UNIVERSITY OF CAMBRIDGE MATHEMATICAL TRIPOS Part III Wednesday, 1 June, 2016 9:00 am to 12:00 pm PAPER 310 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 (a) In lectures we assumed homogeneity and isotropy and used the Einstein equations to obtain the Friedmann equations: 2 à a ä a = 4TTG 3 8TTG 3" - " ? - (p+3P) . Use them to derive the continuity equation, p= - 3ª (p+P). The content of our universe today can be described by three fluids, radiation (Pr = 1/3pr), matter (Pm ~ 0), and dark energy (PA = - PA). Define what we mean by the critical density Pcrit, use it to define the fractional densities 22j, and calculate how they scale with a(t) (b) Rewrite the Friedmann equations in terms of the fractional densities today to obtain à 2 = H3 a 2r.0 + 2m,0 a + 21,0a2 ) - K 2m,0 - 22A,0a ä -H? = + 2ª2 @3 Sr.0 In the absence of dark energy use the above, or otherwise, to name and describe qualitatively the evolution of a in the case of: K > 0, K < 0, and K = 0. (c) In the case of K < 0 make this concrete by deriving the parametric solution for a universe containing only matter, a(n) = 52m,0 (cosh n - 1) 2 (1 - $2m,0) t(n) = 2 (1 - $2m,0) H? 1 52m,0 3/2 (sinh n - n) where n should be determined. [ You may use: sinh-1 y = Z VI+y2 dy sinh2 (%) = 1 (cosh(x) -1) ] Part III, Paper 310
UNIVERSITY OF CAMBRIDGE 3 2 (a) A flat universe which is both homogeneous and isotropic is described by the metric, ds2 = a2 (T) (-dr2 + dij dæ daj) . By considering a spatial gauge transformation ãi = ri + ?i describe the gauge problem. (b) Now, the FRW metric in flat space with only scalar perturbations has the form, ds2 = a2(+) ( - (1 +2A) dr2 + 20;Bdx'd+ + (1 +2?) {j +2 (010} - {j V2 ) E de'daci ) . Consider a general scalar gauge transformation, 4 = " + CH, where §0 = T(T,x) and ¿i = ai L(T, x). Show that the metric transforms as, d㪠dãß dxH dxv Jaß , and use this to show any two of the following: A = A -T' - HT, ? = B+T - L', Õ = C - HT - - V2L, 1 ? = E-L where H is the comoving Hubble parameter State why tensor perturbations are gauge invariant. (c) One way to solve the gauge problem is to work with gauge invariant variables. A key gauge invariant variable is the constant