hudson m. cosmology lecture notes

PHYS 787:COSMOLOGY Winter 2005 Mon & Fri 11:30-12:50 PM Main Link Rooms (W: EIT 2053, G: MacN 101) WWW: http://astro.uwaterloo.ca/~mjhudson/teaching/phys787 Instructor: Mike Hudson mjhudson@uwaterloo.ca Office: Physics 252 (ext 2212) Textbook: The primary textbook is Structure Formation in the Universe, T. Padmanabhan, 1993, Camb. Univ. Press. Other useful references are listed on the P787 WWW references page Prerequisites: None. Some knowledge of General Relativity is advanta- geous but is not required. Syllabus: 1. Observational Overview 2. Homogeneous Universe (a) Metric; Redshift (b) Dynamics (c) Distance; Ages; Volumes 3. Hot Big Bang (a) Thermodynamics (b) Recombination (c) Nucleosynthesis 4. Structure Formation (a) Linear Perturbation Theory (b) Statistics of LSS 1 (c) Nonlinear models 5. Galaxies and Galaxy Formation 6. Cosmic Microwave Background Fluctuations 7. Gravitational Lensing 8. Inflation Grading: Assignments 50% Term Paper & Seminar 50% The course WWW page: http://astro.uwaterloo.ca/~mjhudson/teaching/phys787 will always have the most up-to-date information. 2 Preamble About this course This course aims to give a broad review of modern cosmology. The emphasis is on physical cosmology, i.e. its content, the physical processes in the expanding Universe and the formation of structure from the horizon down to the scale of galaxies. I will focus on the current paradigm, the Big Bang model and structure formation in a Universe dominated by dark matter and dark energy. A deep knowledge of General Relativity is not necessary, although a familiarity with GR will make the course more palatable. Likewise a basic understanding of astrophysical processes and some knowledge of basic particle physics are helpful. In an effort to be broad some depth has necessarily been sacrificed, but I hope that enough background and reference pointers have been provided for the interested student to delve deeper on their own. 1 INTRODUCTION TO COSMOLOGY 1 1 Introduction to Cosmology 1.1 A Very Brief History Early Cosmological Models I will skip the full treatment of early cosmological models — which would cover the Ptolemaic model and the Copernican revolution via Tycho and Kepler — except to note that the “Copernican Principle”, i.e. that we do not live in a special place in the Universe, has proved to be influential. Newton’s cosmology was infinite. Time and space were absolute and independent of the matter in the Universe. Newton’s 1692 Letter to Richard Bentley: It seems to me, that if the matter of our sun and planets, and all the matter of the universe, were evenly scattered through all the heavens, and every particle had an innate gravity towards all the rest, and the whole space throughout which this matter was scattered, was finite, the matter on the outside of this would by its gravity tend towards all the matter on the inside, and by consequence fall down into the middle of the whole space, and there compose one great spherical mass. But, if the matter were evenly disposed throughout an infinite space, it could never convene into one mass, but some of it would convene into one mass and some into another, so as to make an infinite number of great masses, scattered great distances from one to another throughout all that infinite space. And thus might the sun and fixed stars be formed, supposing the matter were of a lucid nature. Problems with Newton’s Universe: • Stability • Olber’s paradox - an infinite universe would produce an infinite amount of light at our posi- tion, so ”why is the night sky dark?” Einstein’s Static Model In 1917, before discovery of cosmological redshifts, Einstein proposed a closed universe with a spherical geometry which was finite in extent, centreless and edgeless. In order to make this model static, Einstein introduced into GR a small repulsive force known as the cosmological constant. Einstein believed in a static Universe – to the extent that he was willing to add an extra parameter to his theory. Why? (Later he referred to the cosmological constant as his “greatest blunder”). 1 INTRODUCTION TO COSMOLOGY 2 Shortly afterward de Sitter discovered an expanding but empty solution of Einstein’s equations - motion without matter. Friedmann (1922) found solutions with both expansion and matter, which Lemaitre (1927) independently rediscovered. Why was the Universe assumed to be homogeneous? Early Extra-galactic Cosmography At the beginning of the twentieth century, it was generally accepted that our galaxy was disk- shaped and isolated. But what were the spiral “nebulae” like M31 (Andromeda) - were they inside or outside the Milky Way? Immanuel Kant had speculated that they were other “island” universes. In 1912, Slipher measured spectra from the nebulae, showing that many were Doppler-shifted. By 1924, 41 nebulae had been measured, and 36 of these were found to be receding. In 1929, Hubble measured the distances to “nebulae”. He measured Cepheid stars in nearby galax- ies such as M31 and then measured the relative distances between M31 and more distant galaxies by assuming that brightest stars were standard candles. Combining these with the known velocities (corrected to the velocity frame of the Milky Way), he obtained the plot shown in Fig. 1.1. Figure 1.1: Hubble’s plot of velocity versus distance Fitting a straight line, v = H 0 r , (1.1) 1 INTRODUCTION TO COSMOLOGY 3 Hubble found H 0 = 500 km/s/Mpc, a value about 7 times too large 1 The outstanding feature, however, is the possibility that the velocity-distance re- lation may represent the de Sitter effect, and hence that numerical data may be intro- duced into discussions of the general curvature of space. (Hubble 1929) 1.2 Review of Observational Cosmology 1.2.1 Preliminary Definitions Ω denotes a density divided by the critical density needed to close the Universe, ρ crit = 3H 2 8πG (1.2) Subscripts m, b, r, v denote the densities of matter, baryonic matter, radiation and vacuum. No subscript indicates the total density 2 Subscript 0 denotes the present-day value of a parameter, e.g. H 0 is the present-day value of the Hubble constant. Units In this section we will use “astronomer” units. 1 Megaparsec (Mpc) = 3.26 × 10 6 light years = 3.1 × 10 22 m 1 year = 3.16 × 10 7 s 1 Solar Mass (M) = 1.99 × 10 30 kg 1.2.2 Expansion of the Universe Fig. 1.2 shows a modern Hubble diagram using Type Ia supernovae as distance indicators. Note the deviations from linearity at large z, we will return to this later. Supernovae in all directions in 1 Hubble made two errors. First, Hubble assumed that the variable stars he observed in nearby galaxies (Cepheids) were the same as a different class of variable stars (W Virginis) in our galaxy. Second, what Hubble thought were bright stars in other galaxies were actually collections of bright stars. These errors were not discovered until the 1950s. 2 This convention is quite recent (and still by no means universal). In many sources Ω implicitly refers to matter. The contribution from the vacuum is often denoted Ω Λ , Λ, or λ depending on how it is normalized. 1 INTRODUCTION TO COSMOLOGY 4 Calan/Tololo (Hamuy et al,  A.J. 1996) Supernova Cosmology Project effective m B (0.5,0.5) (0, 0) ( 1, 0 ) (1, 0) (1.5,–0.5) (2, 0) (Ω Μ, Ω Λ ) =  ( 0, 1 ) Flat Λ = 0 redshift z 14 16 18 20 22 24 26 0.02 0.05 0.1 0.2 0.5 1.00.02 0.05 0.1 0.2 0.5 1.0 Perlmutter, et al. (1998) FAINTER (Farther) (Further back in time) MORE REDSHIFT (More total expansion of universe  since the supernova explosion) In flat universe: Ω M = 0.28 [± 0.085 statistical] [± 0.05 systematic] Prob. of fit to Λ = 0 universe: 1%  Figure 1.2: Hubble diagram for Type Ia Supernovae (Perlmutter et al.) the sky fit the curve: the expansion is indeed isotropic. The Hubble Space Telescope Key Project measured the flux of Cepheid stars in nearby galaxies to allow a calibration of the distance scale and hence the Hubble constant 3 h = H 0 /(100 km/s) = 0.72 ± 0.08 (1.3) 3 In fact, the Hubble constant is neither constant in space n– because of peculiar velocities – nor in time, so it would be better called the Hubble parameter. 1 INTRODUCTION TO COSMOLOGY 5 from (Freedman et al. 2001). 1.2.3 Isotropy and Homogeneity of the Universe The Universe is observed to be isotropic on very large scales. Fig. ?? plots a sample of distant galaxies on the sky: clustering is evident on small angular scales but on the largest scales the distribution looks smooth. Figure 1.3: This picture covers a region of sky about 100 degrees by 50 degrees around the South Galactic Pole. The intensities of each pixel are scaled to the number of galaxies in each pixel, with blue, green and red for bright, medium and faint galaxies (1-mag slices centred on B magnitude 18, 19 and 20). The many small dark ‘holes’ are excluded areas around bright stars, globular clusters etc. (From the APM survey.) By obtaining redshifts of galaxies and using Hubble’s law, we can plot the distribution of galaxies in 3D, as in Fig. 1.4. On the largest scales, the distribution of galaxies is homogeneous. On small scales (1 − 10 Mpc), mass is clumped in galaxies and clusters of galaxies. On intermediate scales (10 − 100 Mpc), clusters are grouped into superclusters and are connected by walls and filaments. 1.2.4 Cosmic Microwave Background (CMB) Gamow predicted relic radiation from a primeval fireball in 1948. Penzias & Wilson (Bell Labs Engineers) discovered the CMB in the radio in the 1960s. The spectrum of the CMB is a perfect black body with a temperature of 2.728 ± 0.004K. 1 INTRODUCTION TO COSMOLOGY 6 Figure 1.4: The 2-Degree-Field Galaxy Redshift Survey 1 INTRODUCTION TO COSMOLOGY 7 Figure 1.5: Three false color images of the sky as seen at microwave frequencies. The orientation of the maps are such that the plane of the Milky Way runs horizontally across the center of each image. The top figure shows the temperature of the microwave sky in a scale in which blue is 0 K and red is 4. Note that the temperature appears completely uniform on this scale. The middle image is the same map displayed in a scale such that blue corresponds to 2.721 Kelvin and red is 2.729 Kelvin. The ”yin-yang” pattern is the dipole anisotropy that results from the motion of the Sun relative to the rest frame of the cosmic microwave background. The bottom figure shows the microwave sky after the dipole anisotropy has been subtracted from the map. This removal eliminates most of the fluctuations in the map: the ones that remain are thirty times smaller. On this map, the hot regions, shown in red, are 0.0002 Kelvin hotter than the cold regions, shown in blue. The band across the centre is emission from our Galaxy. [...]... homogeneity? (This is know as the “horizon problem”) 3 33 CLASSIC OBSERVATIONAL COSMOLOGY 3 Classic Observational Cosmology The classic cosmological tests are based on the global proprties of the FRW model and its expansion history More modern tests based on the growth of structure will be discussed later 3.1 Distance Measures in Cosmology Conceptually, the simplest distance is the proper distance, i.e the... comparison between mass and galaxies yields Ω ≈ 0.3 (1.5) 1 INTRODUCTION TO COSMOLOGY 12 Figure 1.9: The rotation curve of the Milky Way Galaxy from a compilation of data (From Fich & Tremaine 1991) Figure 1.10: Light and mass in the cluster Cl 0024+17 if fluctuations in the mass follow fluctuations in the light 1 13 INTRODUCTION TO COSMOLOGY Figure 1.11: Density fluctuation fields of POTENT mass (left-hand... the orbits of galaxies in clusters and argued that there was evidence for the presence of dark matter The temperature of X-ray emitting plasma in clusters leads to the same conclusion 1 INTRODUCTION TO COSMOLOGY 10 Figure 1.7: Predicted abundances (by mass) of 4 He, D, 3 He and Li Rotation Curves of Spiral Galaxies How large are the DM haloes of galaxies? Fig 1.9 shows the rotational velocity of stars... large radii, well beyond the edge of the disk, the rotation curve is observed to be approximately constant This implies M (< r) ∝ r This believed to be due to a dark matter “halo” 1 11 INTRODUCTION TO COSMOLOGY Figure 1.8: The Baryon “Budget” (Fukugita, Hogan & Peebles, 1998 ApJ, 503,518) Gravitational Lensing Strong gravitational lensing leads to the formation of multiple images and giant arcs Weak...1 INTRODUCTION TO COSMOLOGY 8 Fluctuations in the CMB Dipole Term The CMB has a dipole anisotropy with an amplitude 3.358mK (known since 70s) This is interpreted as a Doppler effect due to the motion of the Sun with respect... as a Schechter function The predicted mass function of virialized (dark-matter dominated) objects has a similar behaviour, but breaks at larger masses corresponding to rich clusters 1 INTRODUCTION TO COSMOLOGY 14 Figure 1.12: Isochrone fit to M92.The best fit age is 15±1 Gyr From Harris et al 1997 What physics causes the break in the galaxy spectrum? Morphologies There are two different types of galaxies:... ellipticals have a weak disk Ellipticals are found mainly in rich clusters of galaxies, whereas spirals are found in low density regions How is galaxies related to mass on different scales? 1 15 INTRODUCTION TO COSMOLOGY 1.3 The “Standard Model” Our Universe is expanding from a hot big bang in which the light elements were synthesized There was a period of inflation which led to a “flat” universe today Structure... predictions for combinations of the latter parameters 4 Of course it is possible that the primordial fluctuation spectrum is not described by a simple power law as a function of scale 1 INTRODUCTION TO COSMOLOGY 16 The simplest viable model contains only a few parameters that need to be measured (h, A, Ωm , Ωb ) A more general model might have as many as a dozen parameters Occam says that we should prefer... Universe Also in agreement with assessments from weak lensing and large-scale flows Errors remain large, though • Big-bang nucleosynthesis gives: Ωb h2 = 0.02 in agreement with above 1 INTRODUCTION TO COSMOLOGY 17 • Hubble constant is: 0.72 ± 0.08 in agreement with the above • These numbers yield an age of the Universe of 12.2 Gyr, in agreement with the globular cluster ages Some outstanding issues... properties as well 2.2 The Metric In General Relativity (GR), space-time is described by a metric ds2 = c2 dt2 − dl2 This is compactly written ds2 = gαβ dxα dxβ where Greek indices run from 0 to 3 Index 0 denotes time (so dx0 could also be written dt) and the indices 1 to 3 indicate the spatial dimensions Light travels along paths with ds = 0 Other particles follow geodesics, which can be thought of as shortest . matter, primarily in groups/intergalactic medium. 1 INTRODUCTION TO COSMOLOGY 9 Figure 1.6: CMB temperature fluctuations as a function of multipole number l, showing data from many recent experiments 787 :COSMOLOGY Winter 2005 Mon & Fri 11:30-12:50 PM Main Link Rooms (W: EIT 2053, G: MacN 101) WWW: http://astro.uwaterloo.ca/~mjhudson/teaching/phys787 Instructor: Mike Hudson mjhudson@uwaterloo.ca Office:. fluctuations in the mass follow fluctuations in the light 1 INTRODUCTION TO COSMOLOGY 13 Figure 1.11: Density fluctuation fields of POTENT mass (left-hand column) vs. IRAS galaxies (middle column),