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Cosmic Microwave Background Cosmic Microwave Background Galaxies 626 Lecture 3: From the CMBR to the first star Galaxies 626 Firstly, some very brief cosmology for background and notation: Summary: Foundations of Cosmology 1. Universe is homogenous and isotropic - need to consider large volumes - very good observational evidence from CMB 2. Universe is expanding - Hubble law compatible with homogenous / isotropic assumptions 3. Universe was once hot - existence of the microwave background with a thermal spectrum 4. Evolution described by General Relativity Expansion of the Universe Expansion of Universe cannot Homogeneity + isotropy alter the relative orientations of galaxies expanding with the Universe Means that if the present separation between two galaxies is d0, then the separation at time t can be written as: d = d0a(t) a(t) is the scale factor - it is dimensionless and depends upon time but not on position. Relative velocity of the two galaxies is: a˙ ! v = d˙ = d a˙ (t) = d 0 a ! Definition of the Hubble parameter is v = H x d, so: a˙ H = a H is a function of time, present value is denoted H0 Sometime useful to define comoving coordinates. If we divide distances! by a(t), then two galaxies which simply recede from each other due to the Hubble expansion always have the same separation in comoving coordinates. We will usually express densities as per comoving volume equivalent to the density they would have after expansion to the present time. Can derive the evolution of a(t) using mostly Newtonian mechanics, provided we accept two results from General Relativity: 1) Birkhoff’s theorem: this states (in part) that for a spherically symmetric system, the force due to gravity at radius r is determined only by the mass interior to that radius. 2) Energy contributes to the gravitating mass density, which equals: u energy density " + -3 m c 2 (ergs cm ) of radiation and relativistic particles density of matter ! Consider the evolution of a spherical volume of the Universe, radius L: Sphere expands with the Universe, L so L = L0a(t) Since expansion is described entirely by a(t), can consider any size sphere we want - if L is small `reasonable’ to assume that space is approximately Euclidean. Expansion of the sphere will slow due to the gravitational force of the matter (+energy) inside: d 2L GM = - dt 2 L2 ! Note: no pressure forces because Universe is homogenous Contributions to the gravitating mass come from matter plus energy density from radiation: • Matter density ρm • Radiation with energy density u has pressure: 1 P = u 3 gravitating mass density is: 3P " = " + ! m c 2 Mass within sphere, radius L, is: 4 3 ! M = "V = #L " 3 ! Substitute into acceleration equation: d 2L G 4 = - " #L3$ dt 2 L2 3 Since L = L0a(t), with L0 a constant, can write this as an equation for the evolution of the scale factor a(t): ! 4"G $ 3P' a˙˙ = - & # + ) a 3 % m c 2 ( (also substituting for ρ in the above expression) • Matter density ρm > 0 !• Pressure of radiation is also positive RHS of the equation is always negative Impossible to have a static Universe Lack of static solutions is not a problem - Universe is expanding. But this was not known in 1917. Einstein therefore modified the equations of General Relativity so the equation becomes: 4"G $ 3P' * a˙˙ = - & # + ) a+ a 3 % m c 2 ( 3 Λ is the cosmological constant (the factor 3 is just convention). A positive cosmological constant tends to ! accelerate the expansion - i.e. as if the Universe is filled with material with a negative pressure. Is a static solution stable? Properties of the cosmological constant Cosmological constant is assumed to be a smooth component… i.e. it does not cluster or clump together in the same way as ordinary matter. Original cosmological constant was… constant in time! This is just an assumption, however - models in which the vacuum energy varies with time are called quintessence. For Λ to be important today, it must have a value comparable to the first term in the equation: %36 -2 (for ρ ~ 10-30 g cm-3) " ~ 4#G$m ~10 s Gh t = =10"43 s `Fundamental’ unit of time is the Planck time: Planck c 5 #2 120 Might guess that " ~ tPlanck …bad guess by factor 10 . ! ! Cosmological constant problem… ! Which terms are most important? 4"G $ 3P' * a˙˙ = - & # + ) a+ a 3 % m c 2 ( 3 • Early times - energy density of radiation is large compared to the energy density of matter !• Later, matter dominates • Finally, if Λ is non-zero, eventually it dominates Radiation dominated Each of these changes Matter dominated in different way as Universe expands - Cosmological constant distinct expansion dominated laws Galaxies 626 The cosmic microwave background Cosmic Microwave Background Following recombination, photons that were coupled to the matter have had very little subsequent interaction with matter. Now observed as the cosmic microwave background. Arguably the most important cosmological probe, because it originates at a time when the universe was very nearly uniform: • Fluctuations were small - easy to calculate accurately (linear rather than non-linear) • Numerous complications associated with galaxy and star formation (cooling, magnetic fields, feedback) that influence other observables not yet important. Basic properties: isotropy, thermal spectrum Anisotropies: pattern of fluctuations Basic Properties of the CMB Excellent first approximation: CMB has a thermal spectrum with a uniform temperature in all directions Thermal spectrum: support for the hot big bang model Isotropy: evidence that the universe is homogenous on the largest observable scales The thermal radiation filling the universe maintains a thermal spectrum as the universe expands Suppose that at recombination the radiation has a thermal spectrum with a temperature T ~ 3000 K. Spectrum is given by the Planck function: 2h" 3 1 B = " c 2 eh" / kT #1 At time t, number of photons in volume V(t) with frequencies between ν and ν + dν is: ! 8"# 2 1 dN(t) = V(t)d# c 3 eh# / kT $1 ! Now consider some later time t’ > t. If there have been no interactions, the number of photons in the volume remains the same: dN(t) = dN(t' ) However, the volume has increased with the expansion of the universe and each photon has been redshifted: a3 (t' ) V(t' ) =V(t) ! a3(t) a(t) "' =" a(t' ) a(t) d"' = d" a(t' ) Substitute for V(t), ν and dν in formula for dN(t), and use fact that dN(t’) = dN(t) ! Obtain: '2 ' # 1 ' ' dN(t ) = 8" 3 ' V(t )d# c e(h# / kT )$(a(t')/ a(t )) %1 which is a thermal spectrum with a new temperature: a(t) T' = T ! a(t' ) Conclude: radiation preserves its blackbody spectrum as the universe expands, but the temperature of the blackbody decreases: ! T " a#1 "(1+z) Recombination happened when T ~ 3000 K, at a redshift z = 1090. ! CMB Anisotropies Universe at the time of recombination was not completely uniform - small over (under)-densities were present which eventually grew to form clusters (voids) etc. In the microwave background sky, fluctuations appear as: • A dipole pattern, with amplitude: "T #10$3 T Origin: Milky Way’s velocity relative the CMB frame. Reflects the presence of local mass concentrations - clusters, !superclusters etc. • Smaller angular scale anisotropies, with ΔT / T ~ 10-5 Experiments detect any cosmic source of microwave radiation - not just cosmic microwave background: • Low frequencies - free-free / synchrotron emission • High frequencies - dust CMB dominates at around 60 GHz Also different spectra - can be separated given measurements at several different frequencies WMAP results K band - 22 GHz W band - 94 GHz Directly `see’ the primordial CMB anisotropy at these frequencies Full sky map from WMAP • Dipole subtracted (recall dipole is much larger than the smaller scale features) • Galactic foreground emission subtracted as far as possible Characterizing the Microwave Background Sky First approximation - actual positions of hot and cold spots in the CMB is `random’ - does not contain useful information Cosmological information is encoded in the statistical properties of the maps: • What is the characteristic size of hot / cold spots? ~ one degree angular scale • How much anisotropy is there on different spatial scales? CMB is a map of temperature fluctuations on a sphere - conventionally described in terms of spherical harmonics Spherical harmonics Any quantity which varies with position on the surface of a sphere can be written as the sum of spherical harmonics: "T (#,$) = almYlm (#,$) T % l,m spherical harmonic measured anisotropy function map as function! of weight - how much spherical polar angles of the signal is θ and φ accounted for by this particular mode The spherical harmonic functions themselves are just (increasingly complicated) trignometric functions, e.g.: 5 2 2i# Y22(",#) = 3sin "e 96$ ! l = 6, m = 0 l = 6, m = 3 Having decomposed the observed map into spherical harmonics, result is a large set of coefficients alm. Next compute the average magnitude of these coefficients as a function of l: 2 Cl " |alm| Plot of Cl as a function of l is described as the “angular power spectrum” of the microwave background. Each Cl measures how much anisotropy there is on a particular angular scale,! given by: 180o " ~ l Angular power spectrum is basic measurement to compare with theory ! Observational determinations of CMB anisotropy Early 2000 amount of anisotropy large scales small scales Red curve is a theoretical model - evidence for a peak but curve is not significantly constrained by the data at high l Compilation of all available data includes WMAP and some ground based / balloon experiments sensitive to smaller angular scales Peak at degree scales Decline toward very small Plateau at scales large scales Want to understand physical origin of each of these features The power spectrum reflects fluctuations in the density at the time of recombination: Photons escape from the overdense region Recombination Consider a slight overdensity collapsing during the radiation dominated phase.
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    ftoc.qrk 5/24/04 1:46 PM Page iii Contents Timeline v de Sitter,Willem 72 Dukas, Helen 74 Introduction 1 E = mc2 76 Eddington, Sir Arthur 79 Absentmindedness 3 Education 82 Anti-Semitism 4 Ehrenfest, Paul 85 Arms Race 8 Einstein, Elsa Löwenthal 88 Atomic Bomb 9 Einstein, Mileva Maric 93 Awards 16 Einstein Field Equations 100 Beauty and Equations 17 Einstein-Podolsky-Rosen Besso, Michele 18 Argument 101 Black Holes 21 Einstein Ring 106 Bohr, Niels Henrik David 25 Einstein Tower 107 Books about Einstein 30 Einsteinium 108 Born, Max 33 Electrodynamics 108 Bose-Einstein Condensate 34 Ether 110 Brain 36 FBI 113 Brownian Motion 39 Freud, Sigmund 116 Career 41 Friedmann, Alexander 117 Causality 44 Germany 119 Childhood 46 God 124 Children 49 Gravitation 126 Clothes 58 Gravitational Waves 128 CommunismCOPYRIGHTED 59 Grossmann, MATERIAL Marcel 129 Correspondence 62 Hair 131 Cosmological Constant 63 Heisenberg, Werner Karl 132 Cosmology 65 Hidden Variables 137 Curie, Marie 68 Hilbert, David 138 Death 70 Hitler, Adolf 141 iii ftoc.qrk 5/24/04 1:46 PM Page iv iv Contents Inventions 142 Poincaré, Henri 220 Israel 144 Popular Works 222 Japan 146 Positivism 223 Jokes about Einstein 148 Princeton 226 Judaism 149 Quantum Mechanics 230 Kaluza-Klein Theory 151 Reference Frames 237 League of Nations 153 Relativity, General Lemaître, Georges 154 Theory of 239 Lenard, Philipp 156 Relativity, Special Lorentz, Hendrik 158 Theory of 247 Mach, Ernst 161 Religion 255 Mathematics 164 Roosevelt, Franklin D. 258 McCarthyism 166 Russell-Einstein Manifesto 260 Michelson-Morley Experiment 167 Schroedinger, Erwin 261 Millikan, Robert 171 Solvay Conferences 265 Miracle Year 174 Space-Time 267 Monroe, Marilyn 179 Spinoza, Baruch (Benedictus) 268 Mysticism 179 Stark, Johannes 270 Myths and Switzerland 272 Misconceptions 181 Thought Experiments 274 Nazism 184 Time Travel 276 Newton, Isaac 188 Twin Paradox 279 Nobel Prize in Physics 190 Uncertainty Principle 280 Olympia Academy 195 Unified Theory 282 Oppenheimer, J.
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