Msc Particle Cosmology Problem Sheet 2 – Relativistic Cosmology (contd) Questions marked (∗) February 14, 2013 c C. Contaldi are a bit tougher. 1. For simplicity assume that our universe is now Λ dominated i.e. ΩΛ = 1. Λ is a cosmological constant. (a) Using the current value of H0 = 70 Km s−1 Mpc−1 calculate the distance to the event horizon (our future horizon). (b) How is the comoving size of the event horizon scaling with time? (c) Estimate how long it will take for Andromeda, our nearest galactic neighbour at a current distance of 1 Mpc from the Milky Way, to drop behind our event Horizon. For simplicity assume both the Milky Way and Andromeda are comoving with the Hubble flow (in reality they are gravitationally bound). (d) What will we observe happening to Andromeda as it approaches the horizon? 2. The Lagrangian density of a scalar field φ with potential V (φ) is given by 1 Lφ = − g µν ∂µ φ∂ν φ − V (φ) . 2 (a) Derive the form of the stress energy tensor defined through the variation of the scalar action with respect the (inverse) metric √ 2 δ( −gLφ ) Tµν = − √ . −g δg µν (b) Derive expressions for the energy density and pressure of the scalar field by identifying with the components of the stress energy of a perfect fluid T µν = (ρ + p)U µ Uν + pδ µν . (c) Obtain the equation of motion for the field φ using T µν;µ = 0. equation which reduces to Consider the ν = 0 ∂T 00 + Γµ0µ T 00 − Γi0i T ii = 0 . ∂t Using Γ000 = 0 and Γi0i = 3 H for the FRW metric we then have ∂ 1 1 1 − φ̇2 − V (φ) − 3 H φ̇2 + V (φ) − 3 H φ̇2 − V (φ) = 0 , ∂t 2 2 2 which reduces to φ̈ + 3 H φ̇ + V,φ = 0 . 3. (∗) Consider the FRW metric dr2 2 2 + r dΩ = −dt2 + a2 (t)g̃ij dxi dxj , ds = −dt + a (t) 1 − Kr2 2 2 2 where g̃ij is the maximally symmetric subspace. (a) Work out the components of the connection Γijk for the subspace for the general case K 6= 0. Msc Particle Cosmology Problem Sheet 2 – Relativistic Cosmology (contd) February 14, 2013 c C. Contaldi (b) The connection and Ricci tensor for the full space can then be obtained using the result that for maximally symmetric subspaces Γijk ≡ Γ̃ijk , and Rij = R̃ij − (aä + 2ȧ2 )g̃ij . Use this to derive the general Friedmann equations for K 6= 0 with perfect fluids. 4. In the lectures I quoted some standard results for equilibrium densities of various species. The derivation of these is straightforward. (a) Starting with the (µ = 0) phase space density for fermi-dirac and bose-einstein particles derive the relationship between number density and temperature in the ultra-relativistic limit (E(p) → pc). You should find useful the following integral representations of the Riemann zeta function ζ(n) for integer n Z ∞ n−1 x 1 dx, ζ(n) = (n − 1)! 0 ex − 1 Z ∞ n−1 1 x ζ(n) = dx, 1−n (1 − 2 )(n − 1)! 0 ex + 1 with ζ(2) = π 2 /6, ζ(3) = 1.202, and ζ(4) = π 4 /90. (b) Similarly show that the entropy density of a relativistic species of bosons is sb = g 2π 2 3 T , 45 and that the relationship between the entropy contribution from relativistic fermion and boson is 7 sf = sb . 8 Page 2
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