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The Big-Bang Theory AST-101, Ast-117, AST-602

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The Big-Bang Theory AST-101, Ast-117, AST-602 AST-101, Ast-117, AST-602 The Big-Bang theory Luis Anchordoqui Thursday, November 21, 19 1 17.1 The Expanding Universe! Last class.... Thursday, November 21, 19 2 Hubbles Law v = Ho × d Velocity of Hubbles Recession Distance Constant (Mpc) (Doppler Shift) (km/sec/Mpc) (km/sec) velocity Implies the Expansion of the Universe! distance Thursday, November 21, 19 3 The redshift of a Galaxy is: A. The rate at which a Galaxy is expanding in size B. How much reader the galaxy appears when observed at large distances C. the speed at which a galaxy is orbiting around the Milky Way D. the relative speed of the redder stars in the galaxy with respect to the blues stars E. The recessional velocity of a galaxy, expressed as a fraction of the speed of light Thursday, November 21, 19 4 The redshift of a Galaxy is: A. The rate at which a Galaxy is expanding in size B. How much reader the galaxy appears when observed at large distances C. the speed at which a galaxy is orbiting around the Milky Way D. the relative speed of the redder stars in the galaxy with respect to the blues stars E. The recessional velocity of a galaxy, expressed as a fraction of the speed of light Thursday, November 21, 19 5 To a first approximation, a rough maximum age of the Universe can be estimated using which of the following? A. the age of the oldest open clusters B. 1/H0 the Hubble time C. the age of the Sun D. the age of the Galaxy E. there is no simple estimate Thursday, November 21, 19 6 To a first approximation, a rough maximum age of the Universe can be estimated using which of the following? A. the age of the oldest open clusters B. 1/H0 the Hubble time C. the age of the Sun D. the age of the Galaxy E. there is no simple estimate Thursday, November 21, 19 7 17.2 Cosmological Models! Thursday, November 21, 19 8 The Early Cosmological Models! Einstein in 1917 constructed the first relativistic cosmological models. Thinking that the universe is static, he introduced the cosmological constant term to balance the force of gravity. This model was unstable.! Willem De Sitter in 1917 also developed a similar model, but also! obtained solutions of Einstein equations for a nearly empty, expanding universe.! In 1932, Einstein & De Sitter jointly developed another, simple cosmological model which bears their names.! Thursday, November 21, 19 9 The Friedmann and Lemaitre Models! ! Alexander Friedmann! In 1922 developed the GR-based, expanding universe model. It was not taken very seriously at the time, since the expansion of the universe has not yet been established.! Georges Lemaitre ! In 1927 independently developed cosmological models like Friedmanns. In 1933, he ran the film backwards to a hot, dense, early state of the universe he called the cosmic egg. This early prediction of the Big Bang was largely ignored.! They used the homogeneity and isotropy to reduce the full set of 16 Einstein equations of GR to one: the Friedmann-Lemaitre eqn.! Thursday, November 21, 19 10 Kinematics of the Universe! We introduce a scale factor, R(t)! commonly denoted as R(t) or a(t): a spatial distance between any two unaccelerated frames which move with their comoving coordinates! This fully describes the evolution of t! a homogeneous, isotropic universe! Computing R(t) and various derived quantities defines the cosmological models. This is accomplished by solving the Friedmann (or Friedmann-Lemaitre) Equation! The equation is parametrized (and thus the models defined) by a set of cosmological parameters! Thursday, November 21, 19 11 Cosmological Parameters! Cosmological models are typically defined through several handy key parameters:! 1 ∆R ∆R = R R H = final − initial ∆t = t t R ∆t final − initial Thursday, November 21, 19 12 Mass-energy determines geometry Geometry determines where mass- energy can go Thursday, November 21, 19 13 Cosmological Parameters! Thursday, November 21, 19 14 Geometry and the Fate of the Universe! Matter and energy content of the universe determines its geometry (curvature of space), and the ultimate fate! ρ < ρcrit, k = −1 ! Possible expansion histories: ! negative curvature! expands for ever! ρ = ρcrit, k = 0! flat (Euclidean) ! expands for ever! ρ> ρcrit, k = +1 ! positive curvature! collapses! Thursday, November 21, 19 15 Hubble Constant Defines the Scale of the Universe! H0 = slope at t0! R0! 0! t0! { 1 / H0 = Hubble time! c / H0 = Hubble length! Thursday, November 21, 19 16 Critical Mass Density for the Universe I We can get an estimate of Potential energy of mass m in how much mass is needed to m gravitational field of M “close” the universe. More V GMm accurately, we calculate the M PE = mean density needed to − R close the universe. R Kinetic energy of mass m 1 KE = mV 2 We balance gravitational 2 1 GMm potential energy and kinetic Total energy: E =KE+PE= mV 2 energy using simple 2 − R Newtonian mechanics. G = Gravitational constant G = 6.67 x 10-8 cm3 g-1s-2 or G = 6.67 x 10-11 m3 kg-1 s-2 E = 0 corresponds to mass 2GM 1/2 m having escape velocity V = esc R from M ✓ ◆ Example: Earth R ~ 6371 km Vesc ~ 11.2 km/s M = 5.97 x 1024 kg Thursday, November 21, 19 17 Critical Mass Density for the Universe II We can get an estimate of We can also solve for E = 0 from how much mass is needed to 1 GMm E =KE+PE= mV 2 “close” the universe. More 2 − R accurately, we calculate the M V 2 mean density needed to to get = close the universe. R 2G We balance gravitational Now we consider m to be any object in potential energy and kinetic the universe and we relate its velocity to energy using simple the Hubble law, V = H0R Newtonian mechanics. 2 2 M V (H0R) Then: = = E = 0 corresponds to mass R 2G 2G m having escape velocity M H2 Or: = 0 from M R3 2G 2 2 M H0 3H0 Volume of sphere = (4π/3)R3 à = = (4π/3)R3 2G(4π/3) 8πG Critical mass This gives a mass density ρc 2 (g / volume) 3H0 density ~ ⇥ = 10-29 g cm-3 c 8πG Thursday, November 21, 19 18 If the density of the Universe is less than critical, then the Universe: A. will ultimately collapse back in on itself B. will expand forever C. must be spherical D. will have a temperature of 2.73K forever E. must be static, with an unknown cause for the redshifts Thursday, November 21, 19 19 If the density of the Universe is less than critical, then the Universe: A. will ultimately collapse back in on itself B. will expand forever C. must be spherical D. will have a temperature of 2.73K forever E. must be static, with an unknown cause for the redshifts Thursday, November 21, 19 20 The ultimate fate of the Universe depends upon the overall geometry of spacetime. Each of these geometries has implications fortherelativesizeoftheUniverseastime progresses. In Figure 2, the relative size of the Universe is graphed as Distance vs. Time for the three primary geometries, labeled A, B, and C. Answer the following three questions about this Figure. Curve A describes which type of Universe: A 61. Curve A describes which type of Universe: A. open, whereA. the Open, Universe where the UniverseWill continue will continue toto expand expand forever forever B. Hyperbolic, where the expansion of the Universe becomes exaggerated B. hyperbolic,C. where Flat, where the parallelexpansion lines never of meet the Universe becomes exaggerated D. Closed, where the Universe will ultimately re-collapse C. flat, whereE. parallel Parallel, wherelines multiple never Universes meet parallel our own evolution B 62. Curve B describes which type of Universe: D. closed, where the Universe will ultimately re-collapse A. Open, where the Universe will continue to expand forever B. Flat, where parallel lines never meet E. parallel, whereC. Parallel, multiple where multiple Universes Universes parallel parallel our our own own evolution evolution D. Hyperbolic, where the expansion of the Universe becomes exaggerated E. Closed, where the Universe will ultimately re-collapse C 63. Curve C describes which type of Universe: A. Flat, where parallel lines never meet B. Parallel, where multiple Universes parallel our own evolution Thursday, November 21, 19C. Closed, where the Universe will ultimately re-collapse 21 D. Hyperbolic, where the expansion of the Universe becomes exaggerated E. Open, where the Universe will continue to expand forever 11 The ultimate fate of the Universe depends upon the overall geometry of spacetime. Each of these geometries has implications fortherelativesizeoftheUniverseastime progresses. In Figure 2, the relative size of the Universe is graphed as Distance vs. Time for the three primary geometries, labeled A, B, and C. Answer the following three questions about this Figure. Curve A describes which type of Universe: A 61. Curve A describes which type of Universe: A. open, whereA. the Open, Universe where the UniverseWill continue will continue toto expand expand forever forever B. Hyperbolic, where the expansion of the Universe becomes exaggerated B. hyperbolic,C. where Flat, where the parallelexpansion lines never of meet the Universe becomes exaggerated D. Closed, where the Universe will ultimately re-collapse C. flat, whereE. parallel Parallel, wherelines multiple never Universes meet parallel our own evolution B 62. Curve B describes which type of Universe: D. closed, where the Universe will ultimately re-collapse A. Open, where the Universe will continue to expand forever B. Flat, where parallel lines never meet E.
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