Structures in the early Universe Particle Astrophysics chapter 8 Lecture 4

Structures lect 42

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 Structures lect 43

PART 1: PROBLEMS IN STANDARD COSMOLOGICAL MODEL problems in Standard Model of Cosmology related to initial conditions required:  horizon problem  flatness problem Structures lect 44

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 radiation Structures lect 45 Lecture 1

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 Structures lect 46 Lecture 1 Q: Which mechanism caused the exponential inflation? A: scalar inflaton field Exponential inflation takes care of horizon and flatness problems Parts 1, 2, 3

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 Structures lect 47 Lecture 1 Structures observed in galaxy surveys and anisotropies in CMB observations: Q: What seeded these structures? A: quantum fluctuations in inflaton field Parts 4, 5, 6

Horizon in static universe Particle horizon = distance over which one can observe a particle by exchange of a light signal Particle and observer are causally connected In a static universe of age t photon emitted at time t=0 would travel Would give as horizon distance today the Hubble radius Structures lect 48 v=c t < <14 Gyr

Horizon in expanding universe In expanding universe, particle horizon for observer at t 0 Expanding, flat, radiation dominated universe Expanding, flat, matter dominated universe expanding, flat, with Ω m =0.24, Ω Λ = Structures lect 49 Lecture 1

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 Angle subtented today by horizon size at decoupling is about 1° We expect that there was thermal equilibrium only within this horizon Why is CMB uniform (up to factor ) over much larger angles, i.e. over full sky? Answer : short exponential inflation before decoupling Structures lect 410

Horizon at time of decoupling Age of universe at matter-radiation decoupling Optical horizon at decoupling Expanded to angle subtended today in flat matter dominated universe Structures lect 411

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 radiation domination matter domination At time of decoupling At GUT time ( s) Structures lect 412

Structures lect 413

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 Structures lect 414 ? todayGUT scale Big Bang Planck scale

Structures lect 415 The solution is INFLATION Horizon problem Flatness problem Non-homogenous universe Questions?

PART 2 EXPONENTIAL EXPANSION The need for an exponential expansion in the very early universe Structures lect 416

Structures lect 417 © Rubakov Exponential expansion inflation phase Steady state expansion Radiation dominated Described by SM of cosmology

When does inflation happen? Structures lect 418 Inflation period

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 Caused by a scalar inflaton field Structures lect 419

Constant energy density & exponential expansion Assume that at given moment vacuum energy dominates over all other terms – and is given by cosmological constant A constant energy density causes an exponential expansion For instance between t 1 and t 2 in flat universe Structures lect 420

How much inflation is needed? 1 Calculate backwards size of universe at GUT scale after GUT time universe is dominated by radiation If today Then we expect On the other hand horizon distance at GUT time was Structures lect 421

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 Inflation needs to increase size of universe by factor Structures lect 422 At least 60 e-folds are needed

horizon problem is solved  Whole space was in causal contact and thermal equilibrium before inflation  Some points got disconnected during exponential expansion  They entered into our horizon again at later stages → horizon problem solved Structures lect 423

Structures lect 424 © Scientific American Feb 2004

Flatness problem is solved If curvature term at start of inflation is roughly 1 Then at the end of inflation the curvature is even closer to Structures lect 425

Conclusion so far Constant energy density yields exponential expansion This solves horizon and flatness problems Which mechanism causes the exponential expansion? Structures lect 426 Questions?

PART 3 INFLATION SCENARIOS Guth (1981): need extremely rapid exponential inflation by huge factor as preliminary stage of Big Bang -> Scalar inflaton field Structures lect 427

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 : ϕ(t) ‘inflaton field’ ϕ describes a scalar particle with mass m Dynamics is analogue to ball rolling from hill Scalar has kinetic and potential energy Structures lect 428

Chaotic inflation scenario (Linde, 1982) Inflation starts at different field values in different locations Inflaton field is displaced by some arbitrary mechanism, probably quantum fluctuations Structures lect 429 Total energy density associated with scalar field During inflation the potential V(ϕ) is only slowly varying with ϕ – slow roll approximation Kinetic energy can be neglected © Lineweaver V(Φ) Φ inflation reheating

Slow roll leads to exponential expansion Total energy in universe = potential energy of field Friedman equation for flat universe becomes Structures lect 430 Exponential expansion

Reheating at the end of inflation Lagrangian energy of inflaton field Mechanics (Euler-Lagrange) : equation of motion of inflaton field = oscillator/pendulum motion with damping field oscilllates around minimum stops oscillating = reheating Structures lect 431 Frictional force due to expansion

Reheating and particle creation Structures lect 432 Inflation ends when field ϕ reaches minimum of potential Transformation of potential energy to kinetic energy = reheating Scalar field decays in particles © Lineweaver V(Φ) Φ inflation reheating

Summary Structures lect 433 Planck era GUT 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 t R kT

Structures lect 434 Questions?

PART 4 PRIMORDIAL FLUCTUATIONS Quantum fluctuations in inflaton field as seed for present- day structures: CMB and galaxies Structures lect 435 © Lineweaver V(Φ) Φ

quantum fluctuations in inflaton field Structures lect 436 Field starting in B needs more time to reach minimum than field starting in A Reaches minimum Δt later Quantum fluctuations set randomly starting value in A or B Potential should vary slowly

Quantum fluctuations as seed Introduce quantum fluctuations in inflaton field Δ ϕ This yields fluctuations in inflation end time Δt Inflation lasts longer in some locations – different ‘bubbles’ - reheating will take place at different times Hence: variations of energy density in space Production of photons and baryons Adiabatic fluctuations in energy density Structures lect 437 Kinetic Energy

Perturbations in the cosmic fluid density fluctuations modify locally the space-time metric and therefore also the curvature of space Modification of space-time metric yields change in gravitation potential Φ Particles and radiation created during reheating will ‘fall’ in gravitational potential wells Fluctuations are adiabatic: same for photons and particles Structures lect 438

Gravitational potential fluctuations gravitational potential Φ due to energy density ρ spherical symmetric case 2 cases – On scale of horizon r hor = 1/H – Potential change due to fluctuation Δρ over arbitrary small scale λ Fractional perturbation in gravitational potential at certain location Structures lect 439

Spectrum of fluctuations During inflation Universe is effectively in stationary state Amplitudes of fluctuations are ‘frozen’ and will not depend on space and time rms amplitude of fluctuation Amplitude is independent of horizon size – independent of epoch -> scale-invariant Structures lect 440

Observations Structures lect 441 CMB lumpy smooth λ (Mpc)

Structures lect 442 Questions?

PART 5 GROWTH OF STRUCTURE Clustering of particles at end of inflation Evolution of fluctuations during radiation dominated era Evolution of fluctuations during matter dominated era Damping effects Structures lect 443

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 lect 444

Gravitational collapse of gas cloud when gas cloud condenses gravitational potential energy is transformed in kinetic (heat) energy of gas particles Hydrostatic equilibrium when pressure of heated gas = inward gravitational pressure Cloud condenses when Example: cloud of molecular H with mass = M  and T=20K : r crit = 0.35 kpc For smaller scale fluctuations in cosmic fluid: use Jeans criterium Structures lect 445

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 ρ and sound speed v s Sound wave in gas = pressure wave Sound velocity in an ideal gas Structures lect 446

Jeans length = critical dimension Compare cloud (density fluctuation) size L with Jeans length when L << λ J : no condensation When L >> λ J : cloud condenses around the density perturbations Structures lect 447

Evolution of fluctuations in radiation era After inflation : photons and relativistic particles Radiation dominated till decoupling at z ≈ 1100 Velocity of sound is relativistic horizon distance ~ Jeans length : no condensation of matter (equ 5.47 in chap 5) Density fluctuations inside horizon survive – fluctuations continue to grow in amplitude as R(t) Structures lect 448

Evolution of fluctuations in matter era After decoupling: dominated by non-relativistic gas Neutral atoms (H) are formed – sound velocity drops Jeans length becomes smaller – Horizon grows – therefore faster gravitational collapse – Matter structures can grow Cold dark matter plays vital role in growth of structures Structures lect 449

Damping by photons Fluctuations are most probably adiabatic: baryon and photon densities fluctuate together Photons travel at light velocity - have most of time 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 lect 450

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 Damping by neutrinos Structures lect 451

CMB - 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 Structures lect 452 C(θ) = correlation between temperature fluctuations in 2 directions (n,m) separated by angle θ average over all pairs with given θ

CMB - Angular spectrum of anisotropies 2 Expand in Legendre polynomials, integrate over ϕ C ℓ describes intensity of correlations Structures lect 453

Structures lect 454 Physics of Young universe Large wavelengths Physics of primordial fluctuations Short wavelengths Silk damping 1° = horizon at decoupling WMAP result

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 Structures lect 455

Examen Book Perkins 2 nd edition chapters 5, 6, 7, 8, + slides Delen die niet gekend moeten zijn wordt opgestuurd per ‘Open book’ examen Structures lect 456

Structures lect 457 Questions?