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Continuation of Part 4: Dynamics of FLRW Spacetimes 1 1. General Relativity 2. Einstein’s equations 3. Kinematics of FLRW 4. Dynamics of FLRW 5. Thermal history 6. Big Bang, horizons, inflation, and all that Continuation of Part 4: Dynamics of FLRW spacetimes 1. General Relativity 2. Einstein’s equations 3. Kinematics of FLRW 4. Dynamics of FLRW 5. Thermal history 6. Big Bang, horizons, inflation, and all that History of cosmic domination log ρ Key puzzles (problems?) a -4 of standard FLRW : radiation (a) Radiation-matter transition: why this much entropy/baryon? (a) (b) Curvature: why so little? (b) Why? (c) Dark energy: why is the age of Why not? cosmic acceleration so close to today? a-2 : curvature a -3 : matter (c) Why? ~a0 : dark energy | | | log a z~104 z~0.5 today 1. General Relativity 2. Einstein’s equations 3. Kinematics of FLRW 4. Dynamics of FLRW 5. Thermal history 6. Big Bang, horizons, inflation, and all that Puzzle #1: Why this amount of radiation relative to matter? ➪ Thermodynamics - coming up! 1. General Relativity 2. Einstein’s equations 3. Kinematics of FLRW 4. Dynamics of FLRW 5. Thermal history 6. Big Bang, horizons, inflation, and all that Puzzle #2: Why is the spatial curvature so small? 3k ρ = k −8π Ga2 k 0 k Ωk = 2 2 Ωk = 2 3H2 −H a ↔ −H0 ρ = c 8π G Present observational limits (i.e., CMB) imply that the “radius of curvature” (k-1/2) of the spatial section is much larger (> 100 x) than the Hubble radius, 3 Gpc. How can this be? d d k d k k 1 a¨ 1 Ω = = =2 = 2Ωk ( 4πG)(ρ +3p) dt k −dt H2 a2 −dt a˙ 2 H2a2 H a − H − d 1 = log Ω = 8πG(ρ +3p) ⇒ da k H2 Ωk always grows, unless ρ +3p < 0! 1. General Relativity 2. Einstein’s equations 3. Kinematics of FLRW 4. Dynamics of FLRW 5. Thermal history 6. Big Bang, horizons, inflation, and all that With “normal” matter (dust and/or radiation), curvature should eventually dominate: Curvature small today ➪ Curvature extremely small in the early Universe! Why? flat open closed 1. General Relativity 2. Einstein’s equations 3. Kinematics of FLRW 4. Dynamics of FLRW 5. Thermal history 6. Big Bang, horizons, inflation, and all that Consequences of a curvature-dominated Universe Suppose there is only radiation, and that the spatial curvature is negative. 1 da H(t)= = H Ω0 a 4 + Ω0 a 2 Ω0 + Ω0 =1 a dt 0 r − k − r k 0 0 Ωr Ω 1 Defining: a = = k − ∗ Ω0 Ω0 − k k 2 da a H0 we rewrite this equation as: = dt 2 2 2 a 1 a /a 1 a− − ∗ − ∗ Contraction! The solution is: 1 1/2 5 a = 1+2H (t t ) H2 (t t )2 0 − 0 − a2 1 0 − 0 Expansion 4 ∗ − a 3 This function has a maximum at: 2 1 2 1 t = t0 +(a 1)H0− a(t )=a 5 10 15 ∗ ∗ − ∗ ∗ t 1. General Relativity 2. Einstein’s equations 3. Kinematics of FLRW 4. Dynamics of FLRW 5. Thermal history 6. Big Bang, horizons, inflation, and all that Puzzle #3: Why has the Universe just recently started to accelerate? log ρ ➪ a -4 No idea. : radiation None. Many maybes. Not sure. Nobody knows. a -3 : matter (c) Why? ~a0 : dark energy | | | log a z~104 z~0.5 today 1. General Relativity 2. Einstein’s equations 3. Kinematics of FLRW 4. Dynamics of FLRW 5. Thermal history 6. Big Bang, horizons, inflation, and all that Useful numbers • Age of the Universe today: T0 = (13.7 ± 0.2) Gy 2 -29 -3 • Density:!! ! ! ρ0 = (1.9 ± 0.15) h x 10 g cm -1 -1 • Hubble parameter: !! H0 = 100 h Km s Mpc 1 pc = 2.36 l.yr. h = 0.70 ± 0.03 1 Mpc = 3.1 x 1024 cm Hubble radius: !! c/H0 = 3 h Gpc 2 2 • Baryon fraction:!!! Ωb h = (ρb /ρtot) h = 0.024 ± 0.003 • Radiation -5 -2 (photons and massless neutrinos):! Ωr = 2.5 x 10 h • Matter (dark matter + baryons)! Ωm = 0.2 - 0.3 • Curvature! |Ωk| < 0.01 With these we can compute, e.g., matter-radiation equality: 1. General Relativity 2. Einstein’s equations 3. Kinematics of FLRW 4. Dynamics of FLRW 5. Thermal history 6. Big Bang, horizons, inflation, and all that Part 5: Thermal history Suggested reading: V. Mukhanov, “Physical Foundations of Cosmology” P. Peter and J.-P. Uzan, “Primordial Cosmology” E. Kolb & M. Turner, “The Early Universe” [Original works by Zwicky; Gamow, Alpher, Herman; Bethe; Penzias, Wilson; Peebles; Schramm; ...] 1. General Relativity 2. Einstein’s equations 3. Kinematics of FLRW 4. Dynamics of FLRW 5. Thermal history 6. Big Bang, horizons, inflation, and all that redshift Brief thermal history of the Universe 109 200 s 1 MeV BBN 103 380.000 yrs 1 eV Decoupling (surf. last scattering) 0 15Gy time energy 1. General Relativity 2. Einstein’s equations 3. Kinematics of FLRW 4. Dynamics of FLRW 5. Thermal history 6. Big Bang, horizons, inflation, and all that Thermodynamics in an FLRW Universe As the Universe expands and becomes less dense, it also cools down. Hence the behavior of the energy density and pressure of the matter content is tied up with its Thermodynamics and Statistical Mechanics. Elementary particles come in two types: bosons (spin=0, 1, 2, ...), and fermions (spin=1/2, 3/2, ...) Because of quantum statistics, each type occupies phase space in a different way. The number of particles in a phase space cell is: d3pd3x p chemical dN = f(p, x) 1 potential 3 fBE = E µ (2π) − e kB T 1 − dN d3p 1 f = n(x)= = 3 f(p, x) FD E µ dV (2π) k −T x e B +1 1. General Relativity 2. Einstein’s equations 3. Kinematics of FLRW 4. Dynamics of FLRW 5. Thermal history 6. Big Bang, horizons, inflation, and all that Let’s forget about the chemical potential (µ) for a moment. Then these distributions are: Units of kB=1 : [Temperature] = [Energy] 1 K = 8.3 x 10-5 eV 1 1 eV = 1.1 x 104 K f = ± eE/T 1 ± There is a beautiful result from Hamiltonian and Statistical mechanics, called the Liouville Theorem (Liouville 1838, Gibbs 1902), which states that the phase-space distribution function is constant on all trajectories. This theorem is equivalent to the collisionless Boltzmann equation, which reads: df ∂f p = + + F f dt ∂t m · ∇x · ∇p In an homogeneous and isotropic (FLRW) Universe, there are no preferred directions, so the phase space function f must be conserved in time! Since E~1/a, we must have that T~1/a , so that the form of the BE or FD distributions are preserved by the expansion of the Universe! 1. General Relativity 2. Einstein’s equations 3. Kinematics of FLRW 4. Dynamics of FLRW 5. Thermal history 6. Big Bang, horizons, inflation, and all that Energy and pressure The energy and momentum of particles are related: E2 p 2 = m2 EdE= pdp − ↔ The stress-energy density for a system of particles can be written in terms of their 4- momenta and 4-velocities: |v| = |p|/E! pν T µν (x)= pµ U ν δ(4)(x x ) = pµ δ(4)(x x ) − n E − n n n d3p The simplest quantity is the particle number density: dn = f (p, x) ± (2π)3 d3p The energy density for this ensemble of particles is: dρ = Ef (p, x) ± (2π)3 1 p2 d3p And if the momenta have no preferred directions, then: dP = f (p, x) 3 E ± (2π)3 1. General Relativity 2. Einstein’s equations 3. Kinematics of FLRW 4. Dynamics of FLRW 5. Thermal history 6. Big Bang, horizons, inflation, and all that The ensemble averages of the number density, energy density and pressure are: T m BE ζ(3) T 3 π2 2 2 1/2 1 (E m ) 3 ζ(3) 3 n = dE E − = T m FD 4 π2 T 2π2 eE/T 1 ± mT 3/2 m/T T m BE/FD 2π Equil.!e− ->1/V 2 T m BE π T 4 30 2 2 1/2 1 2 (E m ) 7 π2 ρ = dE E − = T m FD T 4 2π2 eE/T 1 8 30 ± T m BE/FD mn+ 3 nT 2 T m BE 1 ρ 3 2 2 3/2 1 (E m ) 1 P = dE − = T m FD 3 ρ 6π2 eE/T 1 ± T m BE/FD nT ρ 1. General Relativity 2. Einstein’s equations 3. Kinematics of FLRW 4. Dynamics of FLRW 5. Thermal history 6. Big Bang, horizons, inflation, and all that Energy density of all relativistic matter Let’s collect all the types of particles which are relativistic - i.e., whose masses are much smaller than their equilibrium temperatures. Let’s also allow for some of these particles to be out of equilibrium with other particles, so that their temperatures may be different. The total energy density in the relativistic degrees of freedom is, then: π2 2 2 2 4 BE 1 ∞ x m /T T 30 4 2 i i = T 4 g i ρr = gi Ti 2 dx x x− i 4 2π e 1 T × 2 i mi/Ti i 7 π FD ± 8 30 We can define an effective number of relativistic degrees of freedom as: 2 Ti 7 Tj π 4 g = g + g ρr = g T i j ∗ 30 ∗ T 8 T i: BE j: FD 1. General Relativity 2. Einstein’s equations 3.
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