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 • scenarios – scalar inflaton field

• Primordial fluctuations in inflaton field • Growth of density fluctuations during radiation and 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 • DlDevelopmen 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 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

4 radiationρ c2 ∼ (1+ z) 3 matterρ c2 ∼ ()1+ z vacuumρ c2 ∼ cst 2 curvatureρ c2 ∼ ()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 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 ? 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

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 ρ ==ρ tot Λ 8 G • AconstantA constant energy density causes an exponentialπ expansion

• For instance between t1 and t2 in flat universe

2 2 ⎛⎞RG8πρΛ Λ H ==⎜⎟ = R H 2 R 33 2 t - 1 ⎝⎠ ( t) R = e 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

60 R 2 ()GUT 1m 26 e ≈=≈−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 ⎛⎞ 2H 2 R2 e t1 - ∼ 1 Rt()2 H ⎜⎟ ()t 2 0 2 = 1 kc ⎜⎟= - ⎜ R1 ⎟ 52 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 2 TV+ φ 2 ρ cV==+()φ ρ c ~ V ()φ0 φ R3 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 isrehtiheating phhase • Relativistic particles are created • Expansion is now radiation dominated • Hot Big Bang evolution R starts GUT era t 2011-12 Structures in early Universe 30 Quantum fluctuations in early universe as seed for present- day structures PART 3 PRIMORDIAL FLUCTUATIONS

2011-12 Structures in early Universe 31 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

2011-12 Structures in early Universe 32 Quantum fluctuations as seed

• quantum fluctuations in inflaton field Δφ Δ=t • → fluctuations in inflation end time Δt φ • Energy fluctuations ΔE are possible within ΔE Δt ∼

Δρφ • Exponential inflation: particles and anti-particles are inflated apart outside horizon • Quantum fluctuations are ‘frozen-in’ → can be treated as classical perturbations • → Su perposition of waesaves

2011-12 Structures in early Universe 33 Wavelength is much larger than horizon at end of inflation

2011-12 Structures in early Universe 34 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 • SlScalecanbeanysize within R(t), even larger than hihorizon • Change in curvature of space-time yields change in gravitation potential • Particles created during reheating will ‘fall’ in gravitational potential wells Φ ρ x

2011-12 Structures in early Universe 35 Gravitational potential fluctuations • gravitational potential Φ due to energy density ρ spherical symmetric case 2πGrρ 2 Φ(r)= 3 • 2 cases – Global ‘background’ potential from ‘average’ field f – Potential change due to fluctuation Δρ over arbitrary scale λ 2πGρ λ 2 Φ= 2 ΔΦ = 2πG (Δρ ) 3H 3 • Fractional perturbation in gravitational potential at certain location ΔΦ Δρ = H 22λ Φ ρ

2011-12 Structures in early Universe 36 Spectrum of fluctuations • During inflation Universe is effectively in stationary state • Fluctuation amplitudes will not depend on space and time

ΔΦ ~ cst Hcst2 ~ Φ • Define rms amplitude of fluctuation δ 2 ⎛⎞Δρ λ = ⎜⎟ ⎝⎠ρ rms

ΔΦ 2 2 Δρ 1 = H λ δ 2 ∼ Φ ρ λ λ 2 • Amplitude is independent of scale R(t) at which perturbation enters horizon

2011-12 Structures in early Universe 37 Clustering of particles at end of inflation – role of CDM Evolution of fluctuations during radiation dominated era Evolution of fluctuations during matter dominated era Damppging effects PART 4 GROWTH OF STRUCTURE

2011-12 Structures in early Universe 38 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?

2011-12 Structures in early Universe 39 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 • λ 1 When will this lead to condensation? ⎛⎞π 2 • Compare cloud size L to Jeans length λ J = vs ⎜⎟ J ⎝⎠Gρ Depends on density r and sound speed vs

• Sound wave in gas = pressure wave p Cp 5 vs ===γ γ • Sound velocity in an ideal gas ρ CV 3

2011-12 Structures in early Universe 40 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 • ccoudloud con denses aaoudround ttehe dens i ty pertu r bati on s • Start of

2011-12 Structures in early Universe 41 Need cold dark matter • Start from upward density fluctuation Δρ • Density contrast evolution after decouppgling in flat matter dominated universe

2 Δρ 33ΔRc⎛⎞ δ (t0 ) δ = =− ~ ⎜⎟R()t ∝+()1 z RGM⎝⎠10 δ ()z ρ • AhtdbhAssume photons and baryons have same fluc tua tions at decoupling (adiabatic fluctuations) • z(dec) =1100 and density contrast at decoupling = 10-5 −2 • Expect density contrast in matter today δ (t0 )~10 • Observe from galaxy distributions : much larger value • Need CDM in addition to baryons

2011-12 Structures in early Universe 42 Evolution of fluctuations in radiation era • After inflation : photons and relativistic particles • Radiation dominated till decoupling at z ≈ 1100 • Velocity of sound is relativistic 2 c ρc 5 pp v ∼ p = vs = ~ s 3 3 ρ ρ 3 • horizon distance ~ Jeans length : no condensation of matter 1 1 Dt λ 2 2 H ( )= ct J ⎛⎞ππ⎛⎞32 ==vs ⎜⎟ct ⎜⎟ ⎝⎠Gρ ⎝⎠9 • Density fluc tu ations inside horiz on su rv iv e – fluc tu ations continue to grow as R(t) • But: photon and neutrino components have damping effect

2011-12 Structures in early Universe 43 Evolution of fluctuations in matter era • After decoupling: dominated by non-relativistic matter • Neutral atoms (H) are formed – sound velocity drops 1 55kTp ideal gas 2 ⎛⎞v ⎛⎞5kT −5 vs == s =≈210× 33ρ m ⎜⎟c ⎜⎟2 ⎝⎠mat ⎝⎠3McH

• Jeans length becomes smaller λ 1 ⎛⎞π 2 • therefore faster gravitational collapse J = vs ⎜⎟ ⎝⎠Gρ • Matter structures can grow

• Cold dark matter plays vital role in growth of structures

2011-12 Structures in early Universe 44 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 • RltResult : redtiduction of ditdensitycontttrast → diffusi on didamping (Silk damping)

2011-12 Structures in early Universe 45 Damping by neutrinos • Neutrinos = relativistic hot dkdark 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

2011-12 Structures in early Universe 46 Observations y pp lum

⎛⎞Δρ ⎜⎟ CMB ⎝⎠ρ rms hh moot ss

λ (Mpc) 2011-12 Structures in early Universe 47 From density contrast to temperature anisotropy Angular power spectrum of anisotropies Experimental observations PART 5 OBSERVATIONS OF STRUCTURES

2011-12 Structures in early Universe 48 CMB temperature map

dipole ΔT 10−3 OmK T ≈→()

ΔT 10−5 OµK T ≈→()

Contrast enhanced to make 10-5 variations visible! 2011-12 Structures in early Universe 49 Small angle anisotropies • Adiabatic fluctuations at end of radiation domination : photons & matter fluctuate together • Photons keep imprint of density contrast ⎛⎞ΔΔT ⎛⎞1 ρ ⎜⎟= ⎜⎟ ⎝⎠T Adiab ⎝⎠3 ρ • horizon at decoupling is seen now within an angle of 1° • temperature correlations at angles < 1° are due to primordial density fluctuations • Seen as acousti c peaks in CMB aaguangular pow er di stri buti on • Peaks should all have same amplitude – but: Silk damping by photons

2011-12 Structures in early Universe 50 δ 2 ⎛⎞Δρ 1 λ = ⎜⎟∼ 2 ⎝⎠ρλrms x © Scientific American Feb 2004

2011-12 Structures in early Universe 51 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 27K2.7K Δ=−Tn() Tn () T0

C(θ) = correltilation btbetween temperature fluctuations in 2 directions (nmn,m) separated by angle θ average over all pairs with given θ

2011-12 Structures in early Universe 52 Angular spectrum of anisotropies • Expand in Legendre polynomials, integrate over f

• Cℓ describes amplitude of correlations • TtTemperature fluc tua tions ΔT are superposition o f fluctuations at different scales (λ) • Decompose in spherical harmonics

2011-12 Structures in early Universe 53 Angular spectrum of CMB anisotropies

• Coefficients Cℓ = angular power spectrum - measure level of structure found at angular separation of θ π 200 ≈≈ degrees • Large ℓ means small angles • ℓ = 200 corresponds to 1° = horizon at decoupling as seen today • Amplitudes seen today depend on: 9 primordial density fluctuations Fit of Cℓ as function of ℓ 9 Baryy/pon/photon ratio yields cosmological 9 Amount of dark matter parameters 9 ….

2011-12 Structures in early Universe 54 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

2011-12 Structures in early Universe 55 Physics of Physics of Young universe primordial fluctuations Large wavelengths Short wavelengths

Sachs-Wolfe effect Silk damping

1° = horizon at decoupling

2011-12 Structures in early Universe 56 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 57 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

2011-12 Structures in early Universe 58