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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 2026-10-14

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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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A galaxy contains so many stars that its two-body relaxation time is generally much longer than its age. Individual encounters can therefore be neglected and each star moves in the smooth collective potential. Liouville conservation along these Hamiltonian trajectories gives the Collisionless Boltzmann equation

\frac{d F}{d t} = 0.

(1)

In spherical phase-space coordinates this is

F_{t} + \overset{r}{˙} F_{r} + \dot{θ} F_{θ} + \overset{φ}{˙} F_{φ} + \overset{v}{˙}_{r} F_{v_{r}} + \overset{v}{˙}_{θ} F_{v_{θ}} + \overset{v}{˙}_{φ} F_{v_{φ}} = 0 .

(2)

The spherical line element is

d s^{2} = d r^{2} + r^{2} d θ^{2} + r^{2} sin^{2} θ d φ^{2}

, so

\overset{r}{˙} = v_{r}, \dot{θ} = v_{θ} / r, \overset{φ}{˙} = v_{φ} / (r sin θ) .

(3)

For a unit-mass star in a spherical potential,

L = \frac{1}{2} (\overset{r}{˙}^{2} + r^{2} \dot{θ}^{2} + r^{2} sin^{2} θ \overset{φ}{˙}^{2}) - Φ (r) .

(4)

The Euler--Lagrange equations, followed by differentiating

v_{θ} = r \dot{θ}

and

v_{φ} = r sin θ \overset{φ}{˙}

, give

\overset{v}{˙}_{r} = \frac{v _{θ}^{2} + v _{φ}^{2}}{r} - Φ^{'} (r),

(5)

\overset{v}{˙}_{θ} = \frac{v _{φ}^{2} cot θ - v _{r} v _{θ}}{r}, \overset{v}{˙}_{φ} = - \frac{v _{φ} v _{r} + v _{φ} v _{θ} cot θ}{r} .

(6)

Integrating the Boltzmann equation over velocity space, with vanishing velocity-space boundary terms, gives spherical mass conservation:

ρ_{t} + \partial_{r} (ρ ⟨ v_{r} ⟩) + \frac{2}{r} ρ ⟨ v_{r} ⟩ = 0 .

(7)

Multiplication by

v_{r}

and integration gives the radial Jeans equation. With isotropic dispersion

⟨ v_{r}^{2} ⟩ = σ^{2} + ⟨ v_{r} ⟩^{2}, ⟨ v_{θ}^{2} ⟩ = ⟨ v_{φ}^{2} ⟩ = σ^{2},

(8)

the geometric dispersion terms cancel, and use of continuity yields

\partial_{r} (ρ σ^{2}) + ρ \partial_{t} ⟨ v_{r} ⟩ + ρ ⟨ v_{r} ⟩ \partial_{r} ⟨ v_{r} ⟩ = - ρ Φ^{'} (r) .

(9)

The factor

ρ

on the right is required dimensionally.

In a Lambda-CDM background, the local excess mass contributes

GM (r) / r^{2}

, homogeneous matter inside radius

r

contributes

4 π G ρ_{b} r /3

, and the cosmological constant contributes outward acceleration

Λ r /3

. Hence

Φ^{'} (r) = \frac{GM ( r )}{r ^{2}} + \frac{4 π G ρ _{b}}{3} r - \frac{Λ}{3} r .

(10)

Since

Ω_{m} = 8 π G ρ_{b} / (3 H^{2})

Ω_{Λ} = Λ/ (3 H^{2})

, and

q = Ω_{m} /2 - Ω_{Λ}

Φ^{'} (r) = \frac{GM ( r )}{r ^{2}} + q H^{2} r .

(11)

Write

⟨ v_{r} ⟩ = Hr + v_{p}

, where

v_{p}

is the peculiar velocity. Then

\partial_{t} ⟨ v_{r} ⟩ + ⟨ v_{r} ⟩ \partial_{r} ⟨ v_{r} ⟩ = (\dot{H} + H^{2}) r + v_{p, t} + H v_{p} + Hr v_{p, r} + v_{p} v_{p, r} .

(12)

The acceleration equation gives

\dot{H} + H^{2} = \overset{a}{¨} / a = - q H^{2}

, which cancels the background term in

Φ^{'}

. Therefore

\partial_{r} (ρ σ^{2}) = - ρ [\frac{GM ( r )}{r ^{2}} + S (r, t)],

(13)

S = v_{p} v_{p, r} + H (v_{p} + r v_{p, r}) + v_{p, t} .

(14)

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Past exam of the relativistic Astrophysics and Cosmology course of the University of Cambridge / 2012 / Paper 1 / 3 / b / Solution 2026-10-20

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A positive cosmological constant contributes a constant vacuum energy density and negative pressure. In the time-unit convention of Question 2,

ρ_{Λ} = \frac{Λ}{8 π G}, p_{Λ} = - ρ_{Λ} c^{2}, \frac{R ¨}{R} = - \frac{4 π G}{3} (ρ + \frac{3 p}{c ^{2}}) + \frac{Λ}{3} .

(1)

It therefore causes accelerating expansion once its negative-pressure effect outweighs the decelerating components. Its density stays constant while matter dilutes, so a small early contribution can dominate late. For an isolated positive-vacuum component the scale factor grows exponentially. The cosmological constant problem is the mismatch between the small inferred vacuum density and generic quantum-field vacuum-energy expectations, rather than a failure of the background solution itself.

The key direct expansion evidence came from Type Ia supernova distance-redshift measurements. After light-curve and colour standardization, distant events were fainter than expected in simple decelerating matter-only models, indicating larger luminosity distances. Riess and collaborators' 1998 analysis and Perlmutter and collaborators' independent 42-supernova study favoured an accelerating low-matter-density cosmology with a positive vacuum contribution. Dust, source evolution, selection effects, photometric calibration and peculiar velocities must be controlled: faintness alone is not a model-independent measurement of

Λ

.^[ref]^[ref]

cosmic microwave background acoustic geometry and independent matter-density constraints help break the supernova parameter degeneracy: nearly flat geometry with substantially less than critical matter requires another component within the FLRW model. large-scale structure of the universe and distance measurements provide further consistency tests. The combined evidence supports a dark-energy contribution compatible with a constant; accelerating expansion by itself does not prove exactly constant vacuum energy or rule out every alternative gravity model.

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Past exam of the relativity course of the University of Cambridge / 2022 / Section B / 4 / d / Solution 2026-10-18

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A nearby fundamental observer measures radial proper distance

d ℓ = a d χ

and proper time

d t

. Thus the signed radial peculiar velocity is

v = a \frac{d χ}{d t} = \frac{a χ ˙}{t ˙} .

(1)

Substitute

a^{2} \overset{χ}{˙} = C

and the massive normalization from part (b):

v (a) = \frac{C / a}{1 + C ^{2} / ( a ^{2} c ^{2} )}, aγ v = C .

(2)

Equivalently, for velocity

v_{0}

at scale factor

a_{0}

v (a) = \frac{v _{0} a _{0}}{a ^{2} ( 1 - v _{0}^{2} / c ^{2} ) + a _{0}^{2} v _{0}^{2} / c ^{2}} .

(3)

This is the exact law of redshifting of peculiar momentum. In the nonrelativistic limit,

v \propto a^{- 1}

; an initially relativistic speed stays below

c

and decreases as the universe expands.

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Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2016 / S2 / 3 / 2 / Solution 2026-10-25

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The given calibration yields

M_{V} = - 3.5 lo g_{10} 50 ≃ - 5.946

. The distance modulus is

31.446

, giving

d ≃ 19.5 Mpc

if extinction is negligible. Hubble's law predicts

v_{H} ≃ 1400

km s

^{- 1}

, compared with 1150 km s

^{- 1}

observed. The approximately

- 250

km s

^{- 1}

difference can be a line-of-sight peculiar velocity of the cluster and observer, together with distance-calibration uncertainties. The local flow need not follow a perfectly smooth expansion law.

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Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S2 / 2 / c / Solution 2026-10-25

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Using the same luminosity estimate, the distance modulus is

12.5 - (- 22.49) = 34.99

, implying

D ≃ 99.4

Mpc. Pure Hubble expansion at that distance predicts about

7450 km s^{- 1}

. The measured velocity therefore implies

v_{pec, los} ≃ 6500 - 7450 ≃ - 950 km s^{- 1} .

(1)

Its line-of-sight peculiar velocity is toward us, of the order expected for orbital motions in a rich galaxy cluster. A galaxy falling toward a cluster centre from its far side could have this sign. Its actual front/back location or three-dimensional orbit is not determined without the cluster's mean velocity, distance and projected position. Tully-Fisher scatter and the assumption of identical luminosities also limit that inference.

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Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S26 / 3 / b / Solution 2026-10-25

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A K-star-like integrated spectrum indicates that old or intermediate-age cool stars, often luminous red giants, dominate the observed optical/near-infrared light. It suggests little ongoing formation of very luminous hot O and B stars. It does not imply that every star is K type: faint low-mass dwarfs and remnants can contribute substantial mass while contributing little light.

Using the reference wavelength printed in the problem,

z = \frac{874.0 - 854.4}{854.4} = 0.02294.

(1)

The low-redshift Hubble estimate is

D ≃ \frac{cz}{H _{0}} ≃ 96.9 Mpc .

(2)

Peculiar velocities and the distinction between luminosity and angular-diameter distance introduce percent-level or larger corrections to this simple estimate. The numerical result intentionally uses the supplied calcium reference rather than replacing it with a different tabulated line wavelength.

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Peculiar-velocity equation Created 2026-09-28 Updated 2026-10-14

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For a fluid in an expanding universe, the peculiar-velocity equation is the Euler equations for an inviscid fluid after decomposing physical velocity into Hubble flow and peculiar velocity. It contains the Hubble-drag term

H v

and force

- \nablaΦ/ a

from the peculiar gravitational potential.

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Physical momentum in an FRW universe 2026-10-14

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A freely falling particle's physical momentum is measured in the local orthonormal frame of a comoving observer. With proper peculiar velocity

v = a d x / d t

and Lorentz factor

γ = (1 - v^{2})^{- 1/2}

, it is

mγ v

. It differs by one factor of the scale factor from the conserved comoving momentum. For a nonrelativistic particle, its peculiar velocity therefore decays as

a^{- 1}

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Quasar-cloud redshift separation 2026-10-14

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For nearby redshifts in a homogeneous expanding universe, the comoving radial distance increment is

c δz / H (z)

. Multiplying by the scale factor at the absorber's epoch gives the displayed proper distance in cosmology. The conversion neglects peculiar velocities; observed redshift differences can also contain such velocities and errors in the quasar emission redshift.

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Redshifting of peculiar momentum 2026-10-18

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A freely moving massive test particle in an FLRW metric has physical momentum

p = mγ v

relative to a comoving observer, where

v

is its peculiar velocity. For a radial geodesic, the geodesic Lagrangian gives

C = a^{2} d χ / d τ

constant, and the local radial momentum is

p = ma d χ / d τ = m C / a

. The proper-time normalization gives

d t / d τ = 1 + C^{2} / (a^{2} c^{2})

, so

v = \frac{C / a}{1 + C ^{2} / ( a ^{2} c ^{2} )} .

(1)

Consequently nonrelativistic peculiar velocity scales as

a^{- 1}

, while the exact relativistic law keeps

∣ v ∣ < c

. The momentum magnitude of a free photon also scales as

a^{- 1}

, giving cosmological redshift through

E = c p

. These statements concern force-free propagation; a bound system need not follow the cosmological expansion.

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Redshift-space distortions 2026-10-14

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Redshift-space distortions arise because observed galaxy redshifts include peculiar velocity as well as cosmological redshift. Distances inferred solely from expansion therefore distort clustering along the line of sight. Coherent large-scale motion and small-scale velocity dispersion produce different effects on the inferred matter power spectrum.

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Scale-free potential and bulk-flow variance 2026-10-14

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For a matter power spectrum

P (k) = A k^{n}

at a fixed epoch, the Poisson equation gives potential Fourier amplitudes proportional to

δ_{k} / k^{2}

, and the growing-mode peculiar velocity is proportional to

δ_{k} k / k^{2}

. With a spherical top-hat window function, their variances are proportional to

\int k^{n - 2} W (k R)^{2} d k

and

\int k^{n} W (k R)^{2} d k

, respectively. Rescaling

k R

gives the displayed powers when the integrals converge:

1 < n < 5

for the averaged potential and

- 1 < n < 3

for the bulk velocity. Absolute averaged potential has infrared sensitivity; band-limited fluctuations or potential differences can have different convergence requirements. The density variance converges only for

- 3 < n < 1

, so these three absolute averages cannot all converge for one unbroken power law at every scale.

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Spectroscopic velocity drift 2026-10-14

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For a transition with fixed rest wavelength,

λ_{obs} = (1 + z) λ_{rest}

. The apparent spectroscopic velocity shift is

d v = c d λ_{obs} / λ_{obs} = c d z / (1 + z)

. It is a convenient wavelength-shift unit, not necessarily a change in a source's peculiar velocity. Omitting the denominator

1 + z

overestimates the signal at high redshift.

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Supernova calibration and population tests 2026-10-18

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The High-Z Supernova Search Team analysis fitted multicolour light curves and interstellar extinction, and compared a second decline-rate method. Similar local and distant spectroscopy supported, but did not establish, population stability. Nearly colourless attenuation remained a qualified alternative.^[ref]

The Supernova Cosmology Project compared low- and high-redshift colours, removed red outliers in its primary fit, and tested Malmquist bias with artificial-source recovery.^[ref]

The inference can be organized as a chain of independently testable failure modes. A magnitude zero point offset between nearby and distant observations imitates a distance modulus offset; a common offset instead changes the intercept. An erroneous photometric passband response changes a K correction in a redshift-dependent way because different emitted wavelengths cross the detector band. Residual host-galaxy light changes the recovered radiative flux. Color excess constrains wavelength-dependent interstellar extinction only after intrinsic colour is modeled; grey attenuation is not bounded by colour alone. Changes in stellar population or metallicity could shift standardized absolute magnitude without large changes in the chosen shape parameter. These are logical degeneracies, so increasing the sample size alone cannot remove them.

A useful control strategy therefore compares subsets with different host environments, fits consistent rest-frame photometric passbands, and checks artificial-source recovery through the same image processing and selection. A Malmquist bias tends to select brighter detectable sources, but the relative bias between the nearby and distant samples is what matters. Peculiar velocity mainly perturbs the nearby redshift anchor; weak gravitational lensing changes individual brightnesses and the selection distribution. Neither can be replaced by an arbitrary extra scatter term when its mean or selection dependence is uncertain.

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