Paper59_2 (2) - MATHEMATICAL TRIPOS Part III Friday 4 June 2004 1.30 to 4.30 PAPER 59 ADVANCED COSMOLOGY Attempt THREE questions There are six questions

Paper59_2 (2) - MATHEMATICAL TRIPOS Part III Friday 4 June...

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MATHEMATICAL TRIPOS Part III Friday 4 June, 2004 1.30 to 4.30 PAPER 59 ADVANCED COSMOLOGY Attempt THREE questions. There are six questions in total. The questions carry equal weight.
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You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
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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 DµDνWγDνDµWγ= Wλ(3)Rγνµλ,(∗)for any Wγ∈ Σ3i.e. Wγnγ= 0, show that(3)Rγνµλ= PξλPµαPνβPγδ(4)RδβαξKµγKνλ+ KνγKµλ.You may assume Kµνis symmetric. [Hint: express the left hand side of (∗) in four dimensional terms.] =
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Paper 59
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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
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