Modern Cosmology Final Examination Solutions 60 Pts

Transcription

1 Modern Cosmology Final Examination Solutions 6 Pts Name:... Matr. Nr.:... February,. Observable Universe [4 Pts] 6 Pt: Complete the plot of Redshift vs Luminosity distance in the range < z < and plot (i) the naive Hubble-law, (ii) a pure vacuum Universe (de Sitter), (iii) the LambdaCDM, (iv) the SCDM (Ω Λ = = Ω k ). Give the formula d L (z) for (i), (ii) and (iv). Redshift z.5 Naive Hubble de Sitter Vacuum.5 SCDM LCDM Luminosity Distance [Mpc] Naive Hubble Law, = 7 km/s/mpc, c/ = 45 Mpc: d L (z) = c z () de Sitter Vacuum: d L (z) = c z(+z) () SCDM: d L (z) = c [+z +z] ()

2 6 Pt: The temperature correlation function of the Cosmic Microwave Background can be expanded in Legendre polynomials P l (θ): C(T( n),t( n )) = (l+)c l P l ( n n )/4π. (4) l= Complete the plot l(l+)c l /π vs l for the range l <. Give the relation between l and the angular scale θ on the Sky. Explain the three features in the plot: < l <, < l <, l >. ell(ell + )C_ell/π [µk²] 6 5 CMB Power Spectrum 4 Sachs-Wolfe Plateau ISW Rise Acoustic Peaks Damping Tail Multipole ell Pt: Give different arguments for the existence of Dark Energy in the Universe (if you want to make comments, please use the back of this page):. Luminosity distance for Supernovae Ia data.. Analysis of anisotropies in CMB WMAP7 data. Physical constants: c = 99,79 km/s ; G = 6.67 m /kg/s ; k B =.8 J/K ; h =.5 4 J s

3 . The LCDM Universe [4 Pts] 6 Pt Show that the LCDM ansatz a(t) = (Ω m /Ω Λ ) / sinh / ( Ω Λ t/) solves the Friedmann equation for Ω k =. ȧ = C cosh() Ωm sinh / (5) () ȧ = C cosh () sinh / () ( Ω m)h ( ) / Ωm = H +sinh () Ω m sinh / () ( Ω m) ( ) / ( ) = H Ωm / Ωm Ω m Ω m sinh / () +H sinh 4/ ()( Ω m ) Ω m = H [ Ωm a +( Ω m)a ]. (6) 4 Pt Cosmic time: The age of the LCDM Universe t(z) at a given redshift z follows from the inversion of the above with a(t) = /(+z). Use x(z) [/C(+z)] /. t(z) = arsinh(x(z)) ln(x+ x Ωm H +). (7) Ωm Here we use the identity arsinh(x) = ln(x+ x +).. Express the present age t in terms of the Hubble age t H = /. The age t of the Universe follows from this for z =, i.e. for x = /C / = [( Ω m )/Ω m ] / =.644 for Ω m =.7, and therefore t = ln(x + x.78 +) =.7 =.99 =.7Gyrs. (8) Ωm. Express the age of the Universe at redshif 8.6 in terms of the Hubble age t H = /. The age t of the galaxy at z = 8.6 follows for z = 8.6, i.e. for x g = /[C( + z g )] / = [( Ω m )/Ω m ] / /( + z g ) / =.55 for Ω m =.7, and therefore t(8.6) = Ωm ln(x g + x g +) = =.44 = 6Mioyrs.. Calculate the light travel distance for a Quasar with redhift 6.5 in units of Mpc. Light travel time distance for the quasar at redshift 6.5 follows from this by Ω m =.7, / =.7 Gyrs and C =.4 with t =.7 and t(6.5) =... Mio yrs d T (z = 6.5) = c(t t(z = 6.5)) =.Glyrs. () (9)

4 4 Pt Fundamental Plane of Cosmology: Make a plot of the Fundamental Plane of Cosmology for (Ω Λ,Ω M ) including captions for the axes (on the back).. Mark the positions of the flat models with Ω k = and those with q <?. Mark the position of the standard LCDM.. Include a sketch of the constaints from Supernovae Ia data, WMAP data and Dark Matter measurements in galaxy clusters. No Big Bang Perlmutter, et al. (999) Jaffe et al. () Bahcall and Fan (998) Supernovae vacuum energy density (cosmological constant) CMB expands forever recollapses eventually closed - Clusters open flat mass density 4

5 . The Early Universe and Cosmic Structure [ Pts] 6 Pt Describe the timeline of the Universe from the Planck epoch to galaxy formation. For this, complete the following table for the various key epochs in the evolution of the Universe ( Pt per epoch + 4 Pts for correct g ): Epoch Temperature k B T Duration Redshift Relic Particles in ev, MeV, GeV secs, yrs z Planck epoch 9 GeV 4 s 6 Quantum era Inflation 6 GeV 4 5 s 56 9 Density flucts GUT Phase Transition 6 GeV 5 s 9 Baryon Number SuSy Symmetry Breaking TeV 8 s 4 5 WIMPs, LSP Electro-weak Phase Trans GeV s 5 W, Z Quark-Hadron Phase Trans 5 MeV µs Hadrons Primordial Nucleosynthesis. MeV min 8 D, He, Li Equilibrium Epoch K 77 kyrs Matter Recombination K 8 kyrs 8 CMB, Dark Age K 5 Myr - H Reionisation - 8 K 5 Myr - Gyr - 6 H +, He ++, e Galaxy epoch 8 - K -.7 Gyr < 6 Cosmic Web Calculate the Degrees of Freedom g at k B T =. MeV: g = g B g F = + 7 (++ ) =.75. () 8 Calculate the Degrees of Freedom g at k B T = TeV (without SuSy): g = γ +W ± +Z +8coloredgluons+Higgs?+quarks+leptons = ( + + ) = 8 4 = 69.5.() 5

6 8 Pt Inflation: Describe fundamental problems of the classical Big Bang Cosmology:. The Horizon problem:.... The Flatness problem:.... The Scale problem:... Describe how these problems are solved by the introduction of an inflationary period in the early Universe.. At least N = ln[a(t inf )/a(t in )] = 57 exponential expansion.. dito. Exponential expansion. Give the full evolution equations for the st Friedmann equation and for the inflaton field Φ(t) for a given potential V(Φ) and their slow-roll approximation. Slow-roll approximation H = 8πG [ Φ +V(Φ)] () Φ+H Φ = dv dφ. (4) Slow-roll parameters ǫ = M Pl η = M Pl H = 8πG V(Φ) (5) H Φ = dv dφ. (6) ( V ) V, M Pl = hc 8πG (7) V V V H. (8) 6

7 8 Pt Complete the following table for the minimal 8 Parameters of the Inhomogeneous FRW Universe, determined by WMAP7: Table : Best-fit cosmological parameters from WMAP seven-year results Parameter Value WMAP7 Physical Meaning of the Parameter 7± km/s/mpc Hubble expansion, h = / km/s/mpc A s.9 9 Amplitude of CMB Power spectrum at λ = π/k = 5 Mpc n s.96 Power spectrum spectral index r <. Tensor-to-Scalar ratio Ω B h. Baryon density parameter Ω cdm h. Cold Dark Matter content Ω k. Curvature parameter Ω k = kr H /R τ.87 Optical depth to reionization Power spectrum: = A s ( k k ) ns (9) r = A T () A s zrei cdt τ = dz n e(z)σ T dz. () 7