UNIVERSITY OF CAMBRIDGE MATHEMATICAL TRIPOS Part III Wednesday, 4 June, 2014 9:00 am to 12:00 pm PAPER 49 COSMOLOGY Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS Cover sheet Treasury Tag Script paper SPECIAL REQUIREMENTS None You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
UNIVERSITY OF CAMBRIDGE 2 1 The metric for a homogenous and isotropic FRW spacetime is ds2 = dt2 - a2 (t) 1 - kr2 dr2 + r2(d02 + sin2 0dø2) , where a(t) is the scale factor and k parameterises spatial curvature. The evolution of the scale factor is determined by the Friedmann equation H2 - OG [ pi(a) - az , i where H = a/a is the Hubble parameter and pi denote the relevant energy densities (matter, radiation, etc.). i) Consider a curved FRW universe filled with a single fluid with constant equation of state w = P/p. Derive the evolution equation for the fluid density p from the first law of thermo- dynamics. Hence, or otherwise, show that d?2k dlna =(1+3w)$2k(1-52k), where Sk(a) = (aH)2 k . Use this result to explain the flatness problem of the standard Big Bang cosmology. [10] ii) Now, consider a flat FRW universe with present radiation density $2r,0 = 8TGPr,0/(3H2 0) and matter density 12m,0 = 1 - 52r,0. Write the Friedmann equation in conformal time, dr = dt/a(t), using the Hubble constant Ho = H (To) and setting the scale factor today equal to unity, a(To) = 1. Show that the equation is solved by a(T) = AT2 + BT , where the constants A and B should be determined. For $2r,0 ~ 10-4, calculate the angular scale subtended by the causal horizon at matter-radiation equality. Justify any approximations you make. Given that recombination happens shortly after matter-radiation equality, explain the horizon problem of the standard Big Bang cosmology. [10] Part III, Paper 49
UNIVERSITY OF CAMBRIDGE 3 2 The distribution function for massless neutrinos is 1-1 fv(p) = exp [exp (" +1 , Ty P-Hv) where Ty is the neutrino temperature and uy is the chemical potential. A neutrino species with |v | > Ty is called degenerate. i) Show that the energy density of degenerate neutrinos is Pv ~ 8-2 , where My x Ty in thermal equilibrium. What is the contribution of the correspond- ing anti-neutrinos? [6] ii) Derive a bound on Alv/Ty from the requirement that the present energy density in degenerate neutrinos doesn't exceed the critical density, Pcrit = 6 Ă— 103 T, where T0 is the present CMB temperature. [8] iii) Discuss qualitatively how degenerate neutrinos would affect the production of primordial helium. [6] Part III, Paper 49 [TURN OVER