 Angular spectrum of anisotropies 2 • Expand in Legendre polynomials, integrate over 1 C2 l 1Cl Pl cos 4 l
• Cℓ describes intensity of correlations • Temperature fluctuations ΔT are superposition of fluctuations at different scales λ • Decompose in spherical harmonics T , l aYlm lm , T0 l0 m l
221 Cl aalm lm 21l m
2012-13 Structures in early Universe 60 Angular spectrum of CMB anisotropies
• Coefficients Cℓ = angular power spectrum - measure level of structure found at angular separation of 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 Cℓ as function of ℓ Baryon/photon ratio yields cosmological Amount of dark matter parameters ….
2012-13 Structures in early Universe 61 Physics of Physics of Young universe primordial fluctuations Large wavelengths Short wavelengths
Silk damping
1° = horizon at decoupling WMAP result
2012-13 Structures in early Universe 62 Position and height of peaks • Position and heigth of acoustic peaks depend on curvature, matter and energy content and other cosmological parameters Flat universe Dependence on curvature Dependence on baryon content
2012-13 Structures in early Universe 63 overview • Part 1: problems in Standard Model of Cosmology: horizon and flatness problems – presence of structures
• Part 2 : 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
2012-13 Structures in early Universe 64 Examen • Book Perkins 2nd edition chapters 5, 6, 7, 8, + slides • Few examples worked out : ex5.1, ex5.3, ex6.1, ex7.2 • ‘Open book’ examination
2012-13 Structures in early Universe 65