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The third paper of the advanced cosmology module from the university of cambridge's mathematical tripos exam series, held on june 13, 2005. The paper covers various topics in cosmology, including the conservation of canonical momentum for a point particle in a background metric, the field equations for a massive scalar field in a flat frw background, and the evolution of density perturbations in the early universe. Students are required to answer three questions from the five provided.
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Monday 13 June, 2005 1.30 to 4.
Attempt THREE questions.
There are FIVE questions in total. The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
1 The action for a point particle of mass m in a background metric gμν is
m 2
gμν x˙μ^ x˙ν^ dτ ,
where xμ^ = (t, x), dots denote τ derivatives, and the canonical four-momentum P (^) μc ≡ (^) δδS x˙μ is subject to the constraint gμν^ P (^) μcP (^) νc = −m^2.
For a flat FRW background, gμν = a^2 (t)ημν , show that
(i) the canonical three-momentum Pc^ is conserved, (ii) the special relativistic physical momentum P ≡ γmv, with v = (dx/dt) and γ = 1/
1 − v^2 , decays as a(t)−^1.
Now consider the action for a massive scalar field,
d^4 x
−g(−
∂μφ∂ν φgμν^ −
m^2 φ^2 ).
(iii) Compute S for a flat FRW background metric gμν = a^2 (t)ημν and show that the field equations read
φ′′^ + 2
a′ a
φ′^ − ∇^2 φ = −a^2 m^2 φ , (∗)
where primes denote (^) ∂t∂.
(iv) For massless field, m = 0, in a radiation-dominated universe, a ∝ t, show the general solution to (∗) may be written
φ =
a(t)
k
(Akeik.xe−iwt^ + c.c.) ,
where w = |k| and Ak is constant for each k.
(v) Recalling the quantisation rule [P (^) ic , xj ] = −iℏδij → Pc^ = ℏ i ∇, do the k modes of φ have fixed canonical, or physical momentum? How does this accord with your answers to parts (i) and (ii)?
Paper 64
3 In synchronous gauge for linear perturbations about a flat (k = 0) FRW back- ground, the metric is taken to be
ds^2 = a^2 (τ )
−dτ 2 + (δij + hij )dxidxj^
where | det(hij )| 1 and the relevant connections are Γ^000 = a′/a, Γ^00 i = Γi 00 = 0,
Γ^0 ij =
a′ a
[δij + hij ] +
h′ ij , Γi 0 j =
a′ a
δij +
h′ ij , Γijk =
[hij,k + hik,j − hjk,i] ,
with primes (e.g. a′) denoting differentiation with respect to the conformal time τ.
(i) Assume the universe is filled with a perfect fluid with energy-momentum tensor
T μν^ = (ρ + P )uμuν^ + P gμν^ ,
where uμ^ is the fluid 4-velocity, ρ is the energy density and P is the pressure, with the latter related through an equation of state P = wρ. In a comoving synchronous frame (i.e. uμ^ = a−^1 (1, v) with |v| 1), linearise the energy-momentum tensor to find that in Fourier space we have
a^2
ρ¯(1 + δ) , T 0 i^ =
a^2
(1 + w)¯ρ ikiθ , T ij^ =
a^2
w ρ¯ [(1 + δ)δij − hij ] ,
with ki the components of the comoving wavevector k, the background density ¯ρ = ¯ρ(t), and where suitable definitions should be given for the density perturbation δ and the velocity potential θ. Show that energy-momentum conservation equation T 0 μ;μ = 0 yields
δ′^ − (1 + w)k^2 θ + 12 (1 + w)h′^ = 0 ,
where h = hii. Discuss the relation of a cold dark matter perturbation δc (i.e. P = 0 and comoving v = 0) to the metric perturbation hij , demonstrating that density change is due to volume change in synchronous gauge.
(ii) Now assume that the cold dark matter density perturbation evolution equation is given by
δ′′ c +
a′ a
δ′ c −
a′ a
Ωc δc = 0 ,
where the density parameter is Ω ≡ (8πG/3)¯ρ(a/a′)^2. Why is this equation only valid during the matter-dominated era (when a ∝ τ 2 ), that is, only after equal-matter radiation (τ > τeq) and before the accelerating phase (τ < τΛ)? Find the growing and decaying mode solutions for δc while τeq < τ < τΛ.
During the radiation era (τ < τeq), adiabatic cold dark matter perturbations on superhorizon scales (k < 2 π/τ ) grow as δc ∝ τ 2 , but subhorizon (k > 2 π/τ ) perturbations stagnate, δc ≈ const. Provide brief physical explanations for these different behaviours.
Use these growing mode solutions to derive an approximate transfer function T (k) in Fourier space to project forward to today a primordial set of initial perturbations, that is, δk(τ 0 ) = T (k)δk(τi) ,
Paper 64
where k = |k|, initially τi τeq and, for simplicity, assume that today τ 0 ≈ τΛ (ignore acceleration). For a primordial scale-invariant power spectrum Pi(k) = Ak (A is a constant), use this transfer function to find the power spectrum today. Comment on any significant features in the power spectrum and the lengthscales and observed structures with which they might be associated.
Paper 64 [TURN OVER
5 Consider a scalar field φ with action
d^4 x
−g(−
(∂φ)^2 − V (φ)).
(i) Show this yields a stress-energy tensor
Tμν = ∂μφ∂ν φ − gμν
(∂φ)^2 + V (φ)
[Hint: You may use δ
−g = (^12)
−gδgαβ gαβ^ and δ(gαβ^ ) = −gαμgβν^ δgμν .] (ii) Compute the energy density ρ = −T 00 and pressure P = 13 T (^) ii due to a homogeneous scalar field φ¯(t) in a flat FRW background with line element
a^2 (t)(−dt^2 + dx^2 ) ,
where t is the conformal time.
(iii) If φ is perturbed, φ = φ¯(t) + δφ(t, x) ,
show that the momentum density
δT (^) i^0 = −
a^2
φ¯′∂iδφ.
(iv) For a general scalar metric perturbation, the line element reads
ds^2 = a^2 (t)
−(1 + 2Φ)dt^2 − 2 ∂iβdxidt + [δij (1 − 2Ψ) + 2∂i∂j χ] dxidxj^
Under a coordinate transformation ˜xμ^ = xμ^ + ξμ, we have
˜gμν (x) − gμν (x) = −gμαξα,ν − gναξ,μα − gμν,αξα^ , φ^ ˜(x) − φ(x) = −ξα∂αφ.
In particular, if ξ^0 = T and ξi^ = ∂iL work out the transformation of Ψ and δφ.
Write down a gauge-invariant combination of Ψ and δφ and give its physical interpretation.
Paper 64
