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1 MATHEMATICAL TRIPOS Prt III Mondy 12 June, to 11 PAPER 55 ADVANCED COSMOLOGY Attempt TWO questions. There re THREE questions in totl. The questions crry equl weight. STATIONERY REQUIREMENTS Cover sheet Tresury Tg Script pper SPECIAL REQUIREMENTS None You my not strt to red the questions printed on the subsequent pges until instructed to do so by the Invigiltor.

2 2 1 In synchronous guge (with metric perturbtions h 0µ = 0 bout flt FRW universe with Ω tot = 1), liner perturbtions of multicomponent fluid obey the following evolution equtions δ N + (1 + w N )ik v N (1 + w N)h = 0, w N v N + (1 3w N ) v N + ikδ N = 0, 1 + w N ( ) h + 2 h + 3 (1 + 3w N )Ω N δ N = 0, N where δ N is the density perturbtion, Ω N is the frctionl density, v N is the velocity nd P N = w N ρ N is the eqution of stte of the Nth fluid component, nd k is the comoving wvevector (k = k ), h is the trce of the metric perturbtion nd primes denote differentition with respect to conforml time τ (dτ = dt/). (i) Assume tht the lte universe (t > t eq ) is filled with two components, () comoving non-reltivistic mtter (cold drk mtter) ρ C with no pressure (P C = 0) nd (b) gs of rndomly-oriented cosmic strings ρ S with n verge eqution of stte P S = ρ S /3. Show tht the cold drk mtter string gs equtions rising from ( ) become δ C + δ C 3 ( ) 2 Ω C δ C = 0, 2 ( δ S + 2 δ S 1 ) 3 δ C 1 ( ) 2 3 k2 δ S Ω C δ C = 0. ( ) (ii) By considering new time vrible, η ρ S /ρ C, show tht the dynmicl eqution for the cold drk mtter perturbtion δ C cn be re-expressed s d 2 δ C dη η dδ C 2η(1 + η) dη 3 2η 2 (1 + η) δ C = 0. ( ) [Hint: Recll tht ( ) 2 = 8πG 3 ρ tot 2, ρ N + 3 (ρ N + P N ) = 0.] ( ) 2 = 4πG 3 (ρ tot + P tot ) 2, (iii) Consider erly times η 1 when the cold drk mtter dominted over the string component nd seek power series solution of ( ) of the form δ C = 0 η α + 1 η α Hence or otherwise show tht there is n pproximte growing mode solution of the form δ C A k η (1 4 7η), (η 1). Compre this to the expected growth rte for cold drk mtter perturbtions in mtter dominted universe. (iv) Define the Jens length λ J. Now consider solving the cold drk mtter string gs equtions in the opposite symptotic limit η 1. Show tht the cold drk mtter perturbtion is pproximtely frozen, δ C const. Drw qulittive digrm of the cold

3 3 drk mtter trnsfer function T (k) for wvenumbers coming inside the horizon fter the time of mtter-rdition equlity, t > t eq. Wht is the nlogue here of the dibtic initil condition for rdition δ R = 4 3 δ C when k H? Briefly discuss the pprent qulittive behviour of the string perturbtions δ S on both superhorizon nd subhorizon scles. [TURN OVER

4 4 2 (i) Consider photon with four-momentum p µ (p µ p µ = 0) propgting in perturbed FRW universe (flt Ω = 1) with line element ds 2 = 2 (τ) [ dτ 2 + (δ ij + h ij )dx i dx j] where k is the comoving wvevector nd ˆk i = k i / k. A comoving observer with fourvelocity u µ = 1 (1, 0, 0, 0) mesures the photon energy to be E = u µ p µ = p 0 q/ where q is the comoving momentum. Use the geodesic eqution dpµ dλ + Γµ νσp ν p σ = 0 to show tht for photon trjectory long (unit) direction ˆn i we hve to liner order dq dτ = 1 2 qh ij ˆn iˆn j, dˆn i dτ = O(h ij). [Hint: You my ssume tht Γ 0 00 =, Γ0 0i = 0, Γ0 ij = (δ ij + h ij ) h ij, Γi 0j = δ ij h ij nd Γi jk = 1 2 (h ij,k + h ik,j h jk,i ).] (ii) The photon distribution function f(x, p, τ) cn be expnded bout the Plnck spectrum f 0 (p, τ) = f 0 (q) s f(x, p, τ) = f 0 (q) + f 1 (x, q, ˆn, τ), where the photon momentum p q/. Show tht the collisionless Boltzmnn eqution cn be re-expressed in the form df dλ dxµ dλ f x µ + dpµ dλ f p µ = 0 f 1 τ + f ˆni 1 x i + dq df 0 dτ dq + dq f 1 dτ q + dˆni f 1 dτ ˆn i = 0, which, using the results from prt (i), t liner order reduces to f 1 τ + ikµf 1 = 1 df 0 2 dq h ij ˆn iˆn j, where µ = ˆk ˆn. Finlly, given tht ρ γ is the bckground photon density, rgue tht the brightness function (x, ˆn, τ) 4 T T 4π 4 qf 1 q 2 dq ρ γ must therefore stisfy + ikµ = 2h ij ˆn iˆn j. ( ) (iii) Assume recombintion occurs instntneously t photon decoupling (τ = τ dec ) with the brightness function given in Fourier spce by only the two lowest moments, (k, µ, τ dec ) = δ γ (k, t dec ) + 4ˆn v(k, t dec ), where δ γ is the photon density perturbtion nd v is the verge fluid velocity. Briefly describe the importnt physicl mechnisms which mke this poor pproximtion on smll ngulr scles (with multipole l < 200). Use this ssumption nd eqn ( ) to derive the Schs-Wolfe eqution in rel spce for the temperture fluctution in direction ˆn: T T (x 0, ˆn, τ 0 ) = 1 4 δ γ(x, τ dec ) + ˆn v(x, τ dec ) 1 2 τ0 τ dec h ij ˆn iˆn j dτ.

5 5 3 In the 3+1 formlism, we split spcetime using the line element ds 2 = N 2 dt 2 + (3) g ij (dx i N i dt)(dx j N j dt), with lpse function N(t, x i ), shift vector N i (t, x i ) nd (3) g ij (x i ) the three-metric on constnt time spcelike hypersurfces Σ. The vector n µ = 1 N (1, N i ) is norml to Σ nd defines the extrinsic curvture through K ij n i;j = 1 ( ) (3) g ij,0 + N i j + N j i, 2N where denotes the covrint derivtive in Σ. (i) Consider the conforml 3-metric (3) g ij = ( (3) g) 1/3 (3) g ij where (3) g = det( (3) g ij ) nd, hence or otherwise, tke the trce of the extrinsic curvture expression to find K (3) g ij K ij = 1 [ (3)ġ ] 2N (3) g 2N i i. In the context of n expnding universe (setting N i = 0), rgue tht K/3 cn be interpreted s loclly defined Hubble prmeter H(t, x i ). [Hint: You my ssume tht Tr(A 1 da/dt) = d(ln(det A))/dt for ny mtrix A with non-vnishing determinnt.] (ii) When linerising the 3+1 metric bout flt FRW universe, we define the sclr perturbtions by N(t, x i ) N(t)(1 + Φ(t, x i )), N i 2 B,i, (3) g ij = 2 (t)[(1 2Ψ)δ ij 2E,ij ], nd lso ρ = ρ + δρ nd P = P + δp, where brs denote bckground homogeneous quntities. In synchronous guge, we tke Φ = 0 nd B = 0. Given tht metric perturbtions trnsform s δ g αβ = δg αβ ḡ αβ,γ ξ γ ḡ γβ ξ γ,α ḡ αγ ξ γ,β under (t, xi ) ( t, x i ) = (t + ξ 0, x i + ξ i ), where ξ i i λ, show tht there is residul guge freedom in synchronous guge given by the coordinte trnsformtion, ξ 0 = C(xi ) N, λ = C(xi ) N 2 dt + D(xi ), where C nd D re rbitrry functions of x i only. Briefly discuss the significnce of this guge freedom during () infltion nd (b) the stndrd hot big bng. In longitudinl Newtonin guge we tke insted E = B = 0. Find trnsformtion lw tht expresses the density perturbtion δρ/ρ in Newtonin guge in terms of synchronous guge quntities. (iii) Show tht the quntity ζ = Ψ 1 3 δρ ρ + P, [TURN OVER

6 6 is guge-invrint nd tht it is independent of time on superhorizon scles, tht is, ζ = 0 for k H. [Hint: You my ssume definite eqution of stte P = wρ, tht the perturbed energy density conservtion eqution is δρ/ N = 3H(δρ + δp ) + ( ρ + P )(κ 3HΦ) u, nd tht the metric perturbtion Ψ stisfies Ψ/ N = HΦ+ 1 3 κ+ 1 3 χ, where 2 / 2, u genertes the sclr velocity perturbtion, nd κ nd χ generte the trce nd trceless prt of K ij respectively. ] END OF PAPER