Physical Cosmology Notes Phil Bull, December 12, 2019 Contents 1. Expanding universe 5 1.1. What is cosmology?............................................5 1.2. Brief history of the Universe.......................................6 1.3. The state of the Universe today......................................6 1.4. Olbers’ paradox..............................................7 1.5. Historical discovery of the expanding universe..............................8 1.6. Expansion and redshift..........................................8 1.7. Spectra..................................................9 1.8. Expansion and the Hot Big Bang model................................. 10 2. Geometry and distance 12 2.1. Recession velocity and Hubble’s Law.................................. 12 2.2. Parallax distance............................................. 13 2.3. Proper vs comoving coordinates..................................... 14 2.4. Measuring distances: the space-time metric............................... 14 2.5. Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric......................... 16 2.6. Geometry of space: open, closed, and flat universes........................... 16 2.7. Space-time metric with curvature..................................... 16 2.8. Useful unit conversions.......................................... 17 3. Friedmann equation 18 3.1. The Friedmann equation......................................... 18 3.2. Hubble parameter and expansion rate................................... 18 3.3. Critical density and curvature....................................... 19 3.4. Change in energy density as space expands............................... 20 3.5. Matter-only solution to the Friedmann equations............................. 20 3.6. Interchangeability of time, redshift, and scale factor........................... 21 3.7. Age of the Universe............................................ 21 3.8. Matter, curvature, and the fate of the Universe.............................. 22 3.9. Newtonian derivation of the Friedmann equation............................ 23 4. Distances and horizons 26 4.1. Cosmological distances.......................................... 26 4.2. Distance travelled by a light ray..................................... 26 4.3. Luminosity distance and standard candles................................ 27 4.4. Distance ladder.............................................. 27 4.5. Angular diameter distance........................................ 27 4.6. Cosmological horizon........................................... 28 5. Cosmic acceleration 30 5.1. Conservation equation.......................................... 30 5.2. Equation of state............................................. 30 5.3. Cosmic acceleration and deceleration.................................. 31 5.4. Deceleration parameter.......................................... 31 5.5. Properties of the cosmological constant................................. 32 5.6. Cosmological constant solution...................................... 32 5.7. Age and Hubble radius in an exponentially-expanding space-time................... 33 5.8. The Cosmological Constant problem................................... 33 1 5.9. The fate of our Universe......................................... 34 6. Cosmic Microwave Background Radiation 37 6.1. What was the early universe like?.................................... 37 6.2. Formation of the Cosmic Microwave Background............................ 37 6.3. The surface of last scattering....................................... 38 6.4. Recombination and decoupling...................................... 38 6.5. Recombination.............................................. 38 6.6. Decoupling................................................ 40 6.7. Blackbody spectrum of the CMB..................................... 41 7. Cosmic Microwave Background Anisotropies 43 7.1. CMB anisotropies............................................. 43 7.2. Physical processes that cause anisotropies................................ 44 7.3. Baryon acoustic oscillations....................................... 45 7.4. Diffusion damping............................................ 47 7.5. Secondary anisotropies.......................................... 47 7.6. Spherical harmonics............................................ 49 7.7. Power spectrum of the CMB....................................... 51 7.8. Features in the CMB power spectrum.................................. 51 7.9. Dependence on cosmological parameters................................. 53 8. Inflation 56 8.1. How special is our Universe?....................................... 56 8.2. The horizon problem........................................... 56 8.3. The flatness problem........................................... 58 8.4. The (magnetic) monopole problem.................................... 59 8.5. The inflationary mechanism....................................... 59 8.6. Cosmological Klein-Gordon equation.................................. 61 8.7. Scalar field dynamics........................................... 61 8.8. Slow-roll approximation......................................... 62 8.9. Quantum fluctuations and the primordial power spectrum........................ 63 8.10. Reheating................................................. 63 9. Big Bang Nucleosynthesis 65 9.1. Thermal history of the early Universe.................................. 65 9.2. Neutron decay............................................... 65 9.3. Nucleosynthesis.............................................. 66 10. Dark matter 70 10.1. Observational evidence for dark matter.................................. 70 10.2. Properties of dark matter......................................... 70 10.3. Particle dark matter............................................ 71 10.4. Baryonic dark matter and compact objects................................ 72 10.5. Warm vs cold dark matter......................................... 74 10.6. Hierarchical structure formation..................................... 74 10.7. Dark matter halos............................................. 75 11. Structure formation 78 11.1. Perturbation theory............................................ 78 11.2. Growth of matter fluctuations....................................... 79 11.3. Poisson equation............................................. 80 11.4. Fourier transforms............................................ 80 11.5. Matter power spectrum.......................................... 81 11.6. Correlation function............................................ 82 2 11.7. Peculiar velocities............................................. 84 12. Observational cosmology 85 12.1. Type Ia supernovae............................................ 85 12.2. Galaxy surveys.............................................. 85 12.3. Gravitational lensing........................................... 87 3 About this module Problem sheets and tutorials • There are 10 problem sheets, which will be released each week after the first lecture of the week (on Tuesday). Seven of them are assessed as your coursework, and should be handed in by Wednesday at 4pm the following week, to the School Office on Floor 1. • Each problem sheet includes some maths practice questions, some general exercises to help build your under- standing, and either a practice exam question, or an assessed coursework question. The assessed questions are graded, and contribute 10% of the final grade for the module. Solutions will be released for each problem sheet at the tutorial sessions, and posted online the following week. • Please attempt all of the questions! The problem sheets are designed so that if you can do all of the general exercises, you should have everything you need to complete the practice exam question or assessed question. • There are two tutorial sessions per week, at 11–12pm on Tuesday (Laws: 2.07 (1)) and 11–12pm on Wednesday (ENG: 2.16 (2)). They are run by myself and Rebeca Martinez Carrillho. You are free to attend either one, but it is important that you attend every week! We will help with all of the questions except the assessed ones, which you’re expected to do yourself. You are also encouraged to work in groups with other students to figure out the general questions, but you must attempt the assessed questions on your own. I have a zero tolerance policy for copying/cheating on assessed coursework questions! Mid-term exam There will be a mid-term exam on Tuesday of Week 6 (TBC!), just before Reading Week. This will be worth 10% of the total marks for the module. If you can do all of the questions on the problem sheets up to Week 5, you should be well-prepared for the mid-term. (Students who are unable to attend the mid-term due to extenuating circumstances should discuss re-sits with me.) Final exam There will be a final exam in the January exam period, worth 80% of the total marks for the module. A revision session will be held in Week 12. If you can do all of the questions in the problem sheets, you will be well-prepared for the exam. The exam questions will not be based directly on these problems though, so make sure you understand the material (rather than just practising the problems)! Past exam papers Past exam papers will be made available, but please not that the module content has changed since previous years. In particular, you should expect the exam to be more difficult! (This is why it’s strongly advisable to attend the tutorials, do all the problems etc...) Office hours and extra support My office hours are from 1–2pm on Tuesdays