ASTROPHYSICS I: lecture notes, ETH Zurich 2018 9 Cosmology covers in Choudhuri Section 9.3 and 9.6, and large fractions of Sections 10 and 11 notes provide a simplified treatment for book sections: 11.2 and 11.9. additional topic: the concordance model 9.1 Expansion of the local Universe Doppler shifts, if objects move with radial velocity v relative to the observer { positive Doppler shift = redshift z = motion away from observer { for v ! c one needs to use the relativistic speed { the following is valid for v << c (say v < 0:1c) ∆λ vc v z = z}|{= λ c alternative description λ λ + ∆λ ∆λ vc v obs = em = 1 + = 1 + z z}|{= 1 + λem λem λem c relativistic Doppler effect: v u λobs u1 + v=c = t = 1 + z λem 1 − v=c { (1 + β)=(1 − β) = (1 + β)(1 + β + β2 + β3 + :::) ! (1 + β)2 for β ! 0 { for v > 0:6c ! z > 1 and ∆λ > λ Hubble law { Slipher (1914) noted that most galaxies are redshifted { Hubble (1929) proposed linear relationship between velocity and distance (Hubble law) v = H0 d { H0: Hubble constant (not a constant in time) { later studies confirmed this general expansion law Interpretation: { the whole (observable) universe is expanding { no center can be recognized { we are part of a big explosion or Hubble flow 75 Determination of the Hubble constant Task: distance determination for galaxies { absolute luminosities of galaxies are hard to determine accurately { must be calibrated in many steps, determining and using M for standard candles 5 log10 d[pc] = m − M + 5 problems: { m can be affected by interstellar extinction { bright objects are rare and M is not well determined distance ladder step method reachable calibration problems distance 1. trigonometric paral- 200 pc main sequence lumi- only few stellar clus- laxes nosity ters 2. main sequence fitting 100 kpc stellar clusters / LMC differencies in stellar (LMC) with Cepheids metallicities 3a. Cepheids P-L-relation 20 Mpc normal galaxies a lot of data required 3b. Cepheids P-L-relation 20 Mpc local Hubble law peculiar velocity of galaxies 3c. Cepheids P-L-relation 20 Mpc galaxies with SNIa SNIa are rare light curve 4. e.g. Tully-Fisher rel. 100 Mpc distant objects TF-rel. has large dis- persion 5. SNIa 2 Gpc deviations from linear best large distance Hubble law calibrator Direct methods to determine Hubble law (alternative) gravitational lensing { time delays for individal images of lensed, variable quasar ! distance to source and lens { problem: uncertainties in the gravitational potential of lensing object Hubble constant: Problem for the determination of the Hubble constant: measured velocity: v = H0d + δv { peculiar velocities of galaxies δv ≈ ±1000 km/s { expected systematic effect H0 ≈ 50 − 100 km/s Mpc ! systematic effects dominates only for d > 15 Mpc ! for smaller distances the mean of a (large) sample is required 76 SLIDE: H0 from HST current value for the Hubble constant (HST-key program based on Cepheid distances) −1 −1 H0 = 72 ± 8 km s Mpc radial velocity yields distance for δv v v d = for v c H0 or redshift ! distance relation c d = z Mpc for v c H0 Hubble time assuming that expansion was constant since the beginning of the Universe { time when all galaxies were at the same point 1 17 9 tH = = 4:4 · 10 s = 14 · 10 yr H0 ! rough estimate for the age of the Universe Redshift surveys galaxy distances follow from redshift for z > 0:01 (vr = cz > 3000 km/s) ! 3-dim. galaxy distribution measured galaxy distribution shows: { filaments of enhanced galaxy densities { large empty bubbles (voids) { cluster of galaxies at intersections of filaments galaxy distribution on large scales: { isotropic { homogeneous SLIDES: redshift surveys for galaxies 77 9.2 Newtonian Cosmology Cosmological principle: space is homogeneous and isotropic follows from: { galaxy distribution { 3-K cosmic micro-wave background (CMB) Newtonian theory of gravity cannot describe universe: { g(r) = rΦ(r) = r(G R ρ/(r0 − r)dr0 { homogeneous space has a homogeneous gravitational potential Φ(r) = const { if rΦ = 0, then g = 0, ! wrong ! general relativity is required for description of the gravitation of the Universe Simplification: Newtonian cosmology { important cosmological equation from Newtonian mechanics agree with general relativity ! easy description of cosmology Conceptual differences between Newtonian cosmology and relativistic cosmology: Newtonian cosmology relativistic cosmology we are at the center of the Universe all points in space are equivalent galaxies are moving away in space space is expanding with galaxies redshift cause by Doppler effect due to re- redshift caused by stretching of light wave- cession of galaxies length due to expansion Newtonian cosmology adopts 2 principles of the field theory from general relativity { A. special relativity is valid like in electrodynamics (Maxwell's equation) { information propagates with c (action at a distance is not instantaneous) { B. Einsteins equivalence principle { a gravitational field is equivalent to an accelerated frame { gravity is considered as curvature of space time (gravity ! spacetime is curved) { bodies are deflected from a rectilinear path in a gravitational field 9.3 The metric in general relativity definitions: metric: description of the curvature of spacetime geodesic: shortest path between two points Examples A. no gravitational fields ! space is flat and bodies move on straight lines B. with gravitational fields ! space is curved and a body moves on curved path 78 Curvature for 2-dim. surfaces: A: flat (plane) surface { straight lines are geodesics { sum of the 3 angle in triangles α + β + γ = 180◦ B: curved surface { geodesic are curved { triangle α + β + γ 6= 180◦ { spherical surface: great circles are geodesics metric describes: 2 { distance squared ds between two nearby points (x1; x2; :::) and (x1 + dx1; x2 + dx2; :::) Examples: { for a plane surface in polar coordinates (r; theta): ds2 = a2(dr2 + r2dθ2) { for a spherical surface with radius a (θ; φ): ds2 = a2 (dθ2 + sin2 θdφ2) = a2dΩ2 { general formulation (gα,β is the metric tensor) 2 ds = Σα,β gα,β dxαdxβ 9.4 Metric for the universe cosmological principle: homogeneous and isotropic geometry ! metric has uniform (constant) curvature everywhere ! there are only three cases under these conditions { zero curvature ! flat (plane) geometry { positive curvature ! spherical geometry { negative curvature ! saddle-like geometry SLIDE: spherical and saddle like surfaces Overview on the mathematical formulation of metrics: curvature zero positive negative plane (flat) spherical saddle-like 2-dimensional surface 2 2 2 2 2 2 2 2 2 2 2 2 2 ds = a (dx1 + x1dx2) a (dx1 + sin x1dx2) a (dx1 + sinh x1dx2) 3-dimensional space ds2 = a2 (dχ2 + χ2dΩ2) a2 (dχ2 + sin2 χdΩ2) a2 (dχ2 + sinh2 χdΩ2) 3-dimensional space (alternative formulation) χ = r sin χ = r sinhχ = r ds2 = a2 (dr2 + r2dΩ2) a2(dr2=(1 − r2) + r2dΩ2) a2(dr2=(1 + r2) + r2dΩ2) 3-dimensional space (compact formulation) k = 0 k = +1 k = −1 ds2 = a2((dr2=(1 − kr2) + r2dΩ2) ! 79 Example: Volume of the positive curved Universe positive curved space metric has a finite volume volume element with length: { a dχ, a sin χdθ and a sin χ sin θdφ { yields volume element dV = a3 sin2χdχ sin θdθdφ and total volume (χ extending from 0 to π) Z χ=π Z θ=π Z φ=2π V = a3 sin2 χdχ sin θdθ dφ = 2π2a3 χ=0 θ=0 φ=0 ! comparable to a sphere which has a finite surface area without any edges Spacetime: add a time part according to special relativity ds2 = −c2dt2 + dx2 + dy2 + dz2 ! Robertson-Walker metric = 4-dimensional spacetime metrics of the Universe: dΩ dr2 z }| { ds2 = −c2dt2 + a(t)2 + r2 (dθ2 + sin2 θdφ2) 1 − kr2 { only three values k = 0; +1; −1 for homogeneous and isotropic space { a(t) is the scale factor which can vary with time ! An expanding universe: { coordinates (r; θ; φ) defined with respect to a references (us) { co-moving coordinates (r; θ; φ) are constant in time for a galaxy (peculiar motion of galaxies are neglected) measure of distance d(t): Z r 1 d(t) = a(t) p dr0 0 1 − kr02 speed of the expansion v(t): Z r 1 v(t) =a _(t) p dr0 0 1 − kr02 ! equivalent to the Hubble law v(t) a_(t) H(t) = = d(t) a(t) 80 9.5 Friedmann equation Universe models based on Friedmann equation { old, outdated models { but give a simple picture on basic cosmological models Assumptions: { spherical shell with radius a { located inside a larger volume with constant density: SLIDE: sketch for spherical expansion potential energy per unit shell mass GM (4=3)πGρa3 4 E = − = = − πGρa2 pot a a 3 kinetic energy per unit shell mass 1 E = a_ 2 kin 2 the total mechanical energy E (constant of motion) 1 4 E = a_ 2 − πGρa2 2 3 is equal to the total energy described by the metric of the Universe (follows from Einstein equation): kc2 E = − metric 2 where k = 0; −1; +1 Friedmann equation: a_ 2 kc2 8πG + = ρ a2 a2 3 Analogy with mechanics: { a projectile with positive total energy escapes to infinity { a projectile with negative total energy falls back due to the attraction of gravity k has opposite sign to energy E: { k = −1 ! Universe will expand forever, E > 0 { k = +1 ! Universe expansion stops and collapses, E < 0 { k = 0 ! Universe is exactly on borderline, or expansion stops for a = 1, E = 0 Borderline: { k = 0 defines the critical density of the Universe ρc 3H2 ρ = c 8πG ρc for present day Hubble constant H0 is −26 −3 ρc;0 ≈ 1 · 10 kg m (corresponds to the mass of 6 protons per m3) 81 density parameter: ratio between measured ρ0 and critical ρc;0 ρ Ω = 0 ρc;0 Temporal evolution parameters evolve with time: { a(t),a _(t), H(t), ρ(t) and Ω(t) { values at present epoch: a0,_a0, H0, ρ0 and Ω0 { evolution of energy density: { energy density = matter density, for a matter dominated universe a 3 ρc2 = ρ c2 0 0 a Solutions for the Friedmann equations: A.