Astr 2320 Tues. May 2, 2017 Today s Topics Chapter 23: Cosmology: The Big Bang and Beyond Introduction Newtonian Cosmology Solutions to Einstein s

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1 Astr 0 Tues. May, 07 Today s Topics Chapter : Cosmology: The Big Bang and Beyond Introduction Newtonian Cosmology Solutions to Einstein s Field Equations The Primeval Fireball Standard Big Bang Model

2 Chapter Homework Chapter : #,, 4

3 Chapter : Introduction to Cosmology Principle of Equivalence One of Einstein s greatest insights. Gravitational mass and inertial mass are equivalent Observers cannot distinguish between accelerating reference frames and the gravitational field Light rays must be deflected in gravitational field! If light is mass-less the gravitational field (accelerating reference frame) must result from curved space (space-time)

4 Chapter : Introduction to Cosmology Paths of Light Rays Since photons are mass-less they follow null geodesics They map space-time Spatial curvature = gravitational field Note the two effects: Deflection of starlight (no net redshift) Increased path length (time dialation) Einstein s Predictions Advance of perihelion of mercury Deflection of Starlight (Sun, Galaxy Clusters) Gravitational Redshift (Pound- Rebka, White Dwarfs) Frame dragging (Grav. Probe B) 4

5 Chapter : Introduction to Cosmology Poisson s Equation Recall that the grav. force and grav. field are related Both are vector fields Analogous to the electric field we can write Gauss law for gravity Mass density is a scalar so we can just add contributions from different sources Gravity is a conservative force Grav. field is the gradient of the gravitational potential We can rewrite Gauss law as Poisson s equation F For an imaginary surface surrounding some mass we can integrate (surface integral) for the net flux : g da = 4 GM where da is a vector normal to the surface and M is mass enclosed. V g The differential form is then : g = 4πGρ where is the mass density. In cartesian coords : f f f f = i + j + k (where i, j, k x y z The gravitational field and potential are related through the gradient : g = φ (with (del) as the gradient operator) and so we have : φ = 4πGρ For spherical coordinates Poisson' s equation becomes : r = Mg( r) ) ' r r ( where φ & $ = 4πGρ ( r) r % the divergence operator (gradient dot - ed with vector field) and are the unit vectors in the x, y, z directions) is the Laplacian (second derivative of scalar field) ρ is 5

6 Chapter : Introduction to General Relativity Einstein s Field Equations Analogous to Poisson s Equation for Newtonian Gravity Difference is the Addition of Spatial Curvature (metric) Each tensor is a 4 x 4 matrix equation but only 0 partial differential equations are independent but with 4 space-time coords. only 6 are really needed. General solutions are still almost impossible without introducing symmetries. Specific metrics simplify these even further so analytic solutions are possible. 8πG G = T where G is the space - timecurvature tensor and T is the 4 c energy - momentum tensor. This ( G ) is often written in an expanded form to separate the curvature into a local and global contribution : 8πG R g R + g Λ = T where R 4 c tensor, R is the Ricci scalar tensor, and g is called the Ricci curvature is the metric. T describes the distribution of mass - energy and it's time - dependent flux and the metric describes the nature (symmetries) of space - time. R is like a multi - dimensional Laplacian of the metric while R describes the geometry resuling from a specific metric. 6

7 Chapter : Introduction to General Relativity Metrics Describe Space-Time Geometry Proper distance computed by integrating along a path Minkowski s Metric Simplest description of the line element Applicable to vacuum Schwarzschild Metric Appropriate for a point-mass (spherically symmetric) Schwartzschild Radius Obvious Robertson-Walker Appropriate for uniform, isotropic space-time Explicitly includes time dependence via the timedependence of the scale factor: a(t) Minkowski's Metric (flat spacetime, no mass) : ds ( % & # & 0 0 0# η = & 0 0 0# & # ' $ In spherical coordinates this metric becomes : ds dω = dθ Schwarzschild's metric describes space - timearound a point mass : ( GM % ( GM % ds = & # + & # dr + r dω ' r $ ' r $ The Robertson - Walker metric describes space - time for a homogeneous - isotropic and time - dependent case. There are cases depending on the curvature : ds = = = coordiates. + dx + dr + sin + a( t) + dy + r ds θdφ dω + dz where where ds or in matrix form : represents the spacial 7

8 Chapter : Introduction to General Relativity Friedmann-Lemaitre Equation Field equations with the of Robertson- Walker metric yield the Friedmann- Lemaitre equations Derivation is much easier in Newtonian gravity and so: Let s Play! Mass density (and gravity) drops as universe expands. Thus H(t) constant. dr r( t) = H ( t) r so if we define a( t) = (normalized coords) we have : r( t ) da H ( t) = so once we know a(t) we know r(t). If mass is conserved : a - ρ( t) a( t) = ρ( t ) ρ (density must decrease as r(t) ) Newtonian view : dv F = and since Poisson's equation is F = 4πGρ we can combine : dv d ' dh $ = [ H ( t) r] = r% + H " from the chain rule so the divergence is : & # / dh, - + H r 4 G but since r we have : * = π ρ =. + dh 4πGρ + H = or equivalently : dh 4πGρ ( t) + = so the decelleration depends on the mass density. H H ( t) Substituting for H(t) gives : a da d a 8πG + ρ 0 a 8πG ρ = constant a 0 8πG kc ρ = a 0 d a 4πGρ + = 0 (static universe is impossible unless ρ = 0) a da Substituting for ρ (mass conservation) and multiplying through by a ' da $ % " & # ' % & da $ " # da = 0 (note each term is a perfect differential). Integrating : 0 Introducing an integration constant of kc (Freidmann - Lemaitre Equation) 8 gives : gives :

9 Chapter : Cosmology: The Big Bang and Beyond - II What does it mean? Solution to Einstein s Field Equations for a Homogeneous and Isotropic Universe is Friedmann-Lemaitre Equation: a " $ # da % ' & 8πGρ = kc where k is the curvature constant a For insight into the Friedmann - Lamaitre equation multiply by a " da% $ ' # & 4πGρa = kc or in Newtonian terms : v GM(a) = kc a Kinetic Energy + Potential Energy = Total Energy (Energy Conservation) The critical density which just closes the universe occurs when k = 0 " da% $ ' # & ρ c = H 0 8πG = 4πGρa but since H = " da% $ ' a # & then : (at present time x 0-0 gm /cm for H 0 = 7km /sec/mpc) 9

10 Chapter : Cosmology: The Big Bang and Beyond - III H is a function of time and must decrease for Λ = 0. So: dh/ = d/(/a da/) = H (/H /a d a/ ) so define: q(t) = - /H /a d a/ (deceleration parameter) so dh/ = H (q + ) Dividing the Freidmann equation by H shows that models depend only on H 0 and q 0 (current expansion and deceleration) H 0 a ( q 0 ) = - kc (so q 0 = ½ means k = 0) Thus geometry depends on q 0 q 0 = ½ k = 0 (flat universe) q 0 > ½ k > 0 (closed universe) q 0 < ½ k < 0 (open universe) Since q 0 = ρ/ρ 0 and H = /a da/ we can re-write the Friedmann equation in terms of just H and q: (da/) = H /a[q + a( q)] 0

11 Solutions to Friedman s Equation Friedmann s Equation at Present Time: (da/) = H 0 /a[q 0 + a( q 0 )] Consider two cases: q 0 = 0 q 0 = da/ ~ constant da/ ~ a -/ a(t) ~ t a(t) ~ t / H 0 T 0 = H 0 T 0 = / Time-Scale Test Compare H 0 to Age of Universe (Oldest Globular Clusters) The product H 0 T 0 provides a strong test of cosmological models Also standard candle (Type Ia SN) + inverse square law (see figure on next panel)

12 Solutions to Friedman s Equation

13 Chapter Homework Chapter : #,, 4