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Cosmology & the Big Bang

Cosmology & the Big Bang. AY16 Lecture 19, April 10, 2008 Introduction to Cosmology Basic Principles Fundamental Observations The FRW Metric. “My husband adheres to the Big Bang theory of creation.”. COSMOLOGY. The BIG Picture!

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Cosmology & the Big Bang

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  1. Cosmology & the Big Bang AY16 Lecture 19, April 10, 2008 Introduction to Cosmology Basic Principles Fundamental Observations The FRW Metric

  2. “My husband adheres to the Big Bang theory of creation.”

  3. COSMOLOGY The BIG Picture! The cosmological model dominates much of extragalactic astronomy, in fact much of astronomy & astrophysics, even physics. What is Cosmology? “The study of the large scale structure & evolution of the Universe” What is Cosmogony? “The study of the origin of observable structures” Cosmology is perhaps the oldest real science. Its tied to our “World View.”

  4. Changing Worldviews Age Universe ---------------------------------------------------------------------------------- 10,000 years BC --- The Valley you lived in 1,000 years BC --- Your Kingdom 300 years BC --- The Mediterranean (for Egypto-Greco-Romans, at least!) 100 years AD --- The Earth + Celestial Sphere 400 years ago --- The Solar System 100 years ago --- The Milky Way 75 years ago --- The “Modern” Universe (2 Billion Light -Years in *radius*) Today --- An Infinite Universe (the visible part has a Radius of ~15 Gly)

  5. A Brief History of Extragalactic Astronomy: ~ 1750 Early Rumblings of “Island Universes” from I. Kant, T. Wright, P. Laplace. This seems to have been forgotten soon after. 1800’s Catalogs of Things but no understanding. de la Caille, Messier, Herschel3, Dreyer William, Caroline & John ~ 1875The Discovery of the Galaxy --- Kapteyn’s Universe

  6. 1910 Removal of the Solar-Centric view 1900-1920 Shapley and the Great Debate 1907 Bohlin --- M31 Parallax 1918 Van Maanen --- M31 Parallax 1885 S Andromeda = SN1885a large reflectors + photographic plates 1920 The Shapley-Curtis debate Shapley + Globular Clusters + Cepheids 1924 Hubble & the Hooker --- NGC 6822 Cepheids, eventually M31 Cepheids

  7. 1910-1930 Theory! Einstein, Friedmann, deSitter, Lemaitre, Tolman, Robertson … 1922 Opik’s M31 Mass-to-Light ratio L = 4πr2  GMm/r = ½ mv2 M = ½ v2r/G, so M/L = ½ and Opik estimated D of M31 at 450 kpc. v21 1 G  4πD

  8. Hubble (+Slipher)  Velocity-Distance Law 1930’s Hubble’s Classification Scheme for Galaxies (Tuning Fork Diagram) N.B. Absolutely necessary but wrong interpretation, set galaxy evolution back 20 years! Hubble’s Galaxy Counts (Humason) • Zwicky & Smith DARK MATTER 1940’s Galactic Dust, Stellar Populations, Hubble Diagram

  9. Gamow & the Hot Big Bang 1950’s deVaucouleurs’ Local “Supergalaxy” Rubin: Flows Dicke: CM HMS Velocities + Hubble Diagram Baade & Sandage: H0 revisions Minkowski: Radio galaxies 1960’s The Hubble Constant Debate Tinsley: Stellar Evolution  Galaxy Evolution

  10. Greenstein & Schmidt: Quasars Arp: Peculiar Galaxies Spinrad &Taylor : Population Synthesis Page: Galaxy Masses 1970’s Stability & Halos Starbursts H0!!! q0!!! First Feeble Redshift Surveys CMB Dipole

  11. Galaxy Clusters & X-Rays Gravitational Lenses Galaxy Formation 1980’s Large-Scale Structure Large Scale Flows & Cold Dark Matter Galaxy Counts H0!!!! IRAS & Dusty Starbursts

  12. 1990’s COBE: 2.723 K + fluctuations Biased galaxy Formation Unified AGN Models Λ!!!!! Concordance Cosmology HST and galaxy evolution 2000+ The Cosmic Web Reionization First Light

  13. COSMOLOGY is a modern subject: Today based on Principles & Observables The basic framework for our current view of the Universe rests on ideas and discoveries (mostly) from the early 20th century. Basics: Einstein’s General Relativity The Copernican Principle

  14. Fundamental Principles: • Cosmological Principle: (a.k.a. the Copernican principle). There is no preferred place in space --- the Universe should look the same from anywhere The Universe is HOMOGENEOUS and ISOTROPIC. we believe this is true to zeroth order (i.e. on large scales, yes, on small scales, no)

  15. A variant of the CP is The Perfect Cosmological Principle: The Universe is also the same in time. The STEADY STATE Model (XXX it’s demonstrably wrong) The Anthropic Cosmological Principle: We see the Universe in a preferrred state (time etc.) --- when Humans can exist.

  16. the ACP is almost the opposite of the PCP. it leads to the Goldilocks Universe: Not too hot, Not too cold Not too dense, Not too empty Not too young, Not too old…. Relativistic Cosmological Principle: The Laws of Physics are the same everywhere and everywhen. (!!!) absolutely necessary, often assumed and forgotten. (!!!)

  17. Fundamental Observations: The Sky is Dark at Night (Olber’s P.) this implies there must be some limit to the observable Universe. The Universe is generally Expanding It’s not static. galaxies appear to be moving away from us --- and each other.

  18. Olber’s Paradox

  19. Hubble’s Discovery of Expansion

  20. The Universe is Homogeneous on large scales --- there exists an almost isotropic microwave background (the CMB) of T~3K a.k.a. relic radiation • The Universe is not Empty. It has stuff in it, stuff consistent with a hot origin (the Universe has a temperature), i.e. contents consistent with nuclear physics operating in an initially hot, dense medium

  21. COBE Fluctuations dt/t < 10-5, i.e. much smoother than a baby’s bottom!

  22. Observational Cosmology consists of taking these bases to build a more detailed picture of the structure and evolution of the Universe. Sometimes to (1) Feed Theorists (2) Kill Theories (3) Explore  generally support Gamow’s hot big bang model

  23. COSMOLOGICAL FRAMEWORK: The Friedmann-Robertson-Walker Metric + The Cosmic Microwave Background =THEHOTBIGBANG

  24. The Big Bang WRONG!

  25. WRONG!!!

  26. WRONG ?? T THE TRUTH BEHIND THE BIG BANG THEORY

  27. RIGHT! How my wife describes my job!

  28. Mathematical Cosmology The simplest questions are Geometric. How is Space measured? Standard 3-Space Metric: ds2 = dx2 + dy2 + dz2 = dr2 + r2dθ2 + r2sin2θdf2 In Cartesian or Spherical coordinates in Euclidean Space.

  29. Now make our space Non-Static, but “homogeneous” & “isotropic”  ds2 = R2(t)(dx2+ dy2 + dz2) And then allow transformation to a more general geometry (i.e. allow non-Euclidean geometry) but keep isotropic and homogeneous:

  30. ds2 = (1+1/4kr2)-2 (dx2+dy2+dz2)R2(t) where r2 = x2 + y2 + z2, and k is a measure of space curvature. Note the Special Relativistic Minkowski Metric ds2 = c2dt2 – (dx2 +dy2 + dz2)

  31. So, if we take our general metric and add the 4th (time) dimension, we have: ds2 = c2dt2 – R2(t)(dx2 +dy2 + dz2)/(1+kr2/4) or in spherical coordinates and simplifying, ds2 = c2dt2 – R2(t)[dr2/(1-kr2)+ r2(dq2+sin2q df2)] which is the (Friedmann)-Robertson-Walker Metric, a.k.a. FRW

  32. The FRW metric is the most general, non-static, homogeneous and isotropic metric. It was derived ~1930 by Robertson and Walker. R(t), the Scale Factor, is an unspecified function of time (which is usually assumed to be continuous) and k = 1, 0, or -1 = the Curvature Constant For k = -1 or 0, space isinfinite

  33. K = +1 Spherical c < pr K = -1 Hyperbolic c > pr K = 0 Flat c = pr

  34. Consider Expansion in an Isotropic Universe l = l0 a(t-t0) l0 a(t) = Universal Expansion factor

  35. Expansion is self-similar and produces a change in the frequency of received radiation. If t0 = now, here, observed & t1 = the time at which light is emitted from a distant object in the scaled universe: a(t)δt R(t)l t0+Δt0 t0 te+Δte R0(t) l te= t1

  36. υ0 a(t1) λ1 1/Δte = υ1frequency of emitted radiation  = = soυ1, λ1 = lab or rest frequency/wavelength and υ0, λ0 = observed frequency/wavelength and if we define z ≡ redshift = Then 1 + z = = υ1 a(t0) λ0 λ0 – λ1 λ0 R(t0) a(t0) R(t1) a(t1)

  37. For small z we can interpret this “redshift” in terms of a Doppler shift, cz = v, or Doppler velocity. For small Δt, if r0 = 0 (set origin to us, the observer), we have: r1 = c (t0 - t1)/R(t0) d = r1 R(t0) = c(t0 -t1) the distance and thus cz = c(t0 – t1) 1 dR(t1) R(t0) dt

  38. dR(t0) 1 dR(t1) 1 dt R(t0) dt R(t1) so v = cz = H0d where H0 = ≈ is the definition of Hubble’s Constant. Note that this is true for small z only. This formula for distance is NOT subject to special relativity. The convention is to quote apparent velocities as v = cz.

  39. The apparent (radial) velocity of any object is made up of three parts v = vH + vP + vG vH = the cosmological stretching of the metric, a.k.a. the Hubble Flow vP = the component of the “peculiar” velocity w.r.t. the Hubble Flow that is an actual space velocity. vPis Doppler, so computations of dynamical properties like cluster velocity dispersions that result from vP do require the (1+v2/c2) –1/2 correction. vG = the gravitational redshift, usually tiny

  40. Expansion Age & Hubble Distance The Hubble Constant has units of inverse time; its actually also a measure of the expansion age of the Universe: τH = H0-1 = 9.78x109 h-1 years = 3.09x1017 h-1 s where H0 = 100 h km/s/Mpc And the Hubble Distance is DH = c/H0 = 3000 h-1 Mpc = 9.26x1025 h-1 m

  41. What about the scale factor R(t)? R(t) is specified byPhysics we can use Newtonian Physics (the Newtonian approximation) but now General Relativity holds. Start with Einstein’s (tensor) Field Equations Gmu= 8pTmn + Lgmu and Gmu= Rmn - 1/2 gmu R

  42. Where Tmn is the Stress Energy tensor Rmn is the Ricci tensor gmu is the metric tensor Gmu is the Einstein tensor and R is the scalar curvature  Rmn - 1/2 gmu R = 8pTmn + Lgmu is the Einstein Equation

  43. The vector/scalar terms of the Tensor Field Equations give the linear form Einstein’s Equations: (dR/dt)2/R2 + kc2/R2 = 8pGe/3c2+Lc2/3 energy density CC 2(d2R/dt2)/R + (dR/dt)2/R + kc2/R2 = -8pGP/c3+Lc2 pressure term

  44. And Friedmann’s Equations: (dR/dt)2 = 2GM/R + Lc2R2/3 – kc2  kc2 = Ro2[(8pG/3)ro – Ho]2 if L = 0 (no Cosmological Constant) or (dR/dt)2/R2- 8pGro /3 =Lc2/3 – kc2/R2 which is known as Friedmann’s Equation

  45. 4πG Note that if we assume Λ = 0, we have (d2R/dt2)/R = - (ρ + 3P) and in a matter dominated Universe, ρ >> P So we can define a critical density by combining the cosmological equations: ρC = = 3 . 3H02 3 R2 8πG R2 8πG

  46. And we define the ratio of the density to the critical density as the parameter Ω ≡ ρ/ρC Fora matter dominated, Λ=0 cosmology, Ω > 1 = closed Ω = 1 = flat, just bound Ω < 1 = open There are many possible forms of R(t), especially when Λ and P are reintroduced. Its our job to find the right one!

  47. Λ = 0

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