Structures in the early Universe Particle Astrophysics chapter 8 Lecture 4 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 2011-12 Structures in early Universe 2 •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 2011-12 Structures in early Universe 3 Galactic and intergalactic B fields cosmology model assumes Non-uniformity observed uniform & homogeous in CMB and galaxy medium distribution, • Is cltilustering of matter ithin the universe dtdue to galtilactic and intergalactic magnetic fields? • At small = stellar scales : intense magnetic fldfieldsare onlyimportant in later stages of star evolution • At galtilacticscale: magnetic fie ld in Milky Way ≈ 3G3µG • At intergalactic scale : average fields of 10-7 –10-11 G • → itinterga lac ticmagnetic fie ldsprobblbably have very small role in development of large scale structures • DDlevelopmen toflf large scallleesstttructures mailinly dtdue to gravity 2011-12 Structures in early Universe 4 Horizon in static universe • Particle horizon = distance over which one can observe a particle by exchange of a light signal vvc=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 DctH = • Would give as horizon distance today the Hubble radius SSCTATIC DtctGpcH ( 00) ==424.2 2011-12 Structures in early Universe 5 Horizon in expanding universe • In expanding universe, particle horizon for observer at t0 t0 R(t0 ) DdDH tc0 = dt ( ) ∫0 Rt() • EdiExpanding, fla t, ra dia tion ditddominated universe 1 2 Rt() ⎛⎞t Dt= 2 ct = ⎜⎟ H ( 00) Rt()0 ⎝⎠t0 • Expanding,flat,, flat, matter dominated universe 2 3 Rt() ⎛⎞t DtH ()00= 3 ct = ⎜⎟ Rt( 0 ) ⎝⎠t0 DtH ()00= 3.3 ct • expanding, flat, with Wm=0.24, WΛ=0.76 ~ 14Gpc 2011-12 Expanding Universe 6 Summary lecture 1 2 4 radiationρ c∼ (1+ z) 2 3 matterρ c∼ ()1+ z vacuumρ c2 ∼ cst 2 2 curvatureρ c∼ ()1+ z 2011-12 Structures in early Universe 7 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 abt1bout 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 2011-12 Structures in early Universe 8 Horizon at time of decoupling • Age of universe at matter-radiation decoupling 5 tydec =×410 zdec ≈1100 • Optical horizon at decoupling static Dtγγ ()dec= ct dec Dt=+ ct1 z • This distance measured today γγ ()0 dec ( dec ) • angle subtended today in flat matter dominated universe Dtγγ ()0 ctdec()1+ z dec θdec ∼∼ ∼1° DtHdec()003 ctt()− 2011-12 Structures in early Universe 9 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? kc2 • Friedman equation rewritten Ω−=()t 1 22 Rt( ) Ht( ) radiation domination Ω−()tt1~ ∼ n Rt( ) t 2 matter domination Ω−()tt1~ 3 −16 • At time of decoupling Ω−1~10 • At GUT time (10 -34s) Ω −1101 ~ 10−52 2011-12 Structures in early Universe 10 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 2011-12 Structures in early Universe 11 Horizon Flatness problem problem Non-homogenous universe The solution is INFLATION 2011-12 Structures in early Universe 12 The need for an exponential expansion in the very early universe PART 2 EXPONENTIAL EXPANSION 2011-12 Structures in early Universe 13 Accelerated expansion domination of Dark Energy Steady state expansion Radiation and matter dominate Described by SM of Cosmo Exponential expansion inflation phase © Rubakov 2011-12 Structures in early Universe 14 Constant energy density = exponential expansion • Assume that at Planck scale the energy density is given by a cosmological constant Λ ρtot ==ρ Λ 8πG • AconstantA constant energy density causes an exponential expansion • For instance between t1 and t2 in flat universe 2 2 ⎛⎞RG8πρ Λ H ==⎜⎟ Λ = R H t -t ⎝⎠R 33 2 = e 2 1 R ( ) 1 1 2 • H = constant ⎛⎞ρΛ HH= 0 ⎜⎟ ⎝⎠ρc 2011-12 Structures in early Universe 15 When does inflation happen? Inflation period 2011-12 Structures in early Universe 16 When does inflation happen? • Grand Unification at GUT energy scale kT~1016 GeV ()GUT • GUT time t ~10−34 sec • couplings of GUT – Electromagnetic – Weak – Strong interactions are the same • 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 2011-12 Structures in early Universe 17 How much inflation is needed? • Calculate backwards size of universe at GUT scale • after GUT time universe is dominated by radiation 1 1 2 2 Rt()GUT ⎛⎞tGUT Rt( ) ∼ t ∼ ⎜⎟ R(tt00) ⎝⎠ • If today 17 26 R(tctc00) ~=×( 4 × 10 s) ~ 4 × 10 m • Then we expect 1 −34 2 EXPECTED ⎛⎞10 s R ()tmGUT ~1 Rt()()GUT ~ Rt0 ×⎜⎟17 ⎝⎠410× s • On the other hand horizon distance at GUT time was −26 D~H ()tctmGUT GUT ~10 2011-12 Structures in early Universe 18 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 Rm1 ≤10 • Inflation needs to increase size of universe by factor e60 R 2 ()GUT 1m 26 ≈=≈−26 10 H (tt21−>) 60 RRstart1 ( start inflation) 10 m Ht- t R2 21 At least = e ( ) R1 60 e-folds are needed 2011-12 Structures in early Universe 19 horizon problem is solved ¾ Whole space was in causal contact and thermal equilibrium before inflation ¾ Some points got disconnected during exponential expansion ¾ entered into our horizon again at later stages • → horizon problem solved 2011-12 Structures in early Universe 20 © Scientific American Feb 2004 2011-12 Structures in early Universe 21 Flatness problem is solved • If curvature term at start of inflation is roughly 1 kc2 22= O(1) Rt()11 Ht () • Then at the en d o f in fla tion the curvature is even closer to 1 kc2 1 2 2 ⎛⎞R 2H t -t Rt H2 2 = e 1 2 ∼ 10- 52 ()2 = ⎜⎟1 () kc2 ⎜⎟R ⎜ 1 ⎟ 2 2 ⎝⎠ Rt()1 H Ω=110 ±−−52 at ts =10 34 2011-12 Structures in early Universe 22 Conclusion so far • Constant energy density yields exponential expansion • This solves horizon and flatness problems • What causes the exponential inflation? 2011-12 Structures in early Universe 23 Scalar inflaton field PART 2 INFLATION SCENARIOS 2011-12 Structures in early Universe 24 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≈ 1016 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 2011-12 Structures in early Universe 25 Chaotic inflation scenario (Linde, 1982) • IfliInflation starts at different fie ld valilues in different lilocations • 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 kine ticenergy produces particles = reheating • Different models can fulfill this inflation scenario © Lineweaver 2011-12 Structures in early Universe 26 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 2011-12 Structures in early Universe 27 Equation of motion of scalar field • Lagrangian energy of inflaton field 2 3 ⎡φ ⎤ dφ LTVRV(φφ) =−=⎢⎥ −( ) φ = ⎣ 2 ⎦ dt • MhiMechanics (Eu ler-L)Lagrange) : equation oftiff motion of ifltinflaton field dV φ ++3Ht( )φ = 0 dφ Frictional ffporce due to expansion • • = oscillator/pendulum motion with damping • field oscilllates around minimum • stops oscillating = reheating 2011-12 Structures in early Universe 28 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 TV<< φ ≈=φ0 cst • Total energy in universe = potential energy of field 2 TV 2 2 + φ c ~ V ρ cV==+3 ()φ ρ ()φ0 φ R 2 φ • Friedman equation for flat universe becomes 2 V 8πρG 0 2 2 ⎛⎞R φ 8π (φ ) HtH ~ cst H ==⎜⎟ =2 ⎝⎠RM33PL Exponential expansion 2011-12 Structures in early Universe 29 Summary Planck era kT • When the field reaches the minimum, potential energy is transfdformed in kinetic energy • This