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includes one expansion term from continuity and one from momentum dilution. This equation uses sub-Hubble, pressureless linear perturbations and the stated matter-only source for the gravitational potential.

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Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 55 / 3 / b / Solution Created 2026-10-03 Updated 2026-10-06

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Along the zeroth-order photon path, set

Y = Θ + ψ

and

S = (3/4) \int d \hat{m} Θ (\hat{m}) [1 + (e \cdot \hat{m})^{2}] / (4 π)

. The scalar photon energy redshift and the Photon Boltzmann equation with Thomson scattering give

\frac{d Y}{d η} = ϕ^{'} + ψ^{'} + τ^{'} [Y - ψ - S - e \cdot v_{b}] .

(1)

The integrating factor is

e^{- τ}

, because

τ^{'} < 0

in this convention. Hence

\frac{d}{d η} (e^{- τ} Y) = e^{- τ} (ϕ^{'} + ψ^{'}) + g (η) (ψ + S + e \cdot v_{b}), g = - τ^{'} e^{- τ} .

(2)

Tight coupling before decoupling makes the incident distribution nearly isotropic in the baryon frame. Neglecting its quadrupole, the angular kernel integrates to

S = Θ_{0}

, since

(3/4) ⟨ 1 + (e \cdot \hat{m})^{2} ⟩ = 1

. The primordial term at the optically thick lower endpoint is erased. With instantaneous last scattering, no later scattering and

g = δ (η - η_{*})

, the line-of-sight solution is

Θ (η_{0}, x_{0}, e) + ψ (η_{0}, x_{0}) = (Θ_{0} + ψ + e \cdot v_{b}) (η_{*}, x_{0} - χ_{*} e) + \int_{η_{*}}^{η_{0}} d η (ϕ^{'} + ψ^{'}) (η, x_{0} - (η_{0} - η) e) .

(3)

The approximation is linear, uses an unperturbed ray, neglects the scattering quadrupole and polarization feedback, and idealizes the cosmological visibility function as a delta function. It also excludes reionization or any other post-recombination opacity.

The photon monopole

Θ_{0}

is the intrinsic emission-temperature perturbation. The emission potential

ψ_{*}

gives gravitational redshift on leaving the potential well; together they form the Sachs-Wolfe combination. The baryon peculiar velocity supplies the Doppler CMB anisotropy. The integral is the Integrated Sachs-Wolfe effect, from changes in gravitational potentials during travel. The observer potential is a direction-independent endpoint shift absorbed into the observed monopole. An observer peculiar velocity, if included, contributes an additional kinematic dipole; the stated observer is coordinate-comoving.

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Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 55 / 4 / a / Solution Created 2026-10-03 Updated 2026-10-06

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A scalar cosmological perturbation at fixed wavevector carries no helicity or distinguished transverse direction. Its metric and scalar sources are unchanged by rotations around

k

, and the scalar baryon peculiar velocity is longitudinal. With scalar initial conditions, rotational invariance of the linear evolution therefore makes

Θ

a function only of

μ = \hat{k} \cdot e

. Expand it using the stated normalization, without an extra

(2 ℓ + 1)

factor:

Θ (μ) = \sum_{ℓ \geq 0} (- i)^{ℓ} Θ_{ℓ} P_{ℓ} (μ), Θ_{ℓ} = i^{ℓ} \frac{2 ℓ + 1}{2} \int_{- 1}^{1} d μ P_{ℓ} (μ) Θ (μ) .

(1)

In the Thomson kernel,

1 + v^{2} = 4/3 + (2/3) P_{2} (v)

. The spherical harmonic addition theorem and Orthogonality of Legendre polynomials give

\int \frac{d m ^}{4 π} P_{ℓ} (\hat{k} \cdot \hat{m}) P_{L} (e \cdot \hat{m}) = \frac{δ _{ℓ L}}{2 ℓ + 1} P_{ℓ} (\hat{k} \cdot e) .

(2)

Only the monopole and quadrupole survive; the quadrupole's

(- i)^{2} = - 1

gives the angular scattering integral

\int \frac{d m ^}{4 π} Θ (\hat{m}) [1 + (e \cdot \hat{m})^{2}] = \frac{4}{3} Θ_{0} - \frac{2}{15} Θ_{2} P_{2} (μ) .

(3)

This also explains why an isotropic distribution has zero net collision term.

The streaming term

ik μ Θ

and the Legendre polynomial recurrence relation yield, after division by

(- i)^{ℓ}

k [(ℓ + 1) Θ_{ℓ + 1} / (2 ℓ + 3) - ℓ Θ_{ℓ - 1} / (2 ℓ - 1)]

. With

v_{b} = i \hat{k} v_{b}

, the velocity collision source is

+ τ^{'} v_{b}

in the dipole equation; the gravitational gradient source is

+ k ψ

. Therefore the photon Boltzmann hierarchy is

Θ_{ℓ}^{'} + k [\frac{ℓ + 1}{2 ℓ + 3} Θ_{ℓ + 1} - \frac{ℓ}{2 ℓ - 1} Θ_{ℓ - 1}] = τ^{'} [(1 - δ_{ℓ 0}) Θ_{ℓ} + δ_{ℓ 1} v_{b} - \frac{1}{10} δ_{ℓ 2} Θ_{2}] + δ_{ℓ 0} ϕ^{'} + δ_{ℓ 1} k ψ .

(4)

Set the absent

Θ_{- 1}

contribution to zero. This is algebraically the printed hierarchy. In particular,

Θ_{0}^{'} + \frac{k}{3} Θ_{1} = ϕ^{'}, Θ_{1}^{'} + k (\frac{2}{5} Θ_{2} - Θ_{0}) = τ^{'} (Θ_{1} + v_{b}) + k ψ,

(5)

Θ_{2}^{'} + k (\frac{3}{7} Θ_{3} - \frac{2}{3} Θ_{1}) = \frac{9}{10} τ^{'} Θ_{2} .

(6)

Since

τ^{'} < 0

, these collision terms damp the nonzero multipoles and drive the dipole to

Θ_{1} = - v_{b}

. The latter sign follows from the paper's velocity and

(- i)^{ℓ}

conventions, not a reversed physical photon velocity.

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Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 1 / 9C / Solution Created 2026-09-24 Updated 2026-10-05

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For comoving galaxies their relative comoving vector

χ

is constant, and the physical displacement is

r (t) = a (t) χ

, where

a

is the scale factor. Differentiating gives the Hubble law

v = H r, H = \frac{a ˙}{a} .

(1)

Peculiar velocity relative to this comoving frame is excluded in this relation.

Transport a wave crest between neighbouring comoving observers separated by a small physical distance

d ℓ

. Their recession speed is

H d ℓ

, while the light travel time is

d t = d ℓ / c

to first order. The local Doppler shift therefore changes its wavelength by

d λ / λ = H d t

. Multiplying these infinitesimal shifts along the ray, or integrating their logarithms, yields

lo g \frac{λ _{0}}{λ _{e}} = \int_{t_{e}}^{t_{0}} H (t) d t = lo g \frac{a ( t _{0} )}{a ( t _{e} )}, 1 + z = \frac{a ( t _{0} )}{a ( t _{e} )} .

(2)

The use of the local first-order Doppler law is exact in the limit of infinitesimal steps: the accumulated second-order errors tend to zero. A single special-relativistic recession velocity assigned to widely separated galaxies would not justify this cosmological redshift relation.

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Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 346 / 1 / a / Solution Created 2026-10-03 Updated 2026-10-05

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Write

x

for a comoving coordinate,

u

for the peculiar velocity,

H = \overset{a}{˙} / a

for the Hubble parameter, and

δ = δ ρ / \overset{ρ}{ˉ}

for the density contrast. For a homogeneous self-gravitating fluid, the linearized continuity equation, Euler equations for an inviscid fluid, and cosmological Poisson equation are

\dot{δ} = - \frac{1}{a} \nabla \cdot u, \dot{u} + H u = - \frac{c _{s}^{2}}{a} \nabla δ - \frac{1}{a} \nabla ϕ, \nabla^{2} ϕ = 4 π G a^{2} \overset{ρ}{ˉ} δ .

(1)

Here

c_{s}

is the physical adiabatic sound speed, gradients refer to comoving coordinates, and the homogeneous background force has already been subtracted. Differentiating the first equation and eliminating

u

and

ϕ

gives

\ddot{δ} + 2 H \dot{δ} - \frac{c _{s}^{2}}{a ^{2}} \nabla^{2} δ - 4 π G \overset{ρ}{ˉ} δ = 0.

(2)

A Fourier mode with comoving wavenumber

k

therefore obeys

\ddot{δ}_{k} + 2 H \dot{δ}_{k} + (\frac{c _{s}^{2} k ^{2}}{a ^{2}} - 4 π G \overset{ρ}{ˉ}) δ_{k} = 0.

(3)

The Jeans wavenumber occurs where the coefficient in parentheses vanishes. Since a comoving wavelength is

2 π / k

, the result is

k_{J} = \frac{a 4 π G ρ ˉ}{c _{s}}, λ_{J}^{com} = \frac{c _{s}}{a} \frac{π}{G ρ ˉ} .

(4)

Below this comoving Jeans length, pressure restores a displaced fluid element: there are acoustic waves rather than growing Jeans instability. In a static background they oscillate; expansion changes their amplitude and frequency, and dissipation can damp them. Pressure support alone does not erase them. In the pre-recombination photon-baryon fluid, Silk damping supplies a separate erasure mechanism.

For collisionless matter, replace the sound speed by a characteristic velocity dispersion. For an isotropic Maxwell-Boltzmann velocity distribution, defining

σ^{2} = ⟨ u_{x}^{2} ⟩ = ⟨ ∣ u ∣^{2} ⟩ /3

, the static collisionless marginal-stability calculation gives

λ_{J, collisionless}^{com} = \frac{σ}{a} \frac{π}{G ρ ˉ} = \frac{v _{rms}}{a} \frac{π}{3 G ρ ˉ} .

(5)

The coefficient depends on the distribution and the convention for velocity dispersion. Small-scale suppression is now collisionless free streaming and phase mixing, rather than collisional acoustic waves. An instantaneous collisionless Jeans length must be distinguished from the accumulated comoving distance

\int v d t / a

traveled since decoupling.

For the requested thermal histories, an Einstein-de Sitter universe describes the matter-era limit: it cannot literally also have a radiation-dominated epoch. Interpret the question as a spatially flat universe with negligible cosmological constant, passing from radiation domination,

a \propto t^{1/2}

, to matter domination,

a \propto t^{2/3}

. For the usual schematic dynamical Jeans scale use

\overset{ρ}{ˉ}_{tot}

in the estimate, so

λ_{dyn}^{com} \propto {c_{s} a, c_{s} a^{1/2}, t ≪ t_{eq}, \overset{ρ}{ˉ}_{tot} \propto a^{- 4}, t ≫ t_{eq}, \overset{ρ}{ˉ}_{tot} \propto a^{- 3} .

(6)

Near the particle horizon a relativistic perturbation calculation replaces the Newtonian derivation; the scaling estimate still identifies the relevant sound-crossing or streaming scale.

Before cosmological recombination, adiabatic initial conditions keep the entropy per baryon fixed in the tightly coupled photon-baryon fluid. With

R_{b} = 3 \overset{ρ}{ˉ}_{b} / (4 \overset{ρ}{ˉ}_{γ}) \propto a

, its photon-baryon sound speed is

c_{s}^{2} = \frac{c ^{2}}{3 ( 1 + R _{b} )} .

(7)

For example, this follows from

δ ρ_{γ} / \overset{ρ}{ˉ}_{γ} = (4/3) δ ρ_{b} / \overset{ρ}{ˉ}_{b}

and

δ P = c^{2} δ ρ_{γ} /3

. In the radiation-dominated limit

R_{b} ≪ 1

c_{s} ≃ c / 3

; in the strongly baryon-loaded matter-era limit

R_{b} ≫ 1

c_{s} \propto a^{- 1/2}

. Thus the baryon Jeans length across recombination has the asymptotic history

λ_{J, b}^{com} \propto ⎩ ⎨ ⎧ a \propto t^{1/2}, approximately constant, a^{- 1/2} \propto t^{- 1/3}, t ≪ t_{eq}, t_{eq} ≪ t < t_{rec}, t > t_{rec} with adiabatic gas cooling .

(8)

The transition to the plateau is smooth: retaining

1 + R_{b}

gives

λ_{J, b}^{com} \propto a^{1/2} / 1 + R_{b}

in matter domination. If baryon loading is still small, this intermediate segment instead rises as

a^{1/2}

until loading becomes important.

At photon decoupling, radiation ceases to provide pressure support to the baryons, so the sound speed drops from the coupled-fluid value to that of a nonrelativistic ideal gas,

c_{s, b}^{2} = γ k_{B} T_{b} / (μ m_{p})

, with

γ = 5/3

. The drop in comoving Jeans length at fixed density is the same ratio of sound speeds, typically several orders of magnitude. Subsequently an isentropic process gives

T_{b} \propto \overset{ρ}{ˉ}_{b}^{γ - 1} \propto a^{- 2}

, so

c_{s, b} \propto a^{- 1}

and

λ_{J, b}^{com} \propto a^{- 1/2}

. Residual energy exchange through Compton scattering can initially keep

T_{b} \propto a^{- 1}

, producing a short approximately flat gas segment before the decreasing segment. The idealized sketch assumes immediate adiabatic cooling:

log(comoving baryon Jeans scale)
  ^
  |                   __________________
  |                 /                   |
  |               /                     |  loss of photon pressure
  |             /                       |
  |           /                         +---\
  |         /                                \
  +-------------------|-----------------|----------> log(time)
                    t_eq              t_rec
        slope 1/2          slope ~0        slope -1/3

For the cold-dark-matter kinetic Jeans scale, take the stated toy history to mean that collisions maintain a particle temperature equal to the photon temperature until kinetic decoupling. This is an assumption about thermal coupling; a negligible collision rate after decoupling makes the subsequent treatment collisionless. Assume the ordering

t_{NR} < t_{dec} < t_{eq} < t_{rec}

. While relativistic, the characteristic particle speed is constant; while nonrelativistic but thermally coupled,

σ \propto T \propto a^{- 1/2}

; after kinetic decoupling, particle momentum redshifts as

a^{- 1}

and

σ \propto a^{- 1}

. Combining these with the background dynamical density gives

λ_{dyn, CDM}^{com} \propto ⎩ ⎨ ⎧ a \propto t^{1/2}, a^{1/2} \propto t^{1/4}, constant, a^{- 1/2} \propto t^{- 1/3}, t < t_{NR}, t_{NR} < t < t_{dec}, t_{dec} < t < t_{eq}, t > t_{eq} .

(9)

There is no sudden change at cosmological recombination: cold dark matter has already decoupled. Its small initial velocity dispersion makes its suppression scale much smaller than the pre-recombination baryon scale.

log(comoving CDM dynamical Jeans scale)
  ^
  |                        _______________
  |                    ___/               \
  |                ___/                    \       no jump
  |              /                          \      at t_rec
  |            /                             \
  +-----------|------------|--------------|----|------> log(time)
            t_NR         t_dec          t_eq t_rec
   slope 1/2    slope 1/4       slope 0       slope -1/3

The density convention matters. If one instead defines a species' self-gravitating collisionless Jeans length using only its own nonrelativistic density

\overset{ρ}{ˉ}_{m} \propto a^{- 3}

, its thermally coupled segment is constant and its decoupled segment decreases as

a^{- 1/2}

even during radiation domination. The constant radiation-era segment in the second sketch is the usual instantaneous background dynamical streaming estimate, not an exact single-species self-gravity threshold. A complete multicomponent calculation uses the separate perturbation equations and their combined cosmological Poisson equation.

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Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 346 / 3 / Solution Created 2026-10-03 Updated 2026-10-05

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Assume nonrelativistic pressureless matter, an initial irrotational vector field of growing-mode linear cosmological density perturbations, a homogeneous expanding background, and no shell crossing. All gradients below refer to initial comoving coordinates. Let

ρ_{0} = \overset{ρ}{ˉ}_{m} (t) a^{3} (t) = \overset{ρ}{ˉ}_{m} (t_{i})

, with

a (t_{i}) = 1

, and normalize the linear growth factor by

D (t_{i}) = 1

. The paper calls

D

a growth rate, but it is the dimensionless growth factor;

\dot{D}

is its time derivative.

Writing

x_{i}

for the actual position at

t_{i}

, the Zeldovich approximation is, to first order,

x (t) = x_{i} + [D (t) - 1] s (x_{i}), \dot{x} = \dot{D} s (x_{i}) .

(1)

This explicitly gives

x (t_{i}) = x_{i}

. The usual notation

x = q + D s (q)

instead uses the uniform reference Lagrangian coordinate

q

, with

x_{i} = q + s (q)

. The two labels can be interchanged inside a first-order displacement, but not in the unperturbed position without the initial displacement correction.

To determine

s

, mass conservation between the initial and current positions gives

1 + δ (t) = \frac{1 + δ _{i}}{d e t [ 1 + ( D - 1 ) \nabla s ]} = 1 + δ_{i} - (D - 1) \nabla \cdot s + O (δ_{i}^{2}) .

(2)

Demanding

δ (t) = D (t) δ_{i}

gives

\nabla \cdot s = - δ_{i}

. The irrotational vector field assumption makes

s

a gradient; the initial cosmological Poisson equation is

\nabla^{2} Φ_{i} = 4 π G ρ_{0} δ_{i}

. With the same boundary conditions for these potential equations, and after removing an irrelevant uniform translation,

s = - \frac{\nabla Φ _{i}}{4 π G ρ _{0}}, \dot{x} = - \frac{D ˙}{4 π G ρ ˉ _{m} a ^{3}} \nabla Φ_{i} .

(3)

The evolution of

D

follows from the linearized peculiar-motion equation

\ddot{x} + 2 H \dot{x} = - \nablaΦ/ a^{2}

. Since cosmological Poisson equation gives

Φ (t) = (D / a) Φ_{i}

to this order, substitution yields the linear growth equation

\ddot{D} + 2 H \dot{D} - 4 π G \overset{ρ}{ˉ}_{m} D = 0.

(4)

Thus the approximation extrapolates the linear growing displacement along a fixed initial direction, even as its density mapping becomes nonlinear.

The early spin of a dark-matter halo comes from an external gravitational torque on its nonspherical initial mass region. A uniform external acceleration moves its centre of mass without spinning it; the spatially varying tidal tensor exerts different forces on different parts. The tidal torque theory requires misalignment of the region's shape and the surrounding tidal field. A local irrotational vector field of velocity does not imply zero integrated angular momentum for a nonspherical region.

Put

y = x_{i} - \hat{x}_{i}

and use the physical peculiar velocity

v = a \dot{x}

. The physical lever arm is

a (x - \hat{x})

, while a leading-order mass element is

ρ_{0} d^{3} x_{i}

. Substituting the velocity formula into the supplied definition of angular momentum about the centre of mass gives

J = - \frac{a ^{2} D ˙}{4 π G} \int_{V_{L}} y \times \nabla Φ_{i} d^{3} x_{i} .

(5)

Terms from the perturbed mass measure are higher order. Subtracting the barycentre's velocity also changes nothing because

\int_{V_{L}} y d^{3} x_{i} = 0

at this order.

Apply the divergence theorem componentwise:

ϵ_{jk l} \int_{V_{L}} y_{k} \partial_{l} Φ_{i} d^{3} x_{i} = ϵ_{jk l} \int_{Σ_{L}} Φ_{i} y_{k} d S_{l} - ϵ_{jk l} \int_{V_{L}} Φ_{i} δ_{k l} d^{3} x_{i} = ϵ_{jk l} \int_{Σ_{L}} Φ_{i} y_{k} d S_{l} .

(6)

The antisymmetry of the Levi-Civita symbol kills the last term. Consequently, for the ordinary oriented comoving surface element

d S

J = - \frac{a ^{2} D ˙}{4 π G} \int_{Σ_{L}} Φ_{i} y \times d S .

(7)

The original PDF prints a different coefficient,

- \dot{D} / (4 π G \overset{ρ}{ˉ}_{m} a^{2})

. That coefficient does not follow from its own mass measure and velocity equation and has the wrong dimensions for total angular momentum. The corrected coefficient above also gives the time dependence consistent with the paper's later tensor expression; this is a source typo, not a TeX transcription issue.

For a spherical

V_{L}

centered on its barycentre, the outward normal is parallel to

y

, so

y \times d S = 0

pointwise. Hence

J = 0 for a spherical initial region .

(8)

For an equipotential surface,

Φ_{i} = Φ_{*}

Σ_{L}

, and

ϵ_{jk l} \int_{Σ_{L}} y_{k} d S_{l} = ϵ_{jk l} \int_{V_{L}} \partial_{l} y_{k} d^{3} x_{i} = ϵ_{jk l} δ_{k l} Vol (V_{L}) = 0.

(9)

Therefore

J = 0 if the initial boundary is an equipotential .

(10)

The second conclusion holds for any boundary shape at this order, not just a sphere.

To extract the leading tidal torque, expand the initial peculiar gravitational potential about the barycentre using its Taylor series:

Φ_{i} (\hat{x}_{i} + y) = Φ_{0} + g_{k} y_{k} + \frac{1}{2} H_{km} y_{k} y_{m} + \dots, H_{km} = \partial_{k} \partial_{m} Φ_{i} (\hat{x}_{i}) .

(11)

The constant has no force;

g

has zero integrated torque because the first mass moment vanishes. With the mass second-moment tensor

I_{km} = ρ_{0} \int_{V_{L}} y_{k} y_{m} d^{3} x_{i},

(12)

we obtain

J_{j} = - \frac{a ^{2} D ˙}{4 π G ρ _{0}} ϵ_{jk l} H_{l m} I_{km} = \frac{D ˙}{4 π G ρ ˉ _{m} a} ϵ_{jk l} H_{km} I_{m l} .

(13)

The second equality swaps the indices

k, l

and uses the symmetry of both tensors. Define the initial acceleration tidal tensor by

T_{km} = - \partial_{k} \partial_{m} Φ_{i} (\hat{x}_{i}) = - H_{km} .

(14)

Then the formula in the question has exactly the stated sign:

J_{j} = - \frac{D ˙}{4 π G ρ ˉ _{m} a} ϵ_{jk l} T_{km} I_{m l} .

(15)

If one defines

T

as the positive Hessian matrix of

Φ_{i}

instead, the displayed contraction has a plus sign. Defining the convention is essential. Also, the paper's

I

is a mass second-moment tensor; the mechanical inertia tensor is

tr (I) 1 - I

Only the anisotropic parts contribute: a multiple of the identity produces zero contraction with the Levi-Civita symbol. In axes where

I = diag (I_{1}, I_{2}, I_{3})

, for example,

J_{1} = \frac{D ˙}{4 π G ρ ˉ _{m} a} T_{23} (I_{2} - I_{3}) .

(16)

Thus a spherical region or tensors sharing principal axes have zero leading torque; unequal principal moments together with off-diagonal tidal components create spin. The Taylor series truncation assumes that higher spatial derivatives of the tidal field are sufficiently small across the region.

In an Einstein-de Sitter universe,

a \propto t^{2/3}

D \propto a

, and

\overset{ρ}{ˉ}_{m} \propto a^{- 3}

. The initial

I

and

T

are time independent, so

J \propto a^{2} \dot{D} \propto t^{4/3} t^{- 1/3} \propto t .

(17)

This is early-time growth for fixed initial matter, before collapse invalidates the extrapolation. It does not predict indefinite linear spin growth for a virialized halo.

Two reasons for only approximate agreement with simulations are that nonlinear collapse and shell crossing change the trajectories and the tidal field, invalidating the first-order displacement and fixed leading tidal tensor; and that real halos undergo dark-matter halo mergers, anisotropic accretion, and exchange of material and angular momentum, so the halo identified at a later time need not be the same isolated collection of initial particles. These processes can change both the magnitude and the direction of the spin.

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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 312 / 3 / a / Solution 2026-09-28

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The cosmological continuity equation expresses conservation of dark-matter mass:

δ

is the density contrast,

v

is the peculiar velocity, and a prime denotes a conformal time derivative. The cosmological Euler equation expresses momentum conservation:

H = a^{'} / a

is the conformal Hubble rate,

ϕ

is the peculiar gravitational potential,

ρ

is the density, and

σ_{ij}

is the velocity-dispersion tensor of collisionless matter. The terms are respectively Hubble drag, convective acceleration, gravity, and velocity-dispersion stress.

For curl-free flow, introduce the peculiar-velocity divergence

θ = \nabla \cdot v

and set

σ_{ij} = 0

. A Fourier transform of the nonlinear continuity term gives

δ^{'} (k) + θ (k) = - \int \frac{d ^{3} q}{( 2 π ) ^{3}} α (q, k - q) θ (q) δ (k - q),

(1)

with the alpha mode-coupling kernel

α (k_{1}, k_{2}) = \frac{( k _{1} + k _{2} ) \cdot k _{1}}{k _{1}^{2}} .

(2)

Taking the divergence of the Euler equation gives the quadratic velocity kernel

β (k_{1}, k_{2}) = \frac{∣ k _{1} + k _{2} ∣ ^{2} ( k _{1} \cdot k _{2} )}{2 k _{1}^{2} k _{2}^{2}} .

(3)

This is the beta mode-coupling kernel; its symmetry follows from the two velocity factors.

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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 346 / 1 / Solution 2026-09-29

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Write physical position as

r = a (t) x

and physical velocity as

\dot{r} = H r + v, v = a \dot{x} .

(1)

Subtract the homogeneous expanding background from the pressureless Euler equation. To linear order in peculiar velocity and density contrast, the convective term is negligible and

\dot{v} + H v = - \frac{1}{a} \nablaΦ,

(2)

\frac{d}{d t} (a v) = - \nablaΦ .

(3)

Here

\nabla

differentiates with respect to comoving position and

Φ

is the peculiar gravitational potential.

The linear continuity and Poisson equations are

\dot{δ} + \frac{1}{a} \nabla \cdot v = 0, \nabla^{2} Φ = 4 π G a^{2} \overset{ρ}{ˉ}_{m} δ .

(4)

For the growing mode

δ (x, t) = D (a) δ_{i} (x)

, matter conservation gives

\overset{ρ}{ˉ}_{m} \propto a^{- 3}

, so the potential scales as

Φ (x, t) = \frac{D ( a )}{a} Φ_{i} (x)

(5)

after absorbing the fiducial normalization into

Φ_{i}

. Integrating the linear Euler equation with a negligible decaying mode gives

v = - \frac{\nabla Φ _{i}}{a} \int^{t} \frac{D ( a )}{a} d t^{'} .

(6)

Taking the divergence of Euler and using continuity and Poisson yields the linear growth equation

\ddot{D} + 2 H \dot{D} - 4 π G \overset{ρ}{ˉ}_{m} D = 0.

(7)

In the fiducial normalization used in the question this can be rearranged as

\frac{D ( a )}{a} = \frac{1}{4 π G ρ ˉ _{m, i}} \frac{d}{d t} (a^{2} \frac{d D}{d t}) .

(8)

The cosmological Lagrangian displacement obeys

\dot{ψ} = v / a

. Combining the last two relations and choosing the growing displacement gives the Zeldovich approximation

ψ (t) = - \frac{D ( a )}{4 π G ρ ˉ _{m, i}} \nabla Φ_{i} .

(9)

Define

b (t) = \frac{D ( a )}{4 π G ρ ˉ _{m, i}} .

(10)

Then

x = x_{i} - b \nabla Φ_{i}

and

v = - a \dot{b} \nabla Φ_{i}

. Keeping the lowest nonvanishing order in displacement in the halo angular momentum gives

J = - \overset{ρ}{ˉ}_{m} a^{5} \dot{b} \int_{V_{L}} d^{3} x_{i} (x_{i} - \overset{ˉ}{x}_{i}) \times \nabla Φ_{i} .

(11)

The angular momentum vanishes for a spherical Lagrangian patch, and also whenever the patch's inertia principal axes align with the local tidal-field principal axes. It likewise vanishes in a locally isotropic tidal field.

Taylor-expand

\partial_{j} Φ_{i} (x_{i}) = \partial_{j} Φ_{i} (\overset{ˉ}{x}_{i}) + T_{j l} (x_{i} - \overset{x}{ˉ}_{i})_{l} + \dots, T_{j l} = \partial_{j} \partial_{l} Φ_{i} ∣_{\overset{ˉ}{x}_{i}} .

(12)

The constant-gradient term vanishes by the barycentre definition. With

I_{l k} = \int_{V_{L}} d^{3} x_{i} (x_{i} - \overset{x}{ˉ}_{i})_{l} (x_{i} - \overset{x}{ˉ}_{i})_{k} a^{3} \overset{ρ}{ˉ}_{m},

(13)

one obtains the tidal torque theory result

J_{i} = - a^{2} \dot{b} ϵ_{ijk} T_{j l} I_{l k} .

(14)

I

is the protohalo inertia tensor and

T

is the local tidal, or gravitational Hessian, tensor. Their eigenframe misalignment produces the torque.

In an Einstein-de Sitter universe,

D \propto a \propto t^{2/3}

. Hence

\dot{b} \propto \overset{a}{˙} \propto t^{- 1/3}

and

∣ J ∣ \propto a^{2} \dot{b} \propto t .

(15)

This linear estimate is normally stopped near turnaround. Simulated haloes subsequently gain angular momentum through nonlinear torques, anisotropic accretion, and mergers, including the orbital angular momentum of infalling subhaloes, so the early linear estimate underpredicts the final value.

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 346 / 2 / Solution Created 2026-09-24 Updated 2026-09-25

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