1 Astrophysical Cosmology Andy Taylor Institute for Astronomy, University of Edinburgh, Royal Observatory Edinburgh

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<ul><li> Slide 1 </li> <li> 1 Astrophysical Cosmology Andy Taylor Institute for Astronomy, University of Edinburgh, Royal Observatory Edinburgh </li> <li> Slide 2 </li> <li> 2 Lecture 1 </li> <li> Slide 3 </li> <li> 3 </li> <li> Slide 4 </li> <li> 4 The large-scale distribution of galaxies </li> <li> Slide 5 </li> <li> 5 Temperature Variations in the Cosmic Microwave Background </li> <li> Slide 6 </li> <li> 6 Properties of the Universe Universe is expanding. Components of the Universe are: Universe is 13.7 Billion years old. Expansion is currently accelerating. </li> <li> Slide 7 </li> <li> 7 1920s: The Great Debate Are these nearby clouds of gas? Or distant stellar systems (galaxies) </li> <li> Slide 8 </li> <li> 8 In 1924 Edwin Hubble finds Cepheid Variable stars in M31. Cepheid intrinsic brightness correlate with variability (standard candle), so can measure their distance. Measured 3 million light years (1Mpc) to M31. Eye near far brighter fainter Edwin Hubble 1920s: The Great Debate </li> <li> Slide 9 </li> <li> 9 The Expanding Universe Between 1912 and 1920 Vesto Slipher finds most galaxys spectra are redshifted. Slipher is first to suggest the Universe is expanding ! Vesto Slipher </li> <li> Slide 10 </li> <li> 10 In 1929 Hubble also finds fainter galaxies are more redshifted. Infers that recession velocities increase with distance. Hubbles Law </li> <li> Slide 11 </li> <li> 11 The Expanding Universe 1. Grenade Model: </li> <li> Slide 12 </li> <li> 12 The Expanding Universe 2. Scaling Model: x(t) = R(t) x 0 </li> <li> Slide 13 </li> <li> 13 The Expanding Universe D t Now Hubble Time: t H =1/H H=70 km/s/Mpc t H =14Gyrs tHtH 0 </li> <li> Slide 14 </li> <li> 14 In 1915 Albert Einstein showed that the geometry of spacetime is shaped by the mass-energy distribution. General Theory of Relativity required to describe the evolution of spacetime. Albert Einstein Relativistic Cosmologies </li> <li> Slide 15 </li> <li> 15 Cosmological Coordinates (t, x): How do we lay down a global coordinate system? In general we cannot. Can we lay down a local coordinate system? Yes, can use Special Relativity locally, if we can cancel gravity. We can cancel gravity by free-falling (equivalence principle). Relativistic Cosmologies </li> <li> Slide 16 </li> <li> 16 Relativistic Cosmologies Equivalence Principle: </li> <li> Slide 17 </li> <li> 17 Relativistic Cosmologies Equivalence Principle: </li> <li> Slide 18 </li> <li> 18 Relativistic Cosmologies In free-fall, a Fundamental Observer locally measures the spacetime of Special Relativity. Special Relativity Minkowski-space line element: So all Fundamental Observers will measure time changing at the same rate, dt. Universal cosmological time coordinate, t. </li> <li> Slide 19 </li> <li> 19 Relativistic Cosmologies How can we synchronize this Universal cosmological time coordinate, t, everywhere? With a Symmetry Principle: On large-scales Universe seems isotropic (same in all directions, eg, Hubble expansion, galaxy distribution, CMB). Combine with Copernican Principle (were not in a special place). </li> <li> Slide 20 </li> <li> 20 Relativistic Cosmologies Isotropy + Copernican Principle = homogeneity (same in all places) A 0 B 1 2 So 2 = 1 = 0. So uniform density everywhere </li> <li> Slide 21 </li> <li> 21 Relativistic Cosmologies Isotropy + homogeneity = Cosmological Principle A 0 B 1 2 So 2 = 1 = 0. So uniform density everywhere </li> <li> Slide 22 </li> <li> 22 Relativistic Cosmologies With the Cosmological Principle, we have uniform density everywhere. Density will decrease with expansion, so = (t). So can synchronize all Fundamental Observers clocks at pre-set density, 0, and time, t 0 : </li> <li> Slide 23 </li> <li> 23 Relativistic Cosmologies What is the line element (metric) of a relativistic cosmology? Locally Minkowski line element (Special Relativity): t x Worldline Proper time: d dt Universal time dx/c Spacetime Diagram: </li> <li> Slide 24 </li> <li> 24 Lecture 2 </li> <li> Slide 25 </li> <li> 25 Relativistic Cosmologies A general line element (Pythagoras on curved surface): Minkowski metric tensor: We have Universal Cosmic Time of Special Relativity, t, so </li> <li> Slide 26 </li> <li> 26 What is spatial metric, 3 2 ? From Cosmological Principle (homogeneity + isotropy) spatial curvature must be constant everywhere. Only 3 possibilities: Sphere positive curvature Saddle negative curvature Flat zero curvature. Relativistic Cosmologies </li> <li> Slide 27 </li> <li> 27 What is form of 3 2 ? Consider the metric on a 2-sphere of radius R, 2 2 : Relativistic Cosmologies dd dd dd R </li> <li> Slide 28 </li> <li> 28 Relativistic Cosmologies dr dd dd R The metric on a 2-sphere of radius R: Now re-label as r and as : where r = (0, ) is a dimensionless distance. </li> <li> Slide 29 </li> <li> 29 Can generate other 2 models from the 2-sphere: Relativistic Cosmologies k = +1 k = - 1 k = 0 </li> <li> Slide 30 </li> <li> 30 General 3-metric for 3 curvatures: Relativistic Cosmologies k = +1 k = - 1 k = 0 </li> <li> Slide 31 </li> <li> 31 Different properties of triangles on curved surfaces: Relativistic Cosmologies k = 0 k = +1 r d sin r d r r dd </li> <li> Slide 32 </li> <li> 32 Different properties of triangles on curved surfaces: Relativistic Cosmologies k = +1 k = - 1 k = 0 dd sin r d sinh r d r d r r r </li> <li> Slide 33 </li> <li> 33 Finally add extra compact dimension: Promote a 2-sphere to a 3-sphere So metric of 3-sphere is Relativistic Cosmologies dd dd </li> <li> Slide 34 </li> <li> 34 The Robertson-Walker metric generalizes the Minkowski line element for symmetric cosmologies: Relativistic Cosmologies k = +1 k = - 1 k = 0 - The Robertson-Walker Metric Invariant proper time Universal Cosmic time Scale factor Co-moving radial distance Co-moving angular distance </li> <li> Slide 35 </li> <li> 35 Alternative form of the Robertson-Walker metric: Relativistic Cosmologies k = +1 k = - 1 k = 0 </li> <li> Slide 36 </li> <li> 36 Alternative form of the Robertson-Walker metric: Relativistic Cosmologies k = +1 k = - 1 k = 0 </li> <li> Slide 37 </li> <li> 37 The Robertson-Walker models. Relativistic Cosmologies k = +1 k = - 1 k = 0 k = +1: positive curvature everywhere, spatially closed, finite volume, unbounded. k = - 1: negative curvature everywhere, spatially open, infinite volume, unbounded. k = 0: flat space, spatially open, infinite volume, unbounded. </li> <li> Slide 38 </li> <li> 38 The Robertson-Walker models. Relativistic Cosmologies We have defined the comoving radial distance, r, to be dimensionless. The current comoving angular distance is: d = R 0 S k (r) (Mpc). The proper physical angular distance is: d(t) = R(t)S k (r) (Mpc). </li> <li> Slide 39 </li> <li> 39 Lecture 3 </li> <li> Slide 40 </li> <li> 40 Relativistic Cosmologies Superluminal expansion: The proper radial distance is The proper recession velocity is: What does this mean? Locally things are not moving (just Special Relativity). But distance (geometry) is changing. No superluminal information exchange. </li> <li> Slide 41 </li> <li> 41 Light Propagation How does light propagate through the expanding Universe? Let a photon travel from the pole (r=0) along a line of constant longitude (d =0,d =0). The line element for a photon is a null geodesic (zero proper time): </li> <li> Slide 42 </li> <li> 42 Light Propagation Equation of motion of a photon: The comoving distance light travels. </li> <li> Slide 43 </li> <li> 43 Light Propagation Lets assume R(t)=R 0 (t/t 0 ) : </li> <li> Slide 44 </li> <li> 44 Causal structure Lets assume &gt; 1: t l For t &gt;&gt; t 1, r is constant. This is called an Event Horizon. As t 1 tends to 0, l(t) diverges, everywhere is causally connected. l R t1t1 </li> <li> Slide 45 </li> <li> 45 Causal structure Lets assume &lt; 1: t l At early times all points are causally disconnected. The furthest that light can have travelled is called the Particle Horizon. t=0 </li> <li> Slide 46 </li> <li> 46 Cosmological Redshifts Consider the emission and observation of light: </li> <li> Slide 47 </li> <li> 47 Cosmological Redshifts Consider the emission and observation of light: A bit later: </li> <li> Slide 48 </li> <li> 48 Cosmological Redshifts But the comoving position of an observers is a constant: Say the wavelength of light is = c t: so </li> <li> Slide 49 </li> <li> 49 Cosmological Redshifts Can also understand as a series of small Doppler shifts: d=c t t=0 t= t V=Hd=cH t </li> <li> Slide 50 </li> <li> 50 Decay of particle momentum Every particle has a de Broglie wavelength: So momentum (seen by FOs) is redshifted too: Why? (Hubble drag, expansion of space?) d=Rr, V=Hd t=0 t= t </li> <li> Slide 51 </li> <li> 51 Lecture 4 </li> <li> Slide 52 </li> <li> 52 The Dynamics of the Expansion In 1922 Russian physicist Alexandre Friedmann predicted the expansion of the Universe Rr Newtonian Derivation: m M=4 (Rr) 3 /3 Birkhoffs Theorem </li> <li> Slide 53 </li> <li> 53 The Dynamics of the Expansion In 1922 Russian physicist Alexandre Friedmann predicted the expansion of the Universe Friedmann Equation: Rr m M=4 (Rr) 3 /3 Birkhoffs Theorem V </li> <li> Slide 54 </li> <li> 54 So a low-density model will evolve to an empty, flat expanding universe. There is a direct connection between density &amp; geometry: Geometry &amp; Density </li> <li> Slide 55 </li> <li> 55 So with the right balance between H and , we have a flat model. There is a direct connection between density &amp; geometry: Geometry &amp; Density </li> <li> Slide 56 </li> <li> 56 We can define a critical density for flat models and hence a density parameter which fixes the geometry. Critical density &amp; density parameter k = +1 c &gt;1 k = - 1 c k = 0 c </li> <li> Slide 57 </li> <li> 57 How does evolve with time? Critical density &amp; density parameter t 1 </li> <li> Slide 58 </li> <li> 58 What is present curvature length? Critical density &amp; density parameter Define a dimensionless Hubble parameter: </li> <li> Slide 59 </li> <li> 59 What is present density? Critical density &amp; density parameter Or 1 small galaxy per cubic Mpc. Or 1 proton per cubic meter. </li> <li> Slide 60 </li> <li> 60 The meaning of the expansion of space Consider an expanding empty, spatially flat universe. c.f. a relativistic Grenade Model: Minkowski metric: Let v=Hr, H=1/t so v=r/t. Switch to comoving frame: </li> <li> Slide 61 </li> <li> 61 The meaning of the expansion of space Rewrite in terms of t (comoving time): Hence in the comoving frame: but this is a k=-1 open model with R=ct! So what is curvature? And is space expanding ? </li> <li> Slide 62 </li> <li> 62 The matter dominated universe Consider a universe with pressureless matter (dust, galaxies, or cold dark matter). As Universe expands, density of matter decreases: = 0 (R/R 0 ) -3. Consider a flat model: k=0, =1. R t </li> <li> Slide 63 </li> <li> 63 Lecture 5 </li> <li> Slide 64 </li> <li> 64 The matter dominated universe The spatially flat, matter-dominated model is called the Einstein-de Sitter model. R t </li> <li> Slide 65 </li> <li> 65 The matter dominated universe Consider an open or closed, matter-dominated universe. Define a conformal time, d cdt/R(t). R t </li> <li> Slide 66 </li> <li> 66 The matter dominated universe Consider a closed, matter-dominated universe. Define a conformal time, d cdt/R(t). R t </li> <li> Slide 67 </li> <li> 67 The matter dominated universe Consider an open or closed, matter-dominated universe. Define a conformal time, d cdt/R(t). R t </li> <li> Slide 68 </li> <li> 68 The matter dominated universe Expand forever Eventual recollapse Big Bang Big Crunch k = -1 k = +1 &lt; 1 = 1 &gt; 1 So for matter-dominated models geometry/density=fate. k = 0 </li> <li> Slide 69 </li> <li> 69 The radiation dominated universe As Universe expands, density of matter decreases: m = 0m (R/R 0 ) -3. Radiation energy density: r = 0r (R/R 0 ) -4. At early enough times we have radiation-dominated Universe. Log R Log mm rr For T(CMB)=2.73K, z eq =1000. </li> <li> Slide 70 </li> <li> 70 The radiation dominated universe At early enough times we also have a flat model: k=0 So Particle Horizon! </li> <li> Slide 71 </li> <li> 71 The radiation dominated universe Timescales: Matter-dominated: R~t 2/3 Radiation dominated: R~t 1/2 </li> <li> Slide 72 </li> <li> 72 The radiation dominated universe Spatial flatness at early times: Recall: How close to 1 can this be? At Planck time (t=10 -43 s)? t 1 </li> <li> Slide 73 </li> <li> 73 Energy density and Pressure Thermodynamics and Special Relativity: So energy-density changes due to expansion. </li> <li> Slide 74 </li> <li> 74 Energy density and Pressure For pressureless matter (CDM, dust, galaxies): Radiation pressure: Cf. electromagnetism. Conservation of energy: </li> <li> Slide 75 </li> <li> 75 Lecture 6 </li> <li> Slide 76 </li> <li> 76 Pressure and Acceleration Time derivative of Friedmann equation: Acceleration equation for R: </li> <li> Slide 77 </li> <li> 77 Vacuum energy and acceleration Gravity responds to all energy. What about energy of the vacuum? Two possibilities: 1.Einsteins cosmological constant. 2.Zero-point energy of virtual particles. </li> <li> Slide 78 </li> <li> 78 Einsteins Cosmological Constant Einstein introduced constant to make Universe static. </li> <li> Slide 79 </li> <li> 79 Einsteins Cosmological Constant Problem goes back to Newton (1670s). Einsteins 1917 solution: </li> <li> Slide 80 </li> <li> 80 Einsteins Cosmological Constant Einstein called this: My greatest blunder. But this is not stable to expansion/contraction. </li> <li> Slide 81 </li> <li> 81 British physicist Paul Dirac predicted antiparticles. Werner Heisenbergs Uncertainty Principle: Vacuum is filled with virtual particles. Observable (Casmir Effect) for electromagnetism. Zero-point vacuum energy +- +- </li> <li> Slide 82 </li> <li> 82 The Vacuum Energy Problem So Quantum Physics predicts vacuum energy. But summation diverges. If we cut summation at Planck energy it predicts an energy 10 120 times too big. Density of Universe = 10 atoms/m 3 Density predicted = 1 million x mass of the Universe/m 3 Perhaps the most inaccurate prediction in science? Or is it right? </li> <li> Slide 83 </li> <li> 83 Vacuum energy Vacuum energy is a constant everywhere: V ~ R 0 Thermodynamics: Consider a piston: The equation of state of the vacuum. </li> <li> Slide 84 </li> <li> 84 Vacuum energy and acceleration Effect of negative pressure on acceleration: So vacuum energy leads to acceleration. R t Eddington: is the cause of the expansion. </li> <li> Slide 85 </li> <li> 85 General equation of State In general should include all contributions to energy-density. Log R Log mm rr VV </li> <li> Slide 86 </li> <li> 86 General equation of State In general must solve F.E. numerically. Geometry is still governed by total density: 0 1 2 m 1 V 0 OPEN CLOSED FLAT </li> <li> Slide 87 </li> <li> 87 General equation of State In general must solve F.E. numerically. But in general no geometry-fate relation: 0 1 2 m 1 V 0 OPEN CLOSED FLAT RECOLLAPSE R...</li></ul>

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