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MATHEMATICAL TRIPOSPart IIIFriday 4 June, 2004 1.30 to 4.30PAPER 59ADVANCED COSMOLOGYAttempt THREE questions.There are six questions in total.The questions carry equal weight.
You may not start to read the questionsprinted on the subsequent pages untilinstructed to do so by the Invigilator.
21 In the 3+1 formalism for General Relativity, one selects a set of spacelike surfacesΣ3which foliate spacetime, with a timelike normal nµnormalised so that nµnνgµν= −1. The projection operator onto the tangent space of Σ3is Pνµδνµ+ nµnν, and the extrinsic curvature of Σ3is Kαβ= Pαµrµnβ∈ Σ3, where rµis the four dimensional covariant derivative. The three dimensional covariant derivative Dµis given by applying rµand then projecting all tensor indices into Σ3using Pνµ.Show that PαµPδα= Pδµ, and Pαµnµ= 0. Show that PµαPνβrαPβ= Kµνn . From the identity DµDνWγ−DνDµWγ= Wλ(3)Rγνµλ,(∗)for any Wγ∈ Σ3i.e. Wγnγ= 0, show that(3)Rγνµλ= PξλPµαPνβPγδ(4)Rδβαξ−KµγKνλ+ KνγKµλ.You may assume Kµνis symmetric. [Hint: express the left hand side of (∗) in four dimensional terms.]=
3Consider a massless scalar field φ in a contracting, flat FRW universe. (a) Assuming the background field φ0(τ ) is spatially homogeneous, show that pa(τ ) = (−τ )1/2,φ0(τ ) = 3/2 ln(−τ ),−∞ < τ <0where τ is the conformal time, solves the Friedmann equation and the scalarfield equation in units where 8πG = 1.(b) In conformal Newtonian gauge, the perturbed metric isds,]Φ